Vibration Eigenvalues Research Articles

A level set based shape and topology optimization method for maximizing the simple or repeated first eigenvalue of structure vibration

We present a level set based shape and topology optimization method for maximizing the simple or repeated first eigenvalue of structure vibration. Considering that a simple eigenvalue is Frechet differentiable with respect to the boundary of a structure but a repeated eigenvalue is only Gateaux or directionally differentiable, we take different approaches to derive the boundary variation that maximizes the first eigenvalue. In the case of simple eigenvalue, material derivative is obtained via adjoint method, and variation of boundary shape is specified according to the steepest descent method. In the case of N-fold repeated eigenvalue, variation of boundary shape is obtained as a result of a N-dimensional algebraic eigenvalue problem. Constraint of a structure's volume is dealt with via the augmented Lagrange multiplier method. Boundary variation is treated as an advection velocity in the Hamilton---Jacobi equation of the level set method for changing the shape and topology of a structure. The finite element analysis of eigenvalues of structure vibration is accomplished by using an Eulerian method that employs a fixed mesh and ersatz material. Application of the method is demonstrated by several numerical examples of optimizing 2D structures.

Dynamic profilometry without out-of-plane conversion to measure vibration frequency of a cantilever beam

A new technique to measure oscillation frequencies and modal shapes of an Euler–Bernoulli cantilever beam using dynamic profilometry and phase extraction techniques is presented. The proposed technique does not require a fixed reference or out-of-plane conversion, and works on nonstop vibration. A binary pattern is projected on the cantilever beam surface mechanically forced to vibrate harmonically in a natural mode. The Fourier transform method is employed to obtain the phase difference between two consecutive frames, in particular it is applied to four consecutive frames so that three consecutive phase differences are available. Finally, the three-step temporal phase-shifting technique is applied to measure the vibration's eigenvalues and eigenfunctions. This paper presents the analysis of the underlying theory and the experimental results obtained.

Accurate in-plane free vibration analysis of rectangular orthotropic plates

Accurate analytical type solutions are obtained for the free in-plane vibration eigenvalues and mode shapes of fully clamped orthotropic rectangular plates. Appropriate governing differential equations for orthotropic plates are developed in dimensionless form and employed. Solutions are obtained by the method of superposition. It will be apparent that the solution procedure can be readily extended to analyze plates with other types of classical boundary conditions. The work reported is a natural extension of earlier work related to the in-plane vibration of isotropic plates. This appears to be the first analytical type study related to orthotropic plate in-plane vibration. It is expected that results tabulated here will form a valuable reference against which the results of other researchers may be compared.

252 固有振動数最大化を目的とするシェル構造の形状最適化

This paper presents a numerical optimization method for optimal configuration design of shell structures. It is assumed that the shell surface is varied in the out-of-plane direction to the surface in order to obtain the optimal free-form shape, and the thickness is not changed with respect to the shape variation. A solution to the maximization problem of vibration eigen value subject to a volume constraint is proposed. With this solution, the optimal shape is obtained without shape design parameterization. The problem is formulated as a non-parametric shape optimization problem. The shape gradient function is theoretically derived using the material derivative formulas, Lagrange multiplier method and the adjoint variable method. The Robin type traction method is applied to determine the free-form shell while minimizing the objective functional. The calculated results show the effectiveness of the proposed method for optimal free-form design of shell structures.

Exact free vibration analysis of axially moving viscoelastic plates

In this paper, an exact finite strip method is developed for the free vibration analysis of axially moving viscoelastic plates. Using the differential equation that governs the vibration of plates traveling at a constant axial speed, and utilizing the rheological models to model the viscoelastic behavior of materials, the exact stiffness matrix of a finite strip of plate is extracted in the frequency domain. Terms of this matrix are implicit functions of the eigenvalues of free vibration, in-plane forces, viscoelasticity parameters, axial speed and the geometry of the plate. By assembling the stiffness matrices of these finite strips, the global stiffness matrix of a plate moving on intermediate rollers is obtained, from which the eigenvalues defining the free vibration of the plate are extracted within the domain of complex numbers using a suitable algorithm. These values (that represent the exact frequencies of the plate vibration and damping) can be used as a benchmark to evaluate the accuracy of other numerical techniques. Finally, by presenting a set of numerical examples, the effects of axial speed and viscoelastic parameters on the free vibration of moving plates are examined in this paper.

Quantitative nondestructive evaluation of thin plate structures using the complete frequency information from impact testing

This article deals the theory for solving an inverse problem of plate structures using the frequency-domain information instead of classical time-domain delays or free vibration eigenmodes or eigenvalues. A reduced set of output parameters characterizing the defect is used as a regularization technique to drastically overcome noise problems that appear in imaging techniques. A deconvolution scheme from an undamaged specimen overrides uncertainties about the input signal and other coherent noises. This approach provides the advantage that it is not necessary to visually identify the portion of the signal that contains the information about the defect. The theoretical model for Quantitative nondestructive evaluation, the relationship between the real and ideal models, the finite element method (FEM) for the forward problem, and inverse procedure for detecting the defects are developed. The theoretical formulation is experimentally verified using dynamic responses of a steel plate under impact loading at several points. The signal synthesized by FEM, the residual, and its components are analyzed for different choices of time window. The noise effects are taken into account in the inversion strategy by designing a filter for the cost functional to be minimized. The technique is focused toward a exible and rapid inspection of large areas, by recovering the position of the defect by means of a single accelerometer, overriding experimental calibration, and using a reduced number of impact events.

Vibration analysis of thin-walled curved beams using DQM

The differential quadrature method (DQM) is applied to computation of the eigenvalues of small amplitude free vibration for thin-walled curved beams including a warping contribution. Natural frequencies are calculated for singlespan, curved, wide-flange uniform beams having a range of nondimensional parameters representing variations in warping stiffness, torsional stiffness, radius of curvature, included angle of the curve, polar mass moment of inertia, and various end conditions. Results are compared with existing exact and numerical solutions by other methods for cases in which they are available. It is found that the DQM gives good accuracy even when only a limited number of grid points is used. In addition, results are given for a cantilever beam not previously considered for this problem. Finally, parametric results are presented in dimensionless form.

Dynamic response of arch bridges traversed by high-speed trains

A mechanical model describing the planar elasto-dynamics of arch bridges with general arch profiles is presented. The model is amenable to analytical or semi-analytical treatments and is effective for parametric studies, design of control systems or structural optimizations. The Ritz's energy approach is employed to calculate the solutions of the vibration eigenvalue problem—natural frequencies and mode shapes—and the forced responses to external excitations, namely those induced by the passage of trains. A closed-form solution of the bridge dynamic response to the transit of trains with arbitrary load distributions and running speeds is found and the train-induced resonances are accordingly discussed. In particular, three European high-speed trains—the French TGV, the Italian ETR 500, and the German ICE—traversing a lower-deck steel arch bridge are considered and the ensuing responses are investigated.

On the Free Vibrations of Variable-Arc-Length Beams: Analytical and Experimental

Vibrations of horizontal, variable-arc-length beams considering the effects of large initial static sag deflections due to self weight are investigated. The variability in beam arc length arises from one end being pinned, and the other end being supported by a frictionless roller at a fixed distance from the pinned end. Using Lagrange’s equation, the large amplitude, free-vibration, equation of motion can be derived and simplified for small amplitude motion. These formulations represent small amplitude vibration about the large deformed static sag configurations. A solution is obtained by using the nonlinear finite element method. The vibration eigenvalue problem is solved using inverse iteration techniques to obtain the natural frequencies and corresponding mode shapes. Free and forced vibration experimental studies were conducted. Frequencies and mode shapes are shown for a range of beam specimens to complement the analytical results.

Exact solutions for the free in-plane vibration of rectangular plates with two opposite edges simply supported

Two distinct types of simple support boundary conditions are formulated for rectangular plates undergoing in-plane free vibration. Each type of simple support is shown to be analogous to the well known simple support edge conditions encountered in the free transverse vibration analysis of rectangular plates. It is shown that exact solutions may be obtained for plate free vibration eigenvalues and mode shapes when two opposite plate edges are given either type of simple support, the other two edges being given any combination of classical edge conditions. A complete analysis is provided for plate in-plane vibration with pairs of clamped or free edge conditions imposed on the non-simply supported boundaries and exact eigenvalues are tabulated for a large range of plate aspect ratios. This appears to be the first thorough study of these in-plane vibration problems with exact solutions.

The flutter analysis of an infinitely long orthotropic panel in a rectangular duct

The aeroelastic stability of an orthotropic panel in a duct with rectangular cross section is examined. The panel extends along the flow direction in the duct which is infinite in length. The panel vibration is modeled by linear plate theory and the flow in the duct is modeled by the compressible linearized potential theory. The panel is placed in the mid‐section of the duct and is simply supported at the sides. The material of the panel is assumed to be orthotropic. This would model, for example, unidirectional fibers placed in an isotropic plate, either along or perpendicular to the flow direction. An analytical solution is given for the eigenvalues of the panel vibration and results are presented for a range of parameters in the problem.

Highly accurate free vibration eigenvalues for the completely free orthotropic plate

Highly accurate free vibration eigenvalues for the completely free orthotropic plate

Eigenvalue analysis of Timoshenko beams and axisymmetric Mindlin plates by the pseudospectral method

A study of the free vibration of Timoshenko beams and axisymmetric Mindlin plates is presented. The analysis is based on the Chebyshev pseudospectral method, which has been widely used in the solution of fluid mechanics problems. Clamped, simply supported, free and sliding boundary conditions of Timoshenko beams are treated, and numerical results are presented for different thickness-to-length ratios. Eigenvalues of the axisymmetric vibration of Mindlin plates with clamped, simply supported and free boundary conditions are presented for various thickness-to-radius ratios.

Investigation of Buckling Phenomenon Induced by Growth of Vertebral Bodies Using a Mechanical Spine Model

A hypothesis that idiopathic scoliosis is a buckling phenomenon of the fourth or sixth mode, which is the second or third lateral bending mode, induced by the growth of vertebral bodies was presented in a previous paper by the authors using numerical simulations with a finite-element model of the spine. This paper presents experimental proof of the buckling phenomenon using mechanical spine models constructed with the geometrical data of the finite-element model used in a previous work. Using three spine mechanical models with different materials at intervertebral joints, the change in the natural vibration eigenvalue of the second lateral bending mode with the growth of vertebral bodies was measured by experimental modal analysis. From the result, it was observed that natural vibration eigenvalue decreased with the growth of vertebral bodies. Since the increase in primary factor inducing the buckling phenomenon decreases natural vibration eigenvalue, the obtained result confirms the buckling hypothesis.

FREE VIBRATION ANALYSIS OF CANTILEVER PLATES WITH STEP DISCONTINUITIES IN PROPERTIES BY THE METHOD OF SUPERPOSITION

Utilizing the superposition method, an analytical type solution is obtained for the free vibration eigenvalues and mode shapes of a cantilever plate with step discontinuities in plate properties. Property discontinuity lines run parallel to the clamped edge of the plate. Verification tests are performed for limiting cases by comparing computed eigenvalues with known eigenvalues for plates with uniform properties. Very good agreement is also obtained when computed results are compared with those obtained experimentally utilizing a test plate with discontinuities in thickness. Computed eigenvalues and mode shapes are presented for the benefit of other researchers. Besides the general interest, the problem has an application in the modelling of certain multi-story buildings during seismic studies.

Analytical Study of Vertical and Torsional Free Vibration of Cable Supported Bridge Decks.

The superposition method is employed to obtain an analytical type solution for the free vibration of cable supported bridge decks. Each pair of vertical elastic cables is considered to impart a vertical force to the deck by means of a rigid cross-member passing transversely beneath it. Rigid knife-edge support encountered at bridge towers is handled as well. In this introductory study the deck is treated as a thin isotropic plate. Any number of support cable pairs, of any stiffness, may be handled. Inter cross-member distances are referred to as spans. Free vibration eigenvalues and mode shapes are presented for three and four span illustrative cases.

Vibration Analysis of a Multi-Stage Gear System Including Drive Mechanism Elements. Proposal of Three-Dimentional Vibration Model and Eigenvalue Analysis of Helical Gear System.

We have developed a new method for building a vibration model of an actual gear-drive system. Each gear has six degrees of freedom: three translational and three rotational motions. Stiffness matrixes representing each gear shaft and tooth meshing were made and overlapped into one stiffness matrix for the whole system. This enabled easy modelling of complex gear-drive systems such as multi-stage gears and branched gears. The results are in good agreement with experimental results with respect to the major resonance frequencies and vibration modes.

Stiffening effects on the free vibration behavior of composite plates with PZT actuators

The free vibration behavior of thin composite plates with surface bonded piezoelectric patches including stress stiffening effects is investigated. A finite element formulation is presented, based on the Reissner–Mindlin theory and including non-linear strain–displacement relations, to formulate a free vibration eigenvalue problem in the presence of a geometrical stiffness matrix. The case of symmetric laminates and ideal linear behavior is assumed for the piezoelectric actuation. Due to the absence of membrane-bending coupling, in-phase applied voltages produce only inplane-induced piezoelectric stress resultants, which are assumed proportional to the applied voltages. Examples with numerical results for unconstrained plates equipped with piezoelectric actuators show that inplane induced stresses may significantly affect the free vibration behavior.

Exact eigenvalue correspondences between laminated plate theories via membrane vibration

Based on Reddy’s third-order theory, the first-order theory and the classical theory, exact explicit eigenvalues are found for compression buckling, thermal buckling and vibration of laminated plates via analogy with membrane vibration. These results apply to symmetrically laminated composite plates with transversely isotropic laminae and simply supported polygonal edges. Comprehensive consideration of a Winkler–Pasternak elastic foundation, a hydrostatic inplane force, an initial temperature increment and rotary inertias is incorporated. Bridged by the vibrating membrane, exact correspondences are readily established between any pairs of buckling and vibration eigenvalues associated with different theories. Positive definiteness of the critical hydrostatic pressure at buckling, the thermobukling temperature increment and, in the range of either tension loading or compression loading prior to occurrence of buckling, the natural vibration frequency is proved.

210 半正定値計画法を用いた重複固有値を有するトラスのトポロジー最適化問題

Algorithms based on Semi-Definite Programming (SDP) are proposed for the truss topology optimization problems for specified fundamental eigenvalue of free vibration and linear buckling load factor, and optimal topologies of trusses are computed by using the Semi-Definite Programming Algorithm (SDPA). It is well known that optimizing structures for specified minimum eigenvalue is difficult because of non-differentiability of the minimum eigenvalue for the cases of multimodal solutions. It is shown, in the examples, that the proposed algorithms are applicable to multimodal cases.