Rayleigh’s Quotient of Linear Piezoelectricity and Its Use in the Approximate Calculation of the Natural Frequencies of Piezoelectric Continua

2005 ◽  
Vol 73 (2) ◽  
pp. 189-196 ◽  
Author(s):  
Piotr Cupiał
Keyword(s):  
Approximate Calculation ◽  
Natural Frequencies ◽  
Ritz Method ◽  
Rayleigh Quotient ◽  
Monotonic Convergence ◽  
Linear Piezoelectricity ◽  
Electrical Boundary Condition ◽  
Complete Set ◽  
Stationarity Conditions ◽  
Rayleigh’S Quotient

The paper discusses an extension to the linear theory of piezoelectricity of the Rayleigh quotient used in the analysis of the properties and the approximate calculation of the natural frequencies of elastic continua. It is shown that for a piezoelectric continuum an infinite number of equivalent expressions can be obtained which generalize the classical Rayleigh quotient. The stationarity conditions of any of these quotients under additional constraints imposed by the Gauss equation of electrostatics and the prescribed natural electrical boundary condition are shown to result in the complete set of the governing equations of the free vibration problem of a piezoelectric continuum. The general results discussed in the paper are illustrated by the approximate calculation of the natural frequencies of a piezoelectric rod by the Rayleigh–Ritz method. Unlike in the case of elastic structures, no monotonic convergence of the approximate frequencies is guaranteed for a piezoelectric continuum, the property which can be explained using the introduced Rayleigh quotients.

Effect of Hydrostatic Pressure and Depth of Fluid on the Vibrating Rectangular Plates Partially in Contact with a Fluid

2011 ◽  
Vol 110-116 ◽  
pp. 927-935
Author(s):  
Korosh Khorshidi
Keyword(s):  
Hydrostatic Pressure ◽  
Natural Frequencies ◽  
Ritz Method ◽  
Three Dimensional ◽  
Rayleigh Quotient ◽  
Rectangular Plates ◽  
Element Analysis ◽  
Displacement Potential ◽  
Fluid Displacement ◽  
Three Dimensional Finite Element

This study is focused on vibration analysis of a rectangular plate in partial coupled with a vertical bounded fluid. The fluid displacement potential satisfying the fluid boundary conditions is derived and the wet dynamic modal functions of the plate are expanded in terms of the finite Fourier series for a compatibility requirement along the contacting surface between the plate and the fluid. The natural frequencies of the plate coupled with sloshing fluid modes are calculated using Rayleigh–Ritz method based on minimizing the Rayleigh quotient. The proposed analytical method is verified by comparing the presented results with the results obtained by three–dimensional finite element analysis. Finally, the influence of hydrostatic pressure and fluid depth on the natural frequencies are examined and discussed in details.

Natural Frequencies and Mode Shapes of Elliptic Plates With Boundary Characteristic Orthogonal Polynomials As Assumed Shape Functions

1991 ◽  
Author(s):  
C. Rajalingham ◽  
R. B. Bhat ◽  
G. D. Xistris
Keyword(s):  
Coordinate System ◽  
Orthogonal Polynomials ◽  
Natural Frequencies ◽  
Ritz Method ◽  
Mode Shapes ◽  
Polar Coordinate System ◽  
Free Boundaries ◽  
Natural Modes ◽  
Polar Coordinate ◽  
Boundary Characteristic

Abstract The natural frequencies and natural modes of vibration of uniform elliptic plates with clamped, simply supported and free boundaries are investigated using Rayleigh-Ritz method. A modified polar coordinate system is used to investigate the problem. Energy expressions in Cartesian coordinate system are transformed into the modified polar coordinate system. Boundary characteristic orthogonal polynomials in the radial direction, and trigonometric functions in the angular direction are used to express the deflection of the plate. These deflection shapes are classified into four basic categories, depending on its symmetrical or antisymmetrical property about the major and minor axes of the ellipse. The first six natural modes in each of the above categories are presented in the form of contour plots.

Natural Frequencies of Parallelogrammic Plates Using Characteristic Orthogonal Polynomials in Rayleigh-Ritz Method

1992 ◽  
Author(s):  
L. T. Lee ◽  
W. F. Pon
Keyword(s):  
Boundary Conditions ◽  
Coordinate System ◽  
Orthogonal Polynomials ◽  
Natural Frequencies ◽  
Ritz Method ◽  
Energy Functional ◽  
Comprehensive Treatment ◽  
Energy Functionals ◽  
Simply Supported ◽  
Cartesian Coordinate

Abstract Natural frequencies of parallelogrammic plates are obtained by employing a set of beam characteristic orthogonal polynomials in the Rayleigh-Ritz method. The orthogonal polynomials are generalted by using a Gram-Schmidt process, after the first member is constructed so as to satisfy all the boundary conditions of the corresponding beam problems accompanying the plate problems. The strain energy functional and kinetic energy functionals are transformed from Cartesian coordinate system to a skew coordinate system. The natural frequencies obtained by using the orthogonal polynomial functions are compared with those obtained by other methods with all four edges clamped boundary conditions and greet agreements are found between them. The natural frequencies for parallelogrammic plates with other boundary conditions, such as four edges simply supported, clamped-free and simply supported-free, are also obtained. This method is considered as a better and accurate comprehensive treatment for this type of problems.

Vibration of Triangular Cantilever Plates by the Ritz Method

1954 ◽  
Vol 21 (4) ◽  
pp. 365-370
Author(s):  
B. W. Andersen
Keyword(s):  
Cantilever Beam ◽  
Natural Frequencies ◽  
Ritz Method ◽  
Plan View ◽  
Modes Of Vibration ◽  
Accuracy Of Results ◽  
Isosceles Triangles

Abstract Using the method published by Ritz in 1909, natural frequencies and corresponding node lines have been determined for two symmetric and two antisymmetric modes of vibration of isosceles triangular plates clamped at the base and having length-to-base ratios of 1, 2, 4, and 7 and for the two lowest modes of right triangular plates clamped along one leg and having ratios of the length of the free leg to that of the clamped one of 2, 4, and 7. A nonorthogonal co-ordinate system was used which gave constant limits of integration over the area of the triangle. The co-ordinate transformation made it necessary to modify the functions used by Ritz in approximating deflections and to consider cross products in the integration. The integration was done numerically, using tables compiled by Young and Felgar in 1949. To check the accuracy of results, a solution was obtained to the problem of a vibrating cantilever beam of uniform depth and triangular plan view. The results obtained were found to be consistent with those obtained for the plates by using an eight-term series to approximate the deflections of the symmetric plates (isosceles triangles) and a six-term series to approximate the deflections of the unsymmetric plates (right triangles).

Hydroelastic Vibration of Rectangular Plates

1996 ◽  
Vol 63 (1) ◽  
pp. 110-115 ◽  
Author(s):  
Moon K. Kwak
Keyword(s):  
Boundary Conditions ◽  
Approximate Formula ◽  
Natural Frequencies ◽  
Mass Matrix ◽  
Ritz Method ◽  
Plate Boundary ◽  
Rigid Wall ◽  
Mode Shapes ◽  
Rectangular Plates ◽  
Virtual Mass

This paper is concerned with the virtual mass effect on the natural frequencies and mode shapes of rectangular plates due to the presence of the water on one side of the plate. The approximate formula, which mainly depends on the so-called nondimensionalized added virtual mass incremental factor, can be used to estimate natural frequencies in water from natural frequencies in vacuo. However, the approximate formula is valid only when the wet mode shapes are almost the same as the one in vacuo. Moreover, the nondimensionalized added virtual mass incremental factor is in general a function of geometry, material properties of the plate and mostly boundary conditions of the plate and water domain. In this paper, the added virtual mass incremental factors for rectangular plates are obtained using the Rayleigh-Ritz method combined with the Green function method. Two cases of interfacing boundary conditions, which are free-surface and rigid-wall conditions, and two cases of plate boundary conditions, simply supported and clamped cases, are considered in this paper. It is found that the theoretical results match the experimental results. To investigate the validity of the approximate formula, the exact natural frequencies and mode shapes in water are calculated by means of the virtual added mass matrix. It is found that the approximate formula predicts lower natural frequencies in water with a very good accuracy.

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.

scholarly journals Multiple crack identification in stepped beam by measurements of natural frequencies

2014 ◽  
Vol 36 (2) ◽  
pp. 119-132
Author(s):  
Nguyen Tien Khiem ◽  
Duong The Hung ◽  
Vu Thi An Ninh
Keyword(s):  
Crack Detection ◽  
Natural Frequencies ◽  
Rayleigh Quotient ◽  
Crack Identification ◽  
Multiple Cracks ◽  
Simple Relationship ◽  
Multiple Crack ◽  
Scanning Method ◽  
New Approach ◽  
Efficient Procedure

A new approach is proposed for calculating natural frequencies and crack detection in a stepped cantilever beam with arbitrary number of cracks. This is based an explicit expression of the natural frequencies in term of crack parameter derived in the form similar to the so-called Rayleigh quotient for vibrating beam. The obtained simple relationship between natural frequencies and crack parameters enables not only accurate calculating the natural frequencies but also to develop an efficient procedure for detecting multiple cracks from given natural frequencies. The proposed technique called crack scanning method is illustrated and validated by numerical results.

Maximizing the Fundamental Natural Frequency of Triangular Composite Plates

1996 ◽  
Vol 118 (2) ◽  
pp. 141-146 ◽  
Author(s):  
S. Abrate
Keyword(s):  
Natural Frequency ◽  
Material Properties ◽  
Composite Structures ◽  
Natural Frequencies ◽  
Ritz Method ◽  
Laminated Composite ◽  
Composite Plates ◽  
Mode Shapes ◽  
Fundamental Natural Frequency ◽  
Wide Range

While many advances were made in the analysis of composite structures, it is generally recognized that the design of composite structures must be studied further in order to take full advantage of the mechanical properties of these materials. This study is concerned with maximizing the fundamental natural frequency of triangular, symmetrically laminated composite plates. The natural frequencies and mode shapes of composite plates of general triangular planform are determined using the Rayleigh-Ritz method. The plate constitutive equations are written in terms of stiffness invariants and nondimensional lamination parameters. Point supports are introduced in the formulation using the method of Lagrange multipliers. This formulation allows studying the free vibration of a wide range of triangular composite plates with any support condition along the edges and point supports. The boundary conditions are enforced at a number of points along the boundary. The effects of geometry, material properties and lamination on the natural frequencies of the plate are investigated. With this stiffness invariant formulation, the effects of lamination are described by a finite number of parameters regardless of the number of plies in the laminate. We then determine the lay-up that will maximize the fundamental natural frequency of the plate. It is shown that the optimum design is relatively insensitive to the material properties for the commonly used material systems. Results are presented for several cases.

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.

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