Propagation characteristics in elastic waveguide with discrete nonlinearity

Abstract

Wave propagation in elastic continuum is a subject of interest in many fields of engineering and has been explored for several decades from theoretical, computational and experimental perspective. In this context, a one/two-dimensional elastic continuum is more colloquially called a wave guide since they minimize energy loss by restricting the wave propagation in a specific direction/plane. These one-dimensional elastic wave guides can be dispersion free, for example, an elastic bar propagating longitudinal waves, or can exhibit dispersion, for example, flexural waves in Euler-Bernoulli or Timoshenko beams. However, in practical engineering structures, such wave guides invariably interact with structural components whose spatial scales are much smaller than the elastic continuum and the wavelength of the wave phenomena that they encounter. Such structural components can be modelled as discrete elements by considering point masses, stiffness and damping. The wave propagation characteristics when elastic wave guides interact with linear discrete elements are well known. The present work primarily dwells on the effect of weakly/strongly nonlinear discrete elements on the wave propagation characteristics in nondispersive elastic wave guides. Since, in general, closed form analytical solutions are seldom available for weakly/strongly/essentially nonlinear dynamical systems, in this thesis we propose a systematic approach based on classical perturbation techniques like Method of Multiple Scales (MMS) to find out the solutions. Firstly, we investigate an elastic continuum coupled to a Duffing oscillator with cubic stiffness nonlinearity through a linear spring. The elastic continuum is modelled as a simple, onedimensional bar. The propagation characteristics, the response of the bar and the oscillator motion are found out analytically by using a combination of Multiple Time Scales Analysis and Harmonic Balance method. The excitation pulse is considered to be a sinusoidal displacement function, imposed on the free end of the bar. The closed-form solutions for the responses are determined at different levels of approximation for the cases of primary, superharmonic and subharmonic resonance and also for a general non-resonant case. These closed-form solutions can be used as a reference to check solutions obtained by other standard procedures. Secondly, we study a system comprising of two simple, one-dimensional bars coupled to each other by means of a weakly damped, weakly nonlinear oscillator with cubic stiffness nonlinearity. The input condition is the same as in the previous case, i.e. a sinusoidal pulse applied to one of the bars as a displacement boundary condition. The closed-form solutions for the responses are found out for a general non-resonant case. The procedure for determining the responses analytically in the cases of primary and secondary resonance is also discussed. The same methodology is used to determine the responses. These closed-form solutions can be useful while analysing real-world systems like a two-stage rocket, where the stages can be modelled as simple, elastic bars and the connection between the two systems can be modelled as a nonlinear oscillator, such as the Duffing one or the Van der Pol one. Thirdly, a system comprising of a simple, one-dimensional bar attached with a snap-through truss is studied. The input condition is the same as for the previous systems, i.e. a sinusoidal pulse imposed on the free end of the bar as a displacement boundary condition. This system is essentially nonlinear and it is not possible to find a closed-form solution using standard procedures. However, solutions are provided for small amplitudes of the excitation pulse, taking some relevant assumptions. The same method based on Multiple Time Scales Analysis is used to determine the responses. Apart from providing analytical solutions, the problem is also solved numerically for the same system parameters using a Finite Difference algorithm. The close similarity between the analytical solutions and the numerical solutions is provided. A stability analysis is also done for the snap-through oscillator for sufficiently large amplitudes of the excitation pulse and the stable and unstable regions are shown. This stability diagram can be useful while determining important system input parameters like excitation amplitude and excitation frequency for which the oscillator will snap from one stable equilibrium position to another.