Vibration Eigenvalues Research Articles

Indeterminate shift synthesis of vibration eigenvalue and eigenvector by the use of finite element method.

This paper presents an attempt to formulate a structural synthesis for eigenvalue and eigenvector shift by introducing the indeterminacy to the objective eigenvalue and eigenvenctor. The methodology is based or the iterative approach to the objective expectaions by means of the sensitivity analysis of the finite elment solution and a functional to seek the objective structure as near the baseline structure as possible. The governing equation for the design parameters is rendered to the mean-centered perturbation with respect to the small probabilistic variables representing the indeterminate objective eigenvalue and eigenvector in order to calculate the sensitivity of the parameters at the objective expectations. The sensitivity is employed to evaluate the variance of the objective structure. The numerical example is concerned with the shift of the eigenvalue and eigenvector of the undamped vibration of a truck chasis and shows that a structural synthesis is made possible by setting a few objectives with indeterminacy.

Three-Dimensional Vibration Analysis of a Truncated Quadrangular Pyramid

An analysis is presented for the three-dimensional vibration problem of determining the natural frequencies and the mode shapes of a truncated quadrangular pyramid. For this purpose, the body is transformed into a right quadrangular prism with unit edge lengths by a transformation of variables. With the displacements of the transformed prism assumed in the forms of algebraic polynomials, the dynamical energies of the prism are evaluated, and the frequency equation is derived by the Ritz method. This method is applied to quadrangular pyramids in which the base is clamped and the other sides are free, and the natural frequencies (the eigenvalues of vibration) and the mode shapes are calculated numerically, from which the vibration characteristics arising in the pyramids are studied.

Free vibration of rectangular plates with two symmetrically distributed clamps along one edge

In this paper the method of superposition is employed to compute the free vibration eigenvalues for a thin rectangular plate held in a cantilever configuration by a pair of clamps symmetrically distributed along one edge. Eigenvalues are tabulated for the limiting case when the clamp area approaches zero and the plate is supported by applied concentrated forces and moments. The method is equally applicable to cases where the clamps cover a finite area. Good agreement has been obtained when theoretical results are compared with those obtained experimentally. By means of a small modification to earlier computational practices it has been possible to greatly increase accuracy. The modification is described in detail.

Three-dimensional vibration analysis of a truncated quadrangular pyramid.

An analysis is presented for the three-dimensional vibration problem of determining the natural frequencies and the mode shapes of a truncated quadrangular pyramid. For this purpose, the body is transformed into a right quadrangular prism with unit edge lengths, by a transformation of variables. With the displacements of the transformed prism assumed in the forms of algebraic polynomials, the dynamic energies of the prism are evaluated, and the frequency equation is derived by the Ritz method. This method is applied to quadrangular pyramid-shaped solid bodies in which the base is clamped and the other sides free, and the natural frequencies (the eigenvalues of vibration) and the mode shapes are calculated numerically up to higher modes, from which the characteristics of various types of vibration arising in the bodies are studied. The results are presented in some figures and compared with those obtained by the beam theories.

Finite element reanalysis of real eigenvalue problem. Case study of vibration eigenvalue.

An attempt is made to formulate the finite element reanalysis to cope with the change of vibration eigenvalues and eigenvectors. The methodology is based upon iterative solution of the nonlinear, coupled equations with respect to the change of the eigen pairs. Use is made of the normalization condition of the eigenvectors. Two methods are proposed for the solution, and the first one requires less CPU time while the second is versatile enough to chase complicated change of the eigen pair. The numerical examples show the validity of the proposed reanalysis methods in regard to the vibration of portal frames which are subjected to so large structural modification in terms of the mass density and the modulus of elasticity that the eigenvector is changed markedly.

Nonlinear structural dynamic analysis using a modified modal method

The procedure for predicting the nonlinear dynamic response of structural components subjected to a step loading is presented. The procedure is a modified modal method that involves a change of dependent variables from the unknown nodal degrees of freedom of the finite element model of the structure to a smaller set of generalized coordinates. This change of dependent variables uses a combination of the nonlinear static solution and some selected vibration mode shapes. The vibration mode shapes correspond to the eigenvectors obtained by solving a standard free vibration eigenvalue problem wherein the stiffness matrix is expanded about the nonlinear static solution. A strategy is also presented for determining which and how many vibration mode shapes to include in the transformation. The effect of inaccurate representation of the spatial distribution of the applied load on the nonlinear dynamic response is discussed for two classes of structural behavior. Application of the procedure to structures which exhibit a stiffening behavior and to those with a softening behavior is presented.

Free vibration of an oblique circular cylindrical shell.

An analysis is presented for the free vibration of an oblique circular cylindrical shell. For this purpose, the shell is transformed into a circular cylindrical shell of the unit axial length by a transformation of variables. The deflection displacements of the transformed shell are expressed in a series of the products of the eigenfunctions of an axial beam and the trigonometric functions of angular coordinate. The maximum kinetic and strain energies of the shell based upon the Love theory are evaluated analytically, and the frequency equation is derived by the Ritz method. This method is applied to a free-clamped oblique shell truncated at the free edge, and the natural frequencies (the eigenvalues of vibration) and the mode shapes are calculated numerically up to higher modes.

Free Vibration of Longitudinally Stiffened Prismatic Shells

An analysis is presented for the free vibration of simply supported. longitudinally stiffened prismatic shells, in which the transfer matrix technique is used. The governing equations of vibration of a plate (a side wall) are written in a matrix differential equation by use of the transfer matrix of the plate. The transfer matrix is determined by numerical integration of the equation, and the entire structure matrix of the shell is obtained by the product of the transfer matrix of the plate and the point matrix at the joint between two plates and a stiffener. The frequency equations are derived in therms of the elements of the structure matrix. By the application of the method, the natural frequencies (eigenvalues of vibration) and the mode shapes are calculated numerically for rectangular prismatic shells with structural symmetry, and the effects of the stiffeners on the vibration are discussed.

Complementary variational principles in elastodynamics

Commencing from two forms of virtual work Hamilton's principle and its complementary form are derived for discrete systems. Hamilton's principle is then applied to the elastodynamic problem of continuous media and it is shown how this principle can be modified to yield the generalisations of some well known variational principles of elastostatics to the dynamic régime. The application of the complementary energy procedure is illustrated and some salient features of this method, for the free vibration eigenvalue problem, are discussed.

Vibrations of cross-shaped, I-shaped, and L-shaped membranes and plates

The natural frequencies (the eigenvalues of vibration) and the mode shapes are presented for cross-shaped, I-shaped, and L-shaped membranes and simply supported plates. An irregularly shaped membrane is formed on a rectangular base membrane by fixing several segments. The reaction forces acting on the edges of an actual membrane are expanded into Fourier series with unknown coefficients, and homogeneous equations for the coefficients are derived by the restraint conditions on the edges. The natural frequencies and the mode shapes of the actual membrane are determined by calculating the eigenvalues and eigenvectors of the equations. The numerical values obtained for membranes can be immediately converted into those of simply supported plates by the dynamical analogy between membrane and plate.

Natural frequencies of concavely shaped polygonal plates with simply supported edges

This note presents the natural frequencies (the eigenvalues of vibration) and the mode shapes of concavely shaped polygonal plates with simply supported edges by the dynamical analogy with membranes.

Calculating Natural Frequencies With Extended Tuplin’s Method

In this paper an efficient numerical procedure is described which yields the eigenvalue of a lumped mass torsional vibration directly from the frequency equation of the system. Special characteristics of Tuplin’s frequency equation allow all eigenvalues to be easily located and accurately evaluated from the frequency polynomial. In contrast to the general belief that extracting roots of polynomials is less efficient than matrix reduction methods, this paper demonstrates that the direct solution competes favorably with the modern eigenvalue routines such as QR and tridiagonal methods [15] [16] [17] [21] in torsional vibration problems. A BASIC program FUNG has been written based on the numerical concepts of this paper. The current version is able to solve multiple branch systems of many degrees of freedom subject to the restriction that no branch shall exceed 4 rotors and 4 shafts. The program has been tested for various examples and the output compared with the known results. Within the above range of applicability, this method beats the modern tridiagonal eigenvalue subroutines [16] [21] by a comfortable margin which ranges from 15 times to 120 times faster. The comparisons were made on the basis of solving the same problems on the same computer. FUNG assumes a tight tolerance of convergence for iteration (correct to approximately 14 significant digits).

Vibration and local instability of thermally stressed plates

The vibration and buckling of a double wedge square cantilever plate has been investigated. It is shown that the free vibration modes, which occur at Delta T ref = 0, transition into the buckled modes which occur at Delta T ref = Delta T ref-cr for the respective mode. Delta T ref-cr for a particular mode is defined as the magnitude of thermal load at which the frequency of the particular mode vanishes. The analysis yields the same number of buckling eigenvalues and buckling modes as there are vibration eigenvalues and vibration modes. Gradual application of the load in the analysis permits the change in each vibration frequency of interest and its associated mode to be followed up to the load at which the frequency of the mode becomes zero. This constitutes the limit of linear theory. As the load is increased, the thin edges of the plate begin to deform during vibration. This local deformation, which begins in the vibration mode, is shown to transition into the phenomena of local edge buckling at Delta T ref-cr for the mode.

Free vibration analysis of rectangular plates with symmetrically distributed point supports along the edges

In this paper a highly accurate mathematical technique for establishing the free vibration eigenvalues and mode shapes of rectangular plates with symmetrically distributed point supports is introduced. The method is based on the principle of superposition and it constitutes, in essence, an extension of a technique described earlier for the analysis of completely free plates. Eigenvalues, covering numerous modes and aspect ratios, are tabulated for corner supported rectangular plates. It is seen that with some modifications the method is equally applicable to problems where the point supports are not symmetrically distributed.

Frequency determination and non-dimensionalization for composite cantilever plates

A method for estimating the natural frequencies of composite cantilever plates and for non-dimensionalizing frequency data is developed and demonstrated. The scheme is based on a partial Ritz (Kantorovich) analysis, which reduces the problem to a set of uncoupled ordinary differential equations. The solution of these equations yields both the eigenvalues of free vibration and the proper forms of the non-dimensional frequency. Together these can be used to estimate the natural frequencies of laminated plates. In addition the non-dimensional frequency expressions can be used to reduce numerical or experimental vibration data. To demonstrate this the first five frequencies of a representative group of laminated plates are calculated by using a finite element model. The calculated frequencies are then non-dimensionalized by using the expressions derived from the Ritz analysis. The resulting non-dimensional frequencies not only show more consistency than the original data, but are shown to agree well with the eigenvalues of the Ritz analysis.

Vibration and stability of a non-uniform Timoshenko beam subjected to a follower force

An analysis is presented for the vibration and stability of a non-uniform Timoshenko beam subjected to a tangential follower force distributed over the center line by use of the transfer matrix approach. For this purpose, the governing equations of a beam are written in a coupled set of first-order differential equations by using the transfer matrix of the beam. Once the matrix has been determined by numerical integration of the equations, the eigenvalues of vibration and the critical flutter loads are obtained. The method is applied to beams with linearly, parabolically and exponentially varying depths, subjected to a concentrated, uniformly distributed or linearly distributed follower force, and the natural frequencies and flutter loads are calculated numerically, from which the effects of the varying cross-section, slenderness ration, follower force and the stiffness of the supports on them are studied.

Natural frequencies of stepped thickness rectangular plates

This paper presents the natural frequencies of stepped thickness square and rectangular plates together with the mode shapes of vibration. The transverse deflection of a stepped thickness plate is written in a series of the products of the deflection functions of beams parallel to the edges satisfying the boundary conditions, and the frequency equation of the plate is derived by the energy method. By use of the frequency equation, the natural frequencies (the eigenvalues of vibration) and the mode shapes are calculated numerically in good accuracy for square and rectangular plates with edges simply supported or elastically restrained against rotation, having square, circular or elliptical stepped thickness, from which the effects of the stepped thickness on the vibration are studied.

Vibration and buckling of rotating flexible rods at transitional parameter values

We consider the connection between vibration and buckling problems for a uniform flexible rod which is clamped at one end and rotates in a plane perpendicular to the axis of rotation. The rod is assumed off-clamped, i.e. the axis of rotation does not pass through the rod's clamped end. The resulting fourth-order boundary value problem with a turning point for the free vibrations is solved using uniform approximations in a transitional parameter range where high rotation rates balance small off-clampings. Second approximations to the vibration eigenvalues are used to determine critical buckling rotation rates for the slightly off-clamped rods. Buckling is unexpected in this situation as the rod is wholly under tension.

Bounds to frequencies of shafts in torsional vibration with restraints and attached masses

In this paper upper and lower bounds are given to eigenvalues of torsional vibration of the following uniform structures: free shaft elastically restrained at a point; fixed shaft elastically restrained at a point; free shaft with masses elastically attached at points; fixed shaft with a mass elastically attached over an interval. Upper bounds were obtained by using the well-known Rayleigh-Ritz procedure. The more difficult problem of obtaining lower bounds was resolved by using recently developed methods of Bazley and Fox. The various problems discussed in this paper illustrate the use of these lower bound methods and their effectiveness in producing excellent bounds.

Vibrations of rectangular cantilever plates subjected to in-plane acceleration loads

The vast literature dealing with the free vibrations of plates contains virtually no references for plates which are loaded by acceleration fields or gravity forces in their plane. The present paper studies the case of the rectangular cantilever plate. In-plane forces caused by an arbitrarily oriented acceleration field must first be determined by solving a plane problem of the theory of elasticity. The transverse free vibration eigenvalue problem is then analyzed by the Ritz method using mode shapes which are the sum of products of the eigenfunctions for vibrating beams. The in-plane forces are included in this analysis. The resulting matrix eigenvalue problem is solved by standard techniques on the digital computer to obtain numerical results for the vibration frequencies and mode shapes of loaded plates. Results are presented which show the effects of varying the intensity of loading, the angle of acceleration, and the aspect ratio of the plate.