Matrix Ordinary Differential Equations Research Articles

A Continuous-Time Approach to the Oblique Procrustes Problem

In the paper proposed we will make use of the gradient flow approach to consider a generalization of the well-known oblique Procrustes rotation problem, involving oblique simple structure rotation of both the core and component matrices resulting from three-mode factor analysis. The standard oblique Procrustes rotations to specified factor-structure and factor-pattern follw as special cases. The approach adopted leads to globally convergent algorithm and includes solving of initial value problem for certain matrix ordinary differential equation. Necessary conditions are established for the solution of the problem. The same approach is extended easily to the weighted oblique Procrustes rotation. Finally, some simulated numerical results are given and commented.

Dynamic interaction between the vehicle and simply supported curved bridge deck

Vehicle—guideway interaction for a vehicle on simply-supported curved bridge decks is studied with the effects of vehicle speed and guideway parameters taken into account. In the dynamic model, the vehicle is considered to have four-degrees of freedom, with lumped masses, linear springs and dampers. Although the material and geometric linearities are preserved in the formulation, the consideration of interaction between the vehicle and the bridge results in a nonlinear structural response. The finite element method is used for solving the governing differential equation of the vehicle—bridge system. In this paper, multi-predictor—corrector procedure in connection with the Newmark method, is employed for the solution of the resulting matrix ordinary differential equations. The deflection at the middle of the beam is determined for different vehicle speed and different radius of curvatures of the curved deck.

Further results on light propagation in helicoidal bianisotropic mediums: oblique propagation

The recent fabrication of thin-film helicoidal bianisotropic mediums (HBMs) has created an impetus for understanding their electromagnetic response. Using the Oseen transformation, we have derived a 4×4 matrix ordinary differential equation for oblique (i.e. non–axial) propagation in general linear HBMs. A perturbational technique is presented for almost-axial propagation; and two other techniques—one semi–analytic and the other analytic—are presented for general oblique propagation. All three techniques yield exact closed-form results for axial propagation. Our use of the Oseen transformation amounts to viewing oblique propagation not by itself but as a graft on axial propagation, and is thus responsible for the stability of all three techniques.

Macroscopic model of second-harmonic generation in a dielectric thin-film helicoidal bianisotropic medium

The oblique propagation problem in a dielectric thin-film helicoidal bianisotropic medium (TFHBM), which has a reference uniaxial basis and exhibits second-harmonic generation (SHG), is solved. First, the linear constitutive relation is set up, followed by a constitutive relation to describe SHG. Then, a homogeneous 4 × 4 matrix ordinary differential equation is solved for the linear electromagnetic field phasors, and an inhomogeneous 4 × 4 matrix ordinary differential equation is solved for the second-harmonic electromagnetic field phasors. Extension to more general nonlinear HBMs shall follow the same route, provided the angular frequency of the nonlinear field generated is different from the angular frequencies of the linear fields that are needed to generate the nonlinear field. © 1996 John Wiley & Sons, Inc.

Solution of linear partial differential equations by Lie algebraic methods

A new algorithm is proposed for obtaining explicit solutions of the Cauchy problem defined by a certain class of partial differential equations (PDE) of parabolic type. The algorithm exploits the algebraic structure of the problem to transform the PDE into an ordinary matrix differential equation, which is then solved by Lie algebraic techniques.

Doubly asymptotic approximations for transient poroelastodynamics

A doubly asymptotic approximation (DAA) is an approximate temporal impedance relation at the boundary of a continuous medium. Here, first- and second-order DAAs are formulated for an infinite external poroelastic medium described by Biot’s equations. As with their acoustic, elastodynamic, and electromagnetic predecessors, the poroelastodynamic DAAs approach exactness in both the early-time (high-frequency) and late-time (low-frequency) limits, effecting a smooth transition between. They also lend themselves to straightforward boundary-element discretization, producing matrix ordinary differential equations in time that are readily solved by numerical integration. An initial examination of poroelastodynamic DAA accuracy is presented for two problems with spherical symmetry.

The influence of roadway surface irregularities and vehicle deceleration on bridge dynamics using the method of lines

Abstract In this paper, an analysis of vehicle-bridge interaction problems, taking into account the dynamic effects induced by vehicle bouncing due to roadway surface irregularities and varying vehicle speeds, is presented. The problem formulation and solution methodology include the effects of vehicle suspension systems, the vehicle travelling path, and the arbitrary boundary conditions and geometry of the bridge superstructure. The kinematic coupling reflecting the dynamic interaction between the traversing vehicle and the vibrating bridge superstructure is explicitly treated in the problem. Vehicle models used in this study take into account the stiffness and damping characteristics of vehicle suspension and tires, and permit the inclusion of a variety of truck configurations. The spatial discretization of the governing differential equation of the vehicle-bridge system is based on the finite element method. The resulting matrix ordinary differential equation is then discretized with respect to time on the basis of the Newmark method. The process of reduction of a partial differential equation into a matrix ordinary differential equation through a spatial discretization procedure, and then solving this expression using a time integration technique, is referred to as the numerical method of lines. Since the consideration of the vehicle-bridge interaction introduces non-linearity into the problem formulation, a multipredictor-corrector scheme is adopted in the solution procedure to obtain accurate results. Numerical examples, illustrating the influence of roadway surface irregularities and vehicle deceleration on the dynamic response of bridge structures, are provided to illustrate the validity and efficiency of the proposed formulation and solution methodology.

Numerical method of lines for analysis of vehicle-bridge dynamic interaction

In this paper, a finite element-based formulation and implementation of vehicle-bridge dynamic interaction problems are presented. A variety of vehicle models are considered. The problem formulation takes into account the dynamic effects of multiple vehicles traversing on multiple lanes, with arbitrary and variable speeds and directions. The consideration of various vehicle traversing conditions allows for the interaction between vehicles and the bridge superstructure, as well as among vehicles, to be represented in a more realistic manner. The spatial discretization of the governing partial differential equation of the vehicle-bridge system is based on the finite element method. The resulting matrix ordinary differential equation is then discretized with respect to time on the basis of the Newmark method. The process of reduction of a partial differential equation into a matrix ordinary differential equation through a spatial discretization procedure, and then solving this expression using a time integration technique is referred to as the numerical method of lines. Since the consideration of vehicle-bridge interaction introduces nonlinearity into the problem formulation, a multi-predictor-corrector scheme is adopted in the solution procedure to obtain accurate results. The solution algorithm is implemented into a computer code developed by the authors, and numerical examples are solved to illustrate the validity and efficiency of the proposed formulation and solution methodology. Furthermore, the numerical results are detailed in order to highlight the dynamic effects of multiple vehicles on the behavior of bridge superstructures.

Wave propagation theory in anisotropic periodically layered fluid-saturated porous media

A study is made of harmonic plane-wave propagation through an anisotropic, periodically layered, fluid-saturated, porous medium. Biot’s theory is used to describe the constitutive equations of a fluid-saturated porous solid. Employing the idea of introducing fundamental elasticity tensors, the field equations governing the wave propagation are derived in terms of first-order matrix ordinary differential equations. Propagator matrix formulation is adopted to treat the periodically layered property. Within this framework, the inhomogeneous medium is treated by dividing it into homogeneous layers. This technique is acceptable provided the coefficient matrix of ordinary differential equations consists of piecewise constants. Also, within each layer, the general case of anisotropic, fluid-saturated, porous materials can be considered. Based on this approach, a numerical model was developed and results are presented for a fluid-saturated, fabric material.

Symbolically Precise Solutions to a Homogeneous Second Order Matrix Ordinary Differential Equation with Macsyma

A batch file tube_ode2, written in Macsyma version 309.6 for the SUN 3/60 is presented, which uses Laplace transform theory to solve the homogeneous second order matrix ordinary differential equation F″( t) + RF ( t) = 0, where F( t) is an n by n matrix with entries that are infinitely differentiable functions of the real parameter t, F″( t) is the second derivative of F ( t) with respect to t, and R is an n by n matrix with constant entries. The batch file accepts R, and the initial conditions F(0) and F′(0) as matrices and the output is a matrix with functional entries. It is shown that if R, F(0), and F′(0) have rational entries, symbolically precise solutions are obtainable in all cases up to n = 4. Practice indicates that symbolically precise solutions are often obtainable for n > 4, as well. To accomplish this, the method of the adjoint is used in solving the matrix equation A L ( F( t)) = P arising from the action of the Laplace transform on the given ODE, from which the final solution F ( t) is computed as the inverse transform of L ( F( t)). Macsyma's ilt function is utilized in this regard, with a boost from the user-defined function force_factor, which expresses the determinant of A as the product of quadratic factors.

Wave propagation theory in anisotropic periodically layered fluid-saturated porous medium.

A study is made of harmonic plane wave propagation through an anisotropic periodically layered fluid-saturated porous medium. Biot’s theory is used to describe the constitutive equations of a fluid-saturated porous solid. Employing the idea of introducing fundamental elasticity tensors, the field equations governing the wave propagation are derived in terms of first-order matrix ordinary differential equations. Propagator matrix formulation is adopted to treat the periodically layered property. Within this framework, the inhomogeneous medium is treated by dividing it into homogeneous layers. This technique is acceptable provided the coefficient matrix of ordinary differential equations consists of piece-wise constants. Also, within each layer, the general case of anisotropic fluid-saturated porous materials can be considered. Based on this approach, computer software has been developed and a numerical example is presented using air as the acoustic medium.

Reflection and transmission of elastic waves by a structurally chiral arrangement of identical uniaxial layers

Layered composites are constructed by stacking identical uniaxial plates, their consecutive symmetric axes describing either a right- or a left-handed spiral. A matrix representation method is used for solving the reflection and transmission characteristics of such layered composites. The field equations governing the harmonic motions of the layered composite are written as a first-order matrix ordinary differential equation, where the field variables form a 6-vector consisting of the displacement and traction components. Numerical results of the plane wave reflection and transmission characteristics for chiral arrangements are compared with those for a non-chiral arrangement. It is observed that the co-polarized characteristics are unaffected by the structural chirality, while the cross-polarized reflected and transmitted fields are greatly influenced by it.

Inversion of higher-order matrix difference and differential equations through their dynamical solutions

We discuss matrix finite difference and ordinary differential equations in terms of their dynamical solutions which correspond to Green functions for initial-value problems. Explicit formulas, which make no use of Jordan decompositions, are derived by using the Laplace-Stieltjes transform. The situation for inverting matrix polynomials is also considered.

Linear integral transformations and hierarchies of integrable nonlinear evolution equations

Integrable hierarchies of nonlinear evolution equations are investigated on the basis of linear integral equations. These are (Riemann-Hilbert type of) integral transformations which leave invariant an infinite sequence of ordinary differential matrix equations of increasing order in an (indefinite) parameter k. The potential matrices in these equations obey a set of nonlinear recursion relations, leading to a heirarchy of nonlinear partial differential equations. In decreasing order the same equations give rise to a “reciprocal” hierarchy, associated with Heisenberg ferromagnet type of equations. Central in the treatment is an embedding of the hierarchy into an infinite-matrix structure, which is constructed on the basis of the integral equations. In terms of this infinite-matrix structure the equations governing the hierarchies become quite simple. Furthermore, it leads in a straightforward way to various generalizations, such as to other types of linear spectral problems, multicomponent system and lattice equations. Generalizations to equations associated with noncommuting flows follow as a direct consequence of the treatment. Finally, some results on conserved densities and the Hamiltonian structure are briefly discussed.

The Linear-Quadratic Optimal Control Problem with Delays in State and Control Variables: A State Space Approach

This paper generalizes the results of Vinter and Kwong to a control operator with a more general delay structure than the one in A. Ichikawa and to delay systems with both finite and infinite memories. The linear-quadratic optimal control problem over both finite and infinite time horizons is treated. Several forms of associated abstract Riccati differential and algebraic equations are presented. The properties of the kernel of the solution are studied and a coupled set of matrix ordinary and partial differential equations is obtained.

Derivation of a matrix Painlevé equation germane to wave scattering by a broken corner

The problem of wave scattering by a broken corner is considered. This is a linear problem whose analysis, however, is reduced—in a sense described below—to a nonlinear ordinary matrix differential equation. For scattering from a strip, a Painlevé equation, was derived some time ago. The present work shows that matrix Painlevé equations also arise naturally in physical problems.

Response of Rotor Blades to Random Loads under Forward Flight Conditions

SummaryThis paper contains results for the response of a helicopter rotor blade under stationary random excitation when the helicopter is in forward flight. The blade model takes into account the bending and torsional modes as well as the root-rigidity conditions. The resulting linear dynamical system equations with periodic coefficients are treated within the framework of the filtering theory to yield matrix ordinary differential equations for the required response-statistics themselves.

A Dynamic Modeling Method of Unsteady Flows in Long Fluid Lines With Turbulent Bulk Velocities

An analytical method is presented to solve the equations of unsteady gas flow in a long tube with turbulent bulk-flow velocity. The nonlinearity of the equations is overcome by judicious choice of integrating variables and by subsequent linearization, allowing integration as an ordinary matrix differential equation. The method yields, in operational form and in terms of pure time-delays, attenuations and other linear operators, a transfer matrix of the fluid line. This result can be directly implemented for line simulation or control applications; the integration process is not affected by changing end-conditions. Maximum accuracy of the resulting dynamic model is obtained for long lines (fl/D ≥ 2.2) and for small percent changes of the bulk-flow velocity along the tube. Adiabatic flow assumption and averaging of fluid properties over a cross section of tube restricts the analysis to turbulent flows and precludes the application of the model to small bulk flows.

Rotor Blade Response to Random Loads: A Direct Time-Domain Approach

The response of a rotor blade subject to random loads is treated within the framework of time-varying linear dynamical systems with nonstationary random forcing functions. The statistically meaningful characteristics of the response, namely the covariance and the correlation, are directly obtained from a pair of ordinary matrix differential equations in the time domain. Extensive results for different models of a rigidly flapping blade subject to quasi-stationary random excitations have been obtained by this direct time-domain approach to demonstrate its simplicity and efficiency.

Nonzero-sum differential games

The theory of differential games is extended to the situation where there areN players and where the game is nonzero-sum, i.e., the players wish to minimize different performance criteria. Dropping the usual zero-sum condition adds several interesting new features. It is no longer obvious what should be demanded of asolution, and three types of solutions are discussed:Nash equilibrium, minimax, andnoninferior set of strategies. For one special case, the linear-quadratic game, all three of these solutions can be obtained by solving sets of ordinary matrix differential equations. To illustrate the differences between zero-sum and nonzero-sum games, the results are applied to a nonzero-sum version of a simple pursuit-evasion problem first considered by Ho, Bryson, and Baron (Ref. 1).Negotiated solutions are found to exist which give better results forboth players than the usualsaddle-point solution. To illustrate that the theory may find interesting applications in economic analysis, a problem is outlined involving the dividend policies of firms operating in an imperfectly competitive market.