Search

The feedback control of Hopf bifurcation of nonlinear aeroelastic systems with asymmetric aerodynamic lift force and nonlinear elastic forces of the airfoil is discussed. For the Hopf bifurcation analysis, the eigenvalue problems of the state matrix and its adjoint matrix are defined. The Puiseux expansion is used to discuss the variations of the non-semi-simple eigenvalues, as the control parameter passes through the critical value to avoid the difficulty for computing the derivatives of the non-semi-simple eigenvalues with respect to the control parameter. The method of multiple scales and center-manifold reduction are used to deal with the feedback control design of a nonlinear system with non-semi-simple eigenvalues at the critical point of the Hopf bifurcation. The first order approximate solutions are developed, which include gain vector and input. The presented methods are based on the Jordan form which is the simplest one. Finally, an example of an airfoil model is given to show the feasibility and for verification of the present method.

Based on Neumman series and epsilon-algorithm, an efficient computation for dynamic responses of systems with arbitrary time-varying characteristics is investigated. Avoiding the calculation for the inverses of the equivalent stiffness matrices in each time step, the computation effort of the proposed method is reduced compared with the full analysis of Newmark method. The validity and applications of the proposed method are illustrated by a 4-DOF spring-mass system with periodical time-varying stiffness properties and a truss structure with arbitrary time-varying lumped mass. It shows that good approximate results can be obtained by the proposed method compared with the responses obtained by the full analysis of Newmark method.

Using the convex model theory, the vibration control problem of structures with uncertain parameters is discussed, which is approximated by a deterministic one. A method for estimating the upper and lower bounds of eigenvalues of the closed-loop systems is presented by combining the matrix perturbation and optimization. The present method is applied to a vibration system to illustrate the application. The numerical results show that the present method is effective.

An adaptive iteration algorithm is presented in this paper for structural modal reanalysis of topological modifications. The rules of adaptation are given. The method is based on matrix perturbation and can be implemented easily on the computer. A little computation effort is involved, which is very important for topological modification with the increase of the joints and the number of degrees of freedom. In order to illustrate the method, three examples are given. The results show that the proposed method is effective for structural modal reanalysis of topological modifications. Copyright © 2002 John Wiley & Sons, Ltd.

A new method for solving the uncertain eigenvalue problems of the structures with interval parameters, interval finite element method based on the element, is presented in this paper. The calculations are done on the element basis, hence, the efforts are greatly reduced. In order to illustrate the accuracy of the method, a continuous beam system is given, the results obtained by it are compared with those obtained by Chen and Qiu (1994); in order to demonstrate that the proposed method provides safe bounds for the eigenfrequencies, an undamping spring-mass system, in which the exact interval bounds are known, is given, the results obtained by it are compared with those obtained by Qiu et al. (1999), where the exact interval bounds are given. The numerical results show that the proposed method is effective for estimating the eigenvalue bounds of structures with interval parameters.

This study discusses a modal optimal control procedure for defective systems with repeated eigenvalues. From the view point of mathematics, although near defective close eigenvalues are distinct, the characteristic of the system is also defective. Therefore, we have to transform near defective systems into the defective one, and then modal optimal control procedure for the defective systems can be extended to deal with the corresponding problems for near defective systems with close eigenvalues. Because of the defective characteristic of the system, we have to use an invariant sub-space recursive method with numerical stability to calculate the generalized modes of the defective and near defective systems. The Potter's approach is extended to solve the Riccati equations in the generalized model subspace of the defective system. Because the order of the Jordan block matrix of the defective eigenvalues, m, is much smaller than that of the state matrix, n, i.e.,m ⪡n, the present modal optimal control procedure is very simple and reduces the computing effort for the complex system with large number of degrees of freedom. A numerical example is given to illustrate and verify the validity of the procedure.

This paper deals with obtaining accurate eigenvector derivatives in damped vibration systems with modal truncation method only by system matrices and a few available lower modes. A numerical example shows that the method of this paper is correct and efficient.

A finite element formulation for the structure with distributed piezoelectric sensors and actuators (S/As) is presented. Design sensitivity analysis of eigenmodes and dynamic responses for intelligent structures are investigated. A new method of design sensitivity analysis for intelligent structures, the finite element perturbation method, is developed. As an application of this method, a numerical example is given. The results show that the method proves to be effective.

A finite element formulation is presented for modeling the plate structure containing distributed piezoelectric sensors and actuators (S/As). A new plate bending element for analysis of the plate with distributed piezoelectric S/As is developed. This element saves much memory and computation time. Using the bending plate element, a general method of static shape control for the intelligent structure is put forth. Two examples are given to illustrate the application of the method presented in this paper. The purpose of the first example is to check the accuracy of the finite element method presented in this paper. The second example is to study the problem of the static shape control for the intelligent structure. It is concluded that the shape of the intelligent structure can reach the desired shape through passive control or active control.

The modal superposition method is one of the methods often used to extract the response to periodic excitations. It cannot, however, give an exact solution, and may sometimes lead to meaningless results if the truncating of modes is serious. Fast and accurate computation of the contribution of the truncated higher frequency both of lower and higher frequency modes to the response, without using the unavailable truncated modes, is the objective of this paper. An explicit expression on the contribution from these unavailable modes to the solution is presented in this paper in order to improve the solution accuracy. Using this explicit expression, the response to periodic excitations can be computed accurately and conveniently with little extra computational cost. Numerical examples are given to illustrate the effectiveness of the present method.

This paper deals with the correction of random errors of structural frequencies and the optimal selection of parameters to be modified. The concepts of ideal parameter to be modified, optimal parameter to be modified and the optimal parameter group (in which parameters to be modified are all optimal) are introduced. When the number of the parameters to be modified is given, the optimal parameters are selected from all design parameters so that the expectation of the root-mean-square of the corrected frequency errors takes its minimal value, and the modifications of those optimal parameters are also obtained. The formula to calculate this minimal value is given. Furthermore, the minimal number of parameters to be modified in the optimal parameter group is obtained when the number of parameters to be modified is unknown. Finally, a 15-element truss structure is used to illustrate the application of the proposed methods.

The modal superposition method is often used for computing the perturbation of eigenvectors in structural modification and model correction. However, it will bring about significant errors in the solution when the high-frequency modes are truncated. This paper presents a new method, which uses known modes construct a new basis of the N-dimensional Euclidean space (say, the mixed-basis), to calculate the first and second order perturbations of the known eigenvectors. In the present method only the known modes are used. The accuracy of this method not only has no relation to number of the truncated modes but is better than the truncated modal superposition method, in which only the known modes are employed. A numerical example of a truss structure with 36 degrees of freedom is given to illustrate the effectiveness of the method.

This paper extends the stochastic perturbation method to vector-valued and matrix-valued functions. The numerical method for the response and reliability of uncertain structures is formulated using matrix calculus, Kronecker algebra and perturbation theory. Random variables and system derivatives are conveniently arranged into two-dimensional matrices and generalized mathematical formulae are obtained. The results derived are easily amenable to computational procedures.

An efficient and practical procedure based on the generalized Schmidt orthogonalization is presented to compute the eigenvector derivatives when repeated eigenvalues exist

This paper deals with eigenvalue problems involving uncertain but non-random interval stiffness and/or mass matrices. If one views the deviation amplitude of the interval matrix as a perturbation around the nominal value of the interval matrix, one can solve the standard eigenvalue problem of the interval matrix by applying the interval extension to the matrix perturbation method. In this study, the interval perturbation approximating formula is presented for evaluating interval eigenvalues of the interval matrix. Inextensive computational effort is a characteristic of the proposed method. The illustrative numerical examples are provided.

In this paper, we present a method for computing upper and lower bounds of natural frequencies of the structures with uncertain parameters. These parameters are unknown except for the fact that they belong to given bounded sets. The set of possible system states can be described by interval matrices. By solving the interval matrix problem, we obtain the bounds on frequencies of the structure. The numerical results demonstrates the efficacy of the method.

A structure subjected to interval parameters and interval loads which are unknown, except for the fact that they belong to given intervals, is studied and the displacement bound estimation is given. These parameters are uncertain, and yet they are not treated as being random since no information is available on their probabilistic characteristics. A set of possible states of the system is described by interval matrices. The perturbation method is employed as a simple analytical tool for determining static responses-interval displacements of finite element equilibrium equations with interval parameters. The numerical results show that the proposed method is extremely effective.

This paper presents a new method, termed modal perturbation combined mode superposition and matrix perturbation, of nonlinear response analysis for large deflection forced vibration of beams subjected to periodic loading. Longitudinal displacement and inertia are both considered in the formulation. As an application of the method, a numerical example is given. The results show that the modal perturbation proves to be effective in this class of problem.

When the parameters of structures are uncertain, structural natural frequencies become uncertain. In this paper, we deal with the vibration problem of structural parameters with interval uncertainty; the eigenvalue problem of the structures with interval uncertain parameters is transformed into two different eigenvalue problems to be solved. The Rayleigh quotient iteration method is applied to the vibration problem of the structures with interval parameters; the numerical results show that the proposed method is sufficiently accurate and requires little computational effort.

Since the formulations proposed in this paper are based on the perturbation approach and similar to the deterministic sensitivity analysis formulations, the stochastic sensitivity analysis scheme can easily be adapted to fit into the existing finite element programs. Furthermore, because only the first-order deterministic sensitivity is involved and a set of uncorrelated random variables is employed, the stochastic sensitivity analysis proposed does not greatly increase the computational cost compared to deterministic analysis. The numerical results show that the proposed method is acceptable when the coefficient of variation is not too large.

In this paper, we will investigate the method for computing the upper and lower bounds on frequencies of structures with bounded uncertain (or interval) parameters. The stiffness matrix and the mass matrix of structures, whose elements have the initial errors, are unknown except for the fact that they belong to given bounded matrix sets. The set of possible matrices can be described by the interval matrix. By means of the stationary condition of Rayleigh Qutient and the minimax theorem of eigenvalues, the generalized eigenvalue problem of structures with bounded uncertain parameters can be transformed into two different generalized eigenvalue problems. The numerical results indicate that the proposed method is effective.

The paper presents a probabilistic boundary element method for analysis of structural responses to static loading, when the shape parameters of structures are considered as random variables. We can use this method to evaluate the mean values and standard deviations of the responses, and estimate the stochastic errors of structural character resulting from manufacturing errors.

A method is presented for computing the upper and lower bounds of the eigenvalues for structural vibration systems with interval parameters. As an indication of the effectiveness of this method, a numerical example is presented. The numerical example shows that the method proves to be both effective and valid.

In this study a perturbation method for computing the upper and lower bounds of eigenvalues of structural vibration systems with interval parameters is presented. The eigenproblem of the uncertain (interval) structure is expressed by equations consisting of the uncertainties. If the uncertainity is small in parameters, the eigensolutions of uncertain structures can be obtained efficiently by the perturbation method presented. A numerical example is given to demonstrate the validity of the method.

A description and computation of structural systems containing unknown-but-bounded parameters is considered in this paper. A method of solving the extrema of static responses of a structural system with uncertainties to deterministic forces using optimization techniques is presented. A numerical example to show the validity of the method is also given.

A new eigensolution reanalysis method for modified structures is developed and presented based on the usual first-order perturbation scheme and the Rayleigh quotients. By this method, the accuracies of the eigensolutions of the modified structures computed by perturbation methods are improved. A numerical example is presented and the accuracies of the usual first-order perturbation method, second-order perturbation method, William B. Bickford method and the proposed method are compared.

This paper focuses on how parameter changes in a vibration system expressed by a discrete eigenvalue problem affect vibration modes in the special case of closely spaced eigenvalues, i.e. ‘eigenvalue clusters’. For this class of problems, a perturbation analysis is developed that can be used to compute changes in eigenvalues and eigenvectors in response to small parameter changes. The analysis is applied to a 6-degree-of-freedom spring-mass system containing a single pair of closely spaced eigenvalues.

This paper deals with the non-linear eigenvalue problems arising from dynamic finite element and dynamic reduction in structural analysis. Based on matrix perturbation, a new method, the iterative perturbation method, is presented to solve the non-linear eigenproblem. As an application of the new method, numerical examples are given. The results show that this method proves to be effective.

Regarding the shape changes of a structure as the perturbation of its surface nodes within the neighbourhood of the given values, this method uses the perturbation formula to obtain the eigensolutions of the modified system without solving the generalized eigenvalue problem

Conclusions Aperturbationmethodusingacorrectionintransformationispresented to improve the condensation solution. The method proved to be effective and is recommended as a postprocessor of the condensation. Theoretically, the perturbation solution should approach the exact eigenvalue from above. In practice, however, truncation error in the linear equation causes the approximation to overshoot the exact solution. Inclusion of several of the lowest terms can provide excellent estimations for both the eigenvalue and mode shape changes. One of the most dife cult problems is the rate of convergence in the series expansion. The sequential terms in the ine nite series may indicate the convergence characteristics in an eigenvalue approximation. Some of the nonlinear terms may be included to get more accurate solutions. Rather than iterations with nonlinear perturbation equations, integrated reduction methodssuch as the hybrid dynamic condensation are preferable.