Showing papers on "Equations of motion published in 1972"

Dynamics of Atmospheric Flight

Abstract: : The book treats the dynamics of atmospheric flight, with special reference to the stability and control of airplanes. An extensive set of numerical examples covers STOL airplane, subsonic jet transport, hypersonic airplane, stability augmentation, and wind and density gradients. The book completely covers equations of motion, including effects of round rotating Earth and distortional motion. There are complete chapters on human pilots and handling qualities and flight in turbulence, with numerical examples for a jet transport. Small-perturbation equations for longitudinal and lateral motion are presented in convenient matrix forms, both in time domain and Laplace transforms, dimensional and non-dimensional. (Author)

Governing Equations for Vibrating Constrained-Layer Damping Sandwich Plates and Beams

Abstract: A simple differential equation is derived to describe constrained-layer damping in nonsymmetric sandwich plates and beams composed of isotropic and homogeneous layers. The natural boundary conditions related to this equation are determined and some typical numerical results obtained by this equation are given. The equation is valid within the linear theories of elasticity and viscoelasticity in the absence of any constraints on thicknesses, positions, symmetries, and densities of the layers.

Analytic continuation of classical mechanics for classically forbidden collision processes.

Abstract: Classically forbidden processes are those that cannot take place via ordinary classical dynamics. Within the framework of classical S‐matrix theory, however, classical mechanics can be analytically continued and classical‐limit approximations obtained for these classically forbidden, or weak transition amplitudes (i.e., S‐matrix elements). The most powerful and general way of analytically continuing classical mechanics for a complex dynamical system is to integrate the equations of motion themselves through the classically inaccessible regions of phase space. Success in calculating these analytically continued trajectories is reported in this work; with certain special features of these complex‐valued trajectories recognized and taken account of, it is seen that they are essentially as easy to deal with numerically as ordinary (i.e., real) classical trajectories. Application to the linear A+BC collision (vibrational excitation) gives excellent results; transition probabilities as small as 10−11 (the smallest ones available for comparison) have been obtained, agreement with the exact quantum mechanical values being within a few percent.

Variational Formulation of Dynamics of Fluid-Saturated Porous Elastic Solids

Abstract: Biot's theory of acoustic propagation in saturated porous elastic media has been used to systematically develop a Gurtin type variational principle which has been discretized by the finite element idealization. The resulting matrix equations of motion have been integrated in time by a general step-by-step integration scheme.

On a quadratic first integral for the charged particle orbits in the charged Kerr solution

Abstract: Associated with the charged Kerr solution of the Einstein gravitational field equation there is a Killing tensor of valence two. The Killing tensor, which is related to the angular momentum of the field source, is shown to yield a quadratic first integral of the equation of the motion for charged test particles.

Elastic Dislocation Theory for a Self‐Gravitating Elastic Configuration with an Initial Static Stress Field

Abstract: Summary The linearized equations of motion and linearized boundary and continuity conditions governing small elastic-gravitational disturbances away from equilibrium of an arbitrary, uniformly rotating, self-gravitating, perfectly elastic Earth model with an arbitrary initial static stress field are derived. The appropriate form of Rayleigh's variational principle and of the Betti reciprocal theorem and the Volterra dislocation relation for such a configuration are given. The latter is then used to derive an explicit expression for the equivalent body forces to be applied in the absence of a seismic dislocation in order to produce a dynamical response of the Earth model equivalent to that produced by the dislocation. It is found that if the initial static stress in the vicinity of the dislocation is purely hydrostatic, then a point tangential displacement dislocation has as an exactly equivalent body force the familiar double couple of moment 0, A0s0. If however the hypocentral static stress field has a deviatoric part, then additional equivalent body forces must be used properly to model a seismic dislocation. The necessary additional equivalent forces are explicitly exhibited; theoretically their existence provides a method of estimating hypocentral stresses, but the application of any such method is probably premature.

Summary of transformation equations and equations of motion used in free flight and wind tunnel data reduction and analysis

Abstract: Basic formulations for developing coordinate transformations and motion equations used with free-flight and wind-tunnel data reduction are presented. The general forms presented include axes transformations that enable transfer back and forth between any of the five axes systems that are encountered in aerodynamic analysis. Equations of motion are presented that enable calculation of motions anywhere in the vicinity of the earth. A bibliography of publications on methods of analyzing flight data is included.

Vibration and Stability of a Uniformly Curved Tube Conveying Fluid

Abstract: This paper presents a theoretical study of the vibration and stability of a uniformly curved tube containing flowing fluid. The assumption of the inextensibility of the tube is applied to derive the equation of motion. A solution for the natural frequency is obtained and numerical results are presented. The effects of flow velocity, fluid pressure, and the Coriolis force on the natural frequency are discussed. It is shown that when the flow velocity and fluid pressure exceed a certain value, the tube becomes subject to buckling‐type instability. Critical loads in terms of the flow velocity and fluid pressure are presented for fixed‐fixed, hinged‐hinged, and fixed‐hinged end conditions.

Dynamics of magnetic domain walls

Abstract: Equations of motion for a flexed domain wall are derived. Velocity dependence of wall thickness is neglected. The orientation angle of the wall‐moment is canonically conjugate to the wall position. Walker's uniform‐motion solution is stable. At higher fields, the wall oscillates. A general method of evaluating the non‐linear mobility in the limit of small damping and small field is given.

Electromagnetic Hydrodynamics of Liquid Crystals

Abstract: We have developed a set of differential equations which describe the hydrodynamics of a conducting liquid crystal subjected to external electromagnetic fields. The mechanical forces resulting from the interaction of the fields with the liquid are expressed as operations on the Maxwell stress tensor. The use of this description allows a concise statement of the equations of motion. As an example of the validity of the formalism, we rigorously solve the boundary-value problem associated with the Williams domain (vortex) mode in nematic liquids. Using standard constitutive relations, physical boundary conditions, and experimentally measured material $p$-azoxylanisole constants, we quantitatively reproduce the significant experimental observations.

Dislocation group dynamics III. similarity solutions of the continuum approximation

Abstract: The continuum approximation replaces a group of discrete dislocations by a continuous dislocation density. It is shown that this formalism possesses similarity solutions of the equations of motion with an arbitrary power law stress-velocity relationship in parallel with the similarity motions of the discrete group. The time scale is identical in the two cases. As with the static continuum approximation, the dynamic approximation leads to a much larger range of explicit solutions than is available for the discrete group.

Planck-like Distributions in Classical Nonlinear Mechanics

Abstract: The classical equations of motion for a finite chain of particles with a nearest-neighbor Lennard-Jones interaction are solved numerically and, for a broad class of initial conditions, Planck-like distributions are obtained for the time averages of the energies of the normal modes. For physical values of the parameters in the problem, the action constant entering such a distribution turns out to be of the order of magnitude of Planck's constant.

Resonance Rotation of Celestial Bodies and Cassini's Laws

Abstract: Celestial body rotation about its center of mass, taking into account the body orbit evolution, is considered. Non-linear evolution equations of motion are constructed. Empirical Cassini's laws describing the Moon motion result from these equations as their stationary points. Bifurcation conditions of steady motions are written out and conditions of their stability are investigated. Hypothesis of Mercury's resonance motion analogous to the ‘motion by Cassini’ is discussed. Consequences of this hypothesis are considered.

Quantum Mechanical Studies of the Vibrational Excitation of H2 by Li

Abstract: The quantum mechanical equations of motion for the vibrational excitation of an harmonic are solved via the set of coupled Volterra equations describing that motion. The numerical techniques which allow accurate inclusion of closed channels are developed. The equations are solved for an ab initio interaction potential and for a model potential. The ab initio interaction potential gives results in poor agreement with experiments. A comparison with the results of a two‐parameter model potential, which can give results close to the experimental results, provides some insight into features of the surface which are important for vibration excitation.

Static solutions of the Brans--Dicke equations.

Abstract: It is shown that, given any static solution of the Einstein vacuum equations, a corresponding family of static vacuum solutions of the Brans-Dicke equations can be written down by inspection. Spherically and axially symmetric fields are considered explicitly. It is demonstrated how some solutions of the Brans-Dicke equations may be obtained without having to solve any field equations explicitly at all.

On propagation of long waves in curved ducts

Abstract: A two‐dimensional detailed study of the behavior of long waves in curved ducts and in junctions between straight and curved ducts will be given. The mathematical treatment of the problem utilizes the method of separation of variables. Solutions and expressions for principal mode of the wave are obtained by using the linearized equation of motion solved for its characteristic values. The unavoidable approximations in the numerical solutions of the cylindrical functions are due to use of series expansion of Bessel functions and from restrictions necessary to solve infinite matrices.

Parametric resonance of stiffened rectangular plates

Abstract: Stiffened rectangular plates parametric instability under in-plane sinusoidal dynamic forces, using mathematical model with stiffeners as discrete elements

Investigation of Nonlinear Motion Experienced on a Slender-Wing Research Aircraft

Abstract: A combined experimental and theoretical study is described of the nature of the nonlinear lateral oscillation experienced by a slender-wing aircraft at high angles of attack. Flight tests of the Handley Page 115 research aircraft confirmed the expectation that the Dutch roll oscillation is undamped at high angles of attack, and also showed that a limit cycle develops, with steady amplitude in bank angle of about 30°. The measured stability derivatives are given, together with the responses obtained from the RAE flight dynamics simulator, in which the digital computation of the equations of motion uses the static aerodynamic data directly from a controlled model in a wind tunnel. The motion is also analysed theoretically using a new approximate method for obtaining solutions for nonlinear differential equations. The analysis gives the conditions sufficient for the existence of a sustained oscillation, and its amplitude and frequency in terms of the aerodynamic and inertia properties of the aircraft.

Equations of motion for the sources of asymptotically flat spaces

Abstract: We present an exact calculation that leads to the equations of motion (which naturally contain gravitational radiation reaction terms) of a system subject to no external forces. The novelty of our approach lies in the fact that the system is to be considered as the source of an asymptotically flat space and that all the revelant physical quantities such as the velocity νμ, 4‐momentum pμ, angular momentum‐center of mass tensor Sμν (as well as higher moments) are then defined in terms of surface integrals taken at infinity. A subset of the Einstein equations (equivalent to Bondi's supplemantary conditions) then yields the time‐evolution equations for these variables.

Hydrodynamics of suspensions

Abstract: Several experiments have been reported that indicate a significant difference between the hydrodynamic properties of suspensions and conventional viscous fluids. According to measurements made with different types of viscometers, the viscosity of the suspension in many cases exhibits anomalous behavior that is not compatible with descriptions of the suspension as a conventional Newtonian or non-Newtonian fluid [1–3]. Stratification into solid and liquid phases also plays an important role in the flow of suspensions. In particular, in the significant practical case of Poiseuille flow of an equidense suspension in a circular pipe, wall and core effects were observed. The wall effect, i. e., the migration of suspended particles toward the pipe centerline and the corresponding reduction of the solid phase concentration near the walls, was observed in suspensions of different nature (particles of different nature and form in a viscous liquid [4] and in blood, i.e., suspensions of blood corpuscles in blood plasma [3, 5]) over a wide range of values of average concentration. In contrast with the wall effect, the core effect, which involves an increase of the solid phase concentration in an annular region at a distance of 0.5–0.7 R from the pipe centerline (R is the pipe radius), was observed only for small values of the average concentration [6, 7]. An experiment [8] showed that the presence of the core effect is associated with rotation of the solid particles. When the center of gravity of each of the particles was shifted, their rotation was hindered. In the absence of rotation the particles always migrated toward the tube centerline. Many studies have been devoted to explanation of the wall effect. The magnitude of the transverse force acting on a sphere rotating in a translational viscous liquid flow was calculated in [9] (the limits of applicability of the resulting formula do not permit examination of those values of the parameters for which the core effect is observed in practice). If we substitute the expression for force into the particle equation of motion and integrate, we find that the particle trajectory approaches the tube axis asymptotically [10], regardless of the initial position of the particle. Another approach is the averaged description of the suspension behavior. The Navier-Stokes equation, with a coefficient of viscosity that depends in known fashion on the local solid phase concentration, was written as the equation of motion for the steady flow of a suspension in a circular tube. The variational principle (the principle of minimum energy dissipation) was formulated to find the concentration distribution. The wall effect was also obtained using this approach [11, 12]. An analogous study using the Kessonmodel yielded practically no results [13]. The complete system of equations of two-fluid hydrodynamics was recently constructed in which the fluid and the dispersed phase are considered as two interpenetrating, interacting continua, with rotation and deformation of the dispersed phase particles being taken into consideration (Nguen Van D'ep, Candidate's dissertation: Some questions on the Hydrodynamics of Structured Fluids [in Russian], Voronezh State University, Voronezh, 1968). However, the practical use of this system for solving, for example, the Poiseuille problem appears to be difficult, since it is necessary to know in detail the interaction forces between the phases. In the present study we use the single-fluid approximation, i.e., the quantities introduced (velocity, density, and so on) characterize the motion of the suspensions as a whole, and not any single phase. Only two quantities characterize the solid phase proper: the bulk concentration and the local angular velocity of the solid particles. This approach does not require knowledge of the interaction forces between the phases, and the final equations are considerably simpler than those for the two-fluid description. The thermodynamics of irreversible processes is used to construct a closed system of equations that includes the diffusion equations and the generalized moment of momentum equation, which make it possible to find the three-dimensional distribution of the concentration and the angular velocity of the solid particles. The generalized Fick law, which contains three additional diffusion coefficients, is obtained. In contrast with classical models, the viscous stress tensor is asymmetric, and its antisymmetric part is proportional to the difference in the angular rates of rotation of the solid particles and the suspension as a whole. From these equations follow under certain particular assumptions the equations of the theory of a fluid with internal rotation [14]. As an example of the application of the theory, the Poiseuille problem of flow in a flat channel is solved. The concentration distributions obtained agree well with the experimental data described above.