Analytical study of the transition curves in the bi-linear Mathieu equation
- PDF / 560,430 Bytes
- 13 Pages / 547.087 x 737.008 pts Page_size
- 32 Downloads / 159 Views
(0123456789().,-volV) ( 01234567 89().,-volV)
ORIGINAL PAPER
Analytical study of the transition curves in the bi-linear Mathieu equation K. R. Jayaprakash . Yuli Starosvetsky
Received: 9 March 2020 / Accepted: 4 August 2020 Springer Nature B.V. 2020
Abstract The current work is primarily devoted to the asymptotic analysis of the instability zones existing in the bi-linear Mathieu equation. In this study, we invoke the common asymptotical techniques such as the method of averaging and the method of multiple time scales to derive relatively simple analytical expressions for the transition curves corresponding to the 1:n resonances. In contrast to the classical Mathieu equation, its bi-linear counterpart possesses additional instability zones (e.g. for n [ 2). In this study, we demonstrate analytically the formation of these zones when passing from linear to bi-linear models as well as show the effect of the stiffness asymmetry parameter on their width in the limit of low amplitude parametric excitation. We show that using the analytical prediction devised in this study one can fully control the width of the resonance regions through the choice of asymmetry parameter resulting in either maximum possible width or it’s complete annihilation. Results of the analysis show an
K. R. Jayaprakash (&) Discipline of Mechanical Engineering, Indian Institute of Technology Gandhinagar, Gandhinagar, Gujarat, India e-mail: [email protected] Y. Starosvetsky Faculty of Mechanical Engineering, Israel Institute of Technology-Technion, Haifa, Israel e-mail: [email protected]
extremely good correspondence with the numerical simulations of the model. Keywords Bi-linear Mathieu equation Action– angle variables Instability boundaries
1 Introduction The bi-linear systems frequently model various engineering structures comprising of moving elements with intermittent contacts. As a simplest bi-linear model, one may think of a one degree of freedom (1DOF) oscillator mounted on an elastic spring with different stiffness characteristics in tension and compression. Theoretical understanding of the response regimes of bi-linear oscillatory (BLO) models subjected to various types of external loading is crucial in various engineering applications such as machine tool cutting and milling processes [1, 2], dynamics of cracked structures [3–7], dynamics of suspension bridges [8], modelling of the topological interlocking structures [9, 10] and more. Analysis of the response regimes of these special dynamical systems is quite challenging due to their essential nonlinearity and non-smoothness. Many computational and analytical attempts have been devoted to understanding of the complex dynamics of a 1DOF BLO model subject to various types of external loading such as harmonic and parametric
123
K. R. Jayaprakash, Y. Starosvetsky
forcing. It is worth noting that in the absence of external forcing, analysis of free vibrations of bi-linear oscillator becomes rather simple as the system can be split into two separate linear systems f
Data Loading...
