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.
3. Solution Methodology
In the following, the spreading residue harmonic is used to solve (1). Firstly, by introducing a new variable and substituting it into (1), we can get where represents the first-order derivative with respect to and is the unknown angular frequency of (1).
Since we discuss the existence of a periodic solution, we usually choose the base functionsand we find the expression of the steady state solutionswhere is an order parameter and are unknown.
Next, we mainly analyze the zeroth-order harmonic approximation, the first-order harmonic approximation, and the second-order harmonic approximation.
3.1. The Zeroth-Order Harmonic Approximation
To meet the initial conditions in (2), we can set the following initial guess solution of ; that is,Substituting (5) into (2), the equation is yielded:
denotes the zeroth-order residual term. When , is the exact solution.
Based on the Galerkin procedure, the secular term cannot appear on the right hand side of (6). Equating the term’s coefficient to zero, we obtain a linear equation containing an unknown . Through solving that equation, we are able to work out the unknown frequency : In addition, the zeroth-order approximation solution can be obtained as follows:
When the obtained zeroth-order solution is substituted into (6), the terms of and generally are not zero.
3.2. The First-Order Harmonic Approximations
Substituting (4) into (2), the coefficient of parameter is put forward, and we can obtain
According to (3), we choose the following equation as the solution of the representation:where is the unknown.
Combining (10) with (9), we considerWe can obtainTo increase the accuracy, (11) should be added in the zeroth-order residual term .
Based on the Galerkin procedure, (12) should not contain secular terms. Letting coefficients of and be zeros, hence, we obtain a linear equation set containing two unknowns and . Through solving the equation set, we can get where
Therefore, the first-order harmonic approximation can be procured:
When the obtained first-order solution is substituted into (12), the terms of and commonly are not zero.
3.3. The Second-Order Harmonic Approximations
Substituting (4) to (2) and presenting the coefficient of parameter yield
Observing (16), we know it is linear with respect to and . In accordance with the form of (3), we apply where and are unknowns.
Combining (17) with (16) and calculating the equationwe can obtain
With the purpose of increasing the accuracy, (18) should be put on the first-order residual term .
In order to prevent the right hand side of (19) from exhibiting the secular terms , , and , we make all their coefficients equal to zero. Then, we can get three linear equations containing three unknown parameters , , and . According to the three linear equations, we can solve the three unknowns. Thus, the second-order harmonic approximation is shown:
In conclusion, we draw the th-order harmonic approximation
4. Results and Discussion
In order to make sure of the effectiveness of the current technique, we compare the results from the second-order spreading residue harmonic balance approach with the energy balance method [34], the homotopy analysis method [35], and the exact solution [33], which are presented in Table 1, for different parameters and amplitudes of vibration , where the exact solution is computed using the numerical technique. The relative errors of vibration frequency are tabulated in Table 2.
For Mode 1 in Table 2, we observe that the relative error between and is much less than the relative error between and . The same goes for Mode 2 and Mode 4. However, the relative error between and for Mode 3 is more than the relative error between and . Hence, we conclude that the accuracy of the second-order SRHBM solutions is improved in Mode 1, Mode 2, and Mode 4. Similarly, from Table 2, the SRHBM relative error is smaller than EBM relative error in Mode 1, Mode 2, Mode 3, and Mode 4. These results show that the approximate solutions obtained by the SRHBM are closer to the exact solutions.
To further demonstrate the accuracy of the spreading residue harmonic balance approach, the time history responses and the phase portrait are rendered for four different sets of parameters in Figures 2–5. From the phase portrait diagram, we obviously discover that the second-order residue harmonic balance solutions are very consistent with the exact solutions. From the phase portrait, we observe that the system is a periodic motion. Moreover, in the whole range, the presented approximate solutions converge to the exact solutions. Extraordinarily, we argue for in .
5. Conclusions
In this paper, the spreading residue harmonic balance method is applied to discuss the strongly nonlinear vibration system. Particularly, we take a restrained cantilever beam as an example. The SRHBM does not need to add small parameters in the calculation process. Besides, this approach approximates the exact solution quickly and only the first- or second-order approximations. And by comparing its results with HAM and EBM for various parameters and amplitudes of vibration, it reveals that SRHBM can be used to solve a nonlinear equation with high nonlinearities. According to the figures and tables, it is effective to explain that the presented approximations are more accurate. Therefore, we can conclude that the SRHBM is more available and effective. Ultimately, we consider that the SRHBM can be used to deal with more complex strongly nonlinear vibration problems.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Authors’ Contributions
All the authors contributed equally and significantly to the writing of this paper. All the authors read and approved the final manuscript.
Acknowledgments
The authors gratefully acknowledge the support of the National Natural Science Foundation of China (NNSFC) through Grants nos. 11572288, 11302184, and 11372015.