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This paper is concerned with the computation of the normal form and its application to a viscoelastic moving belt. First, a new computation method is proposed for significantly refining the normal forms for high-dimensional nonlinear systems. The improved method is described in detail by analyzing the four-dimensional nonlinear dynamical systems whose Jacobian matrices evaluated at an equilibrium point contain three different cases, that are, (i) two pairs of pure imaginary eigenvalues, (ii) one nonsemisimple double zero and a pair of pure imaginary eigenvalues, and (iii) two nonsemisimple double zero eigenvalues. Then, three explicit formulae are derived, herein, which can be used to compute the coefficients of the normal form and the associated nonlinear transformation. Finally, employing the present method, we study the nonlinear oscillation of the viscoelastic moving belt under parametric excitations. The stability and bifurcation of the nonlinear vibration system are studied. Through the illustrative example, the feasibility and merit of this novel method are also demonstrated and discussed.

Bifurcation and stability analysis of nonlinear differential equations is one of the challenging problems of mathematicians and engineers. Normal form theory is one of the most important tools for such analysis [

Some recent developments of the theory of normal form can be found in [

On the other hand, Kuznetsov [

The normal form theory plays an important role in the study of bifurcation behavior of differential dynamical systems. Itovich and Moiola [

Some of the above works use the Jordan canonical form of the leading matrix

In this paper, we will develop an efficient method for computing the normal forms directly for general four-dimension systems and apply the method to consider controlling bifurcations. The approach is efficient since it does not require the computation of the Jordan canonical form of

Consider a dynamical system described by the following differential equation:

Without loss of generality,

We take the coordinate transformation as follows:

Then, (

Employing the normal form theory, we introduce a linear operator as follows:

Hence, the

The rationale for the classical normal form theory can be explained by the following theorem (see [

Let the notations be the same as above. Suppose that the decomposition (

By applying the Takens normal form theory [

Consider a four-dimensional generalized averaged system with

At the same time, (

Consider the following three cases of the Jordan matrix

The Jordan matrix

The Jordan matrix

The Jordan matrix

The forms of the Jordan matrix

It is easy to see that the three-order polynomial solutions in four variables can be obtained from (

For the case (i), (

Then, we solve the following equations:

For the case (ii), we obtain

and we need to solve the following equations:

Finally, the case (iii) leads to

Similarly, the following equations must be solved:

Let

For the case of two pairs of pure imaginary eigenvalues, we arrive at

For the case of one nonsemisimple double zero and a pair of pure imaginary eigenvalues, we get

For the case of two nonsemisimple double zero eigenvalues, we derive

It should be clear that

Let

For the case of two pairs of pure imaginary eigenvalues, we have

For the case of one nonsemisimple double zero and a pair of pure imaginary eigenvalues, (

For the case of two nonsemisimple double zero eigenvalues, we deduce

From (

We determine all

In this section, we apply the proposed method in Section

The model of a viscoelastic moving belt and the coordinate system.

In the subsequent analysis, we use the method of multiple scales and Galerkin’s approach in the partial differential governing equations of the viscoelastic moving belt. We introduce the mass, gyroscopic, and linear stiffness operators as follows:

To obtain a system that is suitable for the application of the method of multiple scales, we introduce the scale transformations

Substituting (

The differential operators of the method of multiple scales can be defined as

In this paper, we investigate the case of primary parametric resonance for the

The above algorithm applied to the system (

Comparing the method developed here with other methods given in [

It is known that (

The characteristic equation corresponding to the trivial zero solution is

The eigenvalues of the above equations are

Take a linear transformation of coordinate as follows:

Let

The first two equations are independent of the last two. The last two equations describe rotations in the planes

This system is called the amplitude system. The trivial equilibrium,

The behavior of systems (

For the case of

The trivial and the nontrivial equilibria of (

The nontrivial equilibrium

The numerical simulation result is given in Figure

Periodic solution of the viscoelastic moving belt system.

An efficient method for computing the normal form of high-dimensional nonlinear systems is presented in this paper. This computation method is applied to obtain the normal form of the averaged equation for the viscoelastic moving belt under parametric excitations. Based on the current studies, it is found that the newly developed computation method improves the classical normal form. Therefore, it is a further reduction of the classical normal form. Meanwhile, the normal form derived herein is used to explore the bifurcation and stability analysis of axially viscoelastic moving belts under parametric excitations.

In contrast to earlier works, this article argues that the normal forms of high-dimensional nonlinear systems may always be achieved without computing either the Jordan canonical form of

The coefficients in the averaged system (

The authors declare that there is no conflict of interests regarding the publication of this article.

The authors gratefully acknowledge the support of the National Natural Science Foundations of China (NNSFC), through Grant nos. 11302184 and 11202189, the Scientific Research Fund of Zhejiang Provincial Education Department, through Grant no. Y201121157, and the Scientific Research Foundation of Xiamen University of Technology, through Grant no. 90030631. We are also grateful for helpful comments which were provided by Wei Zhang at the Beijing University of Technology.