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.

Free Vibration of Composite Annular Plates Embedded with SMA Fibers in the Circumferential Direction

2007 ◽  
Vol 353-358 ◽  
pp. 1306-1309
Author(s):  
Shi Rong Li ◽  
Wen Shan Yu ◽  
Liang Liang Fan
Keyword(s):  
Temperature Rise ◽  
Shooting Method ◽  
Natural Frequencies ◽  
Volume Fraction ◽  
Plate Theory ◽  
Constitutive Law ◽  
Axisymmetric Vibration ◽  
Classical Plate Theory ◽  
One Dimensional ◽  
Annular Plates

Based on Brinson’s one-dimensional thermo-constitutive law of SMA and classical plate theory, linear non-axisymmetric vibration of uniform heated composite annular plate embedded with circumferential SMA fibers is investigated. Natural frequencies of the annular plates with immovably clamped boundary condition, depending on the temperature rise are obtained by using shooting method. The characteristic curves of first four natural frequencies of non-axisymmetric vibration versus temperature rise are plotted. Influences of the volume fraction and the initial strain of SMA on the natural frequencies of plate are analyzed. The numerical results show that, the activation of SMA can enhance the natural frequency both in the inverse martensite transformation temperature range and the thermal buckling temperature.

Free Bending Vibrations of a Centrally Bonded Symmetric Double Lap Joint (or Symmetric Double Doubler Joint) With a Gap in Mindlin Plates or Panels

2007 ◽  
Author(s):  
U. Yuceoglu ◽  
O. Gu¨vendik ◽  
V. O¨zerciyes
Keyword(s):  
Boundary Conditions ◽  
Natural Frequencies ◽  
Mode Shapes ◽  
Shear Stresses ◽  
Lap Joint ◽  
Bending Vibrations ◽  
Joint System ◽  
Adhesive Layers ◽  
Bonded Joint ◽  
Free Bending

In this present study, the “Free Bending Vibrations of a Centrally Bonded Symmetric Double Lap Joint (or Symmetric Double Doubler Joint) with a Gap in Mindlin Plates or Panels” are theoretically analyzed and are numerically solved in some detail. The “plate adherends” and the upper and lower “doubler plates” of the “Bonded Joint” system are considered as dissimilar, orthotropic “Mindlin Plates” joined through the dissimilar upper and lower very thin adhesive layers. There is a symmetrically and centrally located “Gap” between the “plate adherends” of the joint system. In the “adherends” and the “doublers” of the “Bonded Joint” assembly, the transverse shear deformations and the transverse and rotary moments of inertia are included in the analysis. The relatively very thin adhesive layers are assumed to be linearly elastic continua with transverse normal and shear stresses. The “damping effects” in the entire “Bonded Joint” system are neglected. The sets of the dynamic “Mindlin Plate” equations of the “plate adherends”, the “double doubler plates” and the thin adhesive layers are combined together with the orthotropic stress resultant-displacement expressions in a “special form”. This system of equations, after some further manipulations, is eventually reduced to a set of the “Governing System of the First Order Ordinary Differential Equations” in terms of the “state vectors” of the problem. Hence, the final set of the aforementioned “Governing Systems of Equations” together with the “Continuity Conditions” and the “Boundary conditions” facilitate the present solution procedure. This is the “Modified Transfer Matrix Method (MTMM) (with Interpolation Polynomials). The present theoretical formulation and the method of solution are applied to a typical “Bonded Symmetric Double Lap Joint (or Symmetric Double Doubler Joint) with a Gap”. The effects of the relatively stiff (or “hard”) and the relatively flexible (or “soft”) adhesive properties, on the natural frequencies and mode shapes are considered in detail. The very interesting mode shapes with their dimensionless natural frequencies are presented for various sets of boundary conditions. Also, several parametric studies of the dimensionless natural frequencies of the entire system are graphically presented. From the numerical results obtained, some important conclusions are drawn for the “Bonded Joint System” studied here.

Split Vibration Modes in Acoustically-Coupled Disk Stacks

1995 ◽  
Author(s):  
Jung-Ge Tseng ◽  
Jonathan Wickert
Keyword(s):  
Natural Frequencies ◽  
Three Dimensional ◽  
System Level ◽  
Thin Plates ◽  
Mode Shapes ◽  
Design Parameters ◽  
Phase System ◽  
Acoustic Coupling ◽  
Annular Plates ◽  
Vibration Model

Abstract Vibration of an array of stacked annular plates, in which adjacent plates couple weakly through an acoustic layer, is investigated through experimental and theoretical methods. Such acoustic coupling manifests itself through split natural frequencies, beating in the time responses of adjacent or separated plates, and system-level modes in which plates in the array vibrate in- or out-of-phase at closely-spaced frequencies. Laboratory measurements, including a technique in which the frequency response function of all in-phase modes but no out-of-phase modes, or visa versa, is measured, demonstrate the contribution of coupling to the natural frequency spectrum, and identify the combinations of design parameters for which it is important. For the lower modes of primary interest here, the natural frequencies of the out-of-phase system modes decrease as the air layer becomes thinner, while those of the in-phase mode remain sensibly constant at the in vacuo values. A vibration model comprising N classical thin plates that couple through the three-dimensional acoustic fields established in the annular cavities between plates is developed, and its results are compared with measurements of the natural frequencies and mode shapes.

Hydroelastic Vibration of Rectangular Plates

1996 ◽  
Vol 63 (1) ◽  
pp. 110-115 ◽  
Author(s):  
Moon K. Kwak
Keyword(s):  
Boundary Conditions ◽  
Approximate Formula ◽  
Natural Frequencies ◽  
Mass Matrix ◽  
Ritz Method ◽  
Plate Boundary ◽  
Rigid Wall ◽  
Mode Shapes ◽  
Rectangular Plates ◽  
Virtual Mass

This paper is concerned with the virtual mass effect on the natural frequencies and mode shapes of rectangular plates due to the presence of the water on one side of the plate. The approximate formula, which mainly depends on the so-called nondimensionalized added virtual mass incremental factor, can be used to estimate natural frequencies in water from natural frequencies in vacuo. However, the approximate formula is valid only when the wet mode shapes are almost the same as the one in vacuo. Moreover, the nondimensionalized added virtual mass incremental factor is in general a function of geometry, material properties of the plate and mostly boundary conditions of the plate and water domain. In this paper, the added virtual mass incremental factors for rectangular plates are obtained using the Rayleigh-Ritz method combined with the Green function method. Two cases of interfacing boundary conditions, which are free-surface and rigid-wall conditions, and two cases of plate boundary conditions, simply supported and clamped cases, are considered in this paper. It is found that the theoretical results match the experimental results. To investigate the validity of the approximate formula, the exact natural frequencies and mode shapes in water are calculated by means of the virtual added mass matrix. It is found that the approximate formula predicts lower natural frequencies in water with a very good accuracy.

scholarly journals An investigation into the free vibrations of carbon nanotubes using analytical and finite element methods

2021 ◽  
Author(s):  
Ishan Ali Khan
Keyword(s):  
Carbon Nanotubes ◽  
Finite Element ◽  
Eigenvalue Problem ◽  
Natural Frequencies ◽  
Nonlinear Eigenvalue Problem ◽  
Dynamic Stiffness ◽  
Free Vibrations ◽  
Mode Shapes ◽  
Wide Range ◽  
Walled Cnts

Since their discovery, immense attention has been given to carbon nanotubes (CNTs), due to their exceptional thermal, electronic and mechanical properties and, therefore, the wide range of applications in which they are, or can be potentially, employed. Hence, it is important that all the properties of carbon nanotubes are studied extensively. This thesis studies the vibrational frequencies of double-walled and triple-walled CNTs, with and without an elastic medium surrounding them, by using Finite Element Method (FEM) and Dynamic Stiffness Matrix (DSM) formulations, considering them as Euler-Bernoulli beams coupled with van der Waals interaction forces. For FEM modelling, the linear eigenvalue problem is obtained using Galerkin weighted residual approach. The natural frequencies and mode shapes are derived from eigenvalues and eigenvectors, respectively. For DSM formulation of double-walled CNTs, a nonlinear eigenvalue problem is obtained by enforcing displacement and load end conditions to the exact solution of single equation achieved by combining the coupled governing equations. The natural frequencies are obtained using Wittrick-Williams algorithm. FEM formulation is also applied to both double and triple-walled CNTs modelled as nonlocal Euler-Bernoulli beam. The natural frequencies obtained for all the cases, are in agreement with the values provided in literature.

scholarly journals Coupled flexural - torsional vibration and stability analysis of pre-loaded beams using conventional and dynamic finite element methods

2021 ◽  
Author(s):  
Heenkenda Jayasinghe
Keyword(s):  
Finite Element ◽  
Eigenvalue Problem ◽  
Axial Force ◽  
Torsional Vibration ◽  
Natural Frequencies ◽  
Nonlinear Eigenvalue Problem ◽  
Compressive Force ◽  
Mode Shapes ◽  
Dynamic Finite Element ◽  
Starting Point

Dynamic Finite Element (DFE) and conventional finite element formulations are developed to study the flexural - torsional vibration and stability of an isotropic, homogeneous and linearly elastic pre-loaded beam subjected to an axial load and end-moment. Various classical boundary conditions are considered. Elementary Euler - Bernoulli bending and St. Venant torsion beam theories were used as a starting point to develop the governing equations and the finite element solutions. The nonlinear Eigenvalue problem resulted from the DFE method was solved using a program code written in MATLAB and the natural frequencies and mode shapes of the system were determined form the Eigenvalues and Eigenvectors, respectively. Similarly, a linear Eigenvalue problem was formulated and solved using a MATLAB code for the conventional FEM method. The conventional FEM results were validated against those available in the literature and ANSYS simulations and the DFE results were compared with the FEM results. The results confirmed that tensile forces increased the natural frequencies, which indicates beam stiffening. On the contrary, compressive forces reduced the natural frequencies, suggesting a reduction in beam stiffness. Similarly, when an end-moment was applied the stiffness of the beam and the natural frequencies diminished. More importantly, when a force and end-moment were acting in combination, the results depended on the direction and magnitude of the axial force. Nevertheless, the stiffness of the beam is more sensitive to the changes in the magnitude and direction of the axial force compared to the moment. A buckling analysis of the beam was also carried out to determine the critical buckling end-moment and axial compressive force.

Natural Frequencies and Normal Modes of a Spinning Timoshenko Beam With General Boundary Conditions

1992 ◽  
Vol 59 (2S) ◽  
pp. S197-S204 ◽  
Author(s):  
Jean Wu-Zheng Zu ◽  
Ray P. S. Han
Keyword(s):  
Boundary Conditions ◽  
Mode Shape ◽  
Timoshenko Beam ◽  
Natural Frequencies ◽  
Normal Modes ◽  
Single Mode ◽  
Mode Shapes ◽  
Simulation Studies ◽  
Classical Boundary ◽  
First Time

A free flexural vibrations of a spinning, finite Timoshenko beam for the six classical boundary conditions are analytically solved and presented for the first time. Expressions for computing natural frequencies and mode shapes are given. Numerical simulation studies show that the simply-supported beam possesses very peculiar free vibration characteristics: There exist two sets of natural frequencies corresponding to each mode shape, and the forward and backward precession mode shapes of each set coincide identically. These phenomena are not observed in beams with the other five types of boundary conditions. In these cases, the forward and backward precessions are different, implying that each natural frequency corresponds to a single mode shape.

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.

Determination of the first n modes of a tall building from a reduced eigenvalue problem

1964 ◽  
Vol 54 (4) ◽  
pp. 1233-1254
Author(s):  
Moshe F. Rubinstein
Keyword(s):  
Eigenvalue Problem ◽  
Natural Frequencies ◽  
Tall Building ◽  
Degree Of Freedom ◽  
Mode Shapes ◽  
Story Building

Abstract The first n natural frequencies and mode shapes of an N degree of freedom structure (n < N) are derived from the solution of a reduced eigenvalue problem of order smaller than N. The reduced eigenvalue problem is formulated by using experience to select approximations to the first n modes desired. Accuracy is improved when more than n modes are selected. The method is illustrated by a study on an 18 story building.

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