Rotating vector solving method applied for nonlinear oscillator

Significant number of procedures for solving of the finite degree-of-freedom forced nonlinear oscillator are developed. For all of them it is common that they are based on the exact solution of the corresponding linear oscillator. For technical reasons, the aim of this paper is to develop a simpler solving procedure. The rotating vector method, developed for the linear oscillator, is adopted for solving of the nonlinear finite degree-of-freedom oscillator. The solution is assumed in the form of trigonometric functions. Assuming that the nonlinearity is small all terms of the series expansion of the function higher than the first are omitted. The rotating vectors for each mass are presented in the complex plane. In the paper, the suggested rotating vector procedure is applied for solving of a three-degree-of-freedom periodically excited oscillator. The influence of the nonlinear stiffness of the flexible elastic beam, excited with a periodical force, on the resonant properties of the system in whole is investigated. It is obtained that the influence of nonlinearity on the amplitude and phase of vibration is more significant for smaller values of the excitation frequency than for higher ones.

Multiple Coexisting Attractors and Hysteresis in the Generalized Ueda Oscillator

A periodically forced nonlinear oscillator called the generalized Ueda oscillator is proposed. The restoring force term of this equation consists of a nonlinear functionsgn(x)and an absolute function with a variant power. Dynamics is investigated by detailed numerical analysis as well as dynamic simulation, including the largest Lyapunov exponent, phase diagrams, and bifurcation diagrams. Multiple coexisting attractors and complex hysteresis phenomenon are observed. The results show that this system has rich dynamical behaviors, and it has a promising application in the fields of science and engineering.

Application of a Weakly Nonlinear Absorber to Suppress the Resonant Vibrations of a Forced Nonlinear Oscillator

A weakly nonlinear vibration absorber is used to suppress the primary resonance vibrations of a single degree-of-freedom weakly nonlinear oscillator with periodic excitation, where the two linearized natural frequencies of the integrated system are not under any internal resonance conditions. The values of the absorber parameters are significantly lower than those of the forced nonlinear oscillator, as such the nonlinear absorber can be regarded as a perturbation to the nonlinear primary oscillator. The characteristics of the nonlinear primary oscillator change only slightly in terms of its new linearized natural frequency and the frequency interval of primary resonances after the nonlinear absorber is added. The method of multiple scales is employed to obtain the averaged equations that determine the amplitudes and phases of the first-order approximate solutions. Selection criteria are developed for the absorber linear stiffness (linearized natural frequency) and nonlinear stiffness in order to achieve better performance in vibration suppression. Illustrative examples are given to show the effectiveness of the nonlinear absorber in suppressing nonlinear vibrations of the forced oscillator under primary resonance conditions.

POINCARÉ PLANES IN NONLINEAR ELECTRONICS

Details of electronic circuitry to define Poincaré planes in the phase space of nonlinear electronic systems are presented. It allows an experimental setup to capture data at every moment the system's orbit crosses the Poincaré plane. We illustrate how the circuit is used in an experimental setup that allows us (i) to reconstruct bifurcation cascades and to disclose induced first return chaotic maps in a harmonically forced nonlinear oscillator, and (ii) to study bistable switching in Chua's oscillator.