Nonlinear vibration of a lumped system with springs-in-series

Applied Mathematical Modelling, 2010

In this paper, the vibration of a mass grounded system which includes two linear and nonlinear springs in series has been considered. Since this system, depending on its parameters can oscillate symmetrically and asymmetrically, both cases have been solved using multiple times scales (MTS) method and some analytical relations have been obtained for natural frequency of oscillations. The results have been compared with previous work and good agreement has been obtained. Also forced vibrations of system in primary and secondary resonances have been studied for the first time and the effects of different parameters on the frequency-response have been investigated.

Journal of Low Frequency Noise, Vibration and Active Control

An analytical technique has been developed based on the harmonic balance method to obtain approximate angular frequencies. This technique also offers the periodic solutions to the nonlinear free vibration of a conservative, couple-mass-spring system having linear and nonlinear stiffnesses with cubic nonlinearity. Two real-world cases of these systems are analysed and introduced. After applying the harmonic balance method, a set of complicated higher-order nonlinear algebraic equations are obtained. Analytical investigation of the complicated higher-order nonlinear algebraic equations is cumbersome, especially in the case when the vibration amplitude of the oscillation is large. The proposed technique overcomes this limitation to utilize the iterative method based on the homotopy perturbation method. This produces desired results for small as well as large values of vibration amplitude of the oscillation. In addition, a suitable truncation principle has been used in which the solutio...

2015

The aim of the paper is analysis of dynamical regular response of the nonlinear oscillator with two serially connected springs of cubic type nonlinearity. Behaviour of such systems is described by a set of differential-algebraic equations (DAEs). Two examples of systems are solved with the help of the asymptotic multiple scales method in time domain. The classical approach has been appropriately modified to solve the governing DAEs. The analytical approximated solution has been verified by numerical simulations.

Arabian Journal for Science and Engineering, 2012

We consider periodic solution for coupled systems of mass-spring. Two practical cases of these systems are explained and introduced. An analytical technique, called the Hamiltonian approach, is applied to calculate approximations to the achieved nonlinear differential oscillation equations. The concept of the Hamiltonian approach is briefly introduced, and its application for nonlinear oscillators is studied. The method introduces an alternative to overcome the difficulty of computing the periodic behavior of the oscillation problems in engineering. The results obtained employing first-order and second-order Hamiltonian approach are compared with those achieved using two other analytical techniques, named the energy balance method and the amplitude frequency formulation, and also to assess the accuracy of solutions, the results were compared with the exact ones. The results indicate that the present analysis is straightforward and provide us a unified and systematic procedure which is simple and more accurate than the other similar methods. In short, this new approach yields extended scope of applicability, simplicity, flexibility in application, and avoidance of complicated numerical integration as compared with the previous approaches such as the perturbation and the classical harmonic balance methods.

2018

Dynamic systems are generally categorized as linear and non-linear systems based on the analysis approaches developed. Bi-linear spring mass systems form an intermediate category which possesses characteristics of both linear and non-linear systems. Analytical solutions to bi-linear systems cannot in general be reduced to a single equation covering the entire motion domain as in the case of linear systems. Nevertheless, the step-wise linear solutions can be obtained within each domain and the solutions can be related to one other from displacement and velocity continuity requirement for mechanical systems. Detailed analytical solution to single degree of freedom bi-linear spring mass system free vibration is presented in this paper towards deriving equations for time period as well as logarithmic decrement. Displacement versus time as well as velocity versus time plots is generated to demonstrate bi-linear system behavior against the linear system behavior. The solution developed is...

Nonlinear Dynamics, 2006

This paper presents a fixed-end two-mass system (TMS) with end constraints that permits uncoupled solutions for different masses. The coupled nonlinear models for the present fixed-end TMS were solved using the continuous piecewise linearization method (CPLM) and detailed investigation on the effect of mass-ratio on the TMS response was conducted. The investigations showed that increased mass-ratio leads to decreased oscillation frequency and an asymptotic response was obtained at very large mass-ratios. Theoretical solutions to determine the asymptotic response were derived. Also, it was observed that distinct responses can be obtained for the same mass-ratio depending on the mass combination in the TMS. The present fixed-end TMS and the analyses presented give a broader understanding of fixed-end TMS.

Mathematical and Computational Applications, 1997

A mathematical model covering many practical vibration problems of continuous systems has been proposed. The equations of motion consist of two nonlinearly coupled partial differential equations. The quadratic and cubic nonlinearities as well as the linear part of the equations are represented by arbitrary operators. A perturbation approach (method of multiple time scales) has been applied directly to the partial differential equations. The responses as well as the amplitude and phase modulation equations are found for the case of primary resonances of the external excitation and one-to-one internal resonances between the natural frequencies of the equations. The coefficients of the amplitude and phase modulation equations are calculated in their most general form. Results are then applied to a nonlinear cable vibration problem having small sagto-span ratios.

Mechanics Research Communications, 2001

Journal of Sound and Vibration, 2009

A general vibrational model of a continuous system with arbitrary linear and cubic operators is considered. Approximate analytical solutions are found using the method of multiple scales. The primary resonances of the external excitation and three-to-one internal resonances between two arbitrary natural frequencies are treated. The amplitude and phase modulation equations are derived. The steady-state solutions and their stability are discussed. The solution algorithm is applied to two specific problems: (1) axially moving Euler-Bernoulli beam, and (2) axially moving viscoelastic beam.

Proceedings of SPIE - The International Society for Optical Engineering