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Results for “reduced eigenvalue problem”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 19 records
All sourcesNASA NTRSDOE OSTI
NASA NTRS1983

Reanalysis and design in structural dynamics

A unified efficient formulation for the static and dynamic reanalysis of locally modified structures is presented. The reanalysis problem is to find the structural responses when some of the element properties are adjusted as a result of design modifications or when additional structures are appended to the original system. The reanalysis problem is formulated as a problem of much lower order than the original system. This is achieved by utilizing the linearity property of the structure using the pseudoload concept together with the solution of the original system. The modifications to the structure are treated as displacement-dependent pseudoloads of the system. By expressing the modified system response as linear combinations of the response of the original system and a term depending on the pseudoload, a reduced set of response equations can be obtained. In the static and sinusoidal steady state analysis, this leads to a set of linear algebraic equations. For a free vibration analysis, this results in a reduced eigenvalue problem. Using these reduced equations, a design problem can be solved; that is, the magnitude of a specified local modification for given dynamic response characteristics can be calculated. Several examples are presented to illustrate the general formulations.

Wang, B. P.↗
NASA NTRS1982

Evaluation and reduction of errors induced by the Guyan transformation

The Guyan reduction refers to a method used to reduce the number of degrees of freedom in a structural model for dynamic analysis. Experience shows that, if the method is properly employed, then this reduction method does in fact provide a "reasonably" accurate approximation of the dynamic characteristics of the unreduced model. A method that provides an accurate estimate of the error in each mode of the reduced eigenvalue problem and, in addition, gives first order correction terms that greatly improve the accuracy of the eigenvectors and eigenvalues is presented. The method is demonstrated by standard COSMIC NASTRAN DAMP alters to rigid format 3, real normal mode analysis. Comparative computer time of modal extraction versus error analysis is discussed for the VAX-11/780 version of COSMIC NASTRAN.

Fox, G. L.↗
NASA NTRS1972

A NASTRAN correlation study for vibrations of a cross-stiffened ship's deck

To evaluate the effectiveness of NASTRAN for predicting the vibration modes of panels with bending-membrane coupling, a cross-stiffened ship deck was analyzed. In correlation with experimental data, one NASTRAN finite element representation gave results slightly more accurate than a previous analytical solution. Computational time was excessively long, due to the Guyan method of reducing the eigenvalue problem. It is recommended that a more efficient method of matrix reduction be implemented for the lumped mass formulation.

Thornton, E. A.↗
NASA NTRS1975

Ellipsometer measurements of epitaxial GaAs layers: A concept

Report discusses analysis involving Maxwell's equations formed in a 6-by-6 matrix. By applying boundary conditions at proper points in sample, equation for propagation of light through anisotropic medium is reduced to eigenvalue problem resulting in 4-by-4 matrix.

Desmet, D. J.↗
NASA NTRS1972

Diffraction of a plane pulse by a three-dimensional corner.

The conical solutions for the incidence of a plane pulse on a three-dimensional corner are presented. The corner is represented by a trihedron with one edge perpendicular to the other two. Both the boundary condition of the first kind and that of the second kind are considered. Outside the characteristic sphere of the vertex of the corner, the solution is represented by the well-known conical solutions in two variables. Inside the characteristic sphere, the problem involves three conical variables. By the separation of variables, the problem is reduced to that of an eigenvalue problem with an irregular boundary which is in turn reduced to a system of homogeneous algebraic equations. The eigenvalues are then determined numerically. By the superposition of the conical solutions for plane pulses, the solution for the incidence of a plane wave is obtained. Numerical examples simulating the incidence of a sonic boom on the corner of a structure are presented.

Ting, L.↗
NASA NTRS1987

Vibration analysis of quasi-symmetric structures

A computational technique is developed to reduce the size of the dynamical model of a large complex quasi-symmetric structure prior to free vibration analysis. The structure is discretized using a two-field mixed FEM model; the response vectors are decomposed into symmetric and antisymmetric components; the eigenvectors of the quasi-symmetric structure are approximated by a linear combination of a few preselected sets of symmetric and antisymmetric modes generated by the reduced-size model; global approximation vectors are selected and generated; and the resulting eigenvalue problem is significantly reduced via a classical Bubnov-Galerkin procedure. Numerical results for a shallow laminated anisotropic panel of quadrilateral planform are presented in extensive tables and graphs, demonstrating the effectiveness of the method.

Noor, Ahmed K.↗
NASA NTRS1988

A comparative study of theoretical methods on Goertler instability

Goertler vortices arise in boundary layers along concave surfaces due to centrifugal effects and these vortices in combination with Tollmien-Schlichting (TS) waves and crossflow vortices may play an important role in triggering early transition. There have been two distinct theoretical approaches to solve the problem. In the classical, normal-mode approach (NMA) the linear development of Goertler vortices is reduced to an eigenvalue problem. In the other method, initiated by Hall, the governing partial differential equations for the perturbations which are parabolic in the streamwise direction, are solved as an initial value problem. This method predicts multiple neutral curves depending on how and where the mean flow is perturbed, unlike the NMA where a unique but different neutral curve is predicted by each investigator. The present paper attempts to compare the two techniques and shows that mutually compatible results can be obtained when physically consistent assumptions are made.

Kalburgi, Vijay↗
NASA NTRSDate not supplied

Towards Formalization of Advanced Linear Algebra with Applications to Dynamical Systems using PVS

Linear Algebra is essential for numerous aerospace problems of interest. Formal reasoning about hybrid systems that contain variables modeled by differential equations rely on concepts from Linear Algebra such as eigenvalues, matrix decompositions, and matrix valued functions. For example, the long-term dynamics of a system of differential equations depend on the stability/instability of its equilibrium points, which often reduces to an eigenvalue problem. This talk will embark on a quest to formalize theorems and results about eigenvalues and eigenvectors using PVS. We shall start our journey with 2 x 2 complex matrices, where we will apply our PVS code to a simple example of a dynamical system. Since it can be difficult or impossible to give simple expressions of eigenvalues for larger matrices (i.e. 5 x 5 or higher), we then move towards specifying the power method for verified computation of eigenvalue approximations in PVS. This effort requires development of multivariate complex arithmetic. At the end of the day, having such additions to the PVS NASA libraries will help move towards the use of formal methods to verify concepts of control theory and system level verification.

Linear Algebra↗
NASA NTRS1982

Application of the Finite Element Method to Rotary Wing Aeroelasticity

A finite element method for the spatial discretization of the dynamic equations of equilibrium governing rotary-wing aeroelastic problems is presented. Formulation of the finite element equations is based on weighted Galerkin residuals. This Galerkin finite element method reduces algebraic manipulative labor significantly, when compared to the application of the global Galerkin method in similar problems. The coupled flap-lag aeroelastic stability boundaries of hingeless helicopter rotor blades in hover are calculated. The linearized dynamic equations are reduced to the standard eigenvalue problem from which the aeroelastic stability boundaries are obtained. The convergence properties of the Galerkin finite element method are studied numerically by refining the discretization process. Results indicate that four or five elements suffice to capture the dynamics of the blade with the same accuracy as the global Galerkin method.

Straub, F. K.↗
NASA NTRS1989

Computation of canonical correlation and best predictable aspect of future for time series

The canonical correlation between the (infinite) past and future of a stationary time series is shown to be the limit of the canonical correlation between the (infinite) past and (finite) future, and computation of the latter is reduced to a (generalized) eigenvalue problem involving (finite) matrices. This provides a convenient and essentially, finite-dimensional algorithm for computing canonical correlations and components of a time series. An upper bound is conjectured for the largest canonical correlation.

Pourahmadi, Mohsen↗
NASA NTRS1976

Determination of poles and zeros of transfer functions for flexible spacecraft attitude control

The transfer function matrix is obtained for a three-input and three-output model of minimum sensors and actuators for the attitude control system of flexible spacecraft, and a method is described for determining the poles and zeros of this transfer function. Three cases are considered: (1) the actuators and the sensors are all attached to the primary body, (2) the actuators are on the primary body and the sensors are on the sub-body, and (3) the actuators are on the sub-body and the sensors are on the primary body. The zero-determination problem is shown to reduce to eigenvalue calculations of a matrix which is constructed from the inertial and modal matrices in a simple fashion.

Ohkami, Y.↗
NASA NTRS1981

Instability of non-zonal baroclinic flows - Multiple-scale analysis

The linear instability of a non-zonal flow can be reduced to an eigenvalue-eigenfunction problem, governed by a nonseparable partial differential equation (Niehaus, 1980). Approximate solutions, found by the method of multiple scales, are derived here and compared with earlier results found using a spectral method. The amplitude maxima are correctly located. The zonal variations of local wavenumber and of amplitude are qualitatively correct, but not sufficiently extreme. Because the method is oversensitive to local conditions, and less sensitive to global constraints, this comparison provides theoretical limits to the possibility of parameterizing transient eddies in terms of the local time mean state of the atmosphere. The method can be extended easily to flows with more realistic vertical structure.

Niehaus, M. C. W.↗
NASA NTRS1990

An efficient design sensitivity analysis of eigenvectors

Subspace iteration has been a major advance in solving large eigen problems when only a subset of eigen-pairs is required. The essence of this method is a transformation from displacement coordinates of an n-th order eigensystem to generalized coordinates of a smaller q-th order. The eigenvalue problem is then solved in the reduced space. The method was first developed by Clint and Jennings for real symmetric systems and was then called 'simultaneous iteration'. The success of the method prompted further research along this line and there have been many improved algorithms developed. This approach has been widely used by structural engineers for extracting the most useful natural frequencies and mode shapes of large-scale dynamic systems. This paper exploits into a new direction which is in the form of iterative process for simultaneously calculating eigenvector derivatives of many eigenvectors with respect to multi-variables. The method fully uses all the available information from preceding eigenvalue solution and, thus, effectively economizes computational efforts. It iterates through two equations derived from the first variation of the two fundamental equations used in subspace iteration method. There is no expensive large matrix decomposition required and the process converges to acceptable solution in a finite number of iterations. Therefore, the procedure increases its efficiency superiority over the others as the system size or the number of interested eigenvectors become larger and larger.

Ting, T.↗
NASA NTRS1985

Numerical Techniques for Scattering from Submerged Objects

To represent the final results in terms of matrices, one expands all appropriate physical quantities in terms of partial wave basis states. This includes expansions for the incident and scattered fields and the surface quantities. The method then utilizes the Huygen-Poincare integral representation for both the exterior and interior solutions, leading to the required matrix equations. One thus deals with matrix equations, the complexity of which depends on the nature of the problem. It is shown that in general a transition matrix T can be obtained relating the incident field A with the scattered field f having the form T = PQ(-1), where f = TA. The structure of Q can be quite complicated and can itself be composed of other matrix inversions such as arise from layered objects. Recent improvements in this method appropriate for a variety of physical problems are focused on, and on their implementation. Results are outlined from scattering simulations for very elongated submerged objects and resonance scattering from elastic solids and shells. The final improvement concerns eigenfunction expansions of surface terms, arising from solution of the interior problem, obtained via a preconditioning technique. This effectively reduces the problem to that of obtaining eigenvalues of a Hermitian operator. This formalism is reviewed for scattering from targets that are rigid, sound-soft, acoustic, elastic solids, elastic shells, and elastic layered objects. Two sets of the more interesting results are presented. The first concerns scattering from elongated objects, and the second to thin elastic spheroids.

Werby, M. F.↗
NASA NTRS1994

Quantum statistics of Raman scattering model with Stokes mode generation

The model describing three coupled quantum oscillators with decay of Rayleigh mode into the Stokes and vibration (phonon) modes is examined. Due to the Manley-Rowe relations the problem of exact eigenvalues and eigenstates is reduced to the calculation of new orthogonal polynomials defined both by the difference and differential equations. The quantum statistical properties are examined in the case when initially: the Stokes mode is in the vacuum state; the Rayleigh mode is in the number state; and the vibration mode is in the number of or squeezed states. The collapses and revivals are obtained for different initial conditions as well as the change in time the sub-Poisson distribution by the super-Poisson distribution and vice versa.

Tanatar, Bilal↗
NASA NTRS1984

Rotordynamic Analysis of the SSME Turbopumps Using Reduced Models

Alternative methods for the rotor-dynamic and sensitivity analysis of large rotor systems are examined. The methods are assessed for their ability to utilize accurate models of reduced size along with effective procedures for describing the dynamic behavior of the systems. Frequency response-based techniques are developed for determining the steady state response to imbalance of the space shuttle main engine (SSME) turbopumps and the related eigenvalue problem. The rotor and housing are represented by reduced receptances associated with their coupling points. The housing may be described by all of its normal modes within a frequency range of interest. The effects of truncated higher and lower modes are accounted for in an approximate manner. A procedure is described for determining the sensitivity of the coupling forces to changes in the coupling elements and rotor speed of the turbopump systems. In addition, an eigenvalue sensitivity analysis technique is adopted for application to the systems. Computer programs were developed for the numerical implementation of the impedance and eigenvalue sensitivity formulated in this study.

Noah, S. T.↗
NASA NTRS1995

Expendable launch vehicle studies

Analytical support studies of expendable launch vehicles concentrate on the stability of the dynamics during launch especially during or near the region of maximum dynamic pressure. The in-plane dynamic equations of a generic launch vehicle with multiple flexible bending and fuel sloshing modes are developed and linearized. The information from LeRC about the grids, masses, and modes is incorporated into the model. The eigenvalues of the plant are analyzed for several modeling factors: utilizing diagonal mass matrix, uniform beam assumption, inclusion of aerodynamics, and the interaction between the aerodynamics and the flexible bending motion. Preliminary PID, LQR, and LQG control designs with sensor and actuator dynamics for this system and simulations are also conducted. The initial analysis for comparison of PD (proportional-derivative) and full state feedback LQR Linear quadratic regulator) shows that the split weighted LQR controller has better performance than that of the PD. In order to meet both the performance and robustness requirements, the H(sub infinity) robust controller for the expendable launch vehicle is developed. The simulation indicates that both the performance and robustness of the H(sub infinity) controller are better than that for the PID and LQG controllers. The modelling and analysis support studies team has continued development of methodology, using eigensensitivity analysis, to solve three classes of discrete eigenvalue equations. In the first class, the matrix elements are non-linear functions of the eigenvector. All non-linear periodic motion can be cast in this form. Here the eigenvector is comprised of the coefficients of complete basis functions spanning the response space and the eigenvalue is the frequency. The second class of eigenvalue problems studied is the quadratic eigenvalue problem. Solutions for linear viscously damped structures or viscoelastic structures can be reduced to this form. Particular attention is paid to Maxwell and Kelvin models. The third class of problems consists of linear eigenvalue problems in which the elements of the mass and stiffness matrices are stochastic. dynamic structural response for which the parameters are given by probabilistic distribution functions, rather than deterministic values, can be cast in this form. Solutions for several problems in each class will be presented.

Bainum, Peter M.↗