Free Vibrations of Viscoelastic Timoshenko Beams

1971 ◽  
Vol 38 (2) ◽  
pp. 515-521 ◽  
Author(s):  
T. C. Huang ◽  
C. C. Huang
Keyword(s):  
Boundary Conditions ◽  
Laplace Transform ◽  
Correspondence Principle ◽  
Free Vibrations ◽  
Mode Shapes ◽  
Viscous Damping ◽  
Complex Frequency ◽  
Timoshenko Beams ◽  
Numerical Illustration ◽  
End Conditions

The correspondence principle has been applied to derive the differential equations of viscoelastic Timoshenko beams with external viscous damping. These equations are solved by Laplace transform and boundary conditions are applied to obtain complex frequency equations and mode shapes for beams of any combination of end conditions. For beams without external damping, the correspondence principle can be applied directly to the available solutions of elastic Timoshenko beams. Numerical illustration is given.

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 In-Plane free Vibration Analysis of an Annular Disk with Point Elastic Support

2011 ◽  
Vol 18 (4) ◽  
pp. 627-640 ◽  
Author(s):  
S. Bashmal ◽  
R. Bhat ◽  
S. Rakheja
Keyword(s):  
Boundary Conditions ◽  
Orthogonal Polynomials ◽  
Ritz Method ◽  
Free Vibration Analysis ◽  
Free Vibrations ◽  
Outer Edge ◽  
Mode Shapes ◽  
Annular Disk ◽  
Different Boundary Conditions ◽  
Boundary Characteristic

In-plane free vibrations of an elastic and isotropic annular disk with elastic constraints at the inner and outer boundaries, which are applied either along the entire periphery of the disk or at a point are investigated. The boundary characteristic orthogonal polynomials are employed in the Rayleigh-Ritz method to obtain the frequency parameters and the associated mode shapes. Boundary characteristic orthogonal polynomials are generated for the free boundary conditions of the disk while artificial springs are used to account for different boundary conditions. The frequency parameters for different boundary conditions of the outer edge are evaluated and compared with those available in the published studies and computed from a finite element model. The computed mode shapes are presented for a disk clamped at the inner edge and point supported at the outer edge to illustrate the free in-plane vibration behavior of the disk. Results show that addition of point clamped support causes some of the higher modes to split into two different frequencies with different mode shapes.

An accurate solution method for the vibration analysis of Timoshenko beams with general elastic supports

2014 ◽  
Vol 229 (13) ◽  
pp. 2327-2340 ◽  
Author(s):  
Dongyan Shi ◽  
Qingshan Wang ◽  
Xianjie Shi ◽  
Fuzhan Pang
Keyword(s):  
Boundary Conditions ◽  
Series Expansion ◽  
Solution Method ◽  
Free Vibrations ◽  
Accurate Solution ◽  
Fourier Series Expansion ◽  
Timoshenko Beams ◽  
End Points ◽  
Solution Algorithms ◽  
Wide Range

In this investigation, an accurate solution method is presented for the free vibrations of Timoshenko beams with general elastic restraints at the end points, a class of problems which are rarely attempted in the literatures. Unlike in most existing studies where solutions are often developed for a particular type of boundary conditions, the current method can be generally applied to a wide range of boundary conditions with no need of modifying solution algorithms and procedures. Under the current framework, the displacement and rotation functions are generally sought, regardless of boundary conditions, as an improved trigonometric series in which several supplementary functions introduced to remove the potential discontinuities with the displacement components and its derivatives at the end points and accelerated the series expansion. Mathematically, the current Fourier series expansion is an exact solution for a class of problems with the Timoshenko beam such that both the governing equations and the boundary conditions simultaneously satisfy any specified degree of accuracy. The effectiveness and reliability of the presented solution are demonstrated by comparing the present results with those results published in literatures and finite element method data, and numerous new results for beams with elastic boundary restraints is presented, which may serve as benchmark solution for future researches.

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.

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.

Axisymmetric Free Vibrations of a Transversely Isotropic Finite Cylindrical Rod

1988 ◽  
Vol 55 (4) ◽  
pp. 855-862 ◽  
Author(s):  
C. P. Lusher ◽  
W. N. Hardy
Keyword(s):  
Boundary Conditions ◽  
Elastic Constants ◽  
Transversely Isotropic ◽  
Acoustic Vibration ◽  
Elastic Cylinder ◽  
Free Vibrations ◽  
Mode Shapes ◽  
Hexagonal Symmetry ◽  
Hexagonal System ◽  
Better Than

The frequencies of free vibration and mode shapes are calculated for axisymmetric modes of an elastic cylinder of finite length having hexagonal symmetry with the crystallographic c-axis coincident with the axis of the cylinder (a transversely isotropic finite cylindrical rod). A series solution is used which satisfies term-by-term the differential equations of linear elasticity and the boundary conditions on the shear stress; the boundary conditions on the normal stresses are satisfied by using an orthogonalization procedure. As an example, the method is applied to sapphire, with one of the six elastic constants (c14) taken to be zero. The other five elastic constants are those of the hexagonal system. The calculated acoustic vibration frequencies agree to better than 1 percent with measurements made on sapphire at room temperature, for a cylinder of half-height to radius ratio ∼ 1.

A new semi-analytical solution method for free vibration analysis of composite rectangular plates with general edge constraints coupled with single piezoelectric layer

2018 ◽  
Vol 29 (20) ◽  
pp. 3873-3889 ◽  
Author(s):  
Mehdi Baghaee ◽  
Amin Farrokhabadi ◽  
Ramazan-Ali Jafari-Talookolaei
Keyword(s):  
Boundary Conditions ◽  
Piezoelectric Layer ◽  
Shear Deformation Theory ◽  
Free Vibration Analysis ◽  
Composite Plates ◽  
Energy Functional ◽  
Free Vibrations ◽  
Mode Shapes ◽  
Rectangular Plates ◽  
Generalized Displacements

In this article, a new approach is presented to study the free vibrations of rectangular composite plates coupled with single piezoelectric layer. The laminated plate with general stacking sequences is subjected to the elastic edge restraints. Based on the first-order shear deformation theory and Hamilton’s principle, the equations of the motion along with boundary conditions of the problem are deduced. To solve the problem, generalized displacements as well as general electric potentials are expanded using the Legendre polynomial series as the base functions. Then, the kinetic and potential energies of the problem are obtained. Afterwards, by means of Lagrange multipliers all the boundary conditions have been added to the energies to form the functional. This energy functional is extremised to get the natural frequencies and mode shapes of the problem through generalized eigenvalue problem. Credibility of the proposed method is verified by comparing the obtained results with those achieved by other theories and finite element method.

Free Vibrations of Symmetric Arch: Boundary Conditions of Stress Resultants Revisited

2020 ◽  
Vol 20 (09) ◽  
pp. 2071008
Author(s):  
Joon Kyu Lee ◽  
Byoung Koo Lee
Keyword(s):  
Boundary Conditions ◽  
Natural Frequency ◽  
Mode Shape ◽  
Natural Frequencies ◽  
Free Vibrations ◽  
Mode Shapes ◽  
All Solutions

This paper deals with analyzing free vibrations of the symmetric arch. The boundary conditions of the stress resultants are newly derived, which can be replaced by the conventional boundary conditions of the deflections. All solutions of the natural frequency with the mode shape, using the new boundary conditions, are the same as those of the conventional deflections. The boundary conditions mixed with new and conventional conditions act correctly to calculate natural frequencies. The mode shapes of the stress resultants using the new boundary conditions are reported in two types: symmetric and anti-symmetric modes.

scholarly journals An Investigation on the Nonlinear Free Vibration Analysis of Beams with Simply Supported Boundary Conditions Using Four Engineering Theories

2011 ◽  
Vol 2011 ◽  
pp. 1-17 ◽  
Author(s):  
R. A. Jafari-Talookolaei ◽  
M. H. Kargarnovin ◽  
M. T. Ahmadian ◽  
M. Abedi
Keyword(s):  
Natural Frequencies ◽  
Free Vibration Analysis ◽  
Free Vibrations ◽  
Geometrically Nonlinear ◽  
Timoshenko Beams ◽  
Analysis Method ◽  
Nonlinear Free Vibration ◽  
End Conditions ◽  
Homotopy Analysis ◽  
Transverse Deflection

The objective of this study is to present a brief survey on the geometrically nonlinear free vibrations of the Bernoulli-Euler, the Rayleigh, shear, and the Timoshenko beams with simple end conditions using the Homotopy Analysis Method (HAM). Expressions for the natural frequencies, the transverse deflection, postbuckling load-deflection relation to, and critical buckling load are presented. The results of nonlinear analysis are validated with the published results, and excellent agreement is observed. The effects of some parameters, such as slender ratio, the rotary inertia, and the shear deformation, are examined as other parameters are fixed.

Vibration and Buckling Analyses of Beams by the Modified Differential Quadrature Method

2000 ◽  
Vol 16 (4) ◽  
pp. 189-195 ◽  
Author(s):  
Y.-T. Chou ◽  
S.-T. Choi
Keyword(s):  
Boundary Conditions ◽  
Present Method ◽  
Differential Quadrature Method ◽  
Free Vibrations ◽  
Differential Quadrature ◽  
Quadrature Method ◽  
Timoshenko Beams ◽  
Buckling Loads ◽  
New Formulation ◽  
Excellent Accuracy

ABSTRACTIn this paper the modified differential quadrature method (MDQM) is proposed for static and vibration analyses of beams. Modified weighting matrices are developed and a new formulation process is presented for incorporating boundary conditions such that the numerical error induced by using the δ-method in the original DQM is reduced. The present method is applied to various beam problems, such as static deflections of Euler beams, buckling loads of columns, and free vibrations of Timoshenko beams. Numerical results of the present method are shown to have excellent accuracy when compared to exact values and are more accurate than those obtained by the original DQM. The accuracy and efficiency of the present method have been demonstrated.

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