design of binary actuators based on shape memory alloy wires

Proceedings of the ASME 2012 Conference on Smart Materials, Adaptive Structures and Intelligent Systems SMASIS2012 September 19-21, 2012, Stone Mountain, Georgia, USA

DRAFT

SMASIS2012-7964 ANALYTICAL AND NUMERICAL MODELLING OF SHAPE MEMORY ALLOY NEGATOR SPRINGS FOR LONG-STROKE CONSTANT-FORCE ACTUATORS Andrea Spaggiari

Eugenio Dragoni

Dept. of Engineering Sciences and Methods, University of Modena and Reggio Emilia, Italy ABSTRACT This paper explores the merits of shape memory Negator springs as powering elements for solid state actuators. A Negator spring is a spiral spring made of strip of metal wound on the flat with an inherent curvature such that, in repose, each coil wraps tightly on its inner neighbour. The unique characteristic of Negator springs is the nearly-constant force needed to unwind the strip for very large, theoretically infinite deflections. Moreover the flat shape, having a high area over volume ratio, grants improved bandwidth compared to any solution with solid wires or helical springs. The SMA material is modelled as elastic in austenitic range while an exponential continuum law is used to describe the martensitic behaviour. The mathematical model of the mechanical behaviour of SMA Negator springs is provided and their performances as active elements in constant-force, long-stroke actuators are assessed. The SMA Negator spring is also simulated in a commercial finite element software, ABAQUS, and its mechanical behaviour is estimated through FE analyses. The analytical and the numerical prediction are in good agreement, both in martensitic and in austenitic range. INTRODUCTION Shape memory alloys are smart materials successfully used in the field of compact solid-state actuation and microactuation, because of their high power over mass and force over mass ratios [1]. Usually, shape memory actuators are based on SMA wires or helical springs. The use of SMA wires ensures a high actuating force, a minimal amount of active material and good mechanical bandwidth (i.e. operating frequency) of the actuator. By contrast, SMA wires are undermined by poor strokes and considerable length of the overall construction [2]. SMA helical springs have complementary performances with respect to SMA wires: the stroke per unit of length of the actuator is greatly enhanced, but only at the cost of much more active material. Moreover, the electric current needed to supply

1

the SMA springs is much higher than for the wires and the bandwidth is drastically reduced. In addition, a SMA spring used as actuator keeps the linear behaviour of a traditional spring and its stroke is reduced by the back-up element necessary to restore the initial position of the actuator. This linear characteristic does not fit well with the fact that the external load is often constant, which results in reduced stroke and output force of the actuator. The ideal force stroke characteristic of an actuator is constant, in order to increase the actuator efficiency and to avoid over-sizing. In the technical literature there are few purely mechanical devices which exhibit a constant force output, often with complex mechanisms [3]. Adding mechanical complexity would contrast with the extreme handiness of integration of SMA devices. This paper aims at overcoming these drawbacks by envisaging SMA elements in the form of Negator constantforce springs [4]. A Negator spring is a spiral spring made of strip of metal wound on the flat with an inherent curvature such that, in repose, each coil wraps tightly on its inner neighbour. In use the strip is extended with the free end loaded and the inner end supported on a drum, as showed in Figure 1a. The unique characteristic of Negator springs is the nearly-constant force needed to unwind the strip for very large deflections [6]. Moreover the flat shape, having a high area over volume ratio, grants quick cooling times and improved bandwidth compared to any solution with solid wires. The stroke of these springs is virtually infinite and the system is also efficient from the energetic standpoint, compared to other SMA devices, since the heating may occur in the small part of the spring where bending occurs. The merits of SMA Negator springs are explored in this paper through an analytical and finite element (FE) model of their mechanical behaviour. The numerical FE model of the Negator spring is provided both in the austenitic and martensitic range. The three-dimensional finite element model is in good agreement with the analytical prediction and confirms the benefits of the SMA Negator spring as a constant

Copyright © 2012 by ASME

force actuator. The findings of the paper demonstrates that it is possible to obtain a constant force actuator by exploiting the peculiarities of the Negator springs combined with a SMA material. NOMENCLATURE A b E Ea Ema Emb Em F F1 F2 I P Pa Pel Pm R Rel Rext Rint t V x y α εg εmax ν ρ σy

Area of the active portion of the Negator spring Width of the cross section of the Negator spring Elastic modulus of the steel Negator spring Austentic modulus of the SMA material Initial slope of SMA σ-ε martensitic curve Post elastic slope of SMA σ-ε martensitic curve Equivalent martensitic modulus of the SMA Net force in the actuation direction Actuator force: Negator spring in martensitic range Actuator force: Negator spring in austenitic range Moment of inertia of the cross section of the spring External load Force of the Negator spring in austenitic phase Force of the backup element Force of the Negator spring in martensitic phase Natural winding radius of the Negator spring Electrical resistance of the Negator spring External radius of the Negator spring Internal radius of the Negator spring Thickness of the cross section of the Negator spring Volume of the active portion of the Negator spring Horizontal coordinate Vertical coordinate R , shape factor of the Negator spring t Yield strain of SMA martensitic curve Maximum admissible strain for the martensitic SMA Poisson's ratio of the material SMA electrical resistivity Yield stress of SMA martensitic curve

METHOD Applications and modelling of elastic Negator springs Negator springs are elastic elements exploited in constantforce, long-stroke actuators. Moreover the Negator springs may be combined in order to form a rotary actuator as well. According to Votta [4] there are three typical configurations, called, extension member, A-motor and B-Motor, shown in Figure 1. In the first configuration the Negator spring stands by itself, loaded by an external load P (Figure 1a). The second one, called A-motor is a unique strip of metal wound on two bushing with different radii and same curvature (Figure 1b). The third configuration, called B-motor is similar to the Amotor but the curvature on the two bushing is opposite (Figure 1c).

2

(a)

(b)

(c) Figure 1 - Configurations of Negator springs, extension member (a), A-motor (b) and B-motor (c) In the first configuration (Figure 1a) an external load is needed to unwound the spring, while in motor springs (Figure 1b-c), the system is not in equilibrium due to the different radii of the two bushings and thus both configurations act as a rotary actuator. The nomenclature used henceforth is defined in Appendix 1. The fundamental work of Votta [4] takes the extension member (shown in Figure 2) as reference configuration and considers the system as perfectly elastic, with Young modulus E. Equating the external work, Pdx, to the internal bending M d work, needed to straighten the curved strip of length 2

Rd = dx, gives [4]: (1) Ebt 3 2 24 R where R is the natural winding radius of the spring, b is the width, and t the strip thickness. It is possible to improve the model prediction of the experimental force by considering the effect of the Poisson's coefficient obtaining the following expression from [4]:

P

P

(2)

Ebt 3  1   2  24 R 2

Equation (1) and (2) must be corrected because the curvature is different. Thus the two expressions become:

P

Ebt 3 24

 1  2  Rint

2  1 1        Rint Rext  

(3)

2 (4) Ebt 3  1  1 1   2  2      1    24  Rint  Rint Rext     Figure 3b reports the comparison between three experimental curves recorded by the tensile machine (solid lines) and the analytical prediction given by literature model [4] considering both equation (3), represented by a dashed line and equation (4), shown as a dotted line. The error in the prediction of the experimental force is around 10% for the equation (3) and -2% using the correction with the Poisson coefficient of equation (4).

P

(a)

Table 1 - Dimensions and material of the Negator spring used in the experimental tests External radius, Rext

21.75 mm

Internal radius Rint

16.75 mm

Thickness of the strip, t

(b) Figure 2 - Scheme of the loading of Negator spring (a) and cross section of the spring (b) This effect is not due to the shift from the beam to the plate theory otherwise the new expression would contain the term (1ν2) in the denominator. The explanation of this correction which is in good agreement with a lot of experimental tests [4] can be found from the energetic standpoint. The decrease in the external load P is due to a certain amount of energy stored in the Negator spring to produce a camber (a synclastic curvature) in the metal strip. Since this correction is mainly based on experimental observations and has no solid mechanical background this analysis considers equation (1) as the reference formula. An experimental test on a Negator spring used as extension member was carried out in order to verify the Votta's theory. The spring was made in harmonic steel by Technospring [5]. The spring dimensions are reported in Table 1. The experimental set up consists in a drum supported by a couple of miniature bearings connected to the frame of the tensile machine. An electromechanical Galdabini SUN 500 was used to pull off the free end of the spring for a 500mm stroke. The set up of the Negator spring under loading is reported in Figure 3a. The synclastic curvature is clearly visible, as well as the additional degree of freedom given by the lower fixture to prevent any transverse loading of the spring. According to [4] the basic relationship can be improved by considering the complete geometry when the external radius, Rext, is very different from the natural or internal radius Rint. 3

0.39 mm

Width of the strip, b

40 mm

Elastic Modulus, E

195 GPa

Poisson's Ratio, ν

0.33

Modelling of SMA Negator springs Modelling of the SMA material. In order to develop an analytical framework for SMA Negator spring it is necessary to properly describe the SMA behaviour. Applying the crude Votta model could be correct only for initial region of the austenitic regime. In order to model the martensitic behaviour of the SMA a pseudoplastic law must be considered. A simple way to model the martensite behaviour of the SMA is to consider a bilinear law, shown in Figure 4 [7]. Even though the SMA properties in Figure 4 come from testing of a specific wire, the model needs only the correspondent properties of the SMA used (e.g. in thin strips) and it can be immediately applied. The material properties retrieved from Figure 4 are reported in Table 2. In order to describe the experimental points with a smoothest law it is possible to interpolate the experimental points in Figure 4 with an exponential law. By ensuring that the curve starts from the origin with a slope Ema, yields at εg and in the post elastic field has the slope Emb, the following stress-strain function is obtained:



  Ema   g   1  e 



g

   Emb   

(5)

A comparison between the experimental points and the exponential law is shown in Figure 4. According to the technical literature the maximum strain of a SMA material εmax should be under 3.5% in order to guarantee a fatigue life beyond 100.000 cycles. Table 2 - Nitinol material properties retrieved by Saes Getters 0.15 mm wires [7]

(a)

Modulus in the austenitc region, Ea

42000MPa

First martensitic elastic modulus, Em

14000MPa

Martensitic pseudoplastic modulus, Emb

980MPa

Martensitic yield strain εg

0.3%

Martensitic yield stress σy

42MPa

Martensitic maximum strain εmax

3.5%

Equivalent bending modulus of the martensitic cross-section. The beam theory gives the first relationship valid under the Euler-Bernoulli assumption on deformations: "plane sections remain plane and perpendicular to the midplane after deformation". This kinematic equation links the strains to the curvature radius, R: (6) y  R The second useful relationship is the definition of the total bending moment needed to bend a straight beam (or to straighten a curved one) which needs no particular hypothesis and is: t /2 (7) M  2 y    y  b  y  dy 0

So long as the material acts elastically equation (7) can be simplified by using the constitutive law and the classical elastic formula is found: (8) EI Ebh 3  R 12 R where the second expression is valid for the cross-section of the Negator spring reported in Figure 2b. In order to develop an analytical relationship for the bending moment of the Negator spring in martensitic range it is necessary to consider the non linear relation given by equation (5). When the bending stress is not linear equation (7) must be used to calculate the bending moment using the true bending stress distribution. The problem can be handled following the same procedure shown by Feodosiev [8] for an elastoplastic cantilever beam under a pure bending moment.

M

(b) Figure 3 - Experimental test of the steel Negator spring (a), experimental and predicted force displacement curves (b).

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t   Em I b  2 R  g 2 2  12  R  Ema   g  (2 R   g  t )  e  (11) R 12 R    b t 3  Emb  3R  Ema   g 8  R 2   g2  t 2    12 R  From which the equivalent modulus is immediately found: t

2

 2 R  g R Em  Emb  12     Ema   g2  e t

R    2 g   1  t  

(12)

  R R Ema   g  1  8      g2    t t    2

3 

This expression can be simplified considering the nondimensional group α=R/t which is a parameter which depends only on the Negator spring geometry. The other three variables in the equation (Ema , Emb , εg) are given once the SMA material is selected (Figure 4a).

Em  Emb  12    Ema    e 2

2 g



1 2  g

 2

g

   1 

(13)

Figure 4 - Experimental stress-strain curve (black dots, [7]) with bilinear approximation (black line, [7]) and proposed exponential interpolation (gray line) for a commercial SMA.

3    Ema   g 1  8   2   g2 

By combining (7) and (5) we obtain the bending moment of the SMA martensitic strip considering exactly the non linear stress-strain curve proposed in Figure 4:

Equation (6), computed for y=t/2 can be used to assess the limit of the Negator spring in terms of maximum martensitic strain εmax (at the surface of the strip). The limit on εmax, reported in Table 2 would ensure a decent fatigue life for the SMA manufact. The engineering limits of the parameter α are:

M  2

t /2

0

y   ( )  bdy 

(9)       g  2  y  b   Ema   g   1  e   Emb   dy 0         Using the kinematic relationship given by (6) and substituting in (9) the integral is readily solved and the moment sustained by a martensitic curved beam of natural radius R, thickness t, width b, made in SMA material acting as in Figure 4 is: t /2

t   b  2 R  g 2 2 M 12  R  Ema   g  (2 R   g  t )  e  12 R    b t 3  Emb  3R  Ema   g 8  R 2   g2  t 2    12 R 

(10)

This general expression is valid for every SMA elastoplastic with a curvature 1/R provided that the martensitic phase can be fairly interpolated with equation (5). It is now possible to retrieve the equivalent martensitic modulus Em , by comparing equation (10) and (8), in order to exploit the Votta's model as if the material was linear even in the martensitic field. Recalling equation (10) and (8) we have:

5

0.035   max  t

 1    14.3 (14) 2R 2 In order to provide a more explicit link between the equivalent modulus and the SMA material properties, the equation (13) can be normalized with respect to Ema as follows: 1

 Em E 2  g  mb  12   2   g2  e  2 g    1  Ema Ema

(15)

3     g 1  8   2   g2 

The curves for some typical values of the elastic martensitic strain εg are reported in Figure 5 as a function of α. The grey band on the left represents the non admissible value of α, given by equation (14). The curves of Figure 5 are reported for the value of Ema Emb given by Table 2 since this ratio slightly affects the behaviour of the system. Applying the SMA material to the Negator spring model. A SMA Negator spring can be easily used to build an actuator, for example as shown in Figure 6. When the SMA Negator spring is cold (martensitic phase), the backup element, always present in SMA actuators, is able to deform the SMA element causing a displacement in right direction. Figure 6a shows e.g. the backup element as an elastic Negator spring, acting with a force Pel.

When the SMA is heated up, the material becomes austenitic with SMA modulus Ea. The force given by the SMA spring is obtained by applying Ea into equation (1) so the Negator SMA spring is able to win the backup force Pel, making the actuator move in left direction (Fig. 4c). E bt 3 (17) F2  Pa  Pel  a 2  Pel 24 R The maximum available force in the actuation direction, F, for a SMA spring of the above described material is obtained upon addition between equations (17) and (16) bt F (18)  Ea  Em  24 2

Finite Element modelling of Negator springs (a) Figure 5 - Relationships between the admissible values of α and the normalized equivalent Young's modulus ( Em / Ema) for typical values of the elastic martensitic strain (g) The force exerted by a SMA Negator spring can be obtained by combining (1) and (12) and thus the force F1 is:

F1  Pel  Pm  Pel 

Em bt 3 24 R 2

(16)

(a)

(b) Figure 6 - Possible configuration of the SMA Negator spring actuator in cold (a) and hot (b) state. 6

Elastic Negator springs. This section describes the numerical finite element modelling of an elastic Negator spring. The FE modelling of a Negator spring is a challenging problem, mainly due to multiple self contacts in the model. The Negator spring has normally multiple wounds which slide only a little, because they are tightly wound and the rotation of the internal drum allows the extension of the spring. Thus in the FE model the contact is defined as frictionless. Since there are no manufacturers of SMA Negator springs, (1) chance to provide an experimental test on the so far there is no real system. However, the FE model can be verified in the elastic range against the experimental tests of Figure 2. The finite element commercial software used is Abaqus 6.10-2 [13]. The model consists of a unique shell, as reported in Figure 7a which is wound with the natural radius R, reported in Table 1. In order to avoid locking of the nodes only two turns are modeled with linear shell elements with reduced integration (S4R). The shell elements are supported by a drum, modeled as perfectly rigid with an analytically rigid feature. The shell are square elements with a side dimension of 2mm, small enough to be coupled with the analytical rigid surface and to obtain a reasonably fast model (around 10000 d.o.f. in the model, less than 20 minutes to be run). The FE model represents only half of the Negator spring due to the longitudinal symmetry plane and a prescribed displacement up to 100 mm was applied to the free end of the spring. Since only two wounds of the Negator spring are represented it is not possible to extend the Negator spring more than this. The only boundary condition applied is due to the symmetry of the system, while the rigid drum on which the Negator spring slides is fixed. Even though in the experimental tests and in the typical extension member configuration the drum rotates along its axis it is not possible to do that in the FE analyses, since a rigid body movement produces numerical instability. The same condition given by the bearings is enforced by using frictionless contact both between the drum and the spring and between the self contact of the wounds. The force displacement curves of the FE elastic analyses, not

reported here for the sake of brevity showed a fair agreement of the FE model with the experimental test. The error is around 10%, and it is consistent with the load predicted by equation (3). Shape memory Negator springs. The model developed for the elastic Negator spring is used for the SMA Negator spring, by changing the constitutive law only. The material curve used in FE exactly reproduces the smooth law of Figure 4 and it was implemented with a hyperelastic law following the Marlow model (uniaxial test data), as already done in [14]. The hyperalsticity helps in overcoming numerical problems due to plasticization of the material, which occurred with the first plastic model (DruckerPrager) initially used. The limit of such approach is that it is suitable in case of monotonic analyses but cannot be recommended of loading and unloading of SMA elements. The application of potential based thermomechanical models [15],[16], is not considered here because those models are too complex for the purpose of this analysis. The results of the FE analyses are reported in Figure 7b-cd, showing the entire geometry by mirroring the results about the symmetry plane. Figure 7b shows the stress contour of the equivalent stress in the Negator spring in the austenitic phase. Figure 7c shows the stress contour of the equivalent stress in the Negator spring in the martensitic phase and Figure 7d shows the plot of the displacement in load direction for the martensitic range. The finite element models can be compared only with the analytical models developed in the previous section, since there are no experimental prototypes of a SMA Negator spring available. The force displacement curves of the FE models are reported in Figure 8 and compared with the force given by equation (1) using Ea or Em respectively. Since the turns are

(a)

(b)

designed in the FE model with the same diameter, equation (1) and not equation (3) must be considered. GENERAL DISCUSSION The merits and the drawback of the SMA Negator spring are described in the following sections, discussing both the analytical and finite element modelling techniques adopted. Analytical modelling of the SMA Negator spring As shown is Figure 5 the non-dimensional parameter α strongly affects the behaviour of the SMA Negator spring. The typical values of α according to steel Negator springs manufacturers are around 100, mainly because the stresses becomes quite high even at low deformations considering the very high modulus of the steel. The same problem may occur in SMA based Negator spring since the deformations are comparable. The choice of α is crucial in developing a Negator spring able to work properly. A first limit is given by equation (14), which guarantees that the admissible strain in martensitic range, εmax, is not overcome. Moreover there are some manufacturing limits in wounding the Negator spring when the thickness becomes too high. 7

(c)

(d) Figure 7 - Finite element model of the Negator spring (a). Contour of the austenitic equivalent stress (b), contour of the martensitic equivalent stress (c) and plot of the martensitic displacement (d) in mm.

It can be noticed from equation (1) for the elastic range and in equation (10) for the SMA spring that the width b acts simply as a gain parameter. Since the width, b, does not affect the spring behaviour, the width can be used to increase the force of the Negator spring without affecting neither the stress state nor the cooling time. Finite element modelling of the SMA Negator spring The FE model developed gives a good assessment of the Negator spring behaviour. Both the austenitic prediction (Figure 7b) and martensitic one (Figure 7c-d) are in good agreement with the theory in both cases. The prediction is consistent of the FE model is consistent with the curve of Figure 8 in which the error are around 5%. Even though the self contact between the turns prevented the complete description of the Negator spring the FE approach is able to provide a reliable prediction of the spring behaviour. Merits of the SMA Negator spring The mechanical characteristic of this actuator exhibits two great advantages compared with typical SMA actuators. The first one is that the stroke can be very long, as it depends only on the physical length of the spring. The stroke length is one of the main issues in SMA actuators, especially when wires are considered. Negator springs made in SMA have the unique property to provide very long strokes which are not dependent on the output force. The second mechanical asset is that output force is constant, the best condition for an actuator. Unless elastic compensation system are considered [7], the typical mechanical characteristic of a SMA actuator is linear regardless the backup element, as shown in [9]. The linear force-stroke characteristic often leads to an over sizing of the actuator in order to guarantee the desired force over the entire travel. There are also relevant electro-thermo-mechanical advantages in this particular shape. First of all the SMA Negator spring can be heated only in a small region, because as shown by Figure 2a the material is deformed by the bending moment only when it comes out from the drum. After that stage, it does not produce any additional force and thus it is not necessary to guarantee the heating of the complete Negator spring. Many of the advantages are reported in a patent [10] which describes several possible architectures with multiple SMA Negator spring coupled together. It is of course possible to consider electrical heating in order to reach the transformation temperature and also in this case the Negator spring is quite performing compared to traditional SMA elements. One of the issues of conventional SMA springs is the low resistance and thus the high current (typically 1-2 Amps) needed to heat them up, especially when high forces are required and the wire diameter is large. In case of Negator springs the main issue is to prevent direct contact between the turns, which is possible by applying an insulator in between. Once the contact impedance can be considered infinite it is quite easy to increase the spring electrical resistance by changing its geometrical parameters. 8

Figure 8 - Analytical (solid lines) finite element (dashed lines) force-displacement curves of the SMA Negator spring for austenitic and martensitic states. For example small thickness and high radii (i.e. high α) a high number of turns and a small width of the SMA strip lead to an increase of the electrical resistance. An approximated formula to calculate Rel is: R2  R2 (19) Rel     ext 2 int . bt where ρ is the SMA resistivity (either austenitic or martensitic range). Another important aspect to consider in developing new shapes for SMA actuator is the cooling time. The cooling time for SMA alloys limits their bandwidth and it is mainly related to the surface to volume ratio. SMA wires are faster than SMA helical springs mainly because their high surface to volume ratio. SMA Wave springs [12] have better activation frequencies than helical springs because of the thin strip of SMA used to build them. A similar thin strip is used in case of Negator springs and, in addition, only a little portion of the material is heated up, leading to faster cooling of the device. The area over volume ratio of the active portion of the SMA Negator spring is immediately found: A 2  b  R  2   . (20) V b  R   t t Since the area over volume ratio depends only on the thickness it is possible to obtain the desired value of cooling time by acting on t, while the desired output force can be obtained by increasing the width b. Helical springs has an area over volume ratio which is in inverse proportion with the wire diameter [12] and thus the cooling time is coupled with the

output force, the higher the force the lower the actuation frequency. Unfortunately there are also two drawbacks for SMA Negator springs. The first one is the complexity of manufacturing, which is the main reason to the absence of such a commercial device. The second one is the high stress level in Negator springs. This could be an important issue for the SMAs which suffer of thermo mechanical fatigue and the S/N curve is not established yet. The main outcomes of this work are the analytical relationship which gives the behaviour of a SMA Negator spring as a function of the spring geometry (15). This equation can be used to design efficient SMA Negator actuators, which provides both constant force and very long strokes. CONCLUSION This paper shows the merits of the Negator spring geometry applied to the field of shape memory actuators. When compared to the traditional SMA elements, like wires or helical springs the Negator geometry exhibits a great advantage: a nearly constant force displacement characteristic. Moreover the flat shape leads to lower cooling times because of the high area to volume ratio and the electrical characteristic are very good as well. The mechanical behaviour of the SMA Negator spring is modelled both analytically and by means of a finite elements software. Analytical relationships which give the behaviour of a SMA Negator spring as a function of the spring geometry are retrieved from the mathematical model and a non dimensional parameter which rules the system performance is found. The FE model implemented the martensitic pseudo-plastic behaviour using an exponential law based on experimental points. The prediction given by FE and the analytical models are in good agreement both in the austenitic and martensitic range. Future works are aimed at verifying experimentally the accuracy of the proposed model and at measuring the SMA Negator mechanical and thermal performances. REFERENCES [1] Funakubo, H., 1986, "Shape Memory Alloys", Gordon and Breach Science Publishers, New York,. [2] Nespoli et al., 2010, "The high potential of shape memory alloys in developing miniature mechanical devices: A review on shape memory alloy miniactuators", Sens. Actuators, A, 158(1), pp. 149-160. [3] Jenuwin, J.G. and Midha, A., 1994, " Synthesis of Single-Input and Multiple-Output Port Mechanisms with Springs for Specified Energy Absorption" J. Mech. Des. 116(3), pp. 937-943. [4] Votta, F.A. Jr. and Lansdale, P.A., 1952, "The theory and design of Long Deflection Constant Force Spring Elements", Transactions of the ASME, 74, pp. 439-50. [5] Technospring Italia Via Puccini, 2 -21010 Besnate (VA) http://www.technosprings.com/en/index.html [6] Wahl, A.M., 1963, "Mechanical Springs", Second edition, McGraw-Hill, New York.

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[7] Scirè Mammano. G. and Dragoni, E., 2011, "Increasing stroke and output force of linear shape memory actuators by elastic compensation", Mechatronics 21(3), pp. 570580. [8] Feodosiev, V.I., 1977, "Selected Problems and Questions in Strength of Materials", MIR Publishers, Moscow. [9] Spaggiari, A., Spinella, I. and Dragoni, E., "Design equations for binary shape memory actuators under arbitrary external forces", J. Intell. Mater. Syst. Struct. (In press) DOI: 10.1177/1045389X12444491 [10] Weems, W. "Constant Force Spring Actuator", International Patent WO99/61792 (Lockheed Martin Corp.). [11] Dynalloy FLEXINOL ® Technical Specifications http://www.dynalloy.com/TechDataWire.php, 14762 Bentley Circle Tustin, CA 92780, USA. [12] Spaggiari, A. and Dragoni, E., 2011, "Multiphysics Modeling and Design of Shape Memory Alloy Wave Springs as Linear Actuators", J. Mech. Des. 133(6) 061008 (8 pages). [13] Abaqus 6.9, Reference book - SIMULIA. 2011. [14] Scirè Mammano, G. and Dragoni, E., 2011, "Modeling of Wire-on-Drum Shape Memory Actuators for Linear and Rotary Motion", J. Intell. Mater. Syst. Struct. 22, pp. 1129-1140. [15] Boyd, J.G. and Lagoudas, D.C., 1996, "A Thermodynamical Constitutive Model for Shape Memory Materials. Part I", Int. J. Plast 12(6), pp. 805-842. [16] Arghavania, J., Auricchio, F., Naghdabadia, R. and Sohrabpoura, S., 2009, "A 3-D phenomenological constitutive model for shape memory alloys under multiaxial loadings", Int. J. Plast. 26(7), pp. 976-991.