scholarly journals Autoparametric vibrations of a nonlinear system with pendulum

2006 ◽  
Vol 2006 ◽  
pp. 1-19 ◽  
Author(s):  
J. Warminski ◽  
K. Kecik
Keyword(s):  
Nonlinear System ◽  
Nonlinear Oscillator ◽  
Equations Of Motion ◽  
Resonance Region ◽  
Harmonic Force ◽  
Strongly Coupled ◽  
Vibration Absorption ◽  
Strongly Nonlinear ◽  
Chaotic Vibrations ◽  
External Harmonic Force

Vibrations of a nonlinear oscillator with an attached pendulum, excited by movement of its point of suspension, have been analysed in the paper. The derived differential equations of motion show that the system is strongly nonlinear and the motions of both subsystems, the pendulum and the oscillator, are strongly coupled by inertial terms, leading to the so-called autoparametric vibrations. It has been found that the motion of the oscillator, forced by an external harmonic force, has been dynamically eliminated by the pendulum oscillations. Influence of a nonlinear spring on the vibration absorption near the main parametric resonance region has been carried out analytically, whereas the transition from regular to chaotic vibrations has been presented by using numerical methods. A transmission force on the foundation for regular and chaotic vibrations is presented as well.

Strongly Nonlinear Vibrations of a Hyperelastic Thin-walled Cylindrical Shell Based on the Modified Lindstedt–Poincaré Method

2019 ◽  
Vol 19 (12) ◽  
pp. 1950160 ◽  
Author(s):  
Jing Zhang ◽  
Jie Xu ◽  
Xuegang Yuan ◽  
Wenzheng Zhang ◽  
Datian Niu
Keyword(s):  
Cylindrical Shell ◽  
Nonlinear System ◽  
Differential Equations ◽  
Nonlinear Vibrations ◽  
Harmonic Excitation ◽  
System Of Differential Equations ◽  
Response Curves ◽  
Strongly Nonlinear ◽  
Hardening Behavior ◽  
Thin Walled

Some significant behaviors on strongly nonlinear vibrations are examined for a thin-walled cylindrical shell composed of the classical incompressible Mooney–Rivlin material and subjected to a single radial harmonic excitation at the inner surface. First, with the aid of Donnell’s nonlinear shallow-shell theory, Lagrange’s equations and the assumption of small strains, a nonlinear system of differential equations for the large deflection vibration of a thin-walled shell is obtained. Second, based on the condensation method, the nonlinear system of differential equations is reduced to a strongly nonlinear Duffing equation with a large parameter. Finally, by the appropriate parameter transformation and modified Lindstedt–Poincar[Formula: see text] method, the response curves for the amplitude-frequency and phase-frequency relations are presented. Numerical results demonstrate that the geometrically nonlinear characteristic of the shell undergoing large vibrations shows a hardening behavior, while the nonlinearity of the hyperelastic material should weak the hardening behavior to some extent.

Chaotic vibrations of carbon nanotubes subjected to a traversing force considering nonlocal elasticity theory

2021 ◽  
pp. 239779142110633
Author(s):  
Reza Ebrahimi
Keyword(s):  
Nonlocal Parameter ◽  
Power Spectra ◽  
Beam Theory ◽  
Nonlocal Elasticity Theory ◽  
Effective Control ◽  
Maximum Lyapunov Exponent ◽  
Harmonic Force ◽  
Spectra Analysis ◽  
Chaotic Vibrations ◽  
Euler Bernoulli Beam

The existence of chaos in the lateral vibration of the carbon nanotube (CNT) can contribute to source of instability and inaccuracy within the nano mechanical systems. So, chaotic vibrations of a simply supported CNT which is subjected to a traversing harmonic force are studied in this paper. The model of the system is formulated by using nonlocal Euler–Bernoulli beam theory. The equation of motion is solved using the Rung–Kutta method. The effects of the nonlocal parameter, velocity and amplitude of the traversing harmonic force on the nonlinear dynamic response of the system are analyzed by the bifurcation diagrams, phase plane portrait, power spectra analysis, Poincaré map and the maximum Lyapunov exponent. The results indicate that the nonlocal parameter, velocity and amplitude of the traversing harmonic force have considerable effects on the bifurcation behavior and can be used as effective control parameters for avoiding chaos.

Active Vibration Absorption Using Delayed Resonator With Relative Position Measurement

1995 ◽  
Author(s):  
Nejat Olgac ◽  
Martin Hosek
Keyword(s):  
Relative Position ◽  
Primary Structure ◽  
Vibration Suppression ◽  
Control Force ◽  
Range Analysis ◽  
Position Measurement ◽  
Harmonic Force ◽  
Stability Range ◽  
Vibration Absorption ◽  
Active Vibration

Abstract A novel active vibration absorption technique, the Delayed Resonator, has been introduced recently as a unique way of suppressing undesired oscillations. It suggests a control force on a mass-spring-damper absorber in the form of a proportional position feedback with a time delay. Its strengths consist of extremely simple implementation of the control algorithm, total vibration suppression of the primary structure against a harmonic force excitation and full effectiveness of the absorber in a semi-infinite range of disturbance frequency, achieved by real-time tuning. All this development work was done using the absolute displacements of the absorber in the feedback. These displacement measurements may be difficult to obtain and for some applications impossible. This paper deals with a substitute and easier measurement: the relative motion of the absorber with respect to the primary structure. Theoretical foundations for the Delayed Resonator (DR) are briefly recapitulated and its implementation on a single-degree-of-freedom primary structure disturbed by a harmonic force is introduced utilizing both absolute and relative position measurement of absorber mass. Methods for stability range analysis and transient behavior are presented. Properties acquired for the same system with these two different feedback are compared. Relative position measurement case is found to be more advantageous in most applications of the Delayed Resonator method.

Computation of the adiabatic invariant of the symmetric nonlinear oscillator $$d^2y \over dt^2+Q(\epsilon t)y^n=R(\epsilon t)y^m$$

1998 ◽  
Vol 9 (2) ◽  
pp. 187-194
Author(s):  
J. HU
Keyword(s):  
Nonlinear Oscillator ◽  
Nonlinear Oscillators ◽  
Adiabatic Invariant ◽  
Strongly Nonlinear ◽  
Strongly Nonlinear Oscillators

In a recent paper, the author showed that for certain symmetric bisuperlinear equations, cosine-like boundary behaviours will not yield symmetric solutions [1]. In this paper, we attack the adiabatic invariant problem by showing that, for these strongly nonlinear oscillators, the adiabatic invariant is intimately related to z′(0;∈) for a family of solutions.

Nonlinear Dynamic of Cantilever Functionally Graded Plates Under the Thermalmechanical Loads

2009 ◽  
Author(s):  
Yu-xin Hao ◽  
Wei Zhang ◽  
Jian-hua Wang
Keyword(s):  
Nonlinear System ◽  
Functionally Graded Materials ◽  
Nonlinear Dynamic ◽  
Equations Of Motion ◽  
Functionally Graded ◽  
Degree Of Freedom ◽  
Mode Shapes ◽  
Governing Equations ◽  
Graded Materials ◽  
Two Degree Of Freedom

An analysis on nonlinear dynamic of a cantilevered functionally graded materials (FGM) plate which subjected to the transverse excitation in the uniform thermal environment is presented for the first time. Materials properties of the constituents are graded in the thickness direction according to a power-law distribution and assumed to be temperature dependent. In the framework of the Third-order shear deformation plate theory, the nonlinear governing equations of motion for the functionally graded materials plate are derived by using the Hamilton’s principle. For cantilever rectangular plate, the first two vibration mode shapes that satisfy the boundary conditions is given. The Galerkin’s method is utilized to discretize the governing equations of motion to a two-degree-of-freedom nonlinear system under combined thermal and external excitations. By using the numerical method, the two-degree-of-freedom nonlinear system is analyzed to find the nonlinear responses of the cantilever FGMs plate. The influences of the thermal environments on the nonlinear dynamic response of the cantilevered FGM plate are discussed in detail through a parametric study.

Dynamics of Oscillator With Soft Impacts

2001 ◽  
Author(s):  
František Peterka
Keyword(s):  
Nonlinear System ◽  
Linear System ◽  
Mechanical System ◽  
Linear Model ◽  
Piecewise Linear ◽  
Impact Oscillator ◽  
Strongly Nonlinear ◽  
Strongly Nonlinear System ◽  
New Phenomena ◽  
The Impact

Abstract The impact oscillator is the simplest mechanical system with one degree of freedom, the periodically excited mass of which can impact on the stop. The aim of this paper is to explain the dynamics of the system, when the stiffness of the stop changes from zero to infinity. It corresponds to the transition from the linear system into strongly nonlinear system with rigid impacts. The Kelvin-Voigt and piecewise linear model of soft impact was chosen for the study. New phenomena in the dynamics of motion with soft impacts in comparison with known dynamics of motion with rigid impacts are introduced in this paper.

Damped Transition of a Strongly Nonlinear System of Coupled Oscillators Into a State of Continuous Resonance Scattering

2011 ◽  
Author(s):  
David Andersen ◽  
Xingyuan Wang ◽  
Yuli Starosvetsky ◽  
Kevin Remick ◽  
Alexander Vakakis ◽  
...  
Keyword(s):  
Equations Of Motion ◽  
Resonance Scattering ◽  
Linear Oscillator ◽  
System Response ◽  
Analytical Treatment ◽  
Strongly Nonlinear ◽  
Damped System ◽  
Initial Excitation ◽  
Strongly Nonlinear System ◽  
Energy Plane

We examine analytically and experimentally a new phenomenon of ‘continuous resonance scattering’ in an impulsively excited, two-mass oscillating system. This system consists of a grounded damped linear oscillator with a light, strongly nonlinear attachment. Previous numerical simulations revealed that for certain levels of initial excitation, the system engages in a special type of response that appears to track a solution branch formed by the so-called ‘impulsive orbits’ of this system. By this term we denote the periodic (under conditions of resonance) or quasi-periodic (under conditions of non-resonance) responses of the system when a single impulse is applied to the linear oscillator with the system being initially at rest. By varying the magnitude of the impulse we obtain a manifold of impulsive orbits in the frequency-energy plane. It appears that the considered damped system is capable of entering into a state of continuous resonance scattering, whereby it tracks the impulsive orbit manifold with decreasing energy. Through analytical treatment of the equations of motion, a direct relationship is established between the frequency of the nonlinear attachment and the amplitude of the linear oscillator response, and a prediction of the system response during continuous scattering resonance is provided. Experimental results confirm the analytical predictions.

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