Transitions and statistical characteristics of vibrations in a bimodular oscillator

Abstract

Regular and stochastic oscillations in a simple periodically forced vibroacoustic system with a piecewise-linear (bimodular) elasticity are considered from the viewpoint of their statistical properties: oscillation spectra, the largest Lyapunov exponents, and fractal dimension. It is shown that a strange attractor exists in a limited range of parameters together with a triple-period dynamic cycle being absorbed by the latter at some finite value of the parameter. The corresponding bifurcation process is followed in detail. (c) 1995 American Institute of Physics.

Nonlinear distortion of signals radiated by vibroseis sources

“…Note that, as stated in the Introduction, the nonlinearity is assumed to be localized at the contact, while the wave propagation from the contact is assumed to be linear. General analysis of oscillatory systems with piecewise-linear parameters has been performed by several workers (e.g., Mahfouz and Badrakhan, 1990;Ostrovsky and Starobinets, 1995). In particular, models with bimodular elasticity are often invoked to describe strong nonlinearity in vibratory systems with backlash, a contact of one piece with another, or behavior of precompressed elements (Mahfouz and Badrakhan, 1990).…”

“…In particular, models with bimodular elasticity are often invoked to describe strong nonlinearity in vibratory systems with backlash, a contact of one piece with another, or behavior of precompressed elements (Mahfouz and Badrakhan, 1990). The bimodular elasticity model (5) provides qualitatively similar time histories z j (t) for any amplitude of the exciting force (Mahfouz and Badrakhan, 1990;Ostrovsky and Starobinets, 1995). This means that the harmonic distortion depends on the ratio K 1 /K 2 only and not on the driving-force level.…”

“…The free vibrations of a piecewise-linear oscillatory system consist of the spliced half-periods of sinusoidal functions, as illustrated in Figure 7. The top half is a sinusoidal signal with amplitude a 1 and half-period T 1 and the bottom half is a sinusoidal signal with amplitude a 2 and half-period T 2 (Ostrovsky and Starobinets, 1995). Figure 7 corresponds to the case of bimodular nonlinearity [equation 5].…”

“…where 1,2 = 2π/T 1,2 (Ostrovsky and Starobinets, 1995). The nature of the shape of the time history is clear: the system Figure 6.…”

On the nonlinear normal modes of free vibration of piecewise linear systems

“…Finite degree of freedom dynamical systems with piecewise linear elastic forces can be used to describe the vibrations of a wide class of machines and mechanisms. Therefore many efforts have been made to analyze the dynamics of such systems [1][2][3][4][5]. Free, forced and parametric vibrations of mechanical systems have been investigated by the Shaw-Pierre nonlinear normal modes [6].…”