# Response Statistics of a Shape Memory Alloy Oscillator with Random Excitation
## 1 Introduction  
Shape memory alloys (SMAs) are included in certain kinds of smart materials that are relatively lightweight, easy to manufacture, and are able to produce high forces or displacements with low power consumptions [1; 2; 3]. The distinctive mechanical behavior of these materials results from the crystalline phase transformation between an austenite (a high symmetry, highly ordered parent phase) and a martensite (a low symmetry, less ordered product phase). The SMAs exhibit strongly nonlinear thermomechanical responses associated with stress- or temperature-induced transformations of their crystalline structure. These reversible transformations lead to the special properties of superelasticity and shape memory. The remarkable properties of SMAs are attracting much technological interest in the fields of engineering, transportation, communication, as well as in electrical, mechanical, medical, and aerospace industries, among others [4; 5; 6]. SMA coil spring [7] is one of the most widely used SMA elements, having been applied to some devices, such as automotive thermostat, fire damper in vent duct, robot claw, overcurrent protector, and other devices as a driver [8]. Over the past few decades, an increasing number of researchers have explored the application of the SMA spring oscillator in dynamical systems. For example, [PERSON] and [PERSON] [9] studied nonlinear dynamics and chaos of a coupled shape memory oscillator, while [PERSON] [10] investigated nonlinear thermo-mechanical oscillations of a shape memory device. Nonlinear dynamics of a non-smooth SMAs oscillator has been conducted [11]. [PERSON] and [PERSON] [12] showed similar periodic and chaotic motions of the SMA model through numerical simulations, which owe to the changing temperature, damping coefficient, and amplitude of the external forces. The SMAs have a wide range of application in engineering practice. Due to their good properties, they can adapt to various complex external environments. However, there may be various random factors in the actual environments, such as thermal disturbance, wind load, and uneven road surface. These uncertain factors may affect the performance of the SMAs and hinder the application of the SMAs in vibration control, energy harvesting, and other related fields. Therefore, it is of great significance to study the dynamic properties of the SMAs under random conditions, which can provide guidance for the design of SMA material properties, especially in the theoretical aspect exploration. [PERSON] et al. [13] studied the responses of a SMA oscillator under white noise excitation via a Preisach formalism. [PERSON] et al. [14] explored the random vibrations of a single-degree-of-freedom SMA oscillator system with an asymmetric hysteresis that is described by the Bouc-Wen model. [PERSON] and [PERSON] [15] employed an equivalent linearization technique to predict the pseduoelastic responses of a SMA oscillator subjected to Gaussian white noise. [PERSON] et al. [16; 17] studied the dynamic behaviors of the SMA oscillator system from the global viewpoint through numerically cell mapping method. The development and wide application of stochastic dynamics theory makes it a trend to explore the dynamic behaviors of SMAs models in the presence of random excitations. It is worth mentioning that, compared with the Gaussian noise, narrow-band random excitation is also important and widely present in actual engineering systems. For example, [PERSON] and [PERSON] [18] studied the primary responses of a Duffing oscillator with the narrow-band random excitation via a multiple-scale method and a stochastic averaging method. [PERSON] et al. [19; 20; 21] extended the multiple-scale method to the analysis of nonlinear systems under narrow-band random excitations. [PERSON] et al. studied the dynamic responses of a series of typical nonlinear system with narrow-band random parameter excitation, including axially moving viscoelastic beam [22], Duffing oscillator [23; 24], and two-degree-of-freedom airfoil system [25].