**3. Model equations**

 

For the above model, the force diagram, as presented in Figure 2, yields the following model equations.

$$\begin{cases} m\ddot{\mathbf{x}}\mathbf{\dot{x}} = -k\mathbf{x}\mathbf{x}\mathbf{i} - c\mathbf{x}\dot{\mathbf{x}}\mathbf{i} - kz(\mathbf{x}\mathbf{i} - \mathbf{x}\mathbf{z}) - cz(\dot{\mathbf{x}}\mathbf{i} - \dot{\mathbf{x}}\mathbf{z}) - m\mathbf{z}\mathbf{g} \\\ m\mathbf{z}\ddot{\mathbf{x}}\mathbf{z} = kz(\mathbf{x}\mathbf{i} - \mathbf{x}\mathbf{z}) + cz(\dot{\mathbf{x}}\mathbf{i} - \dot{\mathbf{x}}\mathbf{z}) - kz(\mathbf{x}\mathbf{z} - \mathbf{x}\mathbf{s}) - cz(\dot{\mathbf{x}}\mathbf{z} - \dot{\mathbf{x}}\mathbf{s}) - mz\mathbf{g} \\\ m\mathbf{z}\ddot{\mathbf{x}}\mathbf{s} = kz(\mathbf{x}\mathbf{z} - \mathbf{x}\mathbf{s}) + cz(\dot{\mathbf{x}}\mathbf{z} - \dot{\mathbf{x}}\mathbf{s}) - kz(\mathbf{x}\mathbf{s} - \mathbf{x}\mathbf{s}) - cz(\dot{\mathbf{x}}\mathbf{s} - \dot{\mathbf{x}}\mathbf{s}) - mz\mathbf{g} \\\ m\mathbf{u}\ddot{\mathbf{x}}\mathbf{u} = kz(\mathbf{x}\mathbf{s} - \mathbf{x}\mathbf{s}) + cz(\dot{\mathbf{x}}\mathbf{s} - \dot{\mathbf{x}}\mathbf{s}) - mz\mathbf{g} \end{cases} \tag{1}$$

with initial conditions:

Modeling the Foot-Strike Event in Running Fatigue via Mechanical Impedances 155

$$\text{fix}(\mathbf{0}) = \mathbf{0} \quad ; \quad \dot{\text{x}}(\mathbf{0}) = \mathbf{z} \\ \mathbf{o} = -\mathbf{1} \text{ } m \text{ / s} \quad ; \quad \dot{\text{i}} = \mathbf{1}, \mathbf{2}, \mathbf{3}, \mathbf{4} \tag{2}$$

and gravitational acceleration

154 Injury and Skeletal Biomechanics

segments.

vertical direction. [13-15,19-20].

**3. Model equations** 

with initial conditions:

equations.

ankle joint, the knee joint and the hip joint.

 

place as a result of running fatigue.

**2. Biomechanical modeling of the lower limb** 

reasonable prediction of the maximal vertical foot/ground reaction force.

The goal of this research was to characterize the heel-strike shock propagation and attenuation in running by means of a biomechanical model, and to examine changes taking

This section deals with the modeling of the heel-strike event. With the development of biomechanical models of human body motion, it has become possible to simulate vertical landing, such as occurring during running, in order to gain insight into intermuscular coordination and to elucidate control strategies of the musculoskeletal system. A common method to deal with this type of problems is to lump together elements of the human body e.g., muscles, tendons, ligaments, bones and joints so that the overall musculoskeletal system is represented as a damped elastic mechanism. Several models describing vertical landing can be found in the literature [12-18]. These models are usually characterized by the presence of elastic springs and viscous dampers, with constant properties and provide a

Indeterminacy of the locomotor system can be addressed by adopting the lumping method, whereby the material elements of the human body e.g., muscles, tendons, ligaments, bones and joints are lumped together so that the overall musculoskeletal system is represented as a multi-degree-of freedom damped elastic mechanism, interconnecting the masses of the body

The foot- or heel-strike period during landing from fall, during hopping or during the stance phase of running has been generally modeled using one-dimensional models along the

In this study we represent the body segments during heel-strike by a four degree-offreedom elastically-damped uni-axial biomechanical model. The model thus includes 4 masses connected by elastic stiffnesses with parallel damping elements, as shown in Figure 1. In more details, the masses *m*j (*j* = 1..4) represent, respectively, the foot + shoe, the shank, the thigh and the rest of the body (including the non-supporting leg). Each of the stiffness *k*<sup>j</sup> and damping *c*j (*j* = 1..4) represent, respectively, the lumped effects of the heel-pad + sole, the

For the above model, the force diagram, as presented in Figure 2, yields the following model

22 2 1 2 2 1 2 3 2 3 3 2 3 2 33 3 2 3 3 2 3 4 3 4 4 3 4 3

 

**g**

**g g**

(1)

*mx k x x c x x k x x c x x m mx k x x c x x k x x c x x m*

 

11 11 11 2 1 2 2 1 2 1

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

*mx kx cx k x x c x x m*

( )( )

44 4 3 4 4 3 4

*mx k x x c x x m*

**g**

<sup>4</sup>

( ) ( )

$$\mathbf{g} = -9.81 \text{ m/s}^2$$

These values rely on reported landing velocities between -0.8 m/s to -1.2 m/s for running speeds of 3.5 m/s (comparable to the speeds of this study), while wearing various types of running shoes [21-23].

The above masses are expressible in terms of the total body mass from anthropometric data [24].

$$\begin{aligned} m\_1 &= 0.0145 \cdot m \\ m\_2 &= 0.0465 \cdot m \\ m\_3 &= 0.100 \cdot m \\ m\_4 &= 1 - (m\_1 + m\_2 + m\_3) = 0.839 \cdot m \end{aligned} \tag{3}$$

From the simultaneous recording of the foot ground reaction forces and accelerations on the masses *mj*, information about the rise time of the peak acceleration can be obtained.

**Figure 1.** Lumped model including 4 masses connected by elastic stiffnesses with parallel dampings. The masses *m*j (*j* = 1..4) represent, respectively, the foot + shoe, the shank, the thigh and the rest of the body (including the non-supporting leg). Each of the stiffness *k*j and damping *c*j (*j* = 1..4) represent, respectively, the lumped effects of the heel-pad + sole, the ankle joint, the knee joint and the hip joint.

Modeling the Foot-Strike Event in Running Fatigue via Mechanical Impedances 157

**Respiratory Data**

sacrum, by means of two elastic belts passed in a horizontal plane around the shank and the waist, respectively. The tensions of the belts were well above the level in which the acceleration trace for a given impact force became insensitive to the accelerometer attachment force, thus ensuring stability of the accelerometer as well as consistency of the

The shank accelerometer was aligned with the axis of the tibia to provide the axial component of the tibial acceleration and the accelerometer on the sacrum was oriented along the spine. These accelerometers allowed us to acquire the shock accelerations propagated in the longitudinal directions of the tibia and the spine. As earlier reported, such attachment is suitable for faithfully measuring the amplitude of shock acceleration [5-8].

Force platforms, type Kistler Z-4035, were used for the simultaneous recordings of the foot-

An overview of the experimental setup is shown in Figure 3. For examining the effect of global fatigue due to running, the subjects were asked to run on a Quinton Q55 treadmill.

**Figure 3.** Description of the experimental apparatus with a subject running on a treadmill. Two accelerometers are attached , one on the tibial tuberosity (Tib. Tub.) and the other on the sacrum (Sac.) ; the accelerometers' data are sampled through an amplifier and A/D converter to a PC. Likewise, the

**Sac.**

Global, or metabolic fatigue is associated with the development of metabolic acidosis following an endurance exercise and is accompanied by a decrease in the end tidal carbon dioxide pressure (PETCO2) [31]. In long distance running metabolic fatigue is reached when

**Treadmill**

**Tib. Tub.**

readings and reproducibility of the data [13,30].

ground reaction forces and acceleration.

**Acceleration - A/D**

**Amplifier**

**PC - Computer**

respiratory data are sampled and collected on-line.

the running speed exceeds the anaerobic threshold [31].

**4.2. Running fatigue tests** 

**Figure 2.** The force diagram for the model elements presented in Figure 1.
