Oscillations of a rigid sphere embedded in an infinite elastic solid

1967 ◽  
Vol 63 (4) ◽  
pp. 1207-1227 ◽  
Author(s):  
P. Chadwick ◽  
E. A. Trowbridge
Keyword(s):  
Small Amplitude ◽  
Elastic Solid ◽  
Time Dependent ◽  
Forced Oscillations ◽  
Graphical Form ◽  
Rigid Sphere ◽  
Impulsive Force ◽  
Free Oscillations ◽  
Time Harmonic ◽  
Two Parameters

AbstractIn this paper, which is a continuation of (1), we study steady (i.e. time-harmonic) and transient rectilinear oscillations of small amplitude of a rigid sphere embedded in an infinite elastic solid. Two types of transient motions are considered, forced oscillations in which the sphere is subject to a prescribed time-dependent force, and free oscillations in which the sphere is set in motion by an impulsive force. For each mode of vibration of the sphere the character of the solution is determined by two parameters, the density contrast between the sphere and its surroundings and a parameter related to the Poisson's ratio of the elastic solid. Numerical results referring to transient rectilinear oscillations are presented in graphical form.

Oscillations of a rigid sphere embedded in an infinite elastic solid

1967 ◽  
Vol 63 (4) ◽  
pp. 1189-1205 ◽  
Author(s):  
P. Chadwick ◽  
E. A. Trowbridge
Keyword(s):  
Elastic Solid ◽  
Forced Oscillations ◽  
Graphical Form ◽  
Rigid Sphere ◽  
Free Oscillations ◽  
Modes Of Vibration ◽  
Fixed Axis ◽  
Steady Oscillations ◽  
Angular Oscillations ◽  
Elastic Shear

AbstractA detailed study is made of angular oscillations of small amplitude about a fixed axis of a rigid sphere embedded in an infinite elastic solid. Three modes of vibration of the sphere are considered: steady oscillations arising from the application of a periodic torque; forced oscillations produced by an arbitrary time-dependent torque; and free oscillations excited by an impulsive torque. Due to the transfer of energy to the surrounding material by the radiation of an elastic shear wave, free oscillations of the sphere are damped, the principal parameter affecting the damping being the density contrast between the sphere and its surroundings. Illustrative numerical results, referring to steady and free oscillations of the sphere, are presented in graphical form.

scholarly journals Response Due To Impulsive Force In Generalized Thermomicrostretch Elastic Solid

2015 ◽  
Vol 20 (3) ◽  
pp. 487-502
Author(s):  
V. Kumar ◽  
R. Singh
Keyword(s):  
Fourier Transforms ◽  
Normal Force ◽  
Elastic Solid ◽  
Integral Transforms ◽  
Normal Displacement ◽  
Impulsive Force ◽  
Field Equations ◽  
Laplace And Fourier Transforms ◽  
Eigen Value ◽  
Plain Strain

Abstract A two dimensional Cartesian model of a generalized thermo-microstretch elastic solid subjected to impulsive force has been studied. The eigen value approach is employed after applying the Laplace and Fourier transforms on the field equations for L-S and G-L model of the plain strain problem. The integral transforms have been inverted into physical domain numerically and components of normal displacement, normal force stress, couple stress and microstress have been illustrated graphically.

scholarly journals Application of hybrid asymptotic methods and modern software for mathematical models developing for nonlinear dynamics of structures with variables in time parameters

2021 ◽  
pp. 66-70
Author(s):  
S. Homeniuk ◽  
S. Grebenyuk ◽  
D. Gristchak
Keyword(s):  
Nonlinear Dynamics ◽  
Numerical Methods ◽  
Nonlinear Systems ◽  
Analytical Solution ◽  
Mathematical Models ◽  
Nonlinear Dynamic ◽  
Numerical Solutions ◽  
Hybrid Approach ◽  
Time Dependent ◽  
Forced Oscillations

The relevance. The aerospace domain requires studies of mathematical models of nonlinear dynamic structures with time-varying parameters. The aim of the work. To obtain an approximate analytical solution of nonlinear forced oscillations of the designed models with time-dependent parameters. The research methods. A hybrid approach based on perturbation methods, phase integrals, Galorkin orthogonalization criterion is used to obtain solutions. Results. Nonlocal investigation of nonlinear systems behavior is done using results of analytical and numerical methods and developed software. Despite the existence of sufficiently powerful numerical software systems, qualitative analysis of nonlinear systems with variable parameters requires improved mathematical models based on effective analytical, including approximate, solutions, which using numerical methods allow to provide a reliable analysis of the studied structures at the stage designing. An approximate solution in analytical form is obtained with constant coefficients that depend on the initial conditions. Conclusions. The approximate analytical results and direct numerical solutions of the basic equation were compared which showed a sufficient correlation of the obtained analytical solution. The proposed algorithm and program for visualization of a nonlinear dynamic process could be implemented in nonlinear dynamics problems of systems with time-dependent parameters.

The Forced Oscillations of Submerged Bodies

2003 ◽  
Vol 125 (4) ◽  
pp. 710-715
Author(s):  
Angel Sanz-Andre´s ◽  
Gonzalo Tevar ◽  
Francisco-Javier Rivas
Keyword(s):  
Rigid Body ◽  
Small Amplitude ◽  
Center Of Mass ◽  
Rigid Body Dynamics ◽  
The Body ◽  
Central Point ◽  
Modal Testing ◽  
Forced Oscillations ◽  
Motion Equations ◽  
Marine Applications

The increasing use of very light structures in aerospace applications are given rise to the need of taking into account the effects of the surrounding media in the motion of a structure (as for instance, in modal testing of solar panels or antennae) as it is usually performed in the motion of bodies submerged in water in marine applications. New methods are in development aiming at to determine rigid-body properties (the center of mass position and inertia properties) from the results of oscillations tests (at low frequencies during modal testing, by exciting the rigid-body modes only) by using the equations of the rigid-body dynamics. As it is shown in this paper, the effect of the surrounding media significantly modifies the oscillation dynamics in the case of light structures and therefore this effect should be taken into account in the development of the above-mentioned methods. The aim of the paper is to show that, if a central point exists for the aerodynamic forces acting on the body, the motion equations for the small amplitude rotational and translational oscillations can be expressed in a form which is a generalization of the motion equations for a body in vacuum, thus allowing to obtain a physical idea of the motion and aerodynamic effects and also significantly simplifying the calculation of the solutions and the interpretation of the results. In the formulation developed here the translational oscillations and the rotational motion around the center of mass are decoupled, as is the case for the rigid-body motion in vacuum, whereas in the classical added mass formulation the six motion equations are coupled. Also in this paper the nonsteady motion of small amplitude of a rigid body submerged in an ideal, incompressible fluid is considered in order to define the conditions for the existence of the central point in the case of a three-dimensional body. The results here presented are also of interest in marine applications.

Thermoelastic Constitutive Equations for Chemically Hardening Materials

1974 ◽  
Vol 41 (3) ◽  
pp. 652-657 ◽  
Author(s):  
Bernard W. Shaffer ◽  
Myron Levitsky
Keyword(s):  
Bulk Modulus ◽  
Strain Rate ◽  
Chemical Reaction ◽  
Constitutive Equations ◽  
Strain Energy ◽  
Elastic Solid ◽  
Time Dependent ◽  
Component Model ◽  
Two Component ◽  
Potential Applications

Thermoelastic constitutive equations are derived for a material undergoing solidification or hardening as the result of a chemical reaction. The derivation is based upon a two component model whose composition is determined by the degree of hardening, and makes use of strain-energy considerations. Constitutive equations take the form of stress rate-strain rate relations, in which the coefficients are time-dependent functions of the composition. Specific results are developed for the case of a material of constant bulk modulus which undergoes a transition from an initial liquidlike state into an isotropic elastic solid. Potential applications are discussed.

Harmonic Oscillations of Nonlinear Two-Degree-of-Freedom Systems

1955 ◽  
Vol 22 (1) ◽  
pp. 107-110
Author(s):  
T. C. Huang
Keyword(s):  
Nonlinear System ◽  
Degrees Of Freedom ◽  
Forced Oscillations ◽  
Viscous Damping ◽  
Free Oscillations ◽  
Response Curves ◽  
Harmonic Oscillations ◽  
Restoring Force ◽  
Nonlinear Restoring Force ◽  
Two Degree Of Freedom

Abstract In this paper an investigation is made of equations governing the oscillations of a nonlinear system in two degrees of freedom. Analyses of harmonic oscillations are illustrated for the cases of (1) the forced oscillations with nonlinear restoring force, damping neglected; (2) the free oscillations with nonlinear restoring force, damping neglected; and (3) the forced oscillations with nonlinear restoring force, small viscous damping considered. Amplitudes of oscillations and frequency equations are derived based on the mathematically justified perturbation method. Response curves are then plotted.

Small oscillations of systems with several degrees of freedom

2020 ◽  
pp. 29-40
Author(s):  
Gleb L. Kotkin ◽  
Valeriy G. Serbo
Keyword(s):  
Magnetic Field ◽  
Degrees Of Freedom ◽  
Forced Oscillations ◽  
Free Oscillations ◽  
Small Oscillations ◽  
Symmetry Properties ◽  
Gyroscopic Forces ◽  
Simple Molecules ◽  
Simple Systems ◽  
Three Degrees Of Freedom

This chapter addresses the free and forced oscillations of simple systems (with two or three degrees of freedom), the free oscillations of systems with the degenerate frequencies, and the eigen-oscillations of the electromechanical systems. This chapter also studies the oscillations of more complex systems using orthogonality of eigenoscillations and the symmetry properties of the system, the free oscillations of an anisotropic charged oscillator moving in a uniform constant magnetic field, and the perturbation theory adapted for the small oscillations. Finally, the chapter addresses oscillations of systems in which gyroscopic forces act and the eigen-oscillations of the simple molecules.

Aerodynamic Flutter and Flight Surface Actuation

2007 ◽  
Author(s):  
S. A. Gadsden ◽  
S. Habibi
Keyword(s):  
Bulk Modulus ◽  
Small Amplitude ◽  
Mechanical Load ◽  
Impedance Control ◽  
Hydraulic Oil ◽  
Effective Bulk Modulus ◽  
Two Parameters

This paper proposes a novel form of impedance control in order to reduce the effects of aerodynamic flutter on a flight surface actuator. The forces generated by small amplitude flutter were studied on an electrohydrostatic actuator (EHA). The effects of flutter were modeled and analyzed. Through analysis, it was found that in EHA systems, two parameters would impact the response of flutter: damping (B) of the mechanical load, and the effective bulk modulus of the hydraulic oil (βe). These can be actively controlled as proposed here in order to provide variable impedance. The results of changing these variables are discussed and presented here.

Dynamic response of an elastic half-space to time-dependent surface tractions over an embedded spherical cavity. III (With comments on a paper by R. D. Gregory)

1969 ◽  
Vol 309 (1498) ◽  
pp. 313-329 ◽  
Keyword(s):  
Dynamic Response ◽  
Half Space ◽  
Rayleigh Waves ◽  
Spherical Cavity ◽  
Elastic Half Space ◽  
Time Dependent ◽  
Plane Boundary ◽  
Surface Motion ◽  
Harmonic Solution ◽  
Time Harmonic

Discussion of the problem of an elastic half-space with spherical cavity is continued in respect of Rayleigh waves on the plane boundary. Displacements in the initial and first group of higher order Rayleigh waves are derived by using the time-harmonic solution developed in part I of this series with attention confined to the case of time-harmonic normal stress at the cavity. These are employed to find also the response to an exponential shock at the cavity and graphs are presented showing the surface motion due to the initial Rayleigh waves. Finally, in an appendix to the paper, some comments are given on a recent paper by R. D. Gregory on the problem of the half-space with cavity.

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