Research paper
A new semi-analytical approach for quasi-periodic vibrations of nonlinear systems

https://doi.org/10.1016/j.cnsns.2021.105999Get rights and content

Highlights

  • The proposed approach can quickly obtain the semi-analytical QP solution.

  • The ETMRM does not need multiple integration operations.

  • The semi-analytical solutions have high accuracy at a much lower computational cost.

  • The “adaptive zero-setting curve” is introduced to accelerate the convergence.

  • The proposed approach has a large convergence region and high computational efficiency.

Abstract

This paper presents a new semi-analytical approach, namely the enhanced time-domain minimum residual method to solve the semi-analytical quasi-periodic solution for nonlinear system. This approach does not require multiple numerical integration and can be applied to strongly nonlinear systems. The approach is mainly three-fold. Firstly, the semi-analytical solution of the nonlinear quasi-periodic system is expanded into a set of trigonometric series with unknown coefficients, i.e., x(t)k=1N[bkcos(ωkt)+cksin(ωkt)]. Then, the problem of solving quasi-periodic solution can be expressed as: determining the coefficients of the trigonometric series so that the residual objective function R=Mx¨+Cx˙+Kx+N(x¨,x˙,x,t)F(t) is minimum over a period, i.e., minaA0TR(a,t)TR(a,t)dt. Finally, the nonlinear minimum optimization problem is solved iteratively through the enhanced response sensitivity approach. Moreover, the “adaptive zero-setting curve” is introduced to accelerate the convergence. Two numerical examples, a van der Pol-Duffing system and a nonlinear energy sink system are adopted to verify the feasibility of the proposed approach.

Introduction

Nonlinear phenomena have been widely studied by many scholars in the past decades [1], [2], [3], [4]. Generally speaking, there are two types of vibration in nonlinear systems, periodic vibration and aperiodic vibration [5], [6]. Periodic vibration refers to the reciprocating motion with a fundamental frequency and a fixed period [7]. In contrast, aperiodic vibration has no fixed period, and it includes quasi-periodic (QP) vibration [8] and chaotic motion [9]. The characteristic of QP vibration is that the response contains multiple irreducible frequencies, and three types of nonlinear systems will appear QP vibrations [10]. The first type is that the system is excited by multiple external excitations [11]. Because the frequencies of these external excitations are irreducible, the system will enter the QP state after stabilization. Of course, the multi-degree-of-freedom (MDOF) system may exhibit QP response even under a single excitation [12]. Due to the nonlinear coupling between the resonance modes of the system, one or more additional frequencies irreducible to the external excitation frequency will appear, which will directly lead to the QP response. The last type generally exists in the self-excited system without external excitation [8]. Due to the negative damping effect and nonlinear coupling between the resonance modes, QP vibration may occur in some special scenarios. As the nonlinear system with QP vibration widely exists in dynamic system [13], optics [14], aeroelasticity [15], electronics [16] and other fields, it is necessary to study the nonlinear QP vibration system in detail.

For the nonlinear system with periodic motion, we have many methods such as numerical or analytical or semi-analytical methods to analyze the system’s various characteristics [17], [18]. However, there are not many analytical or semi-analytical methods besides numerical methods for the QP motion. The QP response can be obtained by numerical methods such as finite difference method [19], Runge-Kutta (RK) method [20], Newmark-β method [21], shooting method [22] and so on. Furthermore, the numerical continuation method is also reported to be used in the bifurcation analysis [23]. However, the numerical methods can not gain the unstable solution of the nonlinear system, and it is difficult to analyze the bifurcation type of numerical solutions.

The shortcomings of the above numerical methods make us pay more attention to analytical or semi-analytical methods. Conventional analytical methods include classical perturbation method [24], [25], harmonic balance method (HBM) [26], incremental harmonic balance (IHB) method [12], homotopy analysis method (HAM) [7], [27], etc. For example, Guennoun et al. [25] have investigated the QP solutions of the weakly damped nonlinear QP Mathieu equation by the multiple-scales method. Fontanela et al. [26] considered the cyclic symmetry in the mechanical components, and studied the QP localized vibrations in nonlinear cyclic and symmetric structures by the harmonic balance methods. Kim and Noah [28] used a modified HBM–Alternating Frequency Time method to analyze the QP responses of a Jeffcott rotor system. Luo [29] proposed a new approach to obtain the periodic flows of quasi-periodic flows in the chaotic systems and nonlinear systems analytically. Besides, Luo and his collaborators [30], [31] also systematically investigated the analytical solution of periodic-m motion in the forced van der Pol system (fvdP), and further analyzed the stability, bifurcation and frequency-amplitude characteristics of this system. By analyzing the frequency-amplitude of the periodically fvdP system, an asymmetric period-1 motion is found [32]. Although the above methods have obtained the QP solutions of some nonlinear systems, they are only limited to weak nonlinear systems or single-degree-of-freedom (SDOF) systems.

The IHB method [8], [33] is a semi-analytical method that is widely used to solve both weakly and strongly nonlinear systems. Lau et al. [6] studied the QP oscillation of an undamped fixed beam by the IHB method with multiple time scales. Ju et al. [12] proposed a modified two-time scales IHB method to calculate the steady-state QP responses of nonlinear systems. Huang and Zhu [10] has analyzed the QP response of the axially moving beam by the IHB method with multiple time-scales. By adding two additional equations and combining the IHB method, Liu et al. [8] analyzed the QP vibration of the airfoil-store system and explained the self-excited system from periodic vibration to QP vibration. Although the IHB method can be used to solve the QP response of some nonlinear system, it still has the following defects:

  • Firstly, the solution of the IHB method converges by the Newton–Raphson iteration. However, for the nonlinear system with QP response, the convergence domain of the IHB method is too small. That is to say, the initial iterative value of the solution must be selected very precisely to converge [34];

  • The IHB method is used to solve the QP solution, which requires multiple integration operations in each iteration. It consumes many computing resources and resulting in slow convergence speed;

  • To the best of our knowledge, none of the semi-analytical techniques including can handle nonlinear QP systems with more than two fundamental frequencies [10].

Aiming at the defect of solving QP problems with the IHB method, a new semi-analytical approach called the enhanced time-domain minimum residual method (ETMRM) is proposed for the nonlinear QP vibration. The convergence rate of ETMRM is further accelerated by introducing the “adaptive zero-setting curve”.

The rest of this paper is structured as follows: The characteristics of QP vibration were summarized in Section 2. The basic ideas of ETMRM, include the “adaptive zero-setting curve” and the enhanced response sensitivity approach (ERSA) are introduced in Section 3. In Section 4, an SDOF van der Pol-Duffing system with multiple external excitations, and an MDOF nonlinear energy sink (NES) system with single external excitation are adopted as numerical examples to exhibit the feasibility of the ETMRM. Finally, the conclusions are given in Section 5.

Section snippets

Characteristics of quasi-periodic vibration

In this section, we take a SDOF van der Pol-Duffing (vdPD) system with two external excitation [11], [27] as the example to illustrate the characteristics of QP system. The original van der Pol (vdP) system was first proposed by the Dutch engineer Balthazar van der Pol in studying the oscillation effect of the triode. The vdP and the vdPD system have been widely studied in the nonlinear field, and lots of academic achievements have emerged. Consider the vdPD system in the form of{x¨+ϵ(1x2)x˙+ω0

Problem statement

In this section, a nonlinear QP system with external excitation is taken as an example to illustrate the basic idea and implementation process of the ETMRM. Consider the following forms of nonlinear dynamic system with n DOFs:Mx¨+Cx˙+Kx+N(x¨,x˙,x,t)=F(t)where M,C,K are the mass matrix, damping matrix and stiffness matrix of the system, respectively. N(x¨,x˙,x,t) is the nonlinear part of the system, which may include the displacement vectors x, velocity vectors x˙ or acceleration x¨ vectors, or

Numerical examples

In this section, we will take the SDOF vdPD system with multiple external excitations mentioned in Section 2, and an MDOF nonlinear energy sink (NES) system with single external excitation as the examples to illustrate the analysis process of the ETMRM.

Conclusion

This paper presents a new semi-analytical approach, namely the enhanced time-domain minimum residual method to archive the semi-analytical quasi-periodic solution for the nonlinear system. The approximate semi-analytical QP solution was expressed as the triangle series and a set of unknown harmonic coefficients. Therefore, the problem of obtaining the semi-analytical QP solutions is changed into determining a set of parameters so as to the residual is minimum in a period. The enhanced response

CRediT authorship contribution statement

Guang Liu: Methodology, Investigation, Formal analysis, Writing – original draft. Ji-ke Liu: Investigation, Resources, Visualization. Li Wang: Validation, Data curation. Zhong-rong Lu: Validation, Supervision, Project administration, Funding acquisition.

Declaration of Competing Interest

The authors declare that they have no conflict of interest.

Acknowledgment

The present investigation was performed under the support of the National Natural Science Foundation of China (No. 11972380), Guangdong Province Natural Science Foundation (No. 2018B030311001), the University stability support program of Shenzhen (No. 20200831164024001).

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    The MATLAB implementation codes in this paper can be downloaded at: Guang Liu’s Website, Guang Liu’s Blog or Guang Liu’s ResearchGate.

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