Chaos in piezoelectric composite laminated beams has significant implications in the design of this model. Some results for this model have been obtained numerically. With the energy-phase method and numerical simulations, global dynamics of piezoelectric composite laminated beams is investigated in this paper. The average equation of the piezoelectric composite laminated beam is obtained by the normal form theory. The existence of multipulse homoclinic orbits for undisturbed and dissipative cases is analyzed by the energy-phase method, and the mechanism of chaotic motion of the system is given. The effect of the dissipation factor on pulse sequence and layer radius is studied in detail. The chaotic motion of the system is verified by numerical simulations.

Since piezoelectric composite materials have good thermal stability and high specific strength and specific stiffness, they can be used to manufacture aircraft wings and satellite antenna shells of large launch vehicles. The composite material has the special vibration damping characteristic, so it can reduce the vibration and noise. Moreover, due to the amazing thermal, electrical, chemical, and mechanical properties of nanowires and nanobeams, they can be utilized in micro- and nano-electro-mechanical systems [

Multipulse chaotic motions are very common in high-dimensional systems. With the extended Melnikov method, Zhang et al. [

In this paper, global dynamics of a piezoelectric composite laminated beam is investigated with the energy-phase method and numerical simulations. The existence of multipulse homoclinic orbits for undisturbed and dissipative cases is analyzed. The effect of the dissipation factor on pulse sequence and layer radius is studied in detail. Numerical simulations verify the analytical results.

Consider the piezoelectric composite material laminated beam shown in Figure

Dynamic model of the piezoelectric laminated beam.

According to Reddy’s high-order shear deformation beam theory, von Karman theory, and Hamilton’s principle, considering the primary resonance and 1 : 9 internal resonance, the average equation can be obtained as follows with the multiscale method [

Using the normal form theory, combined with the Maple [

Introduce the following scale transformation:

The normal form with perturbation terms is obtained as follows:

The Hamilton function of system (

When

The Hamilton function of system (

If

When

If

Define the critical curve

Along this critical line, pitchfork bifurcation will occur. Since

For all

According to the research results of [

Letting

According to Hamilton principle, the energy at saddle point is equal to that on homoclinic orbit, namely,

From equations (

Separating the variables and integrating yields

From (

The undisturbed system restricted to

It is easy to obtain that the resonance value is

The geometric structure of the stable manifold

Geometric structure diagram of manifolds (

Next, phase drift is calculated. Substituting (

Integrating equation (

For a sufficiently small perturbation

System (

We study the dynamical behavior of manifold

Substituting the above transformation into equation (

When

It is easy to know that system (

The equilibrium points of system (

It is obvious that the undisturbed system has a center

(a) Phase diagram of the undisturbed system (

When

The Hamilton function

When

According to the energy-phase method proposed by Haller and Wiggins [

According to equation (

The transversal zero set of the

According to (

So, the transversal zeros of

Introduce the following angle transformation:

Then, the transversal set of

According to (

Define the open set of inner orbits as follows:

It is deduced that as the closer the periodic orbit of

Define the layer sequence inside the track as follows:

It is easy to know that

Layer sequence structure diagram of the undamped system.

When

From (

Substituting (

According to (

Next, define the dissipation factor

It is easy to know the upper bound of the dissipation factor is

When

Therefore, it can be concluded that the upper bound of the maximum number of pulses is inversely proportional to the dissipation factor

Define the transversal zero set of the energy difference function as follows:

For any integer

Introducing the following angle transformation:

Since each multipulse orbit has only one energy function, the energy function corresponding to the center point is denoted by

The set sequence of the inner orbit is denoted as

Layer sequence structure diagram of the system with damping.

According to equation (

Silnikov type 3-pulse homoclinic orbit.

Next, it needs to verify that the meeting point of equation (

If

Equation (

Consequently, one can solve the dissipation factor

When (

Next, we need to verify whether

After simplification, the opposite condition of (

When (

Finally, we need to verify that the regression point of the

Then, it is necessary to find the regression point closest to saddle point

Therefore, for a sufficiently small disturbance

Using the Mathematica software, we draw the relationship between pulse sequence and phase difference and the relationship between layer radius and phase difference. The relationship between pulse number

Relationship between pulse sequence

Relationship between layer radius

In the case of disturbance, select the parameter value

Relationship between pulse sequence

Relationship between layer radius

According to conditions (

Phase portraits and time histories of the system. (a) Phase portrait in space

In this paper, the multipulse chaotic motion of piezoelectric composite laminated beams excited by transverse and longitudinal excitation is studied. The energy-phase method is used to explore the existence of multipulse orbits of piezoelectric composite laminated beams residing in slow manifold under Hamilton resonance, and the existence of the homoclinic orbit will lead to chaos. Numerical simulations prove that chaos will occur in the system, and the relationship between pulse sequence, layer radius, and phase difference is obtained under different dissipation values. It is presented that there may exist multipulse orbits which are homoclinic to fixed points on the slow manifold in the resonant case for this system. The effects of the phase shift and dissipation factor on pulse sequence and layer radius are also analyzed. It is concluded that the number of pulses in multipulse orbit decreases gradually, and the homoclinic tree breaks down with the increase in dissipation factor. Through theoretical research and numerical simulation, one can obtain that the selection of system parameters and initial conditions is the main reason for the occurrence of multipulse chaos for nonlinear dynamic systems. In real life, composite laminated beams may appear chaos, which has some hidden dangers to their security and stability. Therefore, we should choose appropriate parameters to avoid chaotic motion as much as possible.

The data used to support the findings of this study are included within the article.

The authors declare that they have no conflicts of interest regarding the publication of this work.

The research was supported by the National Natural Science Foundation of China (11772148 and 11872201).