Classical Modal Method Research Articles

Application of Fiber Optic Sensing System for Predicting Structural Displacement of a Joined-Wing Aircraft

This work aims to achieve real-time monitoring of strains and structural displacements for the target Joined-Wing aircraft. To this end, a Fiber Optic Sensing System (FOSS) is designed and deployed in the aircraft. The classical modal method, which is used for Strain-to-Displacement Transformation (SDT), is improved to adapt to different boundary conditions by introducing extra constraint equations. The method is first verified by numerical studies on a cantilever beam model and the high-fidelity finite element model of the Joined-Wing aircraft. Ground static tests are then carried out to further demonstrate the capability of the developed FOSS and SDT algorithm in practical application. The results have shown that the improved modal method is able to predict structural deformation under different boundary conditions by using only free–free modes. In addition, the errors between the predicted displacement and the reference in the ground test are within 10%, which proves the FOSS has reasonable accuracy and the potential for future flight tests.

MODAL METHOD BASED ON SUBSECTIONAL GEGENBAUER POLYNOMIAL EXPANSION FOR LAMELLAR GRATINGS: WEIGHTING FUNCTION, CONVERGENCE AND STABILITY

The Modal Method by Gegenbauer polynomials Expan- sion (MMGE) has been recently introduced for lamellar gratings by Edee (8). This method shows a promising potential of outstanding convergence but still sufiers from instabilities when the number of polynomials is increased. In this work, we identify the origin of these instabilities and propose a way to remove them. Among the numerical methods developed for the analysis of lamellar difiraction gratings, modal methods play an important role because of their great versatility and relative efiectiveness. In the classical modal method (1,2), the eigenvalues are obtained by solving a transcendental equation. In other modal methods, the eigenmodes and propagation constants are generally obtained by searching the eigenvalues and eigenvectors (3{8) of a matrix which is derived from the Maxwell's equations by using the method of moments (9). Mathematically speaking, for one-dimensional gratings with piecewise homogeneous media, and plane wave excitation, the eigenmodes are solutions of the Helmholtz equation subject to boundary conditions at the interfaces between two media and to the pseudo-periodicity condition. Numerically, the rate of convergence of the method depends on how the matrix from which eigenvalues are sought takes into account the continuity relations. Indeed, one of the main difierences between

Numerical method using filter banks for vibroacoustic analysis in the medium frequency domain

Efficient numerical tools are needed to predict the vibroacoustic response of a viscoelastic structure in the medium frequency range. The presented method is formulated within the framework of nonuniform modulated filter banks, a signal processing tool. The structure is described using a variational approach. The method consists of dividing the frequency domain of interest into several subdomains and hence solving a time domain equation of motion for each subdomain. The complete displacement field is then reconstructed from the different time domain solutions. This time-frequency computations rely on the design of analysis and synthesis filter banks. Flexural vibrations of plates in vacuo are studied to illustrate the numerical efficiency of the method. According to the variational approach that is used, results show that the proposed method allows the computation of the solution in frequency domains where classical modal methods and direct methods are limited from a numerical point of view. This numerical tool is of great interest for the study of structures having frequency-dependent properties.

Exponentially convergent and numerically efficient solution of Maxwell’s equations for lamellar gratings: comment

The conclusion reached in a recent paper by Morf [ J. Opt. Soc. Am. A12, 1043 ( 1995)], that the algorithm he proposed is several times faster than the R-matrix algorithm presented in an earlier paper of mine [ J. Opt. Soc. Am. A10, 2581 ( 1993)], is erroneous. By providing operation count, I show that the R-matrix algorithm, when it is applied to the classical modal method, is as efficient as Morf’s algorithm.

Highly accurate modal method for calculating eigenvector derivatives in viscous damping systems

A highly accurate modal superposition method for computing complex eigenvector derivatives in viscous damp- ing systems has been developed. When higher modes are truncated, the conventional modal superposition method cannot give an accurate solution, and the errors may become significant. In this paper, calculating the derivatives is regarded as calculating the structural response to harmonic exciting. Using multiple modal accelerations and shifted-poles, highly accurate results would be obtained when only few modes are used. All of the available modal superposition methods would be directly obtained from the proposed method. Numerical examples show that it achieves better calculation efficiency than all of the available modal methods and Nelson's method when more than one eigenvector derivative is of interest. Moreover, the presented method can be used to improve response calculations and substructure syntheses. IGENVECTOR derivatives with respect to system parameters are widely used in structural design, structural stability anal- ysis, finite element model updating, and structural control. Since the earlier work of Fox and Kapoor,1 many methods have been developed.2'3 All of the available methods could be categorized into three groups: modal method,1'3'4 direct method,1'4 and iter- ation method.5'6 Most of the iteration methods suffer from slow convergence rates and are less efficient. Recently, Ting6 improved the effectiveness of the method. Because the direct methods give out the exact solutions, they are widely used. However, when the derivatives of a large number of eigenvectors are demanded, the cal- culation is very time consuming. Here, the modal methods are more suitable. The modal methods employ a modal superposition idea; there- fore, the accuracy is dependent on the number of modes used in calculation. When the complete set of eigenvectors is used, the ex- act solution could be obtained. To guarantee the accuracy, the clas- sical modal method1 needs higher eigenvectors. Recently, Wang4 presented a modified modal method, in which a modal acceleration type approach was used to approximate the contribution of trun- cated higher modes. Therefore, the modified modal method con- verges faster than the classical method. Ma and Hagiwara7). im- proved the modified method with an appropriate shift value, better convergence than that of the modified method is achieved, and sys- tems having zero eigenvalues can be directly analyzed. To further improve the convergence, the improved modified modal method8'10 and accurate modal superposition method9 have been proposed by several researchers. The basic original idea of the aforementioned improved modal methods was found in the Hu's book.11 The idea was used to improve the convergency of dynamic flexibility in series form for no damping systems. All of the aforementioned methods were used for real eigen- vectors. For the cases of complex eigenvectors in viscous damp- ing systems, the three groups of methods were presented.12'13 The classical modal method for the complex eigenvectors, which was similar to the method presented in Ref. 1 for real eigenvectors, suffers from heavy modal truncation errors. Recently, Akgun14 presented a new family of modal methods for the calculation of the derivatives in non-self-adjo int systems. The modal accelera- tion approach was used to improve the convergence. In this paper, calculating the derivatives in viscous damping systems is regarded as calculating a structural response to a harmonic exciting, using multiple modal accelerations and shifted-poles; a highly accurate modal method for calculating the derivatives has been developed. There are clear mathematical and physical meanings in conduct- ing the proposed method, from which all of the available meth- ods would be directly obtained. Furthermore, fewer eigenvectors (even only the specific eigenvector that is being considered) are re- quired for the predetermined accuracy. As a result, the method is more efficient in computation than Akgun's method and Nelson's method when more than one eigenvector derivative is of inter- est. Numerical examples show that the proposed method is cor- rect and drastically improves solution accuracy. The method can directly be applied in systems with zero eigenvalues. Although the method is developed to calculate eigenvectors derivatives, it can be used to improve response calculations and substructure syntheses.

Vibration of a master plate with attached masses using modal sampling method

At medium frequencies, several problems deal with systems having a large number of modes where classical modal methods must cope with the solution of large linear systems. The objective of the modal sampling method is to reduce these systems of equations in order to obtain a simplified model. The vibration response of a structure is reconstructed using a limited number of modes, called a sample, to interpolate the response of other modes. The accuracy of the prediction can be improved by increasing the size of the sample. The method, which has been developed in the case of a long beam, is presented here in the case of a nonhomogeneous structure such as a large, thin plate where point masses have been added. The structure is excited by a harmonic point force or a planar acoustic wave. A study of the convergence of the method shows that global quantities such as space—and/or frequency—averaged energy converge quickly. Thus predictions can be obtained with models having few degrees-of-freedom. Moreover, results show the ability of the method to take into account a master plate with attached point mass heterogeneities.

Prediction of vibrations using a wave sampling decomposition

At medium frequency range (MFR), numerous vibroacoustic problems deal with systems of large number of modes where classical modal methods must cope with resolution of huge linear systems. In the idea of reducing these models, a mode hybridization method [H. J-P. Morand, ‘‘A modal hybridization method for the reduction of dynamic models in the medium frequency range,’’ Proceedings of the European Conference on New Advances in Computational Structural Mechanics (1991)] gives one equivalent mode obtained with modes of the complex structure. Another method [D. Trentin and J. L. Guyader, ‘‘Vibrations of a master plate with attached masses using modal sampling method,’’ J. Acoust. Soc. Am. (to be published)] based on reducing size of dynamic model in the MFR, gives a prediction of the vibration response of a structure by using a limited number of modes of the homogeneous master structure in order to interpolate the response of modes of whole structure with attached heterogeneities. The method presented here gives a prediction of plate vibration using a wave decomposition instead of modal calculation. The corresponding wave function of the Fourier transform of the displacement is approximated the best by achieving the extremalization of the Hamiltonian of the plate. Quadratic velocity of the structure is derived and the convergence study of the method shows that global quantities involving frequency average can be obtained with reduced models of small size. [Work supported by the Institut National de Recherche et de Sécurité and by the Centre National d’Etudes Spatiales.]

Multilayer modal method for diffraction gratings of arbitrary profile, depth, and permittivity: addendum

A numerically stable method is presented for the analysis of diffraction gratings of arbitrary profile, depth, and in conical mountings. It is based on the classical modal method and uses a stack of lamellar gratpermittivity to approximate an arbitrary profile. A numerical procedure known as the R-matrix propagation aling layers gorithm is used to propagate the modal fields through the layers. This procedure renders the implementation of this new method completely immune to the numerical instability that is associated with the conventional algorithm. Numerical examples including diffraction efficiencies of both dielectric and metallic propagation gratings of depths that range from subwavelength to hundreds of wavelengths are presented. Information about the convergence and the computation time of the method is also included.

Décomposition en paquets d'ondes pour le calcul d'énergie vibratoire de structures

At medium frequency range (MFR), numerous vibroacoustic problems deal with systems of large number of mixes where classical modal methods must cope with resolution of huge linear system. In the idea of reducing theses models a mode hybridization method proposed in [1] gives one equivalent mode when modes of the complex structure are known. Another method [2],[4] based on reducing size of dynamic model in the MFR, gives a prediction of the vibration response of a structure by using a limited number of modes of the homogeneous master structure in order to interpolate the response of modes of whole structure with attached heterogeneities. The method presented here gives a prediction of plate vibration using a wave decomposition instead of modal calculation. It allows one to consider homogeneous structure of complex geometry without making mode calculations before. The corresponding wave function of the Fourier transform of the displacement is approximated the best by achieving the extremalization of the Hamiltonian of the plate. Quadratic velocity of the structure is derived and the convergence study of the method shows that global quantities involving frequency average can be obtained with reduced models of small size

Multilayer modal method for diffraction gratings of arbitrary profile, depth, and permittivity

A numerically stable method is presented for the analysis of diffraction gratings of arbitrary profile, depth, and in conical mountings. It is based on the classical modal method and uses a stack of lamellar gratpermittivity to approximate an arbitrary profile. A numerical procedure known as the R-matrix propagation aling layers gorithm is used to propagate the modal fields through the layers. This procedure renders the implementation of this new method completely immune to the numerical instability that is associated with the conventional algorithm. Numerical examples including diffraction efficiencies of both dielectric and metallic propagation gratings of depths that range from subwavelength to hundreds of wavelengths are presented. Information about the convergence and the computation time of the method is also included.

Convergence of the modal sampling method in the medium frequency range

At medium frequencies, several problems deal with systems with a large number of modes. This is encountered, for instance, in the aerospace field. In that case, the evaluation of the vibration behavior using classical modal methods must cope with resolution of huge linear systems and specific methods have to be developed. The modal sampling method, which has been presented in the case of a long beam in longitudinal motion [J. L. Guyader, J. Acoust. Soc. Am. 88, 2269–2276 (1990)], enables one to reconstitute the vibration response of a structure using a limited number of groups of modes, called a sample. The accuracy of the prediction can be improved by increasing the size of the sample. The modal sampling method is presented here in the case of a large, thin, rectangular plate where point masses can be added. The structure is excited by an harmonic point force or a plane acoustic wave and kinetic energy on the structure can be derived. A study of the convergence of the method is presented when the size of the sample is increased and results show the ability of the method to take into account structural heterogeneities. [Work supported by the Institut National de Recherche et de Sécurité and by the Centre National d’Etudes Spatiales.]

Prediction of sound fields in rooms of arbitrary shape: Validity of the image sources method

In order to find the sound pressure field in a rectangular room, the classical modal method may be applied. However, it does not apply to polyhedric rooms of any shape, to which, however, the image sources method may be applied. The two methods are compared for several geometric shapes: two parallel planes (a layer), a rectangular wave guide and a rectangular enclosure. It is shown under what conditions the image sources method can give accurate enough results. A comparison of theoretical and experimental results is presented.