Use of Mass as a Perturbation Parameter in Vibrations

1967 ◽  
Vol 89 (4) ◽  
pp. 639-644 ◽  
Author(s):  
R. Chicurel ◽  
J. Counts
Keyword(s):  
Perturbation Technique ◽  
Excitation Frequency ◽  
Perturbation Parameter ◽  
Harmonic Excitation ◽  
Power Series Solution ◽  
Continuous Systems ◽  
Linear Vibration ◽  
Denominator Polynomial ◽  
Vibration Problems ◽  
Special Case

Linear vibration problems involving harmonic excitation of discrete and continuous systems are solved by using the classical perturbation technique. The perturbation parameter is proportional to a mass and the square of the excitation frequency. The power series solution for the displacement of some point in the system is converted to the quotient of two polynomials by the use of continued fractions. The eigenvalues (natural frequencies) of the problem are calculated by finding the roots of the denominator polynomial. The situation wherein a quantity which cannot vanish at any frequency can be found is treated as a special case.

scholarly journals Higher-order accurate compact difference solutions for vibration problems of one dimensional continuous systems

2011 ◽  
pp. 771-794
Author(s):  
Mondher Yahiaoui
Keyword(s):  
Natural Frequencies ◽  
Continuous Systems ◽  
One Dimensional ◽  
First Order ◽  
Compact Difference ◽  
Transverse Vibrations ◽  
Seventh Order ◽  
One Step ◽  
Vibration Problems ◽  
Difference Methods

In this paper, we present a fourth-order accurate and a seventh-order accurate, one-step compact difference methods. These methods can be used to solve initial or boundaryvalue problems which can be modeled by a first-order linear system of differential equations. It is then shown in detail how these methods can be used to solve vibration problems of onedimensional continuous systems. Natural frequencies of a cantilever beam in transverse vibrations are computed and the results are compared to analytical ones to prove the high accuracy and efficiency of both methods. A comparison was also made to a finite element solution and the results have shown that both compact-difference methods yield more accurate values even with a reduced number of intervals.

Nonlinear Dynamics of a Dielectric Elastomer Membrane Under Compressive Loading

2020 ◽  
Author(s):  
Robert L. Lowe ◽  
Christopher G. Cooley
Keyword(s):  
Nonlinear Dynamics ◽  
Dynamic Response ◽  
Frequency Response ◽  
Compressive Loading ◽  
Excitation Frequency ◽  
Harmonic Excitation ◽  
Dielectric Elastomer ◽  
Large Strains ◽  
Membrane Stretch ◽  
Wide Range

Abstract This paper investigates the nonlinear dynamics of square dielectric elastomer membranes under time-dependent, through-thickness compressive loading. The dielectric elastomer is modeled as an isotropic ideal dielectric, with mechanical stiffening at large strains captured using the Gent hyperelastic constitutive model. The equation of motion for the in-plane membrane stretch is derived using Hamilton’s principle. The static response of the membrane is first investigated, with equilibrium stretches calculated numerically for a wide range of compressive pre-loads and applied voltages. Snap-through instabilities are observed, with the critical snap-through voltage decreasing with increasing compressive pre-load. The dynamic response of the membrane is then investigated under forced harmonic excitation. Frequency response plots characterizing the steady-state vibration reveal primary, subharmonic, and superharmonic resonances. Near these resonances, two stable vibration states are possible, corresponding to upper and lower branches in the frequency response. Significant and practically meaningful differences in the dynamic response are observed when the system vibrates at a fixed frequency about the upper and lower branches, a feature not discussed in previous research.

scholarly journals Research on the Resonance Characteristics of Rock under Harmonic Excitation

2019 ◽  
Vol 2019 ◽  
pp. 1-11
Author(s):  
Siqi Li ◽  
Shenglei Tian ◽  
Wei Li ◽  
Tie Yan ◽  
Fuqing Bi
Keyword(s):  
Resonance Frequency ◽  
Natural Frequency ◽  
Excitation Frequency ◽  
Harmonic Excitation ◽  
Theoretical Research ◽  
Transform Method ◽  
Factors Affecting ◽  
Least Action ◽  
Principle Of Least Action ◽  
Resonance Frequencies

In order to study the resonance characteristics of rock under harmonic excitation, two vibration models have been presented to estimate the natural frequency of rock encountered during the drilling. The first one is a developed single-DOF model which considers the properties and dimensions of the rock. The second one is a multi-DOF model based on the principle of least action. Subsequently, the modal characteristics, as well as the influence of excitation frequency, the mechanical properties, and dimensions of the rock on its resonance frequency, are analyzed by using FEM. Finally, the ultrasonic test on artificial sandstones and materials of drill tools are carried out indoor, and the FFT transform method is adopted to obtain their resonance frequencies. Based on the analysis undertaken, it can be concluded that the natural frequency of the rock increases with the change of vibration mode. For the same kind of rock, the resonance frequency is inversely proportional to mass, while for the different kinds of rocks, the mechanical parameters, such as density, elastic modulus, and Poisson’s ratio, determine the resonance frequency of the rock together. Besides, the shape of the rock is also one of the main factors affecting its resonance frequency. At last, the theoretical research results are further verified by ultrasonic tests.

Vibration Analysis of Nonlinear Isolator’s Characteristics by ADAMS

2012 ◽  
Vol 152-154 ◽  
pp. 1077-1081 ◽  
Author(s):  
Zhao Qi He ◽  
Yu Chao Song ◽  
Hong Liang Yu
Keyword(s):  
Nonlinear Vibration ◽  
Vibration Isolation ◽  
Excitation Frequency ◽  
Vibration Isolator ◽  
Vibration System ◽  
Linear Vibration ◽  
External Excitation ◽  
Mass Model ◽  
Adams Software ◽  
Isolation Systems

A nonlinear spring-mass model is established to study the dynamic characteristics of nonlinear vibration isolator. By use of ADAMS software, the influence of stiffness, foundation displacement excitation and frequency of external excitation on the nonlinear vibration isolation systems are analyzed. Results indicate that the linear vibration system needs 4s to achieve stability, but the nonlinear vibration system only needs 0.1s. The response value increases with the increase of excitation frequency, the response pick value increases by 61.58% and 102.35% and each corresponding stable value increases by 159.35% and 309.87%.

Noise and Contact in Dynamic AFM Operations

2011 ◽  
Author(s):  
Ishita Chakraborty ◽  
Balakumar Balachandran
Keyword(s):  
White Noise ◽  
Evolution Equations ◽  
Stochastic Dynamics ◽  
Gaussian White Noise ◽  
Excitation Frequency ◽  
Period Doubling ◽  
Harmonic Excitation ◽  
Dynamic Mode ◽  
Grazing Impacts ◽  
Representative System

In this article, the authors study the effects of Gaussian white noise on the dynamics of an atomic force microscope (AFM) cantilever operating in a dynamic mode by using a combination of numerical and analytical efforts. As a representative system, a combination of Si cantilever and HOPG sample is used. The focus of this study is on understanding the stochastic dynamics of a micro-cantilever, when the excitation frequencies are away from the first natural frequency of the system. In the previous efforts of the authors, period-doubling bifurcations close to grazing impacts have been reported for micro-cantilevers when the excitation frequency is in between the first and the second natural frequencies of the system. In the present study, it is observed that the addition of Gaussian white noise along with a harmonic excitation produces a near-grazing contact, when there was previously no contact between the tip and the sample with only the harmonic excitation. Moment evolution equations derived from a Fokker-Planck system are used to obtain numerical results, which support the statement that the addition of noise facilitates contact between the tip and the sample.

On analyticity of travelling water waves

2005 ◽  
Vol 461 (2057) ◽  
pp. 1283-1309 ◽  
Author(s):  
David P Nicholls ◽  
Fernando Reitich
Keyword(s):  
Water Waves ◽  
Bifurcation Theory ◽  
Amplitude Ratio ◽  
Travelling Waves ◽  
Three Dimensional ◽  
Perturbation Parameter ◽  
Boundary Perturbation ◽  
Classical Boundary ◽  
Boundary Model ◽  
Special Case

In this paper we establish the existence and analyticity of periodic solutions of a classical free-boundary model of the evolution of three-dimensional, capillary–gravity waves on the surface of an ideal fluid. The result is achieved through the application of bifurcation theory to a boundary perturbation formulation of the problem, and it yields analyticity jointly with respect to the perturbation parameter and the spatial variables. The travelling waves we find can be interpreted as resulting from the (nonlinear) interaction of two two-dimensional wavetrains, giving rise to a periodic travelling pattern. Our analyticity theorem extends the most sophisticated results known to date in the absence of resonance; ‘short crested waves’, which result from the interaction of two wavetrains with unit amplitude ratio are realized as a special case. Our method of proof also sheds light on the convergence and conditioning properties of classical boundary perturbation methods for the numerical approximation of travelling surface waves. Indeed, we demonstrate that the rather unstable numerical behaviour of these approaches can be attributed to the strong but subtle cancellations in the formulas underlying their classical implementations. These observations motivate the derivation and use of an alternative, stable, formulation which, in addition to providing our method of proof, suggests new stabilized implementations of boundary perturbation algorithms.

Imbibition in a cracked porous medium

1969 ◽  
Vol 47 (22) ◽  
pp. 2519-2524 ◽  
Author(s):  
A. P. Verma
Keyword(s):  
Differential Equation ◽  
Porous Medium ◽  
Viscosity Ratio ◽  
Perturbation Technique ◽  
Water Imbibition ◽  
Phase Saturation ◽  
Water Viscosity ◽  
Oil Water ◽  
The Relationship ◽  
Special Case

In this paper, one special case of oil–water imbibition phenomena in a cracked porous medium of a finite length is analytically discussed. The equation for the linear countercurrent imbibition is a nonlinear differential equation whose solution has been obtained by a perturbation technique. For definiteness, specific results have been used for the relationship between relative permeability and phase saturation) impregnation function, oil–water viscosity ratio, and capillary pressure dependence on phase saturation due to Jones, Bokserman et al., Evgen'ev, and Oroveanu, respectively. An expression for the wetting phase saturation has been derived.

scholarly journals The practical use of variation principles in non-linear mechanics

1962 ◽  
Vol 2 (3) ◽  
pp. 357-368 ◽  
Author(s):  
J. M. Blatt ◽  
J. N. Lyness
Keyword(s):  
Initial Conditions ◽  
Variation Principle ◽  
Linear Vibration ◽  
Average Value ◽  
True Motion ◽  
Non Linear ◽  
Vibration Problems ◽  
The Stability ◽  
Non Linear Mechanics ◽  
Small Disturbances

AbstractContrary to the general impression that variation principles are of purely theoretical interest, we show by means of examples that they can lead to considerable practical advantages in the solution of non-linear vibration problems. In this paper, we develop a variation principle for the period of a free oscillation, as a function of the average value of the Lagrangian over one period. Even extremely simple-minded approximations to the true motion lead to excellent values for the period. The stability of such free oscillations against small disturbances of the initial conditions is treated in a previous paper.

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