jmd 2011 spa drago

Andrea Spaggiarie-mail: [email protected]

Eugenio Dragonie-mail: [email protected]

Department of Engineering Sciences and Methods,

University of Modena and Reggio Emilia,

42122 Reggio Emilia, Italy

Multiphysics Modeling andDesign of Shape Memory AlloyWave Springs as LinearActuatorsThis paper explores the merits of shape memory wave springs as powering elements ofsolid-state actuators. Advantages and disadvantages of the wave construction in compar-ison to the traditional helical shape are presented and discussed by means of dimension-less functions. The main assets of the wave springs are the higher electrical resistance(leading to simpler electrical drives) and the lower cooling time (leading to enhancedworking frequency). The wave geometry is also superior in purely mechanical terms tothe helical counterpart when axial space is at a premium. A step-by-step design proce-dure is proposed, leading to the optimal wave spring meeting the multiphysics designspecifications and constraints. A case study is finally reported, showing the application ofshape memory wave springs to the design of a telescopic linear actuator.[DOI: 10.1115/1.4004196]

Keywords: spring design, shape memory alloys, wave spring

1 Introduction

Shape memory alloys (SMAs) can be exploited successfully tobuild solid-state actuators of reduced complexity and lowerweight than conventional competitors. Many SMA actuators usethe material in shape of easily made straight wires or helicalsprings [1–4]. Despite the remarkable force they can develop, ten-sion wires are undermined by very limited stokes, while tradi-tional helical springs, although capable of greater strokes, sufferfrom modest output force [5], poor energetic efficiency [6,7], andlow mechanical bandwidth due to their high cooling times.

This paper explores the merits of SMA wave springs as a meansfor enhancing the mechanical, thermal, and electrical performan-ces of SMA actuators. Fabricated by winding an undulated strip inmultiple, closely packed coils (see Fig. 1) [8], wave springs repre-sent a valid alternative to more traditional springs (e.g., helical orBelleville) to provide high stiffness under substantial loads espe-cially when the axial dimensional constraints are challenging.

The merits of shape memory actuators exploiting the wavespring geometry are still unexplored. The research reported in thispaper was performed to fill this gap, providing an analytical com-parison between the wave spring and the traditional helical spring,used as benchmark. The main advantages in the context of SMAactuators are the lower cooling time of the wave springs due to thehigh area to volume ratio and the higher electrical resistance. Thepaper develops an analytical model for the multiphysics behaviorof SMA wave springs. The constitutive equations provided areexploited to compare the performances of wave springs with con-ventional helical springs. The comparison is made assuming thatthe two springs are made from the same material, have the samemean diameter, have the same outer diameter, receive the samemaximum force, undergo the same maximum deflection (henceexhibit the same spring rate), and develop the same maximumstress.

The analytical comparison between wave and helical springshows that the wave spring outperforms the traditional solutionterms of reduced cooling time and improved electrical resistance,and due to the absence of dead turns can be exploited when theaxial dimensional constraints are tight. In particular, the reducedcooling time can lead to working frequencies more than four timeshigher than helical springs.

The paper also describes a step-by-step design procedure show-ing how the former equations can be used to identify the waveSMA spring that best meets the design specifications (force,deflection, heating and cooling time, power needs, strength condi-tion) for an actuator spring. The design parameters for the wavespring are section width and thickness, mean diameter, number ofwaves, and number of coils. Application of the design procedureis exemplified for a case study.

For all these reasons, the wave spring geometry can be consid-ered a valid technological alternative for the design of SMAactuators with increased dynamic bandwidth and simple electroniccontrol, especially in case of limited axial height and highstiffness.

2 Materials and Methods

2.1 Design of the SMA Springs. The wave spring consid-ered in the analysis is a crest to crest wave spring with multipleturns and no shim ends (Fig. 1(b)). This wave spring can be con-veniently described using the design variables shown in Fig. 2.

The main design variables are the mean diameter, D, the radialwidth of the cross section, b, the thickness of the cross section, t,the number of waves per turn, N, and the number of turns, Z.

2.1.1 Geometric Properties of the Wave Spring. The geomet-ric properties of the wave spring can be easily written as functionsof the main design variables. The area, Sw, and the perimeter, Pw,of the spring cross section are given by

Sw ¼ bt (1)

Pw ¼ 2bþ 2t (2)

By disregarding the wavy shape of the spring turns, the length ofthe metal strip, Lw, can be calculated as

Manuscript received October 7, 2010; final manuscript received May 2, 2011;published online June 16, 2011. Assoc. Editor: Nancy Johnson. Paper presented atthe ASME 2010 Conference on Smart Materials, Adaptive Structures and IntelligentSystems (SMASIS2010), Philadelphia, Pennsylvania, USA, September 28–October1, 2010. Paper No. SMASIS2010-3711.

Journal of Mechanical Design JUNE 2011, Vol. 133 / 061008-1Copyright VC 2011 by ASME

Downloaded 16 Jun 2011 to 155.185.228.237. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm

a

Lw ¼ pDZ (3)

while the exact length is about 5% more than the simple approxi-mation (3) proposed.

Using Eqs. (1)–(3), the lateral area, Aw ¼ PwLw, and the vol-ume, Vw ¼ SwLw, of the spring can be written as

Aw ¼ 2ðbþ tÞpDZ (4)

Vw ¼ btpDZ (5)

The free height of the wave spring depends on the manufactur-ing process and can be assessed by calculating the angle bbetween the spring wire and a plane normal to the spring axis.The height of a single turn is the minor cathetus of a triangle withbasis pD=2N and thus the whole free height hw can be calculatedas

hw ¼ ZpD

2NtanðbÞ (6)

Taking b � 12� produces values of the free height in agreementwith the data reported by the manufacturer for stock springs.

Even though the manufacturer suggests not to use the wavespring over 60% of the free height, for the sake of completenessthe ideal solid height is given by

Hw ¼ Zbt (7)

2.1.2 Mechanical Properties of the Wave Spring. As shownby Dragoni [9] the wave spring can be modeled as a flat circular ringloaded normally to its plane. The model considers axial displacementand torsional rotation of the spring cross section as contributions tothe total deflection. Both contributions are calculated taking intoaccount bending and torsional moments on the spring wire.

For the multiple-turn springs considered here, the contactbetween wave crest of adjacent turns precludes the rotation of thecross section and limits the effects of the torsional moment. Forthe purpose of this work, only the bending moment due to thethrust load P is taken into account.

Under this assumption, from Ref. [9] the maximum bendingstress has the following expression:

rmax ¼3PD

2bt2Ntan

p2N

� �(8)

This expression is more accurate than that provided by Ref. [8],because it takes into account the effect of the curvature on thebending moment.

The deflection formula proposed by Ref. [9] was developed forclosed-ring (stamped), single-turn wave springs. For the open-ring(wound), multiple-turn springs examined here, Smalley’s providesthe following expression for the deflection, containing the empiricalcoefficient, Ks, specifically determined for this particular geometry:

f ¼ ZKsPD3

Ebt3N4

Di

Do(9)

The value of Ks depends on the number of waves N and is givenby Ref. [8] in tabular form (Ks¼ 3.88 for 2.0�N� 4.0; Ks¼ 2.90for 4.5�N� 6.5; Ks¼ 2.30 for 7.0�N� 9.5; Ks¼ 2.13 forN� 10). For convenience of mathematical development, the fol-lowing continuous interpolation, Ka, will be used to replace Ks:

Ka ¼3 arctanð12� 2NÞ þ 15

5(10)

Letting Di ¼ D� b, Do ¼ Dþ b, and using Ref. [10], Eq. (9)becomes

f ¼ ZPD3

Ebt3N4

D� b

Dþ b� Ka (11)

2.1.3 Electrical Properties of the Wave Spring. The electricalresistance of the wave spring depends on the electrical behavior ofthe contact points between the crests of the waves. Two idealcases may be considered, with the contacts behaving as open cir-cuits (infinite electrical resistance) or as a short circuit (zero elec-trical resistance).

Under the first assumption (open circuit), the resistance of thespring coincides with the resistance of the wire ROC ¼ qLw=Sw.Using Eqs. (1) and (3), the open-circuit resistance becomes

ROC ¼ qpDZ

bt(12)

The second assumption considers perfect contact between thecrest of the wave and so we can split the wave spring in elementsof length pD=2N.

The whole resistance is decreased by a factor 2N due to the par-allel connection in the single ring and increased by a factor Z due

Fig. 1 Examples of multiturn, crest-to-crest VR wave springs [8]: with end shims (a)and without end shims (b)

Fig. 2 Schematic drawing of the wave spring showing primarydesign variables

061008-2 / Vol. 133, JUNE 2011 Transactions of the ASME


a

to the series connection between the crests. Thus, the total short-circuit resistance can be written as

RSC ¼ qpD

2Nbt� Z

2N(13)

2.1.4 Thermal Properties of the Wave Spring. If only convec-tive cooling is considered, following Ref. [6] the cooling time,twc, of a body made of shape memory alloy with volume Vw, den-sity n, and external area Aw is given by

twc ¼nVw

Aw

� �cAM

hln

Ms � Tr

Mf � Tr

� �(14)

where cAM is a fictitious transformation specific heat that includesthe enthalpic contribution, h is the coefficient of convection, Ms

and Mf are the martensite start temperature and the martensite fin-ish temperature, respectively, and Tr is the ambient temperature.The martensite start temperature, Ms, appears in Eq. (14) becauseit is assumed that after the achievement of the austenite finish tem-perature, Af, the hysteresis of the shape memory alloy is exploitedto reduce the cooling time and the actuation energy by cooling theSMA down to Ms. In this way, the SMA remains austenitic at alower temperature. The fictitious specific heat, cAM, in Eq. (14) isdefined as

cAM ¼cA þ cM

2þ XAM

Ms �Mf(15)

where cA and cM are the austenitic and the martensitic specificheats, respectively, and XAM is the transformation enthalpy fromaustenite to martensite.

Recalling Eqs. (4) and (5), Eq. (14) becomes

twc ¼bt

2ðbþ tÞ ncAM

hln

Ms � Tr

Mf � Tr

� �(16)

2.2 Comparison Between Wave Spring and HelicalSpring.

2.2.1 Definition of the Problem. This section compares themechanical and physical properties of the wave spring with thecorresponding properties of a traditional helical springs made ofthe same shape memory material. The primary design variables ofthe helical spring are the section diameter, d, the mean coil diame-ter, D, and the number of coils, n.

The comparison is made in terms of the properties calculated inSec. 2.1 and assumes that wave and helical springs:

(a) are made of the same material(b) receive the same force(c) undergo the same deflection for given force(d) develop the same maximum stress for given force(e) have the same mean diameter(f) have the same outer diameter.

Conditions (b) and (c) imply also the same spring rate for bothsprings, while conditions (e) and (f) imply that the wire diameterof the helical spring is equal to the radial wall of the wave spring,

b ¼ d (17)

Under these conditions, analytical relationships between the pri-mary design variables of the two springs can be found (see thefollowing).

The textbook equations (Ref. [10]) for the stress, sh, and thedeflection, fh, of the helical spring are

sh ¼8PD3

pd3Kb (18)

fh ¼8nPD3

Gd4(19)

The curvature correction factor, Kb, used in Eq. (18) is due toBergstrasser [10] and is defined as

Kb ¼4Cþ 2

4C� 3(20)

where C is the spring index,

C ¼ D

d¼ D

b(21)

By reasoning as for the wave spring, the geometric, electrical, andthermal properties of the helical spring can be expressed as listedin the following. Whenever applicable, the equations are devel-oped neglecting the dead end turns of the spring.

Area of the cross section,

Sh ¼pd2

4(22)

Perimeter of the cross section,

Ph ¼ pd (23)

Length of the helical spring,

Lh ¼ pDn (24)

Combing Eqs. (23) and (24) the lateral area of the helical spring is

Ah ¼ p2dDn (25)

Volume of the helical spring, recalling Eqs. (22) and (24),

Vh ¼p2d2Dn

4(26)

Free height,

hh ¼ pDn tanðbÞ (27)

Solid height,

Hh ¼ dn (28)

Electrical resistance,

Rh ¼ q4Dn

d2(29)

Cooling time, obtained using Eqs. (14), (25), and (26),

thc ¼nd

4

� �cAM

hln

Ms � Tr

Mf � Tr

� �(30)

2.2.2 Relationships Between Primary Variables. The firstconstraint to be enforced is the equivalence of the maximumstresses. As helical springs undergo torsional deformation andshear stresses while wave springs are mainly subjected to bending,which produces normal stresses, the Von Mises criterion is chosento equalize the maximum stresses,

smax

ffiffiffi3p¼ rmax (31)

Using Eqs. (8), (18), and (20), the stress condition (31) can becast as

Journal of Mechanical Design JUNE 2011, Vol. 133 / 061008-3


a

Nt2

d2¼ 4C� 3

4Cþ 2

pffiffiffi3p

16tan

p2N

� �(32)

or, alternatively,

t

d¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4C� 3

4Cþ 2

pffiffiffi3p

16Ntan

p2N

� �s(33)

Condition (c), mentioned earlier, of equal deflection is enforcedby equating Eqs. (11) and (19),

8nPD3

Gd4¼ Z

PD3

Ebt3N4

D� b

Dþ b� Ka (34)

Using Eq. (17) and letting G ¼ 0:5E=ð1þ �Þ, Eq. (34) becomes

N3

Ka

n

Z

Nt2

d2

t

d¼ 1

16ð1þ �ÞC� 1

Cþ 1(35)

Recovering the term Nt2/d2 from Eq. (32) and recalling Eq. (10),Eq. (35) gives

Z

n¼ t

d

5pN3ð1þ �Þffiffiffi3p

3 arctanð12� 2NÞ þ 15� 4C� 3

4Cþ 2

Cþ 1

C� 1tan

p2N

� �(36)

Equations (33) and (36) ensure the mechanical equivalencerequested by the above-presented conditions (a)–(f) and representthe key tools for comparison between the two springs. Once thevalues for N and C are decided, from Eq. (21) the dimensionlessthickness t/d is calculated. Next, with the known values for N, C,and t/d, Eq. (36) gives the ratio Z/n.

Since the spring index normally varies between 5 and 12, theeffect of this variable on the results is quite weak and can be disre-garded (see Sec. 3). The following calculations are based on theaverage spring index C¼ 8, leaving the number of waves N as theonly independent variable on which the comparison is built. Basedon this assumption, Fig. 3 reports the dimensionless ratio Z/n and(for the sake of clarity) the inverse of ratio t/d as functions of N.

Due to manufacture restrictions, N can assume only integer orhalf-integer values. Normally, the wave spring producers offersprings with N in the range from 2.5 to 5.5 when there are multiple

turns, but since the single-turn wave springs can have more waves,the normalized diagrams in the following are reported for a widerrange of N.

Starting from the two dimensionless ratios in Fig. 3, it is possi-ble to normalize the properties of the wave with respect to the cor-responding properties of the helical spring. This approach makesthe comparison very straightforward and compact.

2.2.3 Normalized Properties of the Wave Spring. The nor-malized properties of the wave spring are presented normalizedwith respect to the properties of the helical spring and the genericnormalized property X is defined as the ratio X� ¼ Xw=Xh betweenthe wave and the helical properties.

The first property of interest is the normalized mass, expressedas a ratio between the mass of the wave spring over the mass ofthe equivalent helical spring m� ¼ mw=mh. Since the material isthe same, the mass ratio equals the volume ratio m� ¼ Vw=Vh,from which, using Eqs. (5) and (26),

m� ¼ 4

pt

d

Z

n(37)

The normalized free height, h� ¼ hw=hh, is calculated fromEqs. (6) and (27). Assuming a¼ 6� and b¼ 12�, as typical ofcommercial helical and wave spring, respectively, h* becomes

h� ¼ 2:02 � Zn

1

2N(38)

By dividing Eqs. (7) and (28), the normalized solid height,H� ¼ Hw=Hh, can be expressed as

H� ¼ t

d

Z

n(39)

Concerning the electrical resistance of the wave spring, it is neces-sary to distinguish between the two limits of open-circuit andclose-circuit models. From Eqs. (12) and (29), the normalizedopen-circuit resistance R�OC ¼ ROC=Rh is given by

R�OC ¼4

pd

t

Z

n(40)

Likewise, from Eqs. (13) and (29) the short-circuit resistanceR�SC ¼ RSC=Rh takes the following form:

R�SC ¼p

16N2

Z

n

d

t(41)

Finally, using and Eqs. (16) and (30), the cooling time t� ¼ tw=thcan be written as

t�c ¼2

1þ d

t

(42)

It is seen that all the normalized properties depend on one or moreof the three variables N, t/d, and Z/n. Using Fig. 3 to calculate t/dand Z/n for any given N, the normalized properties can be cast asfunctions of the only variable N. Figure 4 shows the charts for theentire set of normalized functions, plotted against N.

The precise numerical values of the dimensionless ratios in Fig. 3and of the normalized properties in Fig. 4 are collected in Table 1for the range of number of waves N considered.

2.3 Design Procedure for SMA Wave Springs. This sectiondescribes the design procedure to be followed in order to optimizethe design of a SMA wave spring.

The input parameters of the procedure are the following:Fig. 3 Dependence of dimensionless geometrical properties Z/nand d/t on the number of waves N



a

P¼ loadf¼ deflectionDo¼maximum outer diameterDi¼minimum inner diameterhw¼maximum axial heightrmax¼maximum stress in the materialE¼Young’s modulus of the materialtwc¼maximum cooling time.

The design procedure leads to the determination of the mean di-ameter, D, the cross-section width, b, the cross-section thickness,t, the number of waves, N, and the number of turns, Z.

2.3.1 Step 1. Geometry Definition. The first step of the designprocedure involves the determination of the geometry of the stripused for the wave spring.

The available volume should be optimized by maximizing thegeometrical variables; thus we have

b ¼ Do � Di

2(43)

D ¼ Do þ Di

2(44)

Recalling expression (11) for the deflection, the number of turnscan be expressed as

Z ¼ fEbt3N4

KaPD3

Dþ b

D� b(45)

Equation (14) can be rewritten letting x ¼ n cAM=hð Þln Ms � Trð Þ=Mf � Tr

� �, a parameter that depends only on the

boundary conditions such as the material and the roomtemperature,

twc ¼bt

2ðbþ tÞx (46)

Equation (46) is used to calculate the maximum thickness of thecross section of the wave spring needed to meet the thermalconstraint,

t ¼ 2btwc

bx� 2twc(47)

2.3.2 Step 2. Waves and Turns, First Attempt. The secondstep allows the designer to assess the two peculiar parameters of awave spring, i.e., the number of turns, Z, and the number ofwaves, N.

By combining Eqs. (47) and (8) we found an expression whichis dependent only on the number of waves,

N

tan p=2Nð Þ ¼3PD

8b3t2wc

ðbx� 2twcÞ2

rmax

(48)

Due to the constraint on N, which must be an integer or half-inte-ger, it is possible to approximate tan(p/2N) � p/2N, thus obtainingfrom Eq. (48) an explicit expression for N,

N ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3pPD

16b3t2wc

ðbx� 2twcÞ2

rmax

s(49)

Once the number of turns N is known, recalling Eqs. (10) and(47) the number of turns can be calculated from Eq. (45),

Z ¼ f9p2PE

16tbKaD

1

r2max

Dþ b

D� b(50)

2.3.3 Step 3. Verification. The number of turns is obtained byrounding the result from Eq. (50) to the nearest integer. Thedesign parameters should be verified after the procedure due tothe approximation given by the rounding of the two discrete varia-bles N and Z. This approximation, which allows N to be explicitlycalculated, may lead to violation of both the constraint on maxi-mum stress and desired deflection.

The design procedure must be iterated increasing N and Z untilthe design parameters do not respect the constraints and the finalwave spring geometry is obtained.

By way of example, the step-by-step procedure is applied inSec. 2.4 to the design of the powering wave springs of a linear tel-escopic actuator.

2.4 Case Study: Telescopic Actuator. This case study con-siders the application of SMA wave springs to a telescopic actua-tor previously presented by Spinella et al. [11]. The wave springs(Fig. 5) replace the hollow helical springs [12,13] used in the for-mer concept of the actuator. The telescopic architecture of the ac-tuator is exploited to enhance the stroke of the wave springs usinga reasonable amount of turns.

The actuator has three concentric stages, each powered by twoequal antagonistic wave springs. Each spring acts as the backupelement for the other spring in the same stage so as to allow sym-metric bidirectional motion of the actuator. The design constraintsfor the new actuator are reported in Table 2 for an actuator with15 mm of total stroke and 10 N of maximum payload F.

Figure 5(a) shows the telescopic actuator in closed position,while Fig. 5(b) shows the actuator in the uppermost position, with

Fig. 4 Dependence of normalized spring wave properties onthe number of waves, N

Table 1 Values of the dimensionless ratios and normalizedproperties of the wave spring for several number of waves N(reference equations in boldface)

N td

Zn m� H� h� R�OC R�SC t�c

2 0.38 6.11 2.96 2.33 3.09 0.79 12.6 0.552.5 0.29 6.63 2.45 1.92 2.68 0.72 17.9 0.453 0.24 7.43 2.24 1.76 2.51 0.69 24.7 0.383.5 0.20 8.37 2.13 1.67 2.42 0.67 32.9 0.334 0.17 9.39 2.07 1.63 2.37 0.66 42.5 0.304.5 0.15 10.51 2.05 1.61 2.36 0.67 53.9 0.275 0.14 11.81 2.06 1.62 2.39 0.68 67.5 0.245.5 0.12 13.59 2.15 1.69 2.50 0.71 85.7 0.226 0.11 17.03 2.47 1.94 2.87 0.82 117.5 0.206.5 0.10 21.78 2.91 2.28 3.39 0.97 163.1 0.197 0.10 25.29 3.13 2.46 3.65 1.04 204.2 0.18Equation (33) (36) (37) (39) (38) (40) (41) (42)



a

the active (extended) springs on the bottom which push up theinactive (compressed) springs on the top.

Before applying the step-by-step procedure described in Sec. 3,some considerations about the peculiar architecture must bedrawn.

The SMA elements work as antagonistic springs, so from a me-chanical consideration the precompression, p, needed to meet thespecifics must be retrieved. Letting KH be the wave spring stiff-ness in the hot (austenitic) phase, KC be the wave spring stiffnessin the cold (martensitic) phase, and considering two identicalsprings yields

KH

KC¼ EA

EM¼ 3 (51)

KHðp� sÞ ¼ KCðpþ sÞ (52)

ðKH � KCÞp ¼ F (53)

By combining Eqs. (51)–(53) the stiffness of a single stage can befound, so KC ¼ 1:25, p ¼ 4 mm.

Thus, considering the SMA wave spring in martensitic phasethe maximum deflection is f ¼ pþ 0:5s ¼ 6 mm and the maxi-mum load is F ¼ 7:5 N, the two input data for the step-by-stepprocedure.

2.4.1 Step 1. Recalling Eqs. (43), (44), and (46) and takingn ¼ 6450 kgm�3, cAM ¼ 230 J K�1 kg�1, a moderate air flowh ¼ 12 Wm�2 K, and a room temperature of 20 �C (Ref. [6]), x is

equal to 148 s mm�1, thus the dimensions of wave spring strip canbe retrieved,

b ¼ Do � Di

2¼ 30� 20

2¼ 5 mm (54)

D ¼ Do þ Di

2¼ 30þ 20

2¼ 25 mm (55)

t ¼ 2btwc

bx� 2twc¼ 0:21 mm (56)

2.4.2 Step 2. The geometry of step 1 is used in combinationwith the material values in the martensite state in order to obtainthe unrounded number of waves and turns by means of Eqs. (10),(49), and (50),

N ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3pFD

16b3t2wc

ðbx� 2twcÞ2

rmax

s

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3p � 7:5 � 25

16 � 53152� 5 � 148� 2 � 15ð Þ2

110

s¼ 4:24 (57)

Z ¼ f9p2FE

16tbKaD

1

r2max

Dþ b

D� b

¼ 6 � 9p27:5 � 28; 000

16 � 0:21 � 5 � 3:77 � 25� 1

1102� 25þ 5

25� 5¼ 10:9 (58)

2.4.3 Step 3. The values calculated in step 2 have to berounded because N can be an integer or half an integer, while thenumber of turns, Z, must be an integer. So N¼ 4 and Z¼ 11 areconsidered as the first attempt values.

So the nearest values are taken and a verification of the con-straints is performed.

Equation (8) gives

rmax ¼3FD

2bt2Ntan

p2N

� �¼ 3 � 7:5 � 25

2 � 5 � 0:212 � 4 tanp

2 � 4� �

¼ 131:1MPa

(59)

Being higher than the maximum stress, the number of waves mustbe increased. With N¼ 4.5 the stress condition is satisfiedrmax ¼ 102:4 MPa, while Eq. (9) is used to verify the deflection

f ¼ ZFD3

Ebt3N4

D� b

Dþ b� Ka

: ¼ 117:5 � 253

28000 � 5 � 0:213 � 4:54� 25� 5

25þ 5� 3:75 ¼ 5:99mm

(60)

which is consistent with the desired deflection.The other two stages are dimensioned following the same pro-

cedure and thus not reported in the paper. The final design varia-bles values are collected in Table 3.

The global force-stroke performance of the actuator is as fol-lows. Maximum stroke of 15 mm at zero force, 10 mm at 6 N, 5mm at 3 N, and a maximum force of 10 N with no availablestroke.

Fig. 5 Scheme of the telescopic actuator powered by SMAwave springs in closed (a) and extended position (b)

Table 2 Design constraints for the new telescopic actuator

Internal stage Middle stage External stage

Do¼ 30 mm 40 mm 50 mmDi¼ 20 mm 30 mm 40 mmS� 4 mm 5 mm 6 mmF� 10 NL0� 20 mmrmax 600 MPa (austenite)� 110 MPa (martensite)E EA¼ 84 GPa (austenite)�EM¼ 28 GPa (martensite)twc� 15s

Table 3 Design variables values for the wave springs in the tel-escopic actuator

Internal stage Middle stage External stage

D 25 mm 35 mm 45 mmb 5 mmt 0.21 mmN 4 4.5 5Z 11 8 5L0 20 mm 19 mm 14 mm



a

The proposed actuator, powered by SMA wave springs, is com-pact, easy to assemble, and shows good performance in terms ofavailable stroke and force.

3 Discussion

The mechanical equivalence of the wave and helical spring iscondensed in the curves of Fig. 3. Enforcing the conditions ofsame deflection and same stress is a quite severe constraint for thewave spring which has a thin section and becomes very stiff byincreasing the number of waves.

The spring index C is disregarded in the analysis of the springsbecause there is only a weak dependence on the variables depictedin Figs. 3 and 4. The main differences are for low values of C(�5) and are below 5% of the values reported in Fig. 4 for all thenormalized properties.

The first consideration about the wave spring is the quite highnumber of turns that comes out from the values of Z/n. Theassumption made upon the dead turns may explain these high val-ues. The analysis is performed considering no dead turns for bothsprings in order to obtain the expressive normalized equations, butthis condition penalizes the wave spring. Due to the cyclic axi-symmetry of the wave spring the shim ends (Fig. 1(a)) are notstrictly necessary and their axial dimension is low. The helicalspring on the contrary needs at least two dead turns, which sub-stantially increase the height of the spring.

The influence of the dead turns becomes important especially inthe case of reduced axial height, which is the main field of appli-cation of wave springs. Moreover, in order to disregard the wind-ing angle effect there is a need for at least two or better threeactive turns for the helical springs. These considerations lead tothe conclusion that the helical springs are not suitable when theaxial dimensional constraint is really tight, while wave springscan fit well.

The main outcomes of the present paper are condensed inFig. 4, which shows the analytical comparison between wavesprings and traditional springs by means of normalized properties.Since the comparison is performed being equal the spring rate, thedeflection, the load and the internal and external diameter dimen-sion, the paper presents a quantitatively fair comparison betweenhelical and wave SMA springs.

The normalized properties of Fig. 4 shows that the mass, thefree height and the solid height of the wave spring are higher com-pared to helical spring, mainly due to the high ratio Z/n. Again, allthis normalized values improve when the dead turns are consid-ered for the helical spring.

Moreover the wave spring manufacturer considers a maximumof N¼ 5.5 in the case of multiple-turn springs, which reduces theproblem and combined with the dead coils to be added to the tra-ditional spring lead to comparable height and mass.

On the other hand, the wave springs show a lower cooling time,which is a key factor in the SMA exploitation. One of the mostcritical parameter in the SMA applications is the limited mechani-cal bandwidth (working frequency) caused by the high coolingtimes. The wave spring exhibits a reduction of the cooling timewhich ranges from nearly one-half to one-fifth (Fig. 4, t�c), whichmeans that an actuator equipped with wave springs can theoreti-cally reach working frequencies five time higher than an helicalspring one.

SMA helical springs have a quite low electrical resistance thatmakes necessary current controls and drivers to command manySMA devices.

Considering the wave spring geometry, the electrical resistanceis strongly affected by the contact of the crests, leading to two verydifferent values for the open-circuit resistance (crests galvanicallyisolated) and short-circuit resistance (perfect galvanic contact) asnoticed by comparing the curves for R�OC and R�SC in Fig. 4.

Disregarding the contact leads to a very high normalized resist-ance (R�OC, scaled down by a factor of 50 in Fig. 4), hundreds oftimes higher than for the helical spring. By contrast, assuming

perfect galvanic contact leads to more or less the same electricalresistance as the helical spring (R�S in Fig. 4). Preliminary experi-mental observations were performed on commercial steel wavesprings [8]. The tests showed that the electrical resistance is notfar from the open-circuit approximation [12], because even whenfully compressed the crests do not lead to a good electrical con-tact. In the case of SMA wave springs this condition would be thesame, considering that often there could be an oxidation of the ti-tanium (a thin layer of nonconductive TiO2) present on the SMAleading to an enhanced galvanic isolation. Since the open-circuitcondition is better, because it leads to faster activation times andsimpler control electronics, it can be convenient in preventingmetal-to-metal contacts by means of flexible insulating coatingsor promoting superficial oxidation.

The case study presented in Sec. 2.4 highlights the merits of theSMA wave springs and illustrates the application of the step-by-step procedure. The wave springs, combined with the telescopicarchitecture, are very compact and exploit the SMA better thantraditional springs, especially when axial constraints are tight.

To sum up, the wave springs exhibit mechanical performancesgenerally lower than helical springs, but in the specific SMA fieldthey have an advantage due to the quicker cooling time and thegreater electrical resistance.

4 Conclusion

This paper evaluates the merits of the wave spring geometryapplied to the field of shape memory actuators. When comparedto the traditional helical design, the wave geometry has two cru-cial advantages in the higher electric resistance (implying simplersupply electronics) and the lower cooling time (leading to higherworking frequency). Although generally longer and heavier thanthe helical counterpart, the wave spring has no match when it iscalled to generate medium-low forces in very tight axial spaces. Astructured design procedure is proposed to identify the optimalwave spring that satisfies multiphysics design requirements andconstraints. The optimal wave spring geometry is exploited todesign a linear telescopic actuator. This case study shows a feasi-ble application of SMA wave springs in the actuation field.

NomenclatureAh ¼ lateral area of the helical springAw ¼ lateral area of the wave springAf ¼ austenite finish temperatureAs ¼ austenite start temperatureB ¼ radial width of the wave spring cross sectionC ¼ spring index of the helical spring

cAM ¼ fictitious transformation specific heatcA ¼ austenitic specific heatcM ¼ martensitic specific heat

d ¼ section diameter of the helical springD ¼ mean coil diameter of helical and wave springsDi ¼ inner diameter of the helical springDo ¼ outer diameter of the helical springE ¼ Young’s modulus of the SMA materialf ¼ generic deflection of the springs

hh ¼ free height of the helical springhw ¼ free height of the wave springP ¼ Payload of the telescopic actuatorG ¼ shear modulus of the SMA materialh ¼ coefficient of convectionI ¼ generic electrical current supplied to the springs

Ka ¼ coefficient of deflection for the wave spring (authors)Kb ¼ Bergstrasser factor of the helical springKC ¼ stiffness of the cold wave spring in (martensitic phase)KH ¼ stiffness of the hot wave spring in (austenitic phase)Ks ¼ coefficient of deflection for the wave spring (Smalley)Lw ¼ length of the wave springLh ¼ length of the helical spring



a

m* ¼ normalized mass of the wave springmh ¼ mass of the helical springmw ¼ mass of the wave springMf ¼ martensite finish temperatureMs ¼ martensite start temperature

n ¼ number of coils of the helical springsN ¼ number of waves per coil of the wave springsp ¼ precompression in the telescopic actuatorP ¼ generic load applied to the springs

Pw ¼ perimeter of the wave spring cross sectionPh ¼ perimeter of the helical spring cross sectionRh ¼ resistance of the helical spring

R�OC ¼ normalized open-circuit electrical resistanceROC ¼ open-circuit electrical resistance of the wave springR�SC ¼ normalized short-circuit electrical resistanceRSC ¼ short-circuit electrical resistance of the wave spring

s ¼ stroke of a stage of the telescopic actuatorSw ¼ cross section of the wave springSh ¼ cross section of the helical springHh ¼ solid height of the helical springHw ¼ solid height of the wave spring

t ¼ thickness of wave spring cross sectiont�c ¼ normalized cooling time of the wave spring

thc ¼ cooling time of the helical springtwc ¼ cooling time of the wave springTr ¼ room temperature

Vw ¼ volume of the wave springVh ¼ volume of the helical spring

XAM ¼ transformation enthalpy from austenite to martensiteZ ¼ number of turns of the wave springa ¼ winding angle of the helical springb ¼ wave angle of the wave spring� ¼ Poisson’s ratio of the SMA material

q ¼ electrical resistivity of the materialn ¼ density of the SMA material

rmax ¼ maximum stress in the wave springsmax ¼ maximum shear stress in the helical springs

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jmd 2011 spa drago

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