On Elastic Continua With Hereditary Characteristics

1951 ◽  
Vol 18 (3) ◽  
pp. 273-279
Author(s):  
Enrico Volterra
Keyword(s):  
Equations Of Motion ◽  
Free Vibrations ◽  
Forced Vibrations ◽  
Degree Of Freedom ◽  
Elastic Media ◽  
Stress Strain ◽  
Radial Vibrations ◽  
Damping Characteristics ◽  
Transverse Vibrations ◽  
Free And Forced Vibrations

Abstract In a previous paper (1) the free and forced vibrations of systems of one degree of freedom with hereditary damping characteristics were discussed. In the present paper the classical equations of motion for elastic media are extended on the basis of the general linear stress-strain law involving hereditary damping. These equations are applied to the case of free radial vibrations of a sphere. Furthermore, the free vibrations of strings, the free transverse vibrations of beams, and the free vibrations of rectangular and circular membranes are studied under the assumption of hereditary damping.

An Asymptotic Method to Analyze the Vibrations of an Elastic Layer

1969 ◽  
Vol 36 (1) ◽  
pp. 65-72 ◽  
Author(s):  
J. D. Achenbach
Keyword(s):  
Power Series ◽  
Boundary Conditions ◽  
Equations Of Motion ◽  
Free Vibrations ◽  
Forced Vibrations ◽  
Wave Number ◽  
Dimensionless Wave Number ◽  
Analogous Manner ◽  
Independent Variable ◽  
Thickness Variable

The displacement components for both free and forced vibrations are sought as power series of the dimensionless wave number ε, where ε = 2π × layer thickness/wavelength. For the free vibration problem the object is to determine the frequencies, which are also sought as power series of the dimensionless wave number. The displacement and frequency expansions are substituted in the displacement equations of motion and in the boundary conditions. By collecting terms of the same order εn, a system of second-order, inhomogeneous, ordinary differential equations of the Helmholtz type is obtained, with the thickness variable as independent variable, and with associated boundary conditions. For free vibrations, subsequent integration yields the coefficients of εn for the displacements and the frequencies for all modes, and in the whole range of frequencies, but in a range of dimensionless wave numbers 0 < ε < ε* < 1, where ε* increases as more terms are retained in the expansions. For forced vibrations, the amplitudes are determined in an analogous manner if the external surface tractions are of sinusoidal dependence on the in-plane coordinates and on time. The response to surface tractions of more general spatial dependence is obtained by Fourier superposition.

Forced Vibrations of Viscoelastic Timoshenko Beams

1976 ◽  
Vol 98 (3) ◽  
pp. 820-826 ◽  
Author(s):  
C. C. Huang ◽  
T. C. Huang
Keyword(s):  
Boundary Conditions ◽  
Laplace Transform ◽  
Attenuation Factor ◽  
Equations Of Motion ◽  
Bending Moment ◽  
Expansion Method ◽  
Free Vibrations ◽  
Forced Vibrations ◽  
Mode Shapes ◽  
Timoshenko Beams

In a previous paper, the correspondence principle has been applied to derive the differential equations of motion of viscoelastic Timoshenko beams with or without external viscous damping. To study free vibrations these equations are solved by Laplace transform and boundary conditions are applied to obtain the attenuation factor and the frequency of the damped free vibrations and mode shapes. The present paper continues to analyze this subject and deals with the responses in deflection, bending slope, bending moment and shear for forced vibrations. Laplace transform and appropriate boundary conditions have been applied. Examples are given and results are plotted. The solution of forced vibrations of elastic Timoshenko beams obtained as a result of reduction from viscoelastic case and by eigenfunction expansion method concludes the paper.

On the Vibrations of Parabolic Membranes

1966 ◽  
Vol 88 (1) ◽  
pp. 45-50 ◽  
Author(s):  
L. L. Fontenot ◽  
D. O. Lomen
Keyword(s):  
Initial Value Problems ◽  
Free Vibrations ◽  
Forced Vibrations ◽  
Initial Value ◽  
Eigenvalues And Eigenfunctions ◽  
Dirichlet Conditions ◽  
Transverse Vibrations ◽  
Orthogonality Relations

This study is concerned with the transverse vibrations of a theoretical membrane which occupies a region bounded by two confocal parabolas. The eigenvalues and eigenfunctions for Dirichlet conditions are determined. Nodal patterns for the free vibrations of the membrane are given for various forms of symmetry. Orthogonality relations are established; initial value problems and forced vibrations also are considered.

Transient Analysis of Forced Vibrations of Complex Structural-Mechanical Systems

1962 ◽  
Vol 66 (619) ◽  
pp. 457-460 ◽  
Author(s):  
S. P. Chan ◽  
H. L. Cox ◽  
W. A. Benfield
Keyword(s):  
Differential Equations ◽  
Numerical Method ◽  
Finite Differences ◽  
Transient Analysis ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Forced Vibrations ◽  
Degree Of Freedom ◽  
Dynamic Forces ◽  
Complex Structural

This paper presents a numerical method, derived directly from the basic differential equations of motion and expressed in the form of recurrence-matrix of finite differences, that can be generally applied to all multi-degree-of-freedom structures subjected to dynamic forces or forced displacements on any masses at any instants of time. The movements of the system may be described by any form of generalised co-ordinates.

scholarly journals Theoretical and comparative study regarding the Galerkin-Vlasov method use for study the free vibrations of wood flat plate

2010 ◽  
pp. 223-228
Author(s):  
Marius Fetea ◽  
Gabriel Cheregi ◽  
Liana Lustun ◽  
Ildico Smit ◽  
Codruta Lucaci ◽  
...  
Keyword(s):  
19Th Century ◽  
18Th Century ◽  
Free Vibrations ◽  
Forced Vibrations ◽  
Theoretical Studies ◽  
Flat Plates ◽  
Free And Forced Vibrations ◽  
First Results ◽  
Historic Record ◽  
The 19Th Century

As for the historic record of the problem related to the study of flat plates, the first results were out for publishing at the end of the 18th century, the beginning of the 19th century, having Chladni E, Strehlke, Konig, R, Tanaka S, Rayleigh L, Ritz W and later on Gontkevich V, Timoshenko S, Leissa as pioneers. Each of the above mentioned authors have had significant contributions regarding the development of methods in order to solve the plates and establish some rigurous solutions of their differential equations of equilibrium. The making of constructions, machines and different high-perfomance appliances, whose functioning should take place in safety conditions, have required theoretical studies of rich complexity, as well as practical experiments, within which the problem of their free and forced vibrations represent an important category in the respective theme of research.

Free and Forced Vibrations of an Infinitely Long Cylindrical Shell in an Infinite Acoustic Medium

1954 ◽  
Vol 21 (2) ◽  
pp. 167-177
Author(s):  
H. H. Bleich ◽  
M. L. Baron
Keyword(s):  
Surrounding Medium ◽  
Low Frequency ◽  
Free Vibrations ◽  
Forced Vibrations ◽  
Acoustic Medium ◽  
Frequency Range ◽  
Stiffened Shells ◽  
Free And Forced Vibrations ◽  
Water Tables ◽  
Radial Forces

Abstract The paper presents a general method for the treatment of free and forced-vibration problems of infinitely long thin cylindrical shells. Surprisingly simple results are obtained by utilizing the known and tabulated modes of the shell in vacuo as generalized co-ordinates describing the response of the shell. The frequencies of free vibrations of submerged shells are obtained, and the response of the shell and medium to sinusoidally distributed, periodic, radial forces is determined. The results indicate that there is a low-frequency range where no radiation occurs and a high-frequency range where energy is radiated. Free vibration, or resonance in the case of forced vibrations, occurs only in the low-frequency range. The results of the paper may be applied to obtain the response to arbitrarily distributed, periodic, or nonperiodic forces by expanding such forces in Fourier series and/or integrals. The results for free and forced vibrations are discussed in general and for the specific case of steel shells in water. Tables are provided to facilitate numerical computations. With limitations the method is also applicable to ring-stiffened shells, and to the case of a static pressure in the surrounding medium.

Transverse Vibrations of Clamped Rectangular Plates of Generalized Orthotropy Subjected To In-Plane Forces

1980 ◽  
Vol 102 (2) ◽  
pp. 399-404 ◽  
Author(s):  
P. A. A. Laura ◽  
L. E. Luisoni
Keyword(s):  
Exact Solution ◽  
Variational Method ◽  
Forced Vibrations ◽  
Stiffened Plates ◽  
Rectangular Plates ◽  
Structural Element ◽  
Transverse Vibrations ◽  
Free And Forced Vibrations ◽  
Title Problem ◽  
Algorithmic Procedure

An exact solution of the title problem is probably out of the question. It is shown in the present study that a very simple solution can be obtained using simple polynomials and a variational method. Free and forced vibrations of the structural element are analyzed in a unified manner. The algorithmic procedure can be implemented in a microcomputer. The problem is of particular interest in certain filamentary plates as well as of obliquely stiffened plates.

scholarly journals Vibrations of a Timoshenko beam on an inertial foundation under a moving load

2018 ◽  
Vol 196 ◽  
pp. 01056
Author(s):  
Magdalena Ataman
Keyword(s):  
Timoshenko Beam ◽  
Moving Load ◽  
Equations Of Motion ◽  
Free Vibrations ◽  
Forced Vibrations ◽  
Governing Equations ◽  
Concentrated Load ◽  
Numerical Example ◽  
Moving Force ◽  
Speed Response

In the paper vibrations of the Timoshenko beam on an inertial foundation subjected to a moving force are discussed. Considered model of the inertial foundation is described by three parameters. They take into account elasticity, shear and inertia of the subgrade. In the literature such model of the subgrade is called Vlasov or Vlasov-Leontiev model. The Timoshenko beam is traversed by a concentrated load, moving with uniform speed. Response of the beam is found from the governing equations of motion of the problem. Problem of forced vibrations and problem of free vibrations of the beam are solved. Damping of the system is taken into consideration. Solution of the problem is illustrated by numerical example.

Vibrations of Elastic Systems Having Hereditary Characteristics

1950 ◽  
Vol 17 (4) ◽  
pp. 363-371
Author(s):  
Enrico Volterra
Keyword(s):  
Dynamic Stress ◽  
Forced Vibrations ◽  
High Rate ◽  
Stress Strain ◽  
Single Degree Of Freedom ◽  
Free And Forced Vibrations ◽  
First Approximation ◽  
Elastic Systems ◽  
Strain Relationship ◽  
Hereditary Function

Abstract Results of experiments carried out on plastics and rubberlike materials at high rate of straining are given. It is shown that the dynamic stress-strain (σ, ϵ) relationship for those materials can be expressed by the formula σ=f(ϵ)+∫0tϕ(t-τ)dϵ(τ)dτdτ The first term represents the static stress-strain relationship, while the second depends on the rate of straining dedt. As a first approximation it is supposed that the materials follow Hooke’s law when statically stressed. Equation [1] then becomes σ=Eϵ+∫0tϕ(t-τ)dϵ(τ)dτdτ Materials which follow the second equation are called materials with “hereditary characteristics.” Vibrations of single-degree-of-freedom systems having hereditary characteristics are considered. Methods of finding the hereditary function ϕ(t) from forced vibrations are given. Free and forced vibrations of simply supported beams having hereditary characteristics are studied.

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