Vibrations of Transversely Isotropic Finite Circular Cylinders

1994 ◽  
Vol 61 (4) ◽  
pp. 964-970 ◽  
Author(s):  
K. T. Chau
Keyword(s):  
Hyperbolic Equations ◽  
Longitudinal Vibration ◽  
Transversely Isotropic ◽  
Frequency Equation ◽  
Circular Cylinders ◽  
Finite Cylinder ◽  
Mode Shapes ◽  
Axisymmetric Vibration ◽  
Vibration Frequencies ◽  
Axisymmetric Mode

This paper investigates the exact frequency equations for all the possible natural vibrations in a transversely isotropic cylinder of finite length. Two wave potentials are used to uncouple the equations of motion; the resulting hyperbolic equations are solved analytically for the vibration frequencies of a finite cylinder with zero shear tractions and zero axial displacement on the end surfaces and with zero tractions on the curved surfaces. In general, the mode shapes and the frequency equations of vibrations depend on both the range of the frequency and the elastic properties of the material. The vibration frequencies for sapphire cylinders are studied as an example. Two limiting cases are also considered: the long bar limit equals the frequency equation for the longitudinal vibration of bars obtained by Morse (1954) and by Lord Rayleigh (1945); and the frequency equation for thin disks (small length/radius ratio) is also obtained. The frequency for the first axisymmetric mode agrees with the experimental observation by Lusher and Hardy (1988) to within one percent. Natural frequencies for the first three longitudinal and circumferential modes are plotted for all cylinder geometries. The lowest frequency always corresponds to the first nonsymmetric mode regardless of the dimension of the cylinder. For axisymmetric vibration modes, numerical plots show that double roots exist in the frequency equations; such doublets were observed experimentally by Booker and Sagar (1971).

Exact Solutions for Longitudinal Vibration of Multi-Step Bars with Varying Cross-Section

1999 ◽  
Vol 122 (2) ◽  
pp. 183-187 ◽  
Author(s):  
Q. S. Li
Keyword(s):  
Natural Frequencies ◽  
Longitudinal Vibration ◽  
Frequency Equation ◽  
Mode Shapes ◽  
Transmission Tower ◽  
Power Functions ◽  
Closed Form Solutions ◽  
High Rise ◽  
One Step ◽  
Area Variation

Using appropriate transformations, the equation of motion for free longitudinal vibration of a nonuniform one-step bar is reduced to an analytically solvable equation by selecting suitable expressions, such as power functions and exponential functions, for the area variation. Exact analytical solutions to determine the longitudinal natural frequencies and mode shapes for a one step nonuniform bar are derived and used to obtain the frequency equation of multi-step bars. The new exact approach is presented which combines the transfer matrix method and closed form solutions of one step bars. A numerical example demonstrates that the calculated natural frequencies and mode shapes of a television transmission tower are in good agreement with the corresponding experimental data, and the selected expressions are suitable for describing the area variation of typical high-rise structures. [S0739-3717(00)00302-0]

AXISYMMETRIC VIBRATIONS OF CIRCULAR AND ANNULAR PLATES WITH VARIABLE THICKNESS

2001 ◽  
Vol 01 (02) ◽  
pp. 195-206 ◽  
Author(s):  
M. EISENBERGER ◽  
M. JABAREEN
Keyword(s):  
Exact Solution ◽  
Power Series ◽  
Variable Thickness ◽  
Natural Frequencies ◽  
Plate Thickness ◽  
Frequency Equation ◽  
Axisymmetric Vibration ◽  
Vibration Frequencies ◽  
Annular Plates ◽  
Infinite Power

In this work the exact axisymmetric vibration frequencies of circular and annular variable thickness plates are found. The solution is obtained using the exact element method developed earlier. It allows for the exact solution of problems with general polynomial variation in thickness using infinite power series. The solution is exact up to the accuracy of the computer. The natural frequencies of vibration are found as the solutions of the frequency equation. Normalized values for the natural frequencies are given for linear, parabolic and cubic variations of the plate thickness, for circular and annular plates, with four types of boundary conditions on the inner and outer boundaries.

Free Vibrations of Thick, Complete Conical Shells of Revolution From a Three-Dimensional Theory

2005 ◽  
Vol 72 (5) ◽  
pp. 797-800 ◽  
Author(s):  
Jae-Hoon Kang ◽  
Arthur W. Leissa
Keyword(s):  
Ritz Method ◽  
Three Dimensional ◽  
Axial Direction ◽  
Free Vibrations ◽  
Mode Shapes ◽  
Vibration Frequencies ◽  
Shells Of Revolution ◽  
Dimensional Theory ◽  
Conical Shells ◽  
Cylindrical Coordinate

A three-dimensional (3D) method of analysis is presented for determining the free vibration frequencies and mode shapes of thick, complete (not truncated) conical shells of revolution in which the bottom edges are normal to the midsurface of the shells based upon the circular cylindrical coordinate system using the Ritz method. Comparisons are made between the frequencies and the corresponding mode shapes of the conical shells from the authors' former analysis with bottom edges parallel to the axial direction and the present analysis with the edges normal to shell midsurfaces.

Vibrations of Spinning Rings With Radial and Circumferential Displacement Constraints

1991 ◽  
Author(s):  
Shyh-Chin Huang ◽  
Chen-Kai Su
Keyword(s):  
Natural Frequencies ◽  
Point Contact ◽  
Frequency Equation ◽  
Mode Shapes ◽  
Complex Conjugate ◽  
Constrained System ◽  
Form Complex ◽  
Circumferential Displacement ◽  
On The Road ◽  
Displacement Constraints

Abstract The frequencies and mode shapes of rolling rings with radial and circumferential displacement constraints are investigated. The displacement constraints practically come from the point contact, e.g., rolling tire on the road, or other applications. The proposed approach to analysis is calculating the natural frequencies and modes of a non-contacted spinning ring, then employing the receptance method for displacement constraints. The frequency equation for the constrained system is hence obtained, and it can be solved numerically or graphically. The receptance matrix developed for the spinning ring is surprisingly found not symmetric as usual. Moreover, the cross receptances are discovered to form complex conjugate pairs. That is a feature that has never been described in literature. The results show that the natural frequencies for the spinning ring in contact, as expected, higher than those for the non-contacted ring. The variance of frequencies to rotational speeds are then illustrated. The analytic forms of mode shapes are also derived and sketched. The traveling modes are then shown for cases.

DAMAGE DETECTION OF RC SLABS USING NONLINEAR VIBRATION FEATURES

2009 ◽  
Vol 09 (04) ◽  
pp. 687-709 ◽  
Author(s):  
XINQUN ZHU ◽  
HONG HAO
Keyword(s):  
Reinforced Concrete ◽  
Nonlinear Vibration ◽  
Structural Damage ◽  
Concrete Structures ◽  
Dynamic Responses ◽  
Mode Shapes ◽  
Reinforced Concrete Structures ◽  
Vibration Frequencies ◽  
Skeleton Curve ◽  
Time Frequency

Studied herein are the signatures of nonlinear vibration characteristics of damaged reinforced concrete structures using the wavelet transform (WT). A two-span RC slab built in 2003 was tested to failure in the laboratory. Vibration measurements were carried out at various stages of structural damage. The vibration frequencies, mode shapes, and damping ratios at each loading stage were extracted and analyzed. It is found that the vibration frequencies are not sensitive to small damages, but are good indicators when damage is severe. The dynamic responses are also analyzed in the time–frequency domain by WT and the skeleton curve is constructed to describe the nonlinear characteristics in the reinforced concrete structures. The results show that the skeleton curves are good indicators of damage in the reinforced concrete structures because they are more sensitive to small damages than vibration frequencies.

Some Results on Axisymmetric Vibration of Spherical Shells of Variable Thickness

1990 ◽  
Vol 112 (4) ◽  
pp. 432-437 ◽  
Author(s):  
A. V. Singh ◽  
S. Mirza
Keyword(s):  
Variable Thickness ◽  
Natural Frequencies ◽  
Mode Shapes ◽  
Axisymmetric Vibration ◽  
Spherical Shells ◽  
Varying Thickness ◽  
Wide Range

Natural frequencies and mode shapes are presented for the free axisymmetric vibration of spherical shells with linearly varying thickness along the meridian. Clamped and hinged edges corresponding to opening angles 30, 45, 60 and 90 deg have been considered in this technical brief to cover a wide range from shallow to deep spherical shells. Variations in thickness are seen to have very pronounced effects on the frequencies and mode shapes.

A Unified Frequency Equation and the Motion Analysis of a Moving Flexible Beam Carrying a Concentrated Mass

1999 ◽  
Author(s):  
S. Park ◽  
J. W. Lee ◽  
Y. Youm ◽  
W. K. Chung
Keyword(s):  
Natural Frequencies ◽  
Equations Of Motion ◽  
Frequency Equation ◽  
Open Loop ◽  
Mode Shapes ◽  
Flexible Beam ◽  
Lumped Mass ◽  
Concentrated Mass ◽  
Forcing Function ◽  
The Mathematical Model

Abstract In this paper, the mathematical model of a Bernoulli-Euler cantilever beam fixed on a moving cart and carrying an intermediate lumped mass is derived. The equations of motion of the beam-mass-cart system is analyzed utilizing unconstrained modal analysis, and a unified frequency equation which can be generally applied to this kind of system is obtained. The change of natural frequencies and mode shapes with respect to the change of the mass ratios of the beam, the lumped mass and the cart and to the position of the lumped mass is investigated. The open-loop responses of the system by arbitrary forcing function are also obtained through numerical simulations.

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