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NASA NTRS1976

The method of Ritz applied to the equation of Hamilton

Without any reference to the theory of differential equations, the initial value problem of the nonlinear, nonconservative double pendulum system is solved by the application of the method of Ritz to the equation of Hamilton. Also shown is an example of the reduction of the traditional eigenvalue problem of linear, homogeneous, differential equations of motion to the solution of a set of nonhomogeneous algebraic equations. No theory of differential equations is used. Solution of the time-space path of the linear oscillator is demonstrated and compared to the exact solution.

Bailey, C. D.↗

NASA NTRS1975

Stability and transient motion of a plain journal mounted in flexible damped supports

Results are presented for an extensive study of the influence of damped flexible supports on the stability-threshold speed of a symmetric rotor mounted in nonlinear fluid-film bearings. A stability analysis is conducted by applying the Routh criteria and calculating the damped eigenvalues of the system. Stability maps are given in dimensionless form for a range of support stiffness, damping, and mass ratios. The effect of rotor imbalance on the stability of rotor systems is examined, and it is shown that time transient orbits of shaft and support motions clearly indicate their dynamic behavior for both stable and unstable operating speeds. Simple design criteria are proposed for choosing the support stiffness and damping on the basis of rotor weight and journal clearance to promote optimum stability. It is concluded that the use of a flexible damped support system may increase the stability threshold of a plain journal bearing and that damper supports should promote smoother and quieter machine operation by suppressing oil whirl and attenuating unbalanced rotational forces.

Kirk, R. G.↗

NASA NTRS1975

Two-electron bond-orbital model. I

The one-electron bond-orbital model of Harrison (1973) for tetrahedrally coordinated solids is modified to a two-electron model using a generalization of Falicov and Harris' (1969) method for treating the hydrogen molecule. Expressions for nuclear-exchange and pseudodipolar interactions are obtained in terms of bond-orbital model parameters. The eigenfunctions and eigenvalues of the Hamiltonian are employed along with standard perturbation theory to calculate the energy shifts resulting from the application of an external electric field and induced by the magnetic interaction between the nuclear spins and the bond-electron spins. The quantitative predictions of the derived formulas are examined for real semiconductors involving elements and compounds from periodic groups III, IV, and V. It is concluded that the present two-electron model makes it possible to assess the quantitative effects of electron correlation in the anion-cation bond for both nonmagnetic properties (electric susceptibility, dielectric constant) and magnetic properties (nuclear-exchange and pseudodipolar interactions) of various materials.

Huang, C.↗

NASA NTRS1976

A study of sound transmission in curved duct bends by the Galerkin method

A computational procedure based on the Galerkin method is developed to study the sound transmission and reflection characteristics of circularly curved bends of rectangular ducts. This procedure is adaptable to routine computer calculation. An important component of the computer program consists of a QR iterative algorithm which solves the matrix eigenvalue problem generated by the Galerkin method. The present procedure produced very satisfactory convergent results when applied even to cases where the incident wave has high frequency and high wave mode numbers. Some properties of the Galerkin solution are investigated. Its relationship to the classical solution by the method of separation of variables is discussed.

Tam, C. K. W.↗

NASA NTRS1976

Natural modes of oscillation of rotating flexible structures about nontrivial equilibrium

This paper is concerned with the vibrational characteristics of rotating flexible structures in the neighborhood of steady rotation. It is assumed that the structure undergoes deformations during the steady rotation, so that the equilibrium is nontrivial. The object of the paper is to formulate the eigenvalue problem associated with small oscillations of the structure about the nontrivial equilibrium and to develop efficient methods to compute the system natural frequencies and modal vectors. A numerical example showing the procedure for the calculation of spacecraft modes is presented.

Meirovitch, L.↗

NASA NTRS1976

Convergence of an iterative procedure for large-scale static analysis of structural components

The paper proves convergence of an iterative procedure for calculating the deflections of built-up component structures which can be represented as consisting of a dominant, relatively stiff primary structure and a less stiff secondary structure, which may be composed of one or more substructures that are not connected to one another but are all connected to the primary structure. The iteration consists in estimating the deformation of the primary structure in the absence of the secondary structure on the assumption that all mechanical loads are applied directly to the primary structure. The j-th iterate primary structure deflections at the interface are imposed on the secondary structure, and the boundary loads required to produce these deflections are computed. The cycle is completed by applying the interface reaction to the primary structure and computing its updated deflections. It is shown that the mathematical condition for convergence of this procedure is that the maximum eigenvalue of the equation relating primary-structure deflection to imposed secondary-structure deflection be less than unity, which is shown to correspond with the physical requirement that the secondary structure be more flexible at the interface boundary.

Austin, F.↗

NASA NTRS1976

On a numerical solution of the supersonic panel flutter eigenproblem

An automated digital computer procedure is presented in this paper which enables efficient solution of the eigenvalue problem associated with the supersonic panel flutter phenomena. The step-by-step incremental solution procedure is based on an inverse iteration technique which effectively utilizes solution results from the previous step in determining such results during the current solution step. Also, the computations are limited to the determination of a few specific roots only, which are expected to contain the flutter mode, and this is achieved at each step without having to compute any other root. The structural discretization achieved by the finite-element method yields highly banded stiffness, mass, and aerodynamic matrices; the aerodynamic matrix evaluated by the linearized piston theory is real but unsymmetric in nature. The solution algorithm presented in this paper fully exploits the banded form of the associated matrices, and the resulting computer program written in FORTRAN V for the JPL UNIVAC 1108 computer proves to be most efficient and economical when compared to existing procedures of such analysis. Numerical results are presented for a two-dimensional panel flutter problem.

Gupta, K. K.↗

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 NTRS1976

Appendage modal coordinate truncation criteria in hybrid coordinate dynamic analysis

The paper examines the validity of the assumption that certain appendage-distributed (modal) coordinates can be truncated from a system model without unacceptable degradation of fidelity in hybrid coordinate dynamic analysis for attitude control of spacecraft with flexible appendages. Alternative truncation criteria are proposed and their interrelationships defined. Particular attention is given to truncation criteria based on eigenvalues, eigenvectors, and controllability and observability. No definitive resolution of the problem is advanced, and exhaustive study is required to obtain ultimate truncation criteria.

Likins, P.↗

NASA NTRS1976

Free vibrations of laminated composite elliptic plates

The free vibrations are studied of laminated anisotropic elliptic plates with clamped edges. The analytical formulation is based on a Mindlin-Reissner type plate theory with the effects of transverse shear deformation, rotary inertia, and bending-extensional coupling included. The frequencies and mode shapes are obtained by using the Rayleigh-Ritz technique in conjunction with Hamilton's principle. A computerized symbolic integration approach is used to develop analytic expressions for the stiffness and mass coefficients and is shown to be particularly useful in evaluating the derivatives of the eigenvalues with respect to certain geometric and material parameters. Numerical results are presented for the case of angle-ply composite plates with skew-symmetric lamination.

Andersen, C. M.↗

NASA NTRS1976

Gaussian ideal impulsive loading of rigid viscoplastic plates

The response of a thin, rigid, viscoplastic plate subjected to a spatially axisymmetric Gaussian ideal impulse loading was studied analytically. The Gaussian ideal impulse distribution instantaneously imparts a Gaussian initial velocity distribution to the plate, except at the fixed boundary. The plate deforms with monotonically increasing deflections until the initial dynamic energy is completely dissipated in plastic work. The simply supported plate of uniform thickness obeys the von Mises yield criterion and a generalized constitutive equation for rigid, viscoplastic materials. For the small deflection bending response of the plate, neglecting the transverse shear stress in the yield condition and rotary inertia in the equations of dynamic equilibrium, the governing system of equations is essentially nonlinear. A proportional loading technique, known to give excellent approximations of the exact solution for the uniform load case, was used to linearize the problem and obtain analytical solution in the form of eigenvalue expansions. The linearized governing equations required the knowledge of the collapse load of the corresponding static problem.

Hayduk, R. J.↗

NASA NTRS1976

Nonlinear periodic waves

Systematic perturbation procedures for the analysis of nonlinear problems are reviewed. The cases when the multiplicity of an eigenvalue is finite or infinite are treated for self-sustained and forced oscillations. The possibility of the formation of shock waves is discussed. Applications to acoustic problems are presented.

Ting, L.↗

NASA NTRS1976

Effects of mean flow on duct mode optimum suppression rates

The nature of the solution to the convected acoustic wave equation and associated boundary conditions for rectangular ducts containing uniform mean flow is examined in terms of the complex mapping between the wall admittance and characteristic mode eigenvalues. It is shown that the Cremer optimum suppression criteria must be modified to account for the effects of flow below certain critical values of the nondimensional frequency parameter of duct height divided by sound wavelength. The implications of these results on the design of low frequency suppressors are considered.

Kraft, R. E.↗

NASA NTRS1976

Analytical comparison of effects of solid-friction and viscous structural damping on panel flutter

A Galerkin modal analysis is presented that accounts for the effects of both solid friction and viscous structural damping on panel flutter, based on unsteady aerodynamic forces from supersonic potential flow. The eigensolutions are made by complex eigenvalue computer routines. Markedly different effects on the flutter boundary of the two types of structural damping are obtained. This result establishes that there is not, in general, an "equivalent viscous" damping for solid-friction damping. For the limiting case of the static-aerodynamic approximation, a substantially different flutter dynamic pressure is obtained for solid friction identically zero compared with solid friction approaching zero as a limit. Use of the quasi-static aerodynamic approximation eliminates that difference.

Cunningham, H. J.↗

NASA NTRS1977

Aerodynamic loads near cranks, apexes, and tips of thin, lifting wings in incompressible flow

The calculation of the incompressible and irrotational flow in the vicinity of tips and corners of thin, lifting wings is considered. It is shown that the important characteristics of the flow are governed by an eigenvalue problem, which is nonlinear at the trailing edge because of the shed wake (assumed to be in the wing plane). A new solution method was devised because either the existing methods were not valid for the trailing edge case or they would have required excessive amounts of computer time. The new method, which is fundamentally different than the previous ones, was used to calculate solutions for a number of cases, including some for which correct answers had not previously been obtained. Two of these solutions were used to determine the validity of drag and leading-edge-suction distributions near the tips of a delta wing and a swept wing as calculated by using both the vortex lattice method and a kernel function method. The calculations for the swept wing resolved the question of whether or not the induced drag should be zero at the wing tip.

Medan, R. T.↗

NASA NTRS1976

NASA's plans for future NASTRAN capability

Improvements in the NASTRAN systems management office are presented. Among the improvements are the following items: (1) the FEER eigenmethod and its extension to complete eigenvalue problems; (2) the addition of a supersonic flutter package which allows gust response calculations and active control effects as membrane In addition to capabilities mentioned, an error correction information system is also introduced.

Weidman, D. J.↗

NASA NTRS1976

Stiffness coupling application to modal synthesis program, users guide

A FORTRAN IV computer program used to perform modal synthesis of structures by stiffness coupling, using the dynamic transformation method is described. The program was named SCAMP (Stiffness Coupling Approach Modal-Synthesis Program). The program begins with the entry of a substructure's physical mode shapes and eigenvalues or a substructure's mass and stiffness matrix. If the mass and stiffness matrices are entered, the eigen problem for the individual substructure is solved. Provisions are included for a maximum of 20 substructures which are coupled by stiffness matrix springs. Each substructure has a number degrees of freedom (DOF), except that for DOF greater than 100; vector sets having maximum row and column size of 100 were generated prior to entering SCAMP. The substructures are then coupled together via coupling springs, and the dynamic transformation is used to reduce the size of the eigen problem.

Kuhar, E. J.↗

NASA NTRS1977

Application of system identification to analytic rotor modeling from simulated and wind tunnel dynamic test data, part 2

An introduction to aircraft state and parameter identification methods is presented. A simplified form of the maximum likelihood method is selected to extract analytical aeroelastic rotor models from simulated and dynamic wind tunnel test results for accelerated cyclic pitch stirring excitation. The dynamic inflow characteristics for forward flight conditions from the blade flapping responses without direct inflow measurements were examined. The rotor blades are essentially rigid for inplane bending and for torsion within the frequency range of study, but flexible in out-of-plane bending. Reverse flow effects are considered for high rotor advance ratios. Two inflow models are studied; the first is based on an equivalent blade Lock number, the second is based on a time delayed momentum inflow. In addition to the inflow parameters, basic rotor parameters like the blade natural frequency and the actual blade Lock number are identified together with measurement bias values. The effect of the theoretical dynamic inflow on the rotor eigenvalues is evaluated.

Hohenemser, K. H.↗