Axisymmetrical Vibrations of Underwater Hemispherical Shells

1976 ◽  
Vol 98 (3) ◽  
pp. 941-947
Author(s):  
F. C. Chen ◽  
T. C. Huang
Keyword(s):  
Equations Of Motion ◽  
Hydrodynamic Pressure ◽  
Free Vibrations ◽  
Frequency Equation ◽  
Mode Shapes ◽  
Legendre Functions ◽  
Water Field ◽  
Fluid Solid Interaction ◽  
Interaction Problem ◽  
Hemispherical Shells

Free vibrations of an underwater elastic hemispherical thin shell with fixed edge have been investigated based on the bending theory. The solution of this fluid-solid interaction problem involves the differential equations of motion of underwater spherical shells, the velocity potential of the water field, the hydrodynamic pressure, and the continuity and boundary conditions. A transcedental frequency equation in terms of Legendre functions is derived and the normal and tangential mode shapes are found. Examples are given and results are plotted for natural frequencies and modes 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.

Forced Vibrations of Viscoelastic Timoshenko Beams

1976 ◽  
Vol 98 (3) ◽  
pp. 820-826 ◽  
Author(s):  
C. C. Huang ◽  
T. C. Huang
Keyword(s):  
Boundary Conditions ◽  
Laplace Transform ◽  
Attenuation Factor ◽  
Equations Of Motion ◽  
Bending Moment ◽  
Expansion Method ◽  
Free Vibrations ◽  
Forced Vibrations ◽  
Mode Shapes ◽  
Timoshenko Beams

In a previous paper, the correspondence principle has been applied to derive the differential equations of motion of viscoelastic Timoshenko beams with or without external viscous damping. To study free vibrations these equations are solved by Laplace transform and boundary conditions are applied to obtain the attenuation factor and the frequency of the damped free vibrations and mode shapes. The present paper continues to analyze this subject and deals with the responses in deflection, bending slope, bending moment and shear for forced vibrations. Laplace transform and appropriate boundary conditions have been applied. Examples are given and results are plotted. The solution of forced vibrations of elastic Timoshenko beams obtained as a result of reduction from viscoelastic case and by eigenfunction expansion method concludes the paper.

scholarly journals Vibration analysis of multi-stepped and multi-damaged parabolic arches using GDQ

2014 ◽  
Vol 2 (1) ◽  
Author(s):  
Erasmo Viola ◽  
Marco Miniaci ◽  
Nicholas Fantuzzi ◽  
Alessandro Marzani
Keyword(s):  
Boundary Conditions ◽  
Cross Sections ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Free Vibrations ◽  
Mode Shapes ◽  
Internal Boundary ◽  
Radius Of Curvature ◽  
Inertia Effects ◽  
Circular Arch

AbstractThis paper investigates the in-plane free vibrations of multi-stepped and multi-damaged parabolic arches, for various boundary conditions. The axial extension, transverse shear deformation and rotatory inertia effects are taken into account. The constitutive equations relating the stress resultants to the corresponding deformation components refer to an isotropic and linear elastic material. Starting from the kinematic hypothesis for the in-plane displacement of the shear-deformable arch, the equations of motion are deduced by using Hamilton’s principle. Natural frequencies and mode shapes are computed using the Generalized Differential Quadrature (GDQ) method. The variable radius of curvature along the axis of the parabolic arch requires, compared to the circular arch, a more complex formulation and numerical implementation of the motion equations as well as the external and internal boundary conditions. Each damage is modelled as a combination of one rotational and two translational elastic springs. A parametric study is performed to illustrate the influence of the damage parameters on the natural frequencies of parabolic arches for different boundary conditions and cross-sections with localizeddamage.Results for the circular arch, derived from the proposed parabolic model with the derivatives of some parameters set to zero, agree well with those published over the past years.

Experimental Modal Analysis of a Fixed-Fixed Beam Under Axial Force

2017 ◽  
Author(s):  
Pezhman A. Hassanpour ◽  
Khaled Alghemlas ◽  
Adam Betancourt
Keyword(s):  
Analytical Model ◽  
Axial Force ◽  
Experimental Approach ◽  
Equations Of Motion ◽  
Frequency Equation ◽  
Mode Shapes ◽  
Governing Equations ◽  
Experimental Procedure ◽  
Resonance Frequencies ◽  
Fixed Beam

In this paper, an experimental procedure is proposed for determining the resonance frequencies and mode shapes of vibration of a fixed-fixed beam. Since it is fixed at both ends, the beam may sustain an axial force due to several factors including the fasteners and/or change of temperature. The analytical governing equations of motion, frequency equation, and mode shapes of vibration are presented. The analytical model is used to justify the experimental approach as well as interpretation of the experiment data. In this study, a hammer is used to excite the beam, and then the vibration of the beam is observed and recorded at two different points on the beam using two laser Doppler vibrometers. The data from the vibrometers are used to extract the resonance frequencies and mode shapes of vibrations. Using the analytical model, the axial force in the beam is estimated.

Vibration Modes and Frequencies of Timoshenko Beams With Attached Rigid Bodies

1995 ◽  
Vol 62 (1) ◽  
pp. 193-199 ◽  
Author(s):  
M. W. D. White ◽  
G. R. Heppler
Keyword(s):  
Boundary Conditions ◽  
Rigid Body ◽  
Timoshenko Beam ◽  
Equations Of Motion ◽  
Frequency Equation ◽  
Rigid Bodies ◽  
The Body ◽  
Mode Shapes ◽  
Timoshenko Beams ◽  
First Moment

The equations of motion and boundary conditions for a free-free Timoshenko beam with rigid bodies attached at the endpoints are derived. The natural boundary conditions, for an end that has an attached rigid body, that include the effects of the body mass, first moment of mass, and moment of inertia are included. The frequency equation for a free-free Timoshenko beam with rigid bodies attached at its ends which includes all the effects mentioned above is presented and given in terms of the fundamental frequency equations for Timoshenko beams that have no attached rigid bodies. It is shown how any support / rigid-body condition may be easily obtained by inspection from the reported frequency equation. The mode shapes and the orthogonality condition, which include the contribution of the rigid-body masses, first moments, and moments of inertia, are also developed. Finally, the effect of the first moment of the attached rigid bodies is considered in an illustrative example.

scholarly journals Simplified Vibration Model and analysis of a single-conductor transmission line with dampers

2016 ◽  
Vol 231 (22) ◽  
pp. 4150-4162 ◽  
Author(s):  
O Barry ◽  
R Long ◽  
DCD Oguamanam
Keyword(s):  
Transmission Line ◽  
Equations Of Motion ◽  
Frequency Equation ◽  
Mode Shapes ◽  
Viscous Damping ◽  
Equivalent Viscous Damping ◽  
Proposed Model ◽  
Vibration Model ◽  
Mass Spring ◽  
Discrete Mass

A novel model is developed for a vibrating single-conductor transmission line carrying Stockbridge dampers. Experiments are performed to determine the equivalent viscous damping of the damper. This damper is then reduced to an equivalent discrete mass-spring-mass and viscous damping system. The equations of motion of the model are derived using Hamilton’s principle and explicit expressions are determined for the frequency equation, and mode shapes. The proposed model is verified using experimental and finite element results from the literature. This proposed model excellently captures free vibration characteristics of the system and the vibration level of the conductor, but performs poorly in regard to the vibration of the counterweights.

Free Vibration Analysis of a Beam Under Axial Load Carrying a Mass-Spring-Mass

2012 ◽  
Author(s):  
O. R. Barry ◽  
Y. Zhu ◽  
J. W. Zu ◽  
D. C. D. Oguamanam
Keyword(s):  
Free Vibration ◽  
Vibration Analysis ◽  
Axial Load ◽  
Equations Of Motion ◽  
Tensile Load ◽  
Free Vibration Analysis ◽  
Frequency Equation ◽  
Mode Shapes ◽  
Mass System ◽  
Mass Spring

This paper deals with the free vibration analysis of a beam subjected to an axial tensile load with an attached in-span mass-spring-mass system. The equations of motion are derived by means of the Hamilton principle and an explicit expression of the frequency equation is presented. The formulation is validated with results in the literature and the finite element method. Parametric studies are done to investigate the effect of the axial load, the magnitude and location of the mass-spring-mass system on the lowest five natural frequencies and mode shapes. The results indicate that the fundamental mode is independent of the tension and the in-span mass. However, a significant change in all modes is observed when the position of the mass-spring-mass is varied.

Nonlinear Dynamics of Stockbridge Dampers

2015 ◽  
Vol 137 (6) ◽  
Author(s):  
O. Barry ◽  
J. W. Zu ◽  
D. C. D. Oguamanam
Keyword(s):  
Nonlinear Dynamics ◽  
Boundary Conditions ◽  
Equations Of Motion ◽  
Frequency Equation ◽  
Mode Shapes ◽  
Nonlinear Frequency ◽  
Modulation Equations ◽  
Proposed Model ◽  
Cantilevered Beams ◽  
Stockbridge Damper

The present paper deals with the nonlinear dynamics of a Stockbridge damper. The nonlinearity is from damping and the geometric stretching of the messenger. The Stockbridge damper is modeled as two cantilevered beams with tip masses. The equations of motion and boundary conditions are derived using Hamilton’s principle. The model is valid for both symmetric and asymmetric Stockbridge dampers. Explicit expressions are presented for the frequency equation, mode shapes, nonlinear frequency, and modulation equations. Experiments are conducted to validate the proposed model.

Analysis of Free Vibrations of Conical Shells Using Donnell’s Approximations

1995 ◽  
Vol 48 (11S) ◽  
pp. S84-S89
Author(s):  
V. C. M. de Souza ◽  
J. M. F. Saraiva
Keyword(s):  
Shell Theory ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Free Vibrations ◽  
Mode Shapes ◽  
Conical Shells ◽  
Classical Shell Theory ◽  
Clamped Edges

The free vibrations of conical shells, having two open rigidly clamped edges, are investigated by using a variational development of the equations of motion based upon the Classical Shell Theory, and results are compared with those obtained by using Donnell’s approximation in the development of these equations. Through suitable examples, the validity of Donnell’s approximation to compute natural frequencies and mode-shapes of conical shells is shown.

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.

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