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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 91 records · Page 5
All sourcesNASA NTRSDOE OSTI
NASA NTRS1989

Nonlinear strain-displacement relations and flexible multibody dynamics

Dynamics of chains of flexible bodies undergoing large rigid body motions, but small elastic deflections are considered. The role of nonlinear strain-displacement relations in the development of the motion equations correct to first order in elastic deflections is investigated. The general form of these equations linearized only in the small elastic deflections is presented, and the relative significance of various nonlinear terms is studied both analytically and through the use of the numerical simulations. Numerical simulations are performed for a two link chain constrained to move in the plane, subject to hinge torques. Each link is modeled as a thin beam. Slew maneuver simulation results are compared for models with and without properly modeled kinematics of deformation. The goal of this case study is to quantify the importance of the terms in the equations of motion which arise from the inclusion of nonlinear strain-displacement relations. It is concluded that unless the consistently linearized equations in elastic deflections and speeds are available and necessary, the inconsistently (prematurely) linearized equations should be replaced in all cases by ruthlessly linearized equations: equations in which all nonlinear terms involving the elastic deflections and speeds are ignored.

Padilla, Carlos E.↗
NASA NTRS1975

The Use of General Purpose Computer Programs to Derive Equations of Motion for Optimal Isolation Studies

Techniques were developed that utilize general purpose structural analysis computer programs to generate the equations of motion necessary for limiting performance studies. The methodology necessary to couple available general purpose finite element structural programs to a limiting performance capability was developed. Primary emphasis was given to the use of the general purpose program to develop equations of motion in a form that can be used by the limiting performance program.

Pilkey, W. D.↗
NASA NTRS1982

Second order non-linear equations of motion for spinning highly flexible line elements

The second order nonlinear equations of motion are formulated for spinning line elements having little or no intrinsic structural stiffness. The derivation is based on the extended Hamilton's principle and includes the effect of initial geometric imperfections (axial, curvature, and twist) on the line element dynamics. For comparison with previous work, the nonlinear equations are reduced to a linearized form frequently found in the literature. The comparison revealed several new spin-stiffening terms that have not been previously identified and/or retained. They combine geometric imperfections, rotary inertia, Coriolis, and gyroscopic terms.

Salama, M.↗
NASA NTRS1977

Equations of motion for rotating finite bodies in the extended PPN formalism

The equations of motion for rotating finite bodies are computed in the perfect fluid metric in the extended parametric post-Newtonian (PPN) formalism of Will and Nordtvedt (1972) and are used to build a model of the solar system consisting of N oblate, homogeneous, stationary, self-gravitating masses of rotating perfect fluid. These equations contain relativistic acceleration terms which are currently observable or may be observable in the future with improved radio and laser ranging techniques.

Dallas, S. S.↗
NASA NTRS1983

Uniaxial aerodynamic attitude control of artificial satellites

Within the context of a simple mechanical model the paper examines the movement of a satellite with respect to the center of masses under conditions of uniaxial aerodynamic attitude control. The equations of motion of the satellite take account of the gravitational and restorative aerodynamic moments. It is presumed that the aerodynamic moment is much larger than the gravitational, and the motion equations contain a large parameter. A two-parameter integrated surface of these equations is constructed in the form of formal series in terms of negative powers of the large parameter, describing the oscillations and rotations of the satellite about its lengthwise axis, approximately oriented along the orbital tangent. It is proposed to treat such movements as nominal undisturbed motions of the satellite under conditions of aerodynamic attitude control. A numerical investigation is made for the above integrated surface.

Sazonov, V. V.↗
DOE OSTI2020

Closed-Form Solutions for the Equations of Motion of the Heavy Symmetrical Top with One Point Fixed

The Equations of Motion (EOM) for the Heavy Symmetrical Top with One Point Fixed are highly non-linear. The literature describes the numerical methods that are used to resolve this Classical system including modern tools i.e. the Runge-Kutta Fourth Order method. It is more difficult to derivate closed-form solutions for the EOM and as mentioned in the literature it is not always possible to find the close-form solution for all the EOM. Fortunately, there are a few examples available that will serve as a guide to move further on in this topic. It is the purpose of this paper to find a methodology that will produce the solutions for a given subset of EOM that fulfill certain requisites. The report is organized as follows: it starts with a very short summary of the literature available on this topic and quickly follows into the derivation of the EOM using the Euler-Lagrange method. The Routhian will be used to reduce the size of the expression. It continues with the formulation of the Classical cubic function (f(u)) through a novel process. The roots of f(u) are of the outmost importance to be able to find the EOM closed-form solution, and once the final roots are selected the general method that will produce the closed-form solutions is presented. Two sets of examples are included to show the validity of the process and comparisons of the results from the closed-form solutions vs. the numerical results for these examples are shown.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2021

Closed-Form Solutions for the Equations of Motion of the Heavy Symmetrical Top with One Point Fixed

The equations of motion (EOM) for the heavy symmetrical top with one point fixed are highly nonlinear. The literature describes the numerical methods that are used to resolve this classical system, including modern tools, such as the Runge–Kutta fourth–order method. Finding the derivate of closed-form solutions for the EOM is more difficult and, as mentioned in the literature, discovering the solution is not always possible for all the EOM. Fortunately, a few examples are available that serve as a guide to move further in this topic. The purpose of this paper is to find a methodology that will produce the solutions for a given subset of EOMs that fulfill certain requisites. This paper summarizes the literature available on this topic and then follows with the derivation of the EOM using the Euler–Lagrange method. The Routhian method will be used to reduce the size of the expression, and it continues with the formulation of the classical cubic function, ƒ(u), through a novel process. The roots of ƒ(u) are of the utmost importance in finding the EOM closed-form solution, and once the final roots are selected, the general method that will produce the closed-form solutions is presented. Two sets of examples are included to show the validity of the process, and comparisons of the results from the closed-form solutions vs. the numerical results for these examples are shown.

Laos, Hector↗
NASA NTRS1974

Nonlinear equations of motion for the elastic bending and torsion of twisted nonuniform rotor blades

The equations of motion are developed by two complementary methods, Hamilton's principle and the Newtonian method. The resulting equations are valid to second order for long, straight, slender, homogeneous, isotropic beams undergoing moderate displacements. The ordering scheme is based on the restriction that squares of the bending slopes, the torsion deformation, and the chord/radius and thickness/radius ratios are negligible with respect to unity. All remaining nonlinear terms are retained. The equations are valid for beams with mass centroid axis and area centroid (tension) axis offsets from the elastic axis, nonuniform mass and stiffness section properties, variable pretwist, and a small precone angle. The strain-displacement relations are developed from an exact transformation between the deformed and undeformed coordinate systems. These nonlinear relations form an important contribution to the final equations. Several nonlinear structural and inertial terms in the final equations are identified that can substantially influence the aeroelastic stability and response of hingeless helicopter rotor blades.

Hodges, D. H.↗
NASA NTRS1977

NSEG: A segmented mission analysis program for low and high speed aircraft. Volume 3: Demonstration problems

Program NSEG is a rapid mission analysis code based on the use of approximate flight path equations of motion. Equation form varies with the segment type, for example, accelerations, climbs, cruises, descents, and decelerations. Realistic and detailed vehicle characteristics are specified in tabular form. In addition to its mission performance calculation capabilities, the code also contains extensive flight envelope performance mapping capabilities. For example, rate-of-climb, turn rates, and energy maneuverability parameter values may be mapped in the Mach-altitude plane. Approximate take off and landing analyses are also performed. At high speeds, centrifugal lift effects are accounted for. Extensive turbojet and ramjet engine scaling procedures are incorporated in the code.

Hague, D. S.↗
NASA NTRS1986

Lower mass limit of an evolving interstellar cloud and chemistry in an evolving oscillatory cloud

Simultaneous solution of the equation of motion, equation of state and energy equation including heating and cooling processes for interstellar medium gives for a collapsing cloud a lower mass limit which is significantly smaller than the Jeans mass for the same initial density. The clouds with higher mass than this limiting mass collapse whereas clouds with smaller than critical mass pass through a maximum central density giving apparently similar clouds (i.e., same Av, size and central density) at two different phases of its evolution (i.e., with different life time). Preliminary results of chemistry in such an evolving oscillatory cloud show significant difference in abundances of some of the molecules in two physically similar clouds with different life times. The problems of depletion and short life time of evolving clouds appear to be less severe in such an oscillatory cloud.

Tarafdar, S. P.↗
NASA NTRS1972

Equations of motion for the X-14 Aircraft, phase 2 study

A study of the control and power requirements of the X-14 VTOL aircraft is presented. The complete equations of motion for X-14 are derived. The fundamental assumption is that the aircraft is a single rigid body. The equations of motion are derived with respect to a set of axes fixed to the aircraft. Additional assumptions used are that any wind disturbances are irrotational that the twin engines used on the aircraft rotate in the same direction at the same speed and that the engine exhaust is diverted by means of vanes to provide a direction varying thrust vector. The equations obtained are subsequently linearized about various reference conditions and numerical values for the trim parameters and the stability derivatives at these conditions are tabulated.

Hoffman, M. A.↗
NASA NTRS1991

A recursive approach to the equations of motion for the maneuvering and control of flexible multi-body systems

Interest lies in a mathematical formulation capable of accommodating the problem of maneuvering a space structure consisting of a chain of articulated flexible substructures. Simultaneously, any perturbations from the 'rigid body' maneuvering and any elastic vibration must be suppressed. The equations of motion for flexible bodies undergoing rigid body motions and elastic vibrations can be obtained conveniently by means of Lagrange's equations in terms of quasi-coordinates. The advantage of this approach is that it yields equations in terms of body axes, which are the same axes that are used to express the control forces and torques. The equations of motion are nonlinear hybrid differential quations. The partial differential equations can be discretized (in space) by means of the finite element method or the classical Rayleigh-Ritz method. The result is a set of nonlinear ordinary differential equations of high order. The nonlinearity can be traced to the rigid body motions and the high order to the elastic vibration. Elastic motions tend to be small when compared with rigid body motions.

Kwak, Moon K.↗
NASA NTRS1985

A Lagrange-D'Alembert formulation of the equations of motion of a helicopter carrying an externally suspended load

The exact nonlinear equations of motion are derived for a helicopter with an extenal load suspended by fore and aft, rigid-link cables. Lagrange's form of D'Alembert's principle is used. Ten degrees of freedom are necessary to represent the motion of this system in an inertial reference frame: six for the helicopter relative to inertial space and four for the load relative to the helicopter.

Weber, J. M.↗
NASA NTRS1980

Subprograms for integrating the equations of motion of satellites. FORTRAN 4

The subprograms for the formation of the right members of the equations of motion of artificial Earth satellites (AES), integration of systems of differential equations by Adams' method, and the calculation of the values of various functions from the AES parameters of motion are described. These subprograms are written in the FORTRAN 4 language and constitute an essential part of the package of applied programs for the calculation of navigational parameters AES.

Prokhorenko, V. I.↗
NASA NTRS1978

Fixed-base and two-body equations of motion for an Annular Momentum Control Device (AMCD)

Fixed base and two body equations of motion for an Annular Momentum Control Device (AMCD) are presented. An AMCD consists of a spinning annular rim which is suspended by noncontacting magnetic bearings and powered by a noncontacting linear electromagnetic motor. The fixed base equations are for a rigid AMCD rim suspended by magnetic bearings attached to a rigid fixed base. The two body equations are for a rigid AMCD rim suspended by magnetic bearings attached to a rigid body spacecraft. The fixed base equations are applicable to any potential ground based AMCD application such as energy storage.

Groom, N. J.↗
NASA NTRS1985

Optimization of cascade blade mistuning. I - Equations of motion and basic inherent properties

Attention is given to the derivation of the equations of motion of mistuned compressor blades, interpolating aerodynamic coefficients by means of quadratic expressions in the reduced frequency. If the coefficients of the quadratic expressions are permitted to assume complex values, excellent accuracy is obtained and Pade rational expressions are obviated. On the basis of the resulting equations, it is shown analytically that the sum of all the real parts of the eigenvalues is independent of the mistuning introduced into the system. Blade mistuning is further treated through the aerodynamic energy approach, and the limiting vibration modes associated with alternative mistunings are identified.

Nissim, E.↗
NASA NTRS1983

Efficient Algorithms for Computing Trim and Small-Disturbance Equations of Motion of Aircraft Coordinated and Uncoordinated, Steady, Steep Turns

The development of computational algorithms that permit efficient calculation of aircraft trim states and of the associated small disturbance equations of motion for a systematic investigation of the statics and dynamics of aircraft in coordinated and uncoordinated, steady, steep turning flight is reported. The efficiency in the trim computation is realized by decoupling the governing equations. The small disturbance equations of motion, which are given in a general body axis system, include aerodynamic acceleration derivatives; they are cast in a familiar first order, vector matrix format of modern system theory. These algorithms were applied to a variety of rotorcraft simulation models. Results pertaining to a simulated hingeless rotor helicopter are also presented

Chen, Robert T. N.↗
NASA NTRS2016

Documenting the NASA Armstrong Flight Research Center Oblate Earth Simulation Equations of Motion and Integration Algorithm

A desire for more complete documentation of the National Aeronautics and Space Administration (NASA) Armstrong Flight Research Center (AFRC), Edwards, California legacy code used in the core simulation has led to this e ort to fully document the oblate Earth six-degree-of-freedom equations of motion and integration algorithm. The authors of this report have taken much of the earlier work of the simulation engineering group and used it as a jumping-o point for this report. The largest addition this report makes is that each element of the equations of motion is traced back to first principles and at no point is the reader forced to take an equation on faith alone. There are no discoveries of previously unknown principles contained in this report; this report is a collection and presentation of textbook principles. The value of this report is that those textbook principles are herein documented in standard nomenclature that matches the form of the computer code DERIVC. Previous handwritten notes are much of the backbone of this work, however, in almost every area, derivations are explicitly shown to assure the reader that the equations which make up the oblate Earth version of the computer routine, DERIVC, are correct.

integrators↗