Bifurcation Analysis for Brake Squeal

2010 ◽  
Author(s):  
Hartmut Hetzler
Keyword(s):  
Bifurcation Analysis ◽  
Sliding Friction ◽  
Multiple Scales ◽  
Time Integration ◽  
Perturbation Approach ◽  
Disk Brake ◽  
System A ◽  
Bifurcation Behaviour ◽  
Multiple Scales Technique ◽  
Tribological Parameters

This article presents a perturbation approach for the bifurcation analysis of MDoF vibration systems with gyroscopic and circulatory contributions, as they naturally arise from problems involving moving continua and sliding friction. Based on modal data of the underlying linear system, a multiple scales technique is utilized in order to find equations for the nonlinear amplitudes of the critical mode. The presented method is suited for an algorithmic implementation using commercial software and does not involve costly time-integration. As an engineering example, the bifurcation behaviour of a MDoF disk brake model is investigated. Sub- and supercritical Hopf-bifurcations are found and stationary nonlinear amplitudes are presented depending on operating parameters of the brake as well as of tribological parameters of the contact.

EFFECT OF ADDED TIP MASS ON THE NONLINEAR FLAPWISE AND CHORDWISE VIBRATION OF CANTILEVER COMPOSITE BEAM UNDER BASE EXCITATION

2012 ◽  
Vol 12 (02) ◽  
pp. 285-310 ◽  
Author(s):  
M. EFTEKHARI ◽  
M. MAHZOON ◽  
S. ZIAEI-RAD
Keyword(s):  
Cantilever Beam ◽  
Multiple Scales ◽  
Equilibrium Points ◽  
Laminated Composite ◽  
Base Excitation ◽  
System A ◽  
Autoparametric Resonance ◽  
The Stability ◽  
Tip Mass ◽  
Made In

In this paper, a comparative study is performed for a symmetrically laminated composite cantilever beam with and without a tip mass under harmonic base excitation. The base is subjected to both flapwise and chordwise excitations tuned to the primary resonances of the two directions and conditions of 2:1 autoparametric resonance. In the literature, the governing nonlinear equations of the same problem without tip mass have been derived using the extended Hamilton's principle. Extension is made in this study to include the effect of a tip mass on the response of the beam. The natural frequencies are obtained numerically using the diversity guided evolutionary algorithm (DGEA). Next, the multiple scales method is applied to determine the nonlinear response and stability of the system. A set of four first-order differential equations describing the modulation of the amplitudes and phases of interacting modes are derived for the perturbation analysis. For verification, the above equations are reduced to the special case of the cantilever beam without tip mass for comparison with existing results. Finally, the effect of the tip mass on the stability of the fixed points and on the amplitude of oscillation about the equilibrium points in both the frequency and force modulation responses is examined.

The Asymptotic Stability of Weakly Perturbed Two Dimensional Hamiltonian Systems

2001 ◽  
Author(s):  
Ludwig Arnold ◽  
Peter Imkeller ◽  
N. Sri Namachchivaya
Keyword(s):  
Lyapunov Exponent ◽  
Equations Of Motion ◽  
Perturbation Approach ◽  
Two Dimensional ◽  
White Noise Process ◽  
System A ◽  
Additive White Noise ◽  
Small Dissipation ◽  
Small Intensity ◽  
Single Well

Abstract The purpose of this work is to obtain an approximation for the top Lyapunov exponent, the exponential growth rate, of the response of a single-well Kramers Oscillator driven by either a multiplicative or an additive white noise process. To this end, we consider the equations of motion as dissipative and noisy perturbations of a two-dimensional Hamiltonian system. A perturbation approach is used to obtain explicit expressions for the exponent in the presence of small intensity noise and small dissipation. We show analytically that the top Lyapunov exponent is positive, and for small values of noise intensity ε and dissipation ε the exponent grows proportional to ε1/3.

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.

The Dynamics of a Ball-Type Balancer System Equipped with a Pair of Free-Moving Balancing Masses

2001 ◽  
Vol 123 (4) ◽  
pp. 456-465 ◽  
Author(s):  
Jaan-Rong Kang ◽  
Chang-Po Chao ◽  
Chun-Lung Huang ◽  
Cheng-Kuo Sung
Keyword(s):  
High Speed ◽  
Multiple Scales ◽  
Design Guidelines ◽  
Disk Drives ◽  
Rotating System ◽  
Radial Vibrations ◽  
First Order ◽  
System A ◽  
Autonomous Differential Equations ◽  
Steady State Solutions

This study is devoted to evaluate the performance of a ball-type balancer system that is installed in high-speed optical disk drives. The ball-type balancer system, composed of a circular runway and free-moving balls inside, is designed for reducing radial vibrations induced by the inherent unbalance of the rotating system. A balancer system equipped with a pair of balls is considered in this study for its capability to reach possible near-elimination of radial vibrations as opposed to the serious sizing problem of a single balancing-ball system. A mathematical model is first established to describe the dynamics of the balls and rotor system. Utilizing the method of multiple scales and assuming the smallness of radial vibrations, the system dynamics on the slow time scale is represented by eight first-order autonomous differential equations, which accommodate the radial vibratory motions and ball behaviors. The steady-state solutions of these slow equations are then solved and their stability analyzed to predict settling ball positions. The residual vibrations are computed to evaluate the performance of the balancer system and then the design guidelines are distilled for engineers to design the balancer system.

scholarly journals Robustness Analysis of the Collective Dynamics of Nonlinear Periodic Structures Under Parametric Uncertainty

2016 ◽  
Author(s):  
Khaoula Chikhaoui ◽  
Diala Bitar ◽  
Najib Kacem ◽  
Noureddine Bouhaddi
Keyword(s):  
Multiple Scales ◽  
Periodic Structures ◽  
Uncertainty Propagation ◽  
Time Integration ◽  
Robustness Analysis ◽  
Parametric Uncertainty ◽  
Latin Hypercube Sampling ◽  
Algebraic Equations ◽  
Collective Dynamics ◽  
Model Combining

In order to ensure more realistic design of nonlinear periodic structures, the collective dynamics of a coupled pendulums system is investigated under parametric uncertainties. A generic discrete analytical model combining the multiple scales method, the perturbation theory and a standing-wave decomposition is proposed and adapted to the presence of uncertainties. These uncertainties are taken into account through a probabilistic modeling implying that the stochastic parameters vary according to random variables of chosen probability density functions. The proposed model leads to a set of coupled complex algebraic equations written according to the number and positions of the uncertainties in the structure and numerically solved using the time integration Runge-Kutta method. The uncertainty propagation through the established model is finally ensured using the Latin Hypercube Sampling method. The analysis of the dispersion, in term of variability of the frequency and amplitude intervals of the multistability domain shows the effects of uncertainties on the stability and nonlinearity of a three coupled pendulums structure. The nonlinear aspect is strengthened, the multistability domain is wider, more stable branches are obtained and thus the multimode solutions are enhanced.

scholarly journals LOCALIZATION IN A STRONGLY DISORDERED SYSTEM: A PERTURBATION APPROACH

2013 ◽  
Vol 27 (13) ◽  
pp. 1350051
Author(s):  
MARCO FRASCA
Keyword(s):  
Scaling Law ◽  
Localization Length ◽  
Perturbation Approach ◽  
Dimensional System ◽  
Two Dimensional ◽  
Disordered System ◽  
System A ◽  
Large Perturbation ◽  
Two Dimensional System ◽  
Diagonal Disorder

We prove that a strongly disordered two-dimensional system localizes with a localization length given analytically. We get a scaling law with a critical exponent ν = 1 in agreement with the Chayes criterion ν ≥ 1. The case we are considering is for off-diagonal disorder. The method we use is a perturbation approach holding in the limit of an infinitely large perturbation as recently devised and the Anderson model is considered with a Gaussian distribution of disorder. The localization length diverges when energy goes to zero with a scaling law in agreement to numerical and theoretical expectations.

A Lumped Parameter Model for Prediction of Temperature in Disk Pair in Dry Friction Contact

2007 ◽  
Author(s):  
A. Elhomani ◽  
K. Farhang ◽  
M. Krkoska
Keyword(s):  
Carbon Composite ◽  
Lumped Parameter Model ◽  
Bulk Temperature ◽  
Interface Temperature ◽  
Thermomechanical Model ◽  
Disk Brake ◽  
System A ◽  
Surface Failure ◽  
Lumped Parameter ◽  
Thermal Capacitance

In applications involving substantial friction, surface failure is an inevitable phenomenon. Friction induced failure typically involves the generation of considerable heat. Existence of significant frictional force leads to relatively high interface temperature as a result of dynamic nature of flash temperatures at the contact areas. A first step in predicting friction induced failure is to develop an accurate thermomechanical model of the friction system. A thermo-mechanical model is developed in this paper based on a lumped parameter representation of a two-disk brake. A disk is viewed as consisting of three main regions, (1) the surface contact, (2) the friction interface, and (3) the bulk. The lumped parameter model is obtained by dividing a disk into a number of concentric rings and stacked layers. The friction layer contains both the interface and contact elements, each include the equivalent thermal capacitance and conductive resistance. The contact capacitance and resistance are described in terms of the elastic contact interaction between the surfaces of the two disks. Therefore they are obtained using the Greenwood and Williamson model for contact of rough surfaces. Each is described as a statistical summation of the micron-scale interaction of the surfaces. The model is shown to provide accurate prediction of bulk temperature using a dynamometer test on a carbon composite disk pair.

The development of wavepackets in unstable flows

1981 ◽  
Vol 373 (1755) ◽  
pp. 457-476 ◽  
Keyword(s):  
Dispersion Relation ◽  
Multiple Scales ◽  
Three Dimensional ◽  
Fourier Integral ◽  
Saddle Point Method ◽  
Alternative Derivation ◽  
Closed Form Solutions ◽  
Linear Evolution ◽  
Multiple Scales Technique ◽  
Complex Dispersion

By using a model form of the complex dispersion relation for unstable flows, the linear evolution of a localized three-dimensional wavepacket is determined. The disturbance is expressed as a double Fourier integral which is evaluated asymptotically by the saddle-point method. On making certain approximations, simple closed-form solutions are obtained, some of which resemble the curved wavepackets observed by Gaster & Grant (1975) and some the ‘ elliptic ’ packet found by Benjamin (1961). The range of validity of theories which lead to an elliptic packet is clarified. An alternative derivation of some of the results is given by using a multiple-scales technique. The relative merits of the two methods are thereby illuminated.

Veering Phenomena in Systems With Gyroscopic Coupling

2004 ◽  
Vol 72 (5) ◽  
pp. 641-647 ◽  
Author(s):  
Stefano Vidoli ◽  
Fabrizio Vestroni
Keyword(s):  
System Parameter ◽  
Critical Value ◽  
Forced Response ◽  
Perturbation Approach ◽  
Continuous Systems ◽  
System A ◽  
Electromechanical Interaction ◽  
Gyroscopic Coupling ◽  
Internal Coupling

The sharp divergence of two root-loci for a critical value of the parameters is called veering. Veering phenomena are interesting since they involve relevant energetic exchanges between the eigenmodes and strongly affect the undamped forced response of the system. A straightforward perturbation approach has already been used in the literature to analyze the dependence of the eigensprectrum on a system parameter and formulate a veering criterion. This perturbation approach and other ideas are generalized to the study of veering in discrete and continuous systems with gyroscopic operators of internal coupling and the results applied to a real electromechanical interaction.

A Perturbation Approach to Bifurcation Analysis and Limit-Cycle Amplitudes for Brake Squeal Problems

2010 ◽  
Author(s):  
Hartmut Hetzler ◽  
Theodore E. Simos ◽  
George Psihoyios ◽  
Ch. Tsitouras
Keyword(s):  
Limit Cycle ◽  
Bifurcation Analysis ◽  
Perturbation Approach ◽  
Brake Squeal
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