Nonlinear vibration of a lumped system with springs-in-series
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MODELLING AND ANALYSIS OF MECHANICAL SYSTEMS DYNAMICS
Nonlinear vibration of a lumped system with springs-inseries Jan Awrejcewicz . Roman Starosta
. Gra_zyna Sypniewska-Kamin´ska
Received: 12 March 2020 / Accepted: 30 October 2020 Ó The Author(s) 2020
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. Keywords Lumped system Nonlinear dynamics Asymptotic analysis Resonance Modulation equations
J. Awrejcewicz Department of Automation, Biomechanics and Mechatronics, Lodz University of Technology, Stefanowskiego Str. 1/15, 90-924 Lodz, Poland R. Starosta (&) G. Sypniewska-Kamin´ska Institute of Applied Mechanics, Poznan University of Technology, ul. Piotrowo 3, 60-965 Poznan, Poland e-mail: [email protected]
1 Introduction The mechanical systems which contain parallel or serially connected massless springs are widely investigated and discussed in the theoretical and applied mechanics. They have found applications in mechanical and civil engineering, mechatronic devices, and more recently in micromechanical systems. Various configurations of the connections between the springs, including also their spatial orientation, can lead to the complex dynamical behavior of those systems, especially when the elastic elements have the nonlinear characteristics. Such systems could exhibit a variety of interesting behaviors, sometimes even surprising which especially concern the resonance states. Models of many real systems demand to introduce rigid body approximation where some springs and dampers are connected in various configurations. The car suspension containing systems of the parallel and serially connected springs is investigated in [1, 2]. The authors showed that such connections have a great impact on the vibration transmissibility from the rough road to the car body. 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. The differential–algebraic equations on which the first approach is employed have been solved numerically. The second model
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Meccanica
based on a single differential equation,
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