Resonance, Parameter Estimation, and Modal Interactions in a Strongly Nonlinear Benchtop Oscillator

Abstract

We study the vibrations of a strongly nonlinear, electromechanically forced, benchtop experimental oscillator. We consciously avoid first-principles derivations of the governing equations, with an eye towards more complex practical applications where such derivations are difficult. Instead, we spend our effort in using simple insights from the subject of nonlinear oscillations to develop a quantitatively accurate model for the single-mode resonant behavior of our oscillator. In particular, we assume an SDOF model for the oscillator; and develop a structure for, and estimate the parameters of, this model. We validate the model thus obtained against experimental free and forced vibration data. We find that, although the qualitative dynamics is simple, some effort in the modeling is needed to quantitatively capture the dynamic response well. We also briefly study the higher dimensional dynamics of the oscillator, and present some experimental results showing modal interactions through a 0:1 internal resonance, which has been studied elsewhere. The novelty here lies in the strong nonlinearity of the slow mode.

Dynamics of zero-Prandtl number convection near onset

“…40,41 The stability of the fixed points are ascertained through an eigenvalue analysis of the Jacobian. To generate the bifurcation diagram, we start first with the fixed points of the system.…”

Nonlinear dynamic behavior and instability of slender frames with semi-rigid connections

“…Various approaches, including perturbation methods and the harmonic balance method, have been developed to study nonlinear oscillators with a small number of degrees of freedom. Here the methodology proposed by Nandakumar and Chatterjee [48] is used to obtain this relation using the finite element method. First, the nonlinear equations of motion are numerically integrated, and the time response of the slightly damped system is obtained for a chosen node.…”

Nonlinear secondary whirl of an overhung rotor

“…Equilibrium solution branches are numerically computed using an arclengthbased branch following scheme (see Nandakumar & Chatterjee 2005), as is varied in parameter space. Saddle-node bifurcations, jump phenomena and hysteresis, familiar in nonlinear resonances, are all observed.…”

“…The slow flow is also linearized about each fixed point, and eigenvalues of the linearized system are used to determine stability (as depicted in figure 2). Equilibrium solution branches are numerically computed using an arclengthbased branch following scheme (see Nandakumar & Chatterjee 2005), as is varied in parameter space. Saddle-node bifurcations, jump phenomena and hysteresis, familiar in nonlinear resonances, are all observed.…”