To describe the hysteretic nonlinear characteristic of the strain-stress relation of shape memory alloy (SMA), a Van-der-Pol hysteretic cycle is applied to simulate the hysteretic loops. Then, the model of a simply supported SMA beam subject to transverse narrow band noise excitation with nonlinear damping was proposed. The deterministic and the stochastic responses are studied, respectively, applying the multiple scale method. The stability of the steady state responses is analyzed by Floquet theory and the moment method. The numerical simulation results quite agree with the theoretical analysis.

Shape memory alloy (SMA) is a kind of smart materials and is applied in the engineering field widely. There are various mechanical models of SMA proposed in recent decades [

In this paper, we focus on a SMA beam under narrow band noise excitation. Narrow band noise is a harmonic function with constant amplitude and random frequency, which is a reasonable model for the random excitation or response to engineering systems compared with the harmonic excitations. In Section

The experimental hysteretic nonlinearity strain-stress curve of SMA [

The experimental stain-stress curve.

Now, Van-der-Pol hysteretic cycle model was introduced to describe the hysteretic characteristic of SMA. Before the numerical simulation, some assumptions must be given. Firstly, because the difference of the curves is small when the cycle number is limited, we take the average stress value as the final experimental data (see in Figure

The fitting diagram of stain-stress curve.

Then the strain-stress curve of SMA obtained by numerical fitting can be shown in Figure

One hinged-hinged SMA beam is shown in Figure

The hinged-hinged SMA beam.

Considering the boundary conditions at

The bending moment

The dynamical motion equation [

Substituting (

A uniformly approximate solution of (

In this paper only the first-order uniform expansion of the solution

The steady state response

Then the Floquet theory is applied to investigate the stability of the steady state response by linearization. Assuming

Assuming

Obviously, in some proper parametrical conditions, there are multiple solutions coexisting of (

For the triple-solution condition, it is necessary to know the stable conditions for the steady state solutions. The Jacoby matrix is a usually used way to judge the stability, which is expressed as follows:

The characteristic equation can be expressed as

In this paper, the parameters are chosen as

The steady state solutions of the noise-free equation (

The numerical simulation is carried out to prove the theoretical analysis. In Figure

Frequency response (black dots: numerical simulations, black line: theoretical solution, red shadow: unstable region, and red lines: no periodical solution region).

When there are two stable steady state solutions coexisting, one of them has bigger amplitude than the other. Which kind of amplitude the system will act is decided by the initial conditions. In Figure

Domain of attraction to the large amplitude.

For the noise included condition, that is,

Tracking expectation on both sides of (

Phase plot for the noise-free and noise included system.

The frequency response with noise (black dots: numerical simulations and line: theoretical results).

Figure

In this paper, a Van-der-Pol hysteretic cycle is used to simulate the hysteretic character of SMA, which is suitable and convenient for modeling. Then, a nonlinear vibration model of simply supported SMA beam under narrow band noise excitation is built and approximately solved by the multiple scale method. The steady state responses are obtained for both noise-free condition and noise included condition, respectively. After that, the stability of the steady state is analyzed. Some complex dynamical phenomena can be observed, which is helpful for industrial application.

When the strength of the disturbing noise is small enough, the original system can be simplified to an Ito-type differential equation. The singularity theory is applied to study how the parameters influence the number of stationary solutions. It is found that, with some proper parameters there exist three steady state solutions; two of them are stable and realizable, while the other one is unstable. The initial conditions decide which kind of amplitude can appear. The moment method is used to obtain the first-order and second-order moments of the random steady state responses. Similar results are found compared with the deterministic condition. But the difference is that the limit cycle is diffused by the noise. The results of numerical simulation approximately agree with the analysis.

The author declares that there is no conflict of interest regarding the publication of this paper.

This work was supported by the National Natural Science Foundation of China (Grant no. 11272229 and Grant no. 11302144) and the Science Foundation of Tianjin education committee (Grant no. 20120902).