Advances in Mathematical Physics

Volume 2017, Article ID 5214616, 8 pages

https://doi.org/10.1155/2017/5214616

## The Spreading Residue Harmonic Balance Method for Strongly Nonlinear Vibrations of a Restrained Cantilever Beam

^{1}College of Mathematics, Physics and Information Engineering, Zhejiang Normal University, Jinhua, Zhejiang 321004, China^{2}College of Mathematics, Xiamen University of Technology, Xiamen 361024, China^{3}College of Mechanical Engineering, Beijing University of Technology, Beijing 100124, China

Correspondence should be addressed to Y. H. Qian; nc.unjz@4002hyq

Received 2 October 2016; Revised 21 February 2017; Accepted 20 March 2017; Published 10 April 2017

Academic Editor: Zhi-Yuan Sun

Copyright © 2017 Y. H. Qian et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The exact solutions of the nonlinear vibration systems are extremely complicated to be received, so it is crucial to analyze their approximate solutions. This paper employs the spreading residue harmonic balance method (SRHBM) to derive analytical approximate solutions for the fifth-order nonlinear problem, which corresponds to the strongly nonlinear vibration of an elastically restrained beam with a lumped mass. When the SRHBM is used, the residual terms are added to improve the accuracy of approximate solutions. Illustrative examples are provided along with verifying the accuracy of the present method and are compared with the HAM solutions, the EBM solutions, and exact solutions in tables. At the same time, the phase diagrams and time history curves are drawn by the mathematical software. Through analysis and discussion, the results obtained here demonstrate that the SRHBM is an effective and robust technique for nonlinear dynamical systems. In addition, the SRHBM can be widely applied to a variety of nonlinear dynamic systems.

#### 1. Introduction

A lot of problems in physical, mechanical, and aeronautical technology and even in structural applications are essentially nonlinear. Majority of the nonlinear dynamical models are mainly composed of a group of differential equations and auxiliary conditions for modeling processes [1]. In general, it is difficult to obtain the exact solution for strongly nonlinear high dimensional dynamic systems. Hence, the analytical approximate solution of the nonlinear problem has become the research object of many scholars in recent years [2–28].

Generally speaking, the fifth-order Duffing type problem with the inertial and static nonlinear terms is sophisticated all the better [29]. Recently, some scholars have tried to study this kind of nonlinear problem. For instance, Telli and Kopmaz [30] and Lai and Lim [31] used the harmonic balance method to study the linear and nonlinear springs. S.-S. Chen and C.-K. Chen [32] dealt with this fifth-order nonlinear problem by applying the differential transformation approach. Subsequently, Ganji et al*. *[33] and Mehdipour et al*. *[34], respectively, brought in the homotopy perturbation method, amplitude-frequency formulation, and the energy balance method. They used these methods to solve this strongly nonlinear problem, and lower-order approximate solutions are yielded. Qian et al. [35] studied the nonlinear vibrations of cantilever beam by the HAM. Latterly, Guo et al. [36, 37] have presented the residue harmonic balance solution procedure to approximate the periodic behavior of different oscillation systems and they have obtained some more accurate results. Ju and Xue [38, 39] proposed the global residue harmonic balance method to study strongly nonlinear systems. Comparing the obtained solutions with the exact one, they discovered that the approximate results excellently agree with the exact one. Lee [40] used the multilevel residue harmonic balance method to solve a nonlinear panel coupled with extended cavity.

The principal intention of this paper is to investigate the utility of the spreading residue harmonic balance method (SRHBM) [36] for the fifth-order strongly nonlinear problem. The paper consists of the following several parts. Section 2 describes how the strongly nonlinear equation is educed from the governing equations of the cantilever beam model in a nutshell. In Section 3, the SRHBM is introduced and the solution process of different order solutions will be presented. The numerical examples of the SRHBM are rendered and compared with other solutions in Section 4. Finally, conclusion of the paper is drawn in Section 5.

#### 2. Mathematical Formulation

An isotropic slender beam with uniform length and mass per unit length is considered, as shown in Figure 1 [25]. It is assumed that the beam thickness is much smaller than the beam length, so the effects of shear deformation and rotary inertia can be ignored. The angle of inclination is and the beam displacement is . For the boundary condition constraints, one of the conditions is hinged at the bottom of a rotational spring with stiffness , and the other condition is independent. Moreover, the intermediate lumped mass is also connected in along the beam span. By the Euler-Lagrange differential equation, the fifth-order Duffing type temporal problem with strongly inertial and static nonlinearities is able to be derived as follows [25]:where is the dimensionless deflection at the tip of the beam, is the maximum amplitude, the overdot indicates the derivative relative to , and , , , and are parameters. For the complete formulation of (1), readers are referred to [25] for details.