Dynamic contact of a thermoelastic Mindlin–Timoshenko beam with a rigid obstacle

2017 ◽  
Vol 23 (3) ◽  
pp. 411-419
Author(s):  
Igor Bock
Keyword(s):  
Timoshenko Beam ◽  
Vector Measure ◽  
Time Derivative ◽  
Thermal Strain ◽  
Dynamic Contact ◽  
Penalization Method ◽  
Unilateral Problem ◽  
Complementarity Conditions ◽  
Rigid Obstacle ◽  
Angle Of Rotation

We concentrate on the dynamics of a thermoelastic Mindlin–Timoshenko beam striking a rigid obstacle. We state classical formulations involving complementarity conditions. Weak formulations are in the form of systems consisting of a hyperbolic variational inequality for a deflection, a hyperbolic and a parabolic equation for an angle of rotation and a thermal strain, respectively. The penalization method is applied to solve the unilateral problem. The time derivative of the function representing the deflection of the beam’s middle line is not continuous due to the hitting the obstacle. The acceleration term has the form of a vector measure.

scholarly journals A vibrating thermoelastic plate in a contact with an obstacle

2015 ◽  
Vol 63 (1) ◽  
pp. 39-52
Author(s):  
Igor Bock ◽  
Jiří Jarušek
Keyword(s):  
Weak Solution ◽  
Variational Inequality ◽  
Parabolic Equation ◽  
Heat Source ◽  
Time Derivative ◽  
Thermal Strain ◽  
Dynamic Contact ◽  
Dynamic Contact Problem ◽  
Penalization Method ◽  
Thermoelastic Plate

Abstract We deal with a dynamic contact problem for a thermoelastic plate vibrating against a rigid obstacle. Dynamics is described by a hyperbolic variational inequality for deflections. The plate is subjected to a perpendicular force and to a heat source. The parabolic equation for the thermal strain resultant contains the time derivative of the deflection. We formulate a weak solution of the system and verify its existence using the penalization method.

scholarly journals Unilateral Problem for a Viscoelastic Beam Equation Type p-Laplacian with Strong Damping and Logarithmic Source

2022 ◽  
Vol 2 ◽  
pp. 5
Author(s):  
Ducival C. Pereira ◽  
Geraldo M. de Araújo ◽  
Carlos A. Raposo
Keyword(s):  
Existence And Uniqueness ◽  
Source Term ◽  
Global Solutions ◽  
Beam Equation ◽  
Viscoelastic Beam ◽  
Penalization Method ◽  
Unilateral Problem ◽  
Strong Damping ◽  
The Stability ◽  
Galerkin’S Approximation

In this manuscript, we investigate the unilateral problem for a viscoelastic beam equation of p-Laplacian type. The competition of the strong damping versus the logarithmic source term is considered. We use the potential well theory. Taking into account the initial data is in the stability set created by the Nehari surface, we prove the existence and uniqueness of global solutions by using the penalization method and Faedo-Galerkin’s approximation.

scholarly journals Global Existence and Exponential Decay for a Dynamic Contact Problem of Thermoelastic Timoshenko Beam with Second Sound

2017 ◽  
Author(s):  
Wenjun Liu ◽  
Dongqin Chen ◽  
Biqing Zhu
Keyword(s):  
Global Existence ◽  
Contact Problem ◽  
Exponential Decay ◽  
Timoshenko Beam ◽  
A Priori Estimates ◽  
Dynamic Contact ◽  
Second Sound ◽  
Dynamic Contact Problem ◽  
Irregular Boundary ◽  
Cattaneo’S Law

In this paper, we study the global existence and exponential decay for a dynamic contact problem between a Timoshenko beam with second sound and two rigid obstacles, of which the heat flux is given by Cattaneo's law instead of the usual Fourier's law. The main difficulties arise from the irregular boundary terms, from the low regularity of the weak solution and from the weaker dissipative effects of heat conduction induced by Cattaneo's law. By considering related penalized problems, proving some a priori estimates and passing to the limit, we prove the global existence of the solutions. By considering the approximate framework, constructing some new functionals and applying the perturbed energy method, we obtain the exponential decay result for the approximate solution, and then prove the exponential decay rate to the original problem by utilizing the weak lower semicontinuity arguments.

Vibrations of Tapered Timoshenko Beams in Terms of Static Timoshenko Beam Functions

2000 ◽  
Vol 68 (4) ◽  
pp. 596-602 ◽  
Author(s):  
D. Zhou ◽  
Y. K. Cheung
Keyword(s):  
Timoshenko Beam ◽  
Ritz Method ◽  
Free Vibrations ◽  
Basis Functions ◽  
Timoshenko Beams ◽  
Cross Sectional ◽  
Wide Range ◽  
Convergence Study ◽  
The Cross ◽  
Angle Of Rotation

In this paper, the free vibrations of a wide range of tapered Timoshenko beams are investigated. The cross section of the beam varies continuously and the variation is described by a power function of the coordinate along the neutral axis of the beam. The static Timoshenko beam functions, which are the complete solutions of a tapered Timoshenko beam under a Taylor series of static load, are developed, respectively, as the basis functions of the flexural displacement and the angle of rotation due to bending. The Rayleigh-Ritz method is applied to derive the eigenfrequency equation of the tapered Timoshenko beam. Unlike conventional basis functions which are independent of the cross-sectional variation of the beam, these static Timoshenko beam functions vary in accordance with the cross-sectional variation of the beam so that higher accuracy and more rapid convergence have been obtained. Some numerical results are presented for both truncated and sharp-ended Timoshenko beams. On the basis of convergence study and comparison with available results in the literature it is shown that the first few eigenfrequencies can be given with quite good accuracy by using a small number of terms of the static Timoshenko beam functions. Finally, some valuable results are presented systematically.

scholarly journals Global Existence and Exponential Decay for a Dynamic Contact Problem of Thermoelastic Timoshenko Beam with Second Sound

2017 ◽  
Author(s):  
Wenjun Liu ◽  
Dongqin Chen ◽  
Biqing Zhu
Keyword(s):  
Global Existence ◽  
Contact Problem ◽  
Exponential Decay ◽  
Timoshenko Beam ◽  
A Priori Estimates ◽  
Dynamic Contact ◽  
Second Sound ◽  
Dynamic Contact Problem ◽  
Irregular Boundary ◽  
Cattaneo’S Law

In this paper, we study the global existence and exponential decay for a dynamic contact problem between a Timoshenko beam with second sound and two rigid obstacles, of which the heat flux is given by Cattaneo's law instead of the usual Fourier's law. The main difficulties arise from the irregular boundary terms, from the low regularity of the weak solution and from the weaker dissipative effects of heat conduction induced by Cattaneo's law. By considering related penalized problems, proving some a priori estimates and passing to the limit, we prove the global existence of the solutions. By considering the approximate framework, constructing some new functionals and applying the perturbed energy method, we obtain the exponential decay result for the approximate solution, and then prove the exponential decay rate to the original problem by utilizing the weak lower semicontinuity arguments.

Investigation of behavior of the dynamic contact angle in a problem of convective flows

2017 ◽  
Vol 5 (4) ◽  
pp. 27-42
Author(s):  
O.N Goncharova ◽  
◽  
I.V. Marchuk ◽  
A.V. Zakurdaeva ◽  
◽  
...  
Keyword(s):  
Contact Angle ◽  
Dynamic Contact Angle ◽  
Dynamic Contact ◽  
Convective Flows
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