Eigenvalue Problem Of Vibration Research Articles

Analytical and meshless DQM approaches to free vibration analysis of symmetric FGM porous nanobeams with piezoelectric effect

The current study is an first attempt to develop a vibrational-based piezoelectrically actuated smart nanoactuator model based on assumptions of Reddy third-order shear deformation beam model together with nonlocal strain gradient theory. The considered model is a sandwich structure consisting of piezoelectric layers under an external electric field, and symmetric to the midplane functionally graded (FG) porous core. Expressed by displacements equations of motion are derived based on a dynamic version of Hamilton's principle. Eigenvalue problem of free vibration of simply supported nanobeam is solved analytically by using properties of Fourier series, and numerically via meshless Differential Quadrature Method (DQM). The study includes a convergence study for the numerical approach, as well detailed discussion on obtained results involving effects of nonlocal parameters, external mechanical and electrical loads, material gradation coupled with diverse porosity distribution, and influence of external environment on dynamic behavior of the nanostructure. The present paper, in view of included optimization study, may positively affect nanodevices reliability and extend its operation by avoiding resonance phenomenon.

A Semianalytical Vibration Analysis of Partially Wet Square Cantilever Plate With Numerical and Experimental Verification: Partially Wet Modeshapes

Abstract A semianalytical study of a uniform homogenous partially submerged square cantilever plate vibration is presented. The structure is assumed to be a Kirchhoff's plate, clamped on one edge and free on the other edges. The lengthwise section of the plate is a cantilever clamped-free (CF) beam, while the widthwise section is a free-free (FF) beam. The plate modeshape is a weighted superposition of the product of the beam modeshapes, with unknown weights. The CF beam has only flexural modes. The FF beam has two rigid-body modes, i.e., translational and rotational modes. Rayleigh–Ritz method (RRM) is used to set up the free vibration eigenvalue problem. The eigenvector gives the unknown weights. The modeshapes generated are further used in the boundary element method (BEM) to calculate the fluid inertia, which participates in the vibration and leads to a consistent drop in frequencies. The dependence of this reduction on the submergence level is studied for the first six frequencies of the plate. The frequencies are also experimentally generated by the impact hammer test, both in the dry state, and under three distinct levels of submergence: 25%, 50%, and 75% from the free edge opposite to the clamped edge. The frequencies and modeshapes are also verified through numerical analysis using the commercial code ansys 16.0. Conclusions are drawn regarding the influence of fluid inertia distribution on the final plate modeshape, leading to insights into sound structural designs.

Local and parallel finite element method for solving the biharmonic eigenvalue problem of plate vibration

In this paper, we establish a new local and parallel finite element discrete scheme based on the shifted‐inverse power method for solving the biharmonic eigenvalue problem of plate vibration. We prove the local error estimation of finite element solution for the biharmonic equation/eigenvalue problem and prove the error estimation of approximate solution obtained by the local and parallel scheme. When the diameters of three grids satisfy H4 = ϑ(w2) = ϑ(h), the approximate solutions obtained by our schemes can achieve the asymptotically optimal accuracy. The numerical experiments show that the computational schemes proposed in this paper are effective to solve the biharmonic eigenvalue problem of plate vibration.

GBT-Based Vibration Analysis Using the Exact Element Method

Generalized Beam Theory (GBT), intended to analyze the structural behavior of prismatic thin-walled members and structural systems, expresses the member deformed configuration as a combination of cross-section deformation modes multiplied by the corresponding longitudinal amplitude functions. The determination of the latter, often the most computer-intensive step of the analysis, is almost always performed by means of GBT-based “conventional” 1D (beam) finite elements. This paper presents the formulation, implementation and application of the so-called “exact element method” in the framework of GBT-based elastic free vibration analyses. This technique, originally proposed by Eisenberger (1990), uses the power series method to solve the governing differential equations and obtains the vibration eigenvalue problem from the boundary terms. A few illustrative numerical examples are presented, focusing mainly on the comparison between the combined accuracy and computational effort associated with the determination of vibration solutions with the exact and conventional GBT-based (finite) elements. This comparison shows that the GBT-based exact element method may lead to significant computational savings, particularly when the vibration modes exhibit large half-wave numbers.

Symmetry Adaptation of the Rotation-Vibration Theory for Linear Molecules

A numerical application of linear-molecule symmetry properties, described by the D ∞ h point group, is formulated in terms of lower-order symmetry groups D n h with finite n. Character tables and irreducible representation transformation matrices are presented for D n h groups with arbitrary n-values. These groups can subsequently be used in the construction of symmetry-adapted ro-vibrational basis functions for solving the Schrödinger equations of linear molecules. Their implementation into the symmetrisation procedure based on a set of “reduced” vibrational eigenvalue problems with simplified Hamiltonians is used as a practical example. It is shown how the solutions of these eigenvalue problems can also be extended to include the classification of basis-set functions using ℓ, the eigenvalue (in units of ℏ) of the vibrational angular momentum operator L ^ z . This facilitates the symmetry adaptation of the basis set functions in terms of the irreducible representations of D n h . 12 C 2 H 2 is used as an example of a linear molecule of D ∞ h point group symmetry to illustrate the symmetrisation procedure of the variational nuclear motion program Theoretical ROVibrational Energies (TROVE).

Symmetry-Adapted Ro-vibrational Basis Functions for Variational Nuclear Motion Calculations: TROVE Approach.

We present a general, numerically motivated approach to the construction of symmetry-adapted basis functions for solving ro-vibrational Schrödinger equations. The approach is based on the property of the Hamiltonian operator to commute with the complete set of symmetry operators and, hence, to reflect the symmetry of the system. The symmetry-adapted ro-vibrational basis set is constructed numerically by solving a set of reduced vibrational eigenvalue problems. In order to assign the irreducible representations associated with these eigenfunctions, their symmetry properties are probed on a grid of molecular geometries with the corresponding symmetry operations. The transformation matrices are reconstructed by solving overdetermined systems of linear equations related to the transformation properties of the corresponding wave functions on the grid. Our method is implemented in the variational approach TROVE and has been successfully applied to many problems covering the most important molecular symmetry groups. Several examples are used to illustrate the procedure, which can be easily applied to different types of coordinates, basis sets, and molecular systems.

Free Response of a Continuous Vibrational System Using Operational Tau Method

In this paper, the general form of the linear vibrational eigenvalue problems with some boundary conditions is considered. The given problem is then formulated by the operational Tau method with arbitrary polynomial bases. Afterward, considering the most common form of the vibrational eigenvalue problems and boundary conditions, some useful relations are derived from the method to solve such problems easily with a rapid rate of convergence. Finally, some numerical samples are solved with Chebyshev, Legendre and Standard basis and compared with the analytical solutions to clarify the efficiency and accuracy of the method in solving continuous vibrational problems.

Analysis of a Monolithic, Nonperiodic Array of Quartz Crystal Microbalances

We study thickness-shear and thickness-twist vibrations of a crystal plate with a nonperiodic array of surface mass layers for mass sensor applications. A theoretical analysis is performed. The equations of anisotropic elasticity are used. An analytical solution is obtained for the free vibration eigenvalue problem using trigonometric series from which the resonant frequencies and mode shapes are calculated. Results show that, depending on the geometric and physical parameters of the mass layers, the vibration may be mainly trapped under some of the mass layers but not under others.

An Integral Equation Method for Free Vibration of Circular Plate with Concentrated Mass at Arbitrary Positions

The paper concerns on the free vibrations of circular plate with arbitrary number of the mounted masses at arbitrary positions by using the integral equation method. A set of complete systems of orthogonal functions, which is constructed by Bessel functions of the first kind, is used to construct the Green's function of circular plates firstly. Then the eigenvalue problem of free vibration of circular plate carrying oscillators and elastic supports at arbitrary positions is transformed into the problem of integral equation by using the superposition theorem and the physical meaning of the Green’s function. And then the eigenvalue problem of integral equation is transformed into a standard eigenvalue problem of a matrix with infinite order. Numerical examples are presented.

Integral Equation Method for Free Vibration of Annular Plates with Elastic Supports

Annular plates are commonly found in the fields of engineering. The present study is concerned with the integral equation method for the free vibrations of annular plates with elastic supports. A set of complete systems of orthogonal functions, which is constructed by Bessel functions of the first and the second kind is used to construct the Green's function of annular plates. The eigenvalue problem of free vibration of annular plates with Elastic Supports is transformed into the eigenvalue problem of integral equation. And then, the problem of integral equation is transformed into a standard eigenvalue problem of a matrix with infinite order. Numerical example shows the significant advantages of the present method.

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.

The Intramolecular Hydrogen-Bond in Malonaldehyde as Seen by Infrared Spectroscopy. A Four-Dimensional Model Study

Abstract The infrared spectrum of the O–H–O fragment of malonaldehyde is studied using a four-dimensional model. This comprises the OH stretching and the two OH bending vibrations as well an O–O ring deformation mode under the assumption of overall Cs symmetry. The full anharmonic potential energy and dipole moment surfaces are calculated using density functional theory and the respective vibrational eigenvalue problem is solved by an iterative Lanczos method. Fundamental and combination transitions are discussed for the normal species and the symmetrically deuterated isotopomer. Special emphasis is paid to the OH/OD stretching region which reveals the signatures of strong mode mixing what renders a simple assignment in terms of fundamental transitions difficult. In addition results for hot transitions are presented which show a rather different OH/OD band due to the topology of the potential energy surface. The influence of H atom tunneling on the spectrum is briefly addressed employing an alternative three-dimensional model which takes into account the in-plane H atom motion as well as the O–O distance.

A GROUP THEORETIC APPROACH TO THE LINEAR FREE VIBRATION ANALYSIS OF SHELLS WITH DIHEDRAL SYMMETRY

This paper deals with a group theoretic approach to the finite element analysis of linear free vibrations of shells with dihedral symmetry. Examples of such shell structures are cylindrical shells, conical shells, shells with circumferential stiffeners, corrugated shells, spherical shells, etc. The group theoretic approach is used to exploit the inherent symmetry in the problem. For vibration analysis, the group theoretic results give the correct symmetry-adapted basis for the displacement field. The stiffness matrix K and the mass matrix M are identically block diagonalized in this basis. The generalized linear eigenvalue problem of free vibration gets split into independent subproblems due to this block diagonalization. The Simo element is used in the finite element formulation of the shell equilibrium equations. Numerical results for natural frequencies and natural modes of vibration of several dihedral shell structures are presented. The results are shown to be in very good agreement with those reported in the literature. The computational advantages and physical insights due to the group theoretic approach are also discussed.

6D vibrational quantum dynamics: Generalized coordinate discrete variable representation and (a)diabatic contraction

A new discrete variable representation (DVR) in generalized vibrational coordinates is proposed together with a new mixed diabatic/adiabatic contraction technique for the treatment of multidimensional vibrational problems up to high vibrational excitations. Formally based on the equidistant Chebyshev DVR in the grid index the new formulation is particularly suitable for multidimensional minimum energy paths. The new Z-matrix DVR proposed in this paper encompasses usual valence coordinates as well as nonlinear maps of coordinates on optimal nonequidistant grids. The pointwise numerical calculation of all kinetic energy terms avoids the algebraic derivation of specialized analytical forms of the kinetic energy adding to the flexibility of the method. With efficient truncation schemes the generalized DVR allows for a compact representation of the time-dependent wave-packet dynamics in up to six dimensions. Vibrationally adiabatic approaches to the detailed modelling of multidimensional quantum-dynamics usually are hampered by the typically large number of (avoided) crossings in dense spectra. This problem is particularly severe for discrete variable representations. A solution is provided by the new technique of diabatic rotations leading to a systematic construction of locally diabatic channels. This allows the treatment of very dense spectra where conventional truncation techniques fail. Applying the new approach to the vibrational problem of tetratomic molecules demonstrates its flexibility and efficiency. The examples of formaldehyde, ammonia, and hydrogen peroxide cover the whole range from semirigid (CH2O) to large amplitude inversion (NH3) and torsional tunnelling dynamics (H2O2). In solving the full six-dimensional vibrational eigenvalue problems for CH2O and NH3 the Z-matrix DVR shows at least comparable if not superior numerical efficiency compared with specialized techniques. In the case of H2O2 the technique of diabatic rotations and adiabatic contraction for the first time allows the treatment of the tunneling dynamics significantly above the dissociation threshold up to the fifth OH stretch overtone. The calculated decrease of the tunneling rate by about one order of magnitude agrees well with experimental observations.

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.

Classical and quantum mechanical studies of the CO vibrations in CO/Cu(100)

We report classical and quantum mechanical calculations of the vibrational energies of the CO/Cu(100) focusing on the CO–CO intermolecular coupling and the anharmonic coupling between the six CO–Cu modes. The normal mode analysis is used for the classical calculation, and the vibrational self-consistent field method is used to solve the vibrational eigenvalue problem for the quantum mechanical calculation. The vibrational frequencies of the six CO–Cu modes are obtained. A significant vibrational frequency change has been found in frustrated translational mode due to CO–CO intermolecular coupling. The agreement with recent experiments is generally quite good. Details of vibrational dynamics will be discussed with respect to the CO–CO coupling and the anharmonic coupling.

Investigation of vibrational states of the ArHCl+ cation in the electronic ground state

Abstract The potential energy surface (PES) of the electronic ground state of the ArHCl + molecular ion is calculated by the multireference single- and double-excitation configuration interaction (MRD-CI) technique. An analytical representation of the potential energy data is obtained in the form of a power series. The vibrational eigenvalue problem is treated with high accuracy on the basis of the variational method using a direct product basis contracted in two steps (truncation/recoupling method). Unexpectedly for such a weakly bound system the spectrum turns out to be of a rather regular nature up to the first dissociation energy, allowing for an assignment of quantum numbers to a major part of the vibrational states. Furthermore, the wave functions are analysed together with several geometrical and energetic characteristics of the various vibrational states.

Application of contracted distributed approximating functions to solving vibrational eigenvalue problems

It has been shown that an approximately band-limited function can be reconstructed by using the function’s values taken at appropriate equidistant grid points and a generalized Hermite-contracted-continuous-distributed-approximating-function (Hermite-CCDAF) as the reconstruction function. A sampling theorem prescribing the possible choices of grid spacing and DAF parameters has been derived and discussed, and discretized-Hermite-contracted DAFs have been introduced. At certain values of its parameters the generalized Hermite-CCDAF is identical to the Shannon–Gabor-wavelet-DAF (SGWDAF). Simple expressions for constructing the matrix of a vibrational Hamiltonian in the discretized-Hermite-contracted DAF approximation have been given. As a special case the matrix elements corresponding to sinc-DVR (discrete variational representation) are recovered. The usefulness and properties of sinc-DVR and discretized-Hermite-contracted-DAF (or SGWDAF) in bound state calculations have been compared by solving the eigenvalue problem of a number of one- and two-dimensional Hamiltonians. It has been found that if one requires that the same number of energy levels be computed with an error less than or equal to a given value, the SGWDAF method with thresholding is faster than the standard sinc-DVR method. The results obtained with the Barbanis Hamiltonian are described and discussed in detail.

A theoretical procedure for determining rovibrational eigenstates of van der Waals complexes

A theoretical procedure for determining rovibrational eigenstates of van der Waals complexes is presented. In this procedure, the self-consistent field method is used to optimize the basis sets for bending and stretching motions in the van der Waals complex; then, the configuration-interaction method is employed to produce converged results. Numerical radial basis functions are used in solving the vibrational eigenvalue problems. Comparison with other approaches for the Ar(SINGLE BOND)HCl complex is made. The results show that it is possible to obtain a comparable level of accuracy by using fewer configurations. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 66: 119–122, 1998