Vibration Suppression of a Harmonically Forced Oscillator via a Parametrically Excited Centrifugal Pendulum

Vibration suppression has been a widely studied topic for a long time, with various modifications in passive vibration mitigation devices to improve the efficacy. One such modification is the addition of the inerter. The inerter has been integrated into various vibration mitigation devices, whose mass amplification effect could be used to enhance the performance of dynamic vibration absorbers. In the current study, we consider an inerter based pendulum vibration absorber (IPVA) system and conduct a theoretical study on vibration suppression of the device. The IPVA system operates based on the principle of nonlinear energy transfer, wherein the energy of the primary structure is transferred into the pendulum vibration absorber. This is the result of parametric resonance of the pendulum, where the primary resonance of the system becomes unstable and a harmonic regime containing a frequency half the resonant frequency emerges (referred to as secondary regime). We use the harmonic balance method along with bifurcation analysis using Floquet theory to study the stability of primary resonance. It is observed that a pitchfork bifurcation and period-doubling bifurcation are necessary for nonlinear energy transfer to occur. Furthermore, we integrate the IPVA with a linear, harmonically forced oscillator to demonstrate its efficacy compared with a linear benchmark. We also examine the effects of various system parameters on the occurrence of the secondary regime. Moreover, we verify the nonlinear energy transfer phenomenon (due to the occurrence of the secondary regime) by numerical Fourier analysis.

An efficient algorithm to solve damped forced oscillator problems by Bernoulli operational matrix of integration

An asymptotic perturbation solution for a linear oscillator of free damped vibrations in fractal medium described by local fractional derivatives was obtained in Yang and Srivastava (Commun Nonlinear Sci Numer Simul 29(1–3):499–504, 2015). In this paper, we obtain the numerical solution of damped forced oscillator problems by employing the operational matrix of integration of Bernoulli orthonormal polynomials. The operational matrix of integration is determined with the help of the integral operator on Bernoulli orthonormal polynomials. Numerical examples of two different problems of spring are given to delineate the performance and perfection of this approach and compared the results with the exact solution.

Vibration Suppression of a Harmonically Forced Oscillator Using a Passive Nonlinear Vibration Absorber

In this paper, the performance of a nonlinear vibration absorber with different nonlinearity is studied. The analytical solutions of periodic motions are obtained using the general harmonic balance method. As the nonlinear strength is weak, the effectiveness of the absorber is discussed. For strong nonlinearities, unstable parodic motions can be obtained and stabilities of the periodic motions are determined through the eigenvalue analysis. The Hopf and saddle bifurcations are observed. Numerical simulations are illustrated for both masses at the resonance peaks. The harmonic amplitude spectrums show the harmonic effects on periodic motions, and the corresponding accuracy of approximate analytical solutions.

A Complete Investigation of the Effect of External Force on a 3D Megastable Oscillator

Multistability is an essential topic in nonlinear dynamics. Recently, two critical subsets of multistable systems have been introduced: systems with extreme multistability and systems with megastability. In this paper, based on a newly introduced megastable system, a megastable forced oscillator is introduced. The effect of adding a forcing term and its parameters on the dynamical behavior of the designed system is investigated. By the help of bifurcation diagram and Lyapunov exponents, it is shown that the modified oscillator can show a variety of dynamical solutions including limit cycle, torus, and strange attractor.

Nonlinear phase coupling functions: a numerical study

Phase reduction is a general tool widely used to describe forced and interacting self-sustained oscillators. Here, we explore the phase coupling functions beyond the usual first-order approximation in the strength of the force. Taking the periodically forced Stuart–Landau oscillator as the paradigmatic model, we determine and numerically analyse the coupling functions up to the fourth order in the force strength. We show that the found nonlinear phase coupling functions can be used for predicting synchronization regions of the forced oscillator. This article is part of the theme issue ‘Coupling functions: dynamical interaction mechanisms in the physical, biological and social sciences’.