Shock and Vibration

Volume 2017, Article ID 7609528, 11 pages

https://doi.org/10.1155/2017/7609528

## Dynamical Behavior of a Pseudoelastic Vibration Absorber Using Shape Memory Alloys

^{1}Department of Mechanical Engineering, Universidade de Brasília, 70910-900 Brasília, DF, Brazil^{2}Center for Nonlinear Mechanics, COPPE, Department of Mechanical Engineering, Universidade Federal do Rio de Janeiro, P.O. Box 68.503, 21941-972 Rio de Janeiro, RJ, Brazil

Correspondence should be addressed to Marcelo A. Savi; rb.jrfu.eppoc.acinacem@ivas

Received 11 April 2017; Accepted 6 July 2017; Published 11 September 2017

Academic Editor: Laurent Mevel

Copyright © 2017 Hugo De S. Oliveira et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The tuned vibration absorber (TVA) provides vibration reduction of a primary system subjected to external excitation. The idea is to increase the number of system degrees of freedom connecting a secondary system to the primary system. This procedure promotes vibration reduction at its design forcing frequency but two new resonance peaks appear introducing critical behaviors that must be avoided. The use of shape memory alloys (SMAs) can improve the performance of the classical TVA establishing an adaptive TVA (ATVA). This paper deals with the nonlinear dynamics of a passive pseudoelastic tuned vibration absorber with an SMA element. In this regard, a single degree of freedom elastic oscillator is used to represent the primary system, while an extra oscillator with an SMA element represents the secondary system. Temperature dependent behavior of the system allows one to change the system response avoiding undesirable responses. Nevertheless, hysteretic behavior introduces complex characteristics to the system dynamics. The influence of the hysteretic behavior due to stress-induced phase transformation is investigated. The ATVA performance is evaluated by analyzing primary system maximum vibration amplitudes for different forcing amplitudes and frequencies. Numerical simulations establish comparisons of the ATVA results with those obtained from the classical TVA. A parametric study is developed showing the best performance conditions and this information can be useful for design purposes.

#### 1. Introduction

Vibration reduction is essential in engineering systems. Critical situations can be related to excessive, undesirable vibrations. Rotor dynamics, robotics, and structural systems are some examples where vibration reduction is essential for a proper and effective performance of the engineering system. The tuned vibration absorber (TVA) is a well-established passive vibration control device for achieving reduction in the vibration of a primary system subjected to external excitation [1]. The TVA consists of a secondary oscillatory system that, once attached to the primary system, is capable of absorbing vibration energy from the primary system. By tuning the natural frequency of the TVA to a chosen excitation frequency, one produces an attenuation of the primary system vibration amplitude for this specific forcing frequency. An alternative for systems where the forcing frequency varies or has a kind of uncertainty is the concept of an adaptive tuned vibration absorber (ATVA). This device is an adaptive-passive vibration control similar to a TVA but with adaptive elements that can be used to change the tuned condition [2, 3].

The remarkable properties of shape memory alloys (SMAs) are attracting technological interest in several science and engineering fields and numerous applications can be imagined including the ones related to applied dynamics. In general, SMAs are being used in order to explore adaptive dissipation associated with hysteresis loop and the mechanical property changes due to phase transformation [4]. Moreover, the dynamical response of systems with SMA actuators presents a unique dynamical behavior due to their intrinsic nonlinear characteristic [4–9]. Another possibility related to SMA dynamical application is its use as constraints exploiting the high dissipation capacity of SMA hysteretic behavior. Hence, SMA can change the system response producing less complex behaviors when compared with classical elastic constraints [10, 11].

In this regard, SMAs have been used in different ways to perform passive structural vibration control [12, 13]. SMA characteristics motivate the concept of an adaptive tuned vibration absorber that is able to change its stiffness depending on temperature [14–17]. This allows one to attenuate primary system vibration amplitudes not only for one specific forcing frequency, as occurs with the TVA, but also for a broad range of frequencies. In general, the literature discusses the effect of property change due to the temperature-induced phase transformation in SMA-ATVA devices, and there are few reports treating the influence of the hysteretic behavior due to stress-induced phase transformation. Savi et al. [18] presented a discussion about the influence of the hysteretic behavior due to stress-induced phase transformation in pseudoelastic regime of the SMA element of the ATVA. The performance of the pseudoelastic SMA-ATVA for different temperatures was investigated showing vibration reduction in different frequency ranges.

This article revisits the SMA-ATVA considering a parametric analysis with constant temperature. The goal is to identify proper parameters for the system, as the mass ratio between primary system and absorber, and suitable forcing amplitude range where the SMA-ATVA presents better performance than the classical linear TVA. Basically, a secondary system composed of a single degree of freedom oscillator with an SMA element, representing the ATVA, is coupled to a primary system represented by a single degree of freedom linear oscillator. The SMA-ATVA performance is evaluated by analyzing primary system maximum vibration amplitudes for different forcing amplitudes and frequencies. The influence of the hysteretic behavior due to stress-induced phase transformation is considered. A proper constitutive description is employed in order to capture the general thermomechanical aspects of the SMAs. All results from the SMA-ATVA are compared with those obtained from the classical TVA, establishing a proper contrast between the devices and their capacity to promote vibration reduction.

#### 2. Constitutive Model

The thermomechanical description of shape memory alloys is the objective of numerous research efforts that try to contemplate all behavior details [19, 20]. Here, a constitutive model proposed by Paiva et al. [21] is employed. This model considers different material properties for each phase and four macroscopic phases, being built upon Fremond’s model.

The SMA description considers as main variables the total strain, , temperature, , and four internal variables that represent the volume fraction of each macroscopic phase: and , related to detwinned martensites, respectively, associated with tension and compression; that represents the volume fraction of austenite; and that represents the volume fraction of twinned martensite. Since there is a constraint criterion based on phase coexistence, , it is possible to use only three volume fractions and the thermomechanical behavior of the SMA is described by the following set of equations:

Here, is the stress and is the elastic modulus, while is related to thermal expansion coefficient. Note that subscript refers to austenitic phase, while refers to martensite. Parameters and are associated with phase transformation stress levels. Parameter defines the horizontal width of the stress-strain hysteresis loop, while controls the height of the same hysteresis loop. The terms () are subdifferentials of the indicator function with respect to . This indicator function is related to a convex set, which provides the internal constraints related to the coexistence of phases. With respect to evolution equations of volume fractions, and represent the internal dissipation related to phase transformations. Moreover, () are subdifferentials of the indicator function with respect to . This indicator function is related to the convex set , which establishes conditions for the correct description of internal subloops due to incomplete phase transformations. Lagrange multipliers associated with the mentioned constraints may replace these subdifferentials.

Concerning parameter definitions, temperature dependent relations are adopted for* Λ* and as follows:

Here is the temperature below which the martensitic phase becomes stable. Usually, experimental tests provide information of and and temperatures of the start and finish of the martensitic formation. This model uses only one temperature that could be an average value or alternatively value. Moreover, , , , and are parameters related to critical stress for phase transformation.

In order to describe the characteristics of phase transformation kinetics, different values of and might be considered during loading, and , and unloading processes, , . For more details about the constitutive model, see Paiva et al. [21]. All constitutive parameters can be matched from stress-strain tests.

#### 3. Equations of Motion

The SMA-ATVA represents a secondary system that is attached to a primary system, which is represented by a linear oscillator. Therefore, the whole system is a 2DOF oscillator with an SMA element, shown in Figure 1 [18, 22]. The governing equations are obtained from the balance of momentum and it is assumed that the restitution force of the SMA element, , is described by the constitutive equations presented in the previous section. Under these assumptions, the equations of motion are given following the same formalism presented in Savi et al. [18]: