scholarly journals Asymptotic analysis and accurate approximate solutions for strongly nonlinear conservative symmetric oscillators

2017 ◽  
Vol 49 ◽  
pp. 243-254 ◽  
Author(s):  
Baisheng Wu ◽  
Weijia Liu ◽  
Xin Chen ◽  
C.W. Lim
Keyword(s):  
Asymptotic Analysis ◽  
Approximate Solutions ◽  
Strongly Nonlinear

Free and Forced Vibrations of the Strongly Nonlinear Cubic-Quintic Duffing Oscillators

2017 ◽  
Vol 72 (1) ◽  
pp. 59-69 ◽  
Author(s):  
M.M. Fatih Karahan ◽  
Mehmet Pakdemirli
Keyword(s):  
Numerical Simulations ◽  
Numerical Solutions ◽  
Multiple Scales ◽  
Approximate Solutions ◽  
Response Curves ◽  
Strongly Nonlinear ◽  
Free And Forced Vibrations ◽  
Approximate Analytical Solutions ◽  
Strongly Nonlinear Oscillators ◽  
Good Agreement

AbstractStrongly nonlinear cubic-quintic Duffing oscillatoris considered. Approximate solutions are derived using the multiple scales Lindstedt Poincare method (MSLP), a relatively new method developed for strongly nonlinear oscillators. The free undamped oscillator is considered first. Approximate analytical solutions of the MSLP are contrasted with the classical multiple scales (MS) method and numerical simulations. It is found that contrary to the classical MS method, the MSLP can provide acceptable solutions for the case of strong nonlinearities. Next, the forced and damped case is treated. Frequency response curves of both the MS and MSLP methods are obtained and contrasted with the numerical solutions. The MSLP method and numerical simulations are in good agreement while there are discrepancies between the MS and numerical solutions.

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

2017 ◽  
Vol 2017 ◽  
pp. 1-8 ◽  
Author(s):  
Y. H. Qian ◽  
J. L. Pan ◽  
S. P. Chen ◽  
M. H. Yao
Keyword(s):  
Exact Solutions ◽  
Nonlinear Vibration ◽  
Harmonic Balance ◽  
Nonlinear Dynamical Systems ◽  
Harmonic Balance Method ◽  
Time History ◽  
Approximate Solutions ◽  
Balance Method ◽  
Lumped Mass ◽  
Strongly Nonlinear

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.

scholarly journals A perturbed algorithm for strongly nonlinear variational-like inclusions

2000 ◽  
Vol 62 (3) ◽  
pp. 417-426 ◽  
Author(s):  
C.-H. Lee ◽  
Q. H. Ansari ◽  
J.-C. Yao
Keyword(s):  
Exact Solution ◽  
General Setting ◽  
Approximate Solutions ◽  
Strongly Nonlinear ◽  
Special Cases ◽  
The One

In this paper, we define the concept of η- subdifferential in a more general setting than the one used by Yang and Craven in 1991. By using η-subdifferentiability, we suggest a perturbed algorithm for finding the approximate solutions of strongly nonlinear variational-like inclusions and prove that these approximate solutions converge to the exact solution. Several special cases are also discussed.

An Asymptotic Analysis of Three-Dimensional Extrusion

1989 ◽  
Vol 56 (3) ◽  
pp. 519-526 ◽  
Author(s):  
N. Aravas ◽  
R. M. McMeeking
Keyword(s):  
Asymptotic Analysis ◽  
Cross Sections ◽  
Three Dimensional ◽  
Constitutive Models ◽  
Approximate Solutions ◽  
The Finite Element Method ◽  
Wide Range ◽  
Metal Extrusion ◽  
Asymptotic Perturbation ◽  
Generalized Plane Strain

A new method of analysis of three-dimensional metal extrusion using asymptotic perturbation methods is presented in this paper. The plasticity model used depends on the first and second invariants of the stress tensor and covers a wide range of constitutive models commonly used for the analysis of metal-forming operations. It is shown that the three-dimensional extrusion problem can be approximated, to leading order, by a problem of generalized plane-strain. The results of the asymptotic analysis together with the finite element method are used to obtain approximate solutions for extrusions of elliptic or square cross-sections from round billets.

scholarly journals Energy Method to Obtain Approximate Solutions of Strongly Nonlinear Oscillators

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Alex Elías-Zúñiga ◽  
Oscar Martínez-Romero
Keyword(s):  
Potential Energy ◽  
Analytical Solutions ◽  
Energy Method ◽  
Equations Of Motion ◽  
Nonlinear Oscillators ◽  
Approximate Solutions ◽  
Equivalent Representation ◽  
Strongly Nonlinear ◽  
Representation Form ◽  
Strongly Nonlinear Oscillators

We introduce a nonlinearization procedure that replaces the system potential energy by an equivalent representation form that is used to derive analytical solutions of strongly nonlinear conservative oscillators. We illustrate the applicability of this method by finding the approximate solutions of two strongly nonlinear oscillators and show that this procedure provides solutions that follow well the numerical integration solutions of the corresponding equations of motion.

scholarly journals A modified averaging operator with some applications

2020 ◽  
Vol 42 (3) ◽  
pp. 343-356
Author(s):  
Anh Tay Nguyen ◽  
N. D. Anh
Keyword(s):  
Approximate Solutions ◽  
Boundary Values ◽  
Remarkable Feature ◽  
New Approach ◽  
Strongly Nonlinear ◽  
Averaging Operator ◽  
Boundary Regulation ◽  
Two Parameters ◽  
Local Value

The paper presents a new approach to the conventional averaging in which the role of boundary values is considered in a more detailed way. It results in a new weighted local averaging operator (WLAO) taking into account the particular role of boundary values. A remarkable feature of WLAO is that this operator contains a parameter of boundary regulation p and depends on a local value $h$ of the integration domain. By varying these two parameters one can regulate the obtained approximate solutions in order to get more accurate ones. It has been shown that the combination of WLAO with Galerkin method can lead to an effective approximate tool for the buckling problem of columns and for the frequency analysis of free vibration of strongly nonlinear systems. 

Study of Energy Methods to Strongly Nonlinear Vibrations in a Loudspeaker Cone-Shaped Shell

2005 ◽  
Author(s):  
F. D. Zong ◽  
Z. L. Zhang ◽  
J. W. Fang ◽  
Y. J. Yu ◽  
Q. Chen
Keyword(s):  
Magnetic Field ◽  
Limit Cycles ◽  
Nonlinear Oscillations ◽  
Low Frequency ◽  
Approximate Solutions ◽  
Energy Methods ◽  
Frequency Range ◽  
Strongly Nonlinear ◽  
Harmonic Vibrations ◽  
Strongly Nonlinear Oscillations

H. F. Olson points out that a loudspeaker cone-shaped shell, as a nonlinear oscillation system, can be described as the Classical Duffing Equation in low frequency range. Yoshinisa, a Japanese scholar, studied the nonlinear phenomena of the loudspeaker cone-shaped shell in low frequency range driven by a stable galvanic source, including the resonance frequency changing with amplitude and leap phenomena. But their research were not taken the influence of nonlinear magnetic field into account. Its work mostly related to getting solution of nonlinear differential equation by the Numerical Calculation, but it didn’t get approximate solutions. Through research and analysis of the experiment on the loudspeaker cone-shaped shell, we obtain the Generalized Duffing Equation that’s a strongly nonlinearity system which is used to describe the loudspeaker cone-shaped shell driven by a stable voltage source, it considers the nonlinearity of mechanical resilience and the magnetic field. This paper focuses on first finding the approximate solutions (limit cycles) of strongly nonlinear oscillations and nonlinear heteronomy of the loudspeaker cone-shaped shell in low frequency range by use of energy methods. They obtained the equation relating to the forced vibration amplitude with frequency and the corresponding relation about phase versus frequency, and analysed particularly complete stability of limit cycles belonged to the strongly nonlinear systems, and drew several important conclusions. (1) As to strongly nonlinear oscillations of the loudspeaker cone-shaped shell in low frequency range, it is only likely to appear main oscillation and odd-order sub-harmonic oscillations. But it cannot appear super-harmonic vibrations and even-order sub-harmonic vibrations. (2) As to strongly nonlinear oscillations of the loudspeaker cone-shaped shell in low frequency range, two cases about main oscillation and one third sub-harmonic oscillation whose approximate solutions accord with numerical solutions very well. (3) It is worthy to study strongly nonlinear oscillations of commonly thin shell structure such as a loudspeaker cone-shaped shell by use of energy methods, and we will continue to carry out this research.

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