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.

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.

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.

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

Nonlinear vibration, stability, and bifurcation analysis of unbalanced spinning pre-twisted beam

2019 ◽  
Vol 24 (11) ◽  
pp. 3514-3536
Author(s):  
Mohsen Tajik ◽  
Ardeshir Karami Mohammadi
Keyword(s):  
Nonlinear Vibration ◽  
Bifurcation Analysis ◽  
Multiple Scales ◽  
Damping Ratio ◽  
Twist Angle ◽  
Equations Of Motion ◽  
Multiple Scales Method ◽  
Vibration Stability ◽  
Stability And Bifurcation ◽  
Stability And Bifurcation Analysis

In this paper, an Euler–Bernoulli model has been used for nonlinear vibration, stability, and bifurcation analysis of spinning twisted beams with linear twist angle, and with large transverse deflections, near the primary and parametric resonances. The equations of motion, in the case of pure single mode motion are analyzed by two methods: directly applying multiple scales method and using multiple scales method after discretization by Galerkin’s procedure. It is observed that the same final relations are obtained in the two methods. Effects of twist angle, damping ratio, longitudinal to transverse stiffness ratio, and eccentricity on the frequency responses are investigated. Then, the results are compared with the results obtained from Runge–Kutta numerical method on ODEs in a steady state, and confirmed with some previous research. Finally, the results show a good correlation, and it shows that with increasing the twist angle from 0 to 90°, the natural frequencies increase in the first two modes.

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