Parametric Vibrations in Discs: Point-Wise and Distributed Loads, Including Rotating Friction

1995 ◽  
Author(s):  
H. Ouyang ◽  
S. N. Chan ◽  
J. E. Mottershead ◽  
M. I. Friswell ◽  
M. P. Cartmell
Keyword(s):  
Multiple Scales ◽  
Equations Of Motion ◽  
Transverse Load ◽  
Transverse Mass ◽  
Follower Load ◽  
Parametric Resonances ◽  
Spring System ◽  
Multiple Scales Analysis ◽  
Mass Spring ◽  
Rotating Load

Abstract This paper is concerned with the parametric resonances in a stationary annular disc when excited by a rotating load system. Two forms of the load system are considered. In the first, the load consists of a discrete transverse mass-spring-damper system and a frictional follower load. Secondly, a distributed mass-spring system (without friction) is studied. In both cases the transverse load is rotated at a uniform speed around the disc. Equations of motion are developed for the two cases, and the results of a multiple scales analysis are presented. The disc is found to exhibit many parametric resonances at subcritical speeds when friction is present.

Parametric Resonances at Subcritical Speeds in Discs with Rotating Frictional Loads

1994 ◽  
Vol 208 (6) ◽  
pp. 417-425 ◽  
Author(s):  
S N Chan ◽  
J E Mottershead ◽  
M P Cartmell
Keyword(s):  
State Space ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Follower Load ◽  
State Space Analysis ◽  
Eigenvalue Method ◽  
Parametric Resonances ◽  
Annular Disc ◽  
Multiple Scales Analysis ◽  
Mass Spring

This paper is concerned with the parametric resonances in a stationary classical annular disc when excited by a rotating mass-spring-damper system together with a frictional follower load. An analysis by the method of multiple scales is performed to reveal the existence of instabilities associated with subcritical parametric resonances, and other instabilities of the backward waves in modes with nodal diameters. The latter are shown to be driven by friction and not to be dependent upon the rotational speed. A state-space analysis, with truncated modes, is used to investigate the effect of varying the friction, stiffness, mass and damping prameters in a series of simulated problems. The results obtained from the state-space eigenvalue method tend to support the conclusions of the multiple scales analysis.

Nonlinear Vibrations of a Beam-Spring-Large Mass System

2017 ◽  
Author(s):  
Mohammad A. Bukhari ◽  
Oumar R. Barry
Keyword(s):  
Natural Frequencies ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Primary Resonance ◽  
Large Mass ◽  
Mode Shapes ◽  
Resonance Excitation ◽  
Response Curves ◽  
Mass System ◽  
Spring System

This paper presents the nonlinear vibration of a simply supported Euler-Bernoulli beam with a mass-spring system subjected to a primary resonance excitation. The nonlinearity is due to the mid-plane stretching and cubic spring stiffness. The equations of motion and the boundary conditions are derived using Hamiltons principle. The nonlinear system of equations are solved using the method of multiple scales. Explicit expressions are obtained for the mode shapes, natural frequencies, nonlinear frequencies, and frequency response curves. The validity of the results is demonstrated via comparison with results in the literature. Exact natural frequencies are obtained for different locations, rotational inertias, and masses.

An Investigation of Rolling Element Vibrations Caused by Local Defects

2008 ◽  
Vol 130 (4) ◽  
Author(s):  
H. Arslan ◽  
N. Aktürk
Keyword(s):  
Equations Of Motion ◽  
Angular Contact Ball Bearing ◽  
Frequency Spectra ◽  
Rolling Element ◽  
Rolling Elements ◽  
Computer Simulation Program ◽  
Localized Defects ◽  
Spring System ◽  
Mass Spring ◽  
Local Defects

In this paper, a shaft-bearing model is developed in order to investigate the rolling element vibrations for an angular contact ball bearing with and without defects. The shaft-bearing assembly is considered as a mass-spring system. The system shows a nonlinear characteristic under dynamic conditions. The equations of motion in radial and axial directions were obtained for shaft and rolling elements, and they were solved simultaneously with a computer simulation program. Additionally, the effect of localized defects on running surfaces (i.e., inner ring, outer ring, and ball) on the vibration of the balls is investigated. Vibration of rolling elements in the radial direction is analyzed in time and frequency domains. Characteristic defect frequencies and their components can be seen in the frequency spectra of rolling element vibrations. Comparison of the obtained results with similar studies available in literature showed reasonable qualitative agreement.

Dynamic Stability of a Spinning Timoshenko Beam Subjected to a Moving Mass-Spring-Damper Unit

2010 ◽  
Author(s):  
T. H. Young ◽  
M. S. Chen
Keyword(s):  
Dynamic Stability ◽  
Timoshenko Beam ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Axial Direction ◽  
Longitudinal Axis ◽  
Moving Mass ◽  
The Stability ◽  
The Galerkin Method ◽  
Mass Spring

This paper investigates the dynamic stability of a finite Timoshenko beam spinning along its longitudinal axis and subjected to a moving mass-spring-damper (MSD) unit traveling in the axial direction. The mass of the moving MSD unit makes contact with the beam all the time during traveling. Due to the moving MSD unit, the beam is acted upon by a periodic, parametric excitation. In this work, the equations of motion of the beam are first discretized by the Galerkin method. The discretized equations of motion are then partially uncoupled by the modal analysis procedure suitable for gyroscopic systems. Finally the method of multiple scales is used to obtain the stability boundaries of the beam. Numerical results show that if the displacement of the MSD unit is equal to only one of the two transverse displacements of the beam, very large unstable regions may appear at main resonances.

Study on Free Oscillations of Systems with Even Nonlinearities by Means of the Method of Multiple Scales

2007 ◽  
Vol 10-12 ◽  
pp. 193-197
Author(s):  
J.M. Wen ◽  
Z.C. Cao
Keyword(s):  
Differential Equations ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Nonlinear Differential Equations ◽  
Approximate Solutions ◽  
Free Oscillations ◽  
Analytical Technique ◽  
Spring System ◽  
Mass Spring ◽  
Good Agreement

An analytical technique, namely the method of multiple scales, is applied to solve the differential equations of free oscillations with even nonlinearities in a mass-spring system. Unlike other perturbation methods, the method of multiple scales is effective in determining the transient response as well as determining the approximation to the frequency of the nonlinear system. In this paper, the periodic solutions of the even nonlinear differential equations have been obtained by using the method of multiple scales. Compared with the numerical examples, the approximate solutions are in good agreement with exact solutions. The numerical and analytical solutions have clearly shown that there exists the so-called drift phenomenon in the free oscillations of systems with even nonlinearities without any external excitation.

Nonlinear Vibrations Analysis of Overhead Power Lines: A Beam With Mass–Spring–Damper–Mass Systems

2018 ◽  
Vol 140 (3) ◽  
Author(s):  
Mohammad A. Bukhari ◽  
Oumar R. Barry
Keyword(s):  
Multiple Scales ◽  
Equations Of Motion ◽  
Nonlinear Frequency ◽  
Response Curves ◽  
Mass System ◽  
Parametric Studies ◽  
Overhead Power Lines ◽  
Nonlinear Equations Of Motion ◽  
Mass Spring ◽  
Stockbridge Damper

This paper examines the nonlinear vibration of a single conductor with Stockbridge dampers. The conductor is modeled as a simply supported beam and the Stockbridge damper is reduced to a mass–spring–damper–mass system. The nonlinearity of the system stems from the midplane stretching of the conductor and the cubic equivalent stiffness of the Stockbridge damper. The derived nonlinear equations of motion are solved by the method of multiple scales. Explicit expressions are presented for the nonlinear frequency, solvability conditions, and detuning parameter. The present results are validated via comparisons with those in the literature. Parametric studies are conducted to investigate the effect of variable control parameters on the nonlinear frequency and the frequency response curves. The findings are promising and open a horizon for future opportunities to optimize the design of nonlinear absorbers.

scholarly journals Modeling and Simulation of Non-Contact Base Excited AFM

2021 ◽  
Vol 285 ◽  
pp. 07035
Author(s):  
Mohammad Reza Bahrami
Keyword(s):  
Single Molecule ◽  
Multiple Scales ◽  
Sample Surface ◽  
Lumped Parameter Model ◽  
Contact Mode ◽  
Functional Behavior ◽  
Lumped Parameter ◽  
Spring System ◽  
Afm Cantilever ◽  
Mass Spring

AFM has some unique properties such as higher spatial resolution, mapping even a single molecule, simple sample preparation, scanning in different types of medium, and can obtain a 3D scan of the sample surface. Therefore, with the help of AFM, one can obtain a unique understanding of the structure and functional behavior of materials. In this article, to construct the mathematical model of the base excited AFM cantilever mass spring system (lumped parameter model) is used and the solution obtained by the method of Multiple scales. Here, in this work, we consider the AFM operates in the non-contact mode. To study the effect of the non linearity, amplitude of excitation, and damping coefficient, frequency response equation obtained.

Modifications to the response of a parametrically excited cantilever beam by means of smart active elements

2009 ◽  
Vol 224 (8) ◽  
pp. 1579-1591 ◽  
Author(s):  
X Su ◽  
M P Cartmell
Keyword(s):  
Natural Frequencies ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Mode Shapes ◽  
Beam Structure ◽  
Active Elements ◽  
Parametric Resonances ◽  
Instability Regions ◽  
Multiple Scales Perturbation Method

This article is concerned with applying active smart material elements for modifying parametric vibration in a flexible composite beam structure. The glass epoxy beam is bonded to two theoretically prestrained shape memory alloy (SMA) strips and fitted with a lumped end mass. In this study, the components of the recovery force generated during the SMA activation are derived with respect to a three-dimensional frame when the structure is undergoing combined bending and torsional motions. In order to employ Lagrangian dynamics, the generalized forces are formulated and the equations of motion are then derived. Three different parametric resonances for the structure are predicted by using the multiple scales perturbation method. In addition, the effects of the SMA strips on the natural frequencies, the mode shapes, and the instability regions of the structure are all investigated. It is shown that the different thresholds of instability for parametric resonances within a composite structure of this sort may be influenced by smart active elements.

Nonlinear Vibrations of Two-Span Composite Laminated Plates with Equal and Unequal Subspan Lengths

2017 ◽  
Vol 9 (6) ◽  
pp. 1485-1505
Author(s):  
Lingchang Meng ◽  
Fengming Li
Keyword(s):  
Multiple Scales ◽  
Equations Of Motion ◽  
Primary Resonance ◽  
Laminated Plates ◽  
Laminated Plate ◽  
Composite Laminated Plate ◽  
Vibration Localization ◽  
Composite Laminated Plates ◽  
Localization Phenomenon ◽  
Harmonic Resonance

AbstractThe nonlinear transverse vibrations of ordered and disordered two-dimensional (2D) two-span composite laminated plates are studied. Based on the von Karman's large deformation theory, the equations of motion of each-span composite laminated plate are formulated using Hamilton's principle, and the partial differential equations are discretized into nonlinear ordinary ones through the Galerkin's method. The primary resonance and 1/3 sub-harmonic resonance are investigated by using the method of multiple scales. The amplitude-frequency relations of the steady-state responses and their stability analyses in each kind of resonance are carried out. The effects of the disorder ratio and ply angle on the two different resonances are analyzed. From the numerical results, it can be concluded that disorder in the length of the two-span 2D composite laminated plate will cause the nonlinear vibration localization phenomenon, and with the increase of the disorder ratio, the vibration localization phenomenon will become more obvious. Moreover, the amplitude-frequency curves for both primary resonance and 1/3 sub-harmonic resonance obtained by the present analytical method are compared with those by the numerical integration, and satisfactory precision can be obtained for engineering applications and the results certify the correctness of the present approximately analytical solutions.

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