Solution of One-Dimensional Elastic Wave Problems by the Method of Characteristics

1967 ◽  
Vol 34 (3) ◽  
pp. 745-750 ◽  
Author(s):  
P. C. Chou ◽  
R. W. Mortimer
Keyword(s):  
Elastic Wave ◽  
Method Of Characteristics ◽  
Hyperbolic Partial Differential Equations ◽  
Unified Approach ◽  
The Method Of Characteristics ◽  
Dependent Variables ◽  
Generalized Displacements ◽  
The Media ◽  
Shell Equations ◽  
Wave Problems

A number of elastic wave problems which involve one space variable are treated, in a unified manner, by a system of second-order hyperbolic partial differential equations, with the generalized displacements as dependent variables. This system of n equations is analyzed by the method of characteristics yielding closed-form equations for the physical characteristics, the characteristic equations, and the propagation of discontinuities. Procedures for numerical integration along the characteristic curves are established. Among the elastic wave problems that may be represented by this unified approach are the Timoshenko beam, plates, bars, and sheets; in all cases, the media may be non-homogeneous. Various approximate shell equations also may be represented. Results of numerical calculations are in agreement with those obtained by other methods.

Solution of Impact-Induced Flexural Waves in a Circular Ring by the Method of Characteristics

1998 ◽  
Vol 65 (3) ◽  
pp. 569-579 ◽  
Author(s):  
V. P. W. Shim ◽  
S. E. Quah
Keyword(s):  
Numerical Scheme ◽  
Method Of Characteristics ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Circular Ring ◽  
Higher Order ◽  
Flexural Waves ◽  
The Method Of Characteristics ◽  
Finite Difference Equations ◽  
Generalized Displacements

A study of elastic wave propagation in a curved beam (circular ring) is presented. The governing equations of motion are formulated in two forms based on Timoshenko beam theory. Solutions are obtained using the method of characteristics, whereby a numerical scheme employing higher-order interpolation is proposed for the finite difference equations. Results obtained are verified by experiments; it is found that use of the higher-order numerical scheme improves correlation with experimental results. Comparison of the relative accuracy between the two mathematical formulations—one in terms of generalized forces and velocities and the other in terms of generalized displacements—shows the former to be mathematically simpler and to yield more accurate results.

Plane Elastic Waves Generated by Dynamical Loading Applied to Edge of Circular Hole

1981 ◽  
Vol 48 (3) ◽  
pp. 577-581 ◽  
Author(s):  
T. B. Moodie ◽  
J. B. Haddow ◽  
A. Mioduchowski ◽  
R. J. Tait
Keyword(s):  
Circular Hole ◽  
Elastic Waves ◽  
Method Of Characteristics ◽  
Numerical Procedure ◽  
Polar Angle ◽  
Independent Variables ◽  
The Method Of Characteristics ◽  
Dependent Variables ◽  
Spatial Coordinates ◽  
Unbounded Body

The generalized plane stress, or plane strain, elastodynamic problem of an unbounded body, subjected to sudden application of tractions at the surface of a circular hole, is considered. In general this problem involves three independent variables, two spatial coordinates, and time, but it is shown how the method of characteristics, for one spatial variable and time, can be applied when the dependent variables are expanded as Fourier series in terms of the polar angle θ. A numerical procedure is proposed for the method of characteristics and numerical results are obtained for a specific example.

Non-symmetric flow in Laval type nozzles

1972 ◽  
Vol 273 (1232) ◽  
pp. 185-235 ◽  
Keyword(s):  
Steady State ◽  
Combustion Chamber ◽  
Method Of Characteristics ◽  
Three Dimensions ◽  
Symmetric Flow ◽  
Gaseous Flow ◽  
The Method Of Characteristics ◽  
Lateral Forces

The equations of the steady state, compressible inviscid gaseous flow are linearized in a form suitable for application to nozzles of the Laval type. The procedure in the supersonic phase is verified by comparing solutions so obtained with those derived by the method of characteristics in two and three dimensions. Likewise, the solutions in the transonic phase are com pared with those obtained by other investigators. The linearized equation is then used to investigate the nat re of non-symmetric flow in rocket nozzles. It is found that if the flow from the combustion chamber into the nozzle is non-symmetric, the magnitude and direction of the turning couple produced by the emergent jet is dependent on the profile of the nozzle and it is possible to design profiles such that the turning couples or lateral forces are zero. The optimum nozzle so designed is independent of the pressure and also of the magnitude of the non-symmetry of the entry flow. The formulae by which they are obtained have been checked by extensive static and projection tests with simulated rocket test vehicles which are described in this paper.

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