Numerical Analysis of Nonlinear Oscillations in Viscously Damped Systems with Finite Degrees of Freedom
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Abstract
Viscously damped finite degrees of freedom nonlinear oscillators appear in a variety of engineering and physical applications, such as vibration control, structural dynamics, and energy dissipation systems. Indeed, classical linear models can provide useful approximations however they do not capture amplitude responsible behavior, internal resonances or transient responses associated with the nonlinear effects of in any time limited time series data. While earlier works highlighted analytical solutions for simple cases or resonant steady state responses, very few numerical studies have addressed the strongly nonlinear case and multi degree of freedom interactions subject to viscous damping.
In response to this dearth of research, this paper provides a systematic framework for numerical analysis of nonlinear oscillatory responses in viscously-damped finite-dimensional systems. The governing nonlinear differential equations are presented and discretized in state space form based on Newtonian mechanics. High order numerical integration schemes are used to carry out time domain simulations, which are followed by phase space analysis, frequency response analysis, and parametric investigations into how much damping coefficients and nonlinear stiffness terms affect dynamic response.
We find strong departures from the linear predictions including frequency shifts that depend on amplitude, nonlinear decay rates, and mode coupling that becomes more pronounced with higher levels of nonlinearity and damping. Through numerical experiments, we show how viscous damping is crucial for the stabilization or suppression of complex oscillatory regimes, which also modifies the transient response properties.
The results shortly elucidate the interplay within the complex non-linear behaviours of such damped systems, thus contributes significantly towards the design, optimization and control of practical mechanical and structural systems where precise predictions of oscillatory response is fundamental.
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