Nonlinear Free Vibration of a Symmetrically Conservative Two-Mass System With Cubic Nonlinearity

In this study, the nonlinear free vibration of conservative two degrees of freedom systems is analyzed using the homotopy analysis method (HAM). The mathematical model of such systems is described by two second-order coupled differential equations with cubic nonlinearities. First, novel approximate analytical solutions for displacements and frequencies are established using HAM. Then, the homotopy Padé technique is applied to accelerate the convergence rate of the solutions. Comparison between the obtained results and those available in the literature shows that the first-order approximation of homotopy Padé technique leads to accurate solutions with a maximum relative error less than 0.068 percent for all the considered cases.

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Chaotic Amplitude Dynamics for Parametrically Excited Systems With Quadratic Nonlinearities

Volume 3A: 15th Biennial Conference on Mechanical Vibration and Noise — Vibration of Nonlinear, Random, and Time-Varying Systems ◽

10.1115/detc1995-0284 ◽

1995 ◽

Author(s):

Bappaditya Banerjee ◽

Anil K. Bajaj

Keyword(s):

Harmonic Balance ◽

Chaotic Dynamics ◽

Degrees Of Freedom ◽

Order Approximation ◽

Amplitude Equations ◽

Analytical Technique ◽

First Order ◽

Numerical Studies ◽

Quadratic Nonlinearities ◽

First Order Approximation

Abstract Dynamical systems with two degrees-of-freedom, with quadratic nonlinearities and parametric excitations are studied in this analysis. The 1:2 superharmonic internal resonance case is analyzed. The method of harmonic balance is used to obtain a set of four first-order amplitude equations that govern the dynamics of the first-order approximation of the response. An analytical technique, based on Melnikov’s method is used to predict the parameter range for which chaotic dynamics exist in the undamped averaged system. Numerical studies show that chaotic responses are quite common in these quadratic systems and chaotic responses occur even in presence of damping.

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Periodic Solutions of the Free Vibration of a Two-Degrees-of-Freedom Nonlinear Spring-Mass System

Transactions of the Japan Society of Mechanical Engineers ◽

10.1299/kikai1938.39.1461 ◽

1973 ◽

Vol 39 (321) ◽

pp. 1461-1475

Author(s):

Michio KURODA

Keyword(s):

Free Vibration ◽

Periodic Solutions ◽

Degrees Of Freedom ◽

Mass System ◽

Nonlinear Spring ◽

Two Degrees Of Freedom

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Accurate approximate analytical solutions for nonlinear free vibration of systems with serial linear and nonlinear stiffness

Journal of Sound and Vibration ◽

10.1016/j.jsv.2007.06.061 ◽

2007 ◽

Vol 307 (3-5) ◽

pp. 720-736 ◽

Cited By ~ 15

Author(s):

S.K. Lai ◽

C.W. Lim

Keyword(s):

Free Vibration ◽

Analytical Solutions ◽

Nonlinear Stiffness ◽

Nonlinear Free Vibration ◽

Approximate Analytical Solutions ◽

Linear And Nonlinear

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Parametric Analysis of the Ride Comfort of Indian Railway Vehicle

International Journal of Vehicle Structures and Systems ◽

10.4273/ijvss.13.4.25 ◽

2021 ◽

Vol 13 (4) ◽

Author(s):

Rakesh Chandmal Sharma ◽

Sono Bhardawaj ◽

Mohd Avesh ◽

Neeraj Sharma

Keyword(s):

Mathematical Model ◽

Parametric Analysis ◽

Degrees Of Freedom ◽

Ride Comfort ◽

Rear Wheel ◽

Railway Vehicle ◽

Mass System ◽

Rail Vehicle ◽

Indian Railway ◽

The Mathematical Model

This paper focuses to the parametric analysis of Indian Railway Rajdhani (LHB) coach. A suitable mathematical model of 40 degrees of freedom (DOF) is formulated by Lagrangian method. The mathematical model of rail-vehicle is modelled by considering eleven mass system containing of backseat support (without cushion), a seat, a car body, two (front and Rear) bolsters, two (front and Rear) bogie frame and four wheelaxles (front bogie front and rear wheel axles and rear bogie front and rear wheel axles. The vehicle is simulated to travel at speed of 100 km/hr on a tangent track. The results from the simulation are validated by comparing the same with the results from experimental data which is acquired from research designs and standards organization (RDSO), Lucknow (India). The parametric analysis is performed to estimate the effect of different parameters of rail-vehicle on the ride behaviour.

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Construction of Approximate Analytical Solutions to Strongly Nonlinear Coupled van der Pol Oscillators

Advances in Mechanical Engineering ◽

10.1155/2014/817570 ◽

2014 ◽

Vol 6 ◽

pp. 817570

Author(s):

Y. H. Qian ◽

W. K. Liu ◽

S. M. Chen

Keyword(s):

Analytical Solutions ◽

Degrees Of Freedom ◽

Van Der Pol Oscillator ◽

Practical Significance ◽

Engineering System ◽

Van Der Pol ◽

Approximate Analytical Solutions ◽

Homotopy Analysis ◽

Practical Engineering ◽

Van Der Pol Oscillators

Using nonlinear theory to research vibration model of engineering system has important theoretical and practical significance. Multi-degree-of-freedom (MDOF) coupled van der Pol oscillator is a typical model in the nonlinear vibration; many complex dynamic problems in practical engineering can be simplified as this model to be solved in the end. This paper discusses a class of two-degrees-of-freedom (2-DOF) coupled van der Pol oscillator, which was divided into three parameters of different situations α1≠α2, β1≠β2, and γ1≠γ2 to discuss. Employing symbolic software such as Mathematica for those problems, the explicit analytical solutions of frequency ω and displacements x1( t) and x2( t) are well formulated. Results showed that the homotopy analysis method (HAM) can effectively deal with this kind of parameter of different coupled vibrators, just request the values of some parameters are not too big. Finally, we got four important theorems to simplify the solution of the nonlinear system.

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Linearization of Multibody Dynamic Systems Using Automatic Differentiation (AD) Tools

19th Design Automation Conference: Volume 1 — Mechanical System Dynamics; Concurrent and Robust Design; Design for Assembly and Manufacture; Genetic Algorithms in Design and Structural Optimization ◽

10.1115/detc1993-0287 ◽

1993 ◽

Author(s):

Tsung-Chieh Lin

Keyword(s):

Dynamic Systems ◽

Degrees Of Freedom ◽

Automatic Differentiation ◽

Robot Manipulator ◽

Order Approximation ◽

General Purpose ◽

Dynamics Of Multibody Systems ◽

Closed Chain ◽

Multibody Dynamic ◽

First Order Approximation

Abstract This paper presents an automatic method to linearize the dynamics of multibody systems that are modeled through a recursive approach. The first-order approximation of the nonlinear dynamic systems is obtained by the use of an automatic differentiation (AD) tool (GRESS) and a 9,700 lines Fortran model for the dynamics. The efficiency and accuracy of this AD implementation is shown by two examples: a five-bar closed-chain robot manipulator and a 18 degrees of freedom tractor-trailer. This study successfully demonstrates how to create a general-purpose numerical tool that can provide accurate solutions and derivatives for multibody dynamic systems.

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The method of the averaged Hamiltonian and the two-stream explosive instability

Journal of Plasma Physics ◽

10.1017/s0022377800019565 ◽

1976 ◽

Vol 15 (1) ◽

pp. 1-14 ◽

Cited By ~ 10

Author(s):

Z. Sedláček

Keyword(s):

Degrees Of Freedom ◽

Order Approximation ◽

Wave Interaction ◽

Electrostatic Waves ◽

Explosive Instability ◽

Complex Wave ◽

First Order Approximation ◽

Averaged Hamiltonian ◽

The Waves ◽

Action Variables

Second-order perturbation calculation shows that an explosive instability of three resonantly interacting coherent electrostatic waves can be limited, and converted into a multiple-periodic process by nonlinear terms of the same order as those that destabilize the waves in the first-order approximation. No higher- order nonlinearities are necessary. The method used is purely classical, and consists in transforming the Hamiltonian of the waves into angle-action variables, and canonical averaging of the Hamiltonian over the proper angles. The number of degrees of freedom is thus reduced to one, which permits one to analyse the wave interaction in the phase plane without using the usual equations for the complex wave amplitudes.

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An accurate beam theory and its first-order approximation in free vibration analysis

Journal of Sound and Vibration ◽

10.1016/j.jsv.2020.115567 ◽

2020 ◽

Vol 485 ◽

pp. 115567

Author(s):

Longtao Xie ◽

Shaoyun Wang ◽

Junlei Ding ◽

J Ranjan Banerjee ◽

Ji Wang

Keyword(s):

Free Vibration ◽

Vibration Analysis ◽

Order Approximation ◽

Free Vibration Analysis ◽

Beam Theory ◽

First Order ◽

First Order Approximation

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Non-linear parametric vibrations of a composite column under uniform compression

Proceedings of the Institution of Mechanical Engineers Part C Journal of Mechanical Engineering Science ◽

10.1177/0954406212438494 ◽

2012 ◽

Vol 226 (8) ◽

pp. 1921-1938 ◽

Cited By ~ 1

Author(s):

Jerzy Warmiński ◽

Andrzej Teter

Keyword(s):

Order Approximation ◽

Linear Equations ◽

Multiple Time Scales ◽

Multiple Time ◽

Parametric Vibrations ◽

Uniform Compression ◽

Composite Column ◽

Approximate Analytical Solutions ◽

Non Linear ◽

First Order Approximation

Parametric oscillations of a prismatic, thin-walled composite column with a channel section are considered in the article. The simply supported column is made of a seven-layer composite with a symmetric ply alignment. The non-linear problem of buckling is solved with Koiter’s asymptotic theory within the first-order approximation by adoption of a plate model. The asymptotic approximation leads to non-linear equations allowing evaluation of the two mode buckling effect. Parametric vibrations, produced by a periodically changing load component, are investigated near the principal parametric resonances. The approximate analytical solutions are determined by the multiple time scales of method. The influence of amplitude and frequency of parametric excitation on the structure response is investigated. An example bifurcation scenario and a possible transition to chaotic oscillations are also presented.

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