| dc.description.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. | en_US |
