Direct Eigenvalue Analysis Research Articles

Parametrically Excited Instability of Periodic Visco-Elastomer Sandwich Plate with Supported Masses Under Quadrilateral Longitudinal Harmonic Excitations

The parametrically excited instability of the spatially periodic visco-elastomer sandwich plate with supported masses under quadrilateral longitudinal temporal harmonic excitations is studied. The improvability of the parametrically excited instability by periodic distribution parameters is explored. The direct eigenvalue analysis approach for solving the parametrically excited stability problem of the periodic sandwich plate system under longitudinal harmonic excitations is proposed. The spatial periodic distribution of facial layer thickness and core layer modulus of the sandwich plate is considered. The non-linear partial differential equations of longitudinal and transverse coupling motions of the periodic visco-elastomer sandwich plate with supported masses under biaxial longitudinal boundary excitations are derived. The longitudinal displacements of the sandwich plate are separated into two parts and the longitudinal boundary excitations relevant to symmetric part are incorporated into the sandwich plate system. Then the partial differential equations with boundary excitations are converted into parametrically excited system equations and further converted into ordinary differential equations with time-varying parameters, which describe the parametrically excited vibration with multi-mode coupling of the periodic sandwich plate system. The fundamental perturbation solution to the equations is expressed as the product of periodic and exponential parts based on the Floquet theorem. The ordinary differential equations with harmonic parameters are converted into a set of algebraic equations using the harmonic balance method. Then the parametrically excited instability of the periodic sandwich plate system is determined directly by matrix eigenvalues. The overall instability characteristics of parametrically excited vibration with multi-mode coupling of the system under longitudinal harmonic excitations are illustrated by numerical results on unstable regions. The parametrically excited instability can be improved by the spatially periodic distribution of geometrical and physical parameters. The proposed approach is applicable to general sandwich structures with spatial distribution parameters in multi-mode-coupling parametrically excited vibrations for overall instability analysis on continuous frequency band.

A multimode perturbation method for frequency response analysis of nonlinearly vibrational beams with periodic parameters

A multimode perturbation method for frequency response analysis of nonlinearly vibrational beams with periodic distribution parameters is proposed. The partial differential equation with spatial varying parameters for nonlinear vibration of beams with periodic parameters under harmonic excitations is derived. The procedure of the multimode perturbation method includes three main steps: first, the nonlinear partial differential equation is transformed into linear partial differential equations with varying parameters by applying perturbation method; second, the linear partial differential equations are transformed into ordinary differential equations with multimode coupling by applying Galerkin method, where multiple vibration modes of the beams are used and the equations are suitable to nonlinear vibration of periodic structures with high parameter-varying wave in wide frequency band; third, the ordinary differential equations are solved by applying harmonic balance method to obtain vibration response of the nonlinear periodic beam, which is used for characteristics analysis of frequency response and spatial mode. Furthermore, the stability problem of nonlinear harmonic vibration as multidegree-of-freedom system with periodic time-varying parameters is solved by applying direct eigenvalue analysis approach. The proposed method can incorporate multiple vibration modes into response analysis of nonlinear periodic structures and consider mode-coupling effects due to structural nonlinearity and parametric periodicity. Finally, a nonlinear beam with periodic supports under harmonic excitations is studied. Numerical results on frequency response of the beam are given to illustrate an application of the proposed method, new frequency response characteristics, and influences of periodic parameters on structural response. The results have potential application to nonlinear structural vibration control and support damage detection of nonlinear structures with periodic supports.

Parametrically Excited Stability of Periodically Supported Beams Under Longitudinal Harmonic Excitations

A direct eigenvalue analysis approach for solving the stability problem of periodically supported beams with multi-mode coupling vibration under general harmonic excitations is developed based on the Floquet theorem, Fourier series and matrix eigenvalue analysis. The transverse periodic supports are considered for improving the parametrically excited stability of beams under longitudinal periodic excitations. The dynamic stability of parametrically excited vibration of the beam with transverse spaced supports under longitudinal harmonic excitations is studied. The partial differential equation of motion of the beam with spaced supports under harmonic excitations is given and converted into ordinary differential equations with time-varying periodic parameters using the Galerkin method, which describe the parametrically excited vibration of the beam with coupled multiple modes. The fundamental solution to the equations is expressed as the product of periodic and exponential components based on the Floquet theorem. The periodic component and periodic parameters are expanded into Fourier series, and the matrix eigenvalue equation is obtained which is used for directly determining the parametrically excited stability. The dynamic stability of parametrically excited vibration of the beam with spaced supports under harmonic excitations is illustrated by numerical results on unstable regions. The influence of the periodic supports and excitation parameters on the parametrically excited stability is explored. The parametrically excited stability of the beam with multi-mode coupling vibration can be improved by the periodic supports. The developed analysis method is applicable to more general period-parametric beams with multi-mode coupling vibration under various harmonic excitations.

Parametrically excited stability of a periodically stiffened beam under axial periodic excitation

The parametrically excited stability of a periodically stiffened beam under general periodic axial excitation is studied and the effect of periodic stiffeners on the beam stability is considered for the first time. The partial differential equation of motion of the beam with periodic stiffeners under axial excitation is given. The Galerkin method is used to convert the partial differential equation into ordinary differential equations with periodic time-varying parameters. The direct eigenvalue analysis method based on the Fourier expansion and generalized eigenvalue analysis is applied to solve the parametrically excited stability problem of the stiffened beam. A simply supported beam with periodic stiffeners under periodic axial excitation is considered for numerical investigation. The parametrically excited stability of the stiffened beam and the effects of stiffeners and excitation on the stability are illustrated by numerical results on unstable regions.

Eigenvalue Analysis Approach to Random Periodic Parameter-Excited Stability: Application to a Stay Cable

AbstractA direct eigenvalue analysis approach to solving the stability problem of randomly and periodically parametrically excited linear systems is developed based on the response moment stability...

Description of Methods for the Eigenvalue Analysis of Railroad Vehicles Including Track Flexibility

The stability analysis of railroad vehicles using eigenvalue analysis can provide essential information about the stability of the motion, ride quality, or passengers’ comfort. The eigenvalue analysis follows three steps: calculation of steady motion, linearization of the equations of motion, and eigenvalue calculation. This paper deals with different numerical methods that can be used for the eigenvalue analysis of multibody models of railroad vehicles that can include deformable tracks. Depending on the degree of nonlinearity of the model and coordinate selection, different methodologies can be used. A direct eigenvalue analysis is used to analyze the vehicle dynamics from the differential-algebraic equations of motion written in terms of a set of constrained coordinates. As an alternative, the equations of motion can be obtained in terms of independent coordinates taking the form of ordinary differential equations. This procedure requires more computations, but the interpretation of the results is straightforward.

A state-space coupling method for fluid-structure interaction analysis of plates

A state-space coupling method is presented for the direct solution of fluid-loaded natural frequencies and mode shapes of plates. This method expands the frequency-dependent term in the Rayleigh integral in a power series on circular frequency. After factoring out the frequency term from the integrands, integration involved in setting up the acoustic impedance coefficient matrices is confined to a frequency-independent part. The acoustic impedance coefficient matrices are therefore frequency-independent and can be coupled directly with the structural matrix into a canonical state-space form to yield the fluid-loaded modes via the direct eigenvalue analysis. A fluid-loaded stiffened plate is involved to demonstrate the method. Numerical results of the fluid-loaded natural frequencies, mode shapes, and modal damping ratios are given to show the efficacy of the state-space coupling method.

An improved state-space method for coupled fluid–structure interaction analysis

This paper presents an improved state-space method for incorporating fluid–structure interaction into structural acoustic analyses. Giordano and Koopmann [J. A. Giordano and G. H. Koopmann, “State-space boundary element-finite element coupling for fluid–structure interaction analysis,” J. Acoust. Soc. Am. 98, 363–372 (1995)] introduced a state-space approach for the coupled analysis of fluid-loaded structures. Giordano’s method employed a third-order interpolation of the fluid impedance, coupled with state-space formalism, permitting the direct eigenvalue analysis of the coupled system, and thereby yielding the fluid-loaded modes and natural frequencies. This is in contrast to the common technique of sweeping through a range of frequencies so as to identify the natural frequencies and mode shapes. The original state-space approach yielded system matrices of order 4N, where N is the number of degrees of freedom in the underlying model. Where Giordano’s method employed an interpolation based on surface velocity, the work presented in this paper uses an interpolation based on surface displacement. This simple modification to the method yields a system order of 3N, with concomitant reduction in the required computational effort, storage, etc. We further present a method to assess the validity of the resonances obtained through this approximation technique. The method is demonstrated here for an infinite 1-D panel in a baffle, using the surface variational principle and the method of assumed modes to develop the coupled fluid–structure problem. This structure and the analysis methods are not fundamental to the state-space development, and are used solely as vehicles to demonstrate the new formulation.

On a direct eigenvalue analysis for locally modified structures

On a direct eigenvalue analysis for locally modified structures