Numerical Investigation of an Adaptive Vibration Absorber...

Numerical Investigation of an Adaptive Vibration AbsorberUsing Shape Memory Alloys

MARCELO A. SAVI,1,* ALINE S. DE PAULA2AND DIMITRIS C. LAGOUDAS

3

1COPPE, Department of Mechanical Engineering, Universidade Federal do Rio de Janeiro,PO Box 68.503, 21.941.972, Rio de Janeiro, RJ, Brazil

2Department of Mechanical Engineering, Universidade de Brasılia, 70.910.900, Brasılia, DF, Brazil

3Department of Aerospace Engineering, Texas A&M University, 77843-3409, College Station, Texas, USA

ABSTRACT: The tuned vibration absorber (TVA) is a well-established passive vibrationcontrol device for achieving vibration reduction of a primary system subjected to externalexcitation. This contribution deals with the non-linear dynamics of an adaptive tuned vibra-tion absorber (ATVA) with a shape memory alloy (SMA) element. Initially, a single-degree offreedom oscillator with an SMA element is analyzed showing the general characteristics of itsdynamical response. Then, the analysis of an ATVA with an SMA element is carried out.Initially, small amplitude vibrations are considered in such a way that the SMA element doesnot undergo a stress-induced phase transformation. Under this assumption, the SMA influ-ence is only caused by stiffness changes corresponding to temperature-induced phase trans-formation. Afterwards, the influence of the hysteretic behavior due to stress-induced phasetransformation is considered. A proper constitutive description is employed in order to cap-ture the general thermomechanical aspects of the SMAs. The hysteretic behavior introducescomplex characteristics to the system dynamics but also changes the absorber response allow-ing vibration reduction in different frequency ranges. Numerical simulations establish com-parisons of the ATVA results with those obtained from the classical TVA.

Key Words: control, vibration absorber, shape memory alloys, non-linear dynamics,hysteresis.

INTRODUCTION

VIBRATION control is an essential task in engineeringbeing related to different applications. The main

goal is vibration reduction by employing either activeor passive procedures. The tuned vibration absorber(TVA) is a well-established passive vibration controldevice for achieving reduction in the vibration of a pri-mary system subject to external excitation (Meirovitch,1986; Inman, 1989). The TVA consists of a secondaryoscillatory system that once attached to the primarysystem is capable of absorbing vibration energy fromthe primary system. By tuning the natural frequency ofthe TVA to a chosen excitation frequency, one producesan attenuation of the primary system vibration ampli-tude for this specific forcing frequency. An alternativefor systems where the forcing frequency varies or has akind of uncertainty is the concept of an adaptive tunedvibration absorber (ATVA). This device is an adap-tive�passive vibration control similar to a TVA but

with adaptive elements that can be used to change thetuned condition (Brennan, 2006; Ibrahim, 2008).

The remarkable properties of shape memory alloys(SMAs) are attracting technological interest in severalscience and engineering fields and numerous applica-tions have been developed exploiting singular character-istics of these alloys (Machado and Savi, 2003; Paiva andSavi, 2006; Lagoudas, 2008). In terms of applied dynam-ics, SMAs are being used in order to explore adaptivedissipation associated with hysteresis loop and themechanical property changes due to phase transforma-tion (van Humbeeck, 2003; Savi et al., 2008). Moreover,the dynamical response of systems with SMA actuatorspresents a unique dynamical behavior due to their intrin-sic non-linear characteristic, presenting periodic, quasi-periodic, and chaotic responses (Savi and Pacheco, 2002;Machado et al., 2003, 2009; Bernardini and Rega, 2005;Savi et al., 2008). Recently, SMA constraints have beenused for vibration reduction since it is expected that thehigh dissipation capacity of SMA changes the systemresponse producing less complex behaviors (Sitnikovaet al., 2008, 2010; Santos and Savi, 2009).

In this regard, SMAs have been used in a number ofapproaches to passive structural vibration control

*Author to whom correspondence should be addressed.E-mail: [email protected] 12 appears in color online: http://jim.sagepub.com

JOURNAL OF INTELLIGENT MATERIAL SYSTEMS AND STRUCTURES, Vol. 22—January 2011 67

1045-389X/11/01 0067�14 $10.00/0 DOI: 10.1177/1045389X10392612� The Author(s), 2011. Reprints and permissions:http://www.sagepub.co.uk/journalsPermissions.nav

(Salichs et al., 2001; Saadat et al., 2002; Lagoudas et al.,2004; Machado et al., 2009). SMA characteristics moti-vate the concept of an ATVA that is able to change itsstiffness depending on the SMA element’s temperature(Elahinia et al., 2005). This property allows one to atten-uate primary system vibration amplitudes not only forone specific forcing frequency, as occurs with the TVA,but for a range of frequencies.Williams et al. (2002) presented an adaptive�passive

vibration controller using SMA elements in an ATVA.The device is manually tuned in order to achieve atten-uation of vibration of the primary system across a rangeof frequencies. Williams et al. (2005) showed a continu-ously tuned SMA�ATVA as a novel adaptive�passivevibration control device. The same idea was used inRustighi et al. (2005a) that treated an SMA�ATVAmade from a beam-like structure supported in itscenter. Numerical and experimental tests were per-formed showing that a change in the stiffness of theATVA can be obtained by a temperature variation,which changes the Young’s modulus, resulting in a var-iation of the tuning frequency. Rustighi et al. (2005b)presented control algorithms for real-time adaptation ofthis SMA�ATVA device both theoretically and experi-mentally. In general, the literature discusses the effect ofproperty change due to the temperature-induced phasetransformation in SMA�ATVA devices, but does nottreat the influence of the hysteretic behavior due tostress-induced phase transformation.This article deals with the non-linear dynamics of

an SMA�ATVA. Initially, a single-degree of freedom(1DOF) oscillator with an SMA element is analyzedshowing the general characteristics of its dynamicalresponse. Then, the analysis of an SMA�ATVA is car-ried out. This analysis starts by considering small ampli-tude vibrations in a way that the SMA element does notexhibit stress-induced phase transformation. Under thisassumption, the SMA’s influence is only due to stiffnesschanges. This kind of analysis is the most common inliterature and is followed by an investigation concerningthe influence of the hysteretic behavior due to stress-induced phase transformation by considering a properdescription of the thermomechanical behavior of SMAs.All results from the SMA�ATVA are compared withthose obtained from the classical TVA, establishing aproper contrast between the devices and their capacityto promote vibration reduction.This article is divided into five sections. After this

introduction, second section discusses the constitutivemodel employed to describe the thermomechanicalbehavior of SMAs. Third section presents an analysisof a 1DOF SMA oscillator discussing some importantaspects of the SMA dynamical response. A discussion ofthe vibration absorber dynamics is presented in thefourth section, investigating the classical TVA and anATVA with an SMA element. The analysis of the

adaptive TVA is treated by considering either tempera-ture-induced or stress-induced phase transformations.The influence of the stress-induced hysteretic behavioris analyzed establishing how it can be useful for vibra-tion reduction. Finally, conclusions are highlighted inthe last section.

CONSTITUTIVE MODEL

The thermomechanical behavior of SMAs may bemodeled either by microscopic or macroscopic phenom-enological points of view. Its description is the objectiveof numerous research efforts that try to contemplate allbehavior details (Paiva and Savi, 2006; Lagoudas, 2008).Here, a constitutive model that is built upon theFremond’s model and previously presented by Oliveiraet al. (2010), Aguiar et al. (2010), Paiva et al. (2005), Saviand Paiva (2005), Baeta-Neves et al. (2004), and Saviet al. (2002) is employed. This model considers differentmaterial properties for each phase and four macroscopicphases for the description of the SMA behavior.Therefore, besides the total strain, ", and temperature,T, it is necessary to define four internal variables thatrepresent the volume fraction of each macroscopicphase: �1 and �2, related to detwinned martensites,respectively associated with tension and compression,�3 that represents the volume fraction of austenite, and�4 that represents the volume fraction of twinned mar-tensite. Since there is a constraint criterion based onphase coexistence, �1þ �2þ �3þ �4¼ 1, it is possible touse only three variables and the thermomechanicalbehavior of the SMA is described by the following setof equations:

� ¼ E"þ E�h þ �½ �ð�2 � �1Þ �� T� T0ð Þ ð1Þ

_�1 ¼1

�1

��"þ�1ðTÞ þ 2��h þ E�2h

� ��2 � �1ð Þ

þ�h E"�� T� T0ð Þ½ � � @�1J��þ @ _�1

J� ð2Þ

_�2 ¼1

�2

��"þ�2ðTÞ � 2��h þ E�2h

� ��2 � �1ð Þ

��h E"�� T� T0ð Þ½ � � @�2J��þ @ _�2

J� ð3Þ

_�3 ¼1

�3�1

2EA � EMð Þ "þ �h �2 � �1ð Þ½ �

2þ�3ðTÞ

�

þ �A ��Mð Þ T� T0ð Þ "þ �h �2 � �1ð Þ½ � � @�3J��

þ @ _�3J� ð4Þ

Here, � is the stress, E ¼ EM þ �3ðEA � EMÞ the elas-tic modulus while � ¼ �M þ �3ð�A ��MÞ related tothermal expansion coefficient. Note that subscript Arefers to austenitic phase, while M refers to martensite.Parameters �1 ¼ �2 ¼ � ¼ �ðTÞ and �3 ¼ �3ðTÞ are

68 M. A. SAVI ET AL.

associated with phase transformation stress levels.Parameter �h defines the horizontal width of thestress�strain hysteresis loop, while � controls theheight of the same hysteresis loop. The terms @nJ�(n¼ �1, �2, �3) are sub-differentials of the indicator func-tion J� with respect to n. This indicator function isrelated to a convex set �, which provides the internalconstraints related to the coexistence of phases. Withrespect to evolution equations of volume fractions,�1 ¼ �2 ¼ � and �3 represent the internal dissipationrelated to phase transformations. Moreover @nJ�(n ¼ _�1; _�2; _�3) are sub-differentials of the indicatorfunction J� with respect to n. This indicator functionis associated with the convex set �, which establishesconditions for the correct description of internal sub-loops due to incomplete phase transformations. Thesesub-differentials may be replaced by Lagrange multipli-ers associated with the mentioned constraints (Saviet al., 2002).Concerning parameter definitions, temperature

dependent relations are adopted for � and �3 asfollows:

� ¼�L0 þ

L

TMT� TMð Þ, if T4TM

�L0, if T � TM

8<: ð5Þ

�3 ¼�LA

0 þLA

TMT� TMð Þ, if T4TM

�LA0 , if T � TM

8<: ð6Þ

Here, TM is the temperature below where the martens-itic phase becomes stable. Usually, experimental testsprovide information of Ms and Mf, temperatures ofthe start and finish of the martensitic formation. Thismodel uses only one temperature that could be an aver-age value or alternatively, Ms or Mf values. Moreover,L0, L, LA

0 , and LA are parameters related to criticalstress for phase transformation.In order to describe the characteristics of phase trans-

formation kinetics, different values of � and �3 might beconsidered during loading, �L and �L3 , and unloadingprocesses, �U and �U3 . For more details about the con-stitutive model, see Paiva et al. (2005) and Savi andPaiva (2005). All constitutive parameters can bematched from stress�strain tests.As it is well-known, SMA devices demonstrate time-

dependence characteristics which means that their ther-momechanical response depends on the loading rate, seefor example, Shaw and Kyriakides (1995) and Yoon(2008). The adequate modeling of this time-dependencycan be performed by considering the thermomechanicalcoupling terms in the energy equation. Monteiro et al.(2009) discussed the thermomechanical coupling andrate-dependency in SMAs. The considered constitutive

model has viscous characteristic that allows the descrip-tion of the thermomechanical coupling avoiding theintegration of the energy equation, presenting usefulresults (Aguiar et al., 2010). Auricchio et al. (2006)explores the same idea showing the difference betweena viscous model and a rate-independent model with ther-momechanical coupling. Both models have the ability todescribe pseudoelastic behavior in SMA wires. Thistime-dependent aspect can be controlled by the properchoice of model parameters.

SINGLE-DEGREE OF FREEDOM SHAPE

MEMORY OSCILLATOR

Initially, the dynamical behavior of SMAs is analyzedby considering a single-degree of freedom (1DOF) oscil-lator, which consists of a mass m attached to a shapememory element of length l and cross-sectional area A,and restitution force FR. A linear viscous damper, char-acterized by a viscous coefficient c, is also considered inorder to represent dissipations different from the dissi-pation associated with the SMA element. Moreover, thesystem is harmonically excited by a force F(t)¼F0 sin(!t). Figure 1 shows an oscillator where the restitutionforce, FR, is provided by a general SMA element.

The equation of motion of this oscillator may be for-mulated by considering the balance of forces acting onthe mass as follows:

m €uþ c _uþ FR ¼ F0 sinð!tÞ ð7Þ

where FR ¼ �A. The restitution force of the oscillatormay be provided by an SMA element described by theconstitutive equations presented in the previous section(Paiva et al., 2005). Therefore, the following equation ofmotion is obtained (Savi et al., 2008):

m €uþ c _uþEA

luþ A�þ EA�hð Þ �2 � �1ð Þ

��A T� T0ð Þ ¼ F0 sinð!tÞ ð8Þ

where volume fractions evolution �1 and �2 aredescribed by the constitutive model presented in the

m

SMA Element

FR

c

F(t )

u

Figure 1. 1DOF oscillator.

Numerical Investigation of an Adaptive Vibration Absorber 69

preceding section and " ¼ u=l. In order to obtain adimensionless equation of motion, the system’s param-eters are defined as follows:

� ¼F0

ml!20

¼F0

ERA; �� ¼

�RATR

ml!20

¼�RTR

ER;

�� ¼�A

ml!20

¼�

ER; ��h ¼

�hERA

ml!20

¼ �h;

!20 ¼

ERA

ml; ¼

c

m!0; E ¼

E

ER;

� ¼�

�R; $ ¼

!

!0ð9Þ

Note that dimensionless parameters and variables aredefined considering some reference values for tempera-ture dependent parameters. This is done by assuming areference temperature, TR, where these parameters areevaluated. Therefore, parameters with subscript R areevaluated at this reference temperature. These defini-tions allow one to define the following dimensionlessvariables, respectively related to mass displacement(U), temperature (�), and time (�):

U ¼u

l; � ¼

T

TR; � ¼ !0t ð10Þ

Therefore, the dimensionless equation of motion hasthe form:

U00 þ U0 þ EUþ ��þ E ��hð Þ �2 � �1ð Þ

� �� 0ð Þ ¼ � sinð$�Þ ð11Þ

where derivatives with respect to dimensionless time arerepresented by ð Þ0 ¼ dð Þ=d�.Besides the non-linear SMA oscillator, an elastic

1DOF oscillator is analyzed by replacing the SMA ele-ment by an elastic element where the restitution forceis given by FR ¼ ku and its natural frequency is!L ¼

ffiffiffiffiffiffiffiffiffik=m

p. The equivalent elastic oscillator is

governed by:

U00 þ U0 þ!2L

!20

U ¼ � sinð$�Þ ð12Þ

Note that in the above equation, the rate !2L=!

20

appears even though it describes the linear elasticsystem. This is due to the definition of the dimensionlesstime, � ¼ ! 0t, where ! 0 is related to the natural fre-quency of the SMA system. This consideration allowsone to establish a proper comparison between the elasticand the SMA systems.

Numerical Simulation

In order to deal with non-linearities of the SMAoscillator equations of motion, an iterative procedurebased on the operator split technique (Ortiz et al.,

1983) is employed. Under this assumption, the fourth-order Runge�Kutta method is used together with theprojection algorithm proposed by Savi et al. (2002) tosolve the constitutive equations. The solution of the con-stitutive equations also employs the operator split tech-nique together with an implicit Euler method. Thecalculation of �n (n¼ 1,2,3) considers that the evolutionequations are solved in a decoupled way. At first, theequations (except for the sub-differentials) are solvedusing an iterative implicit Euler method. If the estimatedresults obtained for �n do not satisfy the imposed con-straints, the orthogonal projection algorithm pulls theirvalue to the nearest point on the domain’s surface (Paivaet al., 2005). On the other hand, the numerical integra-tion of the linear system uses the classical Runge�Kuttaapproach.

The numerical investigation starts by consideringan analysis of the linear system. Values of m¼ 1,!L ¼ 0:72!0, an excitation frequency of ! ¼ 0:72! 0

and a forcing amplitude � ¼ 0:01 are considered. Thenatural frequency of the linear system is chosen insuch way that the resonance frequency occurs in avalue where the SMA system presents high oscillationamplitude. Under these conditions, the system presentsthe well-known resonant response showed in Figure 2for undamped (¼ 0, left side) and damped (¼ 0.05,right side) oscillators. The undamped case presents aresponse where amplitude grows indefinitely while thedamped system response tends to stabilize in specificamplitude.

The same 1DOF oscillatory system is now considered,however, the linear elastic element is replaced by anSMA element. The SMA parameters used in numericalsimulations are given in Table 1. All simulations arecarried out with T¼ 372K, a high temperature whereaustenite is stable for a stress-free state. Moreover,an SMA element with A ¼ 1:96� 10�5 m2 andl ¼ 50� 10�3 m is considered and a reference tempera-ture TR¼TM is assumed.

Both conditions analyzed for the elastic oscillator arenow reproduced for the SMA oscillator. Figures 3 and 4show displacement time history located on the left andthe correspondent stress�strain curve located on theright for the same forcing frequency imposed to thelinear system, with and without external viscous damp-ing. In both cases, the hysteretic behavior of the SMAoscillator tends to attenuate the amplitude responsewhen compared to the linear system.

Due to non-linearities, the SMA oscillator response isfar more complex than the linear oscillator response.Bernardini and Rega (2005) discussed some aspectsrelated to SMA system resonant conditions. In orderto identify some resonant characteristics of the SMAoscillator, non-linear resonant curves are plotted byincreasing (up-sweep) and decreasing (down-sweep)forcing frequency. In both situations, the system is


numerically integrated assessing the maximum steadystate amplitude. Figure 5 shows the maximum ampli-tudes of the system response as a function of the forcingfrequency with different forcing amplitudes: �¼ 0.008and �¼ 0.012. It is evident that non-linearities introducedynamical jumps to the system response and that thevariation of forcing amplitude changes the frequencywhere jumps occur. Moreover, it should be highlightedthat the response of the SMA system tends to presentsmaller vibration amplitude when compared to thoseobtained by the elastic oscillator. This result showssome aspects of the SMA’s dynamical response. The

next section addresses the use of SMA elements in sec-ondary system used in vibration absorbers.

NON-LINEAR VIBRATION ABSORBERS

The ATVA is an adaptive�passive vibration controldevice similar to a TVA but with adaptive elements thatcan be used to change the ATVA tuned condition. Theaim of ATVAs with SMA elements (SMA�ATVA) is toattenuate primary system vibration amplitudes, not onlyfor one specific forcing frequency, as occurs with theTVA, but for a range of frequencies exploring the

1.0

0.5

0.0U

–0.5

–1.0

0 50 100 150t

200 250 300

2x109

1x109

0

–1x109

–2x109

s (M

Pa)

–0.10 –0.05 0.00 0.05 0.10

e

Figure 3. SMA oscillator response for $¼0.72 and ¼0.Note: Left, time response; Right, stress�strain curve.

1.0

0.5

0.0U

–0.5

–1.0

0 50 100 150t

200 250 300

1.0

0.5

0.0U

–0.5

–1.0

0 50 100 150t

200 250 300

Figure 2. Linear system response for !¼ 0.72!0.Note: Left, undamped oscillator (x¼ 0); Right, damped system (x¼ 0.05).

Table 1. SMA parameters.

EA (GPa) EM (GPa) A (MPa) ah L 0 (MPa) L (MPa) LA0 (MPa) LA (MPa)

54 42 330 0.048 0.15 41.5 0.63 185�A (MPa/K) �M (MPa/K) TM (K) �L (MPa.s) �U (MPa.s) �L

3 (MPa.s) �U3 (MPa.s)

0.74 0.17 291.4 1.0 2.7 1.0 2.7


temperature dependent characteristic of the SMAs. Thissection investigates the SMA�ATVA dynamics byestablishing a comparison between two different absor-bers: a TVA and an SMA�ATVA. The analysis of theSMA�ATVA is done by considering two differentapproaches. The first investigates small amplituderesponse that does not induce stress-induced phasetransformation, which is the classical approach in liter-ature. On the other hand, the second approach considersstress-induced phase transformations and therefore, isassociated with energy dissipation due to hystereticbehavior. In addition, temperature variations are of con-cern, assuming the steady state response. Therefore, dif-ferent, constant temperatures are treated in such a waythat slow varying temperatures are of interest.Figure 6 shows a 2DOF system that represents a pri-

mary system, subsystem (m1, c1, k1), that is harmonicallyexcited by an external force F¼F0 sin(! t) and whosevibration amplitudes wish to be reduced by using a sec-ondary system that consists of a concentrated mass, m2,

attached to a viscous damper with coefficient, c2, and toan element with restoring force FR. This element couldbe an SMA element with length l and transversal sectionarea A, defining an SMA�ATVA system or a linearelastic element defining a classical TVA. Once again,all dissipation mechanisms of the system are representedby the viscous damping coefficients (c1 and c2) except forthe hysteretic dissipation of the SMA element.

Both the classical TVA and the SMA�ATVA systemsmay be described by the following equations of motion:

m1 €u1 þ ðc1 þ c2Þ _u1 � c2 _u2 þ ku1 � FR ¼ F0 sinð!tÞm2 €u2 � c2 _u1 þ c2 _u2 þ FR ¼ 0

�

ð13Þ

The description of the restitution force depends on thekind of absorber. The SMA�ATVA has a restitutionforce FR that is described by the constitutive equationpresented in section ‘Constitutive model.’ Therefore, the

0.120.28

0.24

0.20

0.16

0.12

0.08

0.04

0.00

0.08

UM

AX

UM

AX

0.04

0.000.4 0.6 0.8

v

IncreasingDecreasing

IncreasingDecreasing

1.0 0.4 0.6 0.8

v1.0

Figure 5. Maximum amplitudes response increasing (up-sweep) and decreasing (down-sweep) forcing frequency for different forcingamplitudes.Note: Left, �¼ 0.008; Right, �¼ 0.012.

1.02x109

1x109

0

–1x109

–2x109

0.5

0.0U

s (M

Pa)

–0.5

–1.0

0 –0.10 –0.05 0.00 0.05 0.1050 100 150t e

200 250 300

Figure 4. SMA oscillator response for $¼0.72 and ¼0.05.Note: Left, time response; Right, stress�strain curve.


SMA�ATVA system has the following equations ofmotion together with the SMA constitutive equation:

m1 €u1 þ c1 þ c2ð Þ _u1 � c2 _u2 þ ku1 �EA

lu2 � u1ð Þ

� A�þ EA�hð Þ �2 � �1ð Þþ

þ�A T� T0ð Þ ¼ F0 sinð!tÞ

m2 €u2 � c2 _u1 þ c2 _u2 þEA

lu2 � u1ð Þ

þ A�þ EA�hð Þ �2 � �1ð Þ ��A T� T0ð Þ ¼ 0

8>>>>>>>><>>>>>>>>:

ð14Þ

In order to obtain dimensionless equations of motion,system’s parameters are defined as follows:

!201 ¼

k

m1; !2

02 ¼ERA

m2l; 1 ¼

c1m1!02

; 2 ¼c2

m2!02;

! ¼!201

!202

; m ¼m2

m1; �� ¼

�A

m2l!202

¼�

ER;

��h ¼�hERA

m2l!202

¼ �h; � ¼F0

m1l!202

¼m2

m1

F0

ERA

�� ¼�RATR

m2l!202

¼�RTR

ER; E ¼

E

ER; � ¼

�

�R;

$ ¼!

!02ð15Þ

These definitions allow one to define the followingdimensionless variables, respectively related to mass dis-placements (U1,U2), temperature (�) and time (�):

U1 ¼u1l

; U2 ¼u2l

; � ¼T

TR; � ¼ !02t: ð16Þ

Therefore:

U001 þ 1 þ m2ð ÞU01 � m2U02 þ !U1

þ m½�E U2 �U1ð Þ � ��þ E ��hð Þ �2 � �1ð Þ

þ��ð� � �0Þ� ¼ � sinð$�Þ

U002 � 2U01 þ 2U

02 þ E U2 �U1ð Þ

þð ��þ E ��hÞð�2 � �1Þ � �� 0ð Þ ¼ 0

8>>>>><>>>>>:

ð17Þ

The TVA has a linear restitution force,FR ¼ kLðu2 � u1Þ, that results in the following

dynamical system expressed as dimensionless equationsas follows:

U001 þ 1 þm2

m12

� U01 �

m2

m12U

02 þ

!2L1

!202

þm2

m1

!2L2

!202

� U1

� m2

m1

!2L2

!202

U2 ¼ � sinð$�Þ

U002 � 2U01 þ 2U

02 �

!2L2

!202

U1 þ!2L2

!202

U2 ¼ 0

8>>>>>><>>>>>>:

ð18Þ

where !2L1 ¼

k1m1

and !2L2 ¼

kLm2

, respectively represent-

ing the isolated natural frequencies of primary and sec-ondary systems. As presented in the 1DOF system, weare assuming a dimensionless time � ¼ !02t for elasticand SMA systems and therefore, the equations ofmotion present rates !2

L2=!202 and !2

L1=!202.

This article considers a primary system with m1¼ 5 kgand k1 ¼ 5kL. A critical case where the primary systemis under resonant conditions is assumed. Initially, theTVA is of concern and the mass and the stiffnessvalues are chosen in such a way that the natural fre-quency of the primary system is the same as the 1DOFoscillator considered in the last section: m1/m2¼ 5 andkL ¼ 0:722 � ERA=l. Figure 7 shows the response of theTVA primary system showing the maximum amplitudesas a function of the forcing frequency with �¼ 0.01. Twodifferent damping coefficients are analyzed representingan undamped system (1¼ 2¼ 0) and a damped system(1¼ 2¼ 0.2). These results are compared with the1DOF oscillator response. It can be observed that for$ ¼ 0:72, critical resonant situation for the 1DOF oscil-lator, the absorber reduces the amplitude of the primarysystem response. Nevertheless, the new degree of free-dom related to the secondary system introduces tworesonant frequencies, identified in !1 ¼ 0:58$ and!2 ¼ 0:9$. The undamped system presents amplitudesthat increase indefinitely in these frequencies while thedamping tends to limit the amplitude increase as shownin Figure 7.

The forthcoming analysis discusses the SMA�ATVA,showing how it can be useful for vibration reduction.Two different analyses are developed by exploring thehysteresis loop related to stress-induced phase

m1

k1

F1(t)

u1

m2

c2c1

Element with restoring force Vibration

absorber -

secondary

system

u2FR

Figure 6. Vibration absorber connected to a primary system.


transformations and the classical approach where onlythe stiffness change is considered. In addition, tempera-ture variations are of concern, assuming the steady stateresponse. Therefore, different, constant temperaturesare treated in such a way that slow varying temperaturesare of interest. Initially, small amplitude movement isinvestigated and, therefore, the SMA element does notpresent stress-induced phase transformations. Underthis condition, it is possible to alter the absorber naturalfrequencies due to the change of the element’s stiffnesscaused by temperature-induced phase transformations.In this regard, two different ranges of temperatures areconsidered: temperatures where martensite is stable(below TM), and temperatures where austenite is stable(higher than TA¼ 307.5K). The same characteristicsof the primary oscillator (m1¼ 5 kg, k ¼ 5� 0:722�ERA=l) are assumed and the vibration absorber hasm2¼ 1 kg with an SMA element with the same charac-teristics discussed in the previous section (A ¼1:96� 10�5 m2 and l ¼ 50� 10�3 m). Figure 8 showsa comparison between system responses for two differ-ent temperatures, showing that it is possible to changethe range of frequencies where the primary systemamplitude reduction is achieved by changing the tem-perature. This kind of behavior increases the fre-quency range where the ATVA can be efficiently used,which is an essential advantage when compared to theelastic TVA.Although temperature-induced phase transforma-

tions can provide flexibility to the SMA�ATVAresponse, when compared to the classical TVA, thesystem response may be better explored by assuminglarge amplitudes of the SMA element that causesstress-induced phase transformations related to hyster-etic behavior. The following analysis explores both tem-perature-induced phase transformation variation andhysteretic behavior due to stress-induced phase transfor-mations as well.

SMA�ATVA Exploring Stress-Induced Hysteresis Loop

We now consider the general application of theSMA�ATVA where the amplitudes are no longer lim-ited to small oscillations. Under this new condition,stress-induced phase transformations occur correspond-ing to the hysteresis loop. This behavior is related tostrong non-linearities that are usually associated withcomplex responses. The analysis is focused on the influ-ence of temperature on the system response, but consid-ering just high temperatures, which means that the SMAelement is initially at austenitic phase. Figure 9 showsthe SMA element numerical stress�strain curve at 340and 540K, showing typical pseudoelastic behavior forhigh temperatures. Note that the increase in temperaturecauses the increase of the critical stress level where phasetransformation begins to occur. Therefore, we canunderstand that the increase in temperature is relatedto the higher position of the hysteresis loop in the

0.3

0.2

UM

AX

UM

AX

0.1

0.06

0.04

0.02

0.4 0.6 0.8

v

1DOFTVA 1DOF

TVA

1.0 0.4 0.6 0.8

v1.0

Figure 7. Maximum amplitudes response of the 1DOF linear oscillator and of the TVA primary system with �¼0.01.Note: Left, 1¼ 2¼ 0; Right, 1¼ 2 ¼0.2.

2.5x10–5

2.0x10–5

1.0x10–5

5.0x10–6

1.5x10–5

0.40.0

0.6 0.8 1.0 1.2

UM

AX

v

1DOFATVA–290KATVA–330K

Figure 8. SMA�ATVA primary system maximum amplitudesresponse for different temperatures.


stress�strain space, and this behavior is useful for vibra-tion isolation purposes.The system response is now analyzed in order to verify

the influence of temperature variation on vibrationreduction. The previous temperatures are of concern

(340 and 540K) with parameters $ ¼ 0:58, �¼ 0.0075,1¼ 2¼ 0.1. Figure 10 shows the displacement time his-tory of the primary system (left panel) and the stress�-strain curve of the SMA element (right panel). Underthese conditions, the SMA element does not presentphase transformation and, therefore, the amplitudes isnot enough to pass through the hysteresis loop. Thiskind of behavior makes this response similar to the elasticTVA. Decreasing the temperature to T¼ 340K, the hys-teresis loop tends to be in a lower position on the stress�-strain space as shown in Figure 9. Therefore, thedynamical response of the system starts to dissipateenergy due to the hysteresis loop and this large amountof energy dissipation tends to reduce the vibration ampli-tudes of the primary system. Figure 11 shows the systemresponse represented by the displacement time history ofthe primary system and the stress�strain curve of theSMA element, confirming the expected behavior.

Although we can achieve a more efficient vibrationreduction due to hysteretic energy dissipation, dynami-cal jumps can make the system response different.

0.14

0.07

0.00

–0.07

–0.140 50 100

t

150

U

–2x109

–1x109

2x109

1x109

0

–0.10 –0.05 0.00 0.05 0.10

s

U2–U1

Figure 11. Primary system time response and stress�strain curve at T¼ 340 K.

0.14

0.07

0.00

–0.07

–0.140 50 100

t150

U

–2x109

–1x109

2x109

1x109

0

–0.10 –0.05 0.00 0.05 0.10

s

U2–U1

Figure 10. Primary system time response and stress�strain curve at T¼ 540 K.

–3x109

–2x109

–1x109

3x109

2x109

1x109

0

–0.10 –0.05 0.00 0.05 0.10e

s

340K

540K

Figure 9. Stress�strain curve for different temperatures.


A more detailed analysis of the system should be con-sidered for a proper design of the absorber. Figure 12establishes a comparison among the displacement timehistory of the primary system considering three differentconditions: TVA, ATVA at T¼ 540K and ATVA atT¼ 340K. Two different frequencies are of concernshowing how it can change primary system vibrationreduction.The previous results can be better understood by

considering the frequency domain response expressedin terms of the maximum amplitudes as a function offorcing frequency. This analysis assumes the same initialconditions for each forcing frequency: fU1,U

01,

U2, U02g ¼ f0, 0, 0, 0g. Figure 13 shows this frequency

domain response for the same results showed inFigures 10�12. It can be observed that the better perfor-mance of each absorber depends on temperature andexcitation conditions. Note that the ATVA tends to pre-sent better performance but it depends on forcing char-acteristics since the TVA can achieve smaller primarysystem amplitudes responses for some forcing frequen-cies. Moreover, the ATVA at T¼ 340K presents smalleramplitudes responses for forcing frequencies around$¼ 0.6 when compared to its performance atT¼ 540K. This kind of behavior is due to the energydissipation associated with hysteresis loop, which doesnot happen at the higher temperature. It should be high-lighted that differences from the elastic TVA and theSMA�ATVA performances could be altered by temper-ature variations.Different parameters are now of concern considering

a higher damping coefficient (1¼ 2¼ 0.2). This changeis related to a system modification indicating theincrease of dissipation associated with other aspects dif-ferent from SMA hysteretic dissipation. Figure 14 showsATVA primary system maximum amplitudes responsefor �¼ 0.01 and different high temperatures. The equiv-alent 1DOF response is plotted together allowing acomparison of the absorber effect on system dynamics.Once again, this analysis assumes the same initial

conditions for each forcing frequency:fU1, U

01, U2, U

02g ¼ f0, 0, 0, 0g. It might be observed

that the primary system amplitude reduction canbe achieved for different forcing frequencies and differ-ent temperatures. Moreover, the system’s non-linearities

ATVA–340K ATVA–540K TVA ATVA–340K ATVA–540K TVA

U U

0.14 0.14

0.07 0.07

0.00 0.00

–0.07 –0.07

–0.14 –0.140 50 100 150

t0 50 100 150

t

Figure 12. TVA and ATVA primary system time responses.Note: Left, $¼0.58; Right, $¼ 0.72.

UM

AX

1DOF–Linear

ATVA–372KTVA-Primary system

ATVA–400KATVA–420K

0.06

0.04

0.02

0.00

0.4 0.6 0.8 1.0v

Figure 14. TVA and SMA�ATVA primary system maximum ampli-tudes response for �¼ 0.01 and 1¼ 2¼0.2.

UM

AX

1DOFTVAATVA–340KATVA–540K

0.12

0.09

0.06

0.03

0.000.4 0.6 0.8 1.0

v

Figure 13. TVA and ATVA primary system maximum amplitudesresponse for �¼ 0.0075 and 1¼ 2¼ 0.1.


introduce difficulties on the selection of the second-ary system parameters; however, the temperaturevariation can make its application more flexible,allowing an amplitude reduction in a larger range offorcing frequencies when compared to the linearabsorber.

At this point, let us try to elucidate the flexibility thatSMAs can offer to the ATVA. The same parameters ofthe previous case is considered (�¼ 0.01 and1¼ 2¼ 0.2). Initially, a frequency $ ¼ 0:72 is of con-cern representing a resonant condition for the 1DOFoscillator. Under this condition, amplitude reductions

t

U

0.06

0.04

0.02

0.00

–0.02

–0.04

–0.06

0 100 200 300t

U

0.06

0.04

0.02

0.00

–0.02

–0.04

–0.06

0 100 200 300

Figure 15. Primary system time response for $¼ 0.72 (�¼ 0.01 and 1¼ 2¼ 0.2).Note: Left, TVA; Right, ATVA at T¼ 420 K.

t

U

1.5

1.0

0.5

0.0

–0.5

–1.0

–1.5

U

1.5

1.0

0.5

0.0

–0.5

–1.0

–1.5

0 100 200 300 400 500t

0 100 200 300 400 500

Figure 17. Primary system time response for $¼ 0.58 (�¼ 0.01 and 1¼ 2¼ 0).Note: Left, TVA; Right, ATVA at T¼ 372 K.

t

U

0.06

0.04

0.02

0.00

–0.02

–0.04

–0.06

0 100 200 300t

U

0.06

0.04

0.02

0.00

–0.02

–0.04

–0.06

0 100 200 300

Figure 16. Primary system time response for $¼ 0.58 (�¼ 0.01 and 1¼ 2¼ 0.2).Note: Left, TVA; Right, ATVA at T¼ 372 K.


are successfully achieved by the TVA as well as by theSMA�ATVA at T¼ 420K (Figure 15). Nevertheless,the consideration of the vibration absorber togetherwith the primary system represents a new 2DOFsystem being related to two new resonant frequenciesthat must be avoided. Therefore, the attachment of alinear absorber makes it possible for the system to runonly in specific range of frequencies, and large variationsof this frequency may be dramatic for the systemresponse. The SMA�ATVA, however, presents hyster-etic behavior together with temperature dependent prop-erties that may change this scenario making it possibleto attenuate the vibration amplitudes in these new reso-nant frequencies. Figure 16 shows TVA and ATVA pri-mary system response for $¼ 0.58, where the ATVAtemperature is T¼ 372K. The importance of this kindof flexibility is even more clear by considering anundamped system where 1¼ 2¼ 0. Under this condi-tion, the TVA system is related to a classical resonantcondition when $ ¼ 0:58 (Figure 17, left panel). TheATVA, however, can reduce vibration amplitudes inan efficient way (Figure 17, right panel).The analysis of dynamical jumps is essential for a

proper design of related to the SMA�ATVA. This anal-ysis is now of concern by considering the influenceof forcing frequency variation on system response.Figure 18 shows the same analysis developed in

Figure 14 but now investigating increasing (up-sweep)and decreasing (down-sweep) frequencies. This figureplots together the 1DOF and the TVA responses allow-ing a proper comparison among all results. Note thatdynamical jumps are close to the first natural frequencyof the TVA system. Besides, it should be highlighted thatthe jump positions are altered by temperature variations.

CONCLUSIONS

This article discusses the use of SMA elements invibration absorbers. The thermomechanical behaviorof SMAs is described by a proper constitutive model.Initially, a 1DOF oscillator is presented in which therestitution force is given by the SMA element. The anal-ysis of this system shows that the hysteretic behavior isrelated to high energy dissipation, indicating the use ofSMA element in a vibration absorber. The next step ofthis work consists in the dynamical analysis of an adap-tive-passive absorber built with an SMA element(ATVA). The behavior of this system is compared witha passive absorber, composed of a linear elastic element(TVA). The TVA attenuates the primary system vibra-tion amplitudes in a chosen forcing frequency; however,two new resonance frequencies appear and must beavoided. The SMA�ATVA is capable of reducing thesystem response amplitudes not only on the initially

1DOF–Linear

ATVA–372K–IncreasingTVA–Primary system

ATVA–372K–Decreasing

UM

AX

0.4 0.6 0.8 1.0v

0.06

0.04

0.02

0.00

1DOF–Linear



UM

AX

0.4 0.6 0.8 1.0v

0.06

0.04

0.02

0.00

1DOF–Linear



UM

AX

0.4 0.6 0.8 1.0v

0.06

0.04

0.02

0.00

Figure 18. SMA�ATVA primary system maximum amplitudes response increasing (up-sweep) and decreasing (down-sweep) frequencies for�¼ 0.01, 1¼ 2¼ 0.2, and different temperatures: 372, 400, and 420 K.


chosen forcing frequency but also in these two new res-onant frequencies that appear in the TVA. This happensas a consequence of the mechanical properties variationdue to temperature-induced phase transformations andalso to the energy dissipation related to the hystereticbehavior due to stress-induced phase transformation.It should be highlighted that SMA�ATVA attenuationcapacity in a specific frequency can be controlled byaltering the system temperature. This behavior confersflexibility to the ATVA, allowing its use in situationswhere excitation frequency varies. SMA actuatorshave a great potential to be used in vibration control,nevertheless, it is important to mention that the strongnon-linear behavior of SMAs may introduce unusual,complex dynamical responses that should be furtherinvestigated during the design of the application. Theparametric analysis developed in this article shows dif-ferent kinds of possibilities related to the response ofSMA absorbers, indicating some situations that shouldbe investigated during the design. In this regard, it isimportant to mention dynamical jumps and temperaturedependent behavior.

ACKNOWLEDGMENTS

The authors would like to acknowledge the supportof the Brazilian Research Agencies CNPq and FAPERJand through the INCT-EIE (National Institute of Scienceand Technology � Smart Structures in Engineering)the CNPq and FAPEMIG. The Air Force Office ofScientific Research (AFOSR) is also acknowledgedand it is important to mention the inspirational discus-sions with James M. Fillerup and Victor Giurgiutiu.

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