GLOBAL ANALYSIS FOR A NONLINEAR VIBRATION ABSORBER WITH FAST AND SLOW MODES

2001 ◽  
Vol 11 (08) ◽  
pp. 2179-2194 ◽  
Author(s):  
WEI ZHANG ◽  
JING LI
Keyword(s):  
Normal Form ◽  
Nonlinear Vibration ◽  
Chaotic Dynamics ◽  
Multiple Scales ◽  
Global Analysis ◽  
Equations Of Motion ◽  
Global Bifurcation ◽  
Vibration Absorber ◽  
Nonlinear Vibration Absorber ◽  
Averaged Equations

A two-degree-of-freedom model of a nonlinear vibration absorber is considered in this paper. Both the global bifurcations and chaotic dynamics of the nonlinear vibration absorber are investigated. The nonlinear equations of motion of this model are derived. The method of multiple scales is used to find the averaged equations. Based on the averaged equations, the theory of normal form is used to obtain the explicit expressions of normal form associated with a double zero and a pair of pure imaginary eigenvalues by Maple software program. The fast and slow modes may simultaneously exist in the averaged equations. On the basis of the normal form, the global bifurcation and the chaotic dynamics of the nonlinear vibration absorber are analyzed by a global perturbation method developed by Kovacic and Wiggins. The chaotic motion of this model is also found by numerical simulation.

Nonlinear Vibration Absorber Design: An Asymptotic Approach

2015 ◽  
Author(s):  
Arnaldo Casalotti ◽  
Walter Lacarbonara
Keyword(s):  
Nonlinear Vibration ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Sensitivity Analyses ◽  
Vibration Absorber ◽  
Asymptotic Approach ◽  
Nonlinear Vibration Absorber ◽  
The One ◽  
Two Degree Of Freedom ◽  
Good Agreement

The one-to-one internal resonance occurring in a two-degree-of-freedom (2DOF) system composed by a damped non-linear primary structure coupled with a nonlinear vibration absorber is studied via the method of multiple scales up to higher order (i.e., the first nonlinear order beyond the internal/external resonances). The periodic response predicted by the asymptotic approach is in good agreement with the numerical results obtained via continuation of the periodic solution of the equations of motion. The asymptotic procedure lends itself to manageable sensitivity analyses and thus to versatile optimization by which different optimal tuning criteria for the vibration absorber can possibly be found in semi-closed form.

USING ENERGY-PHASE METHOD TO STUDY GLOBAL BIFURCATIONS AND SHILNIKOV TYPE MULTIPULSE CHAOTIC DYNAMICS FOR A NONLINEAR VIBRATION ABSORBER

2012 ◽  
Vol 22 (01) ◽  
pp. 1250001 ◽  
Author(s):  
S. B. LI ◽  
W. ZHANG ◽  
M. H. YAO
Keyword(s):  
Nonlinear Vibration ◽  
Chaotic Dynamics ◽  
Multiple Scales ◽  
Standard Form ◽  
Primary Resonance ◽  
Phase Method ◽  
Vibration Absorber ◽  
Global Bifurcations ◽  
Modulation Equation ◽  
Nonlinear Vibration Absorber

Global bifurcations and Shilnikov type multipulse chaotic dynamics for a nonlinear vibration absorber are investigated by using the energy-phase method for the first time. A two-degree-of-freedom model of a nonlinear vibration absorber is considered. After the nonlinear nonautonomous equations of this model are given, the method of multiple scales is used to derive four first-order nonlinear ordinary differential equations governing the modulation of the amplitudes and phases of the two interacting modes in the presence of 1:1 internal resonance and primary resonance. Using several coordinate transformations to transform the modulation equation into a standard form, we can apply the energy-phase method to show the existence of the multipulse chaotic dynamics by identifying Shilnikov-type multipulse orbits in the perturbed phase space. We are able to obtain the explicit restriction on the damping, forcing excitation and the detuning parameters, under which the multipulse chaotic dynamics is expected. These multipulse orbits represent the repeated departure from purely vertical oscillations for the nonlinear vibration absorber. Numerical simulations also indicate that there exist different forms of the multipulse chaotic responses and jumping phenomena for the nonlinear vibration absorber.

Multi-Pulse Chaotic Dynamics of Eccentric Rotating Composite Laminated Circular Cylindrical Shell by Using the Extended Melnikov Method

2019 ◽  
Author(s):  
Yan Zheng ◽  
Wei Zhang ◽  
Tao Liu
Keyword(s):  
Cylindrical Shell ◽  
Normal Form ◽  
Chaotic Dynamics ◽  
Multiple Scales ◽  
Circular Cylindrical Shell ◽  
Melnikov Method ◽  
Global Dynamics ◽  
Chaotic Motions ◽  
Extended Melnikov Method ◽  
Averaged Equations

Abstract The researches of global bifurcations and chaotic dynamics for high-dimensional nonlinear systems are extremely challenging. In this paper, we study the multi-pulse orbits and chaotic dynamics of an eccentric rotating composite laminated circular cylindrical shell. The four-dimensional averaged equations are obtained by directly using the multiple scales method under the case of the 1:2 internal resonance and principal parametric resonance-1/2 subharmonic resonance. The system is transformed to the averaged equations. From the averaged equation, the theory of normal form is used to find the explicit formulas of normal form. Based on the normal form obtained, the extended Melnikov method is utilized to analyze the multi-pulse global homoclinic bifurcations and chaotic dynamics for the eccentric rotating composite laminated circular cylindrical shell. The analysis of global dynamics indicates that there exist the multi-pulse jumping orbits in the perturbed phase space of the averaged equation. From the averaged equations obtained, the chaotic motions and the Shilnikov type multi-pulse orbits of the eccentric rotating composite laminated circular cylindrical shell are found by using numerical simulation. The results obtained above mean the existence of the chaos for the Smale horseshoe sense for the eccentric rotating composite laminated circular cylindrical shell.

Multi-Pulse Chaotic Dynamics of the Cantilevered Pipe Conveying Pulsating Fluid

2009 ◽  
Author(s):  
Ming-Hui Yao ◽  
Wei Zhang ◽  
Dong-Xing Cao
Keyword(s):  
Normal Form ◽  
Chaotic Dynamics ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Phase Method ◽  
Global Dynamics ◽  
Pipe Material ◽  
Chaotic Motions ◽  
Cantilevered Pipe ◽  
Pulsating Fluid

The multi-pulse orbits and chaotic dynamics of the cantilevered pipe conveying pulsating fluid with harmonic external force are studied in detail. The nonlinear geometric deformation of the pipe and the Kelvin constitutive relation of the pipe material are considered. The nonlinear governing equations of motion for the cantilevered pipe conveying pulsating fluid are determined by using Hamilton principle. The four-dimensional averaged equation under the case of principle parameter resonance, 1/2 subharmonic resonance and 1:2 internal resonance and primary parametric resonance is obtained by directly using the method of multiple scales and Galerkin approach to the partial differential governing equation of motion for the cantilevered pipe. The system is transformed to the averaged equation. From the averaged equation, the theory of normal form is used to find the explicit formulas of normal form. Based on normal form obtained, the energy phase method is utilized to analyze the multi-pulse global bifurcations and chaotic dynamics for the cantilevered pipe conveying pulsating fluid. The analysis of global dynamics indicates that there exist the multi-pulse jumping orbits in the perturbed phase space of the averaged equation. From the averaged equations obtained, the chaotic motions and the Shilnikov type multi-pulse orbits of the cantilevered pipe are found by using numerical simulation. The results obtained above mean the existence of the chaos for the Smale horseshoe sense for the pulsating fluid conveying cantilevered pipe.

Global Bifurcation and Multipulse Type Chaotic Dynamics in the Nonlinear Nonplanar Oscillations of a Cantilever Beam

2009 ◽  
Author(s):  
Wei Zhang ◽  
Shuangbao Li ◽  
Minghui Yao
Keyword(s):  
Cantilever Beam ◽  
Chaotic Dynamics ◽  
Multiple Scales ◽  
Global Bifurcation ◽  
Melnikov Method ◽  
Extended Melnikov Method ◽  
Averaged Equations ◽  
Suitable Form ◽  
Parametric And External Excitations ◽  
Two Degree Of Freedom

The global bifurcation and multipulse type chaotic dynamics for the nonlinear nonplanar oscillations of a cantilever beam are studied using the extended Melnikov method. The cantilever beam studied is subjected to a harmonic axial excitation and transverse excitations at the free end. After the governing nonlinear equations of nonplanar motion with parametric and external excitations are given, the Galerkin procedure is applied to the partial differential governing equation to obtain a two-degree-of-freedom nonlinear system with parametric and forcing excitations. The resonant case considered here is 1:1 internal resonance, principal parametric resonance-1/2 subharmonic resonance. By using the method of multiple scales the averaged equations are obtained. After transforming the averaged equation in a suitable form, the extended Melnikov method is employed to show the existence of chaotic dynamics by identifying Shilnikov-type multipulse orbits in the perturbed phase space. We are able to obtain the explicit restriction conditions on the damping, forcing, and the detuning parameters, under which multipulse-type chaotic dynamics is to be expected. Numerical simulations indicate that there exist different forms of the chaotic responses and jumping phenomena in the nonlinear nonplanar oscillations of the cantilever beam.

Forced Vibration of Systems With Nonlinear, Nonsymmetrical Characteristics

1957 ◽  
Vol 24 (3) ◽  
pp. 435-439
Author(s):  
S. Mahalingam
Keyword(s):  
Approximate Solution ◽  
Linear System ◽  
Nonlinear Vibration ◽  
Forced Vibration ◽  
Free Vibrations ◽  
Frequency Function ◽  
Vibration Absorber ◽  
Single Degree Of Freedom ◽  
Nonlinear Vibration Absorber ◽  
Single Degree

Abstract A one-term approximate solution is given for the amplitudes of steady forced vibration of a single-degree-of-freedom system with a nonlinear (nonsymmetrical) spring characteristic. The method is similar to that of Martienssen (1), but the construction uses a modified curve (or “frequency function”) in place of the actual spring characteristic, the curve being so chosen that it gives the correct frequency for free vibrations. The method is extended to deal with a nonlinear vibration absorber fitted to a linear system.

Broadening the frequency bandwidth of a finite extensibility nonlinear vibration absorber by exploiting its internal resonances

2020 ◽  
Vol 102 (3) ◽  
pp. 1239-1270
Author(s):  
Alex Elías-Zúñiga ◽  
Luis Manuel Palacios-Pineda ◽  
Daniel Olvera-Trejo ◽  
Oscar Martínez-Romero
Keyword(s):  
Nonlinear Vibration ◽  
Vibration Absorber ◽  
Frequency Bandwidth ◽  
Internal Resonances ◽  
Nonlinear Vibration Absorber ◽  
Finite Extensibility
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