A Matrix Solution for the Vibration Modes of Nonuniform Disks

1956 ◽  
Vol 23 (1) ◽  
pp. 109-115
Author(s):  
F. F. Ehrich
Keyword(s):  
Boundary Conditions ◽  
Vibration Frequency ◽  
Outer Edge ◽  
Transverse Force ◽  
Constant Thickness ◽  
Vibration Modes ◽  
Matrix Solution ◽  
Efficient Manner ◽  
Polar Moment ◽  
Matrix Vector

Abstract An arbitrary disk is represented by a simulated disk composed of circumferential strips. Alternate strips are considered to be massless, constant-thickness elements with the average local elastic properties of the actual disk. Intermediate strips are considered to have the properties of local mass and polar moment of inertia, but to have no physical dimensions or elasticity. A matrix vector, formed of the local antinodal value of deflection, slope, moment, and transverse force, may be operated on by matrices representative of the elastic strips and by matrices representative of the vibratory inertia loading, centrifugal inertia loading, internal stress, and external supports at the mass strips. Thus the influence of boundary conditions at the outer edge on conditions at the inner edge may be calculated in a simple efficient manner. Successive guesses of vibration frequency lead to final satisfaction of all boundary conditions. Concise treatment of all types of boundary conditions and numerical values of required matrices are given in tables. The results of a sample calculation are compared with exact analytic results.

Concentrated-Force Problems in Plane Strain, Plane Stress, and Transverse Bending of Plates

1946 ◽  
Vol 13 (3) ◽  
pp. A183-A197
Author(s):  
P. S. Symonds
Keyword(s):  
Boundary Conditions ◽  
Plane Stress ◽  
Biharmonic Equation ◽  
Infinite Plate ◽  
Outer Edge ◽  
Transverse Force ◽  
Concentrated Force ◽  
Transverse Bending ◽  
Complementary Function ◽  
Concentrated Forces

Abstract A general method is described for the solution of problems of transverse bending of thin plates acted on by concentrated normal forces, and of problems of plane stress or plane strain, in which concentrated forces are applied to the boundaries. The solution is taken in two parts: (a) The special functions which give the stresses or deflections in the neighborhood of the concentrated forces. (b) A complementary function, satisfying the appropriate biharmonic equation, such that the complete solution satisfies the boundary conditions of the problem. For certain types of boundaries, this complementary function can be determined by expanding the concentrated-force functions as infinite trigonometric series. Then by addition of general solutions of the appropriate biharmonic equation, the required boundary conditions may be satisfied. The method is first illustrated by solving the plate-bending problem, for which the solution is known, of a clamped circular disk loaded by a transverse force at any point. It is then applied to the problem of an infinite plate fixed at an inner circular boundary, with outer edge free, and loaded by a transverse force at any point. This solution is obtained in finite form, and typical curves of deflection, bending moments, and shear forces are given in Figs. 3 to 8, inclusive. Using this result, solutions are next obtained for ring-shaped plates of finite outer radius, with the force applied either at the outer edge or at any point between the inner clamped edge and the outer free edge. The former case was previously solved by H. Reissner. Curves comparing the maximum moments and shears in the infinite plate with those of the annular plate with force either at the outer edge, or inside the ring are given in Figs. 9 to 12, inclusive. Finally, a solution is given of the problem in plane stress of a large plate containing an elliptical hole, which is loaded by line forces at the ends of the minor axes of the ellipse. Curves showing results of this solution are given in Figs. 14 and 15.

scholarly journals Numerical solution of acoustic scattering by finite perforated elastic plates

2016 ◽  
Vol 472 (2188) ◽  
pp. 20150767 ◽  
Author(s):  
A. V. G. Cavalieri ◽  
W. R. Wolf ◽  
J. W. Jaworski
Keyword(s):  
Boundary Conditions ◽  
Acoustic Scattering ◽  
Free Edge ◽  
Solution Procedure ◽  
Sound Level ◽  
Vibration Modes ◽  
Elastic Plates ◽  
Beneficial Effects ◽  
Plate Length ◽  
Radiated Sound

We present a numerical method to compute the acoustic field scattered by finite perforated elastic plates. A boundary element method is developed to solve the Helmholtz equation subjected to boundary conditions related to the plate vibration. These boundary conditions are recast in terms of the vibration modes of the plate and its porosity, which enables a direct solution procedure. A parametric study is performed for a two-dimensional problem whereby a cantilevered perforated elastic plate scatters sound from a point quadrupole near the free edge. Both elasticity and porosity tend to diminish the scattered sound, in agreement with previous work considering semi-infinite plates. Finite elastic plates are shown to reduce acoustic scattering when excited at high Helmholtz numbers k 0 based on the plate length. However, at low k 0 , finite elastic plates produce only modest reductions or, in cases related to structural resonance, an increase to the scattered sound level relative to the rigid case. Porosity, on the other hand, is shown to be more effective in reducing the radiated sound for low k 0 . The combined beneficial effects of elasticity and porosity are shown to be effective in reducing the scattered sound for a broader range of k 0 for perforated elastic plates.

Free vibration analysis of rectangular and annular Mindlin plates with undamaged and damaged boundaries by the spectral collocation method

2011 ◽  
Vol 18 (11) ◽  
pp. 1722-1736 ◽  
Author(s):  
Ma’en S Sari ◽  
Eric A Butcher
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Collocation Method ◽  
Vibration Analysis ◽  
Natural Vibration ◽  
Numerical Technique ◽  
Free Vibration Analysis ◽  
Mindlin Plate ◽  
Grid Points ◽  
Matrix Vector

The objective of this paper is the development of a new numerical technique for the free vibration analysis of isotropic rectangular and annular Mindlin plates with damaged boundaries. For this purpose, the Chebyshev collocation method is applied to obtain the natural frequencies of Mindlin plates with damaged clamped boundary conditions, where the governing equations and boundary conditions are discretized by the presented method and put into matrix vector form. The damaged boundaries are represented by distributed translational and torsional springs. In the present study the boundary conditions are coupled with the governing equation to obtain the eigenvalue problem. Convergence studies are carried out to determine the sufficient number of grid points used. First, the results obtained for the undamaged plates are verified with previous results in the literature. Subsequently, the results obtained for the damaged Mindlin plate indicate the behavior of the natural vibration frequencies with respect to the severity of the damaged boundary. This analysis can lead to an efficient technique for structural health monitoring of structures in which joint or boundary damage plays a significant role in the dynamic characteristics. The results obtained from the Chebychev collocation solutions are seen to be in excellent agreement with those presented in the literature.

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.

A Penetration-Based Finite Element Method for Hyperelastic 3D Biphasic Tissues in Contact: Part 1-Derivation of Contact Boundary Conditions

2005 ◽  
Vol 128 (1) ◽  
pp. 124-130 ◽  
Author(s):  
Kerem Ün ◽  
Robert L. Spilker
Keyword(s):  
Finite Element Method ◽  
Finite Element Analysis ◽  
Finite Element ◽  
Boundary Conditions ◽  
Solid Phase ◽  
Contact Boundary ◽  
Element Analysis ◽  
Efficient Manner ◽  
Biphasic Tissues ◽  
Element Method

In this study, we extend the penetration method, previously introduced to simulate contact of linear hydrated tissues in an efficient manner with the finite element method, to problems of nonlinear biphasic tissues in contact. This paper presents the derivation of contact boundary conditions for a biphasic tissue with hyperelastic solid phase using experimental kinematics data. Validation of the method for calculating these boundary conditions is demonstrated using a canonical biphasic contact problem. The method is then demonstrated on a shoulder joint model with contacting humerus and glenoid tissues. In both the canonical and shoulder examples, the resulting boundary conditions are found to satisfy the kinetic continuity requirements of biphasic contact. These boundary conditions represent input to a three-dimensional nonlinear biphasic finite element analysis; details of that finite element analysis will be presented in a manuscript to follow.

Finite Element and Modal Analysis of Key Components of Horizontal Vibrating Centrifuge

2012 ◽  
Vol 619 ◽  
pp. 365-369
Author(s):  
Lan Zhu Ren ◽  
Lin Lin Zeng ◽  
Xin Zhang
Keyword(s):  
Finite Element ◽  
Finite Element Model ◽  
Modal Analysis ◽  
Vibration Frequency ◽  
Element Model ◽  
Relative Displacement ◽  
Vibration Modes ◽  
Reciprocating Motion ◽  
Level 1 ◽  
Two Bodies

This article analyzed the finite element on horizontal centrifuge of VM1400, established finite element model of the corresponding parts and gave the vibration modes of the various movements. Through the stress, strain, displacement calculation and modal analysis of the various components, the conclusions include that the level 1 of the vibrating body and the secondary vibrating body do the regular axial horizontal linear reciprocating motion, there is relative displacement between the two bodies, and the vibration frequency is close to the operating frequency.

Large-amplitude vibration of imperfect angle-ply laminated rectangular plate with viscous damping

2015 ◽  
Vol 12 (3) ◽  
pp. 207-214 ◽  
Author(s):  
He Huang ◽  
David Hui
Keyword(s):  
Boundary Conditions ◽  
Vibration Frequency ◽  
Large Amplitude ◽  
Plane Boundary ◽  
Damping Factor ◽  
Viscous Damping ◽  
Damped System ◽  
Amplitude Vibration ◽  
Out Of Plane ◽  
Large Amplitude Vibration

This paper solves the modified-Duffing ordinary differential equation for large-amplitude vibration of imperfect angle-ply rectangular composite plate. Viscous damping is then introduced in the derivation and analyzed under four different boundary conditions (combining two out-of-plane boundary conditions with two in-plane boundary conditions). It has been shown that even a small viscous damping factor, for example 0.1 from an ordinary damped system can largely decrease the vibration amplitude within several periods. Yet at the same time, the vibration frequency only changes slightly. Furthermore, viscous damping is proved to significantly affect the vibration frequency and the vibration mode from nonlinear to much more linear. This effect is irrelevant to boundary conditions and geometric imperfections.

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