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Abstract
The paper deals with the dynamics of a lumped mass mechanical system containing two nonlinear springs connected in series. The external harmonic excitation, linear and nonlinear damping are included into considerations. The mathematical model contains both differential and algebraic equations, so it belongs to the class of dynamical systems governed by the differential–algebraic system of equations (DAEs). An approximate analytical approach is used to solve the initial value problem for the DAEs. We adopt the multiple scales method (MSM) that allows one to obtain the sufficiently correct approximate solutions both far from the resonance and at the resonance conditions. The steady and non-steady resonant vibrations are analyzed by employing the modulation equations of the amplitudes and phases which are yielded by the MSM procedure.
Highlights
The mechanical systems which contain parallel or serially connected massless springs are widely investigated and discussed in the theoretical and applied mechanics
When two time scales are adopted, the modulation equations have the form of the following differential equations of the first order daðsÞ 1⁄4 À 1 caðsÞ; ð25Þ
The asymptotic analysis leads to the following form of the solution to the initial-value problem given by Eqs. (58)–(60): Fig. 13 Time history of the transient and steady non-resonant vibration: solid line—asymptotic solution for two and three time scales (MSM2 and MSM3), and dashed line—numerical solution x2ðsÞ
Summary
The mechanical systems which contain parallel or serially connected massless springs are widely investigated and discussed in the theoretical and applied mechanics. Various configurations of the connections between the springs, including their spatial orientation, can lead to the complex dynamical behavior of those systems, especially when the elastic elements have the nonlinear characteristics. Telli and Kopmaz [3] studied a one-dimensional oscillator mounted via two springs wherein one of them is linear and the second one has nonlinear features They proposed two mathematical models for the system considered. Another application of the similar system one can find in the paper [5], where the springs assembled both in the serial and parallel configurations are used by the authors as a model of the structural equivalent stiffness In that way, they produce a powerful procedure to study structural behavior.
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