Second-order Matrix Differential Equation Research Articles

Oscillation criteria for second-order matrix dynamic equations on a time scale

We obtain oscillation criteria for a second-order self-adjoint matrix differential equation on a measure chain in terms of the eigenvalues of the coefficient matrices and the graininess function. We illustrate our results with some nontrivial examples.

A Remark on the Theory of Periodic Solutions of a Lyapunov Type Second-Order Matrix Differential Equation

A Remark on the Theory of Periodic Solutions of a Lyapunov Type Second-Order Matrix Differential Equation

Transient Response Analysis of Nonlinear Modally Coupled Structural Dynamics Equations of Motion

The solution of the structural dynamics matrix differential equations of motion in the presence of general structural damping, aerodynamic stiffness and damping, and nonlinear generalized forces is discussed. An intuitive and easy-to-use method of Picard iteration by Dawson is shown to provide exponential convergence to the exact solution of quasilinear second-order matrix differential equations with general modal coupling, and the first mathematical proof of convergence is presented. It is demonstrated, for the special case of general structural damping without aerodynamic forces but with forcing functions that depend only on time, that the method of Picard iteration agrees to within machine accuracy with the method of linear modal velocity of Henkel and Mar. Thomson's coupled damping problem is solved using both methods and percent errors resulting from ignoring modal coupling are quantified. Possible areas of future research are discussed.

Transient response analysis of nonlinear modally coupled structural dynamics equations of motion

The solution of the structural dynamics matrix differential equations of motion in the presence of general structuraldamping,aerodynamicstiffnessanddamping,andnonlineargeneralizedforcesisdiscussed.Anintuitive and easy-to-use method of Picard iteration by Dawson is shown to provide exponential convergence to the exact solution of quasilinear second-order matrix differential equations with general modal coupling, and the e rst mathematical proof of convergence is presented. It is demonstrated, for the special case of general structural damping without aerodynamic forces but with forcing functions that depend only on time, that the method of Picard iteration agrees to within machine accuracy with the method of linear modal velocity of Henkel and Mar. Thomson’ s coupled damping problem is solved using both methods and percent errors resulting from ignoring modal coupling are quantie ed. Possible areas of future research are discussed.

Laguerre matrix polynomials and systems of second-order differential equations

In this paper we introduce the class of Laguerre matrix polynomials which appears as finite series solutions of second-order matrix differential equations of the form tX″ +( A + I − λtI) X′ + CX = 0. An explicit expression for the Laguerre matrix polynomials, a three-term matrix recurrence relation, a Rodrigues formula and orthogonality properties are given.

Summability Methods for Oscillation of Linear Second-Order Matrix Differential Equations

Summability Methods for Oscillation of Linear Second-Order Matrix Differential Equations

Solving second-order matrix differential equations with regular singular points avoiding the increase of the problem dimension

In this paper we propose a method for solving second order matrix differential systems of the form t 2X”+tA(t)X′+B(t)X=0 , where A and B are analytic matrix functions in a neighborhood of t = 0. Series solutions of the problem are constructed without increasing the problem dimension. By introducing the concept of a fundamental set of solutions of the above equation, an analogous result to the Frobenius theorem to solve the corresponding scalar equation is presented.

Finite moments perturbations of 𝑦”=0 in Banach algebras

Rigorous asymptotics for a basis of y + g ( x ) y = 0 , x ∈ [ 1 , + ∞ ) y + g(x)y = 0,\;x \in [1, + \infty ) , is derived in the framework of Banach algebras. The key assumption is ∫ 1 + ∞ x k | | g ( x ) | | d x > ∞ \int _1^{ + \infty } {{x^k}} ||g(x)||dx > \infty for k = 1 k = 1 or k = 2 k = 2 . Such results improve and generalize previous work on linear second-order matrix differential equations.

Model reduction and control of flexible structures using Krylov vectors

Krylov vectors and the concept of parameter matching are combined together to develop a model reduction algorithm for a damped structural dynamics system. The obtained reduced-order model matches a certain number of low-frequency moments of the full-order system. The major application of the present method is to the control of flexible structures. It is shown that, in the control of flexible structures, three types of control energy spillover generally exist: control, observation, and dynamic. The formulation based on Krylov vectors can eliminate both the control and observation spillovers while leaving only the dynamic spillover to be considered. Two examples are used to illustrate the efficacy of the Krylov method. I. Introduction A MAJOR difficulty in the control of flexible structures or any other large-scale system is, in the words of Bellman, the curse of dimensionality. A flexible structure is, by nature, a distributed-pa rameter system, and, hence, it has infinitely many degrees of freedom. Even approximate structural models obtained by some discretization approach are generally still too large for use in control design applications. Therefore, model order reduction plays an indispensable role in the control of flexible structures. Usually, model reduction of a structural dynamics system is performed by the RayleighRitz method, which transforms the system equation to a smaller scale by using a projection subspace. It is indisputable that the choice of projection subspace is important to the accuracy of the reduced model. The eigensubspace, or the normal mode subspace, is frequently used for projection because it has a clear physical meaning and can preserve the system natural frequencies. However, with regard to the accuracy of system response, numerical experience has shown that preservation of the natural frequencies is usually not the only • concern. Other than normal modes, there are other static modes, e.g., constraint modes, attachment modes, and inertiarelief modes, which are frequently used in component mode synthesis.1 In this paper, Krylov vectors, which can be considered as static modes, are used for model reduction. There has been quite a bit of research concerning the convergence and efficiency of Krylov vectors in application to eigenvalue analysis and to the structural dynamics model reduction problem.2'5 Krylov vectors are also efficient when employed in general linear system and controller reduction problems.6'7 The major purpose of this paper is to discuss the possible application of Krylov vectors to controller design for flexible structures. The structural dynamics system studied here is described by a second-order matrix differential equation together with an output measurement equation. To perform model reduction to a structural dynamics system in this input-output configuration, the concept of parameter matching for general linear system model reduction is adopted. Parameter matching constitutes a class of efficient methods for model order reduction

An Algorithm for Computing the Matrix Cosine

We present and analyze an algorithm for computing the cosine of a matrix which is based on the double-angle formula $\cos 2A = 2\cos ^2 A - I$. We discuss the relevance of this computation to second-order matrix differential equations. We draw the analogy between this method for the cosine and the familiar scaling and squaring method for computing the matrix exponential. Numerical experiments employing polynomial and rational approximations to the cosine, in conjunction with the double angle technique, are presented.

Explicit Formulas for Solutions of the Second-Order Matrix Differential Equation Y″ = ay

(1975). Explicit Formulas for Solutions of the Second-Order Matrix Differential Equation Y″ = ay. The American Mathematical Monthly: Vol. 82, No. 2, pp. 159-162.