scholarly journals Transient wave analysis of a cantilever Timoshenko beam subjected to impact loading by Laplace transform and normal mode methods

2012 ◽  
Vol 49 (9) ◽  
pp. 1158-1176 ◽  
Author(s):  
Yu-Chi Su ◽  
Chien-Ching Ma
Keyword(s):  
Laplace Transform ◽  
Normal Mode ◽  
Timoshenko Beam ◽  
Impact Loading ◽  
Wave Analysis ◽  
Transient Wave

scholarly journals An Asymptotic Approach for the Elastodynamic Problem of a Plate under Impact Loading

2010 ◽  
Vol 2010 ◽  
pp. 1-10
Author(s):  
Penelope Michalopoulou ◽  
George A. Papadopoulos
Keyword(s):  
Laplace Transform ◽  
Impact Loading ◽  
Step Function ◽  
Infinite Strip ◽  
Asymptotic Approach ◽  
Heaviside Step Function ◽  
Inertia Effects ◽  
Elastodynamic Problem ◽  
Line Force ◽  
The One

An approach is presented for analyzing the transient elastodynamic problem of a plate under an impact loading. The plate is considered to be in the form of a long strip under plane strain conditions. The loading is taken as a concentrated line force applied normal to the plate surface. It is assumed that this line force is suddenly applied and maintained thereafter (i.e., it is a Heaviside step function of time). Inertia effects are taken into consideration and the problem is treated exactly within the framework of elastodynamic theory. The approach is based on multiple Laplace transforms and on certain asymptotic arguments. In particular, the one-sided Laplace transform is applied to suppress time dependence and the two-sided Laplace transform to suppress the dependence upon a spatial variable (along the extent of the infinite strip). Exact inversions are then followed by invoking the asymptotic Tauber theorem and the Cagniard-deHoop technique. Various extensions of this basic analysis are also discussed.

Timoshenko Beam Dynamics

1971 ◽  
Vol 38 (3) ◽  
pp. 591-594 ◽  
Author(s):  
G. M. Anderson
Keyword(s):  
Laplace Transform ◽  
Timoshenko Beam ◽  
General Problem ◽  
Beam Dynamics ◽  
Time Dependent ◽  
Residue Theorem ◽  
Laplace Transform Method ◽  
Transform Method ◽  
Beam Analysis ◽  
The Laplace Transform

The general problem of Timoshenko beam analysis is solved using the Laplace transform method. Time-dependent boundary and normal loads are considered. It is established that the integrands of the inversion integrals are always single-valued for beams of finite length and modal solutions can always be obtained using the residue theorem.

A Crack in an Anisotropic Layered Material Under the Action of Impact Loading

1988 ◽  
Vol 55 (1) ◽  
pp. 120-125 ◽  
Author(s):  
W. T. Ang
Keyword(s):  
Stress Intensity ◽  
Laplace Transform ◽  
Integral Equations ◽  
Stress Intensity Factors ◽  
Impact Loading ◽  
Layered Material ◽  
Fredholm Integral Equations ◽  
Transform Domain ◽  
Intensity Factors ◽  
The Laplace Transform

The problem of a plane crack in an anisotropic layered material under the action of impact loading is considered in this paper. The problem is reduced in the Laplace transform domain to a set of simultaneous Fredholm integral equations of the second kind. Once these integral equations are solved, the crack tip stress intensity factors in the Laplace transform domain may be readily calculated. The dynamic stress intensity factors can then be obtained through the use of a numerical technique for inverting Laplace transforms. Numerical results are given for specific examples involving particular transversely isotropic materials.

Thermodynamical interactions in a two-temperature dual-phase-lag micropolar thermoelasticity with gravity

2018 ◽  
Vol 14 (1) ◽  
pp. 102-124 ◽  
Author(s):  
Sunita Deswal ◽  
Baljit Singh Punia ◽  
Kapil Kumar Kalkal
Keyword(s):  
Normal Mode ◽  
Normal Mode Analysis ◽  
Mode Analysis ◽  
Phase Lag ◽  
Dual Phase ◽  
Transient Wave ◽  
Content Type ◽  
Temperature Parameter ◽  
Physical Quantities ◽  
Two Temperature

Purpose The dual-phase-lag (DPL) model is applied to study the effect of the gravity field and micropolarity on the wave propagation in a two-temperature generalized thermoelastic problem for a medium. The paper aims to discuss this issue. Design/methodology/approach The exact expressions of the considered variables are obtained by using normal mode analysis. Findings Numerical results for the field quantities are given in the physical domain and illustrated graphically to show the effect of angle of inclination. Comparisons of the physical quantities are also shown in figure to study the effect of gravity and two-temperature parameter. Originality/value This paper is concerned with the analysis of transient wave phenomena in a micropolar thermoelastic half-space subjected to inclined load. The governing equations are formulated in the context of two-temperature generalized thermoelasticity theory with DPLs. A medium is assumed to be initially quiescent and under the effect of gravity. An analytical solution of the problem is obtained by employing normal mode analysis. Numerical estimates of displacement, stresses and temperatures are computed for magnesium crystal-like material and are illustrated graphically. Comparisons of the physical quantities are shown in figures to study the effects of gravity, two-temperature parameter and angle of inclination. Some particular cases of interest have also been inferred from the present problem.

The Response of Infinite Strings and Beams to an Initially Applied Moving Force: Analytical Solution

2012 ◽  
Vol 134 (4) ◽  
Author(s):  
André Langlet ◽  
Ophélie Safont ◽  
Jérôme Renard
Keyword(s):  
Fourier Transform ◽  
Laplace Transform ◽  
Stationary Solution ◽  
Timoshenko Beam ◽  
Bernoulli Beam ◽  
Moving Force ◽  
The Laplace Transform ◽  
Bar Velocity ◽  
Infinite Strings ◽  
Euler Bernoulli Beam

This paper presents the analytical solutions for bilaterally infinite strings and infinite beams on which a point force is initially applied, which then moves on the structure at a constant velocity. The solutions are sought by first applying the Fourier transform to the spatial coordinate dependence, and then the Laplace transform to the time variable of dependence, of the governing equations of motion. For the strings, it is necessary to distinguish between the case of a sonic load (a force moving at the phase velocity of transverse waves) and the cases of subsonic and supersonic loads. This is achieved by a suitable expansion in polynomial ratios of the Laplace transform, before going back to the original Fourier transform, whose inverse is obtained by exact calculations of the integrals over the complex infinite domain. For the Euler-Bernoulli beam, the same process leads to the closed-form (exact) formula for the displacement, from which the stress can be deduced. The displacement consists of the sum of two integrals: one representing the transient part, and the other, the stationary part of the solution. The stationary part is observed in the vicinity of the force for a very long travel time. The transient part is observed at a finite position coordinate, in relative proximity to the starting point of the moving force. For the Timoshenko beam, the final step in the calculation of the displacement and rotation, which requires a numerical evaluation of the integrals, leads to Fourier cosine and sine transforms. The response of the beam depends on the load velocity, relative to the two characteristic velocities: those of shear waves and longitudinal waves. This demonstrates that the transient parts of the solutions, in the Euler-Bernoulli beam or in the Timoshenko beam, are quasi identical. However, classical theory fails to forecast high frequency responses, occurring with velocities of the load exceeding twenty per cent of the bar velocity. For a velocity greater than the velocity of the shear waves, classical theory wrongly forecasts the response. In addition, according to the Euler-Bernoulli beam theory, the flexural waves are able to exceed the bar velocity, which is not realistic. If the load moves for a long period, the solution in the vicinity of the load tends towards a stationary solution. It is important to note that the solution to the stationary problem must be completed by the solution to the associated homogeneous system to represent the physical stationary solution.

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