Natural Frequencies and Mode Shapes for Axisymmetric Vibration of Deep Spherical Shells

1965 ◽  
Vol 32 (3) ◽  
pp. 553-561 ◽  
Author(s):  
E. W. Ross
Keyword(s):  
Natural Frequencies ◽  
Large Degree ◽  
Mode Shapes ◽  
Asymptotic Formulas ◽  
Axisymmetric Vibration ◽  
Legendre Functions ◽  
Approximate Analysis ◽  
Axially Symmetric ◽  
Spherical Shells ◽  
Approximate Formulas

This paper contains the results of an approximate analysis of the axially symmetric vibrations of deep spherical elastic shells. The approximation is based on the asymptotic formulas for Legendre functions of large degree. The results are relatively simple approximate formulas for the natural frequencies and mode shapes under a variety of boundary conditions at the shell edge. The results agree very well with values obtained by Kalnins, using numerical methods.

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.

scholarly journals Axisymmetric Natural Frequencies of Statically Loaded Annular Plates

2003 ◽  
Vol 10 (5-6) ◽  
pp. 301-312 ◽  
Author(s):  
Eihab M. Abdel-Rahman ◽  
Waleed F. Faris ◽  
Ali H. Nayfeh
Keyword(s):  
Eigenvalue Problem ◽  
Boundary Conditions ◽  
Shooting Method ◽  
Large Deformations ◽  
Natural Frequencies ◽  
Numerical Procedure ◽  
Mode Shapes ◽  
Axisymmetric Vibration ◽  
Plate Model ◽  
Annular Plates

We present a numerical procedure to solve the axisymmetric vibration problem of statically loaded annular plates. We use the von Kármán nonlinear plate model to account for large deformations and study the effect of static deflections on the natural frequencies and mode shapes for six combinations of boundary conditions. The shooting method is used to solve the resulting eigenvalue problem. Our results show that static deformations have a significant effect on the natural frequencies and small effect on the mode shapes of the plate. Further, the results show that the presence of in-plane stresses has a significant effect on the natural frequencies.

scholarly journals Vibrations of Complete Spherical Shells With Imperfections

2007 ◽  
Vol 129 (3) ◽  
pp. 363-370 ◽  
Author(s):  
Thomas A. Duffey ◽  
Jason E. Pepin ◽  
Amy N. Robertson ◽  
Michael L. Steinzig ◽  
Kimberly Coleman
Keyword(s):  
Finite Element ◽  
Natural Frequencies ◽  
Mode Shapes ◽  
Spherical Shells ◽  
The Past ◽  
Dynamic Holography ◽  
Theoretical Investigations ◽  
Modes Of Vibration ◽  
Geometrical Imperfections ◽  
Theoretical Results

Numerous theoretical investigations on the natural frequencies for complete spherical shells have been reported over the past four decades. However, attempts at correlating the theoretical results with either experimental or simulated results (both for axisymmetric and nonaxisymmetric modes of vibration) are almost completely lacking. In this paper, natural frequencies and mode shapes obtained from axisymmetric and nonaxisymmetric theories of vibration of complete spherical shells and from finite element computer simulations of the vibrations, with and without geometrical imperfections, are presented. Modal tests reported elsewhere on commercially available, thin spherical marine floats (with imperfections) are then utilized as a basis for comparison of frequencies to both the theoretical and numerical results. Because of the imperfections present, “splitting” of frequencies of nonaxisymmetric modes is anticipated. The presence of this frequency splitting phenomenon is demonstrated. In addition, results of a “whole field” measurement on one of the imperfect shells using dynamic holography are presented.

Frequencies and Mode Shapes for Axisymmetric Vibration of Shells

1967 ◽  
Vol 34 (1) ◽  
pp. 73-80 ◽  
Author(s):  
E. W. Ross ◽  
W. T. Matthews
Keyword(s):  
Theoretical Result ◽  
General Class ◽  
Natural Frequencies ◽  
Numerical Results ◽  
Mode Shapes ◽  
Axisymmetric Vibration ◽  
Elastic Shells ◽  
Natural Frequencies And Modes

This paper treats the axisymmetric vibration of thin elastic shells. Estimates of natural frequencies and modes are obtained for a general class of domes by applying the approximations obtained in a previous paper by one of the authors. Numerical results are obtained for ellipsoidal shells, and one new theoretical result is found.

Nonlinear Axisymmetric Free Vibration Analysis of Liquid-Filled Spherical Shell With Volume Constraint

2017 ◽  
Vol 139 (5) ◽  
Author(s):  
W. Jiammeepreecha ◽  
S. Chucheepsakul
Keyword(s):  
Free Vibration ◽  
Spherical Shell ◽  
Internal Pressure ◽  
Vibration Analysis ◽  
Free Vibration Analysis ◽  
Mode Shapes ◽  
Axisymmetric Vibration ◽  
Volume Constraint ◽  
Constraint Condition ◽  
Spherical Shells

Nonlinear axisymmetric free vibration analysis of liquid-filled spherical shells with volume constraint condition using membrane theory is presented in this paper. The energy functional of the shell and contained liquid can be expressed based on the principle of virtual work using surface fundamental form and is written in the appropriate forms. Natural frequencies and the corresponding mode shapes for specified axisymmetric vibration amplitude of liquid-filled spherical shells can be calculated by finite element method (FEM). A nonlinear numerical solution can be obtained by the modified direct iteration technique. The results indicate that the Lagrange multiplier is a parameter for adapting the internal pressure in order to sustain the shell in equilibrium state for each mode of vibration with the volume constraint condition. The axisymmetric mode shapes of the liquid-filled spherical shells under volume constraint condition were found to be in close agreement with those in existing literature for an empty spherical shell. Finally, the effects of support condition, thickness, initial internal pressure, bulk modulus of internal liquid, and elastic modulus on the nonlinear axisymmetric free vibration and change of pressure of the liquid-filled spherical shells with volume constraint condition were demonstrated. The parametric studies showed that the change of pressure has a major impact on the fundamental vibration mode when compared with the higher vibration modes.

Free Vibration of a Point-Supported Spherical Shell

1985 ◽  
Vol 52 (4) ◽  
pp. 890-896 ◽  
Author(s):  
T. Irie ◽  
G. Yamada ◽  
Y. Muramoto
Keyword(s):  
Free Vibration ◽  
Spherical Shell ◽  
Natural Frequencies ◽  
Ritz Method ◽  
Frequency Equation ◽  
Mode Shapes ◽  
Lagrangian Multiplier ◽  
Legendre Functions ◽  
Associated Legendre Functions ◽  
Lagrangian Multiplier Method

An analysis is presented for the free vibration of an elastically or a rigidly point-supported spherical shell. For this purpose, the deflection displacements of the shell are written in a series of the products of the associated Legendre functions and the trigonometric functions. The dynamical energies of the shell are evaluated, and the frequency equation is derived by the Ritz method. For a rigidly point-supported shell, the Lagrangian multiplier method is conveniently employed. The method is applied to a closed spherical shell supported at equispaced four points located along a parallel of latitude; the natural frequencies and the mode shapes are calculated numerically, and the effects of the point supports on the vibration are studied.

Dynamic Characteristics Analysis of Stepped Circular Plates

2012 ◽  
Vol 446-449 ◽  
pp. 3619-3622
Author(s):  
Hai Yong Cai ◽  
Ying Zhang
Keyword(s):  
Dynamic Characteristics ◽  
Natural Frequencies ◽  
The Self ◽  
Mode Shapes ◽  
Axisymmetric Vibration ◽  
Circular Plates ◽  
Characteristics Analysis ◽  
Dynamic Characteristics Analysis

Natural frequencies and mode shapes of axisymmetric and non-axisymmetric vibration of stepped thin circular plates were calculated with self-programmed program. Three-node annular elements were used in the program.Results of axisymmetric vibration obtained by the self-programmed program and those by ANSYS program were compared.It shows that self-programmed program can calculate axisymmetric vibration as well as non-axisymmetric vibration.The accuracy of self-programmed program is higher than ANSYS method.

Tire Vibrations

1977 ◽  
Vol 5 (4) ◽  
pp. 202-225 ◽  
Author(s):  
G. R. Potts ◽  
C. A. Bell ◽  
L. T. Charek ◽  
T. K. Roy
Keyword(s):  
Natural Frequencies ◽  
Standing Waves ◽  
Resonant Mode ◽  
Mode Shapes ◽  
Resonant Vibrations ◽  
Resonant Frequencies ◽  
Radial Tire ◽  
Analysis Techniques ◽  
Thin Ring ◽  
Good Agreement

Abstract Natural frequencies and vibrating motions are determined in terms of the material and geometric properties of a radial tire modeled as a thin ring on an elastic foundation. Experimental checks of resonant frequencies show good agreement. Forced vibration solutions obtained are shown to consist of a superposition of resonant vibrations, each rotating around the tire at a rate depending on the mode number and the tire rotational speed. Theoretical rolling speeds that are upper bounds at which standing waves occur are determined and checked experimentally. Digital Fourier transform, transfer function, and modal analysis techniques used to determine the resonant mode shapes of a radial tire reveal that antiresonances are the primary transmitters of vibration to the tire axle.

Dynamic model for high-speed rotors based on their experimental characterization

2017 ◽  
Vol 2 (4) ◽  
pp. 25
Author(s):  
L. A. Montoya ◽  
E. E. Rodríguez ◽  
H. J. Zúñiga ◽  
I. Mejía
Keyword(s):  
Dynamic Model ◽  
High Speed ◽  
Natural Frequencies ◽  
Mode Shapes ◽  
Experimental Characterization ◽  
Rotating Systems ◽  
Computing Performance ◽  
The Impact ◽  
Stiffness Distribution ◽  
Good Agreement

Rotating systems components such as rotors, have dynamic characteristics that are of great importance to understand because they may cause failure of turbomachinery. Therefore, it is required to study a dynamic model to predict some vibration characteristics, in this case, the natural frequencies and mode shapes (both of free vibration) of a centrifugal compressor shaft. The peculiarity of the dynamic model proposed is that using frequency and displacements values obtained experimentally, it is possible to calculate the mass and stiffness distribution of the shaft, and then use these values to estimate the theoretical modal parameters. The natural frequencies and mode shapes of the shaft were obtained with experimental modal analysis by using the impact test. The results predicted by the model are in good agreement with the experimental test. The model is also flexible with other geometries and has a great time and computing performance, which can be evaluated with respect to other commercial software in the future.

scholarly journals Spatially Resolved Experimental Modal Analysis on High-Speed Composite Rotors Using a Non-Contact, Non-Rotating Sensor

Sensors ◽  
2021 ◽  
Vol 21 (14) ◽  
pp. 4705
Author(s):  
Julian Lich ◽  
Tino Wollmann ◽  
Angelos Filippatos ◽  
Maik Gude ◽  
Juergen Czarske ◽  
...  
Keyword(s):  
High Speed ◽  
Natural Frequencies ◽  
Mode Shapes ◽  
Fiber Reinforced ◽  
Structural Dynamic ◽  
Spatially Resolved ◽  
Glass Fiber Reinforced ◽  
Vibration Responses ◽  
And Performance

Due to their lightweight properties, fiber-reinforced composites are well suited for large and fast rotating structures, such as fan blades in turbomachines. To investigate rotor safety and performance, in situ measurements of the structural dynamic behaviour must be performed during rotating conditions. An approach to measuring spatially resolved vibration responses of a rotating structure with a non-contact, non-rotating sensor is investigated here. The resulting spectra can be assigned to specific locations on the structure and have similar properties to the spectra measured with co-rotating sensors, such as strain gauges. The sampling frequency is increased by performing consecutive measurements with a constant excitation function and varying time delays. The method allows for a paradigm shift to unambiguous identification of natural frequencies and mode shapes with arbitrary rotor shapes and excitation functions without the need for co-rotating sensors. Deflection measurements on a glass fiber-reinforced polymer disk were performed with a diffraction grating-based sensor system at 40 measurement points with an uncertainty below 15 μrad and a commercial triangulation sensor at 200 measurement points at surface speeds up to 300 m/s. A rotation-induced increase of two natural frequencies was measured, and their mode shapes were derived at the corresponding rotational speeds. A strain gauge was used for validation.

Sign in / Sign up

Export Citation Format

Share Document