Calculation of the Natural Frequencies of Transverse Vibration of Complex Beams Using the Differential-Matrix Equations

2011 ◽  
Vol 243-249 ◽  
pp. 284-289
Author(s):  
Yu Zhang
Keyword(s):  
Iterative Method ◽  
Boundary Conditions ◽  
Cross Section ◽  
Transverse Vibration ◽  
Natural Frequencies ◽  
Composite Beams ◽  
Matrix Equations ◽  
Generalized Differential ◽  
Set Up ◽  
Differential Matrix

The generalized differential-matrix equations of transverse vibration of the beams were set up and they were solved by means of Cauchy sequence iterative method. Then according to the boundary conditions at two ends of the beams the natural frequencies of the transverse vibration of the different beams including the complex beams of non-uniform section and composite beams under different boundary conditions were figured out. The form of the differential-matrix is simple. The calculation of the sequence iterations can be accomplished by simple computer program. Using the method in this paper, the amount of work of calculation is reduced greatly and the results are accurate compared with the approximate method in which a beam of non-uniform section is replaced by many small segments of equal cross-section.

Free Vibration of Thin-Walled Open Section Beams With Unconstrained Damping Treatment

1981 ◽  
Vol 48 (1) ◽  
pp. 169-173 ◽  
Author(s):  
S. Narayanan ◽  
J. P. Verma ◽  
A. K. Mallik
Keyword(s):  
Free Vibration ◽  
Cross Section ◽  
Transverse Vibration ◽  
Natural Frequencies ◽  
Vibration Characteristics ◽  
Numerical Results ◽  
Simply Supported ◽  
Clamped Beams ◽  
Thin Walled ◽  
Open Cross Section

Free-vibration characteristics of a thin-walled, open cross-section beam, with unconstrained damping layers at the flanges, are investigated. Both uncoupled transverse vibration and the coupled bending-torsion oscillations, of a beam of a top-hat section, are considered. Numerical results are presented for natural frequencies and modal loss factors of simply supported and clamped-clamped beams.

scholarly journals Analysis of Timoshenko beam with Koch snowflake cross-section and variable properties in different boundary conditions using finite element method

2021 ◽  
Vol 13 (11) ◽  
pp. 168781402110609
Author(s):  
Hossein Talebi Rostami ◽  
Maryam Fallah Najafabadi ◽  
Davood Domiri Ganji
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Boundary Conditions ◽  
Natural Frequency ◽  
Cross Section ◽  
Timoshenko Beam ◽  
Natural Frequencies ◽  
Variable Properties ◽  
The Third ◽  
Koch Snowflake

This study analyzed a Timoshenko beam with Koch snowflake cross-section in different boundary conditions and for variable properties. The equation of motion was solved by the finite element method and verified by Solidworks simulation in a way that the maximum error was about 2.9% for natural frequencies. Displacement and natural frequency for each case presented and compared to other cases. Significant research achievements illustrate that if we change the Koch snowflake cross-section of the beam from the first iteration to the second, the area and moment of inertia will increase, and we have a 5.2% rise in the first natural frequency. Similarly, by changing the cross-section from the second iteration to the third, a 10.2% growth is observed. Also, the hollow cross-section is considered, which can enlarge the natural frequency by about 26.37% compared to a solid one. Moreover, all the clamped-clamped, hinged-hinged, clamped-free, and free-free boundary conditions have the highest natural frequency for the Timoshenko beam with the third iteration of the Koch snowflake cross-section in solid mode. Finally, examining important physical parameters demonstrates that variable density from a minimum value to the standard value along the beam increases the natural frequencies, while variable elastic modulus decreases it.

Analysis of free vibration of an isotropic plate with surface or internal long crack using generalized differential quadrature method

2019 ◽  
Vol 55 (1-2) ◽  
pp. 42-52
Author(s):  
Milad Ranjbaran ◽  
Rahman Seifi
Keyword(s):  
Differential Equations ◽  
Free Vibration ◽  
Boundary Conditions ◽  
Natural Frequencies ◽  
Isotropic Plate ◽  
Differential Quadrature Method ◽  
Differential Quadrature ◽  
Quadrature Method ◽  
Generalized Differential Quadrature Method ◽  
Generalized Differential

This article proposes a new method for the analysis of free vibration of a cracked isotropic plate with various boundary conditions based on Kirchhoff’s theory. The isotropic plate is assumed to have a part-through surface or internal crack. The crack is considered parallel to one of the plate edges. Existence of the crack modified the governing differential equations which were formulated based on the line-spring model. Generalized differential quadrature method discretizes the obtained governing differential equations and converts them into an algebraic system of equations. Then, an eigenvalue analysis was used to determine the natural frequencies of the cracked plates. Some numerical results are given to demonstrate the accuracy and convergence of the obtained results. To demonstrate the efficiency of the method, the results were compared with finite element solutions and available literature. Also, effects of the crack depth, its location along the thickness, the length of the crack and different boundary conditions on the natural frequencies were investigated.

The Transverse Vibration of a Doubly Tapered Beam

1977 ◽  
Vol 44 (1) ◽  
pp. 123-126 ◽  
Author(s):  
D. O. Banks ◽  
G. J. Kurowski
Keyword(s):  
Free Boundary ◽  
Boundary Conditions ◽  
Cross Section ◽  
Transverse Vibration ◽  
Eigenvalues And Eigenfunctions ◽  
Free Boundary Conditions ◽  
Tapered Beam ◽  
Transverse Vibrations ◽  
The Cross ◽  
Homogeneous Beam

We analyze the transverse vibrations of a thin homogeneous beam which is symmetric with respect to the x-y and x-z planes. The cross section of the beam at x is assumed to have the form D(x)={(x,y,z)|x∈[0,1],y=xαy1,z=xβz1,(y1,z1)∈D1} where D1 is the cross section at x = 1. Expressions are obtained from which the eigenvalues and eigenfunctions can be easily found for 0 ≤ α < 2 and all combinations of clamped, hinged, guided, and free boundary conditions at both ends of the beam.

Natural Frequencies of In-Plane Vibration of Arcs

1983 ◽  
Vol 50 (2) ◽  
pp. 449-452 ◽  
Author(s):  
T. Irie ◽  
G. Yamada ◽  
K. Tanaka
Keyword(s):  
Boundary Conditions ◽  
Cross Section ◽  
Natural Frequencies ◽  
Circular Cross Section ◽  
Plane Vibration

The natural frequencies of in-plane vibration are presented for uniform arcs with circular cross section under all combinations of boundary conditions.

An Analytical Method for Free Vibration Analysis of Composite Beams Subjected to Initial Thermal Stresses

2011 ◽  
Vol 482 ◽  
pp. 1-9
Author(s):  
A. Mahi ◽  
E.A. Adda-Bedia ◽  
A. Benkhedda
Keyword(s):  
Aspect Ratio ◽  
Free Vibration ◽  
Boundary Conditions ◽  
Thermal Stresses ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Shear Deformation Theory ◽  
Free Vibration Analysis ◽  
Composite Beams ◽  
Ply Orientation

The purpose of this paper is to present exact solutions for the free vibration of symmetrically laminated composite beams. The present analysis includes the first shear deformation theory and the rotary inertia. The analytical solutions take into account the thermal effect on the free vibration characteristics of the composite beams. In particular, the aim of this work is to derive the exact closed-form characteristic equations for common boundary conditions. The different parameters that could affect the natural frequencies are included as factors (aspect ratio, thermal load-to-shear coefficient, ply orientation) to better perform dynamic analysis to have a good understanding of dynamic behavior of composite beams. In order to derive the governing set of equations of motion, the Hamilton’s principle is used. The system of ordinary differential equations of the laminated beams is then solved and the natural frequencies’ equations are obtained analytically for different boundary conditions. Numerical results are presented to show the influence of temperature rise, aspect ratio, boundary conditions and ply orientation on the natural frequencies of composite beams.

Free Vibration Analysis of Cracked Composite Beams Subjected to Coupled Bending-Torsion Loads Based on a First Order Beam Theory

2013 ◽  
Vol 325-326 ◽  
pp. 1318-1323 ◽  
Author(s):  
A.R. Daneshmehr ◽  
D.J. Inman ◽  
A.R. Nateghi
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Vibration Analysis ◽  
Natural Frequencies ◽  
Free Vibration Analysis ◽  
Beam Theory ◽  
Order Theory ◽  
Composite Beams ◽  
First Order ◽  
First Order Theory

In this paper free vibration analysis of cracked composite beams subjected to coupled bending-torsion loads are presented. The composite beam is assumed to have an open edge crack. A first order theory is applied to count for the effect of the shear deformations on natural frequencies as well as the effect of coupling in torsion and bending modes of vibration. Local flexibility matrix is used to obtain the additional boundary conditions of the beam in the crack area. After obtaining the governing equations and boundary conditions, GDQ method is applied to solve the obtained eigenvalue problem. Finally, some numerical results are given to show the efficacy of the method. In addition, to count for the effect of coupling on natural frequencies of the cracked beams, different fiber orientations are assumed and studied.

Natural Frequencies of Out-of-Plane Vibration of Arcs

1982 ◽  
Vol 49 (4) ◽  
pp. 910-913 ◽  
Author(s):  
T. Irie ◽  
G. Yamada ◽  
K. Tanaka
Keyword(s):  
Boundary Conditions ◽  
Cross Section ◽  
Timoshenko Beam ◽  
Natural Frequencies ◽  
Circular Cross Section ◽  
Beam Theory ◽  
Timoshenko Beam Theory ◽  
Plane Vibration ◽  
Out Of Plane

The natural frequencies of out-of-plane vibration based on the Timoshenko beam theory are calculated numerically for uniform arcs of circular cross section under all combination of boundary conditions, and the results are presented in some figures.

scholarly journals Transverse Vibration of Rotating Tapered Cantilever Beam with Hollow Circular Cross-Section

2018 ◽  
Vol 2018 ◽  
pp. 1-14 ◽  
Author(s):  
Zhongmin Wang ◽  
Rongrong Li
Keyword(s):  
Differential Equation ◽  
Boundary Conditions ◽  
Cantilever Beam ◽  
Cross Section ◽  
Transverse Vibration ◽  
Slenderness Ratio ◽  
Circular Cross Section ◽  
Dimensionless Parameters ◽  
Inner Radius ◽  
Tapered Cantilever Beam

Problems related to the transverse vibration of a rotating tapered cantilever beam with hollow circular cross-section are addressed, in which the inner radius of cross-section is constant and the outer radius changes linearly along the beam axis. First, considering the geometry parameters of the varying cross-sectional beam, rotary inertia, and the secondary coupling deformation term, the differential equation of motion for the transverse vibration of rotating tapered beam with solid and hollow circular cross-section is derived by Hamilton variational principle, which includes some complex variable coefficient terms. Next, dimensionless parameters and variables are introduced for the differential equation and boundary conditions, and the differential quadrature method (DQM) is employed to solve this differential equation with variable coefficients. Combining with discretization equations for the differential equation and boundary conditions, an eigen-equation of the system including some dimensionless parameters is formulated in implicit algebraic form, so it is easy to simulate the dynamical behaviors of rotating tapered beams. Finally, for rotating solid tapered beams, comparisons with previously reported results demonstrate that the results obtained by the present method are in close agreement; for rotating tapered hollow beams, the effects of the hub dimensionless angular speed, ratios of hub radius to beam length, the slenderness ratio, the ratio of inner radius to the root radius, and taper ratio of cross-section on the first three-order dimensionless natural frequencies are more further depicted.

FREE VIBRATION ANALYSIS OF CROSS-PLY LAYERED COMPOSITE BEAMS WITH FINITE LENGTH ON ELASTIC FOUNDATION

2008 ◽  
Vol 05 (01) ◽  
pp. 21-36 ◽  
Author(s):  
RAMAZAN-ALI JAFARI-TALOOKOLAEI ◽  
MOHAMMAD-HOSSEIN KARGARNOVIN ◽  
MOHAMMAD-TAGHI AHMADIAN
Keyword(s):  
Free Vibration ◽  
Elastic Foundation ◽  
Cross Section ◽  
Finite Length ◽  
Vibration Analysis ◽  
Natural Frequencies ◽  
Rectangular Cross Section ◽  
Free Vibration Analysis ◽  
Composite Beams ◽  
Layered Composite

In this paper, free vibration analysis of cross-ply layered composite beams (LCB) with finite length and rectangular cross-section rested on an elastic foundation is investigated by finite element method. Based on the Timoshenko beam theory which includes the shear deformation and rotary inertia, the stiffness and mass matrices of a LCB are obtained using the energy method. Then, the natural frequencies are calculated by employing eigenvalue technique. The obtained results are verified against existing data in the literatures for a LCB with no foundation and uniform cross-section. Good agreements are observed between these cases. In the same way, the natural frequencies of a specific case, i.e. the stepped beam are calculated and finally, free vibrations of a symmetric and non-symmetric LCB are compared with each others.

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