General linearized equations of wave propagation in strained elastic solids of cylindrical shapes using a simple perturbation method

2002 ◽  
Vol 216 (6) ◽  
pp. 683-690
Author(s):  
A Sinaie ◽  
A Ziaie
Keyword(s):  
Wave Propagation ◽  
Perturbation Method ◽  
Equations Of Motion ◽  
Particle Displacement ◽  
Elastic Solids ◽  
Cylindrical Coordinates ◽  
Linearized Equations ◽  
Practical Applications ◽  
Linear Partial Differential Equations ◽  
Static Part

The equations of particle motion in an elastic isotropic stressed medium are first derived in Cartesian coordinates and then transformed into cylindrical coordinates. The three components of the equations of motion are non-linear partial differential equations and cannot be of use in practical applications. However, noting that the particle displacement is composed of a small dynamic part superimposed on a large static part, these equations are linearized via a simple perturbation method. The linearized equations are presented in closed form. They contain variables, which may be measured and experimented upon in practice, in the field of acoustoelasticity.

Cylindrical and Spherical Shear Waves in Nonhomogeneous Isotropic Viscoelastic Media

1973 ◽  
Vol 51 (10) ◽  
pp. 1091-1097 ◽  
Author(s):  
T. Bryant Moodie
Keyword(s):  
Equations Of Motion ◽  
Shear Waves ◽  
Three Dimensions ◽  
Radial Coordinate ◽  
Shear Stresses ◽  
Progressive Wave ◽  
Elastic Solids ◽  
Linearized Equations ◽  
Viscoelastic Media ◽  
Isotropic Media

This paper is concerned with the propagation of viscoelastic shear waves in nonhomogeneous isotropic media. Herein we develop formal methods of solving the linearized equations of viscoelastodynamics in two and three dimensions for nonhomogeneous Maxwell solids whose properties depend continuously on a single radial coordinate. These methods are developed for the linearized equations of motion formulated in terms of shear stresses, and are based on Cooper's and Reiss' extension to linear homogeneous viscoelastic media of the Karal–Keller technique. Shearing stesses are applied to the boundaries of cylindrical and spherical openings in the viscoelastic media, and formal asymptotic wave front expansions of the solutions are obtained. In both cases a modulated progressive wave that propagates with variable velocity is obtained. The modulation depends on the moduli of rigidity and viscosity, whereas the velocity depends only on the modulus of rigidity. When the viscosity parameter in our Maxwell element tends to infinity, the results reduce to the known results for nonhomogeneous elastic solids.

Characteristic Forms of Differential Equations for Wave Propagation in Nonlinear Media

1981 ◽  
Vol 48 (4) ◽  
pp. 743-748 ◽  
Author(s):  
T. C. T. Ting
Keyword(s):  
Wave Propagation ◽  
Differential Equations ◽  
Constitutive Equations ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Compressible Fluids ◽  
Elastic Solids ◽  
Nonlinear Media ◽  
Nonlinear Materials ◽  
One Dimensional

Characteristic forms of differential equations for wave propagation in nonlinear media are derived directly from equations of motion and equations which combine the constitutive equations and the equations of continuity. Both Lagrangian coordinates and Eulerian coordinates are considered. The constitutive equations considered here apply to a large class of nonlinear materials. The characteristic forms derived here clearly show which components of the stress and velocity are involved in the differentiation along the bicharacteristics. Moreover, the reduction to one-dimensional cases from three-dimensional problems is obvious for the characteristic forms obtained here. Examples are given and compared with the known solution in the literature for wave propagation in linear isotropic elastic solids and isentropic compressible fluids.

On the Unified General Solutions of Linear Wave Motions of Thermoelastodynamics and Hydrodynamics With Practical Examples

1967 ◽  
Vol 34 (4) ◽  
pp. 879-887 ◽  
Author(s):  
P. K. Wong
Keyword(s):  
Equations Of Motion ◽  
Viscous Fluids ◽  
Linear Wave ◽  
Elastic Solids ◽  
Cylindrical Coordinates ◽  
General Solutions ◽  
The Media ◽  
Systematic Solution ◽  
Thermally Conducting ◽  
Wave Problems

In analogy to the development of the potential equations of motion of linear elastodynamics, the governing potential equations for linear wave motions of hydrodynamics and thermoelastodynamics are systematically exploited. As a result of these developments, problems of linear wave motions of homogeneous, isotropic, thermally conducting multi-layered elastic solids and viscous fluids can be systematically solved for the media whose boundaries are described by rectangular, circular, parabolic, and elliptic cylindrical coordinates, as well as by spherical and conical coordinates. Five practical examples are analytically solved to illustrate how the use of the potential equations of motion leads to more systematic solution procedures; these examples can be used for the modeling studies of aero and hydrospace vehicles, geological wave problems, and macroscopic biomechanics.

Wave propagation dynamics in a fractional Zener model with stochastic excitation

2020 ◽  
Vol 23 (6) ◽  
pp. 1570-1604
Author(s):  
Teodor Atanacković ◽  
Stevan Pilipović ◽  
Dora Seleši
Keyword(s):  
Wave Propagation ◽  
Constitutive Equation ◽  
Equations Of Motion ◽  
Weak Form ◽  
Stochastic Excitation ◽  
Expansion Theorem ◽  
Zener Model ◽  
Dissipation Inequality ◽  
Regularity Properties ◽  
Fractional Zener Model

Abstract Equations of motion for a Zener model describing a viscoelastic rod are investigated and conditions ensuring the existence, uniqueness and regularity properties of solutions are obtained. Restrictions on the coefficients in the constitutive equation are determined by a weak form of the dissipation inequality. Various stochastic processes related to the Karhunen-Loéve expansion theorem are presented as a model for random perturbances. Results show that displacement disturbances propagate with an infinite speed. Some corrections of already published results for a non-stochastic model are also provided.

scholarly journals New Approach to the Planetary Theory

1979 ◽  
Vol 81 ◽  
pp. 69-72 ◽  
Author(s):  
Manabu Yuasa ◽  
Gen'ichiro Hori
Keyword(s):  
Perturbation Method ◽  
Canonical Form ◽  
Equations Of Motion ◽  
Disturbing Function ◽  
Averaging Principle ◽  
The Sun ◽  
Planetary Theory ◽  
New Approach ◽  
Canonical Perturbation ◽  
Relative Coordinates

A new approach to the planetary theory is examined under the following procedure: 1) we use a canonical perturbation method based on the averaging principle; 2) we adopt Charlier's canonical relative coordinates fixed to the Sun, and the equations of motion of planets can be written in the canonical form; 3) we adopt some devices concerning the development of the disturbing function. Our development can be applied formally in the case of nearly intersecting orbits as the Neptune-Pluto system. Procedure 1) has been adopted by Message (1976).

scholarly journals Assessment of Linearization Approaches for Multibody Dynamics Formulations

2017 ◽  
Vol 12 (4) ◽  
Author(s):  
Francisco González ◽  
Pierangelo Masarati ◽  
Javier Cuadrado ◽  
Miguel A. Naya
Keyword(s):  
Mechanical System ◽  
Multibody Dynamics ◽  
Equations Of Motion ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Practical Applications ◽  
Linearization Methods ◽  
Highly Nonlinear ◽  
Wide Range ◽  
The Way

Formulating the dynamics equations of a mechanical system following a multibody dynamics approach often leads to a set of highly nonlinear differential-algebraic equations (DAEs). While this form of the equations of motion is suitable for a wide range of practical applications, in some cases it is necessary to have access to the linearized system dynamics. This is the case when stability and modal analyses are to be carried out; the definition of plant and system models for certain control algorithms and state estimators also requires a linear expression of the dynamics. A number of methods for the linearization of multibody dynamics can be found in the literature. They differ in both the approach that they follow to handle the equations of motion and the way in which they deliver their results, which in turn are determined by the selection of the generalized coordinates used to describe the mechanical system. This selection is closely related to the way in which the kinematic constraints of the system are treated. Three major approaches can be distinguished and used to categorize most of the linearization methods published so far. In this work, we demonstrate the properties of each approach in the linearization of systems in static equilibrium, illustrating them with the study of two representative examples.

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