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NASA NTRS1981

Some approximations in the linear dynamic equations of thin cylinders

Theoretical analysis is performed on the linear dynamic equations of thin cylindrical shells to find the error committed by making the Donnell assumption and the neglect of in-plane inertia. At first, the effect of these approximations is studied on a shell with classical simply supported boundary condition. The same approximations are then investigated for other boundary conditions from a consistent approximate solution of the eigenvalue problem. The Donnell assumption is valid at frequencies high compared with the ring frequencies, for finite length thin shells. The error in the eigenfrequencies from omitting tangential inertia is appreciable for modes with large circumferential and axial wavelengths, independent of shell thickness and boundary conditions.

El-Raheb, M.↗

NASA NTRS1981

HADY-I, a FORTRAN program for the compressible stability analysis of three-dimensional boundary layers

A computer program HADY-I for calculating the linear incompressible or compressible stability characteristics of the laminar boundary layer on swept and tapered wings is described. The eigenvalue problem and its adjoint arising from the linearized disturbance equations with the appropriate boundary conditions are solved numerically using a combination of Newton-Raphson interative scheme and a variable step size integrator based on the Runge-Kutta-Fehlburh fifth-order formulas. The integrator is used in conjunction with a modified Gram-Schmidt orthonormalization procedure. The computer program HADY-I calculates the growth rates of crossflow or streamwise Tollmien-Schlichting instabilities. It also calculates the group velocities of these disturbances. It is restricted to parallel stability calculations, where the boundary layer (meanflow) is assumed to be parallel. The meanflow solution is an input to the program.

El-Hady, N. M.↗

NASA NTRS1982

Effects of rotation and magnetic field on the onset of convective instability in a liquid layer due to buoyancy and surface tension

Thermocapillary stability characteristics of a horizontal liquid layer heated from below rotating about a vertical axis and subjected to a uniform vertical magnetic field are analyzed under a variety of thermal and electromagnetic boundary conditions. Results based on analytical solutions to the pertinent eigenvalue problems are discussed in the light of earlier work on special cases of the more general problem considered here to show in particular the effects of the heat transfer, nonzero curvature and gravity waves at the two-fluid interface. Although the expected stabilizing action of the Coriolis and Lorentz force fields in this configuration are in evidence the optimal choice of an appropriate range for the relevant parameters is shown to be critically dependent on the interfacial effects mentioned above.

Sarma, G. S. R.↗

NASA NTRS1982

Integrating matrix formulations for vibrations of rotating beams including the effects of concentrated masses

By expressing partial differential equations of motion in matrix notation, utilizing the integrating matrix as a spatial operator, and applying the boundary conditions, the resulting ordinary differential equations can be cast into standard eigenvalue form upon assumption of the usual time dependence. As originally developed, the technique was limited to beams having continuous mass and stiffness properties along their lengths. Integrating matrix methods are extended to treat the differential equations governing the flap, lag, or axial vibrations of rotating beams having concentrated masses. Inclusion of concentrated masses is shown to lead to the same kind of standard eigenvalue problem as before, but with slightly modified matrices.

Lakin, W. D.↗

NASA NTRS1981

Numerical analysis of spectral properties of coupled oscillator Schroedinger operators. I - Single and double well anharmonic oscillators

Several methods for computing many eigenvalues and eigenfunctions of a single anharmonic oscillator Schroedinger operator whose potential may have one or two minima are described. One of the methods requires the solution of an ill-conditioned generalized eigenvalue problem. This method has the virtue of using a bounded amount of work to achieve a given accuracy in both the single and double well regions. Rigorous bounds are given, and it is proved that the approximations converge faster than any inverse power of the size of the matrices needed to compute them. The results of computations for the g:phi(4):1 theory are presented. These results indicate that the methods actually converge exponentially fast.

Isaacson, D.↗

NASA NTRS1982

An integration scheme to determine the dynamic response of a launch vehicle with several payloads

The coupled equations of motion of a system composed of a launch vehicle and multiple payloads are derived. In the process, it is shown how superfluous interface degrees of freedom on the booster side can be accommodated in the formulation. The discrete system equations are directly integrated, avoiding the solution of an expensive system eigenvalue problem. A modified Newmark-Chan-Beta numerical integration scheme is used to obtain the response. The unique form of the equations of motion allow for a quick and accurate solution both for the response and the internal loads. Although completely general, the presented technique is most effective when the number of interface degrees of freedom is relatively small compared to the overall number of degrees of freedom of the system. The technique is applied to the case of the S.T.S.-S.T.-OMS Kit system. Finally, an evaluation of the presented method is included.

Engels, R. C.↗

NASA NTRS1982

Symmetric baroclinic instability of a Hadley cell

A symmetric baroclinic instability is examined in terms of a Boussinesq fluid contained between two horizontal plates to determine the effects of the Ekman and thermal layers. Governing equations are written for a rotating reference frame, taking into account the Rossby, Ekman, and Prandtl numbers. Equations are defined for the perturbation functions, treated as an eigenvalue problem, and a numerical integration of the full eighth order differential system is performed by a shooting technique. An instability is found to occur in the Hadley cell containing both Ekman and thermal boundary layers when the Richardson number is close to unity. If the Prandtl number is fixed the critical Richardson number decreases with an increasing Ekman number until the Ekman number reaches a certain value, at which time the fluid is stable.

Antar, B. N.↗

NASA NTRS1983

Control of structures in space

Various topics related to the control of large space structures are discussed. Equations of motion for distributed systems, eigenvalue problems, modal equations, control implementation, and the Langley beam experiment are discussed.

Meirovitch, L.↗

NASA NTRS1982

Angles of multivariable root loci

A generalized eigenvalue problem is demonstrated to be useful for computing the multivariable root locus, particularly when obtaining the arrival angles to finite transmission zeros. The multivariable root loci are found for a linear, time-invariant output feedback problem. The problem is then employed to compute a closed-loop eigenstructure. The method of computing angles on the root locus is demonstrated, and the method is extended to a multivariable optimal root locus.

Thompson, P. M.↗

NASA NTRS1982

Calculation of exact vibration modes for plane grillages by the dynamic stiffness method

A dynamic stiffness method is developed for the calculation of the exact modal parameters for plane grillages which consist of straight and uniform beams with coincident elastic and inertial axes. Elementary bending-torsion beam theory is utilized, and bending translation is restricted to one direction. The exact bending-torsion dynamic stiffness matrix is obtained for a straight and uniform beam element with coincident elastic and inertial axes. The element stiffness matrices are assembled using the standard procedure of the static stiffness method to form the dynamic stiffness matrix of the complete grillage. The exact natural frequencies, mode shapes, and generalized masses of the grillage are then calculated by solving a nonlinear eigenvalue problem based on the dynamic stiffness matrix. The exact modal solutions for an example grillage are calculated and compared with the approximate solutions obtained by using the finite element method.

Hallauer, W. L., Jr.↗

NASA NTRS1983

Nonlinear structural dynamics analysis using a modified modal method

The procedure for predicting the nonlinear dynamic response of structural components subjected to a step loading is presented. The procedure is a modified modal method that involves a change of dependent variables from the unknown nodal degrees of freedom of the finite element model of the structure to a smaller set of generalized coordinates. This change of dependent variables uses a combination of the nonlinear static solution and some selected vibration mode shapes. The vibration mode shapes correspond to the eigenvectors obtained by solving a standard free vibration eigenvalue problem wherein the stiffness matrix is expanded about the nonlinear static solution. A strategy is also presented for determining which and how many vibration mode shapes to include in the transformation. The effect of inaccurate representation of the spatial distribution of the applied load on the nonlinear dynamic response is discussed for two classes of structural behavior. Application of the procedure to structures which exhibit a stiffening behavior and to those with a softening behavior is presented.

Knight, N. F., Jr.↗

NASA NTRS1983

Linear and Nonlinear Aspects of Rotordynamics

Excessive vibrations of the liquid oxygen pump in the Space Shuttle's Main Engine have been recorded during hot firing ground testing. In order to determine mathematical explanations of this possibility, destructive phenomenon differential equations have been examined which describe the rotordynamics of the pump. Modeling the rotor as a random eigenvalue problem was considered. Analytical expressions were derived for the solution in the case of symmetric damping and stiffness. This enables one to determine accuracy estimates when testing numerical techniques to solve both asymmetric and nonlinear problems. Finally, the rotor model has had nonlinear elements incorporated to improve its simulation of the pump and to expand the corresponding mathematical theory.

Day, W. B.↗

NASA NTRS1984

Vibration and buckling of general periodic lattice structures

A method is presented for vibration and buckling analysis of arbitrary lattice structures having repetitive geometry in any combination of coordinate directions. The approach is based on exact member theory for representing the stiffness of an individual member subject to axial load, and in the case of vibration, undergoing harmonic oscillation. The method is an extension of previous work that was limited to specific geometries. The resulting eigenvalue problem is of the size associated with the repeating element of the structure. A computer program has been developed incorporating the theory and results are given for vibration of rectangular platforms and a large antenna structure having rotational symmetry. Buckling and vibration results for cable-stiffened rings are also given.

Anderson, M. S.↗

NASA NTRS1983

Mechanics of composite materials: Recent advances; Proceedings of the Symposium, Virginia Polytechnic Institute and State University, Blacksburg, VA, August 16-19, 1982

The present conference on the mechanics of composites discusses microstructure's influence on particulate and short fiber composites' thermoelastic and transport properties, the elastoplastic deformation of composites, constitutive equations for viscoplastic composites, the plasticity and fatigue of metal matrix composites, laminate damping mechanisms, the micromechanical modeling of Kevlar/epoxy composites' time-dependent failure, the variational characterization of waves in composites, and computational methods for eigenvalue problems in composite design. Also discussed are the elastic response of laminates, elastic coupling nonlinear effects in unsymmetrical laminates, elasticity solutions for laminate problems having stress singularities, the mechanics of bimodular composite structures, the optimization of laminated plates and shells, NDE for laminates, the role of matrix cracking in the continuum constitutive behavior of a damaged composite ply, and the energy release rates of various microcracks in short fiber composites.

Hashin, Z.↗

NASA NTRS1984

A survey of payload integration methods

Several full-scale and short-cut methods for analyzing a booster/payload system are presented. Two full-scale techniques are considered: (1) a technique that uses a restrained payload together with a free-booster model, the latter being augmented with residual mass and stiffness correction and (2) a technique that uses a restrained payload and booster model. Both techniques determine the 'modal modes', which require the solution of a system eigenvalue problem; the loads usually are then determined via an acceleration approach. A brief description is given of a number of short-cut methods which are of special interest to Shuttle payload design: structural modification, base drive, and interface impedance methods. Directions for further research and development are suggested.

Engels, R. C.↗

NASA NTRS1983

Planar dynamics of a uniform beam with rigid bodies affixed to the ends

The planar dynamics of a uniform elastic beam subject to a variety of geometric and natural boundary conditions and external excitations were analyzed. The beams are inextensible and capable of small transverse bending deformations only. Classical beam vibration eigenvalue problems for a cantilever with tip mass, a cantilever with tip body and an unconstrained beam with rigid bodies at each are examined. The characteristic equations, eigenfunctions and orthogonality relations for each are derived. The forced vibration of a cantilever with tip body subject to base acceleration is analyzed. The exact solution of the governing nonhomogeneous partial differential equation with time dependent boundary conditions is presented and compared with a Rayleigh-Ritz approximate solution. The arbitrary planar motion of an elastic beam with rigid bodies at the ends is addressed. Equations of motion are derived for two modal expansions of the beam deflection. The motion equations are cast in a first order form suitable for numerical integration. Selected FORTRAN programs are provided.

Storch, J.↗

NASA NTRS1985

Differentiating matrices for arbitrarily spaced grid points

Differentiating matrices allow the numerical differentiation of functions defined at points of a discrete grid. A type of differentiating matrix based on local approximation on a sequence of sliding subgrids is considered. Previous derivations of this type of matrix have been restricted to grids with uniformly spaced points, and the resulting derivative approximations have lacked precision, especially at endpoints. The new formulation allows grids which have arbitrarily space points. It is shown that high accuracy can be achieved through use of differentiating matrices on non-uniform grids which include near-boundary points. Use of the differentiating matrix as an operator to solve eigenvalue problems involving ordinary differential equations is also considered.

Lakin, W. D.↗

NASA NTRS1985

Long-time asymptotics of a system for plasma diffusion

A system of parabolic nonlinear equations that describe the diffusion of a fully collisional plasma across a strong magnetic field is discussed. It is demonstrated that the solution to this system tends to a time asymptotic state which is of space-time separable form, theta(t)f(x). Furthermore, f(x) is independent of the initial conditions and theta(t) depends slightly on the initial conditions. The rate of decay of the temporal part is governed by a nonlinear eigenvalue problem. Since the equations are considered in a bounded domain we are able to analyze the effect of boundary conditions on the evolution of the system. Additional effects on radiation, heating, and particle injection can also be accounted for. Essential differences between the behavior of a fully-coupled system and a scalar equation are observed.

Rosenau, P.↗