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 fifthorder 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.
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 [
Generally speaking, the fifthorder Duffing type problem with the inertial and static nonlinear terms is sophisticated all the better [
The principal intention of this paper is to investigate the utility of the spreading residue harmonic balance method (SRHBM) [
An isotropic slender beam with uniform length
Geometry and coordinate system for a beam with a lumped mass.
In the following, the spreading residue harmonic is used to solve (
Since we discuss the existence of a periodic solution, we usually choose the base functions
Next, we mainly analyze the zerothorder harmonic approximation, the firstorder harmonic approximation, and the secondorder harmonic approximation.
To meet the initial conditions in (
Based on the Galerkin procedure, the secular term
When the obtained zerothorder solution is substituted into (
Substituting (
According to (
Combining (
Based on the Galerkin procedure, (
Therefore, the firstorder harmonic approximation can be procured:
When the obtained firstorder solution is substituted into (
Substituting (
Observing (
Combining (
With the purpose of increasing the accuracy, (
In order to prevent the right hand side of (
In conclusion, we draw the
In order to make sure of the effectiveness of the current technique, we compare the results from the secondorder spreading residue harmonic balance approach
Comparison of the SRHBM frequencies, EBM frequencies, HAM frequencies, and the exact frequencies for various parameters.
Mode 










1  1.0  0.326845  0.129579  0.232598  0.087584  1.01235  1.01232  1.01004  1.01015 
2  0.5  1.642033  0.913055  0.313561  0.204297  0.935046  0.938636  0.935072  0.93639 
3  0.2  4.051486  1.665232  0.281418  0.149677  0.965613  0.966516  0.965843  0.96664 
4  0.3  8.205578  3.145368  0.272313  0.133708  0.860678  0.871382  0.863939  0.86426 
Comparison of the SRHBM relative error, EBM relative error, and HAM relative error with the exact frequencies for various parameters.
Mode 









1  1.0  0.326845  0.129579  0.232598  0.087584  0.21789  0.21482  0.010889 
2  0.5  1.642033  0.913055  0.313561  0.204297  0.14353  0.23986  0.140753 
3  0.2  4.051486  1.665232  0.281418  0.149677  0.106244  0.01283  0.08245 
4  0.3  8.205578  3.145368  0.272313  0.133708  0.414459  0.82406  0.037142 
For Mode 1 in Table
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
Comparison of the approximate and exact solutions of Mode 1.
Comparison of the approximate and exact solutions of Mode 2.
Comparison of the approximate and exact solutions of Mode 3.
Comparison of the approximate and exact solutions of Mode 4.
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 secondorder 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.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
All the authors contributed equally and significantly to the writing of this paper. All the authors read and approved the final manuscript.
The authors gratefully acknowledge the support of the National Natural Science Foundation of China (NNSFC) through Grants nos. 11572288, 11302184, and 11372015.