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).

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.

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.

Effect of guy cable constraint on structural damping of open lattice antenna towers

1980 ◽  
Vol 7 (4) ◽  
pp. 614-620
Author(s):  
J. S. Kennedy ◽  
D. J. Wilson ◽  
P. F. Adams ◽  
M. Perlynn
Keyword(s):  
Natural Frequencies ◽  
Damping Ratio ◽  
Field Tests ◽  
Full Scale ◽  
Structural Damping ◽  
Scale Field ◽  
Modes Of Vibration

This paper presents the results of full-scale field tests on two steel guyed latticed towers. The towers were approximately 83 m in height, were guyed at three levels, and were of bolted angle construction. The observed results consist of the natural frequencies of the first two modes of vibration as well as the damping ratio for the first mode. The observed results are compared with analytical predictions and observations made concerning the contributions of structural and cable action to the damping ratio.

Nondestructive Detection of a Crack in a Triangular Cantilever Beam Based on Frequency Measurement

2007 ◽  
Vol 353-358 ◽  
pp. 2285-2288
Author(s):  
Fei Wang ◽  
Xue Zeng Zhao
Keyword(s):  
Cantilever Beam ◽  
Natural Frequencies ◽  
Frequency Measurement ◽  
Crack Size ◽  
Force Sensors ◽  
Rotational Spring ◽  
Cantilever Deflection ◽  
Crack Location ◽  
Small Force ◽  
Point Of Intersection

Triangular cantilevers are usually used as small force sensors in the transverse direction. Analyzing the effect of a crack on transverse vibration of a triangular cantilever will be of value to users and designers of cantilever deflection force sensors. We present a method for prediction of location and size of a crack in a triangular cantilever beam based on measurement of the natural frequencies in this paper. The crack is modeled as a rotational spring. The beam is treated as two triangular beams connected by a rotational spring at the crack location. Formulae for representing the relation between natural frequencies and the crack details are presented. To detect crack details from experiment results, the plots of the crack stiffness versus its location for any three natural modes can be obtained through the relation equation, and the point of intersection of the three curves gives the crack location. The crack size is then calculated using the relation between its stiffness and size. An example to demonstrate the validity and accuracy of the method is presented.

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 Rectangular Plates by the Ritz Method

1950 ◽  
Vol 17 (4) ◽  
pp. 448-453 ◽  
Author(s):  
Dana Young
Keyword(s):  
Ritz Method ◽  
Normal Modes ◽  
Approximate Solutions ◽  
The Other ◽  
Rectangular Plates ◽  
Elastic Plates ◽  
Modes Of Vibration ◽  
Ritz’S Method ◽  
Square Plate ◽  
Set Up

Abstract Ritz’s method is one of several possible procedures for obtaining approximate solutions for the frequencies and modes of vibration of thin elastic plates. The accuracy of the results and the practicability of the computations depend to a great extent upon the set of functions that is chosen to represent the plate deflection. In this investigation, use is made of the functions which define the normal modes of vibration of a uniform beam. Tables of values of these functions have been computed as well as values of different integrals of the functions and their derivatives. With the aid of these data, the necessary equations can be set up and solved with reasonable effort. Solutions are obtained for three specific plate problems, namely, (a) square plate clamped at all four edges, (b) square plate clamped along two adjacent edges and free along the other two edges, and (c) square plate clamped along one edge and free along the other three edges.

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.

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