**3. Design of compensated SMA actuators**

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

0

*F F s s*

01

*SC*

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

*H C*

*pL L L L*

π

01

*C*

*LL x*

*C*

5 *<sup>F</sup>*

0

<sup>1</sup> <sup>1</sup> 2.875 1 0.9375

*<sup>s</sup>* <sup>−</sup> <sup>−</sup> == = (37)

22 1 1 2.875 0.25 1 1.896 *<sup>F</sup> s s ss* == ⋅ += ⋅ += (39)

5 *<sup>F</sup>*

*<sup>F</sup>* = ⋅ =⋅ =

0.25 *<sup>F</sup> Fcr s s* = ≤ (38)

<sup>2</sup> 0.25 0.1

*mm*

(40)

(41)

2 2 *Fcr s*

<sup>2</sup> 0.25 0.1

*<sup>F</sup>* = ⋅ =⋅ = <sup>0</sup>

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

( ) ( )

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

( ) <sup>1</sup>

*F C*

1 2

*s s*

01 01

regarding the spring wire diameter, considering a spring index *C=7*

( )

( )( )

 γ*G C*

*SC C adm C K LL*

*F*

*F*

( ) ( ) \*

0

( )( ) ( ) ( )

<sup>1</sup> 1 01 111

112

*sss*

⋅ +

*s s s ss*

*<sup>F</sup> <sup>K</sup> N mm*

*ss s s pLL mm*

⋅ + +− ⋅ + +− =−⋅ = ⋅ <sup>=</sup> ⋅ ⋅ (43)

( )( ) ( )

2.875 2.875 1.896 5 12.43 2.875 1 0.5 2.875 1 0.1 2.875 1

⋅ + = ⋅ <sup>=</sup> −− ⋅ + + ⋅ −

01

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

( )( ) ( )

⋅ − ⋅ + +− ⋅ + = ⋅ <sup>=</sup> ⋅ +

⋅ − ⋅ + +− ⋅ + −− =−⋅ <sup>=</sup> ⋅ +

2.875 1.896 1.896 0.25 0.25 1 0.1 1.896 1 12.43 8.16 1.896 2.875 1.896

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)

> <sup>1</sup> <sup>01</sup> 2 7 2 1.61 12.43 4 4 0.48 4 3 8000 0.02 4 7 3

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

*d mm*

π

( )

*F*

− =Δ ⋅ <sup>=</sup> −− ⋅ + + ⋅ −

*F F s s*

<sup>5</sup> 1.61 / 0.25 12.43

<sup>1</sup> 2.875 1.896 0.25 1 0.1 12.43 15.01 2.875 1.896

2 min 2 2 1 1.896 1.61 3.052 / *K K sK* = =⋅ = ⋅ = *SC SC N mm* (44)

( )

212

*sss*

12 2 0 2

*F F*

*ss ss s s s*

( )

+ − +⋅ ⋅ == = ⋅ ⋅ ⋅ − ⋅ ⋅ ⋅ ⋅− (46)

( )

( )

*mm*

(45)

1 1

*sLL* = == ⋅ − <sup>⋅</sup> (42)

SMA based actuators develop significant forces but are usually characterized by low strokes. The stroke is primarily limited by the maximum strain that the alloy can withstand for the expected life as shown in [24], [25]. The backup element needed to recover the stroke prevents the SMA element from recovering completely its shape is a further cause of stroke loss. Furthermore, the force delivered by shape memory actuators varies linearly with the displacement while the external load is usually constant. The design of a SMA actuator ensures that the minimum actuator force is sufficient to contrast the external load [18], providing an additional cause of stroke reduction.

This section introduces a compensation system to store available power from the SMA element in high force positions and then return this power in low force positions [19]. The same principle was successfully applied for electroactive polymer actuators by introducing compliant mechanisms [27]. The compensation system adopted has a negative elastic characteristic, generating a decreasing force as the deformation increases.

## **3.1. Principle of elastic compensation**

The force-deflection diagram in Figure 7 shows the characteristic lines of an SMA actuator in the austenitic (hot SMA, solid circles) and martensitic (cold SMA, solid squares) states. When the SMA element is backed up by a linear spring with the linear characteristic shown with hollow triangles, the net stroke under no external force is *Sspring*. Starting from the same maximum deflection and force of the cold SMA, the stroke increases to *Sweight* when the active element is backed up by a constant force (crossed horizontal line). The improvement is consequent upon the reduced stiffness (lower contrasting forces) of the backup element which allows the SMA to recover a greater share of deformation.

By learning from this positive trend, it is easily seen that an even greater stroke (*Scomp*) is achieved if the backup element displays a negative slope (hollow circles) so that the contrast force would decrease with increasing deflection. Energetically, the compensation system accumulates energy from the SMA element in the positions where the SMA force is high (right-hand side in Figure 7) and releases that energy to the actuators in the positions where the SMA force is low (left-hand side in Figure 7).

As shown in the subsequent sections, a backup element with negative slope as in Figure 7 can be achieved by exploiting one of the many spring-assisted mono or bistable mechanisms described in the technical literature. The use of an elastic compensation system requires the introduction of hard stops to prevent the SMA elements from over-straining. In the case of a single-SMA actuator (see Figure 7), a single hard stop is needed and the behaviour in the

absence of power becomes monostable. In the case of a double-SMA actuator, two hard stops are required and the behaviour in the absence of power becomes bistable. Although the advantages of the compensation system in terms of force and stroke apply to both single-SMA and two-SMA actuators, the improvements are more pronounced for the two-SMA actuator. The general theory [21] demonstrates that a single-SMA actuator can generate a truly constant output force on only one direction of motion. By contrast, a two-SMA actuator can achieve a constant-force profile in both directions. Further advantages of the compensated architecture over regular SMA actuators are the existence of definite equilibrium positions when the power is shut off, the enforcement of precise mono (single-SMA) or binary (two-SMA) positioning, and the possibility of easy control strategies. The input data required for the design of a shape memory actuator are the value of the guaranteed minimum useful force in the two directions of activation (*FON*, *FOFF* in the case of a single SMA element, *FON1*, *FON2* when there are two opposing SMA elements), the value of the stroke desired, *S*, and the type of alloy used for the active elements (i.e. *s1*, *sm*, *sg*).

**Figure 7.** Force-deflection curves of a single-SMA actuator backed up by: a conventional spring, a constant force and an elastic compensation system

#### **3.2. Material model: Definitions**

In order to design compensated shape memory actuators the bilinear model for the martensitic state of SMA (Figure 1b) was used. Since *εg* (then *xg*) is very small (0.2% < *εg* < 0.5%) the force-stroke characteristic of the SMA elements can be approximated as a linear trend (D'E in Figure 8), with slope *KMB*, starting from the force *F0m* in correspondence to zero displacement (Figure 8). The force *F0m* is calculated as:

$$F\_{0m} = \left(K\_{MA} - K\_{MB}\right)\chi\_{\mathcal{K}} \tag{48}$$

The force *FSMA\_ON* produced by the generic shape memory element in the activated state (austenite) is given by the line OC in Figure 8 and is:

Optimum Mechanical Design of Binary Actuators Based on Shape Memory Alloys 19

$$F\_{SMA\\_ON} = K\_A \propto \tag{49}$$

In the cold state (martensite), the force *FSMA\_OFF* is given by the line D'E in Figure 8:

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

absence of power becomes monostable. In the case of a double-SMA actuator, two hard stops are required and the behaviour in the absence of power becomes bistable. Although the advantages of the compensation system in terms of force and stroke apply to both single-SMA and two-SMA actuators, the improvements are more pronounced for the two-SMA actuator. The general theory [21] demonstrates that a single-SMA actuator can generate a truly constant output force on only one direction of motion. By contrast, a two-SMA actuator can achieve a constant-force profile in both directions. Further advantages of the compensated architecture over regular SMA actuators are the existence of definite equilibrium positions when the power is shut off, the enforcement of precise mono (single-SMA) or binary (two-SMA) positioning, and the possibility of easy control strategies. The input data required for the design of a shape memory actuator are the value of the guaranteed minimum useful force in the two directions of activation (*FON*, *FOFF* in the case of a single SMA element, *FON1*, *FON2* when there are two opposing SMA elements), the value of

the stroke desired, *S*, and the type of alloy used for the active elements (i.e. *s1*, *sm*, *sg*).

**Figure 7.** Force-deflection curves of a single-SMA actuator backed up by: a conventional spring, a

Weight Compensation system

Conventional Spring

In order to design compensated shape memory actuators the bilinear model for the martensitic state of SMA (Figure 1b) was used. Since *εg* (then *xg*) is very small (0.2% < *εg* < 0.5%) the force-stroke characteristic of the SMA elements can be approximated as a linear trend (D'E in Figure 8), with slope *KMB*, starting from the force *F0m* in correspondence to zero

0.0 0.5 1.0 1.5 2.0 2.5 3.0

**Deflection (mm)** Hot SMA Cold SMA Conventional spring

Compensation System

*Scompensated Sweight*

*Sspring*

Hot SMA

Weight

Cold SMA

The force *FSMA\_ON* produced by the generic shape memory element in the activated state

( ) <sup>0</sup>*m g MA MB F K Kx* = − (48)

constant force and an elastic compensation system

**Force (N)**

displacement (Figure 8). The force *F0m* is calculated as:

(austenite) is given by the line OC in Figure 8 and is:

**3.2. Material model: Definitions** 

$$F\_{SMA\\_OFF} = F\_{0m} + K\_{MB} \chi \tag{50}$$

In addition to *s1* already defined in eq. (1), to facilitate the design of the actuator, it is useful to define two other dimensionless coefficients:

$$s\_m = \frac{E\_{MB}}{E\_{MA}} = \frac{K\_{MB}}{K\_{MA}}\tag{51}$$

$$s\_{\mathcal{g}} = \frac{\mathcal{E}\_{\mathcal{g}}}{\mathcal{E}\_{adm}} = \frac{\mathcal{X}\_{\mathcal{g}}}{\mathcal{X}\_{adm}}\tag{52}$$

The parameter *sm* is the ratio between the two stiffness of the alloy at cold temperatures and it is a characteristic of the shape memory material. The parameter *sg* is the ratio of the deflection, *xg*, at which the change in stiffness in the martensitic state is recorded and the maximum allowable deflection *xadm*, related to the maximum deformation, *εadm*, admissible for the material. To limit the size of the device, it is convenient to assume the *xmaz* = *xadm* so as to completely exploit the material.

**Figure 8.** Force-deflection curves of a two-SMA actuator superimposed on the characteristic lines of the compensator.

#### **3.3. Design of compensated single SMA actuators**

To design compensated actuators with a single SMA element, it is convenient to introduce the ratio, γ, between the stroke and the maximum deflection of the SMA element:

$$\mathcal{Y} = \bigtimes\_{\mathcal{X}\_{adm}}^{\mathcal{S}} \tag{53}$$

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

**Figure 9.** Force-deflection curves of a two-SMA actuator compensated by a generic negative elastic spring (line *ED*)

The axial dimension of the actuator is reduced as the ratio γtends to one because in this case all the deformation that the material can sustain is used to produce a useful stroke. However, the value γ cannot reach the unity because the minimum deflection would be zero and the SMA element would not be able to exert any useful force.

The force delivered by the shape memory element can be calculated with (49), in the austenitic state, and with the approximated expression (50), in the martensitic state. Figure 8 shows that the high-temperature curve (OC) and the low-temperature curve (A'B) diverge. Thus, the most unfavorable position in which the design force is required is where the difference between the two curves (useful force) is minimal. In this position, the sum of the design forces in both directions must equal the distance between the ON straight line and the OFF straight line. Assuming conventionally *FON* > 0 (because it agrees with the force exerted by the SMA element) and *FOFF* < 0 (because it is opposed to the force produced by the SMA) we have (*FON – FOFF*) = *FSMA\_ON* (*x* = *xmin*) - *FSMA\_OFF* (*x* = *xmin*). Recalling (48),(49),(50) and introducing the dimensionless parameters (1),(51),(52) this condition becomes:

$$F\_{ON} - F\_{OFF} = \frac{k\_{MA}}{\mathcal{Y}} \left[ \left( s\_1 - s\_m \right) \left( 1 - \mathcal{Y} \right) - s\_\mathcal{g} \left( 1 - s\_m \right) \right] \tag{54}$$

The stiffness needed to fulfill the minimum force in both directions and to achieve the desired stroke is obtained by solving (54) with respect to the stiffness *kMA*:

$$k\_{MA} = \frac{\mathcal{V}\left(F\_{ON} - F\_{OFF}\right)}{S\left[\left(s\_1 - s\_m\right)\left(1 - \mathcal{V}\right) - s\_\mathcal{g}\left(1 - s\_m\right)\right]}\tag{55}$$

The next step is to set the desired characteristics of the compensation element in terms of stiffness (*Kcomp*) and of force ( min *comp x x <sup>F</sup>* <sup>=</sup> ), delivered at position *x* = *xmin* = *S* (1<sup>−</sup>γ ) /γ.


**Table 2.** Suggested values for the compensation stiffness to achieve specific behaviour of the actuator

From (1) and (55) the stiffness *KA* of the SMA element in the austenitic state is obtained.

Though the compensation stiffness (always negative) could be set as desired in the range −*KMB* ÷ −*KA*, Table 1 shows how to choose the value of *Kcomp* to obtain favourable operating characteristics: for *Kcomp* = −*KA*, the force of the activated actuator is constant and for *Kcomp* = −*KMB* the force of the deactivated actuator is constant. By setting *Kcomp* = − 0.5(*KA* + *KMB*), the deviation from uniformity of both states of actuator is minimized.

The compensator force the at position *x* = *xmin* to obtain the desired behavior must be:

$$F\_{comp}\Big|\_{\mathbf{x}=\mathbf{x}\_{\min}} = F\_{\text{ON}} - K\_A S \left(\frac{1-\mathcal{Y}}{\mathcal{Y}}\right) \tag{56}$$

The compensator always acts with a force increasing with *x* and opposite to the force delivered by the SMA element.

#### *3.3.1. Case study: single SMA compensated actuator*

In this section, the design procedure of a single SMA compensated actuator is carried out numerically for a generic system with the following technical specifications:

• Required stroke: 10mm

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

Force

*F0comp*

*E*

*O*

*C*

The axial dimension of the actuator is reduced as the ratio

and the SMA element would not be able to exert any useful force.

γ

spring (line *ED*)

However, the value

**Figure 9.** Force-deflection curves of a two-SMA actuator compensated by a generic negative elastic

*S=p*

SMA1 ON

*A*

*ka1* 

*Fnet ON1 FON1*

*kcomp* + *ka1*

*kcomp*

*x D*

case all the deformation that the material can sustain is used to produce a useful stroke.

*kmb2*

*xmin* = 0 *xmax* = *xadm*<sup>1</sup> *S*/2 *S*/2

*G*

The force delivered by the shape memory element can be calculated with (49), in the austenitic state, and with the approximated expression (50), in the martensitic state. Figure 8 shows that the high-temperature curve (OC) and the low-temperature curve (A'B) diverge. Thus, the most unfavorable position in which the design force is required is where the difference between the two curves (useful force) is minimal. In this position, the sum of the design forces in both directions must equal the distance between the ON straight line and the OFF straight line. Assuming conventionally *FON* > 0 (because it agrees with the force exerted by the SMA element) and *FOFF* < 0 (because it is opposed to the force produced by the SMA) we have (*FON – FOFF*) = *FSMA\_ON* (*x* = *xmin*) - *FSMA\_OFF* (*x* = *xmin*). Recalling (48),(49),(50)

and introducing the dimensionless parameters (1),(51),(52) this condition becomes:

γ

desired stroke is obtained by solving (54) with respect to the stiffness *kMA*:

min

*MA*

*k*

stiffness (*Kcomp*) and of force (

*ON OFF m g m k S F F ss s s*

The stiffness needed to fulfill the minimum force in both directions and to achieve the

γ

( ) ( )( ) ( ) <sup>1</sup> 1 1 *ON OFF*

γ

*F F*

<sup>−</sup> <sup>=</sup> − −− −

*Ss s s s*

The next step is to set the desired characteristics of the compensation element in terms of

*m gm*

*comp x x <sup>F</sup>* <sup>=</sup> ), delivered at position *x* = *xmin* = *S* (1<sup>−</sup>

( )( ) ( ) <sup>1</sup> 1 1 *MA*

γ

− = − −− − (54)

γ

Displacement, *x*

*F0m2* 

*B*

SMA2 OFF

cannot reach the unity because the minimum deflection would be zero

tends to one because in this

(55)

γ ) /γ.


The material considered is a Nitinol wire, with austenitic Young modulus *Ea*=75GPa, martensitic Young modulus *EMA*=28GPa, bilinear gradient *EMB*=5GPa, strain at the threshold of pseudo-plastic regime *εg*=0.4% maximum deformation *εadm*=4%.

The non dimensional groups are calculated as follows:

$$s\_1 = \frac{E\_A}{E\_{MA}} = \frac{75}{28} = 2.68\tag{57}$$

$$s\_m = \frac{E\_{MB}}{E\_{MA}} = \frac{5}{28} = 0.18\tag{58}$$

$$s\_g = \frac{\varepsilon\_g}{\varepsilon\_{adm}} = \frac{0.004}{0.04} = 0.1\tag{59}$$

An optimal value for the ratio *γ*=0.75 can be chosen as a tradeoff between axial dimension and oversizing of the active SMA elements.

From (55) it is possible to obtain the cold state stiffness of the SMA wire:

$$\begin{split} K\_{MA} &= \frac{\mathcal{V}\left(F\_{ON} - F\_{OFF}\right)}{S\left[\left(s\_1 - s\_m\right)\left(1 - \chi\right) - s\_\mathcal{g}\left(1 - s\_m\right)\right]} = \frac{0.75\left[10 - \left(-5\right)\right]}{10\left[\left(2.68 - 0.18\right)\left(1 - 0.75\right) - 0.1\left(1 - 0.18\right)\right]} = \\ K &= 2.07 \frac{N}{mm} \end{split} \tag{60}$$

Considering (1) and (51) and the last expression the cold state stiffness *KA* and the post elastic martensitic stiffness *KMB* can be obtained:

$$K\_A = s\_1 K\_{MA} = 2.68 \cdot 2.07 = 5.55 \,\text{N} \,\text{/}mm \tag{61}$$

$$K\_{MB} = s\_m K\_{MA} = 0.18 \cdot 2.07 = 0.373 \,\text{N} \,\text{/}mm$$

From (53) is computed *xadm* , the maximum elongation of the SMA wire:

$$
\lambda \chi\_{adm} = \frac{S}{\chi} = \frac{10}{0.75} = 13.33 mm \tag{62}
$$

Taking into account the maximum deformation of the material *εadm* the wire length *l0* is:

$$l\_0 = \frac{\chi\_{adm}}{\varepsilon\_{adm}} = \frac{13.33}{0.04} = \text{333.3mm} \tag{63}$$

From the austenitic stiffness and the wire length the wire diameter *d* is immediately obtained:

$$d = \sqrt{\frac{4K\_A l\_0}{E\_A \pi}} = \sqrt{\frac{4 \cdot 5.55 \cdot 333.3}{75000\pi}} = 0.177\,\text{mm} \tag{64}$$

The compensator system preload is obtained through (56).This value is negative because in its minimum stroke position *x* = *xmin=* 3.33mm, the preload is opposed to SMA active element action:

$$F\_{comp} \Big|\_{\mathbf{x} = \mathbf{x}\_{\min}} = F\_{ON} - K\_A S \left( \frac{1 - \chi}{\chi} \right) = 10 - 5.55 \cdot 10 \left( \frac{1 - 0.75}{0.75} \right) = -8.5 \,\mathrm{N} \tag{65}$$

In order to minimize the force fluctuations in both travel directions the optimum stiffness of compensation system is obtained from (Table 1) and is:

$$k\_{comp} = -\frac{K\_{MB} + K\_A}{2} = -\frac{0.373 + 5.55}{2} = -2.96 \frac{N}{mm} \tag{66}$$

#### **3.4. Design of compensated actuators with two antagonistic SMA elements**

The input data required for the design of a two antagonistic shape memory actuators are: the value of the minimum useful forces in the two directions of activation *FON1*, *FON2* the value of the stroke desired, S, and the active elements material characteristics (i.e. *s1*, *sm* and *sg*). In the following equations, assuming that the axis of the actuator is horizontal, the forces are assumed positive when directed from right to left and negative when directed from left to right. The displacements are assumed positive from left to right.

For a compensated actuator with two antagonistic SMA element it can be shown [21] that it is always possible to set the ratio γ (53) to 1, so that the actuator stroke is equal to the maximum deflection imposed on the SMA elements (*xmin*=0).

If both active elements are constituted by the same alloy (same parameters *s1*, *sm*, *sg*) is possible to demonstrate [21] that the optimum rigidity of the two elements will be equal (*KA1=KA2=KA* and *KMB1=KMB2=KMB*).

The force generated by element 1 in the ON state is:

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

and oversizing of the active SMA elements.

γ

2.07

*N mm*

*MA*

*K*

=

obtained:

element action:

*m*

*g*

*s*

From (55) it is possible to obtain the cold state stiffness of the SMA wire:

From (53) is computed *xadm* , the maximum elongation of the SMA wire:

0

*A A*

*E* π

min

=

0.75 *adm S*

Taking into account the maximum deformation of the material *εadm* the wire length *l0* is:

γ

*adm adm*

ε

( ) ( )( ) ( )

γ

*F F*

*Ss s s s*

elastic martensitic stiffness *KMB* can be obtained:

*ON OFF*

*m gm*

*s*

*MA*

*g*

ε

ε

*adm*

An optimal value for the ratio *γ*=0.75 can be chosen as a tradeoff between axial dimension

*E*

<sup>1</sup> <sup>5</sup> 0. 28 8 *MB*

> 0.004 0.1 0.04

( )( ) ( ) <sup>1</sup>

<sup>−</sup> − − == = − −− − − −−−

Considering (1) and (51) and the last expression the cold state stiffness *KA* and the post

*K sK MB m MA* = =⋅= 0.18 2.07 0.373 / *N mm*

*x mm*

*<sup>x</sup> <sup>l</sup> mm*

From the austenitic stiffness and the wire length the wire diameter *d* is immediately

<sup>0</sup> <sup>4</sup> 4 5.55 333.3 0.177 75000

 π

<sup>1</sup> 1 0.75 10 5.5

*K l <sup>d</sup> mm*

The compensator system preload is obtained through (56).This value is negative because in its minimum stroke position *x* = *xmin=* 3.33mm, the preload is opposed to SMA active

> .75 *comp ON x x <sup>A</sup> F FS <sup>K</sup> <sup>N</sup>* γ

γ

<sup>10</sup> 13.33

13.33 333.3 0.04

1 1 10 2.68 0.18 1 0.75 0.1 1 0.18

*<sup>E</sup>* = == (58)

== = (59)

( )

(60)

0.75 10 5

<sup>1</sup> 2.68 2.07 5.55 / *K sK A MA* = =⋅= *N mm* (61)

== = (62)

== = (63)

⋅ ⋅ == = (64)

0 5 10 8.5

− − = − =− ⋅ = − (65)

$$F\_{\text{SMA}\\_ON} = K\_A \text{x} \tag{67}$$

In the OFF state, using (3) for the martensite behavior, the force in element 1 is:

$$F\_{\text{SMA}\\_OFF} = F\_{0m} + K\_{MB}\chi\tag{68}$$

For element 2, the force output depends on the difference between the prestretch *p* of the two SMA elements and the position of the actuator. Assuming *γ*=1 the prestretch *p* is equal to the stroke *S*, and the force of element 2 in the ON and OFF state is respectively:

$$F\_{\text{SM}A2\\_ON} = -K\_A(\mathbb{S} - \mathfrak{x})\tag{69}$$

$$F\_{\text{SMA\\_}\\_OFF} = -F\_{0m} - K\_{\text{MB}} (\text{S} - \text{x}) \tag{70}$$

The force delivered from the compensator is:

$$F\_{comp} = F\_{0comp} + k\_{comp} \text{x} \tag{71}$$

where *F0comp* indicates the force of the compensator at position *x* = 0.

Figure 9 shows the force displacement diagram of a two-SMA actuator and helps to understand the relationship between the variables involved in the equations.

Line OA in Figure 9 represents the characteristics of the austenitic (hot) SMA1 element. Segment BC represents the martensitic (cold) response of the antagonistic SMA2 element. Line DE is the characteristic of the compensation spring, with point D corresponding to the centerpoint of the total stroke S of the actuator. Line EG represents an ideally constant external load of amplitude *FON1*. The situation beyond point D is obtained by extrapolating linearly all the characteristic lines shown in Figure 9. Line AE corresponds to the characteristic of the SMA1 element and the compensation spring combined. At any position *x*, the difference between lines AE and BC gives the net output force of the actuator (*Fnet ON1*) when element SMA1 is activated. When element SMA1 is disabled and SMA2 is enabled, the chart becomes similar to Figure 9, with all the lines mirrored with respect to the line AD.

The optimal performance of the actuator in Figure 9 is achieved when lines AE is parallel to line BC, so that the net output force of the actuator (*Fnet ON1*) equals the external load (*FON1*) at any position *x*. Scirè and Dragoni [21] demonstrated that the optimal actuator meets the design specifications (i.e. the given output forces *FON1*, *FON2* and the net stroke *S*) when the following relationships hold true:

$$K\_{MA} = \frac{\left(F\_{ON1} - F\_{ON2}\right)}{S\left[\left(s\_1 - s\_m\right) + 2s\_g\left(s\_m - 1\right)\right]}\tag{72}$$

$$k\_{comp} = -\frac{\left(F\_{ON1} - F\_{ON2}\right)\left(s\_1 + s\_m\right)}{S\left[\left(s\_1 - s\_m\right) + 2s\_g\left(s\_m - 1\right)\right]}\tag{73}$$

$$F\_{0\text{comp}} = \frac{s\_g \left(F\_{ON1} + F\_{ON2}\right) \left(s\_m - 1\right) + \left(F\_{ON1}s\_1 - F\_{ON2}s\_m\right)}{\left(s\_1 - s\_m\right) + 2s\_g \left(s\_m - 1\right)}\tag{74}$$

The force generated by the actuator in the case of element 1 ON and element 2 OFF is:

$$F\_{net\\_ON1} = K\_A \mathbf{x} + F\_{0\text{comp}} + k\_{comp} \mathbf{x} - F\_{0m} - K\_{MB} \left(\mathbf{S} - \mathbf{x}\right) \tag{75}$$

while the force generated in the case of element 2 ON and element 1 OFF is:

$$F\_{net\\_ON2} = F\_{0m} + K\_{MB}\mathbf{x} + F\_{0comp} + k\_{comp}\mathbf{x} - K\_A \left(\mathbf{S} - \mathbf{x}\right) \tag{76}$$

From (1) and (55) the stiffness *KA* of the SMA element in the austenitic state is obtained.

The actuator designed by (72)-(74) provides a constant force in both directions. In the compensated actuators with two SMAs, the force of the compensator becomes zero and changes sign at mid-stroke. In the range *0 < x < S/2* the force of the compensator has the same direction as the force exerted by element 1, while in the range *S/2 < x < S* it has the same direction as the force generated by element 2.

#### *3.4.1. Case study: Double SMA compensated actuator*

This section describes the design of double SMA compensated actuator under the hypothesis of using traction Nitinol springs with the following characteristics:


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

*MA*

*comp*

*k*

0

*F*

*comp*

same direction as the force generated by element 2.

*K*

respect to the line AD.

following relationships hold true:

Line OA in Figure 9 represents the characteristics of the austenitic (hot) SMA1 element. Segment BC represents the martensitic (cold) response of the antagonistic SMA2 element. Line DE is the characteristic of the compensation spring, with point D corresponding to the centerpoint of the total stroke S of the actuator. Line EG represents an ideally constant external load of amplitude *FON1*. The situation beyond point D is obtained by extrapolating linearly all the characteristic lines shown in Figure 9. Line AE corresponds to the characteristic of the SMA1 element and the compensation spring combined. At any position *x*, the difference between lines AE and BC gives the net output force of the actuator (*Fnet ON1*) when element SMA1 is activated. When element SMA1 is disabled and SMA2 is enabled, the chart becomes similar to Figure 9, with all the lines mirrored with

The optimal performance of the actuator in Figure 9 is achieved when lines AE is parallel to line BC, so that the net output force of the actuator (*Fnet ON1*) equals the external load (*FON1*) at any position *x*. Scirè and Dragoni [21] demonstrated that the optimal actuator meets the design specifications (i.e. the given output forces *FON1*, *FON2* and the net stroke *S*) when the

> ( ) ( )() 1 2 <sup>1</sup> 2 1 *ON ON*

*F F*

*Ss s ss* <sup>−</sup> <sup>=</sup> −+ −

*Ss s ss* − + = − −+ −

( )( )

1

while the force generated in the case of element 2 ON and element 1 OFF is:

The force generated by the actuator in the case of element 1 ON and element 2 OFF is:

From (1) and (55) the stiffness *KA* of the SMA element in the austenitic state is obtained.

The actuator designed by (72)-(74) provides a constant force in both directions. In the compensated actuators with two SMAs, the force of the compensator becomes zero and changes sign at mid-stroke. In the range *0 < x < S/2* the force of the compensator has the same direction as the force exerted by element 1, while in the range *S/2 < x < S* it has the

*m gm*

( )( ) ( )() 1 21 <sup>1</sup> 2 1 *ON ON m*

( )() 1 2 11 2

*m gm*

*g ON ON m ON ON m*

*sF F s F s F s*

*s s ss*

2 1

( ) *net ON A comp comp m MB* \_1 0 <sup>0</sup> *F Kx F k F K S x* = + −− − + *x* (75)

( ) *net ON m MB comp comp A* \_2 0 <sup>0</sup> *F F K xF* =+ + − + − *k K x S x* (76)

*F F ss*

*m gm*

1 ( )

+ −+ − <sup>=</sup> −+ − (74)

(72)

(73)


The desired output stroke is *S* =75mm, with the maximum possible force in both directions.

In this case the dimensional parameter *γ* can be taken equal to one and consequently the maximum extension of the active elements *xadm* is equal to the desired stroke *S*.

The non dimensional parameters are:

$$s\_1 = \frac{K\_A}{K\_{MA}} = \frac{0.0615}{0.0414} = 1.486\tag{77}$$

$$s\_m = \frac{K\_{MB}}{K\_{MA}} = \frac{0.0156}{0.0414} = 0.377\tag{78}$$

$$s\_g = \frac{\chi\_g}{\chi\_{adm}} = \frac{23}{75} = 0.306\tag{79}$$

Using the above dimensionless parameters, eq. (72) gives the maximum force differential *(FON1 – FON2)* that the actuator can produce in both directions as:

$$\begin{aligned} \left[F\_{ON1} - F\_{ON2} = K\_{MA}S\right] \left[\left(s\_1 - s\_m\right) + 2s\_g \left(s\_m - 1\right)\right] &= \\ \left[1 = 0.0414 \cdot 75\right] \left[\left(1.486 - 0.377\right) + 2 \cdot 0.306 \left(0.377 - 1\right)\right] &= 2.26N \end{aligned} \tag{80}$$

Assuming that the actuator generates the same output force regardless of the direction of motion gives:

$$F\_{\rm ON1} = -F\_{\rm ON2} = 1.13N \tag{81}$$

From (73) the overall stiffness of the compensation system *kcomp* is calculated:

$$\begin{split} k\_{comp} &= -\frac{\left(F\_{ON1} - F\_{ON2}\right)\left(s\_1 + s\_m\right)}{S\left[\left(s\_1 - s\_m\right) + 2s\_g\left(s\_m - 1\right)\right]} = -\frac{2.26\left(1.486 + 0.377\right)}{75\left[\left(1.486 - 0.377\right) + 2 \cdot 0.306\left(0.377 - 1\right)\right]} = \\ k &= -0.0771\,\mathrm{N} \,\mathrm{/mm} \end{split} \tag{82}$$

Likewise, (74) gives the force *F0comp* that the compensator must apply to the SMA elements at the position *x = 0*:

$$\begin{split} F\_{0\text{comp}} &= \frac{s\_g \left( F\_{ON1} + F\_{ON2} \right) \left( s\_m - 1 \right) + \left( F\_{ON1} s\_1 - F\_{ON2} s\_m \right)}{\left( s\_1 - s\_m \right) + 2 s\_g \left( s\_m - 1 \right)} = \\ &= \frac{0.306 \left[ 1.13 + \left( -1.13 \right) \right] \left( 0.377 - 1 \right) + \left[ 1.13 \cdot 1.486 - \left( -1.13 \right) \cdot 0.377 \right]}{\left( 1.486 - 0.377 \right) + 2 \cdot 0.306 \left( 0.377 - 1 \right)} = 2.893N \end{split} \tag{83}$$

#### **3.5. Design of the compensator system**

This section describes the elastic constants of the shape memory elements and the maximum deflections that they will undergo, calculated from the design data (force and stroke required of the actuator) following the procedure of the previous section. Thanks to these parameters the shape memory elements (wires or springs) can be described using classical engineering formulas. The method also provides the properties needed by the elastic compensation to meet the required performance. Two elastic compensation systems are described in detail in this section: 1) a rocker-arm mechanism and 2) a double articulated quadrilateral.

#### *3.5.1. Rocker-arm compensator*

The compensation mechanism shown in Figure 10 is made up of a rocker-arm R hinged in G to the frame T. The (conventional) compensation spring *Sc* (with free length *L0Trad* and spring rate *kTrad*) connects the extremity F of the shortest side of the rocker-arm to point E of the frame. Point O (the output port of the actuator) is used to connect the primary elastic elements of the actuator (SMA1 and SMA2), which are also fixed at points P and Q to the frame. In the case of a single-SMA actuator, the active element (SMA1) is placed at the bottom of the device, element SMA 2 disappears and the contrasting element is made up by the conventional compensation spring, *S*c.

The mechanism in Figure 10 has a position of unstable equilibrium, corresponding to the configuration where the axis EF of the compensation spring *Sc* passes through the hinge G of the rocker-arm. In this position, the force exerted by the compensator on the slider O is null. In the case of an actuator with a single active element, the compensation spring *Sc* is always placed to the right of hinge G in order to exert a contrasting force on the SMA element 1 needed to deform it in the cold state. For an actuator with two opposing SMA elements, the spring *S*c is located in an unstable position (line EF passes through G) when point O is at the center of the stroke (S). In this way, the compensation mechanism helps active element SMA1 for the lower half of the stroke and the element SMA2 for the upper half of the stroke. If in Figure 10 the absolute value of the angle *γ* is small, i.e. if *l0+x-(a+d)<< c* then the following expressions apply with good approximation:

$$F\_{comp}(\mathbf{x}) = -\frac{abF\_{Tnd}\mathcal{B}}{c\left(a+b\right)}\tag{84}$$

**Figure 10.** Oscillating rocker arm compensation mechanism

( )( )

1

( )()

⋅ ⋅ <sup>=</sup> <sup>+</sup>

*m gm*

this section: 1) a rocker-arm mechanism and 2) a double articulated quadrilateral.

*g ON ON m ON ON m*

+ −+ − = = −+ −

*sF F s F s F s*

*s s ss*

1 2 11 2

the position *x = 0*:

0

*F*

*comp*

*3.5.1. Rocker-arm compensator* 

the conventional compensation spring, *S*c.

following expressions apply with good approximation:

**3.5. Design of the compensator system** 

Likewise, (74) gives the force *F0comp* that the compensator must apply to the SMA elements at

1 ( )

2.893

*N*

(83)

( )( ) ( ) ( ) ()

+− − + −− <sup>=</sup> −⋅ −

This section describes the elastic constants of the shape memory elements and the maximum deflections that they will undergo, calculated from the design data (force and stroke required of the actuator) following the procedure of the previous section. Thanks to these parameters the shape memory elements (wires or springs) can be described using classical engineering formulas. The method also provides the properties needed by the elastic compensation to meet the required performance. Two elastic compensation systems are described in detail in

The compensation mechanism shown in Figure 10 is made up of a rocker-arm R hinged in G to the frame T. The (conventional) compensation spring *Sc* (with free length *L0Trad* and spring rate *kTrad*) connects the extremity F of the shortest side of the rocker-arm to point E of the frame. Point O (the output port of the actuator) is used to connect the primary elastic elements of the actuator (SMA1 and SMA2), which are also fixed at points P and Q to the frame. In the case of a single-SMA actuator, the active element (SMA1) is placed at the bottom of the device, element SMA 2 disappears and the contrasting element is made up by

The mechanism in Figure 10 has a position of unstable equilibrium, corresponding to the configuration where the axis EF of the compensation spring *Sc* passes through the hinge G of the rocker-arm. In this position, the force exerted by the compensator on the slider O is null. In the case of an actuator with a single active element, the compensation spring *Sc* is always placed to the right of hinge G in order to exert a contrasting force on the SMA element 1 needed to deform it in the cold state. For an actuator with two opposing SMA elements, the spring *S*c is located in an unstable position (line EF passes through G) when point O is at the center of the stroke (S). In this way, the compensation mechanism helps active element SMA1 for the lower half of the stroke and the element SMA2 for the upper half of the stroke. If in Figure 10 the absolute value of the angle *γ* is small, i.e. if *l0+x-(a+d)<< c* then the

( ) ( ) *Trad*

= − <sup>+</sup>

*ca b*

β

(84)

*abF F x*

*comp*

2 1 0.306 1.13 1.13 0.377 1 1.13 1.486 1.13 0.377

1.486 0.377 2 0.306 0.377 1

**Figure 11.** Double articulated quadrilateral compensation mechanism

$$k\_{Comp} = \frac{\Theta F\_{Comp}}{\partial \mathbf{x}} = -\frac{abF\_{Trad}}{\left(a+b\right)c^2} \tag{85}$$

When lengths *b*,*c*,*d* have been arbitrarily decided along with the stiffness of the traditional spring *kTrad*, from (84) (calculated for *x=xmin*) and (85) the expressions for *a* and *L0Trad* are derived which define the correct sizing of the compensation mechanism:

$$L\_{0Tnd} = \frac{\left[\left.k\_{comp}\left(A - b + d - l\_0\right) + F\_{comp}\right|\_{\chi\_{\min}}\right] \left(c^2 \left.k\_{comp}^2 \left.\chi - B\right)\right.}}{B \left.k\_{comp}\right.}\tag{86}$$

$$a = l\_0 - A - d - \frac{F\_{comp} \Big|\_{x\_{\min}}}{k\_{comp}} \tag{87}$$

with:

$$A = \mathcal{S}\left(\frac{\mathcal{Y} - 1}{\mathcal{Y}}\right) \tag{88}$$

$$B = b \, k\_{Tud} \, \mathcal{Y} \left[ k\_{comp} (d - l\_0 + A) + F\_{comp} \Big|\_{x\_{\min}} \right] \tag{89}$$

In Equations (86) – (89) the term min *comp <sup>x</sup> <sup>F</sup>* is given by (56) or by (74) respectively for the single SMA or two opposing SMAs actuator. Likewise, the stiffness *kcomp*, is obtained from Table 1 or from (73) for single SMA or for the two opposing SMAs actuator respectively.

#### *3.5.2. Double articulated quadrilateral compensator*

Figure 11 illustrates a second elastic compensation system based on the use of two articulated quadrilaterals (I and II), which are connected together in series. The first quadrilateral (I) is made up of four equal rods DT*,* EU*,* RT and SU, hinged together in T and U and fixed to the frame with hinges D and E. In the upper part, the rods are hinged in R and S to the bar PQ, which can translate vertically. The rods DT, EU, RT and SU have length *m*, while the horizontal segments DE and RS have length *f*. The conventional spring *SA*, with stiffness *kA* and free length *L0A*, is stretched between hinges T and U.

The second quadrilateral (II) is made up of four equal rods VP, ZQ, VF and ZG, connected with internal hinges in V and Z and fixed to the frame with hinges F and G. The length of the rods VP, ZQ, VF and ZG is *n*, while the length of the horizontal segments PQ and FG is *g*.

Quadrilateral II contains two conventional springs, *S*B and *S*c. Spring SB, with stiffness *kB* and free length *L0B*, is stretched horizontally between hinges V and Z. Spring *S*c, with stiffness *kC* and free length *L0C*, is stretched vertically between the horizontal sides FG and PQ. Member PRSQ, which is common to the two quadrilaterals, represents the output ports of the actuator. In addition to the conventional springs, *SA*, *SB* and *SC*, the mechanism in Figure 11 contains also the primary active elements of the actuator. Active element SMA1 is hosted by the quadrilateral I and connects the base DE to the output port PRSQ. The second primary element SMA2 (if applicable) is hosted by quadrilateral II and connects the frame FG to the output port PRSQ.

For the sake of simplicity, all the springs shown in Figure 11 (both SMA and conventional) are traction springs. However, the mechanism can work just as well with all compression springs, which can be designed with the same equations presented below.

By imposing the vertical equilibrium of the member PRSQ in Figure 11 and excluding the forces exerted by the shape memory elements, it is possible to obtain the compensation force as a function of the position *x* of the output port.

Imposing that:

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

with:

In Equations (86) – (89) the term

*3.5.2. Double articulated quadrilateral compensator* 

0

γ

stiffness *kA* and free length *L0A*, is stretched between hinges T and U.

*al Ad*

min

*<sup>k</sup>* = − −− (87)

min

*comp <sup>x</sup> <sup>F</sup>* is given by (56) or by (74) respectively for the

(88)

(89)

*comp <sup>x</sup> comp*

*F*

<sup>1</sup> *A S* γ

<sup>0</sup> ( ) *Trad comp comp <sup>x</sup> B bk k d l A F*

min

 <sup>−</sup> <sup>=</sup> 

<sup>=</sup> −+ +

single SMA or two opposing SMAs actuator. Likewise, the stiffness *kcomp*, is obtained from Table 1 or from (73) for single SMA or for the two opposing SMAs actuator respectively.

Figure 11 illustrates a second elastic compensation system based on the use of two articulated quadrilaterals (I and II), which are connected together in series. The first quadrilateral (I) is made up of four equal rods DT*,* EU*,* RT and SU, hinged together in T and U and fixed to the frame with hinges D and E. In the upper part, the rods are hinged in R and S to the bar PQ, which can translate vertically. The rods DT, EU, RT and SU have length *m*, while the horizontal segments DE and RS have length *f*. The conventional spring *SA*, with

The second quadrilateral (II) is made up of four equal rods VP, ZQ, VF and ZG, connected with internal hinges in V and Z and fixed to the frame with hinges F and G. The length of the rods VP, ZQ, VF and ZG is *n*, while the length of the horizontal segments PQ and FG is *g*.

Quadrilateral II contains two conventional springs, *S*B and *S*c. Spring SB, with stiffness *kB* and free length *L0B*, is stretched horizontally between hinges V and Z. Spring *S*c, with stiffness *kC* and free length *L0C*, is stretched vertically between the horizontal sides FG and PQ. Member PRSQ, which is common to the two quadrilaterals, represents the output ports of the actuator. In addition to the conventional springs, *SA*, *SB* and *SC*, the mechanism in Figure 11 contains also the primary active elements of the actuator. Active element SMA1 is hosted by the quadrilateral I and connects the base DE to the output port PRSQ. The second primary element SMA2 (if applicable) is hosted by quadrilateral II and connects the frame FG to the output port PRSQ.

For the sake of simplicity, all the springs shown in Figure 11 (both SMA and conventional) are traction springs. However, the mechanism can work just as well with all compression

By imposing the vertical equilibrium of the member PRSQ in Figure 11 and excluding the forces exerted by the shape memory elements, it is possible to obtain the compensation force

springs, which can be designed with the same equations presented below.

as a function of the position *x* of the output port.

γ

$$L\_{0A} = f \qquad \qquad \qquad L\_{0B} = \mathcal{g} \qquad \qquad \qquad k\_B = k\_C \tag{90}$$

the compensation force becomes:

$$F\_{comp}\left(\mathbf{x}\right) = -k\_A \left(l\_0 + \mathbf{x}\right) + k\_C L\_{0C} \tag{91}$$

i.e. it depends linearly on *x* as desired in an ideal compensation system. The compensation stiffness implied in (91) is:

$$k\_{comp} = \frac{\partial F\_{comp}}{\partial \mathbf{x}} = -k\_A \tag{92}$$

From (91) the force exerted by the compensation system at *x=xmin* is:

$$\left.F\_{\alpha mp}\right|\_{\mathbf{x}=\mathbf{x}\_{\min}} = -k\_A \left(l\_0 + \mathbf{x}\_{\min}\right) + k\_C L\_{0C} \tag{93}$$

Conditions (92) and (93) complete the design of the compensation mechanism in Figure 11. The value *kcomp* is taken from Table 1, for the case of a single SMA, or alternatively from (73) for two SMA elements. Similarly, the value *min <sup>x</sup> co p <sup>x</sup> <sup>m</sup> <sup>F</sup>* <sup>=</sup> is taken from (56) or (74), for one or two SMA elements, respectively.

By solving (93) with respect to the product *kC L0C* , you can get the correct design of spring *SC*. From condition (90), we see that the springs *SB* and *SC* must have the same stiffness and that the free length of spring *SB* must be *g*. It is thus convenient to set to *g* the free length of spring C as well, so as to have two identical springs *SB* and *SC* in quadrilateral II with spring rate:

$$k\_C = k\_B = \left( F\_{comp} \Big|\_{x = x\_{\min}} + k\_A \left( l\_0 + x\_{\min} \right) \right) \Big/ \mathcal{g} \tag{94}$$

#### **4. General discussion**

#### **4.1. Performances of the uncompensated SMA actuators**

The analytical framework described in Paragraph 3 considers both dissipative and conservative forces acting on the system. Closed-form relationships are developed, which are the basis of a step-by-step procedure for an optimal design of the entire actuator (both with antagonist SMA, bias elastic element or backup force). Specific formulas for dimensioning of the SMA element in the form of straight wires and helical traction springs are also presented.

The physical meaning of dimensionless design parameters *s1, s2, sF* and *s0* deserves specific comments. Parameter *s1* is an intrinsic property of the SMA material, the higher *s1*, the better the SMA behaviour. Parameter *s2* is a measure of the relative stiffness between the bias spring (be it traditional or SMA) and the primary SMA spring. Optimal values of parameter

*s2* maximize the overall travel. Low values of *s2* lead to an increase in stroke and in the actuator dimension. Parameter *sF* quantifies the influence of dissipative forces on the stroke of the actuator. For low values of *sF* the dissipative force is negligible in comparison with the forces expressed by the backup systems and the actuator moves freely. The upper limit of *sF* is reached when the dissipative force becomes high enough to prevent the movement of the actuator. Parameter *s0*, is the equivalent of *sF* for the conservative force. Since the conservative force helps the backup system, high values of *s0* improve the actuator performances.

Also for the three backup systems analyzed, some specific comments are reported. Using a force as back up for the primary spring may be convenient when there is important conservative force acting on the system which helps the restoring process. Often this solution is not feasible due to dimensional constraint which prevents dead loads to be used. The binary actuator with a traditional spring backup gives the worst performances in terms of output characteristic, due to disadvantages of the elastic slope of the traditional spring. By analyzing Section 3.2 equations it is found that for a given cursor stroke, Δ*x* the optimal value of *s2* provided by (21) minimizes the pre-stretch of the system, *p*, and tends to minimize the total size of the actuator. Eq.(18) shows that the pre-stretch of the system, *p*, acts on the stroke, Δ*x*, as a pure gain. Figure 12a displays the normalized stroke, Δ*x/p*, as a function of the other variables: *s0* (used as parameter of the family of surfaces), *s2* and *sF* (these two variables are plotted along the axes), taking *s1=3* because this is a typical value for the elastic moduli ratio of commercial shape memory alloy. Eq. (18) and Figure 12a also highlight that for each combination of *s0* and *sF* exists an optimal value, \* <sup>2</sup>*s* , of parameter *s2* which maximizes the normalized stroke. Eq. (19) shows that for each combination of *s1* and *sF* the stroke increases monotonically when parameter *s2* decreases, implying a more flexible bias spring. Both equations (18) and (19) prove that meaningful strokes (*Δx>0*) are only possible if parameter *sF* is lower than *sFcr* of equation (10).

Section 3.3 shows that the binary actuator with a SMA spring backup gives the best performances in terms of output characteristic, both respect to elastic backup and bias force backup. Equation (30) shows that also in this case the pre-stretch of the system, *p*, acts on the stroke, Δ*x*, as a pure gain. Figure 12b displays the normalized stroke, Δ*x/p*, as a function of the other variables: *s0* (used as a parameter of the family of surfaces), *s2* and *sF* (these two variables are plotted along the axes), taking *s1=3* (a typical value for elastic moduli in austenitic and martensitic phase of a commercial SMA). Equation (30) and Figure 12b also highlight that for each combination of *s1* and *sF* an optimal value, \* <sup>2</sup>*s* , of parameter *s2* exists that maximizes the normalized stroke. Equation (30) shows that for each combination of *s1* and *sF* the stroke increases monotonically when parameter *s2* decreases, implying a more flexible SMA bias spring. Correspondingly, equation (29) shows that the pre-stretch of the system, *p*, tends to infinity. If (31) is used to choose *s2*, the global pre-stretch, *p*, is minimized and the total length of the actuator tends to be reduced, but it is not minimized. To obtain a value of *sF* and a value of *s2* able to minimize the total length of the actuator, expressions that depend on the embodiment of the springs are needed.

**Figure 12.** Normalized stroke of a shape memory actuator backed up by a traditional spring (a) or by an antagonist SMA spring (b) as a function of the parameters *s2* , *sF* and *s0*.

#### **4.2. Performances of the compensated SMA actuators**

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

performances.

the stroke,

Δ

*s2* maximize the overall travel. Low values of *s2* lead to an increase in stroke and in the actuator dimension. Parameter *sF* quantifies the influence of dissipative forces on the stroke of the actuator. For low values of *sF* the dissipative force is negligible in comparison with the forces expressed by the backup systems and the actuator moves freely. The upper limit of *sF* is reached when the dissipative force becomes high enough to prevent the movement of the actuator. Parameter *s0*, is the equivalent of *sF* for the conservative force. Since the conservative force helps the backup system, high values of *s0* improve the actuator

Also for the three backup systems analyzed, some specific comments are reported. Using a force as back up for the primary spring may be convenient when there is important conservative force acting on the system which helps the restoring process. Often this solution is not feasible due to dimensional constraint which prevents dead loads to be used. The binary actuator with a traditional spring backup gives the worst performances in terms of output characteristic, due to disadvantages of the elastic slope of the traditional spring. By analyzing Section 3.2 equations it is found that for a given cursor stroke, Δ*x* the optimal value of *s2* provided by (21) minimizes the pre-stretch of the system, *p*, and tends to minimize the total size of the actuator. Eq.(18) shows that the pre-stretch of the system, *p*,

acts on the stroke, Δ*x*, as a pure gain. Figure 12a displays the normalized stroke,

highlight that for each combination of *s0* and *sF* exists an optimal value, \*

possible if parameter *sF* is lower than *sFcr* of equation (10).

highlight that for each combination of *s1* and *sF* an optimal value, \*

that depend on the embodiment of the springs are needed.

function of the other variables: *s0* (used as parameter of the family of surfaces), *s2* and *sF* (these two variables are plotted along the axes), taking *s1=3* because this is a typical value for the elastic moduli ratio of commercial shape memory alloy. Eq. (18) and Figure 12a also

which maximizes the normalized stroke. Eq. (19) shows that for each combination of *s1* and *sF* the stroke increases monotonically when parameter *s2* decreases, implying a more flexible bias spring. Both equations (18) and (19) prove that meaningful strokes (*Δx>0*) are only

Section 3.3 shows that the binary actuator with a SMA spring backup gives the best performances in terms of output characteristic, both respect to elastic backup and bias force backup. Equation (30) shows that also in this case the pre-stretch of the system, *p*, acts on

of the other variables: *s0* (used as a parameter of the family of surfaces), *s2* and *sF* (these two variables are plotted along the axes), taking *s1=3* (a typical value for elastic moduli in austenitic and martensitic phase of a commercial SMA). Equation (30) and Figure 12b also

that maximizes the normalized stroke. Equation (30) shows that for each combination of *s1* and *sF* the stroke increases monotonically when parameter *s2* decreases, implying a more flexible SMA bias spring. Correspondingly, equation (29) shows that the pre-stretch of the system, *p*, tends to infinity. If (31) is used to choose *s2*, the global pre-stretch, *p*, is minimized and the total length of the actuator tends to be reduced, but it is not minimized. To obtain a value of *sF* and a value of *s2* able to minimize the total length of the actuator, expressions

*x*, as a pure gain. Figure 12b displays the normalized stroke,

Δ*x/p*, as a

<sup>2</sup>*s* , of parameter *s2*

*x/p*, as a function

<sup>2</sup>*s* , of parameter *s2* exists

Δ

The performances of SMA compensated actuators are improved compared to traditional SMA actuators. A comparison between a traditional and a compensated SMA actuators, being equal the active elements and their maximum deformations, shows that the compensated system has longer strokes. Conversely, being equal the output stroke, the deformations in the compensated SMA system are lower than the traditional one. In general the compensated SMA actuator is able to obtain the same performances of a traditional system with lower maximum deformation in the material, while for a given admissible deformation the active elements of a compensated SMA are more stressed because they work against both the external load and the compensation system during the first part of the stroke. The energy stored in the compensator is released in the second part of the stroke helping in keeping the output force constant.

Experimental comparative tests on a SMA actuator with or without a rocker-arm compensator [28] demonstrated that the stroke of the compensated system is always greater regardless the external load. The stroke incremented by 2.5 times with no load up to 22 times with the maximum design force applied. Furthermore, the net stroke of the compensated actuator is not dependent on the applied force. This peculiarity simplifies both the design and the selection of the actuator for a given application. Other remarkable advantages of the compensated actuator lie in the nearly-constant value of the output force and in the fastest response compared to traditional systems, even with longer output strokes. The fastest response is due to the lower difference between internal external forces in the end positions. By contrast, the response time (the time needed to achieve an incipient displacement) is considerably longer for the compensated actuator. This behaviour is due to the high stress at which the SMA springs operate before activation, which requires a greater degree of martensite-austenite transformation (hence a greater temperature) to start the motion. In general, the compensated actuator has no intrinsic positions of stable equilibrium so mechanical hard stops are needed to prevent dangerous over deformation in the SMA. One hard stop provides a monostable behaviour for the single SMA, the two hard stops needed for the double SMA produces a bistable behaviour. The characteristic of mono or bistability leads to great operative advantages because the actuator is able to: maintain a specific position even without power; achieve precise and repetitive positioning; facilitate control strategies of the overall system.
