**1.2. Challenges and issues in SMA actuators design**

The three main challenges in SMA actuator design are: obtaining a simple and reliable material model, increasing the stroke of the actuator and finding design equations to guide the engineer in dimensioning the actuator. To design SMA actuators, a material model must describe the mechanical behaviour of the alloy in two temperature ranges: below the temperature *Mf*, at which the austenite-martensite transformation is finished (OFF or deactivated or cold state) and above the temperature, *Af*, at which the martensite-austenite transformation is completed (ON or activated or hot state). In these two conditions, inside the shape memory material there is only one stable crystalline phase and therefore the macroscopic mechanical properties are known.

The authors proposed two simple material models to describe SMA behaviour, both of them describing the mechanical behaviour of a SMA element at high temperatures (austenitic phase) as linear, characterized by the elastic modulus *EA*. The first one approximates the martensitic behaviour as a linear one [18] (Figure 1a). This behaviour is by no means obvious, thus an explanation is needed. The typical stress-strain curve of a shape memory material, shown in Figure 1a, has two characteristic paths corresponding to a low temperature (martensitic curve) and to a high temperature (austenitic curve). Both curves consist of an initial elastic portion (OA and OC), followed by a constant-stress plateau. In the practical use of shape memory alloys for making actuators, the materials remain within the linear elastic range at the higher temperature but are strained beyond the elastic limit when cooled to the lower temperature. The maximum strain, *εadm* is small enough to ensure the desired fatigue life [24-25] but large enough to maximize the stroke of the actuator for a given amount of material involved. Since this analysis is aimed at dimensioning the binary actuator for the extreme positions, irrespective of the intermediate state, the behaviour of the material is approximated by segments OA' (martensite) and OC (austenite) in Figure 1a. The reference elastic moduli are *EM* and *EA*, respectively. This is also true if the shear behaviour of the shape memory material is considered, when the maximum shear strain, *γadm*, replaces *εadm* and the shear moduli *GM* and *GA* replace *EM* and *EA*.

4 Smart Actuation and Sensing Systems – Recent Advances and Future Challenges

for realistic case studies.

**1.2. Challenges and issues in SMA actuators design** 

macroscopic mechanical properties are known.

The three main challenges in SMA actuator design are: obtaining a simple and reliable material model, increasing the stroke of the actuator and finding design equations to guide the engineer in dimensioning the actuator. To design SMA actuators, a material model must describe the mechanical behaviour of the alloy in two temperature ranges: below the temperature *Mf*, at which the austenite-martensite transformation is finished (OFF or deactivated or cold state) and above the temperature, *Af*, at which the martensite-austenite transformation is completed (ON or activated or hot state). In these two conditions, inside the shape memory material there is only one stable crystalline phase and therefore the

The authors proposed two simple material models to describe SMA behaviour, both of them describing the mechanical behaviour of a SMA element at high temperatures (austenitic phase) as linear, characterized by the elastic modulus *EA*. The first one approximates the martensitic

actuators are presented. Microrobots can be developed using SMA as shown in [7] where there is a basic method to design the SMA spring based on a thermo-electromechanical approach. Reynaerts et al. [8] present design considerations concerning the choice of the active element to evaluate SMA actuator efficiency. Lu et al. [9] design a high strain shape memory actuator taking into account pseudoplasticity and compared its performance with traditional actuators. Due to SMA high non linearity, design curves relying on experiments are proposed in [10] to assess the SMA actuator geometry. A comprehensive review of applications of SMAs in the field of mechanical actuation has recently been published in [11]. Jansen et al [12] develop a linear actuator used as a drive module in an angular positioning mini-actuator. This architecture allows both large force and long strokes to be obtained. Strittmatter et al. [13] propose a SMA actuator for the activation of a hydraulic valve, biased by a conventional spring. Bellini et al. [14] propose a linear SMA actuator able to vary the air inflow for internal combustion engines, improving gas combustion and leading to higher efficiency. Haga et al. [15] propos a mini-actuator to be used in Braille displays. Elwaleed et al. [16] develop a SMA beam actuator able to amplify the SMA actuator strain using elastically instable beams. Among the proposed actuator there is lack of simple design instruments to provide basic information to the designer, either due to specific constraints of application or due to the high complexity of the thermomechanical material models used. In order to answer for an analytical design methodology, the author described in several technical publications a set of equations useful for linear [18] and rotary application [19]. The authors developed the design equations both for SMA actuators under a general system of external forces [20] and produced the design formulae to increase the output stroke thanks to negative stiffness compensators [21]. Moreover two peculiar systems were designed and developed: a telescopic actuator [22] and a wire on drum system [23]. The present work reviews and improves the design rules developed by the authors and set them in a coherent formal analytical framework. Design examples are provided to illustrate the step-by-step application of the design optimization procedures

The second material model describes the behaviour at cold temperatures (martensitic phase) with a bilinear law [21] as shown in the stress-strain diagram of Figure 1b. The model is defined by a first leg OD with an elastic modulus *EMA* and a second leg DE with a gradient *EMB*. We define *εg* as the deformation of the occurrence of the change of slope. Due to the bilinear stress-strain response, the SMA elements used in the actuators (springs or wires) also have bilinear forcedisplacement behaviour when disabled. The elastic moduli *EA, EMA* and *EMB* are replaced by the stiffness *KA*, *KMA*, *KMB*, while the deformation ε*<sup>g</sup>* is replaced by the displacement *xg*. Assuming that the geometric changes related to the deformations of the springs do not influence the elastic constant value, a parameter of merit of the SMA material, *s1*, can be defined as follows:

$$s\_1 = \frac{E\_A}{E\_M} = \frac{G\_A}{G\_M} = \frac{k\_a}{k\_m} \tag{1}$$

This non dimensional group expresses for both models the shape memory capability of the alloy, the larger *s1* the better the material is. The only difference is that in case of the simplest linear model (Figure 1a) the denominator is the secant modulus, while in case of bilinear model (Figure 1b) the denominator is *EMA* the modulus of the first linear martensitic region. In order to balance the active SMA element authors evaluates the influence of three backup elements: a constant force (Section 3.1) a conventional spring (Section 3.2) and an antagonistic SMA (Section 3.3). Moreover the authors propose two compensator systems in order to increase the stroke of the actuator. The systems are either based on a leverage (rocker-arm, Section 4.2.1) or on articulated mechanisms (double quadrilateral, Section 4.2.2), which can be considered as backup elements with negative stiffness.

#### **2. Design procedure of binary SMA actuators**

In this section a design procedure for binary SMA actuators is described and discussed. The system is made up by a generic actuator which moves the output port by means of an elastic system containing an active SMA element and a bias (backup) element.

**Figure 1.** Linear (a) and bilinear (b) material model for martensitic phase of SMA elements

According to the particular means of applying the bias force, the three cases shown in Figure 2 are analyzed:


Without loss of generality, each spring in Figure 2 is modeled as a traction spring exhibiting a linear force-deflection relationship. While the assumption of linearity is obvious for the traditional spring in Figure 2b, the linear behaviour for the SMA spring is an approximation needed to use the model in Figure 1a.

Each actuator presented in Figure 2 is intended to move the output port E through a total useful stroke *Δx* when working against an external dissipative force *FF* and an external conservative force *F0*. The force *FF* is always opposite to the velocity of the cursor and is assumed to be constant. For example, if the cursor is subject to dry friction forces characterized by a static value *FS* and a dynamic value *FD* < *FS*, the design dissipative force *FF* = MAX (*FS* , *FD*) = *FS* will be adopted for the calculation.

This approach comprises every possible external constant load, as exemplified by the membrane pump shown in Figure 3a. This example represents the most general case of external constant forces the actuator has to deal with. The SMA actuator undergoes the following external loads: two generic dissipative forces *F1* (force during aspiration) and *F2* (force during pumping) and a conservative force *Fu* due to the gravity force on the piston. The dissipative force *F1* and the force *Fu* act together against the primary spring, when the piston is moving towards the primary SMA spring (inlet of the fluid). By contrast, the dissipative force *F2* acts on the cursors against the force *Fu* and the primary spring, when the cursor moves away from the primary SMA spring (outlet of the fluid). This force system can be always reduced to the two forces considered in the method: a conservative one, *F0* and two symmetric dissipative ones, *FF*, rearranging the forces as follows:

$$F\_0 = F\_u + \frac{F\_1 - F\_2}{2} \quad , \quad F\_F = \frac{F\_1 + F\_2}{2} \tag{2}$$

The direction of the forces depends on the piston speed, *v*, as shown in Figure 3b-c.

6 Smart Actuation and Sensing Systems – Recent Advances and Future Challenges

**Figure 1.** Linear (a) and bilinear (b) material model for martensitic phase of SMA elements

• two antagonist SMA springs (when one is hot, the other one is cold, Figure 2c).

• a primary SMA spring biased by a constant force (Figure 2a); • a primary SMA spring biased by a traditional spring (Figure 2b);

*FF* = MAX (*FS* , *FD*) = *FS* will be adopted for the calculation.

Figure 2 are analyzed:

needed to use the model in Figure 1a.

According to the particular means of applying the bias force, the three cases shown in

(a) (b)

Without loss of generality, each spring in Figure 2 is modeled as a traction spring exhibiting a linear force-deflection relationship. While the assumption of linearity is obvious for the traditional spring in Figure 2b, the linear behaviour for the SMA spring is an approximation

Each actuator presented in Figure 2 is intended to move the output port E through a total useful stroke *Δx* when working against an external dissipative force *FF* and an external conservative force *F0*. The force *FF* is always opposite to the velocity of the cursor and is assumed to be constant. For example, if the cursor is subject to dry friction forces characterized by a static value *FS* and a dynamic value *FD* < *FS*, the design dissipative force

This approach comprises every possible external constant load, as exemplified by the membrane pump shown in Figure 3a. This example represents the most general case of external constant forces the actuator has to deal with. The SMA actuator undergoes the following external loads: two generic dissipative forces *F1* (force during aspiration) and *F2* (force during pumping) and a conservative force *Fu* due to the gravity force on the piston. The dissipative force *F1* and the force *Fu* act together against the primary spring, when the piston is moving towards the primary SMA spring (inlet of the fluid). By contrast, the dissipative force *F2* acts on the cursors against the force *Fu* and the primary spring, when the cursor moves away from the primary SMA spring (outlet of the fluid). This force system can A second non dimensional parameter, *s2*, useful in the configurations with two springs, is defined as the ratio between the minimum value assumed by the stiffness of bias spring 2 and the stiffness of active spring 1 in the cold state:

$$s\_2 = \frac{K\_{2\text{ min}}}{K\_{1\text{SC}}} \tag{3}$$

**Figure 2.** Three cases of the shape memory actuator biased by: a constant force (a), a traditional spring (b), a shape memory spring (c).

**Figure 3.** Example of generic conservative and dissipative forces acting on the system (a), equivalent forces when SMA is inactive (b) and when SMA is active (c)

The minimum value assumed by the stiffness of spring 2, *K2min*, coincides with the only stiffness of spring 2, *K2C*, if spring 2 is a traditional one, while it coincides with *K2SC* if spring 2 is an active one. A third dimensionless parameter, *sF*, is introduced in order to consider the influence of the dissipative forces in the motion of the SMA actuator. This parameter is defined as the ratio between the dissipative force *FF* and the maximum force sustained by the primary spring in the cold state, calculated as the product between the cold spring stiffness, *K1SC*, and the maximum deflection, *LC-L01*, in the cold state:

$$\mathbf{s}\_F = \frac{F\_F}{\mathbf{K}\_{\rm{1SC}} \cdot \left(L\_\odot - L\_{01}\right)}\tag{4}$$

The fourth dimensionless parameter, *s0*, is introduced in order to consider the influence of the conservative force in the motion of the SMA actuator. This parameter is defined as the ratio between the conservative force *F0* and the maximum force sustained by the primary spring in the cold state, calculated as the product between the cold spring stiffness, *K1SC*, and the maximum deflection, *LC-L01*, in the cold state:

$$s\_0 = \frac{F\_0}{K\_{1SC} \cdot \left(L\_C - L\_{01}\right)}\tag{5}$$

The combination of definitions (4) and (5) gives the relationship between *sF* and *s0*:

$$\mathbf{s}\_0 = \frac{F\_0}{F\_F} \cdot \mathbf{s}\_F \tag{6}$$

#### **2.1. SMA actuator backed up by a constant force**

Figure 4a shows the actuator biased by a constant force in three characteristic positions. Figure 4b shows the relationship between the applied force and the spring deflection during the travel of the actuator between these positions.

The procedure can be retrieved from [20]. The useful stroke of the actuator is obtained as:

$$
\Delta \mathbf{x} = \frac{F\_B}{K\_{\rm{1SC}}} \cdot \frac{\left(s\_1 - 1 - 2s\_F\right)}{s\_1 \cdot \left(s\_F + 1 - s\_0\right)}\tag{7}
$$

The bias force is:

$$F\_B = F\_F \cdot \frac{s\_F + 1 - s\_0}{s\_F} \tag{8}$$

Thus, the stroke can be cast as:

$$
\Delta \mathbf{x} = \left( L\_{\mathbf{C}} - L\_{01} \right) \frac{\mathbf{s}\_1 - \mathbf{1} - \mathbf{2} \mathbf{s}\_{\mathbf{F}}}{\mathbf{s}\_1} \tag{9}
$$

Eq. (7) shows that meaningful strokes ( Δ > *x* 0 ) are only possible if parameter *sF* is lower than the following critical value.

$$s\_{Fcr} = \frac{s\_1 - 1}{2} \tag{10}$$

Moreover, a possible way to choose the value of *sF* consists on studying the free length of the active spring, *L01*, and its wire diameter, *d1*, as described below in eq. 14.

#### *2.1.1. Embodiment of SMA actuator*

8 Smart Actuation and Sensing Systems – Recent Advances and Future Challenges

the maximum deflection, *LC-L01*, in the cold state:

**2.1. SMA actuator backed up by a constant force** 

the travel of the actuator between these positions.

The bias force is:

Thus, the stroke can be cast as:

than the following critical value.

*F*

0

The combination of definitions (4) and (5) gives the relationship between *sF* and *s0*:

*s*

*x*

*s*

( ) <sup>1</sup> <sup>01</sup> *F*

> ( ) 0

1*SC C* 01 *F*

0 0 *F F F s s*

Figure 4a shows the actuator biased by a constant force in three characteristic positions. Figure 4b shows the relationship between the applied force and the spring deflection during

The procedure can be retrieved from [20]. The useful stroke of the actuator is obtained as:

− − Δ= ⋅

*B F*

*C*

active spring, *L01*, and its wire diameter, *d1*, as described below in eq. 14.

( ) ( )

⋅ +−

<sup>0</sup> 1 *<sup>F</sup>*

*F*

*s*

*s*

1 2 1

1 1 1 0

*K ss s*

*s s F F*

( ) <sup>1</sup> 01

*s s xLL*

Eq. (7) shows that meaningful strokes ( Δ > *x* 0 ) are only possible if parameter *sF* is lower

2 *Fcr s*

Moreover, a possible way to choose the value of *sF* consists on studying the free length of the

<sup>1</sup> 1

*F s s*

*B F SC F*

*K LL* <sup>=</sup> ⋅ − (4)

*K LL* <sup>=</sup> ⋅ − (5)

*<sup>F</sup>* = ⋅ (6)

+ − = ⋅ (8)

*<sup>s</sup>* <sup>−</sup> <sup>=</sup> (10)

− − Δ= − (9)

(7)

*SC C F*

The fourth dimensionless parameter, *s0*, is introduced in order to consider the influence of the conservative force in the motion of the SMA actuator. This parameter is defined as the ratio between the conservative force *F0* and the maximum force sustained by the primary spring in the cold state, calculated as the product between the cold spring stiffness, *K1SC*, and If helical springs or wires are considered, two fundamental expressions can be written for the diameter or the wire and for the length of the springs, as a function of the stiffness, *K* and of the deflection of the SMA element, *f.*

$$d = m\_d \cdot \sqrt{\mathbf{K} \cdot f} \tag{11}$$

$$L\_0 = m\_l \cdot f$$

The constant *md* and *mf* depends on the embodiment and can be calculated as shown in the examples in Section 3.1.2 and Section 3.2.2.

A generic expression of the free length, *L01*, can be written as follows:

$$L\_{01} = m\_l \cdot \Delta \mathbf{x} \cdot \frac{\mathbf{s}\_1}{\mathbf{s}\_1 - \mathbf{2} \cdot \mathbf{s}\_F - 1} \tag{13}$$

In (13) the free length, *L01*, depends on the particular embodiment of the spring, but it is always minimized if *sF* is chosen as small as possible. The minimum value of *sF* can be determined by fixing the maximum allowable wire diameter, *d1*, of the SMA element. This maximum value can be determined, for example, using cooling time considerations, the bigger the wire, the slower the cooling, [8], [18].

**Figure 4.** Actuator model (a) and force-deflection diagram (b) of the shape memory actuator biased by a constant force

Combining expression (11) with (1) and (3), the following relationship can be used to determine the minimum value of *sF*.

$$d\_1 = m\_{d1} \cdot \sqrt{\frac{F\_F}{s\_F}} \tag{14}$$

Where *md1* depends only on the embodiment of the SMA primary spring.

#### *2.1.2. Case study: SMA wire based actuator*

The equations (11) and (12) represent the relationships between stiffness, *K*, and free length, *L0*, either for a spring or a wire.

In particular the stiffness of a SMA actuator in wire form is obtained using the coefficient:

$$m\_{dw} = \sqrt{\frac{4}{\pi \cdot E\_m \cdot \varepsilon\_{adm}}}\tag{15}$$

where it is shown that the limit is given by the maximum strain of the SMA wire in martensitic phase. The free length is obtained from eq. (11) by considering the following coefficient for a SMA wire, which comes out simply from the definition of axial strain in a rod.

$$m\_{lw} = \bigvee\_{\mathbf{c}\_{adm}} \tag{16}$$

#### **2.2. SMA actuator backed up by an elastic spring**

Figure 5a shows the actuator biased by a traditional spring in three characteristic positions. Figure 5b displays the relationship between the applied force and the deflection of each spring during the thermal operation of the actuator between those positions.

The detailed procedure can be retrieved from [20]. The pre-stretch is

$$p = \left(L\_{\odot} - L\_{01}\right) \frac{s\_2 + s\_F + 1 - s\_0}{s\_2} \tag{17}$$

$$
\Delta \mathbf{x} = p \frac{s\_2 \cdot \left(s\_1 - \mathbf{2} \cdot \mathbf{s}\_F - \mathbf{1}\right)}{\left(s\_1 + s\_2\right) \cdot \left(s\_2 + s\_F + \mathbf{1} - s\_0\right)} \tag{18}
$$

The following alternative expression of the overall travel can be written

$$
\Delta \mathbf{x} = \left( L\_{\mathbb{C}} - L\_{01} \right) \frac{s\_1 - 1 - 2s\_F}{s\_1 + s\_2} \tag{19}
$$

Lastly, Figure 5b shows that the maximum deflection in the cold state of the bias spring amounts to *p-(LH-L01)*. This expression can be written as:

Optimum Mechanical Design of Binary Actuators Based on Shape Memory Alloys 11

$$p - \left(L\_H - L\_{01}\right) = \left(L\_C - L\_{01}\right) \cdot \frac{s\_1 \cdot \left(s\_2 + s\_F + 1\right) - s\_2 \cdot s\_F - s\_0 \cdot \left(s\_1 + s\_2\right)}{s\_2 \cdot \left(s\_1 + s\_2\right)}\tag{20}$$

#### *2.2.1. Embodiment of the SMA actuator*

10 Smart Actuation and Sensing Systems – Recent Advances and Future Challenges

*2.1.2. Case study: SMA wire based actuator* 

*L0*, either for a spring or a wire.

rod.

1 1

Where *md1* depends only on the embodiment of the SMA primary spring.

*d*

The equations (11) and (12) represent the relationships between stiffness, *K*, and free length,

4

*m adm*

 ε

In particular the stiffness of a SMA actuator in wire form is obtained using the coefficient:

π

where it is shown that the limit is given by the maximum strain of the SMA wire in martensitic phase. The free length is obtained from eq. (11) by considering the following coefficient for a SMA wire, which comes out simply from the definition of axial strain in a

1

Figure 5a shows the actuator biased by a traditional spring in three characteristic positions. Figure 5b displays the relationship between the applied force and the deflection of each

> ( ) 2 0 01

*ss s pLL*

2 1

( ) <sup>1</sup> 01

− − Δ= −

*s s xLL*

Lastly, Figure 5b shows that the maximum deflection in the cold state of the bias spring

*C*

⋅ −⋅ − Δ =

*ss s*

2 1 *<sup>F</sup>*

2 1

1 2 1 2 *<sup>F</sup>*

*s s*

1 *F F*

+ +− = − (17)

+ ⋅ + +− (18)

<sup>+</sup> (19)

*s*

( ) ( )( )

12 2 0

*ss ss s*

ε

*adm*

*lw*

*m*

spring during the thermal operation of the actuator between those positions.

*C*

The detailed procedure can be retrieved from [20]. The pre-stretch is

*x p*

The following alternative expression of the overall travel can be written

amounts to *p-(LH-L01)*. This expression can be written as:

**2.2. SMA actuator backed up by an elastic spring** 

*dw*

*m*

*<sup>F</sup> d m*

*F*

*<sup>s</sup>* = ⋅ (14)

*<sup>E</sup>* <sup>=</sup> ⋅ ⋅ (15)

= (16)

*F*

If helical springs or wires are considered, the two fundamental expressions (11) and (12) can be written for the diameter of the wire and for the length of each spring. Following [20] a generic expression of the total length can be obtained, which depends on the dimensionless parameters *s1*, *s2*, *s0* and *sF* and on the embodiment of the springs.

**Figure 5.** Actuator model (a) and force-deflection diagram (b) of the shape memory actuator biased by a traditional spring

This expression is minimized if *sF* is chosen as small as possible and if a precise value of \* 2*s* is chosen, obtained by equating to zero the derivative of the total length of the actuator. The optimal value \* <sup>2</sup>*s* can be expressed as:

$$s\_2^\* = \sqrt{\frac{s\_1 \cdot \left(m\_{l2} + 1\right) \cdot \left(s\_F + 1 - s\_0\right)}{\left(m\_{l1} + 1\right)}}\tag{21}$$

As a first attempt, it is possible to determine the *ml* coefficients considering the desired embodiment of each spring and a value of *C* = 7, because the value of \* <sup>2</sup>*s* is not greatly affected by *C*.

#### *2.2.2. Case study: SMA spring based actuator*

The equations (11) and (12) represent the relationships between stiffness, *K*, and free length, *L0*, either for a spring or a wire.

In particular the wire diameter of a helical spring can be cast as follows [26]:

$$d = \sqrt{\frac{4 \cdot K \cdot f \cdot K\_b}{\pi \cdot G \cdot \mathcal{V}\_{adm}}} \tag{22}$$

Were *K* is the stiffness of the spring, *f* is the deflection of the spring and *Kb* is the coefficient of Bergstrasser, given by the following relationship:

$$K\_b = \frac{4 \cdot C + 2}{4 \cdot C - 3} \tag{23}$$

Thus for a spring the *mds* coefficient in expression (11) can be expressed as:

$$m\_{ds} = 4\sqrt{\frac{\left(\text{C} + 2\right)}{\pi \cdot \text{G} \cdot \text{ ${}^{\circ}$ }\_{\text{adm}} \cdot \left(\text{4C} - 3\right)}}\tag{24}$$

The external diameter of a helical spring can be calculated using the definition of the spring index (*C* = *D*/*d*) and the number of active coils is given by:

$$N = \frac{G \cdot d}{8 \cdot \mathbb{C}^3 \cdot K} \tag{25}$$

The fully compressed length can be calculated using expression:

$$L\_{fc} = Nd = \frac{G \cdot d^2}{8 \cdot \mathbb{C}^3 \cdot K} \tag{26}$$

The free length is:

$$L\_0 = 1.15 \cdot Nd + f = 1.15 \cdot \frac{G \cdot f \cdot d^2}{8 \cdot C^3 \cdot F} + f \tag{27}$$

Thus the coefficient to calculate the free length in eq. (12) is:

$$m\_{ls} = \frac{1.15 + \pi \cdot \mathcal{Y}\_{adm} \cdot \mathbb{C}^2}{\pi \cdot \mathcal{Y}\_{adm} \cdot \mathbb{C}^2} \tag{28}$$

#### **2.3. SMA actuator backed up by an antagonist SMA spring**

Figure 6a shows the actuator biased by a second SMA spring in three characteristic positions. Figure 6b shows the relationship between the applied force and the deflection of each spring during the thermal operation of the actuator between these positions.

Following the procedure described in [20], the prestretch *p* can be written:

$$p = \left(L\_{\text{C}} - L\_{01}\right) \cdot \frac{s\_1 \cdot s\_2 + s\_F + 1 - s\_0}{s\_1 \cdot s\_2} \tag{29}$$

The stroke can be cast as:

$$
\Delta \mathbf{x} = p \cdot \frac{\mathbf{s}\_2 \cdot \left\{ \left( \mathbf{s}\_1 - \mathbf{1} - \mathbf{s}\_F \right) \cdot \left( \mathbf{s}\_1 + \mathbf{1} \right) - \mathbf{s}\_0 \cdot \left( \mathbf{s}\_1 - \mathbf{1} \right) \right\}}{\left( \mathbf{s}\_1 + \mathbf{s}\_2 \right) \cdot \left( \mathbf{s}\_1 \cdot \mathbf{s}\_2 + \mathbf{s}\_F + \mathbf{1} - \mathbf{s}\_0 \right)} \tag{30}
$$

#### Optimum Mechanical Design of Binary Actuators Based on Shape Memory Alloys 13

The following expression for the optimal value \* <sup>2</sup>*s* is found:

12 Smart Actuation and Sensing Systems – Recent Advances and Future Challenges

Thus for a spring the *mds* coefficient in expression (11) can be expressed as:

4

π

*m*

The fully compressed length can be calculated using expression:

Thus the coefficient to calculate the free length in eq. (12) is:

index (*C* = *D*/*d*) and the number of active coils is given by:

The free length is:

The stroke can be cast as:

of Bergstrasser, given by the following relationship:

Were *K* is the stiffness of the spring, *f* is the deflection of the spring and *Kb* is the coefficient

4 2 4 3 *<sup>b</sup> <sup>C</sup> <sup>K</sup> C*

( )

4 3 *ds adm C*

 γ*G C*

The external diameter of a helical spring can be calculated using the definition of the spring

<sup>3</sup> 8 *G d <sup>N</sup>*

<sup>3</sup> 8 *fc G d L Nd*

<sup>0</sup> <sup>3</sup> 1.15 1.15

*ls*

each spring during the thermal operation of the actuator between these positions.

01

Following the procedure described in [20], the prestretch *p* can be written:

*C*

*x p*

Δ= ⋅

*m*

**2.3. SMA actuator backed up by an antagonist SMA spring** 

*Gfd L Nd f <sup>f</sup> C F* ⋅ ⋅ = ⋅ += ⋅ + ⋅ ⋅

1.15 *adm*

Figure 6a shows the actuator biased by a second SMA spring in three characteristic positions. Figure 6b shows the relationship between the applied force and the deflection of

( ) 1 2 <sup>0</sup>

*ss s s pLL*

*F*

*s s s s ss*

⋅ + +− =−⋅ ⋅ (29)

*s s*

{( )( ) ( )} ( )( ) 2 1 1 01 1 2 12 0

*s s ss s s* ⋅ −− ⋅ + − ⋅ −

+ ⋅ ⋅ + +−

111

*F*

1

π γ

+ ⋅ <sup>=</sup> <sup>⋅</sup>

π γ

*adm*

*C*

<sup>⋅</sup> = = ⋅ ⋅

*C K* <sup>⋅</sup> <sup>=</sup> ⋅ ⋅

2

*C K*

8

2 2

*C*

⋅ ⋅

2

2

( )

⋅ + <sup>=</sup> ⋅ − (23)

(25)

(26)

(28)

(27)

(30)

<sup>+</sup> <sup>=</sup> ⋅⋅ ⋅ − (24)

$$s\_2^\* = \sqrt{s\_F + 1 - s\_0} \tag{31}$$

The following alternative expression of the overall travel can be obtained:

$$
\Delta\chi = \left(L\_C - L\_{01}\right) \cdot \frac{\left(s\_1 - 1 - s\_F\right) \cdot \left(s\_1 + 1\right) + s\_0 \cdot \left(s\_1 - 1\right)}{s\_1 \cdot \left(s\_1 + s\_2\right)}\tag{32}
$$

Both Equations (30) and (32) demonstrate that meaningful strokes ( Δ > *x* 0 ) are only possible if parameter *sF* is lower than the following critical value.

$$s\_{Fcr} = \frac{F\_F \cdot \left(s\_1 - 1\right)^2}{F\_0 \cdot \left(s\_1 - 1\right) + F\_F \cdot \left(s\_1 + 1\right)}\tag{33}$$

Figure 6b shows that the maximum deflection of the bias spring in the cold state amounts to ( ) *<sup>H</sup>* <sup>01</sup> *pL L* − − . This expression can be written as:

$$p - \left(L\_H - L\_{01}\right) = \left(L\_C - L\_{01}\right) \cdot \frac{s\_1 \cdot s\_2 - s\_2 \cdot s\_F + s\_F + 1 - s\_0 \cdot \left(s\_2 + 1\right)}{s\_2 \cdot \left(s\_1 + s\_2\right)}\tag{34}$$

#### *2.3.1. Embodiment of SMA antagonist actuator*

If helical springs or wires are considered, the two fundamental expressions (11) and (12) can be written for the diameter of the wire and for the length of each spring. Following [20] a generic expression of the total length can be obtained, which depends on the dimensionless parameters *s1*, *s2*, *s0* and *sF* and on the embodiment of the springs.

**Figure 6.** Actuator model (a) and force-deflection diagram (b) of the agonist antagonist shape memory actuator

This expression is minimized if *sF* is chosen as small as possible and if an optimal value for \* <sup>2</sup>*s* is chosen, obtained equating to zero the derivative of the total length of the actuator. The optimal value \* <sup>2</sup>*s* can be expressed as:

$$s\_2^\* = \sqrt{\frac{(m\_{l2} - 1) \cdot \left(s\_F + 1 - s\_0\right)}{\left(m\_{l1} - 1\right)}}\tag{35}$$

Expression (35) reduces to expression (31) if the same spring embodiment is considered for spring 1 and spring 2. As a first attempt, it is possible to determine the *ml* coefficients considering the desired embodiment of each spring and a value of *C* = 7, because the value of \* <sup>2</sup>*s* is not greatly affected by *C*. The minimum value of *sF* can be determined by fixing the maximum allowable wire diameter, *d1*, of the SMA element. This maximum value can be determined, for example, using cooling time considerations. Relationship (14) can be used to determine the minimum value of *sF*.

#### **2.4. Design procedures for a binary SMA actuator**

The step by step procedures which guide the designer to apply the above design methods are described in Table 1. The first four steps are the same irrespective of the backup element used in the actuator. From step 5, the procedure is threefold.

These procedures ensure that any actuator biased by a one of the above described back up elements, with the calculated stiffness *KC* and containing whatever SMA spring, with the selected material parameter *s*1 and the calculated cold stiffness *K1SC*, satisfies the design problem (useful stroke Δ*x* , design dissipative force *FF* and design conservative force *F*0) when assembled with the calculated pre-stretch *p*.

#### *2.4.1. Case study: SMA based swing louver*

In this section, the complete design procedure is carried out numerically for the actuation of a swing louver exploited to direct the air flow in domestic air conditioners. An actuator made up of an SMA spring biased by an antagonist SMA spring is designed here as a possible alternative solution to conventional electric motors and linkages. The SMA actuator acts on the louver with a known arm to make the louver swing. The design parameters are: required stroke: 5mm; dissipative force: 5N, conservative force: 2N. The material considered is Nitinol, with the following properties, *γadm* = 0.02, to ensure a fatigue limit over 500 thousand cycles, austenitic shear modulus *Ga*=23000 MPa, equivalent martensitic shear modulus *Gm*=8000 MPa.

The procedure starts calculating the non dimensional groups from eq. (1), (10), (31) and (5):

$$s\_1 = \frac{G\_A}{G\_M} = \frac{23000}{8000} = 2.875\tag{36}$$


**Table 1.** Step by step procedure for each Backup element considered

2

*s*

<sup>2</sup>*s* can be expressed as:

**2.4. Design procedures for a binary SMA actuator** 

used in the actuator. From step 5, the procedure is threefold.

when assembled with the calculated pre-stretch *p*.

*2.4.1. Case study: SMA based swing louver* 

modulus *Gm*=8000 MPa.

determine the minimum value of *sF*.

optimal value \*

of \*

This expression is minimized if *sF* is chosen as small as possible and if an optimal value for \* <sup>2</sup>*s* is chosen, obtained equating to zero the derivative of the total length of the actuator. The

> *l F l m ss*

*m*

Expression (35) reduces to expression (31) if the same spring embodiment is considered for spring 1 and spring 2. As a first attempt, it is possible to determine the *ml* coefficients considering the desired embodiment of each spring and a value of *C* = 7, because the value

( )( ) ( ) \* 2 0

− ⋅ +− <sup>=</sup> <sup>−</sup> (35)

1 1 1 1

<sup>2</sup>*s* is not greatly affected by *C*. The minimum value of *sF* can be determined by fixing the maximum allowable wire diameter, *d1*, of the SMA element. This maximum value can be determined, for example, using cooling time considerations. Relationship (14) can be used to

The step by step procedures which guide the designer to apply the above design methods are described in Table 1. The first four steps are the same irrespective of the backup element

These procedures ensure that any actuator biased by a one of the above described back up elements, with the calculated stiffness *KC* and containing whatever SMA spring, with the selected material parameter *s*1 and the calculated cold stiffness *K1SC*, satisfies the design problem (useful stroke Δ*x* , design dissipative force *FF* and design conservative force *F*0)

In this section, the complete design procedure is carried out numerically for the actuation of a swing louver exploited to direct the air flow in domestic air conditioners. An actuator made up of an SMA spring biased by an antagonist SMA spring is designed here as a possible alternative solution to conventional electric motors and linkages. The SMA actuator acts on the louver with a known arm to make the louver swing. The design parameters are: required stroke: 5mm; dissipative force: 5N, conservative force: 2N. The material considered is Nitinol, with the following properties, *γadm* = 0.02, to ensure a fatigue limit over 500 thousand cycles, austenitic shear modulus *Ga*=23000 MPa, equivalent martensitic shear

The procedure starts calculating the non dimensional groups from eq. (1), (10), (31) and (5):

<sup>23000</sup> 2.875 8000

*<sup>G</sup>*== = (36)

1

*s*

*A M G*

$$s\_{Fcr} = \frac{s\_1 - 1}{2} = \frac{2.875 - 1}{2} = 0.9375\tag{37}$$

$$s\_F = 0.25 \le s\_{Fcr} \tag{38}$$

$$s\_2 = s\_2^\* = \sqrt{s\_1 \cdot \left(s\_F + 1\right)} = \sqrt{2.875 \cdot \left(0.25 + 1\right)} = 1.896\tag{39}$$

$$s\_0 = \frac{F\_0}{F\_F} \cdot s\_F = \frac{2}{5} \cdot 0.25 = 0.1 \cdot s\_0 = \frac{F\_0}{F\_F} \cdot s\_F = \frac{2}{5} \cdot 0.25 = 0.1\tag{40}$$

the maximum deflection of the primary spring, *LC-L01*, is retrieved from (32);

$$\begin{aligned} \left(L\_C - L\_{01}\right) &= \Delta x \cdot \frac{s\_1 \cdot (s\_1 + s\_2)}{\left(s\_1 - 1 - s\_F\right) \cdot \left(s\_1 + 1\right) + s\_0 \cdot \left(s\_1 - 1\right)} = \\ &= 5 \cdot \frac{2.875 \cdot \left(2.875 + 1.896\right)}{\left(2.875 - 1 - 0.5\right) \cdot \left(2.875 + 1\right) + 0.1 \cdot \left(2.875 - 1\right)} = 12.43 \, mm \end{aligned} \tag{41}$$

while eq. (4) allows the SMA spring cold stiffness to be calculated

$$K\_{1SC} = \frac{F\_{\text{F}}}{s\_{\text{F}} \cdot \left(L\_{\text{C}} - L\_{01}\right)} = \frac{5}{0.25 \cdot 12.43} = 1.61 \,\text{N} \,/\text{mm} \tag{42}$$

the overall pre-stretch of, *p*, is calculated from (29) and *KC*, is calculated from eq. (3)

$$p = \left(L\_{\odot} - L\_{01}\right) \cdot \frac{s\_1 \cdot s\_2 + s\_F + 1 - s\_0}{s\_1 \cdot s\_2} = 12.43 \cdot \frac{2.875 \cdot 1.896 + 0.25 + 1 - 0.1}{2.875 \cdot 1.896} = 15.01 mm \tag{43}$$

$$K\_{2\,\mathrm{min}} = K\_{2\,\mathrm{SC}} = s\_2 \cdot K\_{1\,\mathrm{SC}} = 1.896 \cdot 1.61 = 3.052 \,\mathrm{N} \,/\,\mathrm{mm} \tag{44}$$

$$\begin{split} &p - \left(L\_H - L\_{01}\right) = \left(L\_C - L\_{01}\right) \cdot \frac{s\_1 \cdot s\_2 - s\_2 \cdot s\_F + s\_F + 1 - s\_0 \cdot \left(s\_2 + 1\right)}{s\_2 \cdot \left(s\_1 + s\_2\right)} = \\ &= 12.43 \cdot \frac{2.875 \cdot 1.896 - 1.896 \cdot 0.25 + 0.25 + 1 - 0.1 \cdot \left(1.896 + 1\right)}{1.896 \cdot \left(2.875 + 1.896\right)} = 8.16 \, mm \end{split} \tag{45}$$

the maximum deflection of the secondary SMA spring, *p-(LH-L01)*, is calculated from eq. (34).

The detailed design of primary SMA spring is given by combination of eq. (11) and (24) regarding the spring wire diameter, considering a spring index *C=7*

$$d = 4 \sqrt{\frac{\left(\text{C} + 2\right) K\_{1\text{SC}} \left(L\_{\text{C}} - L\_{01}\right)}{\pi \cdot \text{G} \cdot \chi\_{adm} \cdot \left(4\text{C} - 3\right)}} = 4 \sqrt{\frac{\left(7 + 2\right) \cdot 1.61 \cdot 12.43}{\pi \cdot 8000 \cdot 0.02 \cdot \left(4 \cdot 7 - 3\right)}} = 0.48 \, mm \tag{46}$$

the free length of the primary spring is given by the combination of eq. (12) and (28)

Optimum Mechanical Design of Binary Actuators Based on Shape Memory Alloys 17

$$L\_0 = \frac{1.15 + \pi \cdot \mathcal{Y}\_{\text{adm}} \cdot \text{C}^2}{\pi \cdot \mathcal{Y}\_{\text{adm}} \cdot \text{C}^2} \cdot \left(L\_\odot - L\_{01}\right) = \frac{1.15 + \pi \cdot 0.02 \cdot \mathcal{T}^2}{\pi \cdot 0.02 \cdot \mathcal{T}^2} \cdot 12.43 = 17.07 \text{ mm} \tag{47}$$

The same procedure is applied to calculate the design parameter of the antagonist spring.
