Abstract

The nonlinear random vibration of axially moving shape memory alloy (SMA) laminated beam under transverse loads is investigated. Considering the effects of axial movement and random perturbation, the dynamic equation of the SMA laminated beam is established by means of physical equation, force balance conditions, deformation compatibility equation, and the constitutive relation by polynomial function. The transverse vibration differential equation of axially moving SMA laminated beam with two simply-supported edges is obtained through the Galerkin method. The amplitude-frequency response equation under primary resonance and the probability density function under random perturbation are derived by using the averaging method and stochastic averaging method, respectively, and the theoretical results are numerically validated. In addition, under the combination of the harmonic excitation and random perturbation, the effects of axial movement velocity and random perturbation intensity on system steady-state response are analyzed.

1. Introduction

Shape memory alloy (SMA), a material widely used in machinery, electronics, aerospace, civil engineering, medicine, energy and everyday production, and life with special shape memory and pseudoelasticity, is more sensitive to stress and temperature changes, more deformable and resilient than its ordinary metal counterparts. The thermoelastic martensitic transformation due to the temperature or external load changes in SMA can change the properties of the material; hence, SMA is often made into composite structures to achieve structural control. In practical application, SMA particles, wires, or strips are often inserted into other matrix materials, or the SMA membrane is applied over the surface of beam or plate matrices to form an SMA composite structure [1].

SMA is the surface layer, and the linear elastic material is the sandwich layer, which constitutes the laminated beam structure, and these are common composite structures. Collet et al. [2] considered the symmetry hypothesis of SMA under tension, compression, and temperature, and the dynamic behavior of SMA beams is studied by applying external loads. Machado et al. [3] studied the dynamic characteristics and chaotic behavior of coupled SMA oscillators through numerical methods. Savi et al. [4] concerned with the dynamic response of a shape memory two-bar truss, a polynomial constitutive model is assumed to describe the behavior of the shape memory bars, and numerical simulations show that the system can easily reach a chaotic response. Ren et al. [57] carried out a series of work on SMA composite beams, it analyzed the influence of fiber laying angle and the content percentage of SMA on the equivalent damping ratio of the beam, and the vibration frequency response characteristics of the beam structure are studied. Under the influence of stress-induced martensitic transformation, the superelastic nonlinearity of SMA makes the structure generate complex dynamic characteristics. The effects of external excitation amplitude and ambient temperature on the nonlinear dynamic behavior of the beam are analyzed. Then, the semiactive control of laminated beams is analyzed, and the vibration response is reduced by adjusting the parameters of the controlled structural beams. The results show that semiactive control can effectively suppress the occurrence of resonance for structural vibration control [810].

In Reference [11], a more general constitutive relation of SMA fibers is proposed, and a theoretical bending model of SMA fibre reinforced composite laminated beams is established. The stress relationship between the SMA layer and matrix layer is not considered in the modeling process. Zhang [12] and Wu et al. [13] adopted the Tanaka–Liang–Brinson model, described the hysteretic constitutive relation of shape memory alloy with two-flag piecewise linear function, and developed singularity theory suitable for multipiecewise bifurcation problem analysis. Based on piecewise linear constitutive relation, a dynamic model of shape memory alloy laminated beam was established to study the nonlinear vibration of the beam. In reference [14], the free vibration of composite laminated beams embedded with SMA fibers is studied. The Brinson constitutive model is used to study the effect of temperature change on restoring force and fundamental frequency of composite laminated beams. In reference [15], the Brinson constitutive model is used to establish the governing equations of SMA fiber laminated beams, considering the effects of thermal and aerodynamic loads. It is found that the thermal flutter characteristics of the laminated beams can be significantly enhanced by embedding SMA fibers, and the stability boundary of the composite beams can be improved by increasing the volume fraction of SMA fibers.

In addition, on the problem of random vibration, Wang et al. [16] studied the random bifurcation and first passage of the SMA beam under random excitation. Liu et al. [17, 18] analyzed the principal resonance responses of nonlinear single-degree-of-freedom (SDOF) systems with lightly fractional derivative damping of order subject to the narrow-band random parametric excitation. As an example, the stochastic Duffing oscillator with fractional derivative damping is considered. The corresponding theoretical results are well verified through direct numerical simulations. Huan et al. [19, 20] studied the stationary response of a class of nonlinear stochastic systems undergoing Markovian jumps.

In the previous nonlinear vibration analysis of the SMA laminated beam, the SMA constitutive relation is a piecewise linear double-flag model with hysteresis loops. The process is cumbersome and complex, and the influence of structural parameters on vibration has not been studied. In this paper, considering the effect of axial motion and random disturbance, the transverse vibration equation of the SMA laminated beam under transverse load is established by using a simple continuous smooth polynomial constitutive model. The steady-state response behavior of the system is studied, and the influence of parameters such as the axial velocity, the amplitude of external excitation, temperature, and structural parameters (the thickness ratio of the SMA layer to the elastomer layer) is all analyzed.

2. Dynamic Modeling of SMA Laminated Beam in Axial Movement under Uniform Load

2.1. Polynomial Constitutive Relationship of SMA

Based on the Landau–Devonshire thermodynamics theory, Falk proposed a polynomial free energy equation, and then, Paiva and Savi obtained the coefficients of the free energy function, as follows [21]:

The free energy is defined as a sixth-order polynomial equation in a way that the minima and maxima points represent stability and instability of each phase of the SMA. Hence, the form of the free energy is chosen such that, for high temperatures (), it has only one minimum at vanishing strain, representing the equilibrium of the austenitic phase. For intermediate temperatures (), there are three minima corresponding to three stable phases-austenite. Lastly, at low temperatures (), martensite is stable, and the free energy must have two minima at nonvanishing strains.

Derivation of equation (1) to strain, the constitutive equation is given bywhere and are positive material constants and . is the temperature above which austenite is stable, and is the temperature below which martensite is stable. , , , and obtained from experiments in Reference [21], and the stress-strain curve is shown in Figure 4-2 of Reference [21].

The merits of SMA polynomial constitutive relation lie in its simplicity. Paiva and Savi’s [21] research also shows that the polynomial model can qualitatively describe the dynamic behavior of SMA. For the SMA laminated beam with complex structure and complicated stress conditions, it is sometimes difficult to obtain the dynamic equation of the system by using other constitutive models, and the nonlinear dynamics characteristics of the system can be easily obtained and analyzed by using this constitutive model.

2.2. Dynamic Modeling of SMA Laminated Beam

Figure 1 gives the structural diagram of the SMA laminated beam. The laminated beam that subject to both uniform load treated and axial load, where includes harmonic excitation and random disturbance . The beam’s length is , and its width is ; the height of the matrix beam is , and the thickness of the upper and lower SMA layers is . The axial movement velocity of simply supported SMA laminated beam is , where is the stationary coordinate system, with traverse displacement being recorded as , the beam traverse movement velocity is , and the acceleration is . Figure 2 shows the loads on the microbody. The virtual inertia force on the matrix beam is , and is the length of the microbody.

Assuming that the microbody is hypothetically balanced can obtain a balance equation as follows:

Upper (SMA layer):

Elastic matrix:

From the bending moment-deflexion relationship and the stress-internal force relationship, we obtained the following equation:

Lower:

To achieve displacement compatibility on the interface between the SMA layer and the matrix beam, the following conditions must be satisfied:where is the axial displacement of the neutral layer of the matrix beam and and are the axial displacements of the symmetrical surfaces between the upper and lower SMS layers. As only considering the transverse vibration, the effects of the axial deformation of the fast moving beam are ignored. Hence, the axial displacement of the neutral layer in the matrix beam is zero, and . From the equation and the stress-strain relationship of SMA, we have

Substituting equations (5), (7), (8), and (11) into the transverse vibration equation (6), we have

2.3. Vibration Differential Equation

As , in the case of the first-order modal and simply supported at both ends, the displacement solution can be set as

Discretize the continuous simply supported beam by the Galerkin method and introduce nondimensional parameters: , , , , , , , and . And the nondimensional quantities of the differential equation are processed:where

3. Considering Simple Harmonic Excitation

3.1. Solving the System Primary Resonance Amplitude-Frequency Response Equation

Only the effects of simple harmonic excitation are considered, and . In the case of primary resonance, its simple approximation can be expressed aswhere . Using averaging method, we can obtain a simple approximation that satisfies

, by eliminating , we can get the system amplitude-frequency response equation as

3.2. Stability Analysis

To analyze stability, we setwhere and are steady-state solutions, and and are minor perturbations. Substituting equation (19) into equation (17), keeping the linear terms, we obtain the coefficient matrix on which steady-state movement stability is based:

According to the Routh–Huriwitz stability criterion, the solution will be stable when all the characteristic values have a negative real part. Hence, the steady-state movement is unstable when

3.3. Numerical Validation and Parametric Influence Analysis

The amplitude-frequency response equation for the system steady-state response is obtained. Figure 3 validates the theoretical calculation with the fourth-order Runge–Kutta method (, , , , and ). Here, the red solid lines represent the stability boundary yielded from the Routh–Hurwitz criterion equation, within which the response is unstable with distinct jumps occurring in the amplitude-frequency response curves. The blue asterisks are numerical calculations. The analytical solution agrees well with the numerical simulation.

From Figures 4(a) and 4(b), as the temperature and axial velocity change, no qualitative change takes place in the amplitude-frequency curve except for some translation in the corresponding frequency in the resonance region, and changes in the temperature and axial velocity only make some difference to the resonance frequency—as the temperature and velocity increase, the system resonance frequency decreases. From Figure 4(c), as the external excitation amplitude increases, the resonance amplitude increases largely with distinct jumps occurring in the amplitude-frequency response curve. Figure 4(d) shows a remarkable lag as a result of a thickness increase of the surface SMA layer. The vibration amplitude decreases largely, and the multisolution region becomes broader.

4. Considering Random Perturbation

4.1. Solution with Stochastic Averaging Method

Considering the effects of random perturbation only, .

First, introduce transforms:

Substituting it and using stochastic averaging method will obtain the corresponding Ito differential equations:where and are two individual unit Wiener processes and is independent of changes. Hence, we can get the steady-state probability density of vibration amplitude aswhere is a normalized constant.

4.2. Parametric Influence Analysis

Given that , , and , we can get the probability density function (PDF) and probability spectral density (PSD) curves of a system steady-state amplitude under different parameters.

From Figure 5, when only random perturbation is considered, as the random perturbation intensity increases, the PDF curve peak of the vibration amplitude becomes farther from the original point. The probability of large vibration increases. No qualitative change takes place in the shape of the PDF curves, and no second distinct peak appears. Also, comparison of the PSD curves of the traverse displacement of the SMA laminated beam under the three sets of parameters also revealed that, at a small excitation intensity, the system energy is mainly concentrated on one single frequency component; with the increase of the random perturbation intensity , there appear more than one frequency component on which energy is concentrated, though the energy on the first frequency component is more prominent than that on the following ones.

As the axial velocity increases, the following transforms take place in the energy distribution inside the system: the spectrum width narrows sharply and then broadens gradually with the maximum remaining at the system natural frequency, as shown in Figure 6; as the axial velocity increases, the PSD level at the high frequency becomes smaller.

5. Combined Effects of Simple Harmonic Excitation with Random Excitation

Given that , , , , , and , give the initial conditions and assume that the system steady-state response is a periodic movement when , as shown in Figure 7(a). As the random perturbation intensity increases, the periodic movement disappears. The phase diagram changes into a diffused limit cycle. The traverse vibration amplitude varies randomly in the proximity of 0.51. Comparison of the three sets of limit cycles and time histories revealed that, as random perturbation appears, there appear considerable vibration amplitudes for a relatively deterministic system, and the time history becomes a disorder.

When either random or simple harmonic perturbation is considered, the peak of the PSD curve will definitely appear at the system natural frequency (as shown in Figures 4 and 5) and will not change with the random perturbation noise intensity or external excitation amplitude; when both random perturbation and simple harmonic excitation are considered, comparison of the PSD curves under the three different sets of parameters will reveal that, as the random perturbation intensity increases, a second frequency component with prominent energy appears near each order of the natural frequency, and further increase in the random perturbation intensity will also increase the energy on this second frequency component, as shown in Figure 8.

6. Conclusions

The traverse vibration equation for the SMA laminated beam under traverse loads is established based on the polynomial constitutive relationship of SMA, taking into account the effects of axial movement, axial forces, and random perturbation as a means to examine the influences of parameters.

Under simple harmonic excitation alone, the primary resonance amplitude-frequency response curves of the SMA laminated beam show some jumps. Axial velocity and axial loads only make a difference to resonance frequency—as axial velocity increases, resonance frequency decreases; increase in external excitation amplitude results in a remarkable lag and an increase in resonance frequency. Increase in the SMA layer thickness leads to significant vibration reduction and the narrowing of the multivalue region. At a large external excitation amplitude, both temperature and SMA thickness increases can reduce the system vibration amplitude.

Under random perturbation alone, no qualitative change takes place in the shape of the PDF curves. As the random perturbation intensity increases, however, the PDF curve peak of the vibration amplitude becomes farther from the original point, and an increase in the axial velocity at this time will reduce the PSD level at the high frequency.

Under the combined effects of harmonic excitation and random perturbation, the steady-state response phase diagram changes into a diffused limit cycle. A second frequency component with prominent energy appears near each order of the natural frequency in the response PSD curves, and further increase in the random perturbation intensity will also increase the energy on this second frequency component.

Nomenclature

:Cross-sectional area of the matrix beam
:Elastic modulus of the matrix beam
:Density of the matrix beam
:Per unit length damping
:Medium normal angle of the matrix beam
:Bending moment
:Axial internal force
:Cross-sectional shear
:Transverse uniform load
:Harmonic excitation
:Random disturbance
:Axial load.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this manuscript.

Authors’ Contributions

Hao Ying drafted the paper and completed the theoretical derivation, data analysis, and other tasks. After receiving the revision comments, Gao Minglei helped to complete the subsequent revisions, especially the grammatical errors and the introduction. So, the other author Gao Minglei was added after acceptance.

Acknowledgments

This research was supported by the Science and Technology Research Project of Colleges and Universities in Hebei Province, China (no. 2016045), and the Doctoral Starting-Up Foundation of Yanshan University, China (B938).