Free Vibrations of Thin Cylindrical Shells Having Finite Lengths With Freely Supported and Clamped Edges

1955 ◽  
Vol 22 (4) ◽  
pp. 547-552
Author(s):  
Yi-Yuan Yu
Keyword(s):  
Natural Frequencies ◽  
Cylindrical Shells ◽  
Free Vibrations ◽  
Frequency Equation ◽  
The Other ◽  
Lateral Vibration ◽  
Normal Functions ◽  
First Case ◽  
End Conditions ◽  
Good Agreement

Abstract Free vibrations of thin cylindrical shells having finite lengths are investigated on the basis of a set of three differential equations which are derived in a similar manner as Donnell obtained his equations for the bending and buckling problems. The equations can be solved readily after a simplifying assumption is introduced. In this manner the frequency equations are obtained for cylindrical shells with both edges freely supported, with both edges clamped, and with one edge freely supported and the other edge clamped. It is found that the lowest frequency given by the frequency equation is the smallest in the first case, larger in the third, and the largest in the second. The other two frequencies yielded by the frequency equation are approximately the same in all cases. As a result of the approximations, the characteristic equations for the three cases are found to be similar to the frequency equations for the lateral vibration of beams with similar end conditions. For the case of freely supported edges the normal functions obtained are identical in form with those assumed by Flügge and by Arnold and Warburton. For the same case, natural frequencies of one numerical example are computed by means of the present method, and the results are in good agreement with those obtained by these previous authors.

Free Vibrations of Constrained Beams

1952 ◽  
Vol 19 (4) ◽  
pp. 471-477
Author(s):  
Winston F. Z. Lee ◽  
Edward Saibel
Keyword(s):  
Natural Frequencies ◽  
Degree Of Approximation ◽  
Free Vibrations ◽  
Frequency Equation ◽  
Continuous Beam ◽  
Concentrated Mass ◽  
Simple Manner ◽  
Natural Modes ◽  
Modes Of Vibration ◽  
Rigid Supports

Abstract A general expression is developed from which the frequency equation for the vibration of a constrained beam with any combination of intermediate elastic or rigid supports, concentrated masses, and sprung masses can be found readily. The method also is extended to the case where the constraint is a continuous elastic foundation or uniformly distributed load of any length. This method requires only the knowledge of the natural frequencies and natural modes of the beam supported at the ends in the same manner as the constrained beam but not subjected to any of the constraints between the ends. The frequency equation is obtained easily and can be solved to any desired degree of approximation for any number of modes of vibration in a quick and simple manner. Numerical examples are given for a beam with one concentrated mass, for a beam with one sprung mass, and a continuous beam with one sprung mass.

Free Vibrations of Beams on Viscoelastic Pasternak Foundations

2015 ◽  
Vol 744-746 ◽  
pp. 1624-1627
Author(s):  
Li Peng ◽  
Ying Wang
Keyword(s):  
Natural Frequencies ◽  
Free Vibrations ◽  
Mode Analysis ◽  
Analysis Method ◽  
Governing Equations ◽  
Complex Mode ◽  
Transverse Vibrations ◽  
Quadrature Methods ◽  
Foundation Parameters ◽  
Good Agreement

This paper investigates free transverse vibrations of finite Euler–Bernoulli beams resting on viscoelastic Pasternak foundations. The differential quadrature methods (DQ) are applied directly to the governing equations of the free vibrations. Under the simple supported boundary condition, the natural frequencies of the transverse vibrations are calculated, and compared with the results of the complex mode analysis method. The numerical results obtained by using the DQ and the complex mode methods are in good agreement for the first seven order natural frequencies, but with the growth of the orders, the small quantitative differences between them increase. The effects of the foundation parameters on the natural frequencies are also studied in numerical examples.

The Free Vibrations of Stiffened Drill Strings With Static Curvature

1967 ◽  
Vol 89 (1) ◽  
pp. 23-29 ◽  
Author(s):  
D. A. Frohrib ◽  
R. Plunkett
Keyword(s):  
Natural Frequencies ◽  
Free Vibrations ◽  
Drill String ◽  
Bending Stiffness ◽  
Mode Shapes ◽  
Static Tension ◽  
Lateral Vibration ◽  
Governing Equations ◽  
Deflection Curve ◽  
Wide Range

The natural frequencies of lateral vibration of a long drill string in static tension under its own weight are primarily the same as those of the equivalent catenary. These frequencies and the mode shapes are affected to a certain extent by the bending stiffness and to a greater extent by the static deflection curve due to lateral deflection of the bottom end. In this paper, the governing equations are derived and general solutions are given in an asymptotic expansion with the bending stiffness as the parameter. Specific numerical results are given in dimensionless form for the first three natural frequencies for a very wide range of horizontal tension and several appropriate values of bending stiffness for zero vertical static force at the bottom.

Theoretical Analysis of Free Vibrations Based on a New High Order Shell Theory for Cylindrical Shells Conveying Fluid

2017 ◽  
Author(s):  
Ming Ji ◽  
Kazuaki Inaba
Keyword(s):  
Boundary Conditions ◽  
Shell Theory ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Cylindrical Shells ◽  
Fluid Pressure ◽  
Free Vibrations ◽  
High Order ◽  
Out Of Plane ◽  
Conveying Fluid

The natural frequencies of free vibrations for thick cylindrical shells with clamped-clamped ends conveying fluid are investigated. Equations of motion and boundary conditions are derived by Hamilton’s principle based on the new high order shell theory. The hydrodynamic force is derived from the linearized potential flow theory. Besides, fluid pressure acting on the shell wall is gotten by the assumption of non-penetration condition. The out-of-plane and in-plane vibrations are coupled together due to the existence of fluid-solid-interaction (FSI). Under the assumption of harmonic motion, the dispersion relationships are presented. Using the method of frequency sweeping, the natural frequencies of symmetric modes and asymmetric modes corresponding to each flow velocity are found by satisfying the dispersion relationship equations and boundary conditions. Several numerical examples with different flow velocities and thickness are presented compared with previous thin shell theory and FEM results and show reasonable agreement. The effects of thickness are discussed.

Dynamic Beam Modification Using Dimples

2008 ◽  
Vol 130 (4) ◽  
Author(s):  
W. N. Cheng ◽  
C. C. Cheng ◽  
G. H. Koopmann
Keyword(s):  
Natural Frequencies ◽  
Design Method ◽  
Frequency Equation ◽  
Search Method ◽  
The Other ◽  
Impedance Method ◽  
Dimple Size ◽  
Straight Beam ◽  
Novel Method ◽  
Force Equilibrium

In this paper, a design method to modify the vibration characteristics of a beam by creating cylindrical dimples on its surface is investigated. In particular, the vibration response of a beam with several dimples is formulated using the impedance method. The dimpled beam is divided into two kinds of structural segments: one, a curved beam that is modeled as the dimple and the other, a straight beam. The frequency equation is derived by assembling the impedance of each structure segment based on conditions of force equilibrium and velocity compatibility. Then a novel method for shifting the natural frequencies of a beam to preassigned values by creating cylindrical dimples on this structure is introduced. The dimple size and its location on the structure can be determined analytically, so the time consuming process using the traditional optimal search method is thereby avoided. Several examples using this technique are demonstrated.

A Theoretical and Experimental Study of Vibrations of Thick Circular Cylindrical Shells and Rings

1988 ◽  
Vol 110 (4) ◽  
pp. 533-537 ◽  
Author(s):  
R. K. Singal ◽  
K. Williams
Keyword(s):  
Cylindrical Shells ◽  
Three Dimensional ◽  
Free Vibrations ◽  
Theory Of Elasticity ◽  
Frequency Equation ◽  
Experimental Investigations ◽  
Resonant Frequencies ◽  
Modes Of Vibration ◽  
Circular Cylindrical Shells ◽  
Experimental Values

The free vibrations of thick circular cylindrical shells and rings are discussed in this paper. The well-known energy method, which is based on the three-dimensional theory of elasticity, is used in the derivation of the frequency equation of the shell. The frequency equation yields resonant frequencies for all the circumferential modes of vibration, including the breathing and beam-type modes. Experimental investigations were carried out on several models in order to assess the validity of the analysis. This paper first describes briefly the method of analysis. In the end, the calculated frequencies are compared with the experimental values. A very close agreement between the theoretical and experimental values of the resonant frequencies for all the models was obtained and this validates the method of analysis.

FREE VIBRATIONS OF COMBINED HEMISPHERICAL–CYLINDRICAL SHELLS OF REVOLUTION WITH A TOP OPENING

2013 ◽  
Vol 14 (01) ◽  
pp. 1350023 ◽  
Author(s):  
JAE-HOON KANG
Keyword(s):  
Present Method ◽  
Thin Shell ◽  
Natural Frequencies ◽  
Cylindrical Shells ◽  
Ritz Method ◽  
Three Dimensional ◽  
Free Vibrations ◽  
Two Dimensional ◽  
Algebraic Polynomials ◽  
Shells Of Revolution

A three-dimensional (3D) method of analysis is presented for determining the free vibration frequencies of joined hemispherical–cylindrical shells of revolution with a top opening. Unlike conventional shell theories, which are mathematically two-dimensional (2D), the present method is based upon the 3D dynamic equations of elasticity. Displacement components ur, uθ and uz in the radial, circumferential, and axial directions, respectively, are taken to be periodic in θ and in time, and algebraic polynomials in the r and z directions. Potential (strain) and kinetic energies of the joined shells are formulated, and the Ritz method is used to solve the eigenvalue problem, thus yielding upper bound values of the frequencies by minimizing the frequencies. As the degree of the polynomials is increased, frequencies converge to the exact values. Convergence to four-digit exactitude is demonstrated for the first five frequencies. Natural frequencies are presented for different boundary conditions. The frequencies from the present 3D method are compared with those from 2D thin shell theories.

On Vibrations of Pretwisted Rectangular Plates

1962 ◽  
Vol 29 (1) ◽  
pp. 30-32 ◽  
Author(s):  
R. P. Nordgren
Keyword(s):  
Shallow Shell ◽  
Shell Theory ◽  
Free Vibrations ◽  
Frequency Equation ◽  
Numerical Results ◽  
The Other ◽  
Torsional Vibrations ◽  
Rectangular Plates ◽  
Simply Supported

This paper contains an analysis of the free vibrations of uniformly pretwisted rectangular plates, utilizing the exact equations of classical shallow-shell theory. Specifically, solutions are given (a) for two opposite edges simply supported and the other two free, and (b) for all four edges simply supported. Numerical results obtained for case (b) are compared with previous results for the torsional vibrations of pretwisted beams. A simple frequency equation is obtained for case (b), permitting a detailed study of the effects of both pretwist and longitudinal inertia.

scholarly journals Frequency Analysis for Functionally Graded Material Cylindrical Shells: A Significant Case Study

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Rabia Anwar ◽  
Madiha Ghamkhar ◽  
Muhammad Imran Khan ◽  
Rabia Safdar ◽  
Muhammad Zafar Iqbal ◽  
...  
Keyword(s):  
Functionally Graded Materials ◽  
Natural Frequencies ◽  
Constitutive Relations ◽  
Cylindrical Shells ◽  
Frequency Equation ◽  
Functionally Graded ◽  
Graded Materials ◽  
Simply Supported ◽  
Love’S Shell Theory ◽  
Graded Material

Cylindrical shells play an important role for the construction of functionally graded materials (FGMs). Functionally graded materials are valuable in order to develop durable materials. They are made of two or more materials such as nickel, stainless steel, zirconia, and alumina. They are extremely beneficial for the manufacturing of structural elements. Functionally graded materials are broadly used in several fields such as chemistry, biomedicine, optics, and electronics. In the present research, vibrations of natural frequencies are investigated for different layered cylindrical shells, those constructed from FGMs. The behavior of shell vibration is based on different parameters of geometrical material. The problem of the shell is expressed from the constitutive relations of strain and stress with displacement, as well as it is adopted from Love’s shell theory. Vibrations of natural frequencies (NFs) are calculated for simply supported-simply supported (SS-SS) and clamped-free (C-F) edge conditions. The Rayleigh–Ritz technique is employed to obtain the shell frequency equation. The shell equation is solved by MATLAB software.

scholarly journals Dynamic stiffness method for free vibrations analysis of partial fluid-filled orthotropic circular cylindrical shells

2015 ◽  
Vol 37 (1) ◽  
pp. 43-56
Author(s):  
Tran Ich Thinh ◽  
Nguyen Manh Cuong ◽  
Vu Quoc Hien
Keyword(s):  
Natural Frequencies ◽  
Dynamic Stiffness ◽  
Cylindrical Shells ◽  
Computational Cost ◽  
Shear Deformation Theory ◽  
Free Vibrations ◽  
Various Boundary Conditions ◽  
Stiffness Method ◽  
Dynamic Stiffness Method ◽  
Circular Cylindrical Shells

Free vibrations of partial fluid-filled orthotropic circular cylindrical shells are investigated using the Dynamic Stiffness Method (DSM) or Continuous Element Method (CEM) based on theFirst Order Shear Deformation Theory (FSDT) and non-viscous incompressible fluid equations. Numerical examples are given for analyzing natural frequencies and harmonic responses of cylindrical shells partially and completely filled with fluid under various boundary conditions. The vibration frequencies for different filling ratios of cylindrical shells are obtained and compared with existing experimental and theoretical results which indicate that the fluid filling can reduce significantly the natural frequencies of studiedcylindrical shells. Detailed parametric analysis is carried out to show the effects of some geometrical and material parameters on the natural frequencies of orthotropic cylindrical shells. The advantages of this current solution consist in fast convergence, low computational cost and high precision validating for all frequency ranges.

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