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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.

This paper proposes a hypothesis for explaining Pt/C particles’ coarsening inside the catalyst layers of a PEM fuel cell. The hypothesis includes the two parts: (1) due to capillary condensation a water-bridge could be formed between two neighboring nano-scale Pt/C particles at relative humidity under 100% when the surfaces of the Pt/C particles are hydrophilic; (2) the capillary force of the water-bridge tends to pull together the Pt/C particles. The relation is derived in this paper between the capillary force and the factors including the diameter of Pt/C particles, relative humidity, temperature, distance between the two neighboring Pt/C particles and water contact angle. A parametric study is performed showing some details about water-bridge formation. Finally, the stress level induced by the capillary force inside the Nafion thin-film connecting with the Pt/C particles is calculated. The result shows that the capillary force could be large enough to break apart the Nafion thin-film, facilitating the movement of Pt/C particles towards each other.

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.

The Electro-rheological (ER) fluid is a new kind of smart material which has been applied to various engineering problems. Using the viscous and bilinear hysteretic damping model, this paper discusses the nonlinear frequency spectrum by iterative perturbation method for nonlinear vibration system with ER damper. Iterative perturbation is based on the matrix perturbation. The initial frequencies are found by subspace iteration. The subsequent frequency spectrum of nonlinear states, which is based upon the tangent stiffness of the structure, is calculated by the iterative perturbation method. A numerical example is given to illustrate the method presented by this paper.

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.

Determining the steady state responses of a structural system, to a harmonic excitation, is an important problem. The modal superposition method is often used, although it cannot always give an exact solution when the mode truncation is serious or the excitation frequency is larger than the frequencies of the truncated modes. Fast and accurately evaluating the contribution due to the truncated modes to the response of harmonic excitation is put forward in reference by Liu et al. [Modal Anal: Int J Exp Modal Anal 9 (1994) 2] without using the truncated modes. The present paper suggests a modification to the approach of Liu et al., in that, it uses eigenvalue shift approach to enhance convergence when the frequency of the harmonic excitation is larger than the frequency of the truncated modes. Three numerical examples are given to demonstrate the validity and the effectiveness of the present method.

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.

In this paper, we focus on the control problem of defective systems with repeated eigenvalues. Using the singular-value decomposition of input matrix B and output matrix C in the modal subspace, the quantitative measures of modal controllability and observability of defective systems are discussed. For a near-defective system with close eigenvalues, we first transform it into a defective one, and then apply the same method to deal with near-defective systems. Numerical examples for a general damping system and an airfoil are given to illustrate the applications of the present method.

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.

Based on the usual perturbation and Pade approximation, a new eigenvalue reanalysis method for modified structures is developed in this paper. By this method, the accuracy of the eigenvalues and varying ranges of the parameters of the structures are improved. As an application of the method, a numerical example is given and the second-order perturbation method and the proposed method are compared in the accuracy of eigenvalues and range of parameters. The results show that the present method is effective.

A technique for determining the locations of piezoelectric sensors and actuators (S/As) of intelligent structures on the basis of element sensitivities of the singular-values of the controllability and observability is presented. The element sensitivity of the singular-values is used as the measure to identify the best locations of S/As, which is defined from the 1st order perturbations of singular-values of the controllability and observability. A criterion and a threshold value are used to determine which actuator/sensor elements need to be kept/removed based on the sensitivity information. Locations of S/As are determined as the point where the element sensitivity of the singular-values is the maximum. Two numerical examples, the cantilever beam and the model of solar panels, are given to illustrate the applications of the present method.

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.

The dynamic mechanical properties of an electro-rheological (ER) fluid composed of polyaniline particles in silicon oil is studied under the shearing state. By applying sinusoidal signal with different frequencies and different shear strain amplitudes, the response of ER fluid damping force has been measured and analyzed. The result shows that the damping force of ER fluid is linear without electric field, and is nonlinear when an electric field is applied. By using the frequency spectrum analysis of the response, we found that the harmonics frequencies of the damping force are odd multiples of the fundamental frequency, and the damping force has bilinear hysteresis property. A simple damping model of ER fluid is given in this paper. The comparisons of the experimental results with those of the numerical simulation show that the model of ER fluid setting up in this paper is reasonable.

A new approach for simultaneous optimization of intelligent structures and the control system is presented. Conventional structural sizing variables and elements of the feedback gain matrix are both treated as independent design variables in design. The mass and the vibration control performance index of intelligent structures are chosen as objective functions, and constraints on real parts of eigenvalue, imaginary parts of eigenvalue of closed-loop system and actuator control forces are included. Simultaneous optimization of the intelligent structure and the control system is converted into multiobjective optimization. A numerical example is presented to prove the validity of the present approach.

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.

A finite element formulation for vibration control and suppression of intelligent structures with a new piezoelectric plate element is presented. On the basis of a negative velocity feedback control law, a general method of active vibration control and suppression for intelligent structures is put forth. Dynamic stability and the effect of vibration control for intelligent structures are investigated by introducing the state space equations of intelligent structures. The damped frequencies as well as the damping ratio are derived by state space analysis. The procedure is illustrated with the help of two numerical examples. The purpose of the first example is to check the accuracy of the present finite element solution with the analytical one. The second example is to study the problem of active vibration control and suppression for intelligent structures.

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 probabilistic finite element method (PFEM) to vector-valued and matrixvalued functions using techniques from matrix calculus and Kronecker algebra, and presents the PFEM in Kronecker notation for linear and nonlinear continua. The results derived are easily amenable to computational procedures.

For laminated composite plates having a relatively low transverse shear modulus, Kirchhoff theory is inadequate for calculating the eigenproblems and sensitivity of laminated plates. In this paper the first-order shear deformation theory is used for symmetrically laminated plates. The orientation angle of the fibres, material parameter and the vectorial distance from the middle-surface to upper surface of each ply are treated as design variables, and sensitivity of eigensolutions for the symmetrically laminated plates with respect to the design parameters are studied.

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.

A probabilistic algorithm for calculating the statistical properties of response due to uncertainties in geometry and material properties for geometrically nonlinear structural dynamics is presented. It is shown here that this technique is readily applicable to geometrically nonlinear small strain dynamic problems in which the equations of motion of structures are derived using the finite element method. A general formulation for problems of this type is presented. The efficiency of the method is illustrated by application to nonlinear vibration of a cantilever beam with random parameters.

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 Technical Brief presents a new method for vibration analysis of a non-classically damped system. The basic idea is to introduce a transformation, which bears clear physical meaning, so that the original non-classical damped system is transformed into a new 2nd-order system that does not have the damping term. The transformed system not only provides an alternative of calculating response, but also reveals more clearly vibration behaviors of the original system.

A computational method for design sensitivity analysis of the non-linear forced vibrations response of beams is presented. The large deflection leads to geometric non-linearity. Longitudinal displacement and inertia are considered in the non-linear vibration formulation. As an application of this method, two numerical examples are given. The results show that the method proves to be effective.

A method to estimate the statistical properties of a geometrically nonlinear mode spectrum for large amplitude forced vibration systems is presented. It is shown here that this technique is readily applicable to geometrically nonlinear small strain problems in which the equations of motion of structures are derived using the finite element method. A general formulation for problems of this type is presented. The efficiency of the method is illustrated by its application to a nonlinear vibration of an elastic beam.

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.

Plastic instability and flow localization in the plane-strain tension of ductile metals are studied numerically using the J2 deformation theory. The effects of the strain-hardening parameter n and of an initial imperfection φ0 on the rupture patterns are investigated in detail. A rupture-mechanism diagram is presented, giving the scheme of diffusing and localizing instabilities with changes of the values of n and φ0

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.

The sensitivity analysis of generalized modes of linear vibration defective systems is considered. By using the Puiseux expansion of perturbed eigensolutions of the defective systems, an efficient numerical method of the sensitivity analysis of the generalized modes is presented. An application of this method is illustrated by an example.

A practical and effective method, which is used to predict the errors due to manufacture of the structural parameters, is presented using sensitivity analysis, random perturbation, and generalized inverse matrix theory. This method is based on the statistics of the natural frequencies of the structure and can be applied to the complex structure which includes both distinct and multiple eigenvalues. A numerical example is given to illustrate the application of the method.

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.