**Physical Insights Into Dynamic Similarity in Animal Locomotion. I. Theoretical Principles and Concepts**

Valery B. Kokshenev *Departamento de Física, Universidade Federal de Minas Gerais, Belo Horizonte Brazil*

### **1. Introduction**

24 Will-be-set-by-IN-TECH

266 Theoretical Biomechanics

von Stengel, S., Kemmler, W., Kalender, W. A., Engelke, K. & Lauber, D. (2007). Differential

von Stengel, S., Kemmler, W., Pintag, R., Beeskow, C., Weineck, J., Lauber, D., Kalender,

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discussion 655.

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URL: *http://link.aip.org/link/?MSE/93/268/1*

effects of strength versus power training on bone mineral density in postmenopausal women: a 2-year longitudinal study, *British journal of sports medicine* 41(10): 649–55;

W. A. & Engelke, K. (2005). Power training is more effective than strength training for maintaining bone mineral density in postmenopausal women, *Journal of Applied*

them about their balance–future fracture risk in a nationwide cohort study of twins,

Comparative zoologists, evolutionary biologists, experimental biologists, and mechanical engineers make wonderful generalizations about the movements of different-sized running bipeds and quadrupeds (Heglund et al., 1974; McMahon, 1975; Alexander, 1976; Hoyt & Taylor, 1981; Garland, 1982; McMahon et al., 1987; Rome et al., 1988; Gatesy & Biewener, 1991; Farley et al.,1993; Cynthia & Farley, 1998; Bullimore & Burn, 2006), flying birds and swimming fish (Hill, 1950; Alexander, 2003; Taylor et al., 2003). The dynamic similarity across body mass and taxa of animals maintaining a certain gait in locomotion has been thoroughly investigated (Alexander, 1976, 1985, 1989, 2005; Alexander & Bennet-Clark, 1976; Alexander & Jayes, 1983; Marden & Allen, 2002; Biewener, 2005; Bejan & Marden, 2006a, b; Bullimore & Donelan, 2008). Based on integrative approach to animal locomotion (e.g., review by Dickinson et al., 2000) and using simple physical ideas (e.g., reviews by Lin, 1982 and Alexander, 2003), many cited above researches have demonstrated the importance of scaling biomechanics via reliably established scaling relations for gait characteristics with changes of speed and body mass. Among well known empirical findings of the dynamic similarity in animals observed across body mass are scaling relations established for stride (or stroke) speed and/or frequency (Hill, 1950; Heglund et al., 1974 Greenewalt, 1975; Garland, 1982; Heglund & Taylor, 1988; Gatesy & Biewener, 1991; Farley et al., 1993; Bullen & McKenzie, 2002; Bejan & Marden, 2006a), duty factor and relative stride length (Alexander & Jayes, 1983; Gatesy & Biewener, 1991), body force output (Alexander, 1985; Marden & Allen, 2002; Bejan & Marden, 2006a), limb stress and stiffness (Rubin & Lanyon, 1984; McMahon & Cheng, 1990; Farley & Gonzalez, 1996; Biewener, 2005; Bullimore & Burn, 2006).

Although the concept of *mechanical similarity*, well known in analytical mechanics (e.g., Duncan, 1953; Landau & Lifshitz, 1976), was accurately re-formulated in application to the *dynamic similarity* (Alexander, 1976, 1989, 2003, 2005; Bullimore & Donelan, 2008), its exploration in biomechanics is often controversial. For example, Hill's seminal observation of dynamic similarity through the optimal speeds of birds in flight gaits (Hill, 1950) was found in sharp disagreement (McMahon, 1975) with that revealed through transient (trot-to-gallop) speeds in quadrupeds (Heglund et al., 1974). Then, comparing anatomic consequences of McMahon's elastic similarity (McMahon, 1975) for the stride length, Alexander noted

**2. Theory**

motion.

similarity.

(*q*, *v*) = K�

L�

**2.1 Similarity in analytical mechanics**

in Animal Locomotion. I. Theoretical Principles and Concepts

and potential energy U scale as

K�

(*v*) − U�

*amplitude F* and U), namely

(1976, Eqs. (10.2) and (10.3)).

dimensional method.<sup>1</sup>

(*v*) = K

According to the key variational principle of Hamiltonian's classical mechanics, the requirement of minimum mechanical action between two fixed points of the conceivable trajectory of an arbitrary mechanical system determines the Lagrangian function *L*(*q*, *v*) through time-dependent coordinates *q*(*t*) and instant velocities *v*(*t*) = *dq*/*dt*. One of the most pronounced properties of the *closed* mechanical systems is preservation of the total energy and momentum, arising respectively from the temporal and spatial homogeneities of the Lagrangian function. The *mechanical similarity* between frictionlessly moving systems also arises from the property of spatiotemporal homogeneity (e.g., Duncan, 1953; Landau & Lifshitz, 1976). Since the property of homogeneity guarantees that the multiplication of Lagrangian on an arbitrary constant does not affect the resulting equations of motion, the *scaling laws* of mechanical similarity can be established without consideration of equations of

<sup>269</sup> Physical Insights Into Dynamic Similarity

More specifically, let us consider the uniform transformation of mechanical trajectories due

amplitude (positive) linear-transformation factors *a* and *b*, resulting in changed velocities *v*� = (*a*/*b*)*v*. The overall-system basic mechanical characteristics, *period T* and *speed V* change as *T*� = *bT* and *V*� = (*a*/*b*)*V*, whereas the kinetic energy K, as a quadratic function of velocities,

K(*v*) and U�

where the *dynamic exponent λ* is introduced to distinguish distinct cases of mechanical

The self-consistency of exploration of the property of homogeneity, determining the property of similarity, requires the proportionality in changes of both the energies of the Lagrangian

(1). Hence, the frictionless propagation of a classical system obeys the *scaling rules* imposed on the overall-system dynamic characteristics (*T* and *V*) and mechanical characteristics (*force*

where *L* is a characteristic linear size of the *trajectory* , as suggested by Landau & Lifshitz

The well known examples of distinct mechanically similar systems distinguished by dynamic exponent readily follow from Eq. (2): (i) *λ* = −1; the case resulting in the third law for planets *T*<sup>2</sup> ∝ *L*<sup>3</sup> that anticipated Newton's theory through interplanetary coupling force *F* ∝ *M*2*L*−2; (ii) *λ* = 2; the system is driven by the elastic-strain field and has the elastic energy <sup>U</sup>*elast* <sup>=</sup> *<sup>K</sup>*Δ*L*<sup>2</sup> with <sup>Δ</sup>*<sup>L</sup>* � *<sup>L</sup>*, and (iii) *<sup>λ</sup>* <sup>=</sup> 1; the system moves in the uniform *gravitational field g* = *Fg*/*M* and has the potential energy U*<sup>g</sup>* = *MgH*, with *H* � *L*. The dimensional factors of proportionality *M* and *K*, required by the scaling relations, are analyzed below by

<sup>1</sup> Hereafter I distinguish two symbols of proportionality: � and ∝ , supporting and not supporting

dimensional units that should be read as "is proportional to" and "scales as", respectively.

� = *bt* performed via arbitrary in

(*v*) = <sup>U</sup>(*aq*) = *<sup>a</sup>λ*U(*q*), (1)

, thus, (*a*/*b*)<sup>2</sup> = *aλ*, or *b* = *a*1−*<sup>λ</sup>*/2, as follows from Eq.

*<sup>T</sup><sup>λ</sup>* <sup>∝</sup> *<sup>L</sup>*1−*<sup>λ</sup>*/2, *<sup>V</sup><sup>λ</sup>* <sup>∝</sup> *<sup>L</sup><sup>λ</sup>*/2, *<sup>F</sup><sup>λ</sup>* <sup>∝</sup> *<sup>L</sup>λ*−1, and <sup>U</sup>*<sup>λ</sup>* <sup>∝</sup> *<sup>L</sup>λ*, (2)

to linear changing of all coordinates *q*� = *aq* and times *t*

 *a b v* = *a b* 2

(*q*), *i.e.*, K� � U�

the conceptual inconsistency between the elastic and dynamic similarities (Alexander, 1989, p.1212). Biewener (2005) also claimed confusing biomechanical consequences of McMahon's scaling relations. More recent example is the observation of a new kind of similarity in running humans, simulated on the basis of spring-mass model, no matching the dynamic similarity in running animals (Delattre et al., 2009), reliably established by the same biomechanical model (Farley et al., 1993).

Another example of conceptual controversy concerns the long term standing problem of the origin of empirical scaling laws in biomechanics. The constructal theory by Bejan & Marden (2006a), unifying running, flying, and swimming animals suggested fundamental explanations of intriguing statistically established universal scaling relations for the optimal speed and stride frequency. As a matter of fact, the authors have clearly demonstrated that their principle of minimum useful energy does not provide unifying scaling laws. Instead of searching for new principles, exemplified by recently rediscovered "least-action principle" in human walking (Fan et al., 2009), the design principle of flow systems (Bejan & Marden, 2006a, b), and the basic theorem of dimensional analysis (Bullimore & Donelan, 2008), seemingly resolving the origin of scaling laws in biomechanics, I have addressed the fundamentals of classical analytical mechanics.

In theoretical physics, the mechanical similarity arises from the key principle of minimum mechanical action closely related to Lagrangian's formalism. Examples of successful applications of Lagrangian's method to the dynamics of human walk and the dynamic similarity in animals are explicit descriptions providing, respectively, (i) the conditions of dynamic instability during a walk-to-run crossover obtained regardless of inverted-pendulum modeling (Kokshenev, 2004) and (ii) the whole spectrum of observable scaling laws inferred without recourse to equations of motion (Kokshenev, 2010).

In biomechanics, the dynamic similarity hypothesis (Alexander, 1976; Alexander & Jayes, 1983; Alexander, 1989) stands that similarly in running terrestrial animals should equal their *Froude numbers* (the squared speeds divided by hip heights times the gravitation constant) when tend to change locomotion modes at certain "equivalent speeds" (Heglund et al., 1974) or maintain a certain gait at "preferred speeds" (Heglund & Taylor, 1988). In contrast to *in vivo* established dynamic similarity in the horses trotting at equal Froude numbers (Bullimore & Burn, 2006), the spring-mass model analysis has indicated that the Froude number alone does not yet guarantee the observation of dynamic similarity in the running animals (Donelan & Kram, 1997; Bullimore & Donelan, 2008). Such kind of controversial findings begged a number of questions: whether the sole Froude number (Alexander & Jayes, 1983; Vaughan & O'Malleyb, 2005; Bullimore & Burn, 2006; Delattre et al., 2009), or the sole *Strouhal number* (the limb length, or wing length divided by the speed times stride period, or stroke period) as hypothesized by Whitfield (2003), Taylor et al. (2003) and, likely, Bejan & Marden (2006a), or both the numbers (Delattre et al., 2009), when taken in a certain algebraic combinations (Delattre & Moretto, 2008), e.g., presented by the Groucho number (Alexander, 1989), may warrant the dynamic similarity in animal locomotion? Moreover, when other dimensionless parameters are chosen as the determinants of dynamic similarity, what is the minimal set of independent physical quantities underlying the principle of similarity (Bullimore & Donelan, 2008)?

In this research, a model-independent theoretical framework basically employing Lagrangian's method suggests to establish the validation domains, conditions of observation, criterion, and the minimal set of determinants of dynamic similarity empirically established in different-sized animals.

### **2. Theory**

2 Will-be-set-by-IN-TECH

the conceptual inconsistency between the elastic and dynamic similarities (Alexander, 1989, p.1212). Biewener (2005) also claimed confusing biomechanical consequences of McMahon's scaling relations. More recent example is the observation of a new kind of similarity in running humans, simulated on the basis of spring-mass model, no matching the dynamic similarity in running animals (Delattre et al., 2009), reliably established by the same biomechanical model

Another example of conceptual controversy concerns the long term standing problem of the origin of empirical scaling laws in biomechanics. The constructal theory by Bejan & Marden (2006a), unifying running, flying, and swimming animals suggested fundamental explanations of intriguing statistically established universal scaling relations for the optimal speed and stride frequency. As a matter of fact, the authors have clearly demonstrated that their principle of minimum useful energy does not provide unifying scaling laws. Instead of searching for new principles, exemplified by recently rediscovered "least-action principle" in human walking (Fan et al., 2009), the design principle of flow systems (Bejan & Marden, 2006a, b), and the basic theorem of dimensional analysis (Bullimore & Donelan, 2008), seemingly resolving the origin of scaling laws in biomechanics, I have addressed the fundamentals of

In theoretical physics, the mechanical similarity arises from the key principle of minimum mechanical action closely related to Lagrangian's formalism. Examples of successful applications of Lagrangian's method to the dynamics of human walk and the dynamic similarity in animals are explicit descriptions providing, respectively, (i) the conditions of dynamic instability during a walk-to-run crossover obtained regardless of inverted-pendulum modeling (Kokshenev, 2004) and (ii) the whole spectrum of observable scaling laws inferred

In biomechanics, the dynamic similarity hypothesis (Alexander, 1976; Alexander & Jayes, 1983; Alexander, 1989) stands that similarly in running terrestrial animals should equal their *Froude numbers* (the squared speeds divided by hip heights times the gravitation constant) when tend to change locomotion modes at certain "equivalent speeds" (Heglund et al., 1974) or maintain a certain gait at "preferred speeds" (Heglund & Taylor, 1988). In contrast to *in vivo* established dynamic similarity in the horses trotting at equal Froude numbers (Bullimore & Burn, 2006), the spring-mass model analysis has indicated that the Froude number alone does not yet guarantee the observation of dynamic similarity in the running animals (Donelan & Kram, 1997; Bullimore & Donelan, 2008). Such kind of controversial findings begged a number of questions: whether the sole Froude number (Alexander & Jayes, 1983; Vaughan & O'Malleyb, 2005; Bullimore & Burn, 2006; Delattre et al., 2009), or the sole *Strouhal number* (the limb length, or wing length divided by the speed times stride period, or stroke period) as hypothesized by Whitfield (2003), Taylor et al. (2003) and, likely, Bejan & Marden (2006a), or both the numbers (Delattre et al., 2009), when taken in a certain algebraic combinations (Delattre & Moretto, 2008), e.g., presented by the Groucho number (Alexander, 1989), may warrant the dynamic similarity in animal locomotion? Moreover, when other dimensionless parameters are chosen as the determinants of dynamic similarity, what is the minimal set of independent physical quantities underlying the principle of similarity (Bullimore & Donelan,

In this research, a model-independent theoretical framework basically employing Lagrangian's method suggests to establish the validation domains, conditions of observation, criterion, and the minimal set of determinants of dynamic similarity empirically established

(Farley et al., 1993).

classical analytical mechanics.

2008)?

in different-sized animals.

without recourse to equations of motion (Kokshenev, 2010).

### **2.1 Similarity in analytical mechanics**

According to the key variational principle of Hamiltonian's classical mechanics, the requirement of minimum mechanical action between two fixed points of the conceivable trajectory of an arbitrary mechanical system determines the Lagrangian function *L*(*q*, *v*) through time-dependent coordinates *q*(*t*) and instant velocities *v*(*t*) = *dq*/*dt*. One of the most pronounced properties of the *closed* mechanical systems is preservation of the total energy and momentum, arising respectively from the temporal and spatial homogeneities of the Lagrangian function. The *mechanical similarity* between frictionlessly moving systems also arises from the property of spatiotemporal homogeneity (e.g., Duncan, 1953; Landau & Lifshitz, 1976). Since the property of homogeneity guarantees that the multiplication of Lagrangian on an arbitrary constant does not affect the resulting equations of motion, the *scaling laws* of mechanical similarity can be established without consideration of equations of motion.

More specifically, let us consider the uniform transformation of mechanical trajectories due to linear changing of all coordinates *q*� = *aq* and times *t* � = *bt* performed via arbitrary in amplitude (positive) linear-transformation factors *a* and *b*, resulting in changed velocities *v*� = (*a*/*b*)*v*. The overall-system basic mechanical characteristics, *period T* and *speed V* change as *T*� = *bT* and *V*� = (*a*/*b*)*V*, whereas the kinetic energy K, as a quadratic function of velocities, and potential energy U scale as

$$\mathcal{K}'(v) = \mathcal{K}\left(\frac{a}{b}v\right) = \left(\frac{a}{b}\right)^2 \mathcal{K}(v) \text{ and } \mathcal{U}'(v) = \mathcal{U}(aq) = a^\lambda \mathcal{U}(q),\tag{1}$$

where the *dynamic exponent λ* is introduced to distinguish distinct cases of mechanical similarity.

The self-consistency of exploration of the property of homogeneity, determining the property of similarity, requires the proportionality in changes of both the energies of the Lagrangian L� (*q*, *v*) = K� (*v*) − U� (*q*), *i.e.*, K� � U� , thus, (*a*/*b*)<sup>2</sup> = *aλ*, or *b* = *a*1−*<sup>λ</sup>*/2, as follows from Eq. (1). Hence, the frictionless propagation of a classical system obeys the *scaling rules* imposed on the overall-system dynamic characteristics (*T* and *V*) and mechanical characteristics (*force amplitude F* and U), namely

$$T\_{\lambda} \propto L^{1-\lambda/2}, V\_{\lambda} \propto L^{\lambda/2}, F\_{\lambda} \propto L^{\lambda-1}, \text{and } \mathcal{U}\_{\lambda} \propto L^{\lambda} \tag{2}$$

where *L* is a characteristic linear size of the *trajectory* , as suggested by Landau & Lifshitz (1976, Eqs. (10.2) and (10.3)).

The well known examples of distinct mechanically similar systems distinguished by dynamic exponent readily follow from Eq. (2): (i) *λ* = −1; the case resulting in the third law for planets *T*<sup>2</sup> ∝ *L*<sup>3</sup> that anticipated Newton's theory through interplanetary coupling force *F* ∝ *M*2*L*−2; (ii) *λ* = 2; the system is driven by the elastic-strain field and has the elastic energy <sup>U</sup>*elast* <sup>=</sup> *<sup>K</sup>*Δ*L*<sup>2</sup> with <sup>Δ</sup>*<sup>L</sup>* � *<sup>L</sup>*, and (iii) *<sup>λ</sup>* <sup>=</sup> 1; the system moves in the uniform *gravitational field g* = *Fg*/*M* and has the potential energy U*<sup>g</sup>* = *MgH*, with *H* � *L*. The dimensional factors of proportionality *M* and *K*, required by the scaling relations, are analyzed below by dimensional method.<sup>1</sup>

<sup>1</sup> Hereafter I distinguish two symbols of proportionality: � and ∝ , supporting and not supporting dimensional units that should be read as "is proportional to" and "scales as", respectively.

(and stroke) length *L*, represented as

*W <sup>L</sup>* <sup>=</sup> *<sup>W</sup>*<sup>1</sup> *<sup>L</sup>* <sup>+</sup>

in Animal Locomotion. I. Theoretical Principles and Concepts

total loss function results in the optimal speed *Vopt* � <sup>3</sup>

and postulating for the frictionless vertical motion that

the constructal theory suggests relations

 *ρ<sup>b</sup> ρa*

1/3

*Vopt* ≈

**2.2.3 Minimum muscular action**

Marden (2006a).

& Allen, 2002).

relations, namely

Δ*L <sup>L</sup>* � <sup>Δ</sup>*Lb Lb*

*t* = *topt* =

*g*1/2*ρ*−1/6

power consumption) realized at the *resonant propagation frequency*.

<sup>=</sup> *<sup>ε</sup><sup>b</sup>* � <sup>Δ</sup>*Lm Lm*

*W*<sup>2</sup>

 2*H <sup>g</sup>* �

*<sup>b</sup> <sup>M</sup>*1/6 and *<sup>T</sup>*−<sup>1</sup>

for the optimal speed and frequency, in the case of fast running and flying animals, as shown, respectively, in Eqs. (5) and (22) and Eq. (23) in Bejan & Marden (2006a). Likewise, the frictionless vertical motion has been postulated in the case of optimal swimming, for which the energy loss *W*<sup>1</sup> � *MgLb* was adopted in Eq. (6). For further details, see Eq. (26) in Bejan &

Aiming to apply the principle of mechanical similarity formulated for closed inanimate systems to musculoskeletal systems of different-sized animals, one should consider conditions of the observation of effectively frictionless propagation of almost-closed (weekly open) animate systems. These are all cases of *efficient locomotion* (maximum useful work at minimum

Since the animal locomotion is substantially muscular (e.g., McMahon, 1984; Rome et al., 1988; Dickinson et al., 2000; Alexander, 2003), the body-system relations shown in Eqs. (4) and (5) can be generalized to the muscle subsystem presented by a synergic group of locomotory muscles of the *effective muscle length Lm* and cross-sectional area *Am*. In the approximation of fully activated muscle states, the maximum muscle stiffness *Km* = *Em Am*/*Lm* is controlled by the geometry-independent muscle rigidity, i.e., the elastic modulus amplitude *Em*, defined by ratio of the peak muscle stress *σ<sup>m</sup>* = Δ*Fm*/*Am* to the peak muscle strain *ε<sup>m</sup>* = Δ*Lm*/*Lm* (e.g., McMahon, 1975, 1984). Let us treat the mass specific muscular (or relative) force output defined by *μ<sup>m</sup>* ≡ Δ*Fm*/*Mm* as the *muscular field* generating active force Δ*Fm* through the *muscle mass Mm*, which being a source of the force field *μ<sup>m</sup>* plays the role of the motor mass (Marden

Dynamic motion parameters of the trajectory of body's center of mass are linked to the characteristic static body's and muscle's lengths via the common in biomechanics *linear*

> Δ*T <sup>T</sup>* <sup>=</sup> <sup>Δ</sup>*Tm Tm*

= *εm*,

*<sup>L</sup>* <sup>=</sup> *MgH*

<sup>271</sup> Physical Insights Into Dynamic Similarity

Here *W*<sup>1</sup> is the vertical loss of energy associated with fall from the height *H* in the gravitational field and *W*<sup>2</sup> is the horizontal loss of energy related to the friction with air of density *ρ<sup>a</sup>* (for further details see Eqs. (1)-(4) in Bejan & Marden, 2006a). The minimization procedure of the

choice of the dynamic variables *t* and *H*. Using the linear relation *H* � *Lb* discussed in Eq. (3)

 *Lb*

*opt* ≈

 *ρ<sup>b</sup> ρa* 1/3

*Vt* <sup>+</sup> *CDρaV*2*L*<sup>2</sup>

*MH*/*L*<sup>2</sup>

*<sup>b</sup>*. (6)

*<sup>b</sup>t*, clearly leaving uncertain a

*<sup>b</sup> <sup>M</sup>*−1/6 (8)

*<sup>g</sup>* <sup>∝</sup> *<sup>M</sup>*1/6, (7)

= *β*, and Δ*F* � Δ*Fm*. (9)

*g*1/2*ρ*1/6

### **2.2 Similarity in biomechanics**

The exploration of the concept of dynamic similarity generally assumes model-independent relations validated for slow and fast gaits and supporting linear transformations in changing dynamic and mechanical characteristics, namely

$$
\Delta L \sim L \sim H \sim L\_b \text{ and } \Delta T \sim T \text{, } \Delta F = K \Delta L\_b. \tag{3}
$$

Here Δ*L* is a change of the amplitude of *dynamic length L*, e.g., stride length *L*, or the maximum vertical displacement *H* of body's center of mass that may be chosen for periodic terrestrial locomotion, or the stroke amplitude of flying and swimming animals (e.g., Alexander, 2003). The dynamic length and its change should be distinguished from the *static length Lb* and its change Δ*Lb*, that may be also chosen either as of the body length *Lb* or the corresponding limb, wing, or tail length. Respectively, *T*−<sup>1</sup> is the stride, wingbeat, or tailbeat rate attributed to the *frequency* of locomotion. The dynamic length is commonly determined by the measured speed *V* (*mean* cycle forward velocity) and frequency, i.e., as *L* = *V*/*T*−1, whereas the body *propulsion force* Δ*F* introduces the *body stiffness K* = Δ*F*/Δ*Lb*. 2

### **2.2.1 Dimensional analysis**

Broadly speaking, the mechanical similarity may be tested by two uniform linear transformations, scaling simultaneously all spatial and temporal characteristics. Extending these two degrees of freedom of *biomechanical mechanical systems* by their *body masses M*, let us introduce the corresponding *ltm class of units* through the independent dimensions [*L*] = *l*, [*T*] = *t*, and [*M*] = *m*, in accord with the standard scaling theory (e.g., Barenblatt, 2002). One can show that the three body-system mechanical quantities

$$
\Delta F \sim \text{LT}^{-2} M\_\text{,} V \sim \text{LT}^{-1} \text{, and } K \sim T^{-2} M \tag{4}
$$

are mutually independent. Indeed, following the method of dimensional analysis (e.g., Barenblatt, 2002), let us assume the converse, i.e., that numbers *x* and *y* exist such that *F VxKy*. Substituting *F*, *V*, and *K* from Eq. (4) and equating exponents in the *ltm* class, one finds the solutions *x* = 1 and *y* = 1 not matching with 2*x* + *y* = 2. This proves that the considered set (Δ*F*, *V*, *K*) consists of three physically independent quantities.

The application of the mathematical concept of geometric similarity (Hill, 1950, Rashevsky, 1948; McMahon, 1975; Lin, 1982) introduced between cylindric-shape bodies through the constraint *ρbALb* = *M* (*A* is body's *cross-sectional area*), where *ρ<sup>b</sup>* is the invariable *body density* (McMahon, 1975, 1984; Alexander, 2003; Bejan & Marden, 2006a, b), allows one to suggested another equivalent set of candidates to the mutually independent *determinants of mechanical similarity*, namely

$$T^{-1} \sim \sqrt{\frac{\Delta F}{A}} L\_{\flat}^{-1}, V \sim \sqrt{\frac{\Delta F}{A}}, \text{and } \Delta F \sim KL\_{b\prime} \tag{5}$$

chosen in the *ltf* class of unites and also depending neither on locomotor gaits nor biomechanical models.

### **2.2.2 Unifying contsructal theory revisited**

For the case of a running (or flying) animal of mass *M* with the *constant horizontal speed V* = *L*/*t*, the constructal law calls for the minimization of the total destruction of work *W* per stride

<sup>2</sup> These last two and other similar model-independent equations play the role of the definitive basic equations in the scaling theory.

(and stroke) length *L*, represented as

4 Will-be-set-by-IN-TECH

The exploration of the concept of dynamic similarity generally assumes model-independent relations validated for slow and fast gaits and supporting linear transformations in changing

Here Δ*L* is a change of the amplitude of *dynamic length L*, e.g., stride length *L*, or the maximum vertical displacement *H* of body's center of mass that may be chosen for periodic terrestrial locomotion, or the stroke amplitude of flying and swimming animals (e.g., Alexander, 2003). The dynamic length and its change should be distinguished from the *static length Lb* and its change Δ*Lb*, that may be also chosen either as of the body length *Lb* or the corresponding limb, wing, or tail length. Respectively, *T*−<sup>1</sup> is the stride, wingbeat, or tailbeat rate attributed to the *frequency* of locomotion. The dynamic length is commonly determined by the measured speed *V* (*mean* cycle forward velocity) and frequency, i.e., as *L* = *V*/*T*−1, whereas the body

Broadly speaking, the mechanical similarity may be tested by two uniform linear transformations, scaling simultaneously all spatial and temporal characteristics. Extending these two degrees of freedom of *biomechanical mechanical systems* by their *body masses M*, let us introduce the corresponding *ltm class of units* through the independent dimensions [*L*] = *l*, [*T*] = *t*, and [*M*] = *m*, in accord with the standard scaling theory (e.g., Barenblatt, 2002). One

are mutually independent. Indeed, following the method of dimensional analysis (e.g., Barenblatt, 2002), let us assume the converse, i.e., that numbers *x* and *y* exist such that *F VxKy*. Substituting *F*, *V*, and *K* from Eq. (4) and equating exponents in the *ltm* class, one finds the solutions *x* = 1 and *y* = 1 not matching with 2*x* + *y* = 2. This proves that the

The application of the mathematical concept of geometric similarity (Hill, 1950, Rashevsky, 1948; McMahon, 1975; Lin, 1982) introduced between cylindric-shape bodies through the constraint *ρbALb* = *M* (*A* is body's *cross-sectional area*), where *ρ<sup>b</sup>* is the invariable *body density* (McMahon, 1975, 1984; Alexander, 2003; Bejan & Marden, 2006a, b), allows one to suggested another equivalent set of candidates to the mutually independent *determinants of mechanical*

Δ*F*

chosen in the *ltf* class of unites and also depending neither on locomotor gaits nor

For the case of a running (or flying) animal of mass *M* with the *constant horizontal speed V* = *L*/*t*, the constructal law calls for the minimization of the total destruction of work *W* per stride

<sup>2</sup> These last two and other similar model-independent equations play the role of the definitive basic

considered set (Δ*F*, *V*, *K*) consists of three physically independent quantities.

Δ*L L H Lb* and Δ*T T*, Δ*F* = *K*Δ*Lb*. (3)

2

Δ*F LT*−2*M*, *V LT*<sup>−</sup>1, and *K T*−2*M* (4)

*<sup>A</sup>* , and <sup>Δ</sup>*<sup>F</sup> KLb*, (5)

**2.2 Similarity in biomechanics**

**2.2.1 Dimensional analysis**

*similarity*, namely

biomechanical models.

equations in the scaling theory.

dynamic and mechanical characteristics, namely

*propulsion force* Δ*F* introduces the *body stiffness K* = Δ*F*/Δ*Lb*.

can show that the three body-system mechanical quantities

*T*−<sup>1</sup>

**2.2.2 Unifying contsructal theory revisited**

Δ*F <sup>A</sup> <sup>L</sup>*−<sup>1</sup> *<sup>b</sup>* , *V*

$$\frac{W}{L} = \frac{W\_1}{L} + \frac{W\_2}{L} = \frac{M\mathfrak{g}H}{Vt} + \mathbb{C}\_D \rho\_a V^2 L\_b^2. \tag{6}$$

Here *W*<sup>1</sup> is the vertical loss of energy associated with fall from the height *H* in the gravitational field and *W*<sup>2</sup> is the horizontal loss of energy related to the friction with air of density *ρ<sup>a</sup>* (for further details see Eqs. (1)-(4) in Bejan & Marden, 2006a). The minimization procedure of the total loss function results in the optimal speed *Vopt* � <sup>3</sup> *MH*/*L*<sup>2</sup> *<sup>b</sup>t*, clearly leaving uncertain a choice of the dynamic variables *t* and *H*. Using the linear relation *H* � *Lb* discussed in Eq. (3) and postulating for the frictionless vertical motion that

$$t = t\_{opt} = \sqrt{\frac{2H}{g}} \sim \sqrt{\frac{L\_b}{g}} \propto M^{1/6} \,\text{s}.\tag{7}$$

the constructal theory suggests relations

$$V\_{opt} \approx \left(\frac{\rho\_b}{\rho\_a}\right)^{1/3} g^{1/2} \rho\_b^{-1/6} M^{1/6} \text{ and } T\_{opt}^{-1} \approx \left(\frac{\rho\_b}{\rho\_a}\right)^{1/3} g^{1/2} \rho\_b^{1/6} M^{-1/6} \tag{8}$$

for the optimal speed and frequency, in the case of fast running and flying animals, as shown, respectively, in Eqs. (5) and (22) and Eq. (23) in Bejan & Marden (2006a). Likewise, the frictionless vertical motion has been postulated in the case of optimal swimming, for which the energy loss *W*<sup>1</sup> � *MgLb* was adopted in Eq. (6). For further details, see Eq. (26) in Bejan & Marden (2006a).

### **2.2.3 Minimum muscular action**

Aiming to apply the principle of mechanical similarity formulated for closed inanimate systems to musculoskeletal systems of different-sized animals, one should consider conditions of the observation of effectively frictionless propagation of almost-closed (weekly open) animate systems. These are all cases of *efficient locomotion* (maximum useful work at minimum power consumption) realized at the *resonant propagation frequency*.

Since the animal locomotion is substantially muscular (e.g., McMahon, 1984; Rome et al., 1988; Dickinson et al., 2000; Alexander, 2003), the body-system relations shown in Eqs. (4) and (5) can be generalized to the muscle subsystem presented by a synergic group of locomotory muscles of the *effective muscle length Lm* and cross-sectional area *Am*. In the approximation of fully activated muscle states, the maximum muscle stiffness *Km* = *Em Am*/*Lm* is controlled by the geometry-independent muscle rigidity, i.e., the elastic modulus amplitude *Em*, defined by ratio of the peak muscle stress *σ<sup>m</sup>* = Δ*Fm*/*Am* to the peak muscle strain *ε<sup>m</sup>* = Δ*Lm*/*Lm* (e.g., McMahon, 1975, 1984). Let us treat the mass specific muscular (or relative) force output defined by *μ<sup>m</sup>* ≡ Δ*Fm*/*Mm* as the *muscular field* generating active force Δ*Fm* through the *muscle mass Mm*, which being a source of the force field *μ<sup>m</sup>* plays the role of the motor mass (Marden & Allen, 2002).

Dynamic motion parameters of the trajectory of body's center of mass are linked to the characteristic static body's and muscle's lengths via the common in biomechanics *linear* relations, namely

$$
\frac{\Delta L}{L} \sim \frac{\Delta L\_b}{L\_b} = \varepsilon\_b \sim \frac{\Delta L\_m}{L\_m} = \varepsilon\_m \frac{\Delta T}{T} = \frac{\Delta T\_m}{T\_m} = \beta, \text{ and } \Delta F \sim \Delta F\_m. \tag{9}
$$

Dynamic regime *λ* = 1 Frequency Length Speed Force Mass

<sup>273</sup> Physical Insights Into Dynamic Similarity

1 2 <sup>1</sup> · *L* − 1 2

> 1 2 <sup>1</sup> · *<sup>L</sup>* <sup>1</sup>

*f ast <sup>A</sup> <sup>T</sup>*<sup>0</sup> *<sup>L</sup>*<sup>0</sup> *<sup>V</sup>*<sup>0</sup> *<sup>F</sup>*<sup>0</sup> *<sup>ρ</sup>*

*<sup>b</sup>* <sup>=</sup> *LbL*−<sup>1</sup> *<sup>T</sup>*<sup>0</sup> *Lb* · *<sup>L</sup>*−<sup>1</sup> *<sup>V</sup>*<sup>0</sup> *<sup>F</sup>*<sup>0</sup> *<sup>M</sup>*<sup>0</sup>

*f ast* /*gM* <sup>∼</sup> *<sup>μ</sup>*1/*<sup>g</sup> <sup>T</sup>*<sup>0</sup> *<sup>L</sup>*<sup>0</sup> *<sup>V</sup>*<sup>0</sup> *<sup>F</sup>*<sup>0</sup> *<sup>M</sup>*<sup>0</sup>

Table 2. The scaling rules of dynamic similarity between animals moving with gradually changing speeds within the dynamically similar fast gaits. The corresponding dynamic regime *λ* = 1 include optimal-speed stationary states and continuous transient dynamic states, all associated with activation of the fast locomotory muscles. The calculations are provided through Eqs. (11), (12), (5), and some other definitive basic equations discussed in

Bejan & Marden (2006a, b) employed the principle of generation of the turbulent flow structure to unify gait patterns of running, swimming, and flying. Specifically, the dynamic similarity between animals across taxa is suggested as an optimal balance achieved between the vertical loss of useful energy (lifting the bodyweight, which later drops) and the horizontal loss caused by friction against the surrounding medium. Broadly speaking, the minimization procedure of total energy losses, being consistent with the concept of minimum cost of locomotion (e.g., Alexander, 2005), is underlaid by the minimization of energy consumption, treated here in terms of the efficient locomotion required by resonance conditions. Consequently, it is not surprising that the contsructal theory has demonstrated its general consistency with scaling rules attributed to the special case of dynamic similarity (Table 2). On the other hand, the optimization approach exemplified in Eq. (6), clearly demonstrating that the empirical scaling relations for speed and frequency should be hold in optimal running, flying and swimming, suggests that solely the gravitational field may explain scaling factors in scaling rules shown in Eq. (8). Therefore, a delicate question on the origin of basic scaling rules in the dynamic similarity remains unanswered in contsructal

However, the major disadvantage of the proposed principle is that the proper scaling relations

underlying the theoretical findings in Eq. (8) could straightforwardly be derived from the postulate adopted in Eq. (7), without recourse to the principle of minimum useful energy.

regardless of the minimization procedure. Indeed, the desired relations

*V*(max)

*<sup>b</sup> <sup>μ</sup>*<sup>1</sup> · *<sup>V</sup>*−<sup>1</sup> (*ρbμ*<sup>2</sup>

<sup>2</sup> *<sup>V</sup>*<sup>1</sup> (*ρbA*)<sup>−</sup> <sup>1</sup>

<sup>1</sup>*A*) 1 <sup>2</sup> ·*F*<sup>−</sup> <sup>1</sup> <sup>2</sup> *ρ* 1 6 *b μ* 1 2 <sup>1</sup> ·*M*<sup>−</sup> <sup>1</sup> 6

*<sup>m</sup> <sup>A</sup>*−<sup>1</sup> *<sup>m</sup>* · *Fm <sup>ρ</sup>*

*<sup>b</sup>* ·*<sup>L</sup> <sup>V</sup>*<sup>0</sup> *<sup>F</sup>*<sup>0</sup> *<sup>M</sup>*<sup>0</sup>

*<sup>m</sup>*-*f ast* and *<sup>μ</sup>*<sup>1</sup> <sup>=</sup> *<sup>μ</sup>*(max)

*opt* were in fact incorporated into contsructal theory

*opt* <sup>∝</sup> *Topt* <sup>∝</sup> <sup>√</sup>*<sup>L</sup>* � *Lb* <sup>∝</sup> *<sup>M</sup>*1/6, (13)

<sup>2</sup> · *<sup>F</sup>* <sup>1</sup> <sup>2</sup> *ρ* − 1 6 *<sup>b</sup> μ* 1 2 <sup>1</sup> ·*M*<sup>1</sup> 6

− 1 3 *<sup>b</sup>* · *<sup>M</sup>*<sup>1</sup> 3

− 1 3 *<sup>b</sup> <sup>μ</sup>*1·*M*<sup>2</sup> 3

2 3 *mμm*1·*M*<sup>1</sup> 3 *m*

*<sup>b</sup>*-*f ast* for, respectively,

<sup>1</sup> · *<sup>V</sup>*<sup>2</sup> (*ρbμ*1*A*)−<sup>1</sup> · *<sup>F</sup> <sup>ρ</sup>*

*T*−<sup>1</sup>

*L*(max) *opt* , *<sup>L</sup>*(max)

*V*(max) *opt* , *<sup>V</sup>*(max)

*K*(max)

*<sup>σ</sup>*(max)

*F*(max)

theory.

*<sup>b</sup>*-*f ast* <sup>∼</sup> *<sup>ρ</sup>bμ*(max)

*Stf ast* = *<sup>T</sup>*−1/*VL*−<sup>1</sup>

**3. Results and discussion 3.1 Minimum useful energy**

*opt*, *<sup>T</sup>*−<sup>1</sup> *res* , *<sup>T</sup>*−<sup>1</sup> <sup>∼</sup> (*K*/*M*)1/2 *<sup>T</sup>*−<sup>1</sup> *<sup>μ</sup>*

in Animal Locomotion. I. Theoretical Principles and Concepts

*Frf ast* <sup>=</sup> *<sup>V</sup>*2/*gLb* <sup>∼</sup> *<sup>μ</sup>*1/*<sup>g</sup> <sup>T</sup>*<sup>0</sup> *<sup>L</sup>*−<sup>1</sup>

the text, all taken at *<sup>λ</sup>* <sup>=</sup> 1. The abbreviations *<sup>μ</sup>m*<sup>1</sup> <sup>=</sup> *<sup>μ</sup>*(max)

muscle subsystem and body system are adopted.

for optimal speed *Vopt* and frequency *T*−<sup>1</sup>

*trans* , *<sup>V</sup>* <sup>=</sup> *LT*−<sup>1</sup> *<sup>μ</sup>*<sup>1</sup> · *<sup>T</sup>*<sup>1</sup> *<sup>μ</sup>*

*trans* , *<sup>L</sup> <sup>μ</sup>*<sup>1</sup> · *<sup>T</sup>*<sup>2</sup> *<sup>L</sup> <sup>μ</sup>*−<sup>1</sup>

*<sup>m</sup>*-*f ast* <sup>=</sup> <sup>Δ</sup>*Fm*/*Am <sup>ρ</sup>mμm*<sup>1</sup> · *<sup>T</sup>*<sup>2</sup> *<sup>ρ</sup>mμm*1·*Lm <sup>ρ</sup><sup>m</sup>* · *<sup>V</sup>*<sup>2</sup>


Table 1. The scaling rules of dynamic similarity in different-sized animals moving in the *stationary dynamic regime*, associated with the activation of *slow locomotor muscles*. The calculations are made on the basis of Eqs. (11) and (12) for muscular subsystem taken at *<sup>λ</sup>* <sup>=</sup> 0 and then extended to the body system via Eq. (5). The abbreviations *Em*<sup>0</sup> <sup>=</sup> *<sup>E</sup>*(max) *m*-*slow* and *<sup>E</sup>*<sup>0</sup> <sup>=</sup> *<sup>E</sup>*(max) *<sup>b</sup>*-*slow* for, respectively, muscle subsystem and body system are adopted.

Here Δ*T* is the limb ground contact time or the timing Δ*Tm* of an activated effective locomotory muscle during maximal shortening (or lengthening) resulted in the *duty factor β*. Considering effective muscles and bones (e.g., Kokshenev, 2007) of the musculoskeletal system as whole, one may ignore the relatively small effects in scaling of muscle and bone masses to body mass and thus introduce in Eqs. (3) and (9) the simplified (isometric) approximation by scaling relations

$$L \sim L\_{\mathfrak{b}} \sim L\_{\mathfrak{m}} \propto M^{1/3} \sim M\_{\mathfrak{m}}^{1/3}.\tag{10}$$

When the principle of mechanical similarity is applied to the different-sized *elastic* body musculoskeletal systems, including muscles, tendons, and bones, and moving at resonance in a certain regime *λ* along geometrically similar body's center of mass trajectories, the following three scaling relations, namely

$$T\_{m\lambda}^{-1} = T\_{\lambda}^{-1} \propto \sqrt{E\_{m\lambda}} L\_m^{-1}, \ V\_{m\lambda} \sim V\_{\lambda} \propto \sqrt{E\_{m\lambda}} \text{ and } \Delta F\_{m\lambda} \propto M\_m E\_{m\lambda} L\_m^{-1} \tag{11}$$

are suggested as the three possible determinants of dynamic similarity discussed in Eq. (5). One can see that the self-consistency between the body system, including locomotor muscle subsystem, activated in the same dynamic regime *λ*, and the principle of mechanical similarity formulated in Eq. (2), requires the dynamic elastic modulus of locomotory muscles in Eq. (11) to be adjusted with the muscle length *Lm* and body length *Lb* through the scaling relations

$$E\_{m\lambda} \propto (L\_{\mathfrak{m}})^{\lambda} \propto (L\_{\mathfrak{b}})^{\lambda} \text{ and } \mu\_{m\lambda} \propto (L\_{\mathfrak{m}})^{\lambda - 1} \sim \mu\_{\lambda} \propto (L\_{\mathfrak{b}})^{\lambda - 1}.\tag{12}$$

Thereby, the dynamic process of generation of the active force Δ*Fm<sup>λ</sup>* = *μmλMm* during muscle contractions at the resonant frequency *T*−<sup>1</sup> *<sup>m</sup><sup>λ</sup>* and the optimized contraction velocity *Vm<sup>λ</sup>* is patterned by the single dynamic exponent *λ*. In Tables 1 and 2, the scaling rules prescribed by the minimum muscular action in efficiently moving animals are provided for two well distinguished patterns of the dynamic similarity regimes *λ* = 0 and *λ* = 1.


Table 2. The scaling rules of dynamic similarity between animals moving with gradually changing speeds within the dynamically similar fast gaits. The corresponding dynamic regime *λ* = 1 include optimal-speed stationary states and continuous transient dynamic states, all associated with activation of the fast locomotory muscles. The calculations are provided through Eqs. (11), (12), (5), and some other definitive basic equations discussed in the text, all taken at *<sup>λ</sup>* <sup>=</sup> 1. The abbreviations *<sup>μ</sup>m*<sup>1</sup> <sup>=</sup> *<sup>μ</sup>*(max) *<sup>m</sup>*-*f ast* and *<sup>μ</sup>*<sup>1</sup> <sup>=</sup> *<sup>μ</sup>*(max) *<sup>b</sup>*-*f ast* for, respectively, muscle subsystem and body system are adopted.

### **3. Results and discussion**

6 Will-be-set-by-IN-TECH

Dynamic regime *λ* = 0 Frequency Length Speed Force Mass

<sup>2</sup> *F*<sup>0</sup> *ρ*

<sup>2</sup> *F*<sup>0</sup> *ρ*

<sup>0</sup> *<sup>F</sup>*<sup>0</sup> *<sup>M</sup>*<sup>0</sup>

<sup>2</sup> *<sup>L</sup>*−<sup>1</sup> · *<sup>F</sup> <sup>ρ</sup>*

<sup>2</sup> *F*<sup>0</sup> *g*−1*ρ*

*<sup>m</sup>* . (10)

*<sup>m</sup> F*<sup>0</sup>

− 1 6 *<sup>b</sup> E* 1 2 <sup>0</sup> · *<sup>M</sup>*<sup>−</sup> <sup>1</sup> 3

− 1 3 *<sup>b</sup>* · *<sup>M</sup>*<sup>1</sup> 3

− 1 3 *<sup>b</sup> <sup>E</sup>*<sup>0</sup> · *<sup>M</sup>*<sup>1</sup> 3

− 2 3 *<sup>b</sup> <sup>ε</sup>bE*<sup>0</sup> · *<sup>M</sup>*<sup>2</sup>

> − 2 3 *<sup>b</sup> <sup>E</sup>*0·*M*−1/3

*m*

*m*-*slow*

*<sup>m</sup>* , (11)

3

*<sup>m</sup> M*<sup>0</sup>

− 1 2 *<sup>b</sup> E* 1 2 <sup>0</sup> · *<sup>L</sup>*−<sup>1</sup> *<sup>b</sup> ρ* − 1 4 *<sup>b</sup> E* 1 4 <sup>0</sup> · *<sup>V</sup>*<sup>−</sup> <sup>1</sup>

<sup>0</sup> *<sup>A</sup>* · *<sup>T</sup>*−<sup>1</sup> *<sup>E</sup>*0*<sup>A</sup>* · *<sup>L</sup>*−<sup>1</sup>

<sup>0</sup> ·*T*−<sup>1</sup> *<sup>g</sup>*−1*ρ*−<sup>1</sup>

<sup>0</sup> · *<sup>T</sup>*<sup>1</sup> *<sup>L</sup>*<sup>1</sup> *<sup>ρ</sup>*

*<sup>b</sup>*-*slow <sup>T</sup>*<sup>0</sup> *<sup>L</sup>*<sup>0</sup> *<sup>V</sup>*<sup>0</sup> *<sup>F</sup>*<sup>0</sup> *<sup>ρ</sup>*

*<sup>b</sup>*-*slow* for, respectively, muscle subsystem and body system are adopted.

Here Δ*T* is the limb ground contact time or the timing Δ*Tm* of an activated effective locomotory muscle during maximal shortening (or lengthening) resulted in the *duty factor β*. Considering effective muscles and bones (e.g., Kokshenev, 2007) of the musculoskeletal system as whole, one may ignore the relatively small effects in scaling of muscle and bone masses to body mass and thus introduce in Eqs. (3) and (9) the simplified (isometric)

*<sup>L</sup>* � *Lb* � *Lm* ∝ *<sup>M</sup>*1/3 � *<sup>M</sup>*1/3

When the principle of mechanical similarity is applied to the different-sized *elastic* body musculoskeletal systems, including muscles, tendons, and bones, and moving at resonance in a certain regime *λ* along geometrically similar body's center of mass trajectories, the following

are suggested as the three possible determinants of dynamic similarity discussed in Eq. (5). One can see that the self-consistency between the body system, including locomotor muscle subsystem, activated in the same dynamic regime *λ*, and the principle of mechanical similarity formulated in Eq. (2), requires the dynamic elastic modulus of locomotory muscles in Eq. (11) to be adjusted with the muscle length *Lm* and body length *Lb* through the scaling relations

Thereby, the dynamic process of generation of the active force Δ*Fm<sup>λ</sup>* = *μmλMm* during muscle

patterned by the single dynamic exponent *λ*. In Tables 1 and 2, the scaling rules prescribed by the minimum muscular action in efficiently moving animals are provided for two well

distinguished patterns of the dynamic similarity regimes *λ* = 0 and *λ* = 1.

*<sup>m</sup>* , *Vm<sup>λ</sup>* � *<sup>V</sup><sup>λ</sup>* <sup>∝</sup> *Emλ*, and <sup>Δ</sup>*Fm<sup>λ</sup>* <sup>∝</sup> *MmEmλL*−<sup>1</sup>

*Em<sup>λ</sup>* ∝ (*Lm*)*<sup>λ</sup>* ∝ (*Lb*)*<sup>λ</sup>* and *<sup>μ</sup>m<sup>λ</sup>* ∝ (*Lm*)*λ*−<sup>1</sup> � *μλ* ∝ (*Lb*)*λ*−1. (12)

*<sup>m</sup><sup>λ</sup>* and the optimized contraction velocity *Vm<sup>λ</sup>* is

*<sup>b</sup> <sup>E</sup>*0·*L*−<sup>1</sup>

*Stslow* <sup>=</sup> *<sup>T</sup>*−<sup>1</sup>*Lb*/*<sup>V</sup>* <sup>=</sup> *LbL*−<sup>1</sup> *<sup>T</sup>*<sup>0</sup> *Lb* · *<sup>L</sup>*−<sup>1</sup> *<sup>V</sup>*<sup>0</sup> *<sup>F</sup>*<sup>0</sup> *<sup>M</sup>*<sup>0</sup>

Table 1. The scaling rules of dynamic similarity in different-sized animals moving in the *stationary dynamic regime*, associated with the activation of *slow locomotor muscles*. The calculations are made on the basis of Eqs. (11) and (12) for muscular subsystem taken at *<sup>λ</sup>* <sup>=</sup> 0 and then extended to the body system via Eq. (5). The abbreviations *Em*<sup>0</sup> <sup>=</sup> *<sup>E</sup>*(max)

− 1 4 *<sup>b</sup> E* 1 4 *<sup>m</sup>*<sup>0</sup> · *<sup>V</sup>* <sup>1</sup>

*<sup>b</sup> ρ* 1 4 *<sup>b</sup> <sup>E</sup>*<sup>−</sup> <sup>1</sup> 4 <sup>0</sup> · *<sup>V</sup>*<sup>−</sup> <sup>1</sup>

*<sup>m</sup> V*<sup>0</sup>

*<sup>b</sup> <sup>g</sup>*−1*<sup>ρ</sup>* − 3 4 *<sup>b</sup> E* 3 4 <sup>0</sup> ·*V*<sup>−</sup> <sup>1</sup>

− 1 2 *<sup>b</sup> E* 1 2

*<sup>T</sup>*−<sup>1</sup> *res* , *<sup>T</sup>*−<sup>1</sup> *<sup>m</sup>* <sup>=</sup> *<sup>T</sup>*−<sup>1</sup> <sup>∼</sup> (*K*/*M*)1/2 *<sup>T</sup>*−<sup>1</sup> *<sup>ρ</sup>*

*<sup>m</sup>*-*slow ρ*

− 1 2 *<sup>b</sup> E* 1 2

*<sup>m</sup>*<sup>0</sup> *<sup>T</sup>*<sup>0</sup> *<sup>L</sup>*<sup>0</sup>

− 1 2 *<sup>b</sup> E* 1 2

<sup>0</sup> , *Vm*-*slow <sup>T</sup>*<sup>0</sup> *<sup>L</sup>*<sup>0</sup> *<sup>ρ</sup>*

*dyn* ∼ *Lb*, *L* = *VT ρ*

*Frslow* = *V*2/*gLb g*−1*ρ*

approximation by scaling relations

three scaling relations, namely

*<sup>λ</sup>* <sup>∝</sup> *EmλL*−<sup>1</sup>

contractions at the resonant frequency *T*−<sup>1</sup>

*T*−<sup>1</sup> *<sup>m</sup><sup>λ</sup>* <sup>=</sup> *<sup>T</sup>*−<sup>1</sup>

− 1 2 *<sup>b</sup> E* 1 2

*<sup>b</sup>*-*slow* <sup>∼</sup> *<sup>K</sup>*(max)

*<sup>m</sup>*-*slow* <sup>=</sup> *<sup>ε</sup>mE*(max)

*<sup>b</sup>*-*slow* <sup>∼</sup> *<sup>ε</sup>bAE*(max)

and *<sup>E</sup>*<sup>0</sup> <sup>=</sup> *<sup>E</sup>*(max)

*L*(max)

*V*(max) *<sup>b</sup>*-*slow* = *ρ*

*K*(max)

*<sup>σ</sup>*(max)

*F*(max)

### **3.1 Minimum useful energy**

Bejan & Marden (2006a, b) employed the principle of generation of the turbulent flow structure to unify gait patterns of running, swimming, and flying. Specifically, the dynamic similarity between animals across taxa is suggested as an optimal balance achieved between the vertical loss of useful energy (lifting the bodyweight, which later drops) and the horizontal loss caused by friction against the surrounding medium. Broadly speaking, the minimization procedure of total energy losses, being consistent with the concept of minimum cost of locomotion (e.g., Alexander, 2005), is underlaid by the minimization of energy consumption, treated here in terms of the efficient locomotion required by resonance conditions. Consequently, it is not surprising that the contsructal theory has demonstrated its general consistency with scaling rules attributed to the special case of dynamic similarity (Table 2). On the other hand, the optimization approach exemplified in Eq. (6), clearly demonstrating that the empirical scaling relations for speed and frequency should be hold in optimal running, flying and swimming, suggests that solely the gravitational field may explain scaling factors in scaling rules shown in Eq. (8). Therefore, a delicate question on the origin of basic scaling rules in the dynamic similarity remains unanswered in contsructal theory.

However, the major disadvantage of the proposed principle is that the proper scaling relations for optimal speed *Vopt* and frequency *T*−<sup>1</sup> *opt* were in fact incorporated into contsructal theory regardless of the minimization procedure. Indeed, the desired relations

$$V\_{opt}^{(\text{max})} \propto T\_{opt} \propto \sqrt{L} \sim \sqrt{L\_b} \propto M^{1/6} \text{ \,\,\,}\tag{13}$$

underlying the theoretical findings in Eq. (8) could straightforwardly be derived from the postulate adopted in Eq. (7), without recourse to the principle of minimum useful energy.

required a generalization of Lagrangian's formalism from the closed mechanical systems to

<sup>275</sup> Physical Insights Into Dynamic Similarity

As the outcome of analytical study, the *slow-walk* and *fast-walk* modes in bipeds emerge as the free-like body's center of mass propagation composed by the forward translation and the elliptic-cyclic backward rotations (Kokshenev, 2004). The optimal-speed stationary regime (*λ* = 0) has been found to be consistent with a slow-walk-to-fast-walk *continuous mode transition* between the two walk modes indicated by the highest symmetry (circular) trajectory of body's center of mass (Kokshenev, 2004, Fig. 2). In contrast, the discontinues in humans *fast-walk-to-slow-run* transition is indicated by the absolute instability of the walk-gait trajectory (Kokshenev, 2004, Fig. 2), signaling on the muscular field amplitude exceeding

humans started run (Alexander & Bennet-Clark, 1976), may be treated as two *indicators* of the model-independent walk-to-run continuous transition, generally uncommon to terrestrial animals. Hence, the provided above estimates for the transient muscle field amplitude suggest the walk-to-run transition as a smoothed crossover between the slow-regime and the

The natural ability of muscles to be tuned to various dynamic regimes is incorporated in animate mechanical systems through the elastic active-force muscle modulus *E*(max)

shown in Eqs. (11) and (12). Thereby, the muscle modulus, most likely sensitive to the intrinsic dynamic muscle length (Kokshenev, 2009), establishes an additional dynamic degree of freedom, not existing in skeletal bone subsystem and other inanimate elastic mechanical systems. Conventionally, the stationary slow-speed dynamics (*λ* = 0) and optimal and transient fast-speed dynamics (*λ* = 1) are attributed to the activation of the slow-twitch-fiber muscles and fast-twitch-fiber muscles respectively recruited by animals during slow and fast locomotion (Rome et al., 1980). As illustrative example in animal swimming, the studies of gait patterns in fish (Videler & Weihs, 1982) revealed that slowly swimming and quickly swimming fish exploit, respectively, red (slow fibre) muscles or white (fast fibre) muscles, showing those contraction velocities at which recruited muscles work most

Broadly speaking, the dynamic similarity observed through the universal scaling exponents in scaling biomechanics is intimately related to the geometric similarity that can directly be observed in animals of the same taxa through the body shape, including body's locomotor appendages, i.e., limbs, wings, and tails or fins. Mathematically, the geometric similarity in animals is due to adopted spatial uniformity, preserving body shapes under arbitrary linear transformations of linear dimensions of animals (Rashevsky, 1948; McMahon, 1975; Lin, 1982). Mechanically, the dynamic similarity between animals across taxa arises from the similarity established between the geometric and kinematic parameters of the body's

Following the formalism of analytical mechanics discussed in Eq. (2), the concept of mechanical similarity has been discussed in physics (e.g., Duncan, 1953) and biomechanics (Alexander & Jayes, 1983; Alexander, 1989, 2005) in terms of the three arbitrary linear-transformation factors (*a*, *b*, and, say, *c*) preserving the homogeneity of all spatial (*L*), temporal (*T*) and force (*F*) mechanical characteristics of animals moving in a certain fashion or gait. Although no conceptual gap exists between the similarities in classical mechanics and biomechanics, the fundamental constraints imposed on the initially chosen arbitrary factors

*walk*-*run* = 0.5, discussed by Ahlborn & Blake (2002) on the basis of the data for

*walk*-*run* = *g*, completed by that for the limb

*<sup>m</sup><sup>λ</sup>* , as

the weakly open, moving at resonance biomechanical systems (Kokshenev, 2010).

gravitation, i.e., *<sup>μ</sup>run* <sup>&</sup>gt; *<sup>g</sup>*. The formal condition *<sup>μ</sup>*(mod)

in Animal Locomotion. I. Theoretical Principles and Concepts

**3.2.2 Mechanical similarity against geometric similarity**

point-mass trajectories and driving forces.

duty factor *<sup>β</sup>*(mod)

fast-regime dynamic resonant states.

efficiently (Alexander, 1989).

Strictly, the postulated basic equation *topt* = 2*H*/*g* discussed in Eq. (7) has been borrowed from frictionless Newtonian's mechanics, arising from the spatiotemporal homogeneity of the gravitational field. Indeed, one can see that Eqs. (7) and (13) is the special case discussed in Eq. (2) at *λ* = 1. The proposed theory of dynamic similarity explains that instead of the gravitational field, in fact adopted on *ad hoc* basis in Eq. (7), the muscular field determines spatiotemporal homogeneity through the universal scaling exponents established for dynamic characteristics of animals naturally tuned to the dynamic similarity regime *λ* = 1 (Table 2).

Without doubts, the gravity is important in terrestrial locomotion and the Froude number related to potential gravitational energy (*Fr* = *MV*(max)<sup>2</sup> *opt* /*MgLb*) plays important role in understanding of dynamic similarity in animate and inanimate systems (e.g.,Vaughan & O'Malleyb, 2005). After Alexander & Jayes (1983), it is widely adopted that instead of scaling relations shown in Eq. (13) the dynamic similarity between fast moving animals may be determined by the requirement for Froude numbers to be constants. Indeed, the universality of Froude numbers, i.e., *Fr* ∝ *M*0, straightforwardly provides the basic scaling rule for optimal speed, namely

$$V\_{opt}^{(\max)} = (Fr \cdot gL\_b)^{1/2} \propto L\_b^{1/2} \propto M^{1/6}.\tag{14}$$

With regard to the second basic scaling rule shown in Eq. (13), it follows from the definitive basic equation for dynamic length *L* = *VT* and Eq. (14) providing

$$T\_{opt}^{-1} = V\_{opt}^{(\text{max})} L\_{opt}^{-1} \sim (Fr \cdot gL\_b^{-1})^{1/2} \propto M^{-1/6}, \text{ with } L\_{opt}/L\_b \propto M^0. \tag{15}$$

It is noteworthy that the optimal frequency is obtained under an additional requirement of the relative dynamic length to be invariable with body mass, as shown in Eq. (15).

### **3.2 Maximum muscular efficiency**

### **3.2.1 Realization of resonant states**

During animal locomotion, *chemical energy* released by muscles in the form of muscular force field and *potential energy* of the gravitational field, both being able to be stored in body's system in the corresponding forms of active-force and reactive-force *elastic energy*, are eventually transformed into useful external body work and partially lost as a heat due to internal work and external frictional effects. In a constant-speed walk, run, flight, and swim, attributed to the dynamic similarity regime *λ* = 0, the total mechanical energy is almost unchanged and the animate mechanical system is almost closed. In the non-stationary dynamic similarity regime *λ* = 1 characteristic of the gradual change in speeds caused by the steady muscular field, the total mechanical energy may be also unchanged because of the permanent consumption of metabolic energy. The efficient propulsion of humans and other animals, in contrast to human-made engines, is accompanied by the tuning of musculoskeletal system to natural (resonant) propagation frequency (McMahon, 1975; Ahlborn & Blake, 2002), resulted in the reduction to minimum the oxygen (Hoyt & Taylor, 1981) and metabolic energy (e.g., Ahlborn & Blake, 2002; Ahlborn et al., 2006) consumption. It has been demonstrated above, that the requirement of minimum action of musculoskeletal system in animals provides the major constraint in realization of dynamically equivalent (similar) states. Nevertheless, the applicability of the key principle in analytical mechanics, driving frictionless systems, to real animate systems characteristic of non-conservative muscle forces required a special physical analysis. In the special case of the stationary (*λ* = 0) human walking (Kokshenev, 2004), the speed-dependent frictional effects were shown to be weak and therefore effectively excluded within the scope of a special dynamic perturbation theory. Likewise, the case of efficient locomotion *λ* = 1, including *moderate run* and *fast run* modes in the fast gaits of animals, has 8 Will-be-set-by-IN-TECH

Strictly, the postulated basic equation *topt* = 2*H*/*g* discussed in Eq. (7) has been borrowed from frictionless Newtonian's mechanics, arising from the spatiotemporal homogeneity of the gravitational field. Indeed, one can see that Eqs. (7) and (13) is the special case discussed in Eq. (2) at *λ* = 1. The proposed theory of dynamic similarity explains that instead of the gravitational field, in fact adopted on *ad hoc* basis in Eq. (7), the muscular field determines spatiotemporal homogeneity through the universal scaling exponents established for dynamic characteristics of animals naturally tuned to the dynamic similarity regime *λ* = 1 (Table 2). Without doubts, the gravity is important in terrestrial locomotion and the Froude number

understanding of dynamic similarity in animate and inanimate systems (e.g.,Vaughan & O'Malleyb, 2005). After Alexander & Jayes (1983), it is widely adopted that instead of scaling relations shown in Eq. (13) the dynamic similarity between fast moving animals may be determined by the requirement for Froude numbers to be constants. Indeed, the universality of Froude numbers, i.e., *Fr* ∝ *M*0, straightforwardly provides the basic scaling rule for optimal

With regard to the second basic scaling rule shown in Eq. (13), it follows from the definitive

It is noteworthy that the optimal frequency is obtained under an additional requirement of

During animal locomotion, *chemical energy* released by muscles in the form of muscular force field and *potential energy* of the gravitational field, both being able to be stored in body's system in the corresponding forms of active-force and reactive-force *elastic energy*, are eventually transformed into useful external body work and partially lost as a heat due to internal work and external frictional effects. In a constant-speed walk, run, flight, and swim, attributed to the dynamic similarity regime *λ* = 0, the total mechanical energy is almost unchanged and the animate mechanical system is almost closed. In the non-stationary dynamic similarity regime *λ* = 1 characteristic of the gradual change in speeds caused by the steady muscular field, the total mechanical energy may be also unchanged because of the permanent consumption of metabolic energy. The efficient propulsion of humans and other animals, in contrast to human-made engines, is accompanied by the tuning of musculoskeletal system to natural (resonant) propagation frequency (McMahon, 1975; Ahlborn & Blake, 2002), resulted in the reduction to minimum the oxygen (Hoyt & Taylor, 1981) and metabolic energy (e.g., Ahlborn & Blake, 2002; Ahlborn et al., 2006) consumption. It has been demonstrated above, that the requirement of minimum action of musculoskeletal system in animals provides the major constraint in realization of dynamically equivalent (similar) states. Nevertheless, the applicability of the key principle in analytical mechanics, driving frictionless systems, to real animate systems characteristic of non-conservative muscle forces required a special physical analysis. In the special case of the stationary (*λ* = 0) human walking (Kokshenev, 2004), the speed-dependent frictional effects were shown to be weak and therefore effectively excluded within the scope of a special dynamic perturbation theory. Likewise, the case of efficient locomotion *λ* = 1, including *moderate run* and *fast run* modes in the fast gaits of animals, has

*opt* = (*Fr* · *gLb*)1/2 <sup>∝</sup> *<sup>L</sup>*1/2

the relative dynamic length to be invariable with body mass, as shown in Eq. (15).

*opt* /*MgLb*) plays important role in

*<sup>b</sup>* <sup>∝</sup> *<sup>M</sup>*1/6. (14)

*<sup>b</sup>* )1/2 <sup>∝</sup> *<sup>M</sup>*<sup>−</sup>1/6, with *Lopt*/*Lb* <sup>∝</sup> *<sup>M</sup>*0. (15)

related to potential gravitational energy (*Fr* = *MV*(max)<sup>2</sup>

*V*(max)

basic equation for dynamic length *L* = *VT* and Eq. (14) providing

*opt* � (*Fr* · *gL*−<sup>1</sup>

speed, namely

*T*−<sup>1</sup>

**3.2 Maximum muscular efficiency 3.2.1 Realization of resonant states**

*opt* <sup>=</sup> *<sup>V</sup>*(max)

*opt <sup>L</sup>*−<sup>1</sup>

required a generalization of Lagrangian's formalism from the closed mechanical systems to the weakly open, moving at resonance biomechanical systems (Kokshenev, 2010).

As the outcome of analytical study, the *slow-walk* and *fast-walk* modes in bipeds emerge as the free-like body's center of mass propagation composed by the forward translation and the elliptic-cyclic backward rotations (Kokshenev, 2004). The optimal-speed stationary regime (*λ* = 0) has been found to be consistent with a slow-walk-to-fast-walk *continuous mode transition* between the two walk modes indicated by the highest symmetry (circular) trajectory of body's center of mass (Kokshenev, 2004, Fig. 2). In contrast, the discontinues in humans *fast-walk-to-slow-run* transition is indicated by the absolute instability of the walk-gait trajectory (Kokshenev, 2004, Fig. 2), signaling on the muscular field amplitude exceeding gravitation, i.e., *<sup>μ</sup>run* <sup>&</sup>gt; *<sup>g</sup>*. The formal condition *<sup>μ</sup>*(mod) *walk*-*run* = *g*, completed by that for the limb duty factor *<sup>β</sup>*(mod) *walk*-*run* = 0.5, discussed by Ahlborn & Blake (2002) on the basis of the data for humans started run (Alexander & Bennet-Clark, 1976), may be treated as two *indicators* of the model-independent walk-to-run continuous transition, generally uncommon to terrestrial animals. Hence, the provided above estimates for the transient muscle field amplitude suggest the walk-to-run transition as a smoothed crossover between the slow-regime and the fast-regime dynamic resonant states.

The natural ability of muscles to be tuned to various dynamic regimes is incorporated in animate mechanical systems through the elastic active-force muscle modulus *E*(max) *<sup>m</sup><sup>λ</sup>* , as shown in Eqs. (11) and (12). Thereby, the muscle modulus, most likely sensitive to the intrinsic dynamic muscle length (Kokshenev, 2009), establishes an additional dynamic degree of freedom, not existing in skeletal bone subsystem and other inanimate elastic mechanical systems. Conventionally, the stationary slow-speed dynamics (*λ* = 0) and optimal and transient fast-speed dynamics (*λ* = 1) are attributed to the activation of the slow-twitch-fiber muscles and fast-twitch-fiber muscles respectively recruited by animals during slow and fast locomotion (Rome et al., 1980). As illustrative example in animal swimming, the studies of gait patterns in fish (Videler & Weihs, 1982) revealed that slowly swimming and quickly swimming fish exploit, respectively, red (slow fibre) muscles or white (fast fibre) muscles, showing those contraction velocities at which recruited muscles work most efficiently (Alexander, 1989).

### **3.2.2 Mechanical similarity against geometric similarity**

Broadly speaking, the dynamic similarity observed through the universal scaling exponents in scaling biomechanics is intimately related to the geometric similarity that can directly be observed in animals of the same taxa through the body shape, including body's locomotor appendages, i.e., limbs, wings, and tails or fins. Mathematically, the geometric similarity in animals is due to adopted spatial uniformity, preserving body shapes under arbitrary linear transformations of linear dimensions of animals (Rashevsky, 1948; McMahon, 1975; Lin, 1982). Mechanically, the dynamic similarity between animals across taxa arises from the similarity established between the geometric and kinematic parameters of the body's point-mass trajectories and driving forces.

Following the formalism of analytical mechanics discussed in Eq. (2), the concept of mechanical similarity has been discussed in physics (e.g., Duncan, 1953) and biomechanics (Alexander & Jayes, 1983; Alexander, 1989, 2005) in terms of the three arbitrary linear-transformation factors (*a*, *b*, and, say, *c*) preserving the homogeneity of all spatial (*L*), temporal (*T*) and force (*F*) mechanical characteristics of animals moving in a certain fashion or gait. Although no conceptual gap exists between the similarities in classical mechanics and biomechanics, the fundamental constraints imposed on the initially chosen arbitrary factors

indicates stabilization of the relative dynamic length, as shown in Eqs. (15) and (16). Hence, experimental observation of the constant Strouhal number in animals moving efficiently in any similar gaits determines them as dynamically similar. When solely the fast gaits are considered, the universal Strouhal number should be revealed along with the universal Froude number *Fr*, whose stabilization in animals across body mass is equivalent to the observation of scaling rule for the maximum amplitude of optimal and transient speeds

<sup>277</sup> Physical Insights Into Dynamic Similarity

each may play the role of the dynamic similarity criterion, are chosen for the corresponding cases *λ* = 0 and *λ* = 1, other universal determinants, such as muscle stress, speed, in the first case, and the Strouhal number and relative muscular field, in the second case, play the role of the indicators of stabilization of the universal similar dynamic states. Moreover, in both cases the *indicators of dynamic similarity* can be extended by the universal duty factors *β* ∝ *M*<sup>0</sup> and

The stationary-state mode patterns of flight gaits were likely first noted by Hill (1950). He had established that the wing frequencies of *hovering birds* are in inverse proportionality with the

for wing muscles contracting in the stationary regime *λ* = 0. Hill's pioneering observation of the fundamental frequency-length scaling law, determining dynamic similarity in the efficient flyers solely via the universal (speed-, period-, and mass-independent) *slow-muscle* elastic

*<sup>m</sup>*<sup>0</sup> (Table 1) can be compared with seminal Kepler's law *<sup>T</sup>*(exp)−<sup>1</sup>

equivalent to observation of the universal Strouhal numbers *Shover* ∝ *M*<sup>0</sup> (Table 1).

*<sup>m</sup>*-*slow* <sup>∝</sup> *<sup>M</sup>*2/3

larger birds flap their wings more slowly than smaller ones (Hill, 1950).

(see Eq. (2) taken at *λ* = −1), determining the mechanical similarity between trajectories of planets driven solely by gravitation. Later, the hovering flight motors were also recognized in

When an animal travels or cruises slowly for long distances, maintaining constant the optimal

throughout the terrestrial, air, or water environment resisting drag forces, the limbs, wings, and fins are expected to be tuned via elastic muscle modulus to maintain universal muscular pressure (Table 1). In turn, this effect gives rise to the constant limb-muscle *safety factor* (ratio of muscle strength to peak functional stress), foreseeing by Hill (1950). Moreover, the peak

swimming by animals ranged over nine orders of body mass, was documented by Alexander (1985). The theoretical suggestion that the stationary-state mechanics, equilibrating all drag forces, is due to slow locomotory muscles, is corroborated by the statistically regressed

The optimum-speed regime *λ* = 1 has been recognized through the equilibration of the air drag by wings of *flapping birds*, manifesting the basic scaling rule for wing frequencies

*flap* <sup>∝</sup> *<sup>M</sup>*−1/6 (Ellington, 1991), also corresponding to the observation of *Sflap* <sup>∝</sup> *<sup>M</sup>*<sup>0</sup> (Table 2). Earlier, the non-stationary flight regime has been foreseen by Hill's notion that

As explicitly shown in Eq. (16), the stabilization of the uniformity in Strouhal numbers in both dynamic regimes *λ* = 0 and *λ* = 1 explains empirically puzzled animal flight and

<sup>0</sup> /*ρ<sup>b</sup>* invariable with bodyweight and frequency (Table 1), or moves

*body* <sup>∝</sup> *<sup>M</sup>*2/3, exerted on the environment during running, flying, and

*<sup>b</sup>* , as follows from Eqs. (14) and (16). If the principal determinants *St* or *Fr*,

*hover* �

 *E*(max)

*hover* <sup>∝</sup> *<sup>M</sup>*−1/3 (Ellington, 1991), that is also

*<sup>m</sup>* remarkably established in both biological and

*<sup>m</sup>*<sup>0</sup> /*ρmL*−<sup>1</sup> *<sup>m</sup>* (Table 1)

*planet* <sup>∝</sup> *<sup>L</sup>*(exp)−3/2 *planet*

*V*(max)

*opt* <sup>∝</sup> *<sup>L</sup>*1/2

swept angles Θ ∝ *M*0.

modulus *E*(max)

speed *V*(max)

*<sup>T</sup>*(exp)−<sup>1</sup>

*cruis* �

body force output *F*(exp)

data on the force output *F*(max)

 *E*(max)

**3.2.4 Mechanically efficient slow and fast flyers**

in Animal Locomotion. I. Theoretical Principles and Concepts

birds by observation of the wing frequencies *<sup>T</sup>*(exp)−<sup>1</sup>

human-made *slow motors* by Marden & Allen (2002).

linear size, that corresponds to the predicted frequency *T*(*pred*)−<sup>1</sup>

of linear transformation, i.e., *b* = *a*1−*λ*/2 and *c* = *aλ*−<sup>1</sup> , underlying Eq. (2) and providing basic scaling rules of the dynamic similarity, namely

$$T\_{\lambda}^{-1} \propto L\_{b}^{\lambda/2 - 1}, V\_{\lambda} \propto L\_{b}^{\lambda/2}, \text{and } \Delta F\_{\lambda} \propto L\_{b}^{\lambda - 1}, \text{with } L/L\_{b} \propto M^{0}. \tag{16}$$

are generally ignored in experimental biology and even violated in some theoretical studies mentioned in the Introduction. This analysis shows how the spatial uniformity, achieved via the stabilization of dynamic length in relation to static length, determines the observation of dynamic similarity in any dynamic regime *λ*.

The provided analysis explains why the model-dependent requirements of strict geometric similarity, e.g., the requirement of the equality of joint angles in running animals (Alexander, 1989) or swinging angles in dynamically similar pendulums (Alexander, 2005), do not reduce the dynamic similarity concept to the so-called *strict dynamic similarity* (Alexander, 1989, 2005). First, one can see that both the angles may be expressed in terms of the *swept angle* Θ = arcsin(*β*/2*St*) introduced in biomechanics of animal locomotion as a relative angle between the leg spring and vertical (see, e.g., Fig. 1 in Farley et al., 1993), which is also known as the maximum *compass angle* (Cynthia & Farley, 1998), modeling in turn the *protraction-retraction angle* in bipeds (Gatesy & Biewener, 1991, Fig. 1). Second, the requirement of observation of the dynamic similarity through scaling laws of the mechanical similarity reproduced in Eq. (16) implies that the Strouhal number *St* (= *Lb*/*L*), as well as some other dimensionless numbers, including the duty factor *β* (= Δ*T*/*T*), should be invariable across different-sized animals, i.e., *St* � *β* ∝ *M*0. One may infer that a rigorous requirement of the strict geometric similarity on that the swing or other related angles (as well as relative stride lengths) must be equal constants is the overestimated constraint of the dynamic similarity concept. In other words, the requirement of dynamic angles to be mass independent, i.e., Θ ∝ *M*0, arising from the the requirement of stabilization of relative dynamic lengths, i.e., *St* ∝ *M*<sup>0</sup> , unambiguously determines observation of the *perfect* dynamic similarity (Alexander, 1989).

### **3.2.3 Criterion, determinants, and indicators of dynamic similarity**

Being the major requirement in realization of both universal dynamic regimes unifying animals in a certain gait, the high mechanical efficiency of the musculoskeletal system prescribed by minimum muscular action plays the role of the unique *criterion* of observation of dynamic similarity.

The concept of mechanical similarity in biomechanics, consistent with that in analytical mechanics, allows linear transformations of two dynamic (spatial and temporal) and one mechanical (force or mass) characteristics through the three independent scaling factors. Hence, the set (*T*−1, *V*, Δ*F*) of mutually independent and model-independent quantities, chosen in Eq. (5) in the *ltf* class of units, can be treated as tentative candidates for the determinants of dynamic similarity. The optimization of muscle-field interactions by the minimum mechanical action (i) introduces new state-dependent scaling relations for *T*−<sup>1</sup> *<sup>λ</sup>* , *Vλ*, and Δ*F<sup>λ</sup>* , which determine three scaling rules of the dynamic similarity, and (ii) reduces the number of independent determinants from three to one, as discussed in Eq. (16). Since the choice of the optimal speed *<sup>V</sup><sup>λ</sup>* by an animal is accomplished by the resonant frequency *<sup>T</sup>*−<sup>1</sup> *<sup>λ</sup>* , the suggested principal set of determinants (*T*−<sup>1</sup> *<sup>λ</sup>* , *Vλ*) does not generally excludes another set, including the uniform muscular field or the uniform relative dynamic length, both required by the high level mechanical efficiency. However, the observation of just only one of the two basic scaling rules guarantees the observation of other features of dynamic similarity in animals. Indeed, the observation of the scaling rule for stride frequency *T*−<sup>1</sup> *<sup>λ</sup>* <sup>∝</sup> *<sup>L</sup>λ*/2−<sup>1</sup> *b*

10 Will-be-set-by-IN-TECH

of linear transformation, i.e., *b* = *a*1−*λ*/2 and *c* = *aλ*−<sup>1</sup> , underlying Eq. (2) and providing

are generally ignored in experimental biology and even violated in some theoretical studies mentioned in the Introduction. This analysis shows how the spatial uniformity, achieved via the stabilization of dynamic length in relation to static length, determines the observation of

The provided analysis explains why the model-dependent requirements of strict geometric similarity, e.g., the requirement of the equality of joint angles in running animals (Alexander, 1989) or swinging angles in dynamically similar pendulums (Alexander, 2005), do not reduce the dynamic similarity concept to the so-called *strict dynamic similarity* (Alexander, 1989, 2005). First, one can see that both the angles may be expressed in terms of the *swept angle* Θ = arcsin(*β*/2*St*) introduced in biomechanics of animal locomotion as a relative angle between the leg spring and vertical (see, e.g., Fig. 1 in Farley et al., 1993), which is also known as the maximum *compass angle* (Cynthia & Farley, 1998), modeling in turn the *protraction-retraction angle* in bipeds (Gatesy & Biewener, 1991, Fig. 1). Second, the requirement of observation of the dynamic similarity through scaling laws of the mechanical similarity reproduced in Eq. (16) implies that the Strouhal number *St* (= *Lb*/*L*), as well as some other dimensionless numbers, including the duty factor *β* (= Δ*T*/*T*), should be invariable across different-sized animals, i.e., *St* � *β* ∝ *M*0. One may infer that a rigorous requirement of the strict geometric similarity on that the swing or other related angles (as well as relative stride lengths) must be equal constants is the overestimated constraint of the dynamic similarity concept. In other words, the requirement of dynamic angles to be mass independent, i.e., Θ ∝ *M*0, arising from the the requirement of stabilization of relative dynamic lengths, i.e., *St* ∝ *M*<sup>0</sup> , unambiguously

*<sup>b</sup>* , and <sup>Δ</sup>*F<sup>λ</sup>* <sup>∝</sup> *<sup>L</sup>λ*−<sup>1</sup>

*<sup>b</sup>* , with *<sup>L</sup>*/*Lb* <sup>∝</sup> *<sup>M</sup>*0, (16)

*<sup>λ</sup>* , *Vλ*,

*<sup>λ</sup>* <sup>∝</sup> *<sup>L</sup>λ*/2−<sup>1</sup> *b*

*<sup>λ</sup>* , *Vλ*) does not generally excludes another set,

*<sup>λ</sup>* ,

basic scaling rules of the dynamic similarity, namely

*<sup>b</sup>* , *<sup>V</sup><sup>λ</sup>* <sup>∝</sup> *<sup>L</sup>λ*/2

determines observation of the *perfect* dynamic similarity (Alexander, 1989).

Being the major requirement in realization of both universal dynamic regimes unifying animals in a certain gait, the high mechanical efficiency of the musculoskeletal system prescribed by minimum muscular action plays the role of the unique *criterion* of observation

The concept of mechanical similarity in biomechanics, consistent with that in analytical mechanics, allows linear transformations of two dynamic (spatial and temporal) and one mechanical (force or mass) characteristics through the three independent scaling factors. Hence, the set (*T*−1, *V*, Δ*F*) of mutually independent and model-independent quantities, chosen in Eq. (5) in the *ltf* class of units, can be treated as tentative candidates for the determinants of dynamic similarity. The optimization of muscle-field interactions by the minimum mechanical action (i) introduces new state-dependent scaling relations for *T*−<sup>1</sup>

and Δ*F<sup>λ</sup>* , which determine three scaling rules of the dynamic similarity, and (ii) reduces the number of independent determinants from three to one, as discussed in Eq. (16). Since the choice of the optimal speed *<sup>V</sup><sup>λ</sup>* by an animal is accomplished by the resonant frequency *<sup>T</sup>*−<sup>1</sup>

including the uniform muscular field or the uniform relative dynamic length, both required by the high level mechanical efficiency. However, the observation of just only one of the two basic scaling rules guarantees the observation of other features of dynamic similarity

in animals. Indeed, the observation of the scaling rule for stride frequency *T*−<sup>1</sup>

**3.2.3 Criterion, determinants, and indicators of dynamic similarity**

the suggested principal set of determinants (*T*−<sup>1</sup>

*<sup>λ</sup>* <sup>∝</sup> *<sup>L</sup>λ*/2−<sup>1</sup>

dynamic similarity in any dynamic regime *λ*.

*T*−<sup>1</sup>

of dynamic similarity.

indicates stabilization of the relative dynamic length, as shown in Eqs. (15) and (16). Hence, experimental observation of the constant Strouhal number in animals moving efficiently in any similar gaits determines them as dynamically similar. When solely the fast gaits are considered, the universal Strouhal number should be revealed along with the universal Froude number *Fr*, whose stabilization in animals across body mass is equivalent to the observation of scaling rule for the maximum amplitude of optimal and transient speeds *V*(max) *opt* <sup>∝</sup> *<sup>L</sup>*1/2 *<sup>b</sup>* , as follows from Eqs. (14) and (16). If the principal determinants *St* or *Fr*, each may play the role of the dynamic similarity criterion, are chosen for the corresponding cases *λ* = 0 and *λ* = 1, other universal determinants, such as muscle stress, speed, in the first case, and the Strouhal number and relative muscular field, in the second case, play the role of the indicators of stabilization of the universal similar dynamic states. Moreover, in both cases the *indicators of dynamic similarity* can be extended by the universal duty factors *β* ∝ *M*<sup>0</sup> and swept angles Θ ∝ *M*0.

### **3.2.4 Mechanically efficient slow and fast flyers**

The stationary-state mode patterns of flight gaits were likely first noted by Hill (1950). He had established that the wing frequencies of *hovering birds* are in inverse proportionality with the linear size, that corresponds to the predicted frequency *T*(*pred*)−<sup>1</sup> *hover* � *E*(max) *<sup>m</sup>*<sup>0</sup> /*ρmL*−<sup>1</sup> *<sup>m</sup>* (Table 1) for wing muscles contracting in the stationary regime *λ* = 0. Hill's pioneering observation of the fundamental frequency-length scaling law, determining dynamic similarity in the efficient flyers solely via the universal (speed-, period-, and mass-independent) *slow-muscle* elastic modulus *E*(max) *<sup>m</sup>*<sup>0</sup> (Table 1) can be compared with seminal Kepler's law *<sup>T</sup>*(exp)−<sup>1</sup> *planet* <sup>∝</sup> *<sup>L</sup>*(exp)−3/2 *planet* (see Eq. (2) taken at *λ* = −1), determining the mechanical similarity between trajectories of planets driven solely by gravitation. Later, the hovering flight motors were also recognized in birds by observation of the wing frequencies *<sup>T</sup>*(exp)−<sup>1</sup> *hover* <sup>∝</sup> *<sup>M</sup>*−1/3 (Ellington, 1991), that is also equivalent to observation of the universal Strouhal numbers *Shover* ∝ *M*<sup>0</sup> (Table 1). When an animal travels or cruises slowly for long distances, maintaining constant the optimal speed *V*(max) *cruis* � *E*(max) <sup>0</sup> /*ρ<sup>b</sup>* invariable with bodyweight and frequency (Table 1), or moves throughout the terrestrial, air, or water environment resisting drag forces, the limbs, wings, and fins are expected to be tuned via elastic muscle modulus to maintain universal muscular pressure (Table 1). In turn, this effect gives rise to the constant limb-muscle *safety factor* (ratio of muscle strength to peak functional stress), foreseeing by Hill (1950). Moreover, the peak body force output *F*(exp) *body* <sup>∝</sup> *<sup>M</sup>*2/3, exerted on the environment during running, flying, and swimming by animals ranged over nine orders of body mass, was documented by Alexander (1985). The theoretical suggestion that the stationary-state mechanics, equilibrating all drag forces, is due to slow locomotory muscles, is corroborated by the statistically regressed data on the force output *F*(max) *<sup>m</sup>*-*slow* <sup>∝</sup> *<sup>M</sup>*2/3 *<sup>m</sup>* remarkably established in both biological and human-made *slow motors* by Marden & Allen (2002).

The optimum-speed regime *λ* = 1 has been recognized through the equilibration of the air drag by wings of *flapping birds*, manifesting the basic scaling rule for wing frequencies *<sup>T</sup>*(exp)−<sup>1</sup> *flap* <sup>∝</sup> *<sup>M</sup>*−1/6 (Ellington, 1991), also corresponding to the observation of *Sflap* <sup>∝</sup> *<sup>M</sup>*<sup>0</sup> (Table 2). Earlier, the non-stationary flight regime has been foreseen by Hill's notion that larger birds flap their wings more slowly than smaller ones (Hill, 1950).

As explicitly shown in Eq. (16), the stabilization of the uniformity in Strouhal numbers in both dynamic regimes *λ* = 0 and *λ* = 1 explains empirically puzzled animal flight and

(from smallest bipedal rodents to largest quadrupedal elephants) indicates observation of the

<sup>279</sup> Physical Insights Into Dynamic Similarity

The dynamically similar continuous resonant states were clearly revealed by Farley et al. (1993) in a trotting rat, dog, goat, horse and a hopping tammar wallaby and red kangaroo. The

provided the following scaling equations for the stride frequency *<sup>T</sup>*(exp)−<sup>1</sup> *run* , peak force output *<sup>F</sup>*(exp) *run* , maximum body stiffness *<sup>K</sup>*(exp) *run* , swept angle <sup>Θ</sup>(exp) *run* , and dynamic length change

*<sup>T</sup>*−<sup>1</sup> *run* � <sup>Δ</sup>*T*−<sup>1</sup> *run* <sup>∝</sup> *<sup>M</sup>*−0.19±0.06 , *<sup>F</sup>*(exp) *run* <sup>=</sup> 30.1*M*0.97±0.14, *<sup>K</sup>*(exp) *run* <sup>∝</sup> *<sup>M</sup>*0.67±0.15,

One can see that all the observed scaling exponents are consistent (within the experimental error) with those predicted by the dynamic similarity regime *λ* = 1 described in Table 2.

Following the concept of mechanical similarity, underlaid by the key principle of minimum action in analytical mechanics, the theory of dynamic similarity in animal locomotion is proposed. Exploring the intrinsic property of locomotory muscles to be tuned, via the variable muscle elasticity, to the natural cyclic frequency characteristic of high level efficiency of locomotion, the scaling rules driving the dynamic similarity in inanimate mechanical elastic systems are suggested for the special case of active-force animate elastic systems. The linear-displacement dynamic approach to contracting locomotory muscles, whose resonant frequencies are required by the principle of minimum mechanical action, establishes two different universal patterns of the dynamic regimes of similarity in different-sized animals distinguished by the dynamic scaling exponent *λ*. The determinants of the stationary locomotion of animals moving at optimal constant speeds (the case *λ* = 0) and the non-stationary locomotion at gradually changing speeds (*λ* = 1), including the transient-mode speed transitions, are self-consistently inferred and described in Tables 1 and 2, respectively. Exemplified by the non-stationary dynamic regime *λ* = 1, the two principal sets of determinants of the dynamic similarity are suggested by the universal exponents for the speed and frequency scaled with body mass, which may be equivalently presented by the corresponding Froude and Strouhal numbers or by other universal dimensionless numbers

The primary determinant, playing the role of the unique criterion of the linear dynamic similarity, is shown (in Eq. (16)) to be the Strouhal number, whose universality in animals across body mass indicates establishing of the linearity between the stride or stroke length and the body length in each animal, falling into one or other dynamically similar regime. In the special case of non-stationary dynamic similarity controlled by fast locomotory muscles, the Froude number may be equivalently chosen as a unique criterion of similarity, as hypothesized by Alexander (Alexander, 1976; Alexander & Jayes, 1983; Alexander, 1989). Since the scaling theory of similarity deals only with scaling relations, but not with scaling equations, Alexander's strict requirement that dynamic similarly between running animals should equal Froude numbers is not generally required by the theory. Instead, the theory of dynamic similarity stands only that changing with speed Froude numbers should be

These data introduce the dynamic similarity pattern of *efficient trotters and hoppers*.

determining the states of dynamic similarity in different-sized animals.

*leg* (� *<sup>L</sup>*(exp)

*<sup>b</sup>* <sup>∝</sup> *<sup>M</sup>*0.30±0.15. (17)

*<sup>b</sup>* )

realistic modeling on the basis of leg-spring model of animals of leg length *L*(exp)

<sup>Θ</sup>(exp) *run* <sup>∝</sup> *<sup>M</sup>*−0.03±0.1, and <sup>Δ</sup>*L*(exp) *run* � *<sup>L</sup>*(exp)

pattern of *efficient runners in mammals*.

in Animal Locomotion. I. Theoretical Principles and Concepts

<sup>Δ</sup>*L*(exp) *run* , namely

**4. Concluding remarks**

swim (Whitfield, 2003). Indeed, Taylor et al. (2003) experimentally established such a kind of the universal similarity through the almost constant Strouhal numbers (laying between 0.2 and 0.4) in cruising with a high power efficiency dolphins, flapping birds, and bats. This observation suggests two patterns of *efficient flyers* distinguished by hovering (*λ* = 0) and flapping (*λ* = 1) modes of slow and fast flight gaits. The corresponding examples of slow-swimming and fast-swimming motors are the gaits to swim established through different swimming techniques for the same fish, using their pectoral fins to swim slowly, but undulating the whole body to swim fast (Alexander, 1989). Unifying flying and swimming animals in *fast gaits*, the data by Taylor et al. (2003) suggest the dynamic similarity pattern of *efficient flapping flyers* and *undulating swimmers*. These and other introduced patterns of dynamic similarity are studied in Kokshenev (2011).

### **3.2.5 Mechanically efficient fast animals**

In experimental biology, it is well known that forces required for fast gaits in animals are proportional to body weight, but since the force generation is more expensive of metabolic energy in faster muscles, small animals show apparently low efficiencies in running (Alexander, 1989). Efficient fast biological motors in running, flying, and swimming animals were established by Marden & Allen (2002) through the scaling equation *F*(max) *opt* <sup>=</sup> *<sup>μ</sup>*(max) *f ast M* (Table 2), where the relative force amplitude *<sup>μ</sup>*(max) *f ast* = 2*g* was re-estimated by Bejan & Marden (2006a, Fig. 2C). Given that the muscular field in running, flying, and swimming animals is twice as many as the gravitational field, adopted above for the lower threshold of a slow run in the walk-to-run transition, the statistical data *<sup>μ</sup>*(max) *f ast* = 2*g* may be conventionally adopted as a universal threshold of fast modes in fast gaits. This threshold associated with the slow-run-to-fast run transient state *λ* = 1 may in turn determine the pattern of efficient fast animals, including fast running mammals, reptiles, insects; flapping birds, bats, and insects; undulating fish and crayfish, according to Marden & Allen (2002) and Bejan & Marden (2006a). The pioneering observations of the transient-state speeds *V*(exp) *trans* and frequencies *<sup>T</sup>*(exp)−<sup>1</sup> *trans* at the trot-to-gallop continuous transition in quadrupeds (*V*(exp) *trot*-*gall* <sup>∝</sup> *<sup>M</sup>*0.22±0.05 and *<sup>T</sup>*(exp)−<sup>1</sup> *trot*-*gall* ∝ *M*−0.15±0.03; Heglund et al., 1974; Heglund & Taylor, 1988), make evidence for, within the experimental error, the predicted stabilization of the uniform (body mass independent) muscular field, i.e., *<sup>μ</sup>*(exp) *run* <sup>∝</sup> *<sup>M</sup>*−0.07±0.08, when tested by the scaling relation *<sup>μ</sup>*(*pred*) *trans* � *VtransT*−<sup>1</sup> *trans* prescribed by dynamic regime *λ* = 1 (Table 2). Likewise, the same generic dynamic regime explains stabilization of the uniformity in the muscular field activated in 13 running animals (from a mice to horses) observed indirectly by Heglund &Taylor (1988) at experimental conditions of the *preferred* trotting speeds and the preferred galloping speeds, providing respectively the determinants of similarity *<sup>μ</sup>*(exp) *trot* <sup>∝</sup> *<sup>M</sup>*0.09±0.07 and *<sup>μ</sup>*(exp) *gall* ∝ *M*0.02±0.07. The revealed large experimental error is most likely caused by small quadrupedal species (one laboratory mice, two chipmunks, three squirrels, and three white rats of bodyweight not exceeding one *kg*), which should be excluded from the dynamic similarity pattern, as potentially having low mechanical efficiency (Alexander, 1989).

Bipeds, showing the resonant frequency *T*(exp)−<sup>1</sup> *trans* ∝ *<sup>M</sup>*−0.178 near the slow-walk-to-fast-run transition (Gatesy & Biewener, 1991), indicate the dynamic similarity pattern of *efficient fast walkers*. The scaling rule *<sup>V</sup>*(exp) *opt* ∝ *<sup>M</sup>*0.17 empirically established by Garland (1983) for maximal speeds in running terrestrial mammals ranging in five orders in body mass (from smallest bipedal rodents to largest quadrupedal elephants) indicates observation of the pattern of *efficient runners in mammals*.

The dynamically similar continuous resonant states were clearly revealed by Farley et al. (1993) in a trotting rat, dog, goat, horse and a hopping tammar wallaby and red kangaroo. The realistic modeling on the basis of leg-spring model of animals of leg length *L*(exp) *leg* (� *<sup>L</sup>*(exp) *<sup>b</sup>* )

provided the following scaling equations for the stride frequency *<sup>T</sup>*(exp)−<sup>1</sup> *run* , peak force output *<sup>F</sup>*(exp) *run* , maximum body stiffness *<sup>K</sup>*(exp) *run* , swept angle <sup>Θ</sup>(exp) *run* , and dynamic length change <sup>Δ</sup>*L*(exp) *run* , namely

$$T\_{\rm run}^{-1} \sim \Delta T\_{\rm run}^{-1} \propto M^{-0.19 \pm 0.06}, F\_{\rm run}^{(\rm exp)} = 30.1 M^{0.97 \pm 0.14}, K\_{\rm run}^{(\rm exp)} \propto M^{0.67 \pm 0.15},$$

$$\Theta\_{\rm run}^{(\rm exp)} \propto M^{-0.03 \pm 0.1}, \text{and } \Delta L\_{\rm run}^{(\rm exp)} \sim L\_b^{(\rm exp)} \propto M^{0.30 \pm 0.15}.\tag{17}$$

One can see that all the observed scaling exponents are consistent (within the experimental error) with those predicted by the dynamic similarity regime *λ* = 1 described in Table 2. These data introduce the dynamic similarity pattern of *efficient trotters and hoppers*.

### **4. Concluding remarks**

12 Will-be-set-by-IN-TECH

swim (Whitfield, 2003). Indeed, Taylor et al. (2003) experimentally established such a kind of the universal similarity through the almost constant Strouhal numbers (laying between 0.2 and 0.4) in cruising with a high power efficiency dolphins, flapping birds, and bats. This observation suggests two patterns of *efficient flyers* distinguished by hovering (*λ* = 0) and flapping (*λ* = 1) modes of slow and fast flight gaits. The corresponding examples of slow-swimming and fast-swimming motors are the gaits to swim established through different swimming techniques for the same fish, using their pectoral fins to swim slowly, but undulating the whole body to swim fast (Alexander, 1989). Unifying flying and swimming animals in *fast gaits*, the data by Taylor et al. (2003) suggest the dynamic similarity pattern of *efficient flapping flyers* and *undulating swimmers*. These and other introduced patterns of

In experimental biology, it is well known that forces required for fast gaits in animals are proportional to body weight, but since the force generation is more expensive of metabolic energy in faster muscles, small animals show apparently low efficiencies in running (Alexander, 1989). Efficient fast biological motors in running, flying, and swimming animals

(2006a, Fig. 2C). Given that the muscular field in running, flying, and swimming animals is twice as many as the gravitational field, adopted above for the lower threshold of a slow

adopted as a universal threshold of fast modes in fast gaits. This threshold associated with the slow-run-to-fast run transient state *λ* = 1 may in turn determine the pattern of efficient fast animals, including fast running mammals, reptiles, insects; flapping birds, bats, and insects; undulating fish and crayfish, according to Marden & Allen (2002) and Bejan & Marden (2006a).

∝ *M*−0.15±0.03; Heglund et al., 1974; Heglund & Taylor, 1988), make evidence for, within the experimental error, the predicted stabilization of the uniform (body mass independent) muscular field, i.e., *<sup>μ</sup>*(exp) *run* <sup>∝</sup> *<sup>M</sup>*−0.07±0.08, when tested by the scaling relation *<sup>μ</sup>*(*pred*)

*M*0.02±0.07. The revealed large experimental error is most likely caused by small quadrupedal species (one laboratory mice, two chipmunks, three squirrels, and three white rats of bodyweight not exceeding one *kg*), which should be excluded from the dynamic similarity

transition (Gatesy & Biewener, 1991), indicate the dynamic similarity pattern of *efficient*

for maximal speeds in running terrestrial mammals ranging in five orders in body mass

*trans* prescribed by dynamic regime *λ* = 1 (Table 2). Likewise, the same generic dynamic regime explains stabilization of the uniformity in the muscular field activated in 13 running animals (from a mice to horses) observed indirectly by Heglund &Taylor (1988) at experimental conditions of the *preferred* trotting speeds and the preferred galloping speeds,

*opt* <sup>=</sup> *<sup>μ</sup>*(max)

*f ast* = 2*g* was re-estimated by Bejan & Marden

*f ast* = 2*g* may be conventionally

*trans* and frequencies *<sup>T</sup>*(exp)−<sup>1</sup>

*trot*-*gall* <sup>∝</sup> *<sup>M</sup>*0.22±0.05 and *<sup>T</sup>*(exp)−<sup>1</sup>

*trot* <sup>∝</sup> *<sup>M</sup>*0.09±0.07 and *<sup>μ</sup>*(exp)

*trans* ∝ *<sup>M</sup>*−0.178 near the slow-walk-to-fast-run

*opt* ∝ *<sup>M</sup>*0.17 empirically established by Garland (1983)

*f ast M*

*trans* at

*trot*-*gall*

*trans* �

*gall* ∝

were established by Marden & Allen (2002) through the scaling equation *F*(max)

dynamic similarity are studied in Kokshenev (2011).

(Table 2), where the relative force amplitude *<sup>μ</sup>*(max)

run in the walk-to-run transition, the statistical data *<sup>μ</sup>*(max)

The pioneering observations of the transient-state speeds *V*(exp)

the trot-to-gallop continuous transition in quadrupeds (*V*(exp)

providing respectively the determinants of similarity *<sup>μ</sup>*(exp)

Bipeds, showing the resonant frequency *T*(exp)−<sup>1</sup>

*fast walkers*. The scaling rule *<sup>V</sup>*(exp)

pattern, as potentially having low mechanical efficiency (Alexander, 1989).

**3.2.5 Mechanically efficient fast animals**

*VtransT*−<sup>1</sup>

Following the concept of mechanical similarity, underlaid by the key principle of minimum action in analytical mechanics, the theory of dynamic similarity in animal locomotion is proposed. Exploring the intrinsic property of locomotory muscles to be tuned, via the variable muscle elasticity, to the natural cyclic frequency characteristic of high level efficiency of locomotion, the scaling rules driving the dynamic similarity in inanimate mechanical elastic systems are suggested for the special case of active-force animate elastic systems. The linear-displacement dynamic approach to contracting locomotory muscles, whose resonant frequencies are required by the principle of minimum mechanical action, establishes two different universal patterns of the dynamic regimes of similarity in different-sized animals distinguished by the dynamic scaling exponent *λ*. The determinants of the stationary locomotion of animals moving at optimal constant speeds (the case *λ* = 0) and the non-stationary locomotion at gradually changing speeds (*λ* = 1), including the transient-mode speed transitions, are self-consistently inferred and described in Tables 1 and 2, respectively. Exemplified by the non-stationary dynamic regime *λ* = 1, the two principal sets of determinants of the dynamic similarity are suggested by the universal exponents for the speed and frequency scaled with body mass, which may be equivalently presented by the corresponding Froude and Strouhal numbers or by other universal dimensionless numbers determining the states of dynamic similarity in different-sized animals.

The primary determinant, playing the role of the unique criterion of the linear dynamic similarity, is shown (in Eq. (16)) to be the Strouhal number, whose universality in animals across body mass indicates establishing of the linearity between the stride or stroke length and the body length in each animal, falling into one or other dynamically similar regime. In the special case of non-stationary dynamic similarity controlled by fast locomotory muscles, the Froude number may be equivalently chosen as a unique criterion of similarity, as hypothesized by Alexander (Alexander, 1976; Alexander & Jayes, 1983; Alexander, 1989). Since the scaling theory of similarity deals only with scaling relations, but not with scaling equations, Alexander's strict requirement that dynamic similarly between running animals should equal Froude numbers is not generally required by the theory. Instead, the theory of dynamic similarity stands only that changing with speed Froude numbers should be

important mechanical characteristics expose the extreme (minimum or maximum) behavior. This misleading principle of the existence of the symmetrical point, even though consistent with the well-known symmetrical nature of a walk and a run, due to which each equivalent leg moves half a stride cycle out of other leg (e.g., Alexander & Jayes, 1983, p.142) was in fact incorporated, likely unconsciously, into the studied model through symmetric mechanics of

<sup>281</sup> Physical Insights Into Dynamic Similarity

The constructal theory of dynamic similarity by Bejan & Marden (2006a, b) treats the potential energy of the body falling in the gravitational field *g* as a useful energy of terrestrial locomotion. Considering only the aerial phase in a stride cycle, the theory excludes the ground contact and thereby all muscular forces providing the body propulsion force *<sup>μ</sup>*(exp)

It is surprisingly that the same theoretical framework excluding muscle forces has resulted

relations, shown in Eq. (8), are weighted solely by the gravitational field. Hence, it has been demonstrated by the authors that consistency between the principle of destruction of minimum useful energy in the gravitational field (the case *λ* = 1 in Eq. (2)) may exist under the additional condition postulated in Eq. (7), in fact borrowed from another, more general

Following the requirement of equality of Froude numbers, experimental biologists mostly study the discrete-state dynamic similarity in animals. For example, Bullimore & Burn (2006) have remarkably established (see their Table 4) the universal criterion of dynamic

(acknowledged by Alexander, 1989) on that besides the Froude number the Strouhal number should be simultaneously constant, as followed from the universality of the Groucho number (McMahon et al., 1987). In contrast, the study by Bullimore & Donelan (2008) of the criteria of dynamic similarity in spring-mass modeled animals suggested four independent determinants, at least. Given that the authors have clearly convinced the reader of that the equality of only two dimensionless numbers is not sufficient for establishing of the dynamic similarity between in-plane modeled animals (Bullimore & Donelan, 2008, Fig. 4), a question arises about what kind of dynamic similarity was reported by Bullimore & Burn (2006)

Bullimore & Donelan (2008, Table 2) have analyzed the well known solutions of the planar spring-mass model through Buckingham's Π-theorem of the dimensional method (e.g., Barenblatt, 2002) and claimed that minimum four independent dimensionless numbers following from the set of mechanical quantities (*V*, *K*, *Vz*, Θ0) are required for the observation of dynamic similarity in bouncing modes of animals. First, one can see that the landing angle Θ<sup>0</sup> should be excluded from the proposed set of physically independent quantities, since the horizontal landing speed *V* and the vertical landing speed *Vz* definitively determine the angle Θ<sup>0</sup> = *arctg*(*V*/*Vz*), as can be inferred from Fig. 1 by Bullimore & Donelan (2008). Then, the requirement to control vertical speed via the model-independent relation *Vz μβT* allows one to reduce the proposed set to the equivalent set (*T*−1, *V*, *μ*, *β*), where the body stiffness is substituted by muscular field via the body stiffness *K ρbAμ*. The resulted set of four quantities (*St*, *Fr*, *μ*, *β*) is dynamically equivalent to the originally suggested set (*V*, *K*, *Vz*, Θ0), but among three determinants (*St*, *Fr*, *μ*) the only one is physically independent. Two other dimensionless numbers play the role of auxiliary determinants in the dynamic similarity, whereas the duty factor may indicate transient-mode and crossover-gait universal states of the same dynamic regime *λ* = 1. Hence, it has been repeatedly demonstrated that the

established in real trotting horses through the only one determinant *Fr*?

*trot* (= 0.70, 0.67, and 0.60) from 21 horses trotting at arbitrary chosen fixed

*trot* (= 0.5, 0.75, and 1). This finding corroborates McMahon's suggestion

*f ast* = 2*g*, since the suggested scaling factors in the basic scaling

*f ast M*.

the supposedly equivalent human legs.

in Animal Locomotion. I. Theoretical Principles and Concepts

in the muscular field *<sup>μ</sup>*(exp)

mechanical principle.

similarity *St*(exp)

Froude numbers *Fr*(exp)

invariable with body mass in animals considered in a certain dynamic state or domain of dynamically equivalent states. A generalization of the proposed theory of the discrete-state dynamic similarity to continuous-state similarity in animals, determining, respectively, by discrete equal and different variable magnitudes of the Froude numbers, will be discussed in the next part of this study (Kokshenev, 2011).

The two kinds of dynamic similarity regimes in animals, well distinguished by the scaling rules established for a number of mechanical characteristics, may explain seemingly controversial experimental observations as well as illuminate some theoretical principles conceptually inconsistent with the mechanical similarity principle of analytical mechanics.

Hill's pioneering observation of bodyweight independence of optimal speeds in a hover flight mode of sparrows and humming birds (Hill, 1950), showing a sharp inconsistency (McMahon, 1975; Jones & Lindstedt, 1993) with the scaling rules for speeds in quadrupeds established at the trot-to-gallop transition (Heglund et al., 1974) can readily be understood by the observations of two distinct dynamic similarity regimes *λ* = 0 and *λ* = 1. Likewise, a more recent claim on that the similarity between humans running at *fixed* speeds, accurately simulated under the requirement of equal Froude numbers, was surprisingly found (Delattre et al., 2009) to be in sharp disagreement with the scaling rules of dynamic similarity in fast running animals reliably established by Farley et al. (1993). A new kind of similarity discovered in running humans arrived the authors to a puzzle conclusion that neither of Froude and Strouhal numbers is appropriate as determinant of dynamic similarity. In this special case, the proposed theory tells us that the dynamic similarity between humans running at the stationary-speed conditions (Table 1) cannot be constrained by constant Froude numbers, as erroneously was adopted in the study by Delattre et al. (2009).

Another Hill's surmise on the constant limb muscle stress, resulted in the universality of the limb safety factor in animals efficiently moving in slow gaits, has been generalized without grounds to all fast gaits by a number of researches. For example, in attempting to introduce "equivalent speed" states *λ* = 1 during trot-to-gallop transition McMahon postulated a constant stress in homologous muscles (McMahon, 1975, Table 4), when suggested the uniform muscle stress *σm*-*slow* ∝ *M*0, corresponding to the case of *λ* = 0. The postulated stress evidently contrasted with the already existing data on peak isometric stress, linearly varying with sarcomere length (Huxley & Neidergerke, 1954), i.e., *σm*-*f ast* ∝ *Lm* (Table 2), and the data on muscle stress later revealed the linearity to fiber length in running and jumping animals (Alexander & Bennet-Clark, 1976). Likewise, when the axial-displacement dynamic similarity (i.e., Δ*Lm*-*f ast* ∝ *Lm*) discussed for fast locomotory muscles is generalized

to non-axial-displacement elastic similarity in long limb mammalian bones (Δ*L*(*bend*) *bone* ∝ *Dbone*, where *Dbone* is bone's diameter; Kokshenev, 2007, Eq. (15)) new puzzled consequences of biomechanical scaling may be revealed. One impressive example is the axial compressive stress *<sup>σ</sup>*(*axial*) *bone* <sup>∝</sup> *Lbone*, estimated as the peak limb bone stress *<sup>σ</sup>*(exp) *bone* <sup>∝</sup> *<sup>M</sup>*0.28 for the avian taxa, matching well the spring-leg data *<sup>σ</sup>*(exp) *leg* <sup>∝</sup> *<sup>M</sup>*0.30 from running quadrupeds following from Eq. (17), has been shown to provide the anecdotal small largest terrestrial giant weight, no much greater than 20 *kg* (Biewener, 2005). This puzzle was understood by that instead of axial stress, which is in fact non-critical, the bending stress *<sup>σ</sup>*(*bend*) *bone* ∝ *Dbone*/*Lbone*, having small but non-zero positive exponent, i.e., *<sup>σ</sup>*(*bend*) *bone* <sup>∝</sup> *<sup>M</sup>*0.08, likely establishes the critical mass of terrestrial giants (Kokshenev & Christiansen, 2011).

One more "least-action principle" in biomechanics was recently declared for walk gaits in humans (Fan et al., 2009). The standard variational procedure was worked out to establish a symmetric point (*T*/2) in the middle of the two-step stride cycle in human gaits, at which all 14 Will-be-set-by-IN-TECH

invariable with body mass in animals considered in a certain dynamic state or domain of dynamically equivalent states. A generalization of the proposed theory of the discrete-state dynamic similarity to continuous-state similarity in animals, determining, respectively, by discrete equal and different variable magnitudes of the Froude numbers, will be discussed in

The two kinds of dynamic similarity regimes in animals, well distinguished by the scaling rules established for a number of mechanical characteristics, may explain seemingly controversial experimental observations as well as illuminate some theoretical principles conceptually inconsistent with the mechanical similarity principle of analytical mechanics. Hill's pioneering observation of bodyweight independence of optimal speeds in a hover flight mode of sparrows and humming birds (Hill, 1950), showing a sharp inconsistency (McMahon, 1975; Jones & Lindstedt, 1993) with the scaling rules for speeds in quadrupeds established at the trot-to-gallop transition (Heglund et al., 1974) can readily be understood by the observations of two distinct dynamic similarity regimes *λ* = 0 and *λ* = 1. Likewise, a more recent claim on that the similarity between humans running at *fixed* speeds, accurately simulated under the requirement of equal Froude numbers, was surprisingly found (Delattre et al., 2009) to be in sharp disagreement with the scaling rules of dynamic similarity in fast running animals reliably established by Farley et al. (1993). A new kind of similarity discovered in running humans arrived the authors to a puzzle conclusion that neither of Froude and Strouhal numbers is appropriate as determinant of dynamic similarity. In this special case, the proposed theory tells us that the dynamic similarity between humans running at the stationary-speed conditions (Table 1) cannot be constrained by constant Froude

numbers, as erroneously was adopted in the study by Delattre et al. (2009).

Another Hill's surmise on the constant limb muscle stress, resulted in the universality of the limb safety factor in animals efficiently moving in slow gaits, has been generalized without grounds to all fast gaits by a number of researches. For example, in attempting to introduce "equivalent speed" states *λ* = 1 during trot-to-gallop transition McMahon postulated a constant stress in homologous muscles (McMahon, 1975, Table 4), when suggested the uniform muscle stress *σm*-*slow* ∝ *M*0, corresponding to the case of *λ* = 0. The postulated stress evidently contrasted with the already existing data on peak isometric stress, linearly varying with sarcomere length (Huxley & Neidergerke, 1954), i.e., *σm*-*f ast* ∝ *Lm* (Table 2), and the data on muscle stress later revealed the linearity to fiber length in running and jumping animals (Alexander & Bennet-Clark, 1976). Likewise, when the axial-displacement dynamic similarity (i.e., Δ*Lm*-*f ast* ∝ *Lm*) discussed for fast locomotory muscles is generalized

to non-axial-displacement elastic similarity in long limb mammalian bones (Δ*L*(*bend*)

*bone* <sup>∝</sup> *Lbone*, estimated as the peak limb bone stress *<sup>σ</sup>*(exp)

of axial stress, which is in fact non-critical, the bending stress *<sup>σ</sup>*(*bend*)

taxa, matching well the spring-leg data *<sup>σ</sup>*(exp)

small but non-zero positive exponent, i.e., *<sup>σ</sup>*(*bend*)

of terrestrial giants (Kokshenev & Christiansen, 2011).

where *Dbone* is bone's diameter; Kokshenev, 2007, Eq. (15)) new puzzled consequences of biomechanical scaling may be revealed. One impressive example is the axial compressive

from Eq. (17), has been shown to provide the anecdotal small largest terrestrial giant weight, no much greater than 20 *kg* (Biewener, 2005). This puzzle was understood by that instead

One more "least-action principle" in biomechanics was recently declared for walk gaits in humans (Fan et al., 2009). The standard variational procedure was worked out to establish a symmetric point (*T*/2) in the middle of the two-step stride cycle in human gaits, at which all

*bone* ∝ *Dbone*,

*bone* <sup>∝</sup> *<sup>M</sup>*0.28 for the avian

*bone* ∝ *Dbone*/*Lbone*, having

*leg* <sup>∝</sup> *<sup>M</sup>*0.30 from running quadrupeds following

*bone* <sup>∝</sup> *<sup>M</sup>*0.08, likely establishes the critical mass

the next part of this study (Kokshenev, 2011).

stress *<sup>σ</sup>*(*axial*)

important mechanical characteristics expose the extreme (minimum or maximum) behavior. This misleading principle of the existence of the symmetrical point, even though consistent with the well-known symmetrical nature of a walk and a run, due to which each equivalent leg moves half a stride cycle out of other leg (e.g., Alexander & Jayes, 1983, p.142) was in fact incorporated, likely unconsciously, into the studied model through symmetric mechanics of the supposedly equivalent human legs.

The constructal theory of dynamic similarity by Bejan & Marden (2006a, b) treats the potential energy of the body falling in the gravitational field *g* as a useful energy of terrestrial locomotion. Considering only the aerial phase in a stride cycle, the theory excludes the ground contact and thereby all muscular forces providing the body propulsion force *<sup>μ</sup>*(exp) *f ast M*. It is surprisingly that the same theoretical framework excluding muscle forces has resulted in the muscular field *<sup>μ</sup>*(exp) *f ast* = 2*g*, since the suggested scaling factors in the basic scaling relations, shown in Eq. (8), are weighted solely by the gravitational field. Hence, it has been demonstrated by the authors that consistency between the principle of destruction of

minimum useful energy in the gravitational field (the case *λ* = 1 in Eq. (2)) may exist under the additional condition postulated in Eq. (7), in fact borrowed from another, more general mechanical principle. Following the requirement of equality of Froude numbers, experimental biologists mostly

study the discrete-state dynamic similarity in animals. For example, Bullimore & Burn (2006) have remarkably established (see their Table 4) the universal criterion of dynamic similarity *St*(exp) *trot* (= 0.70, 0.67, and 0.60) from 21 horses trotting at arbitrary chosen fixed Froude numbers *Fr*(exp) *trot* (= 0.5, 0.75, and 1). This finding corroborates McMahon's suggestion (acknowledged by Alexander, 1989) on that besides the Froude number the Strouhal number should be simultaneously constant, as followed from the universality of the Groucho number (McMahon et al., 1987). In contrast, the study by Bullimore & Donelan (2008) of the criteria of dynamic similarity in spring-mass modeled animals suggested four independent determinants, at least. Given that the authors have clearly convinced the reader of that the equality of only two dimensionless numbers is not sufficient for establishing of the dynamic similarity between in-plane modeled animals (Bullimore & Donelan, 2008, Fig. 4), a question arises about what kind of dynamic similarity was reported by Bullimore & Burn (2006) established in real trotting horses through the only one determinant *Fr*?

Bullimore & Donelan (2008, Table 2) have analyzed the well known solutions of the planar spring-mass model through Buckingham's Π-theorem of the dimensional method (e.g., Barenblatt, 2002) and claimed that minimum four independent dimensionless numbers following from the set of mechanical quantities (*V*, *K*, *Vz*, Θ0) are required for the observation of dynamic similarity in bouncing modes of animals. First, one can see that the landing angle Θ<sup>0</sup> should be excluded from the proposed set of physically independent quantities, since the horizontal landing speed *V* and the vertical landing speed *Vz* definitively determine the angle Θ<sup>0</sup> = *arctg*(*V*/*Vz*), as can be inferred from Fig. 1 by Bullimore & Donelan (2008). Then, the requirement to control vertical speed via the model-independent relation *Vz μβT* allows one to reduce the proposed set to the equivalent set (*T*−1, *V*, *μ*, *β*), where the body stiffness is substituted by muscular field via the body stiffness *K ρbAμ*. The resulted set of four quantities (*St*, *Fr*, *μ*, *β*) is dynamically equivalent to the originally suggested set (*V*, *K*, *Vz*, Θ0), but among three determinants (*St*, *Fr*, *μ*) the only one is physically independent. Two other dimensionless numbers play the role of auxiliary determinants in the dynamic similarity, whereas the duty factor may indicate transient-mode and crossover-gait universal states of the same dynamic regime *λ* = 1. Hence, it has been repeatedly demonstrated that the

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application of Buckingham's theorem only provides a way of generating sets of dimensionless parameters, but does not indicate or even substitute most physically meaningful relations. This well known statement is also illustrated by the provided above study of two scaling equations Eq. (2) and (5), suggesting the same set of three possible determinants of dynamic similarity, respectively provided by the physical concept and the dimensional method.

When comparing the frameworks of dynamic similarity and elastic similarity, respectively elaborated to scale the patterns of fast locomotion gaits and the patterns of primary functions (motor, brake, strut, or spring) of locomotory muscles (Kokshenev, 2008, Table 1), belonging to the same body's elastic system, one can see that in both cases the muscle contractions fall into the same dynamic similarity regime *λ* = 1 generally governed by the same uniform muscular field. However, the two distinct (gait and function) muscle patters should not provide the same scaling rules for dynamic muscle characteristics, including the dynamic length noted by Alexander (1989, p.1212), since the dynamic conditions of muscle cycling are distinct. Indeed, the dynamic cycling in similar locomotion is synchronized with the collective muscle dynamics, corresponding to the condition of maximum overall-body mechanical efficiency, whereas the elastic similarity between individual muscles specialized to a certain mechanical function is likely governed by the requirement of maximum power, generally not matching the condition of minimal oxygen consumption. The observation by Hill (1950, Fig. 1) that the muscle power and efficiency maxima are rather blunt and close in space makes it possible to work at maximum power with nearly maximum efficiency. Further analysis of non-linear dynamic similarity in muscle functions and animal locomotion, including powering intermittent gaits (Alexander, 1989, p.1200), will be discussed elsewhere.

### **5. Acknowledgments**

The author is grateful to Dr. in Engineering Sciences Illya Kokshenev for discussion and critical comments. Financial support by the national agency FAPEMIG is also acknowledged.

### **6. References**

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Ahlborn, B.K., Blake, R.W. & Megill, W.M. (2006). Frequency tuning in animal locomotion, *Zoology* 109: 43-53.

Alexander, R.McN. (1976). Estimates of speeds of dinosaurs, *Nature* 261: 129-130.


16 Will-be-set-by-IN-TECH

application of Buckingham's theorem only provides a way of generating sets of dimensionless parameters, but does not indicate or even substitute most physically meaningful relations. This well known statement is also illustrated by the provided above study of two scaling equations Eq. (2) and (5), suggesting the same set of three possible determinants of dynamic similarity, respectively provided by the physical concept and the dimensional method. When comparing the frameworks of dynamic similarity and elastic similarity, respectively elaborated to scale the patterns of fast locomotion gaits and the patterns of primary functions (motor, brake, strut, or spring) of locomotory muscles (Kokshenev, 2008, Table 1), belonging to the same body's elastic system, one can see that in both cases the muscle contractions fall into the same dynamic similarity regime *λ* = 1 generally governed by the same uniform muscular field. However, the two distinct (gait and function) muscle patters should not provide the same scaling rules for dynamic muscle characteristics, including the dynamic length noted by Alexander (1989, p.1212), since the dynamic conditions of muscle cycling are distinct. Indeed, the dynamic cycling in similar locomotion is synchronized with the collective muscle dynamics, corresponding to the condition of maximum overall-body mechanical efficiency, whereas the elastic similarity between individual muscles specialized to a certain mechanical function is likely governed by the requirement of maximum power, generally not matching the condition of minimal oxygen consumption. The observation by Hill (1950, Fig. 1) that the muscle power and efficiency maxima are rather blunt and close in space makes it possible to work at maximum power with nearly maximum efficiency. Further analysis of non-linear dynamic similarity in muscle functions and animal locomotion, including

powering intermittent gaits (Alexander, 1989, p.1200), will be discussed elsewhere.

Alexander, R.McN. (1976). Estimates of speeds of dinosaurs, *Nature* 261: 129-130.

quadrupedal mammals, *Journal of Zoology* 201: 135–152.

The author is grateful to Dr. in Engineering Sciences Illya Kokshenev for discussion and critical comments. Financial support by the national agency FAPEMIG is also acknowledged.

Ahlborn, B.K. & Blake, R.W. (2002). Walking and running at resonance, *Zoology* 105: 165–174. Ahlborn, B.K., Blake, R.W. & Megill, W.M. (2006). Frequency tuning in animal locomotion,

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University Press, Cambridge.

**6. References**


**0**

**13**

*Brazil*

*b* ,

Valery B. Kokshenev

*de Minas Gerais, Belo Horizonte,*

**Physical Insights Into Dynamic Similarity**

In previous studies, the physical theory of dynamic similarity in animals was deduced from the concept of mechanical similarity in analytical mechanics (Kokshenev, 2010, 2011). Unlike a mechanistic approach to the problem of dynamic similarity in biomechanics of locomotion treating the animate systems as freely falling in the gravitational field and resisting drag forces (Bejan & Marden, 2006), the model-independent physical theory, unifying running,

Broadly speaking, animals of different body size *Lb* are dissimilar in their forward propagation speeds *V* and stride or stroke frequencies *T*−1. However, when traveling or cruising for long distances at the proper body's resonant frequency *<sup>T</sup>*−<sup>1</sup> *res* <sup>∼</sup> *<sup>μ</sup>slow*/*Lb*, the animals are generally exploit the stationary dynamic regime characteristic of the constant uniform body pressure generated by slow locomotory muscles. The relevant muscular field *<sup>μ</sup>slow* ∝ *<sup>L</sup>*−<sup>1</sup>

scaling with body length by the same way as reaction elastic-force field, was observed indirectly (Kokshenev, 2011, Table 1) through the wingbeat frequency in hovering birds *T*−<sup>1</sup> *res* , first by Hill (1950) and then by Ellington (1991). The first direct observation of the body's *slow muscular field* scaling with body mass as *μslow* ∝ *M*−1/3 in flying, running, and swimming animals has been established through the body force output by Alexander (1985) and Marden & Allen (2002). When an animal maintains the resonant cyclic conditions (in frequency and phase; Kokshenev, 2010) with increasing speed, the mechanical efficiency of the properly recruited fast locomotory muscles is higher than that of acting slow muscles, at least in adult species (e.g., Alexander, 1989). Moreover, the mechanically universal (i.e., frequency-, speed-, and force-independent) muscle stiffness *Kf ast* ∝ *M*2/3 results in the body's *fast muscular field μf ast* ∼ *g*, where *g* is the gravitational field, whose uniformity determines the dynamic similarity between animals across body mass through the basic scaling rules *Vf ast* ∝ *M*1/6

The *dynamic similarity patterns*, unifying different-sized animals at a certain dynamically similar slow or fast regime, are unambiguously determined by the reliable observation of just one of the basic scaling rules shown respectively in Tables 1 and 2 in Kokshenev (2011). Alternatively, the equivalent *theoretical observation* of just one *uniform* (across body mass) dimensionless determinant, such as the Strouhal number (*Stslow* ∼ *Stf ast* ∝ *M*0), the Froude

swimming, and flying animals, can be outlined as follows.

**1. Introduction**

and *T*−<sup>1</sup>

*f ast* <sup>∝</sup> *<sup>M</sup>*−1/6.

**in Animal Locomotion. II. Observation**

**of Continuous Similarity States**

*Departamento de Física, Universidade Federal*


## **Physical Insights Into Dynamic Similarity in Animal Locomotion. II. Observation of Continuous Similarity States**

Valery B. Kokshenev

*Departamento de Física, Universidade Federal de Minas Gerais, Belo Horizonte, Brazil*

### **1. Introduction**

18 Will-be-set-by-IN-TECH

284 Theoretical Biomechanics

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Kokshenev, V.B. (2009). Scaling functional patterns of skeletal and cardiac muscles: New

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Rome, L.C., Funke, R.P., Alexander, R.McN., Lutz, G., Aldridge, H., Scott F. & Freadman, M. (1988). Why animals have different muscle fibre types?, *Nature* 335: 824-829. Rubin, C.T. & Lanyon, L.E. (1984). Dynamic strain similarity in vertebrates; an alternative to allometric limb bone scaling, *Journal of Theoretical Biology* 107: 321-327. Taylor G.K., Nudds, R.L. & Thomas, A. L. (2003). Flying and swimming animals cruise at a Strouhal number tuned for high power efficiency, *Nature* 425: 707-710. Vaughan, C.L. & O'Malleyb, M. J. (2005). Froude and the contribution of naval architecture to our understanding of bipedal locomotion, *Gait* & *Posture* 21: 350-362. Videler, J.J. & Weihs, D. (1982). Energetic advantages of burst-and coast swimming of fish at

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Observation of continuous similarity states, In: *Theoretical Biomechanics*, Vaclav Klika

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(ed.), pp. 285-302, ISBN 978-953-307-851-9.

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and New Jersey.

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93: 208101-1-4.

*Physica A* 322: 491-505.

In previous studies, the physical theory of dynamic similarity in animals was deduced from the concept of mechanical similarity in analytical mechanics (Kokshenev, 2010, 2011). Unlike a mechanistic approach to the problem of dynamic similarity in biomechanics of locomotion treating the animate systems as freely falling in the gravitational field and resisting drag forces (Bejan & Marden, 2006), the model-independent physical theory, unifying running, swimming, and flying animals, can be outlined as follows.

Broadly speaking, animals of different body size *Lb* are dissimilar in their forward propagation speeds *V* and stride or stroke frequencies *T*−1. However, when traveling or cruising for long distances at the proper body's resonant frequency *<sup>T</sup>*−<sup>1</sup> *res* <sup>∼</sup> *<sup>μ</sup>slow*/*Lb*, the animals are generally exploit the stationary dynamic regime characteristic of the constant uniform body pressure generated by slow locomotory muscles. The relevant muscular field *<sup>μ</sup>slow* ∝ *<sup>L</sup>*−<sup>1</sup> *b* , scaling with body length by the same way as reaction elastic-force field, was observed indirectly (Kokshenev, 2011, Table 1) through the wingbeat frequency in hovering birds *T*−<sup>1</sup> *res* , first by Hill (1950) and then by Ellington (1991). The first direct observation of the body's *slow muscular field* scaling with body mass as *μslow* ∝ *M*−1/3 in flying, running, and swimming animals has been established through the body force output by Alexander (1985) and Marden & Allen (2002). When an animal maintains the resonant cyclic conditions (in frequency and phase; Kokshenev, 2010) with increasing speed, the mechanical efficiency of the properly recruited fast locomotory muscles is higher than that of acting slow muscles, at least in adult species (e.g., Alexander, 1989). Moreover, the mechanically universal (i.e., frequency-, speed-, and force-independent) muscle stiffness *Kf ast* ∝ *M*2/3 results in the body's *fast muscular field μf ast* ∼ *g*, where *g* is the gravitational field, whose uniformity determines the dynamic similarity between animals across body mass through the basic scaling rules *Vf ast* ∝ *M*1/6 and *T*−<sup>1</sup> *f ast* <sup>∝</sup> *<sup>M</sup>*−1/6.

The *dynamic similarity patterns*, unifying different-sized animals at a certain dynamically similar slow or fast regime, are unambiguously determined by the reliable observation of just one of the basic scaling rules shown respectively in Tables 1 and 2 in Kokshenev (2011). Alternatively, the equivalent *theoretical observation* of just one *uniform* (across body mass) dimensionless determinant, such as the Strouhal number (*Stslow* ∼ *Stf ast* ∝ *M*0), the Froude

further generalizations of the patterns of dynamic similarity revealed in animals across speed

<sup>287</sup> Physical Insights Into Dynamic Similarity

The available experimental data on gait-dependent characteristics of walking and running terrestrial animals based on the study of dimensionless numbers of dynamic similarity, are conventionally separated in two groups, represented by the discrete-state constant and

The study by Farley et al. (1993) of the dynamic similarity in trotting quadrupeds (form a rat to a horse through a dog and a goat) extended by hopping bipeds (tammar wallaby and red kangaroo) has been performed near the trot-to-gallop transient states, whose dynamic

The statistically averaged data for the mean Froude number and the limb duty factor were reported by Farley et al. (1993, p. 74). The relative force output is estimated here on the

More recently, the discrete-state dynamic similarity was established in 21 trotting horses by

their Table 4, the authors have shown that besides the stabilization across body mass of two

Treating the estimated Froude numbers as a dynamic variable, changing with speed in walking and running terrestrial animals, the data on the measured gait-dependent mechanical characteristics were obtained by a number of researches. The available systematic data on continuous dynamic similarity in quadrupeds (from a small rodent to a rhinoceros, Alexander & Jayes, 1983), modern elephants (Hutchinson et al., 2006), land-dwelling birds (from a bobwhite to an ostrich) and two humans (Gatesy & Biewener, 1991) are provided in Table

*trans* discussed in Eq. (17) in Kokshenev (2011) and the failing Strouhal number is

0.50 0.75 1.00 0.70 0.67 0.60 0.47 0.44 0.39 *trans* (= *<sup>V</sup>*<sup>2</sup>

*trans* (= Δ*T*/*T* , with Δ*T* is the ground contact time of one foot,

*trans* <sup>=</sup> 0.41 <sup>±</sup> 0.2, with *<sup>μ</sup>*(max)

*trans* , represented here as

*trot* ∝ *<sup>M</sup>*0, the similar dynamic states are also indicated by

*trot* ∝ *<sup>M</sup>*0. The omitted scaling factors in statistically observed

⎞

*trans*/*gLb*, where *Lb* is

*trans* /*g* = 3.10. (1)

*trot* (= 0.5, 0.75, and 1). In

<sup>⎠</sup> · *<sup>M</sup>*0. (2)

*gait* and the Strouhal similarity function

and taxa.

**2. Materials and methods**

**2.1.1 Discrete similarity data**

*Fr*(exp)

predicted below (Fig. 1).

determinants *Fr*(exp)

the invariable duty factor *<sup>β</sup>*(exp)

**2.1.2 Continuous similarity data**

*gait* scaled by Froude variable *Fr*.

basis of *<sup>F</sup>*(exp)

*St*(exp)

hip height), limb duty factors *<sup>β</sup>*(exp)

**2.1 Systematic experimental data**

continuous-state variable Froude parameters.

*trans* <sup>=</sup> 2.25 <sup>±</sup> 0.13, *St*(*pred*)

similarity was revealed by closely spaced Froude numbers *Fr*(exp)

in Animal Locomotion. II. Observation of Continuous Similarity States

Bullimore & Burn (2006) at three constant Froude numbers *Fr*(exp)

*Fr*(exp) *trot St*(exp) *trot <sup>β</sup>*(exp) *trot*

⎞

⎟⎠ <sup>=</sup>

⎛ ⎝

*trans* <sup>=</sup> 0.40, *<sup>β</sup>*(exp)

and *T* is the stride period), and peak force output *F*(exp)

*trot* <sup>∝</sup> *<sup>M</sup>*<sup>0</sup> and *St*(exp)

scaling relations are schematically reproduced here as

⎛

⎜⎝

1 in terms of the duty-factor similarity function *<sup>β</sup>*(exp)

number (*Frf ast* ∝ *M*0), the force-output number (*μf ast*/*g* ∝ *M*0), the limb duty factor (*βslow* ∼ *β f ast* ∝ *M*0), or the angle (Θ*slow* ∼ Θ*f ast* ∝ *M*0) sweeping by legs, wings, or tails during a stride or a stroke, implies the establishing of the discrete (isolated) dynamically similar states in animals moving in certain fast or slow gait.

In the previous chapter, the pattern of *efficient runners in quadrupeds*, unifying trotting and galloping quadrupeds (from a mice to a horse), was theoretically observed by both the basic scaling rules at well experimentally distinguished "equivalent" speeds (Heglund et al., 1974), "preferred" trotting and galloping speeds (Heglund & Taylor, 1988; Perry et al., 1988), as well as trot-to-gallop transient speeds (Heglund et al., 1974, Heglund & Taylor, 1988). When extended by bipeds and described by a whole spectrum of scaling rules, the latter pattern of similarity called by *efficient trotters and hoppers* was been studied in the domain of "physiologically equivalent" speeds (Farley et al., 1993), unifying near the trot-to-gallop transition running quadrupeds (from a rat to a horse) and hopping bipeds (tammar wallaby and red kangaroo).1 It has been also demonstrated that the direct experimental observation of both two basic scaling rules is equivalent to the theoretical observation of the uniformity of the underlaid muscular field, i.e., *<sup>μ</sup>f ast* � *Vf astT*<sup>−</sup><sup>1</sup> *f ast* <sup>∝</sup> *<sup>M</sup>*0. Other patterns of the established transient (walk-to-run) state and optimum-speed state, called by *efficient walkers in bipeds* and *efficient runners in mammals*, were theoretically observed through the scaling rules for, respectively, frequency and speed measured in bipeds, including humans (Gatesy & Biewener, 1991) and other terrestrial mammals (from the smallest bipedal rodent to the largest quadrupedal elephant, Garland, 1983). Likewise, the dynamic similarity in *efficient flyers and swimmers* associated with the fast dynamic regime of locomotion was established for dolphins, birds, and bats cruising with a high power efficiency (Taylor et al., 2003).

In the previous chapter, Alexander's hypothesis on the dynamic similarity between running different-sized animals, equalling Froude numbers (Alexander, 1976; Alexander & Jayes, 1983; Alexander, 1989), has been studied in the context of *discrete-state* dynamic similarity. These dynamically similar states of animate systems, moving in either the stationary-speed slow-muscle regime or the non-stationary fast-muscle regime, have been specified by optimal-speed, transient-mode, and transient-gait states, resulted in the mentioned above dynamic similarity patterns. Besides the central hypothesis on discrete-state dynamic similarity, the possible existence of gait-dependent characteristic similarity functions (of the *Froude variable Fr*) unifying animals across body mass, speed, and taxa at a certain gait, was also proposed by Alexander & Jayes (1983). From the theoretical point of view, the discrete-state similarity determined by equal Froude constants has been therefore suggested to be generalized to the *continuous-state dynamic similarity*, determined by the *dynamic similarity functions* provided by continuous data on the limb duty factors *β*(*Fr*) and the Strouhal numbers *St*(*Fr*) obtained by Alexander & Jayes (1983).

In this study, the theory of continuous dynamic similarity is developed and tested by the continuous dynamic similarity states revealed in walking and running mammals on the basis of systematic experimental data on the limb duty factor and relative stride length available for quadrupedal (Alexander & Jayes, 1983; Hutchinson et al., 2006) and bipedal (Gatesy & Biewener, 1991) animals. The developed concept of dynamic similarity hopefully provides

<sup>1</sup> Earlier, the continues dynamically similar states were established by Alexander & Jayes (1983) in trotting and galloping cursorial quadrupeds through the Froude numbers lying in the domain 2 ≤ *Fr* ≤ 3. The corresponding local-state trot-gallop transition was estimated at *Frtrot*−*gall* = 2.5 (Alexander & Jayes, 1983; Alexander, 2003).

further generalizations of the patterns of dynamic similarity revealed in animals across speed and taxa.

### **2. Materials and methods**

2 Will-be-set-by-IN-TECH

number (*Frf ast* ∝ *M*0), the force-output number (*μf ast*/*g* ∝ *M*0), the limb duty factor (*βslow* ∼ *β f ast* ∝ *M*0), or the angle (Θ*slow* ∼ Θ*f ast* ∝ *M*0) sweeping by legs, wings, or tails during a stride or a stroke, implies the establishing of the discrete (isolated) dynamically

In the previous chapter, the pattern of *efficient runners in quadrupeds*, unifying trotting and galloping quadrupeds (from a mice to a horse), was theoretically observed by both the basic scaling rules at well experimentally distinguished "equivalent" speeds (Heglund et al., 1974), "preferred" trotting and galloping speeds (Heglund & Taylor, 1988; Perry et al., 1988), as well as trot-to-gallop transient speeds (Heglund et al., 1974, Heglund & Taylor, 1988). When extended by bipeds and described by a whole spectrum of scaling rules, the latter pattern of similarity called by *efficient trotters and hoppers* was been studied in the domain of "physiologically equivalent" speeds (Farley et al., 1993), unifying near the trot-to-gallop transition running quadrupeds (from a rat to a horse) and hopping bipeds (tammar wallaby and red kangaroo).1 It has been also demonstrated that the direct experimental observation of both two basic scaling rules is equivalent to the theoretical observation of the uniformity

established transient (walk-to-run) state and optimum-speed state, called by *efficient walkers in bipeds* and *efficient runners in mammals*, were theoretically observed through the scaling rules for, respectively, frequency and speed measured in bipeds, including humans (Gatesy & Biewener, 1991) and other terrestrial mammals (from the smallest bipedal rodent to the largest quadrupedal elephant, Garland, 1983). Likewise, the dynamic similarity in *efficient flyers and swimmers* associated with the fast dynamic regime of locomotion was established for

In the previous chapter, Alexander's hypothesis on the dynamic similarity between running different-sized animals, equalling Froude numbers (Alexander, 1976; Alexander & Jayes, 1983; Alexander, 1989), has been studied in the context of *discrete-state* dynamic similarity. These dynamically similar states of animate systems, moving in either the stationary-speed slow-muscle regime or the non-stationary fast-muscle regime, have been specified by optimal-speed, transient-mode, and transient-gait states, resulted in the mentioned above dynamic similarity patterns. Besides the central hypothesis on discrete-state dynamic similarity, the possible existence of gait-dependent characteristic similarity functions (of the *Froude variable Fr*) unifying animals across body mass, speed, and taxa at a certain gait, was also proposed by Alexander & Jayes (1983). From the theoretical point of view, the discrete-state similarity determined by equal Froude constants has been therefore suggested to be generalized to the *continuous-state dynamic similarity*, determined by the *dynamic similarity functions* provided by continuous data on the limb duty factors *β*(*Fr*) and the Strouhal

In this study, the theory of continuous dynamic similarity is developed and tested by the continuous dynamic similarity states revealed in walking and running mammals on the basis of systematic experimental data on the limb duty factor and relative stride length available for quadrupedal (Alexander & Jayes, 1983; Hutchinson et al., 2006) and bipedal (Gatesy & Biewener, 1991) animals. The developed concept of dynamic similarity hopefully provides

<sup>1</sup> Earlier, the continues dynamically similar states were established by Alexander & Jayes (1983) in trotting and galloping cursorial quadrupeds through the Froude numbers lying in the domain 2 ≤ *Fr* ≤ 3. The corresponding local-state trot-gallop transition was estimated at *Frtrot*−*gall* = 2.5 (Alexander &

dolphins, birds, and bats cruising with a high power efficiency (Taylor et al., 2003).

*f ast* <sup>∝</sup> *<sup>M</sup>*0. Other patterns of the

similar states in animals moving in certain fast or slow gait.

of the underlaid muscular field, i.e., *<sup>μ</sup>f ast* � *Vf astT*<sup>−</sup><sup>1</sup>

numbers *St*(*Fr*) obtained by Alexander & Jayes (1983).

Jayes, 1983; Alexander, 2003).

### **2.1 Systematic experimental data**

The available experimental data on gait-dependent characteristics of walking and running terrestrial animals based on the study of dimensionless numbers of dynamic similarity, are conventionally separated in two groups, represented by the discrete-state constant and continuous-state variable Froude parameters.

### **2.1.1 Discrete similarity data**

The study by Farley et al. (1993) of the dynamic similarity in trotting quadrupeds (form a rat to a horse through a dog and a goat) extended by hopping bipeds (tammar wallaby and red kangaroo) has been performed near the trot-to-gallop transient states, whose dynamic similarity was revealed by closely spaced Froude numbers *Fr*(exp) *trans* (= *<sup>V</sup>*<sup>2</sup> *trans*/*gLb*, where *Lb* is hip height), limb duty factors *<sup>β</sup>*(exp) *trans* (= Δ*T*/*T* , with Δ*T* is the ground contact time of one foot, and *T* is the stride period), and peak force output *F*(exp) *trans* , represented here as

$$Fr\_{\rm trans}^{(\text{exp})} = 2.25 \pm 0.13 \,\text{J} \,\text{ft}\_{\rm trans}^{(\text{prod})} = 0.40 \,\text{J} \,\text{(exp)}\\ = 0.41 \pm 0.2 \,\text{with } \mu\_{\rm trans}^{(\text{max})} / \text{g} = 3.10. \tag{1}$$

The statistically averaged data for the mean Froude number and the limb duty factor were reported by Farley et al. (1993, p. 74). The relative force output is estimated here on the basis of *<sup>F</sup>*(exp) *trans* discussed in Eq. (17) in Kokshenev (2011) and the failing Strouhal number is predicted below (Fig. 1).

More recently, the discrete-state dynamic similarity was established in 21 trotting horses by Bullimore & Burn (2006) at three constant Froude numbers *Fr*(exp) *trot* (= 0.5, 0.75, and 1). In their Table 4, the authors have shown that besides the stabilization across body mass of two determinants *Fr*(exp) *trot* <sup>∝</sup> *<sup>M</sup>*<sup>0</sup> and *St*(exp) *trot* ∝ *<sup>M</sup>*0, the similar dynamic states are also indicated by the invariable duty factor *<sup>β</sup>*(exp) *trot* ∝ *<sup>M</sup>*0. The omitted scaling factors in statistically observed scaling relations are schematically reproduced here as

$$
\begin{pmatrix} Fr\_{\text{rot}}^{(\text{exp})} \\ St\_{\text{rot}}^{(\text{exp})} \\ \mathcal{J}\_{\text{rot}}^{(\text{exp})} \end{pmatrix} = \begin{pmatrix} 0.50 \ 0.75 \ 1.00 \\ 0.70 \ 0.67 \ 0.60 \\ 0.47 \ 0.44 \ 0.39 \end{pmatrix} \cdot M^{0}. \tag{2}
$$

### **2.1.2 Continuous similarity data**

Treating the estimated Froude numbers as a dynamic variable, changing with speed in walking and running terrestrial animals, the data on the measured gait-dependent mechanical characteristics were obtained by a number of researches. The available systematic data on continuous dynamic similarity in quadrupeds (from a small rodent to a rhinoceros, Alexander & Jayes, 1983), modern elephants (Hutchinson et al., 2006), land-dwelling birds (from a bobwhite to an ostrich) and two humans (Gatesy & Biewener, 1991) are provided in Table 1 in terms of the duty-factor similarity function *<sup>β</sup>*(exp) *gait* and the Strouhal similarity function *St*(exp) *gait* scaled by Froude variable *Fr*.

Strictly speaking, the theory developed in the previous chapter tells us that the observation across body mass of a certain dynamically universal state through equal determinants is a sufficient, but not necessary condition for realization of the same or dynamically similar state in different-sized animals. Indeed, it has been demonstrated that the continues-state similarity pattern of *efficient runners in quadrupeds* can be equally observed through the close spaced discrete similar states indicated by non-equal Froude numbers in the domain1 2 ≤ *Fr* ≤ 3. Since the spectrum of scaling rules of the dynamic similarity theory establishes the universality for solely scaling exponents, leaving aside the problem of scaling factors, the scaling equations empirically established for speed and frequency in dogs and horses may broadly differ in scaling factors, eventually providing different magnitudes in the set of determinants of discrete-state similarity (*Fr*, *St*, *β*) discussed above. Hence, this qualitative analysis suggests the dynamically equivalent states established in different-sized animals can be observed not by solely equal, but also different, generally closely spaced dimensionless numbers unified by the domains of continuous equivalent states distinguished via continues similarity patterns. The developed below theory is discussed in terms of the continuously similar states revealed via the extended set of determinants and indicators of similarity, such as the set (*Fr*, *St*, *μ*, *β*) exemplified by the transient mode-state *continuous similarity pattern*

<sup>289</sup> Physical Insights Into Dynamic Similarity

in Animal Locomotion. II. Observation of Continuous Similarity States

In this study, the discrete states of similarity unified by the generic regime of *fast* locomotion,

*<sup>b</sup> <sup>μ</sup>* · *<sup>V</sup>*−<sup>1</sup> (*ρbμ*2*A*)

<sup>2</sup> *<sup>V</sup>* (*ρbA*)<sup>−</sup> <sup>1</sup>

1 <sup>2</sup> ·*F*<sup>−</sup> <sup>1</sup> <sup>2</sup> *ρ* 1 6 *b μ* 1

<sup>2</sup> · *<sup>F</sup>* <sup>1</sup> <sup>2</sup> *ρ* − 1 6 *<sup>b</sup> <sup>μ</sup>* <sup>1</sup>

<sup>2</sup> · *<sup>M</sup>*<sup>−</sup> <sup>1</sup> 6

> <sup>2</sup> · *<sup>M</sup>*<sup>1</sup> 6

− 1 3 *<sup>b</sup>* · *<sup>M</sup>*<sup>1</sup> 3

− 1 3 *<sup>b</sup> <sup>μ</sup>* · *<sup>M</sup>*<sup>2</sup> 3

Fast gait characteristics Frequency Length Speed Force Mass

<sup>2</sup> · *L* − 1 2

<sup>2</sup> · *<sup>L</sup>* <sup>1</sup>

*gait* /*<sup>g</sup>* <sup>=</sup> <sup>Δ</sup>*F*/*gM <sup>T</sup>*<sup>0</sup> *<sup>L</sup>*<sup>0</sup> *<sup>V</sup>*<sup>0</sup> *<sup>F</sup>*<sup>0</sup> *<sup>M</sup>*<sup>0</sup> *Stgait* = *Lb*/*L T*<sup>0</sup> *LbL*−<sup>1</sup> *V*<sup>0</sup> *F*<sup>0</sup> *M*<sup>0</sup> *Frgait* = *V*2/*gLb T*<sup>0</sup> *L*<sup>0</sup> *V*<sup>0</sup> *F*<sup>0</sup> *M*<sup>0</sup>

Table 2. The determinants suggested for dynamically similar states in animals moving in fast similar gaits. The data are reproduced from Table 2 in Kokshenev (2011). Here *L* is the dynamic (stride or stroke) length and *Lb* is the static (leg or wing) length in animals; *A* is the body's cross-sectional area; *<sup>ρ</sup><sup>b</sup>* is the body density; *<sup>K</sup>*(max) *run* is the maximum amplitude of body stiffness. Other notations are described in the text. The abbreviation for the fast muscular

Before further advancement of the discrete-state similarity theory, it is noteworthy that when a theoretical concept is applied to the real animate systems, several precautions should be taken. The relevant perfect and imperfect qualification in the application of dynamic similarity to real animals was discussed by Alexander (1989). From the theoretical point of view, deviations from the "perfect" uniformity in the dimensionless determinants and the indicators of discrete dynamically similar states may be exemplified by weak body-mass dependence of the swept

*gait* <sup>=</sup> *<sup>L</sup> <sup>μ</sup>* · *<sup>T</sup>*<sup>2</sup> *<sup>L</sup> <sup>μ</sup>*−<sup>1</sup> · *<sup>V</sup>*<sup>2</sup> (*ρbμA*)−<sup>1</sup> · *<sup>F</sup> <sup>ρ</sup>*

*gait* <sup>=</sup> <sup>Δ</sup>*F*/Δ*Lb <sup>T</sup>*<sup>0</sup> *<sup>L</sup>*<sup>0</sup> *<sup>V</sup>*<sup>0</sup> *<sup>F</sup>*<sup>0</sup> *<sup>ρ</sup>*

are going to be generalized to continuous dynamic states described in Table 2.

shown in Eq. (1).

*T*−<sup>1</sup>

*L*(max)

*V*(max)

*K*(max)

*<sup>μ</sup>*(max)

field *<sup>μ</sup>* <sup>=</sup> *<sup>μ</sup>*(max) *run* is adopted.

**2.2.2 Continuous similarity states**

*opt*, *<sup>T</sup>*−<sup>1</sup> *res* <sup>∼</sup> <sup>√</sup>*K*/*<sup>M</sup> <sup>T</sup>*−<sup>1</sup> *<sup>μ</sup>* <sup>1</sup>

*gait* <sup>=</sup> *LT*−<sup>1</sup> *<sup>μ</sup>* · *<sup>T</sup> <sup>μ</sup>* <sup>1</sup>


Table 1. The equations of linear least squares regression for the limb duty factor (*β*) and relative stride length (= *St*−1, see, e.g., Table 2 below) obtained from walking and running animals. The data on African and Asian elephants are by Hutchinson et al. (2006, Table 3). The data on cursorial quadrupeds (a dog, a sheep, a camel, and a rhinoceros) are by Alexander & Jayes (1983, Table II). For quadrupeds, the limb duty factor data are the arithmetic means of forelimb and hindlimb data. The dynamic similarity functions for birds and humans are obtained here by the standard least squares regression of the continuous-state data reproduced from Figs. 5 and 7 in Gatesy & Biewener (1991). The scaling equations for bipeds are approximated by the geometric mean of those for birds and humans.2

### **2.2 Theory of continuous similarity**

### **2.2.1 Statement of the problem**

When Alexander's central hypothesis exemplified in Eq. (2) is applied to the efficiently trotting dogs, a question arises whether the dogs observed at the same three constant Froude numbers will show the same Strouhal numbers and limb duty factors as in the horses? In other words, whether the equivalent dynamic similarity states chosen at certain fixed numbers *Fr* determine the same dynamic similarity states in horses indicated by the same equalized magnitudes of *St* and *β*? Moreover, if a chosen small trotting horse and an estimated big trotting dog show the same Froude number, does it mean that other corresponding determinants and indicators of the dynamic similarity should be equal in magnitudes? All these questions are concern with the concept of continuous-state dynamic similarity justified by the universal gait-dependent dynamic similarity functions hypothesized by Alexander & Jayes (1983).

<sup>2</sup> For example, for the two similarity functions *y*<sup>1</sup> = *a*1*xb*<sup>1</sup> and *y*<sup>2</sup> = *a*2*xb*<sup>2</sup> the geometric mean function results in *<sup>y</sup>* <sup>=</sup> <sup>√</sup>*y*1*y*2, with the corresponding scaling factor *<sup>a</sup>* <sup>=</sup> <sup>√</sup>*a*1*a*<sup>2</sup> and exponent *<sup>b</sup>* = (*b*<sup>1</sup> <sup>+</sup> *<sup>b</sup>*2)/2.

Strictly speaking, the theory developed in the previous chapter tells us that the observation across body mass of a certain dynamically universal state through equal determinants is a sufficient, but not necessary condition for realization of the same or dynamically similar state in different-sized animals. Indeed, it has been demonstrated that the continues-state similarity pattern of *efficient runners in quadrupeds* can be equally observed through the close spaced discrete similar states indicated by non-equal Froude numbers in the domain1 2 ≤ *Fr* ≤ 3. Since the spectrum of scaling rules of the dynamic similarity theory establishes the universality for solely scaling exponents, leaving aside the problem of scaling factors, the scaling equations empirically established for speed and frequency in dogs and horses may broadly differ in scaling factors, eventually providing different magnitudes in the set of determinants of discrete-state similarity (*Fr*, *St*, *β*) discussed above. Hence, this qualitative analysis suggests the dynamically equivalent states established in different-sized animals can be observed not by solely equal, but also different, generally closely spaced dimensionless numbers unified by the domains of continuous equivalent states distinguished via continues similarity patterns. The developed below theory is discussed in terms of the continuously similar states revealed via the extended set of determinants and indicators of similarity, such as the set (*Fr*, *St*, *μ*, *β*) exemplified by the transient mode-state *continuous similarity pattern* shown in Eq. (1).

### **2.2.2 Continuous similarity states**

4 Will-be-set-by-IN-TECH

(exp) *walk* ±Δ*b*

(exp) *walk r*

(exp) *Af rican r*

(exp) *Af rican r*

(exp) *walk a*

*gait* 0.52 −0.16 0.07 0.53 −0.28 0.03

*gait* 2.4 0.34 0.10 1.9 0.40 0.03

*gait* 0.60 −0.09 0.69 0.56 −0.20 0.69

*gait* 0.58 −0.08 0.93 0.40 −0.29 0.90

*gait* 0.59 −0.09 −− 0.47 −0.25 −−

*gait* 2.34 0.16 0.72 2.42 0.31 0.72

*gait* 2.48 0.28 0.80 2.48 0.38 0.76

*gait* 2.41 0.22 −− 2.45 0.35 −−

*eleph* 0.53 −0.16 0.89 0.51 −0.16 0.84

*eleph* 2.0 0.18 0.84 2.1 0.17 0.86

(exp)2 *Af rican a*

(exp)2 *Af rican a*

(exp)2 *walk a*

(exp) *run <sup>b</sup>*

(exp) *run <sup>b</sup>*

(exp) *Asian b*

(exp) *Asian b*

(exp) *Asian r*

(exp) *Asian r*

(exp) *run* <sup>±</sup>Δ*<sup>b</sup>*

(exp) *run <sup>r</sup>*

(exp) *run*

(exp)<sup>2</sup> *run*

(exp)2 *Asian*

(exp)2 *Asian*

(exp) *walk b*

(exp) *walk b*

(exp) *Af rican b*

(exp) *Af rican b*

and humans are obtained here by the standard least squares regression of the

Table 1. The equations of linear least squares regression for the limb duty factor (*β*) and relative stride length (= *St*−1, see, e.g., Table 2 below) obtained from walking and running animals. The data on African and Asian elephants are by Hutchinson et al. (2006, Table 3). The data on cursorial quadrupeds (a dog, a sheep, a camel, and a rhinoceros) are by Alexander & Jayes (1983, Table II). For quadrupeds, the limb duty factor data are the arithmetic means of forelimb and hindlimb data. The dynamic similarity functions for birds

continuous-state data reproduced from Figs. 5 and 7 in Gatesy & Biewener (1991). The scaling equations for bipeds are approximated by the geometric mean of those for birds and

When Alexander's central hypothesis exemplified in Eq. (2) is applied to the efficiently trotting dogs, a question arises whether the dogs observed at the same three constant Froude numbers will show the same Strouhal numbers and limb duty factors as in the horses? In other words, whether the equivalent dynamic similarity states chosen at certain fixed numbers *Fr* determine the same dynamic similarity states in horses indicated by the same equalized magnitudes of *St* and *β*? Moreover, if a chosen small trotting horse and an estimated big trotting dog show the same Froude number, does it mean that other corresponding determinants and indicators of the dynamic similarity should be equal in magnitudes? All these questions are concern with the concept of continuous-state dynamic similarity justified by the universal gait-dependent dynamic similarity functions hypothesized by Alexander &

<sup>2</sup> For example, for the two similarity functions *y*<sup>1</sup> = *a*1*xb*<sup>1</sup> and *y*<sup>2</sup> = *a*2*xb*<sup>2</sup> the geometric mean function results in *<sup>y</sup>* <sup>=</sup> <sup>√</sup>*y*1*y*2, with the corresponding scaling factor *<sup>a</sup>* <sup>=</sup> <sup>√</sup>*a*1*a*<sup>2</sup> and exponent *<sup>b</sup>* = (*b*<sup>1</sup> <sup>+</sup> *<sup>b</sup>*2)/2.

Animals *a*(*Fr*)*<sup>b</sup> a*

Bipeds *a*(*Fr*)*<sup>b</sup> a*

Quadrupeds curs. *<sup>β</sup>*(exp)

Birds *<sup>β</sup>*(exp)

Humans *<sup>β</sup>*(exp)

Bipeds *<sup>β</sup>*(exp)

Birds *St*( exp )−<sup>1</sup>

Humans *St*( exp )−<sup>1</sup>

Bipeds *St*( exp )−<sup>1</sup>

Elephants *<sup>β</sup>*(exp)

Elephants *St*−<sup>1</sup>

**2.2 Theory of continuous similarity 2.2.1 Statement of the problem**

humans.2

Jayes (1983).

Elephants *a*+*b*log(*Fr*) *a*

Elephants *a*(*Fr*)*<sup>b</sup> a*

Quadrupeds curs. *St*( exp )−<sup>1</sup>

In this study, the discrete states of similarity unified by the generic regime of *fast* locomotion, are going to be generalized to continuous dynamic states described in Table 2.


Table 2. The determinants suggested for dynamically similar states in animals moving in fast similar gaits. The data are reproduced from Table 2 in Kokshenev (2011). Here *L* is the dynamic (stride or stroke) length and *Lb* is the static (leg or wing) length in animals; *A* is the body's cross-sectional area; *<sup>ρ</sup><sup>b</sup>* is the body density; *<sup>K</sup>*(max) *run* is the maximum amplitude of body stiffness. Other notations are described in the text. The abbreviation for the fast muscular field *<sup>μ</sup>* <sup>=</sup> *<sup>μ</sup>*(max) *run* is adopted.

Before further advancement of the discrete-state similarity theory, it is noteworthy that when a theoretical concept is applied to the real animate systems, several precautions should be taken. The relevant perfect and imperfect qualification in the application of dynamic similarity to real animals was discussed by Alexander (1989). From the theoretical point of view, deviations from the "perfect" uniformity in the dimensionless determinants and the indicators of discrete dynamically similar states may be exemplified by weak body-mass dependence of the swept

functions, determining continuous similarity. Using the experimental data for the exponents,

<sup>291</sup> Physical Insights Into Dynamic Similarity

The continuous dynamic similarity theory is applied below to the data from running quadrupeds and bipeds (Table 1) on the basis of determinants (Table 2) already specified by the optimum-speed, crossover-gait (walk-to-run), and transient-run (trot-to-gallop) discrete states. The analysis is provided in terms of the continuous-state similarity determinants

As mentioned,<sup>1</sup> the continuous dynamic similarity was first observed by Alexander & Jayes (1983) within the domain 2 ≤ *Fr* ≤ 3 of transient trot-to-gallop dynamically similar states in running cursorial quadrupeds (Table 1). Following the authors, quadrupeds move in dynamically similar way, changing their fashion of locomotion at equal Froude numbers. The trot-to-gallop dynamic transitions were determined by the phase difference between the fore feet, indicating a transition between symmetrical (trot or pace) and asymmetrical (gallop or canter) modes of the run gait (Alexander & Jayes, 1983, Fig. 1). Although all transitions between mode patterns of locomotion are indicated by quantities at which one or more change discontinuously (Alexander & Jayes, 1983; Alexander, 1989, 2003), the *symmetry-mode transitions* are considered here as continuous, at least within the scope of relative stride length and limb duty factor functions, for which no abrupt changes were indicated. The continuous *trot-to-gallop transition*, associated in the cursorial quadrupeds (Table 1) with constant1 *Frtrot*−*gall* <sup>=</sup> 2.5, was re-discovered in quadrupeds (form a rat to a horse) by Farley et al. (1993) as shown in Eq. (1) and re-analyzed in Eq. (17) in Kokshenev (2011) by a number

of scaling rules equally validating for discrete and continuous similar states (Table 2).

In contrast to run-mode transitions, a walk-to-run crossover in quadrupeds is associated with *discontinues* transient-gait states. In quadrupeds, excluding elephants, a gradual increasing of speed provokes abrupt changes in both the relative stride length and limb duty factors.

independent) discontinues transition from a walk to a run determined by the dynamic

the scaling data on continuous dynamic similarity in quadrupeds by Alexander & Jayes (1983)

*unstable-state* points 1 and 2 (shown by closed stars), whereas the continuous trot-to-gallop

In the case of cursorial quadrupeds, the duty-factor similarity function (Fig. 1) established for trotting and galloping quadrupeds by Alexander & Jayes (1983) is remarkably consistent with the trot-to-gallop continuous transition states statistically described by Farley et al. (1993) and shown in Eq. (1). These transient-mode states, likely first revealed by Heglund et al. (1974), suggest a stabilization of the characteristic Strouhal number (Table 3) unifying quadrupeds and bipeds within the domain 2 ≤ *Fr* ≤ 3 associated with the continuous similarity pattern of

*walk* <sup>=</sup> *Fr*(exp) *run* <sup>=</sup> 0.40 (Alexander & Jayes, 1983, Figs. 3 and 4). In Fig. 1,

*walk*−*run* <sup>≤</sup> 0.67 indicate the universal (mass

*walk*−*run* <sup>=</sup> 0.40 is shown by

*gait* (*Fr*) <sup>∼</sup> *gFrb*<sup>3</sup> , with *<sup>b</sup>*<sup>3</sup> <sup>=</sup> <sup>1</sup> <sup>−</sup> *<sup>b</sup>*1, (7)

one obtains a scaling prediction for the muscle-field determinant

*<sup>μ</sup>*(*pred*)

in Animal Locomotion. II. Observation of Continuous Similarity States

straightforwardly following from Eqs. (5) and (6).

(*Fr*, *St*, *β*, *μ*) exemplified in Eq. (1) and in part in Eq. (2).

These changes revealed in the domain 0.61 <sup>≤</sup> *<sup>β</sup>*(exp)

are reproduced and the discontinues walk-to-run transition at *Fr*(exp)

transition is shown by the *continuous-state* points 4 and 5 (shown by open stars).

similarity variable *Fr*(exp)

**3. Results and discussion**

**3.1 Quadrupeds including elephants**

angle <sup>Θ</sup>(exp) *run* (<sup>=</sup> 0.60*M*−0.03) and the *relative body length change <sup>ε</sup>* (exp) *run* (<sup>=</sup> 0.17*M*−0.04), revealed by the leg-spring model in quadrupeds and bipeds (Farley et al., 1993). The plausible reasons of deviations from the universality in dimensionless determinants of the similarity are (i) uncompensated external ground reaction forces and internal body reaction forces, resulting in deviation in body's rigidity, (ii) deviations from the condition of maximal locomotor efficiency and (iii) from the isometrically approximated scaling rules for muscle and bone masses with body mass. The latter effects are caused by small but finite mass allometric exponents (Prange et al., 1979) discussed in the problem of the primary locomotor functions of long skeletal bones (Kokshenev et al. 2003; Kokshenev, 2003, 2007) and striated and cardiac muscles (Kokshenev, 2008, 2009).

Following the original version of the dynamic similarity theory (Kokshenev, 2011), the similarity between equivalent discrete states established by the dimensionless determinants is also indicated by the uniform dimensionless scaling parameters, e.g., *β*, *ε*, or Θ, generally omitted in the scaling rules (e.g., Table 2). In the version of continuous-state dynamic similarity driven by continuous Froude variable, all other dimensionless parameters are also state-dependent continuous functions, e.g., *St*(*Fr*), *β*(*Fr*), or Θ(*Fr*). Following the concept of the unique similarity criterion (Kokshenev, 2011), each of these functions ambiguously determines the dynamic similarity between animals moving in certain gait. Hence, extending scaling relations from the discrete states to continues states, the omitted in Table 2 dimensionless parameters should be restored.

Using the basic definitive equations for resonant frequency and optimum speed (Table 2), one obtains the basic gait-dependent relations of the continuous dynamic similarity, namely

$$T\_{res}^{-1} \sim \sqrt{\frac{\mu\_{gai}}{\varepsilon\_b}} L\_b^{-\frac{1}{2}},\\ V\_{gai} \sim \sqrt{\frac{\mu\_{gai}}{\varepsilon\_b}} L\_{gai} L\_b^{-\frac{1}{2}},\\ \text{with } \varepsilon\_b = \frac{\Delta L\_b}{L\_b},\tag{3}$$

where *Lgait* is the dynamic length. In turn, two corresponding dimensionless numbers

$$Fr\_{gail} \sim \varepsilon\_b^{-1} \frac{\mu\_{gail}}{g} \left(\frac{L\_{gail}}{L\_b}\right)^2, St\_{gail} = \left(\frac{L\_{gail}}{L\_b}\right)^{-1}, \text{with } \varepsilon\_b \sim \frac{L\_{gail}}{L\_b},\tag{4}$$

follow from Table 2, along with the relative static length change *ε<sup>b</sup>* approximated here by the basic relation Δ*Lb* ∼ *L*, common in linear scaling theory.

When the determinant *Fr* is chosen as an independent dynamic variable of continuous similarity, Eq. (4) yields *muscle-field similarity function*

$$
\mu\_{\text{gail}}(Fr) \sim gFrSt\_{\text{gail}}(Fr),\tag{5}
$$

where the omitted numerical factor is not universal and therefore insignificant in scaling theory. Bearing in mind the experimental data (Table 1), the *Strouhal similarity function St*(exp) *gait* (*Fr*) and the *duty-factor similarity function <sup>β</sup>*(exp) *gait* (*Fr*) can be presented as

$$\mathcal{S}t\_{gail}^{(\text{exp})}(Fr) \sim (Fr)^{-b\_1} \text{ and } \mathcal{\beta}\_{gail}^{(\text{exp})}(Fr) \sim (Fr)^{b\_2}.\tag{6}$$

where the exponents *b*<sup>1</sup> and *b*<sup>2</sup> (observed experimentally in Table 1) are generally constrained by the basic definitive equations. One can therefore infer that the dynamic similarity between animate systems is controlled by the scaling exponents of the gait-dependent similarity functions, determining continuous similarity. Using the experimental data for the exponents, one obtains a scaling prediction for the muscle-field determinant

$$
\mu\_{g\_{\text{ait}}}^{(pred)}(Fr) \sim gFr^{b\_3}, \text{with } b\_3 = 1 - b\_1. \tag{7}
$$

straightforwardly following from Eqs. (5) and (6).

### **3. Results and discussion**

6 Will-be-set-by-IN-TECH

by the leg-spring model in quadrupeds and bipeds (Farley et al., 1993). The plausible reasons of deviations from the universality in dimensionless determinants of the similarity are (i) uncompensated external ground reaction forces and internal body reaction forces, resulting in deviation in body's rigidity, (ii) deviations from the condition of maximal locomotor efficiency and (iii) from the isometrically approximated scaling rules for muscle and bone masses with body mass. The latter effects are caused by small but finite mass allometric exponents (Prange et al., 1979) discussed in the problem of the primary locomotor functions of long skeletal bones (Kokshenev et al. 2003; Kokshenev, 2003, 2007) and striated and cardiac muscles (Kokshenev,

Following the original version of the dynamic similarity theory (Kokshenev, 2011), the similarity between equivalent discrete states established by the dimensionless determinants is also indicated by the uniform dimensionless scaling parameters, e.g., *β*, *ε*, or Θ, generally omitted in the scaling rules (e.g., Table 2). In the version of continuous-state dynamic similarity driven by continuous Froude variable, all other dimensionless parameters are also state-dependent continuous functions, e.g., *St*(*Fr*), *β*(*Fr*), or Θ(*Fr*). Following the concept of the unique similarity criterion (Kokshenev, 2011), each of these functions ambiguously determines the dynamic similarity between animals moving in certain gait. Hence, extending scaling relations from the discrete states to continues states, the omitted

Using the basic definitive equations for resonant frequency and optimum speed (Table 2), one obtains the basic gait-dependent relations of the continuous dynamic similarity, namely

> *<sup>μ</sup>gait εb*

where *Lgait* is the dynamic length. In turn, two corresponding dimensionless numbers

, *Stgait* =

follow from Table 2, along with the relative static length change *ε<sup>b</sup>* approximated here by the

When the determinant *Fr* is chosen as an independent dynamic variable of continuous

where the omitted numerical factor is not universal and therefore insignificant in scaling theory. Bearing in mind the experimental data (Table 1), the *Strouhal similarity function*

where the exponents *b*<sup>1</sup> and *b*<sup>2</sup> (observed experimentally in Table 1) are generally constrained by the basic definitive equations. One can therefore infer that the dynamic similarity between animate systems is controlled by the scaling exponents of the gait-dependent similarity

*gait* (*Fr*) <sup>∼</sup> (*Fr*)−*b*<sup>1</sup> and *<sup>β</sup>*(exp)

2

*LgaitL* − 1 2

 *Lgait Lb*

<sup>−</sup><sup>1</sup>

*μgait*(*Fr*) ∼ *gFrStgait*(*Fr*), (5)

*gait* (*Fr*) can be presented as

*gait* (*Fr*) <sup>∼</sup> (*Fr*)*b*<sup>2</sup> , (6)

*<sup>b</sup>* , with *<sup>ε</sup><sup>b</sup>* <sup>=</sup> <sup>Δ</sup>*Lb*

, with *<sup>ε</sup><sup>b</sup>* <sup>∼</sup> *Lgait*

*Lb*

*Lb*

, (3)

, (4)

(exp) *run* (<sup>=</sup> 0.17*M*−0.04), revealed

angle <sup>Θ</sup>(exp) *run* (<sup>=</sup> 0.60*M*−0.03) and the *relative body length change <sup>ε</sup>*

in Table 2 dimensionless parameters should be restored.

*<sup>μ</sup>gait εb L* − 1 2 *<sup>b</sup>* , *Vgait* ∼

> *μgait g*

basic relation Δ*Lb* ∼ *L*, common in linear scaling theory.

*gait* (*Fr*) and the *duty-factor similarity function <sup>β</sup>*(exp)

*St*(exp)

similarity, Eq. (4) yields *muscle-field similarity function*

 *Lgait Lb*

−1 *b*

*<sup>T</sup>*−<sup>1</sup> *res* ∼

*Frgait* ∼ *ε*

2008, 2009).

*St*(exp)

The continuous dynamic similarity theory is applied below to the data from running quadrupeds and bipeds (Table 1) on the basis of determinants (Table 2) already specified by the optimum-speed, crossover-gait (walk-to-run), and transient-run (trot-to-gallop) discrete states. The analysis is provided in terms of the continuous-state similarity determinants (*Fr*, *St*, *β*, *μ*) exemplified in Eq. (1) and in part in Eq. (2).

### **3.1 Quadrupeds including elephants**

As mentioned,<sup>1</sup> the continuous dynamic similarity was first observed by Alexander & Jayes (1983) within the domain 2 ≤ *Fr* ≤ 3 of transient trot-to-gallop dynamically similar states in running cursorial quadrupeds (Table 1). Following the authors, quadrupeds move in dynamically similar way, changing their fashion of locomotion at equal Froude numbers. The trot-to-gallop dynamic transitions were determined by the phase difference between the fore feet, indicating a transition between symmetrical (trot or pace) and asymmetrical (gallop or canter) modes of the run gait (Alexander & Jayes, 1983, Fig. 1). Although all transitions between mode patterns of locomotion are indicated by quantities at which one or more change discontinuously (Alexander & Jayes, 1983; Alexander, 1989, 2003), the *symmetry-mode transitions* are considered here as continuous, at least within the scope of relative stride length and limb duty factor functions, for which no abrupt changes were indicated. The continuous *trot-to-gallop transition*, associated in the cursorial quadrupeds (Table 1) with constant1 *Frtrot*−*gall* <sup>=</sup> 2.5, was re-discovered in quadrupeds (form a rat to a horse) by Farley et al. (1993) as shown in Eq. (1) and re-analyzed in Eq. (17) in Kokshenev (2011) by a number of scaling rules equally validating for discrete and continuous similar states (Table 2).

In contrast to run-mode transitions, a walk-to-run crossover in quadrupeds is associated with *discontinues* transient-gait states. In quadrupeds, excluding elephants, a gradual increasing of speed provokes abrupt changes in both the relative stride length and limb duty factors. These changes revealed in the domain 0.61 <sup>≤</sup> *<sup>β</sup>*(exp) *walk*−*run* <sup>≤</sup> 0.67 indicate the universal (mass independent) discontinues transition from a walk to a run determined by the dynamic similarity variable *Fr*(exp) *walk* <sup>=</sup> *Fr*(exp) *run* <sup>=</sup> 0.40 (Alexander & Jayes, 1983, Figs. 3 and 4). In Fig. 1, the scaling data on continuous dynamic similarity in quadrupeds by Alexander & Jayes (1983) are reproduced and the discontinues walk-to-run transition at *Fr*(exp) *walk*−*run* <sup>=</sup> 0.40 is shown by *unstable-state* points 1 and 2 (shown by closed stars), whereas the continuous trot-to-gallop transition is shown by the *continuous-state* points 4 and 5 (shown by open stars).

In the case of cursorial quadrupeds, the duty-factor similarity function (Fig. 1) established for trotting and galloping quadrupeds by Alexander & Jayes (1983) is remarkably consistent with the trot-to-gallop continuous transition states statistically described by Farley et al. (1993) and shown in Eq. (1). These transient-mode states, likely first revealed by Heglund et al. (1974), suggest a stabilization of the characteristic Strouhal number (Table 3) unifying quadrupeds and bipeds within the domain 2 ≤ *Fr* ≤ 3 associated with the continuous similarity pattern of

Point Mode states and patterns in animals *Fr St*(exp)

in Animal Locomotion. II. Observation of Continuous Similarity States

 unstable walk in quadrupeds 0.40 0.57 0.61 0.55 unstable run in quadrupeds 0.40 0.76 0.67 1.10 efficient runners in quadrupeds 1.00 0.53 0.52 1.91 trans. moderate run in quadrupeds 2.00 0.40 0.43 2.90 transient fast run in quadrupeds 3.00 0.34 0.38 3.69 →<sup>5</sup> efficient trotters and hoppers(∗) 2.25 0.40 0.41 3.10 successful fast walkers (elephants) 1.00 0.49 0.52 1.00 efficient walkers in bipeds (birds) 0.52 0.48 0.64 1.00 unstable fast walk in humans 0.40 0.52 0.62 0.85 unstable fast walk in bipeds 0.45 0.50 0.63 0.93 unstable slow run in humans 0.60 0.49 0.46 1.00 unstable slow run in bipeds 0.55 0.50 0.56 1.00 efficient runners in bipeds 1.00 0.41 0.47 1.47 efficient fast runners in bipeds 1.61 0.35 0.43 2.00 successful hoppers in bipeds 4.66 0.24 0.33 4.00

<sup>293</sup> Physical Insights Into Dynamic Similarity

Table 3. The characteristic points, determining continuous and discontinues dynamically similar states in different-sized and different-taxa animals, are analyzed on the basis of the walk similarity and run similarity functions from cursorial quadrupeds (Figs. 1-5). The

> (*pred*) *walk b*

with the help of Eqs. (13) and (14) in Kokshenev & Christiansen (2011).

*trot* (*Fr*) = 0.4*β*(exp) *run* (*Fr*) and *St*(exp)

which differ in insignificant scaling parameters discussed in Eq. (6).

(*pred*) *walk a*

Quadrupeds cursorial 1.00 0.72 1.91 0.60 Fig. 2 Birds 1.73 0.84 1.57 0.69 Fig. 5 Humans 1.57 0.72 1.37 0.62 Fig. 5 Bipeds 1.65 0.78 1.47 0.65 Fig. 5 Elephants modern 1.00 0.82 1.00 0.83 Fig. 2 Elephants extinct −− −− 0.29 0.30 Fig. 2 Table 4. The predicted relative force-output similarity function determined in Eq. (7) and described in the text. The equations for bipeds are approximated by geometric mean of the data for birds and humans.2 The data on the body mobility for extinct elephants are obtained

As mentioned in the Introduction, the revealed continuous dynamic similarity states is expected to provide further generalizations about movements of animals of different size and taxa. The continuous similarity patterns of *efficient flapping flyers* and *efficient hovering flyers* established within the domain 0.2 ≤ *St* ≤ 0.4 for fast and slow flight modes by Taylor et al. (2003, Table 2) in cruising birds are schematically shown in Fig. 1, using, respectively,

predictions for relative force output *<sup>μ</sup>*(*pred*)

*gait* /*<sup>g</sup>* <sup>=</sup> *<sup>a</sup>*(*Fr*)*<sup>b</sup> <sup>a</sup>*

scaling equations unifying trotting quadrupeds, namely

pattern introduced via Eq. (1).

*<sup>μ</sup>*(*pred*)

*<sup>β</sup>*(exp)

the run-mode *St*(exp) *run* and walk-mode *St*(exp)

*gait <sup>β</sup>*(exp)

*gait* /*<sup>g</sup>* are from in Table 4. (∗)The transient-state

(*pred*) *run* Discussed in

*trot* (*Fr*) = 1.6*St*(exp) *run* (*Fr*), (8)

*walk* universal Strouhal functions from running and

(*pred*) *run <sup>b</sup>*

*gait <sup>μ</sup>*(*pred*) *gait* /*g*

Fig. 1. The comparative analysis of the run-mode and walk-mode dynamic similarity functions in quadrupeds. The *solid lines* are regression data by Alexander & Jayes (1983) described in Table 1. The *dashed lines* are continuously extrapolated data. The *dashed-dotted* lines are the smoothed discrete-state data from trotting horses (shown by *open squares*) by Bullimore & Burn (2006) shown in Eq. (2). The *closed squares* represent those data by Farley et al. (1993, Table 1) for trotting quadrupeds and hopping bipeds (distinguished by *italic style*), which are limited by the experimental error shown in Eq. (1). The *open squares* tentatively represent the data on Strouhal numbers in cruising birds and fish by Taylor et al. (2003). The *closed triangles* are the averaged data from Asian and African elephants (Table 1) by Hutchinson et al. (2006). The open and closed *stars* indicate, respectively, continuous (stable) and discontinues (unstable) dynamic states. For more complete description of these and other characteristic points, see Table 3.

*efficient trotters and hoppers* (Table 3), evidently including the transient and optimal continuous states established at preferred trotting and preferred galloping speeds (Heglund & Taylor, 1988; Perry et al., 1988).

In the special case of trotting horses, the discrete-state data by Bullimore & Burn (2006), discussed in Eq. (2) and shown Fig. 1 by the open squares, are extended here by the data from a horse by Farley et al. (1993), as shown in Fig. 1 by the closed square. Although the smoothed data on trotting horses (shown by dashed point line in Fig. 1) coincide neither with the duty-factor function *<sup>β</sup>*(exp) *run* nor the Strouhal function *St*(exp) *run* , which were suggested to be universal (Alexander & Jayes, 1983), the dynamic continuous similarity between trotting quadrupeds (excluding non-trotting elephants) and trotting horses can theoretically be established. Indeed, the corresponding smoothed lines in Fig. 1 are drawn by the same 8 Will-be-set-by-IN-TECH

100 1

<sup>3</sup> <sup>4</sup>

efficient trotters & *biped hoppers*

efficient hovering flyers

5

Dog

Horse

5

Goat

4

0.1

Strouhal numbers, *St* 

=*Lb/L*

*St*

*Fr*

Walk

2

successful fast walkers, efficient runners, & fast animals

Walk

Quadrupeds

1

efficient runners

Elephants

1

E

2

0.2 0.3 0.4 0.5 0.6 0.7 0.8

Limb duty factors,

Fig. 1. The comparative analysis of the run-mode and walk-mode dynamic similarity functions in quadrupeds. The *solid lines* are regression data by Alexander & Jayes (1983) described in Table 1. The *dashed lines* are continuously extrapolated data. The *dashed-dotted* lines are the smoothed discrete-state data from trotting horses (shown by *open squares*) by Bullimore & Burn (2006) shown in Eq. (2). The *closed squares* represent those data by Farley et al. (1993, Table 1) for trotting quadrupeds and hopping bipeds (distinguished by *italic style*), which are limited by the experimental error shown in Eq. (1). The *open squares* tentatively represent the data on Strouhal numbers in cruising birds and fish by Taylor et al. (2003). The

*closed triangles* are the averaged data from Asian and African elephants (Table 1) by

Hutchinson et al. (2006). The open and closed *stars* indicate, respectively, continuous (stable) and discontinues (unstable) dynamic states. For more complete description of these and

*efficient trotters and hoppers* (Table 3), evidently including the transient and optimal continuous states established at preferred trotting and preferred galloping speeds (Heglund & Taylor,

In the special case of trotting horses, the discrete-state data by Bullimore & Burn (2006), discussed in Eq. (2) and shown Fig. 1 by the open squares, are extended here by the data from a horse by Farley et al. (1993), as shown in Fig. 1 by the closed square. Although the smoothed data on trotting horses (shown by dashed point line in Fig. 1) coincide neither with the duty-factor function *<sup>β</sup>*(exp) *run* nor the Strouhal function *St*(exp) *run* , which were suggested to be universal (Alexander & Jayes, 1983), the dynamic continuous similarity between trotting quadrupeds (excluding non-trotting elephants) and trotting horses can theoretically be established. Indeed, the corresponding smoothed lines in Fig. 1 are drawn by the same

fast walk

*Red kangaroo Tammar wallaby*

3

0.1

other characteristic points, see Table 3.

1988; Perry et al., 1988).

1

10

Run

efficient flapping flyers & undulating swimmers

*Kangaroo rat*

Froude numbers, *Fr*


Table 3. The characteristic points, determining continuous and discontinues dynamically similar states in different-sized and different-taxa animals, are analyzed on the basis of the walk similarity and run similarity functions from cursorial quadrupeds (Figs. 1-5). The predictions for relative force output *<sup>μ</sup>*(*pred*) *gait* /*<sup>g</sup>* are from in Table 4. (∗)The transient-state pattern introduced via Eq. (1).


Table 4. The predicted relative force-output similarity function determined in Eq. (7) and described in the text. The equations for bipeds are approximated by geometric mean of the data for birds and humans.2 The data on the body mobility for extinct elephants are obtained with the help of Eqs. (13) and (14) in Kokshenev & Christiansen (2011).

scaling equations unifying trotting quadrupeds, namely

$$\mathcal{A}\_{\text{trot}}^{(\text{exp})}(Fr) = 0.4 \mathcal{A}\_{\text{run}}^{(\text{exp})}(Fr) \text{ and } \mathcal{S}t\_{\text{trot}}^{(\text{exp})}(Fr) = 1.6 St\_{\text{run}}^{(\text{exp})}(Fr), \tag{8}$$

which differ in insignificant scaling parameters discussed in Eq. (6).

As mentioned in the Introduction, the revealed continuous dynamic similarity states is expected to provide further generalizations about movements of animals of different size and taxa. The continuous similarity patterns of *efficient flapping flyers* and *efficient hovering flyers* established within the domain 0.2 ≤ *St* ≤ 0.4 for fast and slow flight modes by Taylor et al. (2003, Table 2) in cruising birds are schematically shown in Fig. 1, using, respectively, the run-mode *St*(exp) *run* and walk-mode *St*(exp) *walk* universal Strouhal functions from running and

0 0.5 1 1.5 2 2.5 3 3.5 4

near-critical run Largest extinct elephants

*walk*-*run* = 1 is employed as a reference point (Table 4). Likewise, in

*eleph* (*Fr*) by Hutchinson et al. (2006) shown in Table 1, with the

Froude variable, *Fr*

the case of slow running modern elephants, the relative-force similarity function is obtained

help of the same transient state in the walk-to-run continuous transition, also employed

therefore observe (Fig. 2) the dynamic similarity between poorly running (no trotting) modern elephants (shown by the closed triangles) and other hypothetical quadrupeds using the inverse gradient in the fore-hind limb locomotion function, common to both extant and extinct elephants (Kokshenev & Christiansen, 2010, Fig. 5). The data on low level *mobility* of largest extinct elephants (Kokshenev & Christiansen, 2011, Figs. 4 and 5), shown by open triangles in Fig. 2, are obtained by evaluation of the torsional and bending limb bone elastic forces, acting

In humans, a discontinues walk-to-run crossover is generally observed at Froude numbers

*walk*-*run* ≈ 0.5 (Thorstensson & Roberthson, 1987, Kram et al., 1997; Alexander, 1989;

as the reference point determining the dynamic similarity at *Fr*(mod)

successful fast walkers

Fig. 2. Locomotor ability similarity function predicted for quadrupeds by the relative force-output. The *solid lines* are estimates for walking and running quadrupeds obtained via Eq. (7) with the help of the corresponding experimental data on the Strouhal similarity function (Tables 1 and 4). The *dashed line* extrapolates the data on slow-walk to fast-walk modes. The *dashed-dotted line* adjusted with *open squares* is a sketched estimate for trotting horses, also discussed in Fig. 1. The characteristic points and the pattern points are those shown in Fig. 1 and described in Table 3. The comparative data for bipeds are indicated by *italic style*. The *closed triangles* and *open triangles* are respectively the predicted data on the body ability in modern elephants and the body mobility in extinct elephants (Table 4). to the fast-walk mode, where a transient state for the walk-to-run determinant *<sup>μ</sup>*(*pred*)

4

<sup>295</sup> Physical Insights Into Dynamic Similarity

Goat Dog

*Red kangaroo Tammar wallaby* 5

*Kangaroo rat*

*walk*-*run* = *g*

*walk*-*run* = 1. One can

0

predicted in Fig. 1 at *Fr*(mod)

on the basis of the data *St*(exp)

in running giants (Table 4).

*Fr*(exp)

**3.2 Bipeds including humans**

1

2

fast animals

<sup>1</sup> Walk

2

Locomotor ability,

P */g*

3

3

Quadrupeds

in Animal Locomotion. II. Observation of Continuous Similarity States

efficient trotters & *biped hoppers*

3

4

5

walking quadrupeds by Alexander & Jayes (1983). As the result, the theoretically observed *muscle* duty factor *<sup>β</sup>*(*pred*) *flap* <sup>≈</sup> 1/3 predicting in turn the muscle timing <sup>Δ</sup>*T*(*pred*) *flap* ≈ *Tflap*/3 (see Eq. (9) in Kokshenev, 2011), indicates that the wing muscle, driving the flap mode in a flight, is activated during one third time of the cycle period *Tflap*. This finding can be explained by the three-step cycling of the flight-motor individual muscles in birds, as clearly revealed by workloop technics (see e.g., Fig. 3B in Dickinson et al., 2000).

Coming back to terrestrial animals, the dynamic similarity predicted by inverted-pendulum model for the *walk-to-run crossover* was expected to be determined by *Fr*(mod) *walk*-*run* = 1 (e.g., Hildebrand, 1980; Cartmill et al., 2002). Also, the universal discrete state indicated by the model-independent duty factor *<sup>β</sup>*(*pred*) *walk*-*run* = 0.5 (e.g., Alexander & Maloiy, 1989; Alexander, 1992; Usherwood, 2005) could be generally expected, besides the muscular transient field *<sup>μ</sup>*(*pred*) *walk*-*run* = *g* (Kokshenev, 2011). However, such a continuous-state transition has been observed neither in bipeds, including humans, nor in quadrupeds, excluding elephants. As indicated by the walk-instability and run-instability points 1 and 2 in Fig. 1, the fast walk mode is avoided by cursorial quadrupeds. But if the fast walk mode would conventionally be activated by the continuous extrapolation of the universal walk-mode similarity function, the desirable continuous *walk-to-run transition* might be theoretically observable, as shown by the dashed line in Fig. 1. Surprisingly, the hypothetical continuous transition was remarkably *in vivo* established by Hutchinson et al. (2006) in modern African and Asiatic elephants, through the dynamic similarity determinant *Fr*(exp) *walk*-*run* ≈ 1 and the limb duty factor *<sup>β</sup>*(exp) *walk*-*run* ≈ 0.5, indicating dynamic similarity in elephants during the fast-walk-to-moderate-run continuous-state transition. In contrast with other quadrupeds, elephants, having the hindlimb more compliant than the forelimb (Kokshenev & Christiansen, 2010, Fig. 5) are able to avoid abrupt changes in the mean-limb duty factor during the fast-walk-to-run dynamic crossover. Moreover, elephants, being good walkers, most likely can achieve gradually the highest instability point in the fast-walk trajectory of body's center of mass, indicated in the theory of similarity by the transient muscular field *<sup>μ</sup>*(*pred*) *walk*-*run* = *g*. Hence, the transient similarity state, shown by point 3 in Fig. 1, unifying a gait-crossover transient continuous state in the elephants and a mode-transient continuous state in other quadrupeds, suggests two different patterns of similarity: *successful walkers* and *efficient runners*. The last pattern in quadrupeds is in addition indicated by the universal muscular field evaluated as *<sup>μ</sup>*(exp) *run* <sup>≈</sup> <sup>2</sup>*<sup>g</sup>* (for details, see Table 3) characteristic of more extent pattern of efficient fast animals, including fast running mammals, reptiles, insects; flapping birds, bats, and insects; swimming fish and crayfish, as reported by Bejan & Marden (2006, Fig. 2C). When searching for continuous dynamic similarity states through the muscle-force field on the

basis of Eq. (7), one obtains the scaling exponent *b* (*pred*) *run* <sup>=</sup> 0.60 using *<sup>b</sup>* (exp) *run* <sup>=</sup> 0.40 from Table 1, in the case of running cursorial quadrupeds. The corresponding scaling factor *a* (*pred*) *run* <sup>=</sup> 1.91 follows from the pattern *efficient trotters and hoppers* (Table 3) reliably established by Taylor et al. (1993) and employed as a reference point in Eq. (7). The resulted universal similarity function *<sup>μ</sup>*(*pred*) *run* (*Fr*) shown in Fig. 2 in turn predicts the data *<sup>μ</sup>*(*pred*) *run* <sup>=</sup> 1.91*<sup>g</sup>* failing for *efficient runners in quadrupeds* (Table 3).

In the case of *fast* walking (virtual) quadrupeds, the relative-force continuous similarity function *<sup>μ</sup>*(*pred*) *walk* (*Fr*)/*g* is obtained by a continuous extrapolation of the moderate-walk mode 10 Will-be-set-by-IN-TECH

walking quadrupeds by Alexander & Jayes (1983). As the result, the theoretically observed

Eq. (9) in Kokshenev, 2011), indicates that the wing muscle, driving the flap mode in a flight, is activated during one third time of the cycle period *Tflap*. This finding can be explained by the three-step cycling of the flight-motor individual muscles in birds, as clearly revealed by

Coming back to terrestrial animals, the dynamic similarity predicted by inverted-pendulum

Hildebrand, 1980; Cartmill et al., 2002). Also, the universal discrete state indicated by the

1992; Usherwood, 2005) could be generally expected, besides the muscular transient field

*walk*-*run* = *g* (Kokshenev, 2011). However, such a continuous-state transition has been observed neither in bipeds, including humans, nor in quadrupeds, excluding elephants. As indicated by the walk-instability and run-instability points 1 and 2 in Fig. 1, the fast walk mode is avoided by cursorial quadrupeds. But if the fast walk mode would conventionally be activated by the continuous extrapolation of the universal walk-mode similarity function, the desirable continuous *walk-to-run transition* might be theoretically observable, as shown by the dashed line in Fig. 1. Surprisingly, the hypothetical continuous transition was remarkably *in vivo* established by Hutchinson et al. (2006) in modern African

fast-walk-to-moderate-run continuous-state transition. In contrast with other quadrupeds, elephants, having the hindlimb more compliant than the forelimb (Kokshenev & Christiansen, 2010, Fig. 5) are able to avoid abrupt changes in the mean-limb duty factor during the fast-walk-to-run dynamic crossover. Moreover, elephants, being good walkers, most likely can achieve gradually the highest instability point in the fast-walk trajectory of body's center

Hence, the transient similarity state, shown by point 3 in Fig. 1, unifying a gait-crossover transient continuous state in the elephants and a mode-transient continuous state in other quadrupeds, suggests two different patterns of similarity: *successful walkers* and *efficient runners*. The last pattern in quadrupeds is in addition indicated by the universal muscular field evaluated as *<sup>μ</sup>*(exp) *run* <sup>≈</sup> <sup>2</sup>*<sup>g</sup>* (for details, see Table 3) characteristic of more extent pattern of efficient fast animals, including fast running mammals, reptiles, insects; flapping birds, bats, and insects; swimming fish and crayfish, as reported by Bejan & Marden (2006, Fig. 2C). When searching for continuous dynamic similarity states through the muscle-force field on the

follows from the pattern *efficient trotters and hoppers* (Table 3) reliably established by Taylor et al. (1993) and employed as a reference point in Eq. (7). The resulted universal similarity function *<sup>μ</sup>*(*pred*) *run* (*Fr*) shown in Fig. 2 in turn predicts the data *<sup>μ</sup>*(*pred*) *run* <sup>=</sup> 1.91*<sup>g</sup>* failing for *efficient*

In the case of *fast* walking (virtual) quadrupeds, the relative-force continuous similarity

*walk* (*Fr*)/*g* is obtained by a continuous extrapolation of the moderate-walk mode

of mass, indicated in the theory of similarity by the transient muscular field *<sup>μ</sup>*(*pred*)

1, in the case of running cursorial quadrupeds. The corresponding scaling factor *a*

*walk*-*run* = 0.5 (e.g., Alexander & Maloiy, 1989; Alexander,

*walk*-*run* ≈ 0.5, indicating dynamic similarity in elephants during the

(*pred*) *run* <sup>=</sup> 0.60 using *<sup>b</sup>*

model for the *walk-to-run crossover* was expected to be determined by *Fr*(mod)

and Asiatic elephants, through the dynamic similarity determinant *Fr*(exp)

workloop technics (see e.g., Fig. 3B in Dickinson et al., 2000).

*flap* <sup>≈</sup> 1/3 predicting in turn the muscle timing <sup>Δ</sup>*T*(*pred*)

*flap* ≈ *Tflap*/3 (see

*walk*-*run* = 1 (e.g.,

*walk*-*run* ≈ 1 and the

*walk*-*run* = *g*.

(exp) *run* <sup>=</sup> 0.40 from Table

(*pred*) *run* <sup>=</sup> 1.91

*muscle* duty factor *<sup>β</sup>*(*pred*)

limb duty factor *<sup>β</sup>*(exp)

*<sup>μ</sup>*(*pred*)

model-independent duty factor *<sup>β</sup>*(*pred*)

basis of Eq. (7), one obtains the scaling exponent *b*

*runners in quadrupeds* (Table 3).

function *<sup>μ</sup>*(*pred*)

Fig. 2. Locomotor ability similarity function predicted for quadrupeds by the relative force-output. The *solid lines* are estimates for walking and running quadrupeds obtained via Eq. (7) with the help of the corresponding experimental data on the Strouhal similarity function (Tables 1 and 4). The *dashed line* extrapolates the data on slow-walk to fast-walk modes. The *dashed-dotted line* adjusted with *open squares* is a sketched estimate for trotting horses, also discussed in Fig. 1. The characteristic points and the pattern points are those shown in Fig. 1 and described in Table 3. The comparative data for bipeds are indicated by *italic style*. The *closed triangles* and *open triangles* are respectively the predicted data on the body ability in modern elephants and the body mobility in extinct elephants (Table 4).

to the fast-walk mode, where a transient state for the walk-to-run determinant *<sup>μ</sup>*(*pred*) *walk*-*run* = *g* predicted in Fig. 1 at *Fr*(mod) *walk*-*run* = 1 is employed as a reference point (Table 4). Likewise, in the case of slow running modern elephants, the relative-force similarity function is obtained on the basis of the data *St*(exp) *eleph* (*Fr*) by Hutchinson et al. (2006) shown in Table 1, with the help of the same transient state in the walk-to-run continuous transition, also employed as the reference point determining the dynamic similarity at *Fr*(mod) *walk*-*run* = 1. One can therefore observe (Fig. 2) the dynamic similarity between poorly running (no trotting) modern elephants (shown by the closed triangles) and other hypothetical quadrupeds using the inverse gradient in the fore-hind limb locomotion function, common to both extant and extinct elephants (Kokshenev & Christiansen, 2010, Fig. 5). The data on low level *mobility* of largest extinct elephants (Kokshenev & Christiansen, 2011, Figs. 4 and 5), shown by open triangles in Fig. 2, are obtained by evaluation of the torsional and bending limb bone elastic forces, acting in running giants (Table 4).

### **3.2 Bipeds including humans**

In humans, a discontinues walk-to-run crossover is generally observed at Froude numbers *Fr*(exp) *walk*-*run* ≈ 0.5 (Thorstensson & Roberthson, 1987, Kram et al., 1997; Alexander, 1989;

0.1 1 10

4

Birds & humans

4

5

Froude variable, *Fr*

(the closed stars and point 2 in Fig. 3). However, when the *birds* are presented by the smoothed experimental data, they exhibit a common point of dynamic-state stabilization

open star at point 1 in Fig. 3). Although elephants do not show such a kink at the walk-to-run continuous crossover (Fig. 1), the birds as group likely also exhibit a higher leg compliance than humans and most likely also may continuously pass the absolute instability point in the walk-mode trajectory (Kokshenev, 2011), as generally was expected by Ahlborn & Blake

*walk*-*run* = *g*. Since the experimental scaling data on stride frequency in fast walking and

of dynamic similarity (Table 2), a new similarity pattern *efficient walkers* may be suggested for

Specifying the muscle-field similarity functions *<sup>μ</sup>*(*pred*) *run* (*Fr*) through Eq. (7), the continuous similarity transient state determined by *<sup>μ</sup>*(*pred*) *run* <sup>=</sup> *<sup>g</sup>* is adopted as a reference point, observed in starting to run humans, birds, and bipeds at *Fr*(exp) *run* <sup>=</sup> 0.60, 0.52, and 0.55, respectively

uses a kink point equaling both the run and walk similarity functions. In the case of bipeds, the obtained run mode similarity function was interpolated to the corresponding walk mode

*trans* ∝ *<sup>M</sup>*−0.178, Gatesy & Biewener, 1991) obey the scaling criterion

Fig. 4. The comparative analysis of the walk–to-run crossover in bipeds and quadrupeds. The *pointed lines* are the experimental data on duty-factor similarity functions (Table 1). Other

3

3

3

Slow run

<sup>297</sup> Physical Insights Into Dynamic Similarity

1

1

2

1

Slow run

0.5

point 1 in Fig. 3), whereas the run modes show features of stabilization at *Fr*(exp)

determined by a kink in the duty-factor similarity function revealed at *Fr*(exp)

(2002) from the elastic-type-pendulum model considered at the conditions *Fr*(mod)

birds continuously passing the universal walk-to-run crossover (Table 3).

(Table 3). In order to obtain the slow-muscle-field similarity function *<sup>μ</sup>*(*pred*)

2

in Animal Locomotion. II. Observation of Continuous Similarity States

Fast walk

Run

Cursorial quadrupeds

5

*biped* = 0.55 ± 0.05

*bird* = 0.52 (the

*walk*-*run* = 0.5

*walk* (*Fr*) for birds, one

0.35

and *<sup>μ</sup>*(*pred*)

slow running birds (1/*T*(exp)

0.4

0.45

0.5

0.55

Duty factor function,

E

0.6

0.65

0.7

Walk

Birds & humans

Cursorial quadrupeds

notations are the same as in Figs. 1 and 3.

0.75

Fig. 3. The limb duty-factor similarity functions in bipeds. The *points* are the data by Gatesy & Biewener (1991) explained in the inset. The *solid lines* are statistical data from walking and running humans, birds, and bipeds as group (Table 1). The *dotted line* represents the continuous-state data from running cursorial quadrupeds (Fig. 1), excluding those for a slow run; other discrete-state data for quadrupeds are shown in *italic style*. The *dashed lines* indicate regions of the unstable slow run. The open and closed *stars* indicate, respectively, stable and unstable transient dynamic states. These and other patterns of similarity are described in the text and Table 3.

Ahlborn & Blake, 2002) and stabilization of the run mode is likely determined by the vertical component of muscular field *<sup>μ</sup>*(*pred*) *walk*-*run* = *g* (Ahlborn & Blake, 2002). Unlike elephants and other quadrupeds (Fig. 1), humans and birds expose similar behavior in a number of dynamic characteristics with changes in speed and gait (Gatesy & Biewener, 1991). Although a general consistency between patterns of gaits in humans and birds has been generally established, no clear characterization of the walk-to-run crossover was revealed (Gatesy & Biewener, 1991). The standard least squares analysis of the experimental data on limb duty factor and relative stride length (Gatesy & Biewener, 1991, Figs. 5 and 7) provided here (Table 1) is expected to shed light on the problem of the walk-to-run crossover in bipeds.

In Fig. 3, evident dynamic similarity between the walk modes in humans and birds is observed through the duty-factor similarity functions. Within the domains of continuous states (the solid lines in Fig. 3) treated within context of scaling relations shown in Eq. (6), the dynamic similarity between running humans and running birds can approximately be established on the basis of corresponding experimental data on duty-factor similarity functions (Table 3). As for the walk-to-run discontinues crossover in *bipeds*, the instability of walk modes generally occurs at Froude numbers *Fr*(exp) *biped* = 0.45 ± 0.05 (the closed stars and 12 Will-be-set-by-IN-TECH

1

1

efficient runners

efficient fast runners

slow run

*efficient trotters &* biped hoppers 2

2

1

0.01 0.1 1 10

0.5

0.05 5

*running quadrupeds* & hopping bipeds

Froude variable, *Fr*

Fig. 3. The limb duty-factor similarity functions in bipeds. The *points* are the data by Gatesy & Biewener (1991) explained in the inset. The *solid lines* are statistical data from walking and

continuous-state data from running cursorial quadrupeds (Fig. 1), excluding those for a slow

Ahlborn & Blake, 2002) and stabilization of the run mode is likely determined by the vertical

other quadrupeds (Fig. 1), humans and birds expose similar behavior in a number of dynamic characteristics with changes in speed and gait (Gatesy & Biewener, 1991). Although a general consistency between patterns of gaits in humans and birds has been generally established, no clear characterization of the walk-to-run crossover was revealed (Gatesy & Biewener, 1991). The standard least squares analysis of the experimental data on limb duty factor and relative stride length (Gatesy & Biewener, 1991, Figs. 5 and 7) provided here (Table 1) is expected to

In Fig. 3, evident dynamic similarity between the walk modes in humans and birds is observed through the duty-factor similarity functions. Within the domains of continuous states (the solid lines in Fig. 3) treated within context of scaling relations shown in Eq. (6), the dynamic similarity between running humans and running birds can approximately be established on the basis of corresponding experimental data on duty-factor similarity functions (Table 3). As for the walk-to-run discontinues crossover in *bipeds*, the instability

running humans, birds, and bipeds as group (Table 1). The *dotted line* represents the

run; other discrete-state data for quadrupeds are shown in *italic style*. The *dashed lines* indicate regions of the unstable slow run. The open and closed *stars* indicate, respectively, stable and unstable transient dynamic states. These and other patterns of similarity are

shed light on the problem of the walk-to-run crossover in bipeds.

of walk modes generally occurs at Froude numbers *Fr*(exp)

Humans

*walk*-*run* = *g* (Ahlborn & Blake, 2002). Unlike elephants and

Run

*biped* = 0.45 ± 0.05 (the closed stars and

Birds

4 3

Bipeds

efficient walkers

0.2

described in the text and Table 3.

component of muscular field *<sup>μ</sup>*(*pred*)

0.3

0.4

0.5

Human 1 Human 2 Bobwhite Guineafowl Turkey Rhea Ostrich

Walk

All bipeds

Duty factor function,

E

0.6

0.7

0.8

Fig. 4. The comparative analysis of the walk–to-run crossover in bipeds and quadrupeds. The *pointed lines* are the experimental data on duty-factor similarity functions (Table 1). Other notations are the same as in Figs. 1 and 3.

point 1 in Fig. 3), whereas the run modes show features of stabilization at *Fr*(exp) *biped* = 0.55 ± 0.05 (the closed stars and point 2 in Fig. 3). However, when the *birds* are presented by the smoothed experimental data, they exhibit a common point of dynamic-state stabilization determined by a kink in the duty-factor similarity function revealed at *Fr*(exp) *bird* = 0.52 (the open star at point 1 in Fig. 3). Although elephants do not show such a kink at the walk-to-run continuous crossover (Fig. 1), the birds as group likely also exhibit a higher leg compliance than humans and most likely also may continuously pass the absolute instability point in the walk-mode trajectory (Kokshenev, 2011), as generally was expected by Ahlborn & Blake (2002) from the elastic-type-pendulum model considered at the conditions *Fr*(mod) *walk*-*run* = 0.5 and *<sup>μ</sup>*(*pred*) *walk*-*run* = *g*. Since the experimental scaling data on stride frequency in fast walking and slow running birds (1/*T*(exp) *trans* ∝ *<sup>M</sup>*−0.178, Gatesy & Biewener, 1991) obey the scaling criterion of dynamic similarity (Table 2), a new similarity pattern *efficient walkers* may be suggested for birds continuously passing the universal walk-to-run crossover (Table 3).

Specifying the muscle-field similarity functions *<sup>μ</sup>*(*pred*) *run* (*Fr*) through Eq. (7), the continuous similarity transient state determined by *<sup>μ</sup>*(*pred*) *run* <sup>=</sup> *<sup>g</sup>* is adopted as a reference point, observed in starting to run humans, birds, and bipeds at *Fr*(exp) *run* <sup>=</sup> 0.60, 0.52, and 0.55, respectively (Table 3). In order to obtain the slow-muscle-field similarity function *<sup>μ</sup>*(*pred*) *walk* (*Fr*) for birds, one uses a kink point equaling both the run and walk similarity functions. In the case of bipeds, the obtained run mode similarity function was interpolated to the corresponding walk mode

One may infer from the provided tentative analysis of continuous similarity states (Table 3) that animals patterned through the one similarity function can be observed slightly different through another one similarity function. Such deviations can be understood by the absence of the reliably established continuous similarity functions, as shown in Fig. 4. Further experimental studies of the domains of equivalent continuous similarity states in animals, moving in dynamically equivalent gaits, statistically searching for the similarity functions

<sup>299</sup> Physical Insights Into Dynamic Similarity

Strictly speaking, different-sized animals are not geometrically similar. Moreover, different-taxa animals moving in a certain (slow or fast) gait cannot be patterned by the concept of dynamic similarity until the dynamic conditions unifying running, flying, and swimming are clearly established. In the case of terrestrial animals, when the experimental data from bipeds and quadrupeds, moving in two well distinguished (walk and run) gaits, are compared as two groups of biomechanical systems, no complete dynamic similarity neither inside of each group nor between the two groups can be established. Indeed, inside the bipeds as group, humans contrast to birds, since they are likely unable to explore the slow-run mode continuously near the walk-to-run crossover (Fig. 3). Likewise, cursorial mammals contrast to elephants, compared inside the group of geometrically similar quadrupeds (Fig. 1). When the dynamic similarity is considered between bipeds and quadrupeds, the crossover from a walk gait to a run gait does not occur at the dynamic conditions common to all terrestrial animals (Fig. 4). Less experimental data are available on locomotor mode patterns from flying animals and much less is known about swimming animals. One of the goals of the provided research is to clarify conditions of observation of the dynamic similarity in animals in light of the basic

As demonstrated in the first part of this study (Kokshenev, 2011), the dynamic similarity in animals arises from the mechanical similarity existing between frictionlessly moving closed inanimate systems. When mapped onto efficiently moving weakly open dynamically similar biomechanical systems, the animals are conventionally represented (i) as dimensionless center of mass points, possessing a certain body mass *M*, (ii) propagating with speeds of

similarity concept, the geometric similarity between different-sized animals is introduced through the body static length *Lb*, related (isometrically) to body mass as *Lb* ∝ *M*1/3, that in turn introduces the geometric similarity between linear dimensions of all body's appendages (leg, wing, or tail). The dynamic similarity between animals is thus not due to nonexisting geometric similarity between body's shapes, but is determined by the dynamic-parameter similarity between the compared body's center of mass trajectories (Kokshenev, 2011). When attributed to the similarity temporal uniformity is already established by the resonant conditions, unifying moving animals across mass and taxa, the unique requirement, determining the observable similarity criterion, emerges as a condition of the linearity between the dynamic and static lengths, i.e., *L* ∼ *Lb*, where *L* plays the role of the primary "geometric" parameter of dynamic trajectories. The accomplished uniformity in the spatial space of dynamic variables is therefore observed through the universal Strouhal numbers *Sgait* = *Lb*/*L*. Moreover, the scaling rules controlling the dynamic parameters of

*gait* optimized by the variational principle of minimum muscle-field action

*gait*. Within the framework of dynamic

controlled by fixed scaling exponents, are needed.

in Animal Locomotion. II. Observation of Continuous Similarity States

concepts proposed in theoretical biomechanics.

and (iii) accomplishing rotations at the resonant rate *T*−<sup>1</sup>

**4. Conclusion**

amplitudes *V*(max)

(the dashed line in Fig. 3), providing scaling parameter in the desirable function *<sup>μ</sup>*(*pred*) *walk* (*Fr*). Then, the relevant data for humans have been obtained as the data adjusting these from birds and bipeds by the geometric mean (Table 4).

The discussed above features of duty-factor similarity functions obtained from walking and running bipeds, including humans, and quadrupeds, including elephants, are reproduced in Fig. 4. One also may infer that bipeds and quadrupeds, showing overall (qualitative) dynamic similarity in a walk and a run gaits, are also generally similar when showing the activation of a slow-run mode within the transient domain, approximated by 0.4 ≤ *Fr* ≤ 1.1. The exception is not running in a classical sense elephants (Hutchinson et al., 2006), continuously exploiting a fast-walk mode during the walk-to-run crossover.

In Fig. 5, the *efficient fast runners in bipeds* (Table 3) expose transient universal dynamic states close to those in the efficient fast runners in quadrupeds (point 4 in Fig. 5), thereby indicating the universal continuous slow-run-to-fast-run dynamic transition in terrestrial animals. These two patterns are therefore represent the fast running mammals, which belong to the generalized similarity pattern of efficient fast animals (Figs. 1 and 2). Likewise, another dynamic pattern of the continuous fast-walk-to-slow-run dynamic transition is presented by efficient walkers in birds shown by the transient-state point 1 in the inset in Fig. 5.

Fig. 5. The observation of the relative force-output similarity function in bipeds. The *solid lines* are the data on continuous dynamic similarity states from walking and running bipeds drawn by the corresponding scaling equations (Table 4). The *experimental points*, corresponding to those in Fig. 3, are drawn through the scaling equations for a run and a walk gaits in birds and humans (Table 4). The *characteristic points* are described in Table 3. The *inset* reproduces the enlarge data near the walk-to-run crossover approximated by continuous similarity states.

One may infer from the provided tentative analysis of continuous similarity states (Table 3) that animals patterned through the one similarity function can be observed slightly different through another one similarity function. Such deviations can be understood by the absence of the reliably established continuous similarity functions, as shown in Fig. 4. Further experimental studies of the domains of equivalent continuous similarity states in animals, moving in dynamically equivalent gaits, statistically searching for the similarity functions controlled by fixed scaling exponents, are needed.

### **4. Conclusion**

14 Will-be-set-by-IN-TECH

Then, the relevant data for humans have been obtained as the data adjusting these from birds

The discussed above features of duty-factor similarity functions obtained from walking and running bipeds, including humans, and quadrupeds, including elephants, are reproduced in Fig. 4. One also may infer that bipeds and quadrupeds, showing overall (qualitative) dynamic similarity in a walk and a run gaits, are also generally similar when showing the activation of a slow-run mode within the transient domain, approximated by 0.4 ≤ *Fr* ≤ 1.1. The exception is not running in a classical sense elephants (Hutchinson et al., 2006), continuously exploiting

In Fig. 5, the *efficient fast runners in bipeds* (Table 3) expose transient universal dynamic states close to those in the efficient fast runners in quadrupeds (point 4 in Fig. 5), thereby indicating the universal continuous slow-run-to-fast-run dynamic transition in terrestrial animals. These two patterns are therefore represent the fast running mammals, which belong to the generalized similarity pattern of efficient fast animals (Figs. 1 and 2). Likewise, another dynamic pattern of the continuous fast-walk-to-slow-run dynamic transition is presented by

2 4 6 8

1

Walk

0.2 0.4 0.6 0.8

<sup>2</sup> <sup>1</sup>

Froude variable, *Fr*

drawn by the corresponding scaling equations (Table 4). The *experimental points*,

Fig. 5. The observation of the relative force-output similarity function in bipeds. The *solid lines* are the data on continuous dynamic similarity states from walking and running bipeds

corresponding to those in Fig. 3, are drawn through the scaling equations for a run and a walk gaits in birds and humans (Table 4). The *characteristic points* are described in Table 3. The *inset* reproduces the enlarge data near the walk-to-run crossover approximated by

0

0.5

1

1.5

P */g*

2

Run

Run

efficient walkers

*Fr*

3

efficient walkers in birds shown by the transient-state point 1 in the inset in Fig. 5.

*walk* (*Fr*).

(the dashed line in Fig. 3), providing scaling parameter in the desirable function *<sup>μ</sup>*(*pred*)

and bipeds by the geometric mean (Table 4).

a fast-walk mode during the walk-to-run crossover.

0

continuous similarity states.

1

2

3

2

Locomotor ability,

P */g*

4

efficient fast runners & fast animals

*efficient trotters* & biped hoppers

4

Bipeds

Walk

6

Strictly speaking, different-sized animals are not geometrically similar. Moreover, different-taxa animals moving in a certain (slow or fast) gait cannot be patterned by the concept of dynamic similarity until the dynamic conditions unifying running, flying, and swimming are clearly established. In the case of terrestrial animals, when the experimental data from bipeds and quadrupeds, moving in two well distinguished (walk and run) gaits, are compared as two groups of biomechanical systems, no complete dynamic similarity neither inside of each group nor between the two groups can be established. Indeed, inside the bipeds as group, humans contrast to birds, since they are likely unable to explore the slow-run mode continuously near the walk-to-run crossover (Fig. 3). Likewise, cursorial mammals contrast to elephants, compared inside the group of geometrically similar quadrupeds (Fig. 1). When the dynamic similarity is considered between bipeds and quadrupeds, the crossover from a walk gait to a run gait does not occur at the dynamic conditions common to all terrestrial animals (Fig. 4). Less experimental data are available on locomotor mode patterns from flying animals and much less is known about swimming animals. One of the goals of the provided research is to clarify conditions of observation of the dynamic similarity in animals in light of the basic concepts proposed in theoretical biomechanics.

As demonstrated in the first part of this study (Kokshenev, 2011), the dynamic similarity in animals arises from the mechanical similarity existing between frictionlessly moving closed inanimate systems. When mapped onto efficiently moving weakly open dynamically similar biomechanical systems, the animals are conventionally represented (i) as dimensionless center of mass points, possessing a certain body mass *M*, (ii) propagating with speeds of amplitudes *V*(max) *gait* optimized by the variational principle of minimum muscle-field action and (iii) accomplishing rotations at the resonant rate *T*−<sup>1</sup> *gait*. Within the framework of dynamic similarity concept, the geometric similarity between different-sized animals is introduced through the body static length *Lb*, related (isometrically) to body mass as *Lb* ∝ *M*1/3, that in turn introduces the geometric similarity between linear dimensions of all body's appendages (leg, wing, or tail). The dynamic similarity between animals is thus not due to nonexisting geometric similarity between body's shapes, but is determined by the dynamic-parameter similarity between the compared body's center of mass trajectories (Kokshenev, 2011). When attributed to the similarity temporal uniformity is already established by the resonant conditions, unifying moving animals across mass and taxa, the unique requirement, determining the observable similarity criterion, emerges as a condition of the linearity between the dynamic and static lengths, i.e., *L* ∼ *Lb*, where *L* plays the role of the primary "geometric" parameter of dynamic trajectories. The accomplished uniformity in the spatial space of dynamic variables is therefore observed through the universal Strouhal numbers *Sgait* = *Lb*/*L*. Moreover, the scaling rules controlling the dynamic parameters of

similarity functions (Figs. 2 and 5). One may figure out that gait-dependent similarity functions in animals are generally piecewise functions, whose continuous-state domains are separated by discontinues transient dynamically similar states attributed to the gait-crossover transient states (Fig. 4). The suggested universal features revealed by continuous similarity approach challenge further experimental and theoretical studies of dynamic similarity in

<sup>301</sup> Physical Insights Into Dynamic Similarity

walking, running, hopping, flying, and swimming animals.

in Animal Locomotion. II. Observation of Continuous Similarity States

The author acknowledges financial support by the national agency FAPEMIG.

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**6. References**

similarly moving animals are eventually scaled by the body static length, as explicitly shown in Eq. (16) in Kokshenev (2011).

In this chapter, the discrete-state theoretical approach to the dynamic similarity in animals is generalized to the continuous similarity concept on the basis of continuous Froude variable, playing the role of scaling (similarity) parameter. In the discrete-state similarity theory, treating the body mass as scaling parameter, the scaling rules for fundamental observables and some part of dimensionless universal parameters are the determinants of discrete similarity. Another part of dimensionless universal parameters, playing the role of the indicators of discrete-state similarity, is commonly omitted in basic scaling relations of the theory, besides other insignificant scaling factors. In the continuous-state similarity theory, scaling relations should be substituted by more detail dynamically equivalent scaling equations, because all dimensionless indicators, being functions of continuous states through the Froude variable, now turn into the functions of determinants of continuous similarity. Given that the only one determinant is necessary for the observation of dynamically similar states in animals, each observable function of the only one continuous universal variable can determine the dynamic similarity behavior. It has been therefore shown that the surmise by Alexander & Jayes (1983) on the existence of the gait-similarity functions, determining the dynamic similarity in animals at a certain gait, is generally corroborated by the dynamic similarity theory.

The dynamic similarity patterns established in the previous chapter for running, flying, and swimming animals considered in certain localized dynamic states, are re-analyzed here through the experimental data on continuous dynamic similarity functions established for walking and running bipeds and quadrupeds. The three well distinguished domains of continuous dynamic states in terrestrial animals are now suggested to be explored in animals across taxa. These are the domains of optimum-speed (preferred-run, -flight, or -swim) continuous states, the crossover-gait domain of (walk-to-run, hover-to-flap) transient states, and transient-run domain of (moderate-run-to-fast-run, trot-to-gallop etc.) transient states. The analysis of available experimental data in terms of the four-parameter set of determinants of dynamic similarity (Table 3) resulted in further generalizations in patterns of continuous similarity. One evidently generalized pattern suggests to unify efficient trotters in horses and other trotters in cursorial quadrupeds (Fig. 1). Less evident patterns of dynamic similarity are (i) the efficient trotters unified with bipedal hoppers (Figs. 1, 2, 3, and 5) and (ii) the efficient runners in bipeds unified with those in quadrupeds (Fig. 2) and with all other fast flying and swimming animals (Figs. 1, 2, and 5). Another one tentative pattern is proposed for the efficient flapping flyers in birds and the efficient undulating swimmers unified with efficient gallopers in quadrupeds (Fig. 1).

As a predictive preliminary theoretical result, the dynamic similarity function for locomotor ability in animals (associated with the related body force output) is suggested for quadrupeds, including elephants and bipeds, including humans. The locomotor ability in large quadrupeds is twice as many as that in giant quadrupeds, presented by African and Asian elephants (Fig. 2). In turn, the locomotor ability of the 3-5-*ton* efficient modern adult elephants is expected to be three-five times as many as the corresponding body mobility for 10-20-*ton* low-level efficient extinct elephants, running at critical conditions, when compared at close spaced Froude variables (Fig. 2). Likewise, the birds, humans, cursorial and saltatorial mammals show surprisingly close in magnitude muscle-field similarity (Fig. 5), which is twice as many as poorly running modern elephants (Fig. 2).

The discontinuous changes revealed in continuous similarity functions from quadrupeds and bipeds (Figs. 1, 3, and 4) can also be expected in the corresponding relative force-output similarity functions (Figs. 2 and 5). One may figure out that gait-dependent similarity functions in animals are generally piecewise functions, whose continuous-state domains are separated by discontinues transient dynamically similar states attributed to the gait-crossover transient states (Fig. 4). The suggested universal features revealed by continuous similarity approach challenge further experimental and theoretical studies of dynamic similarity in walking, running, hopping, flying, and swimming animals.

## **5. Acknowledgments**

The author acknowledges financial support by the national agency FAPEMIG.

### **6. References**

16 Will-be-set-by-IN-TECH

similarly moving animals are eventually scaled by the body static length, as explicitly shown

In this chapter, the discrete-state theoretical approach to the dynamic similarity in animals is generalized to the continuous similarity concept on the basis of continuous Froude variable, playing the role of scaling (similarity) parameter. In the discrete-state similarity theory, treating the body mass as scaling parameter, the scaling rules for fundamental observables and some part of dimensionless universal parameters are the determinants of discrete similarity. Another part of dimensionless universal parameters, playing the role of the indicators of discrete-state similarity, is commonly omitted in basic scaling relations of the theory, besides other insignificant scaling factors. In the continuous-state similarity theory, scaling relations should be substituted by more detail dynamically equivalent scaling equations, because all dimensionless indicators, being functions of continuous states through the Froude variable, now turn into the functions of determinants of continuous similarity. Given that the only one determinant is necessary for the observation of dynamically similar states in animals, each observable function of the only one continuous universal variable can determine the dynamic similarity behavior. It has been therefore shown that the surmise by Alexander & Jayes (1983) on the existence of the gait-similarity functions, determining the dynamic similarity in animals

The dynamic similarity patterns established in the previous chapter for running, flying, and swimming animals considered in certain localized dynamic states, are re-analyzed here through the experimental data on continuous dynamic similarity functions established for walking and running bipeds and quadrupeds. The three well distinguished domains of continuous dynamic states in terrestrial animals are now suggested to be explored in animals across taxa. These are the domains of optimum-speed (preferred-run, -flight, or -swim) continuous states, the crossover-gait domain of (walk-to-run, hover-to-flap) transient states, and transient-run domain of (moderate-run-to-fast-run, trot-to-gallop etc.) transient states. The analysis of available experimental data in terms of the four-parameter set of determinants of dynamic similarity (Table 3) resulted in further generalizations in patterns of continuous similarity. One evidently generalized pattern suggests to unify efficient trotters in horses and other trotters in cursorial quadrupeds (Fig. 1). Less evident patterns of dynamic similarity are (i) the efficient trotters unified with bipedal hoppers (Figs. 1, 2, 3, and 5) and (ii) the efficient runners in bipeds unified with those in quadrupeds (Fig. 2) and with all other fast flying and swimming animals (Figs. 1, 2, and 5). Another one tentative pattern is proposed for the efficient flapping flyers in birds and the efficient undulating swimmers unified with efficient

As a predictive preliminary theoretical result, the dynamic similarity function for locomotor ability in animals (associated with the related body force output) is suggested for quadrupeds, including elephants and bipeds, including humans. The locomotor ability in large quadrupeds is twice as many as that in giant quadrupeds, presented by African and Asian elephants (Fig. 2). In turn, the locomotor ability of the 3-5-*ton* efficient modern adult elephants is expected to be three-five times as many as the corresponding body mobility for 10-20-*ton* low-level efficient extinct elephants, running at critical conditions, when compared at close spaced Froude variables (Fig. 2). Likewise, the birds, humans, cursorial and saltatorial mammals show surprisingly close in magnitude muscle-field similarity (Fig. 5), which is twice

The discontinuous changes revealed in continuous similarity functions from quadrupeds and bipeds (Figs. 1, 3, and 4) can also be expected in the corresponding relative force-output

at a certain gait, is generally corroborated by the dynamic similarity theory.

in Eq. (16) in Kokshenev (2011).

gallopers in quadrupeds (Fig. 1).

as many as poorly running modern elephants (Fig. 2).

Alexander, R.McN. (1976). Estimates of speeds of dinosaurs, *Nature* 261: 129-130.


**14** 

*Japan* 

Masaya Hirashima

**Induced Acceleration Analysis of** 

*Graduate School of Education, The University of Tokyo* 

**Three-Dimensional Multi-Joint Movements** 

A knowledge of how muscle forces produce joint rotations is fundamental in all fields of human movement science, including rehabilitation and sports biomechanics. Such knowledge is necessary for improving the diagnosis and treatment of persons with movement disabilities and analyzing the techniques used by high-performance athletes (Zajac and Gordon, 1989). However, it is difficult to intuitively understand how muscle forces produce joint rotations in multi-joint movements because of the complexity of interjoint interactions, especially in three-dimensional (3D) movements. Therefore, although muscle activities, joint torques, and joint rotations themselves have been examined extensively in many sports movements (Barrentine et al., 1998; Elliott et al., 2003; Feltner and Dapena, 1986; Fleisig et al., 1995; Fleisig et al., 1996b; Glousman et al., 1988; Hirashima et al., 2002; Marshall and Elliott, 2000; Matsuo et al., 2001; Nunome et al., 2002; Putnam, 1991; Sakurai et al., 1993; Sakurai and Ohtsuki, 2000; Sprigings et al., 1994), the knowledge about

The purpose of this chapter is to provide the framework to properly understand the causeand-effect relationship between joint torques and rotations during sports movements. In fast sports movements such as baseball pitching and soccer kicking, joint rotations occur sequentially from proximal joints to distal joints. This kinematic sequence itself or the underlying kinetic mechanism is often called the "proximal-to-distal sequence," "kinetic chain," or "whip-like effect" (Atwater, 1979; Feltner, 1989; Fleisig et al., 1996a; Kibler, 1995; Kindall, 1992; Putnam, 1991). However, the kinetic mechanism has not been properly understood, because previous studies on sports movements have not focused on the fact that a joint rotation is caused by *two different mechanisms*: instantaneous and cumulative effects. The instantaneous effect is an instantaneous angular acceleration induced by a joint torque at that instant, whereas the cumulative effect is an angular acceleration induced by the entire joint torque and gravity torque history until that instant (Hirashima et al., 2008; Zajac et al., 2002). Because the mechanical causes are clearly different between the two effects in terms of time, clear differentiation is necessary to understand the original cause of

In section 2, I will explain the instantaneous effects produced by a joint torque by systematically presenting examples of single-joint movements, multi-joint movements in a two-dimensional (2D) space, and multi-joint movements in a three-dimensional (3D) space. I

the cause-and-effect relationships between these variables is insufficient.

each joint rotation and develop effective training programs.

**1. Introduction**

**and Its Application to Sports Movements** 

