scholarly journals Coupling between Transverse Vibrations and Instability Phenomena of Plates Subjected to In-Plane Loading

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
D. V. Bambill ◽  
C. A. Rossit
Keyword(s):  
Transverse Vibration ◽  
Natural Frequencies ◽  
Eigenvalue Problems ◽  
Mechanical Performance ◽  
Ritz Method ◽  
Normal Modes ◽  
Thin Plates ◽  
Transverse Vibrations ◽  
Edge Conditions ◽  
Distributed Forces

As it is known, the problems of free transverse vibrations and instability under in-plane loads of a plate are two different technological situations that have similarities in their approach to elastic solution. In fact, they are two eigenvalue problems in which we analyze the equilibrium situation of the plate in configurations which differ very slightly from the original, undeformed configuration. They are coupled in the event where in-plane forces are applied to the edges of the transversely vibrating plate. The presence of forces can have a significant effect on structural and mechanical performance and should be taken into account in the formulation of the dynamic problem. In this study, distributed forces of linear variation are considered and their influence on the natural frequencies and corresponding normal modes of transverse vibration is analyzed. It also analyzes their impact for the case of vibration control. The forces' magnitude is varied and the first natural frequencies of transverse vibration of rectangular thin plates with different combinations of edge conditions are obtained. The critical values of the forces which cause instability are also obtained. Due to the analytical complexity of the problem under study, the Ritz method is employed. Some numerical examples are presented.

scholarly journals Buckling and Pre Stressed Dynamics Analysis of Laminated Composite Plate with Different Boundary Conditions

2019 ◽  
Vol 15 (1) ◽  
pp. 46-55
Author(s):  
Widad I. Majeed ◽  
Zainab Abdul Kareem Abed
Keyword(s):  
Composite Plate ◽  
Eigenvalue Problems ◽  
Virtual Work ◽  
Plate Theory ◽  
Ritz Method ◽  
Laminated Composite ◽  
Thin Plates ◽  
Design Parameters ◽  
Present Solution ◽  
Edge Conditions

Critical buckling and natural frequencies behavior of laminated composite thin plates subjected to in-plane uniform load is obtained using classical laminated plate theory (CLPT). Analytical investigation is presented using Ritz- method for eigenvalue problems of buckling load solutions for laminated symmetric and anti-symmetric, angle and cross ply composite plate with different elastic supports along its edges. Equation of motion of the plate was derived using principle of virtual work and solved using modified Fourier displacement function that satisfies general edge conditions. Various numerical investigation were studied to exhibit a convergence and accuracy of the present solution for considering some design parameters such as edge conditions, aspect ratio, lamination angle, thickness ratio, orthotropic ratio, the results obtained gives good agreement with those published by other researchers.  

The Natural Frequencies of Free and Constrained Non-Uniform Beams

1960 ◽  
Vol 64 (599) ◽  
pp. 697-699 ◽  
Author(s):  
R. P. N. Jones ◽  
S. Mahalingam
Keyword(s):  
Degrees Of Freedom ◽  
Natural Frequencies ◽  
Ritz Method ◽  
Normal Modes ◽  
Iteration Process ◽  
Conservative System ◽  
Basic System ◽  
Orthogonal Functions ◽  
Added Masses ◽  
The Given

The Rayleigh-Ritz method is well known as an approximate method of determining the natural frequencies of a conservative system, using a constrained deflection form. On the other hand, if a general deflection form (i.e. an unconstrained form) is used, the method provides a theoretically exact solution. An unconstrained form may be obtained by expressing the deflection as an expansion in terms of a suitable set of orthogonal functions, and in selecting such a set, it is convenient to use the known normal modes of a suitably chosen “ basic system.” The given system, whose vibration properties are to be determined, can then be regarded as a “ modified system,” which is derived from the basic system by a variation of mass and elasticity. A similar procedure has been applied to systems with a finite number of degrees of freedom. In the present note the method is applied to simple non-uniform beams, and to beams with added masses and constraints. A concise general solution is obtained, and an iteration process of obtaining a numerical solution is described.

Combinations for the Free Vibration Behaviors of Anisotropic Rectangular Plates Under General Edge Conditions

1999 ◽  
Author(s):  
Yoshihiro Narita
Keyword(s):  
Free Vibration ◽  
Structural Design ◽  
Natural Frequencies ◽  
Ritz Method ◽  
Technical Information ◽  
Mode Shapes ◽  
Rectangular Plates ◽  
Plate Models ◽  
Edge Conditions ◽  
Anisotropic Rectangular Plates

Abstract The free vibration behavior of rectangular plates provides important technical information in structural design, and the natural frequencies are primarily affected by the boundary conditions as well as aspect and thickness ratios. One of the three classical edge conditions, i.e., free, simple supported and clamped edges, may be used to model the constraint along an edge of the rectangle. Along the entire boundary with four edges, there exist a wide variety of combinations in the edge conditions, each yielding different natural frequencies and mode shapes. For counting the total number of possible combinations, the present paper introduces the Polya counting theory in combinatorial mathematics, and formulas are derived for counting the exact numbers. A modified Ritz method is then developed to calculate natural frequencies of anisotropic rectangular plates under any combination of the three edge conditions and is used to numerically verify the numbers. In numerical experiments, the number of combinations in the free vibration behaviors is determined for some plate models by using the derived formulas, and are corroborated by counting the numbers of different sets of the natural frequencies that are obtained from the Ritz method.

AXISYMMETRIC VIBRATION OF POLAR ORTHOTROPIC CIRCULAR PLATES OF QUADRATICALLY VARYING THICKNESS RESTING ON ELASTIC FOUNDATION

2005 ◽  
Vol 05 (03) ◽  
pp. 387-408 ◽  
Author(s):  
N. BHARDWAJ ◽  
A. P. GUPTA
Keyword(s):  
Elastic Foundation ◽  
Ritz Method ◽  
Three Dimensional ◽  
Normal Modes ◽  
Mode Shapes ◽  
Axisymmetric Vibration ◽  
Circular Plates ◽  
Varying Thickness ◽  
Edge Conditions ◽  
Polar Orthotropic

This paper is concerned with the axisymmetric vibration problem of polar orthotropic circular plates of quadratically varying thickness and resting on an elastic foundation. The problem is solved by using the Rayleigh–Ritz method with boundary characteristic orthonormal polynomials for approximating the deflection function. Numerical results are computed for frequencies, nodal radii and mode shapes. Three-dimensional graphs are also plotted for the first four normal modes of axisymmetric vibration of plates with free, simply-supported and clamped edge conditions for various values of taper, orthotropy and foundation parameters.

The Normal Modes of Vibrations of Beams Having Noncollinear Elastic and Mass Axes

1940 ◽  
Vol 7 (3) ◽  
pp. A97-A105
Author(s):  
Clyne F. Garland
Keyword(s):  
Physical Properties ◽  
Natural Frequencies ◽  
Ritz Method ◽  
Normal Modes ◽  
Longitudinal Axis ◽  
Cantilever Beams ◽  
Amplitude Ratios ◽  
Numerical Example ◽  
Modes Of Vibration ◽  
Axis Passing

Abstract This analysis deals with vibration characteristics of cantilever beams in which the longitudinal axis, passing through the mass centers of the elementary sections, is not collinear with the longitudinal axis about which the beam tends to twist under the influence of an applied torsional couple. Expressions are derived from which the natural frequencies and normal modes of vibration of such a beam can be determined. The Rayleigh-Ritz method is employed to determine the frequencies and amplitude ratios. Following the development of the general expressions, more specific equations are derived which express the natural frequencies and relative amplitudes of motion in each of two normal modes of vibration. The theoretical relationships of the several physical properties of the beam to the natural frequencies of vibration are shown graphically. Finally a numerical example is presented for a particular beam, and the computed natural frequencies and normal modes are compared with those determined experimentally.

Free Vibration of Plates of Various Shapes with Intermediate Point Supports by the Hp-Cloud Method and Lagrange Multiplier

2016 ◽  
Vol 16 (09) ◽  
pp. 1550055 ◽  
Author(s):  
Sajad Jamshidi ◽  
Mojtaba Azhari ◽  
Hossein Amoushahi
Keyword(s):  
Lagrange Multiplier ◽  
Natural Frequencies ◽  
Cell Structure ◽  
Ritz Method ◽  
Thin Plates ◽  
Plate Boundaries ◽  
Shape Functions ◽  
Intermediate Point ◽  
Simply Supported ◽  
Kronecker Delta

The Hp-Cloud meshless method was developed to study the dynamic analysis of arbitrarily shaped thin plates with intermediate point supports. By proposing a special pattern for the influence radius of nodes and a polynomial type of enrichment function, the Hp-Cloud shape functions with Kronecker delta property were constructed. They can satisfy the zero deflection conditions for the field nodes at the point supports. The results obtained from these shape functions agree well with the previous ones, showing good accuracy and convergence. For plates with sharp corners, it is not possible to construct the Hp-Cloud shape function with Kronecker delta property. To this end, the Lagrange multiplier method was used for enforcing the boundary conditions. The computations were carried out by the Ritz method, and the cell structure method is refined to improve the speed and accuracy of numerical integration on the subscription surface of clouds intersecting with the plate boundaries. Using the algorithm developed, the natural frequencies of plates of various shapes and support patterns were computed. By increasing the number of point supports on the plate edges, the natural frequencies computed of the plate tend to those of the simply supported plate. Appropriate pattern of point supports distribution was presented for modeling the simply supported plates of various shapes by comparing the corresponding natural frequencies.

Combinations for the Free-Vibration Behaviors of Anisotropic Rectangular Plates Under General Edge Conditions

1999 ◽  
Vol 67 (3) ◽  
pp. 568-573 ◽  
Author(s):  
Y. Narita
Keyword(s):  
Free Vibration ◽  
Natural Frequencies ◽  
Numerical Study ◽  
Ritz Method ◽  
Mode Shapes ◽  
Rectangular Plates ◽  
Vibration Behavior ◽  
Edge Conditions ◽  
Important Field ◽  
Anisotropic Rectangular Plates

The free-vibration behavior of rectangular plates constitutes an important field in applied mechanics, and the natural frequencies are known to be primarily affected by the boundary conditions as well as aspect and thickness ratios. Any one of the three classical edge conditions, i.e., free, simply supported, and clamped edges, may be used to model the constraint along an edge of the rectangle. Along the entire boundary with four edges, there exist a wide variety of combinations in the edge conditions, each yielding different natural frequencies and mode shapes. For counting the total number of possible combinations the present paper introduces the Polya counting theory in combinatorial mathematics. Formulas are derived for counting the exact numbers. A modified Ritz method is then developed to calculate natural frequencies of anisotropic rectangular plates under any combination of the three classical edge conditions and is used to numerically verify the numbers. In this numerical study the number of combinations in the free-vibration behavior is determined for some plate models by using the derived formulas. Results are corroborated by counting the numbers of different sets of the natural frequencies that are obtained from the modified Ritz method. [S0021-8936(00)02203-0]

Benchmark Vibration Frequencies of Square Thin Plates with all Possible Combinations of Classical Boundary Conditions

2019 ◽  
Vol 19 (11) ◽  
pp. 1950131
Author(s):  
Aharon Deutsch ◽  
Joseph Tenenbaum ◽  
Moshe Eisenberger
Keyword(s):  
Boundary Conditions ◽  
Elastic Foundation ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Point Load ◽  
Thin Plates ◽  
Vibration Frequencies ◽  
Winkler Elastic Foundation ◽  
Edge Conditions ◽  
Classical Boundary

In this work, a new method is used for the exact vibration analysis of plates with classical boundary conditions. Four classical edge conditions are included: C — clamped, S — Simply supported, F — free, and G — guided. For square plates, all the possibilities add up to 55 cases. The solutions for the natural frequencies of the plates are found in this paper using static analysis. Starting from the equations of motion of an isotropic rectangular thin plate supported on Winkler elastic foundation, with a positive or negative value, the solution for the vibration frequencies of the plate is equivalent to finding the values of the negative elastic foundation that will yield infinite deflection under a point load on the plate. The solution is composed of three parts, the sum of which satisfies exactly both the field equation and the boundary conditions. For zero force, the vibration frequencies are found up to the desired accuracy. Benchmark results of the first six normalized natural frequencies, of isotropic square plates, for all possible 55 combinations of classical boundary conditions are given, many for the first time.

Rotary Inertia and Shear in Beam Vibration Treated by the Ritz Method

1968 ◽  
Vol 72 (688) ◽  
pp. 341-344 ◽  
Author(s):  
B. Dawson
Keyword(s):  
Shear Deformation ◽  
Natural Frequencies ◽  
Ritz Method ◽  
Normal Modes ◽  
Rotary Inertia ◽  
Characteristic Functions ◽  
Cantilever Beams ◽  
Dynamic Displacement ◽  
Modes Of Vibration ◽  
Good Agreement

Summary The natural frequencies of vibration of a cantilever beam allowing for rotary inertia and shear deformation are obtained by the approximate Ritz method. The workability of the method is dependent upon the approximating functions chosen for the dynamic displacement curves. A series of characteristic functions representing the normal modes of vibration of cantilever beams in simple flexure is used as the approximating functions for both deflections due to flexure and shear deformation. Good agreement is shown between frequencies obtained by the Ritz method and those resulting from an analytical solution. The effect upon the natural frequencies of allowing for rotary inertia alone is shown and it is seen to increase rapidly with mode number.

Transverse Vibration of a Rectangularly Orthotropic Spinning Disk, Part 1: Formulation and Free Vibration

1999 ◽  
Vol 121 (3) ◽  
pp. 273-279 ◽  
Author(s):  
A. Phylactopoulos ◽  
G. G. Adams
Keyword(s):  
Transverse Vibration ◽  
Natural Frequencies ◽  
Normal Modes ◽  
Circular Disk ◽  
Polar Coordinates ◽  
Fourier Series Expansion ◽  
Vibration Modes ◽  
Classical Formulation ◽  
Modes Of Vibration ◽  
Orthotropic Disk

The transverse vibration of a spinning circular disk with rectangular orthotropy is investigated. Two dimensionless parameters are established in order to characterize the degree of disk anisotropy and solutions are sought for a range of these parameters. The orthotropic bending stiffness is transferred into polar coordinates and is found to differ from a classical formulation for a stationary disk. A Fourier series expansion is used in the circumferential direction. Unlike the isotropic disk, the Fourier components determining the transverse vibration modes of the orthotropic disk do not separate. This condition results in an eigenvalue problem involving a coupled set of ordinary differential equations which are solved by a combination of numerical integration and iteration. Thus the natural frequencies and normal modes of vibration are determined. Because each eigenfunction contains contributions from more than one Fourier component, the normal modes do not possess distinct nodal diameters or nodal circles. Furthermore, disk orthotropy causes the natural frequencies corresponding to the sine and cosine modes to split; the degree of splitting decreases as the rotational speed increases.

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