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Fixed interface and free interface methods of substructure coupling for dynamic analysis are discussed. Three methods for reducing the number of coordinates required by fixed interface methods are introduced. Matrix ordinary differential equations are employed to improve accuracy in free interface substructure coupling methods.
The paper describes methods for extracting unknown state variables and parameters from dynamic rotor model tests given transient cyclic pitch stirring inputs, blade root flap-bending measurements, and the form of the dynamic rotor equations, including a rotor dynamic inflow description, when none of the physical parameters are known. A simplified version of the maximum likelihood method seems best suited for this purpose. The measurement equation error covariance matrix is assumed constant during each iteration, but updated for the subsequent iteration. A detailed analysis of the suitability of the derived techniques for studying various rotor dynamic inflow effects is provided.
A parameter-estimation method is described for verifying the mathematical model of mixed (combined interactive components from various engineering fields) dynamic systems against pertinent experimental data. The model verification problem is divided into two separate parts: defining a proper model and evaluating the parameters of that model. The main idea is to use differences between measured and predicted behavior (response) to adjust automatically the key parameters of a model so as to minimize response differences. To achieve the goal of modeling flexibility, the method combines the convenience of automated matrix generation with the generality of direct matrix input. The equations of motion are treated in first-order form, allowing for nonsymmetric matrices, modeling of general networks, and complex-mode analysis. The effectiveness of the method is demonstrated for an example problem involving a complex hydraulic-mechanical system.
Problem involves design of controls for linear time-invariant system disturbed by white noise. Solution is Kalman filter coupled through set of optimal regulator gains to produce desired control signal. Key to solution is solving matrix Riccati differential equation. LSOCE effectively solves problem for wide range of practical applications. Program is written in FORTRAN IV for batch execution and has been implemented on IBM 360.
The minimal polynomial and reduced rank extrapolation algorithms are two acceleration of convergence methods for sequences of vectors. In a recent survey these methods were tested and compared with the scalar, vector, topological epsilon algorithms, and were observed to be more efficient than the latter. It was also observed that the two methods have similar convergence properties. The convergence and stability properties of these methods are analyzed and the performance of the acceleration methods when applied to a class of vector sequences that includes those sequences obtained from systems of linear equations by using matrix iterative methods is discussed.
The computational aspects of modelling material failure in structural wood members are presented with particular reference to vector processing aspects. Wood members are considered to be highly orthotropic, inhomogeneous, and discontinuous due to the complex microstructure of wood material and the presence of natural growth characteristics such as knots, cracks and cross grain in wood members. The simulation of strength behavior of wood members is accomplished through the use of a special purpose finite element/fracture mechanics routine, program STARW (Strength Analysis Routine for Wood). Program STARW employs quadratic finite elements combined with singular crack tip elements in a finite element mesh. Vector processing techniques are employed in mesh generation, stiffness matrix formation, simultaneous equation solution, and material failure calculations. The paper addresses these techniques along with the time and effort requirements needed to convert existing finite element code to a vectorized version. Comparisons in execution time between vectorized and nonvectorized routines are provided.
The stability of a decelerating boundary-layer flow is investigated experimentally and numerically. Experimentally, a flat plate having a Blasius boundary layer is decelerated in an 18 m towing tank. The boundary layer becomes unstable to two-dimensional waves, which break down into three-dimensional patterns, hairpin vortices, and finally turbulent bursts when the vortices lift off the wall. The unsteady boundary-layer equations are solved numerically to generate instantaneous velocity profiles for a range of boundary and initial conditions. A quasi-steady approximation is invoked and the stability of local velocity profiles is determined by solving the Orr-Sommerfeld equation using Chebyshev matrix methods. Comparisons are made between the numerical predictions and the experimentally observed instabilities.
Much of the computational effort of the finite element process involves the solution of a system of linear equations. The coefficient matrix of this system, known as the global stiffness matrix, is symmetric, positive definite, and generally sparse. An important technique for reducing the time required to solve this system is substructuring or matrix partitioning. Substructuring is based on the idea of dividing a structure into pieces, each of which can then be analyzed relatively indepenently. As a result of this division, each point in the finite element discretization is either interior to a substructure or on a boundary between substructures. Contributions to the global stiffness matrix from connections between boundary points from the K(bb) matrix are reported. The triangularization of a general K(bb) matrix on a parallel machine is specifically discussed.
This paper associates the Riccati group and its group action on linear-quadratic optimal control problems to the action of a Lie transformation group on a set of Hamiltonian matrices. In this Lie theoretic setting results are presented concerning the associated orbit structure and the structure of the group itself. These results are of importance in understanding the solution structure of matrix Riccati differential equations, and thus also of importance in linear-quadratic optimal control.
This paper concerns a numerical model of a regenerator running at very low temperatures. The model consists of the usual three equations for a compressible fluid with an additional equation for a matrix temperature. The main difficulty with the model is the very low Mach number (approximately 1.E-3). The divergence of the velocity is not small, the pressure divergence is small, and the pressure fluctuation in time is not small. An asymptotic expansion based on the bounded derivative method of Kreiss is used to give a reduced model which eliminates acoustic waves. The velocity is then determined by a two-point boundary value problem which does not contain a time derivative. The solution obtained from the reduced system is compared with the numerical solution of the original system.
Polynomial interpolation methods are applied both to the approximation of functions and to the numerical solutions of hyperbolic and elliptic partial differential equations. The derivative matrix for a general sequence of the collocation points is constructed. The approximate derivative is then found by a matrix times vector multiply. The effects of several factors on the performance of these methods including the effect of different collocation points are then explored. The resolution of the schemes for both smooth functions and functions with steep gradients or discontinuities in some derivative are also studied. The accuracy when the gradients occur both near the center of the region and in the vicinity of the boundary is investigated. The importance of the aliasing limit on the resolution of the approximation is investigated in detail. Also examined is the effect of boundary treatment on the stability and accuracy of the scheme.
A model of the truss-beam has been developed and a set of algorithms provided to determine the flexibility matrix. Using lumped masses at the panel juncture, and applying the flexibility matrix in the equation of motion for the system, the dynamic characteristics are readily determined. An interactive computer program has been developed both in Basic and in Fortran for the IMB PC and other compatible micro-computers. This method is used to determine the lower natural frequencies of two truss-beams, and the results compared with those found by others.
Development of a steady flow solver for use with LINFLO was the objective of this report. The solver must be compatible with LINFLO, be composed of composite mesh, and have transonic capability. The approaches used were: (1) steady flow potential equations written in nonconservative form; (2) Newton's Method; (3) implicit, least-squares, interpolation method to obtain finite difference equations; and (4) matrix inversion routines from LINFLO. This report was given during the NASA LeRC Workshop on Forced Response in Turbomachinery in August of 1993.
A suboptimal Kalman filter system which evolves error covariances in terms of a truncated set of wavelet coefficients has been developed for the assimilation of chemical tracer observations of CH4. This scheme projects the discretized covariance propagation equations and covariance matrix onto an orthogonal set of compactly supported wavelets. Wavelet representation is localized in both location and scale, which allows for efficient representation of the inherently anisotropic structure of the error covariances. The truncation is carried out in such a way that the resolution of the error covariance is reduced only in the zonal direction, where gradients are smaller. Assimilation experiments which last 24 days, and used different degrees of truncation were carried out. These reduced the covariance size by 90, 97 and 99 % and the computational cost of covariance propagation by 80, 93 and 96 % respectively. The difference in both error covariance and the tracer field between the truncated and full systems over this period were found to be not growing in the first case, and growing relatively slowly in the later two cases. The largest errors in the tracer fields were found to occur in regions of largest zonal gradients in the constituent field. This results indicate that propagation of error covariances for a global two-dimensional data assimilation system are currently feasible. Recommendations for further reduction in computational cost are made with the goal of extending this technique to three-dimensional global assimilation systems.
A systematic procedure developed for the calculation of the structural response of aircraft flying through a gust by use of difference equations in the solution of dynamic problems is first illustrated by means of a simple-damped-oscillator example. A detailed analysis is then given which leads to a recurrence matrix equation for the determination of the response of an airplane in a gust. The method takes into account wing bending and twisting deformations, fuselage deflection, vertical and pitching motion of the airplane, and some tail forces. The method is based on aerodynamic strip theory, but compressibility and three-dimensional aerodynamic effects can be taken into account approximately by means of over-all corrections. Either a sharp-edge gust or a gust of arbitrary shape in the spanwise or flight directions may be treated. In order to aid in the application of the method to any specific case, a suggested computational procedure is included. The possibilities of applying the method to a variety of transient aircraft problems, such as landing, are brought out. A brief review of matrix algebra, covering the extent to which it is used in the analysis, is also included. (author)
Using the Newtonian method, the equations of motion are developed for the coupled bending-torsion steady-state response of beams rotating at constant angular velocity in a fixed plane. The resulting equations are valid to first order strain-displacement relationships for a long beam with all other nonlinear terms retained. In addition, the equations are valid for beams with the mass centroidal axis offset (eccentric) from the elastic axis, nonuniform mass and section properties, and variable twist. The solution of these coupled, nonlinear, nonhomogeneous, differential equations is obtained by modifying a Hunter linear second-order transfer-matrix solution procedure to solve the nonlinear differential equations and programming the solution for a desk-top personal computer. The modified transfer-matrix method was verified by comparing the solution for a rotating beam with a geometric, nonlinear, finite-element computer code solution; and for a simple rotating beam problem, the modified method demonstrated a significant advantage over the finite-element solution in accuracy, ease of solution, and actual computer processing time required to effect a solution.
The empirical data used in the 1969 Smithsonian Standard Earth (2) are presented. The variance-covariance matrix, or the normal equations, used for correlation analysis, are considered. The format and contents of the matrix, available on magnetic tape, are described and a sample printout is given.
Transverse single-bunch instabilities of space-charge-dominated coasting beams with round and flat transverse geometries are studied using a unified dispersion-relation framework. The analysis combines the Burov-Lebedev formalism, which captures space-charge tune spread, Landau damping, and instability threshold behavior, with Yokoya’s projection method for representing coherent transverse mode structure and its dependence on beam aspect ratio. In the rigid-beam limit, the formulation reduces to a scalar dispersion relation of Burov-Lebedev paper. For non-rigid transverse oscillations, truncation of Yokoya’s Hermite-based expansion yields a finite-dimensional matrix eigenvalue problem in which space-charge and coupling impedance effects enter through Burov-Lebedev–type denominators. This approach provides a consistent basis for comparing rigid and non-rigid instability behavior in round and flat beams and for assessing the role of beam ellipticity in modifying coherent mode structure and stability thresholds.