On the Kirchhoff plates equations with thermal effects and memory boundary conditions

2009 ◽  
Vol 213 (1) ◽  
pp. 25-38 ◽  
Author(s):  
C.C.S. Tavares ◽  
M.L. Santos
Keyword(s):  
Boundary Conditions ◽  
Thermal Effects ◽  
Kirchhoff Plates

An Analytical Model for a Clamped Isotropic Beam Under Thermal Effects

2002 ◽  
Author(s):  
Rodrigo F. A. Marques ◽  
Daniel J. Inman
Keyword(s):  
Boundary Conditions ◽  
Analytical Model ◽  
Thermal Effects ◽  
Natural Frequencies ◽  
Dynamical Behavior ◽  
Beam Theory ◽  
Optimization Procedure ◽  
Temperature Changes ◽  
Thermally Expand ◽  
Spring Constants

Structures and industrial equipment often operate in environments where temperature variations take place. Although thermal effects may be negligible in some cases, they have caused the unexpected failure of mechanical systems many times. Whether or not temperature has significant effects on the dynamical behavior of such machines and structures depends upon several aspects, amongst which are geometry, material properties and boundary conditions. In this paper we investigate the dynamical behavior of a clamped beam under the influence of a uniform, quasi-statically varying temperature field. An analytical model was used, based on Euler-Bernoulli’s beam theory with the introduction of the proper boundary conditions. Temperature effects are included in terms of an axial force that shows up when the beam tends to thermally expand, but this expansion is restrained by the clamping. Preliminary results do not agree with experimental data, since perfect clamping is difficult to achieve in practice. Finally the model is updated with the inclusion of axial and torsional springs connecting the beam to the support. The spring constants were calculated through optimization procedure to minimize the differences between the natural frequencies obtained from the analytical model and the corresponding experimental ones. Agreement with experimental results is reasonable up to the 4th mode of the beam. In the future, this analytical model is to be used for design and simulation of an active controller that accounts for temperature changes in the structure.

FREE VIBRATION OF KIRCHHOFF PLATES WITH SECTOR SHAPES BY THE METHOD OF DISCRETE SINGULAR CONVOLUTION

2010 ◽  
Vol 07 (02) ◽  
pp. 229-240 ◽  
Author(s):  
M. GÜRSES ◽  
E. KUZU ◽  
Ö. CÍVALEK
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Analytical Solutions ◽  
Natural Frequencies ◽  
Plate Theory ◽  
Kirchhoff Plate ◽  
Kirchhoff Plates ◽  
Vibration Problems ◽  
Discrete Singular Convolution ◽  
Sector Angle

The free vibration of sector plates based on the classical Kirchhoff plate theory is analyzed by the method of discrete singular convolution using the Regularized Shannon delta (RSD) kernel. This method is applied to sector plates with a combination of boundary conditions, and the natural frequencies are calculated. The effects of the sector angle, boundary conditions and mode numbers on the frequency parameters are investigated. Comparisons are made with existing numerical and analytical solutions in the literature. This method is very effective for the study of vibration problems of sector plates.

Spectral modeling of vibrating plates with general shape and general boundary conditions

2011 ◽  
Vol 18 (11) ◽  
pp. 1607-1623 ◽  
Author(s):  
Giuseppe Catania ◽  
Silvio Sorrentino
Keyword(s):  
Boundary Conditions ◽  
Vibration Analysis ◽  
Polynomial Interpolation ◽  
Free Vibration Analysis ◽  
Transverse Displacement ◽  
General Shape ◽  
Analysis And Design ◽  
Kirchhoff Plates ◽  
Vibrating Plates ◽  
Interpolation Functions

The analysis and design of lightweight plate structures require efficient computational tools, because exact analytical solutions for vibrating plates are currently known only for some standard shapes in conjunction with a few basic boundary conditions. This paper deals with vibration analysis of Kirchhoff plates of general shape with non-standard boundary conditions, adopting a Rayleigh-Ritz approach. Three different coordinate mappings are considered, using different kinds of functions: 1) trigonometric and polynomial interpolation functions for mapping the shape of the plate, 2) trigonometric and polynomial interpolation functions for mapping a constraint domain of general shape, 3) products of linearly independent eigenfunctions evaluated from a standard beam in flexural vibration for describing the transverse displacement field of the plate. Flexural free vibration analysis of different shaped plates is then performed using the same approach: skew, trapezoid and triangular plates, plates with parabolic curved edges, sectors of circular plates, circular and elliptic annular plates. Purely elastic plates are considered, but the method may also be applied to the analysis of viscoelastic plates. The results are compared with those available in the literature and using standard finite element analysis.

scholarly journals Stability for the Kirchhoff Plates Equations with Viscoelastic Boundary Conditions in Noncylindrical Domains

2013 ◽  
Vol 2013 ◽  
pp. 1-12
Author(s):  
Jum-Ran Kang
Keyword(s):  
Global Existence ◽  
Boundary Conditions ◽  
Cylindrical Domain ◽  
Uniqueness Of Solutions ◽  
Kirchhoff Plates ◽  
Noncylindrical Domain

We study Kirchhoff plates equations with viscoelastic boundary conditions in a noncylindrical domain. This work is devoted to proving the global existence, uniqueness of solutions, and decay of the energy of solutions for Kirchhoff plates equations in a non-cylindrical domain.

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