Dynamics of Nonlinear Oscillators under Simultaneous Internal and External Resonances (original) (raw)

Dynamic Modeling and Stability Analysis of a Nonlinear System with Primary Resonance

International Journal for Innovation Education and Research, 2020

In recent years, there has been growing interest in the study of nonlinear phenomena. This is due to the modernization of structures related to the need of using lighter, more resistant and flexible materials. Thus, this work aims to study the behavior of a mechanical system with two degrees of freedom with nonlinear characteristics in primary resonance. The structure consists of the main system connected to a secondary system to act as a Nonlinear Dynamic Vibration Absorber, which partially or fully absorbs the vibrational energy of the system. The numerical solutions of the problem are obtained using the Runge-Kutta methods of the 4th order and approximate analytical solutions are obtained using the Multiple Scales Method. Then, the approximation error between the two solutions is analyzed. Using the aforementioned perturbation method, the responses for the ordinary differential equations of the first order can be determined, which describe the modulation amplitudes and phases. Th...

Near the resonance behavior of a periodicaly forced partially dissipative three-degrees-of-freedom mechanical system

Latin American Journal of Solids and Structures, 2018

In this paper, a nonlinear three-degrees-of-freedom dynamical system consisting of a variable-length pendulum mass attached by a massless spring to the forced slider is investigated. Numerical solution is preceded by application of Euler-Lagrange equation. Various techniques like time histories, phase planes, Poincaré maps and resonance plots are used to observe and identify the system responses. The results show that the variable-length spring pendulum suspended from the periodically forced slider can exhibit quasi-periodic, and in a resonance state, even chaotic motions. It was concluded that near the resonance the influence of coupling of bodies on the system dynamics can lead to unpredictable dynamical behavior.

Nonlinear Vibrations, Stability Analysis, and Control

Mathematical Problems in Engineering, 2010

Important advances in mathematics, physics, biology, economics, and engineering science have shown the importance of the analysis of nonlinear vibrations, instabilities, and strongly coupled dynamical behavior.

TRANSIENT RESONANCE OSCILLATIONS OF A MECHANICAL SYSTEM WITH REGARD TO NON-LINEAR DAMPING EFFECTS

Proceedings of DETC’99 1999 ASME Design Engineering Technical Conferences September 12-15, 1999, Las Vegas, Nevada, USA

Transient, forced vibrations of a mechanical system are considered. Dissipative effects such as material damping, aerodynamic damping and damping at interfaces are taken into account. For the modeling of these effects the set of isolated weakly non-linear single degree of freedom oscillators with damping dependent non-linearities is used. The superposition of their responses approximates the resulting response of the system. The justification of this assumption is discussed. The Krylov-Bogoljubov asymptotic method is applied for the investigation of the transient resonance response. Numerical calculations are provided to demonstrate the validity of the Krylov-Bogoljubov first approximation.

The Stability Analysis of a Vibrating Auto-Parametric Dynamical System Near Resonance

Applied Sciences, 2022

This paper examines a new vibrating dynamical motion of a novel auto-parametric system with three degrees of freedom. It consists of a damped Duffing oscillator as a primary system attached to a damped spring pendulum as a secondary system. Lagrange’s equations are utilized to acquire the equations of motion according to the number of the system’s generalized coordinates. The perturbation technique of multiple scales is applied to provide the solutions to these equations up to a higher order of approximations, with the aim of obtaining more accurate novel results. The categorizations of resonance cases are presented, in which the case of primary external resonance is examined to demonstrate the conditions of solvability of the steady-state solutions and the equations of modulation. The time histories of the achieved solutions, the resonance curves in terms of the modified amplitudes and phases, and the regions of stability are outlined for various parameters of the considered system...

Transient Resonance Oscillations of a Slow-Variant System with Small Non-Linear Damping—Modelling and Prediction

Journal of Sound and Vibration, 2000

The transient response of a single-degree-of-freedom oscillator with a slow-variant natural frequency and a small non-linear damping is under consideration. The damping is modelled as a sum of elementary power functions with respect to the system velocity. The passage through a resonance which is induced by a sweep of the excitation frequency during run-up or run-down is studied using the Krylov–Bogoljubov asymptotic method. Numerical calculations are presented to demonstrate the validity of the first asymptotic approximation. Asymptotic approximations for the maximum transient response and the corresponding excitation frequency are derived analytically in the particular case of a system with linear viscous damping. The obtained formulae are tested numerically and compared to known approximations.

Free and forced vibration analysis of a nonlinear system with cyclic symmetry: Application to a simplified model

Journal of Sound and Vibration, 2012

This work program is devoted to studying the nonlinear dynamics of a structure with cyclic symmetry under conditions of geometric nonlinearity, through the use of the harmonic balance method (HBM). In order to study the influence of nonlinearity due to the large deflection of blades, a simplified model has been developed. This approach leads to a system of linearly coupled, second-order nonlinear differential equations, in which nonlinearity appears via cubic terms. Periodic solutions, in both the free and forced cases, are sought by applying HBM coupled with an arc-length continuation method. Solution stability has been investigated using Floquet's theorem. In addition to featuring similar and nonsimilar nonlinear modes, the unforced system is known to contain localized nonlinear modes that arise from branching point bifurcation at certain vibration amplitudes. In the forced case, these nonlinear modes give rise to a complex dynamic behavior. Many bifurcations can take place, thus leading to strong or weak localization that may or may not be stable. In this study, special attention has been paid to the influence of excitation on dynamic responses. Several cases of excitation have been analyzed herein: localized excitation, and low-engine-order excitation. In the case of low-engine-order excitation, sensitivity of the response to a perturbation of this excitation type has been investigated, and it has been shown that for a localized, or sufficiently detuned excitation, several solutions can coexist, some of which are represented by closed curves in the Frequency-Amplitude domain. These various solutions overlap when increasing the force amplitude, leading to forced nonlinear localization. Because closed curves are not tied up with the basic nonlinear solution, they can easily be overlooked. In this study, they have been calculated using a sequential continuation with the force amplitude as a parameter.

Qualitative analysis of a mechanical system of coupled nonlinear oscillators

Electronic Journal of Qualitative Theory of Differential Equations

In this paper we investigate nonlinear systems of second order ODEs describing the dynamics of two coupled nonlinear oscillators of a mechanical system. We obtain, under certain assumptions, some stability results for the null solution. Also, we show that in the presence of a time-dependent external force, every solution starting from sufficiently small initial data and its derivative are bounded or go to zero as the time tends to + ∞ , provided that suitable conditions are satisfied. Our theoretical results are illustrated with numerical simulations.

A general solution procedure for the forced vibrations of a continuous system with cubic nonlinearities: Primary resonance case

Journal of Sound and Vibration, 2009

A general vibrational model of a continuous system with arbitrary linear and cubic operators is considered. Approximate analytical solutions are found using the method of multiple scales. The primary resonances of the external excitation and three-to-one internal resonances between two arbitrary natural frequencies are treated. The amplitude and phase modulation equations are derived. The steady-state solutions and their stability are discussed. The solution algorithm is applied to two specific problems: (1) axially moving Euler-Bernoulli beam, and (2) axially moving viscoelastic beam.