Showing papers on "Equations of motion published in 1984"
TL;DR: In this article, the behavior of a free electron laser in the high gain regime and the conditions for the emergence of a collective instability in the electron beam-undulator-field system were studied.
1,224 citations
01 Aug 1984
Abstract: Stability in dynamical systems subject to some law of force is considered. This leads to a set of differential equations which govern the motion. (AIP)
896 citations
TL;DR: A transition from the Kadanoff-Baym Green's function equations of motion to the Boltzmann equation, and specifications of the respective limit, are examined in detail in this paper.
675 citations
TL;DR: In this article, the authors consider a single degree of freedom elastic system undergoing frictional slip, where the system is represented by a block (slider) slipping at speed V and connected by a spring of stiffness k to a point at which motion is enforced at speedV 0.
Abstract: We consider quasistatic motion and stability of a single degree of freedom elastic system undergoing frictional slip. The system is represented by a block (slider) slipping at speed V and connected by a spring of stiffness k to a point at which motion is enforced at speed V 0 We adopt rate and state dependent frictional constitutive relations for the slider which describe approximately experimental results of Dieterich and Ruina over a range of slip speeds V . In the simplest relation the friction stress depends additively on a term A In V and a state variable θ; the state variable θ evolves, with a characteristic slip distance, to the value − B In V , where the constants A, B are assumed to satisfy B > A > 0. Limited results are presented based on a similar friction law using two state variables. Linearized stability analysis predicts constant slip rate motion at V 0 to change from stable to unstable with a decrease in the spring stiffness k below a critical value k cr . At neutral stability oscillations in slip rate are predicted. A nonlinear analysis of slip motions given here uses the Hopf bifurcation technique, direct determination of phase plane trajectories, Liapunov methods and numerical integration of the equations of motion. Small but finite amplitude limit cycles exist for one value of k , if one state variable is used. With two state variables oscillations exist for a small range of k which undergo period doubling and then lead to apparently chaotic motions as k is decreased. Perturbations from steady sliding are imposed by step changes in the imposed load point motion. Three cases are considered: (1) the load point speed V 0 is suddenly increased; (2) the load point is stopped for some time and then moved again at a constant rate; and (3) the load point displacement suddenly jumps and then stops. In all cases, for all values of k :, sufficiently large perturbations lead to instability. Primary conclusions are: (1) ‘stick-slip’ instability is possible in systems for which steady sliding is stable, and (2) physical manifestation of quasistatic oscillations is sensitive to material properties, stiffness, and the nature and magnitude of load perturbations.
489 citations
TL;DR: In this paper, a study is made of the wave disturbance generated by a localized steady pressure distribution travelling at a speed close to the long-water-wave phase speed on water of finite depth.
Abstract: A study is made of the wave disturbance generated by a localized steady pressure distribution travelling at a speed close to the long-water-wave phase speed on water of finite depth The linearized equations of motion are first used to obtain the large-time asymptotic behaviour of the disturbance in the far field; the linear response consists of long waves with temporally growing amplitude, so that the linear approximation eventually breaks down owing to finite-amplitude effects A nonlinear theory is developed which shows that the generated waves are actually of bounded amplitude, and are governed by a forced Korteweg-de Vries equation subject to appropriate asymptotic initial conditions A numerical study of the forced Korteweg-de Vries equation reveals that a series of solitons are generated in front of the pressure distribution
282 citations
TL;DR: In this article, the distortion and harmonic generation in the near field of a finite amplitude sound beam are considered, assuming time-periodic but otherwise arbitrary on-source conditions, and the amplitude and phase of the fundamental and first few harmonics are calculated along the beam axis, and across the beam at various ranges from the source.
Abstract: Distortion and harmonic generation in the nearfield of a finite amplitude sound beam are considered, assuming time‐periodic but otherwise arbitrary on‐source conditions. The basic equations of motion for a lossy fluid are simplified by utilizing the parabolic approximation, and the solution is derived by seeking a Fourier series expansion for the sound pressure. The harmonics are governed by an infinite set of coupled differential equations in the amplitudes, which are truncated and solved numerically. Amplitude and phase of the fundamental and the first few harmonics are calculated along the beam axis, and across the beam at various ranges from the source. Two cases for the source are considered and compared: one with a uniformly excited circular piston, and one with a Gaussian distribution. Various source levels are used, and the calculations are carried out into the shock region. The on‐axis results for the fundamental amplitude are compared with results derived using the linearized solution modified with various taper functions. Apart from a nonlinear tapering of the amplitude along and near the axis, the results are found to be very close to the linearized solution for the fundamental, and for the second harmonic close to what is obtained from a quasilinear theory. The wave profile is calculated at various ranges. An energy equation for each harmonic is obtained, and shown to be equivalent within our approximation to the three‐dimensional version of Westervelt’s energy equation. Recent works on one‐dimensional propagation are reviewed and compared.
267 citations
01 Jan 1984
TL;DR: In this article, the Hamiltonian formalism of general onedimensional systems of hydrodynamic type was developed and the equations of motion of this system have the form of first order equations, linear in the derivatives.
Abstract: I. Riemannian geometry of multidimensional Poisson brackets of hydrodynamic type. In [1] we developed the Hamiltonian formalism of general onedimensional systems of hydrodynamic type. Now suppose that a system of the type of an ideal fluid (possibly with internal degrees of freedom) is described by a set of field variables u = (u(x)), i = 1, . . . , N , in a space with coordinates x = (x), α = 1, . . . , n. The equations of motion of this system have the form of first order equations, linear in the derivatives:
258 citations
TL;DR: In this article, the authors derived an expression for the average stress tensor, which is non-isotropic although the local stress tensors in the constituent phases are isotropic (viscosity is neglected).
Abstract: Equations of motion correct to the first order of the gas concentration by volume are derived for a dispersion of gas bubbles in liquid through systematic averaging of the equations on the microlevel. First, by ensemble averaging, an expression for the average stress tensor is obtained, which is non-isotropic although the local stress tensors in the constituent phases are isotropic (viscosity is neglected). Next, by applying the same technique, the momentum-flux tensor of the entire mixture is obtained. An equation expressing the fact that the average force on a massless bubble is zero leads to a third relation. Complemented with mass-conservation equations for liquid and gas, these equations appear to constitute a completely hyperbolic system, unlike the systems with complex characteristics found previously. The characteristic speeds are calculated and shown to be related to the propagation speeds of acoustic waves and concentration waves.
246 citations
TL;DR: In this article, the dynamic behavior of rigid-block structures resting on a rigid foundation subjected to horizontal harmonic excitation is examined, and several possible modes of steady-state response are detected, and analytical procedures are developed for determining the amplitudes of the predominant modes and for performing stability analyses.
Abstract: The dynamic behavior of rigid-block structures resting on a rigid foundation subjected to horizontal harmonic excitation is examined. For slender structures, the nonlinear equation of motion is approximated by a piecewise linear equation. Using this approximation for an initially quiescent structure, safe or no-toppling and unsafe regions are identified in an excitation amplitude versus excitation frequency plane. Furthermore, several possible modes of steady-state response are detected, and analytical procedures are developed for determining the amplitudes of the predominant modes and for performing stability analyses. It is shown that the produced stability diagrams can be beneficial to assessing the toppling potential of a rigid-block structure under a given amplitude-frequency combination of harmonic excitation; in this manner the integration of the equation of motion is circumvented.
225 citations
TL;DR: In this paper, the authors introduce a class of models for the motion of a boundary between time-dependent phase domains in which the interface itself satisfies an equation of motion, and formulate local equations of motion as tractable simplifications of the complex nonlocal dynamics that govern moving-interface problems.
Abstract: We introduce a class of models for the motion of a boundary between time-dependent phase domains in which the interface itself satisfies an equation of motion. The intended application is to systems for which competing stabilizing and destabilizing forces act on the phase boundary to produce irregular or patterned structures, such as those which occur in solidification. We discuss the kinematics of moving interfaces in two or more dimensions in terms of their intrinsic geometric properties. We formulate local equations of motion as tractable simplifications of the complex nonlocal dynamics that govern moving-interface problems. Special solutions for dendritic crystal growth and their stability are analyzed in some detail.
205 citations
TL;DR: In this article, a phenomenological stochastic model of the process of thermal and quantal fluctuations of a damped harmonic oscillator is presented, and the divergence of the momentum dispersion associated with the Markovian limit is removed by a Drude regularization.
Abstract: A phenomenological stochastic modelling of the process of thermal and quantal fluctuations of a damped harmonic oscillator is presented. The divergence of the momentum dispersion associated with the Markovian limit is removed by a Drude regularization. The variances of position and momentum are evaluated in closed form at arbitrary temperature and for arbitrary damping. Properties of real and imaginary time correlation functions are discussed, and a spectral decomposition of the equilibrium density matrix is given.
TL;DR: In this paper, a non-linear equation of the free motion of a heavy elastic cable about a deformed initial configuration is developed, which is obtained via a Galerkin procedure, an approximate solution is pursued through a perturbation method.
Abstract: Continuum non-linear equations of free motion of a heavy elastic cable about a deformed initial configuration are developed. Referring to an assumed mode technique one ordinary equation for the cable planar motion is obtained via a Galerkin procedure, an approximate solution of which is pursued through a perturbation method. Suitable nondimensional results are presented for the vibrations in the first symmetric mode with different values of the cable properties. Which procedure is the proper one to account consistently for the non-linear kinematical relations of the cable in one ordinary equation of motion is discussed.
TL;DR: In this article, a simple Lorentz transformation is used to measure the velocity and acceleration of an object in a four-dimensional space, and then the transformation is applied to the velocity of the object in the four dimensions of the space.
Abstract: 1. Kinematics in Inertial Axes.- 1.1 The "Aether" in the Nineteenth Century.- 1.2 Some Experimental Evidence.- 1.3 Einstein's Relativity Postulates.- 1.4 Time and Length Standards. Synchronization.- 1.5 The "Simple" Lorentz Transformation.- 1.6 More General Lorentz Transformations.- 1.7 Time Dilatation and Proper Time.- 1.8 Length Measurements.- 1.9 Volume and Surface Elements.- 1.10 Visual Perception of Objects in Motion.- 1.11 Transformation of Velocities and Accelerations.- 1.12 Four-Vectors.- 1.13 Kinematics in Four Dimensions.- Problems.- 2. Dynamics in Inertial Axes.- 2.1 Equation of Motion of a Point Mass.- 2.2 Mass and Energy.- 2.3 A Few Simple Trajectories.- 2.4 Transformation Equations for Force, Energy, and Momentum.- 2.5 Four-Dimensional Dynamics.- 2.6 Systems of Points.- 2.7 Elastic Collisions.- 2.8 Motion of a Point with Variable Rest Mass.- 2.9 Rocket Acceleration.- 2.10 Inelastic Collisions.- 2.11 Incoherent Matter.- 2.12 The Kinetic Energy-Momentum Tensor.- 2.13 The Total Energy-Momentum Tensor.- Problems.- 3. Vacuum Electrodynamics in Inertial Axes.- 3.1 Transformation Formulas for the Sources.- 3.2 Transformation Equations for the Fields.- 3.3 Force on a Charged Particle.- 3.4 Four-Currents.- 3.5 The Electromagnetic Tensors.- 3.6 Potentials.- 3.7 Transformation of a Plane Wave: The Doppler Effect.- 3.8 The Lienard-Wiechert Fields.- 3.9 Fields of a Charge in Uniform Motion.- 3.10 Fields of a Static Dipole in Uniform Motion.- 3.11 Radiation from an Antenna in Uniform Motion.- 3.12 Radiation from a Moving Oscillation Dipole.- 3.13 Doppler Spectrum from a Moving Source.- Problems.- 4. Fields in Media in Uniform Translation.- 4.1 Polarization Densities.- 4.2 Constitutive Equations.- 4.3 Some Useful Forms of Maxwell's Equations.- 4.4 Point Charge Moving Uniformly in a Dielectric Medium.- 4.5 The Cerenkov Effect.- 4.6 Waves in a Moving Dielectric. The Fresnel Dragging Coefficient.- 4.7 Green's Dyadic for a Moving Dielectric.- 4.8 Electric Dipole Radiating in a Moving Dielectric.- Problems.- 5. Boundary-Value Problems for Media in Uniform Translation.- 5.1 Boundary Conditions.- 5.2 Dielectric Slab Moving in Time-Independent Fields.- 5.3 The Wilsons' Experiment.- 5.4 Sliding Contacts. A Simple Problem.- 5.5 Material Bodies Moving at Low Velocities.- 5.6 Conductors Moving in a Pre-Existing Static Magnetic Field.- 5.7 Circuit Equations.- 5.8 Motional E.M.F..- 5.9 Normal Incidence of a Time-Harmonic Plane Wave on a Moving Mirror.- 5.10 Arbitrary Time-Dependence of the Incident Plane Wave.- 5.11 Oblique Incidence of a Time-Harmonic Plane Wave on a Moving Mirror.- 5.12 A Time-Harmonic Plane Wave Incident on a Dielectric Medium.- 5.13 Reflection of a Plane Wave on a Moving Medium of Finite Conductivity.- 5.14 Revisiting the Boundary Conditions at a Moving Interface.- 5.15 Scattering by a Cylinder Moving Longitudinally.- 5.16 Scattering by a Cylinder Moving Transversely.- 5.17 Three-Dimensional Scattering by Moving Bodies.- 5.18 The Quasistationary Method.- Problems.- 6. Electromagnetic Forces and Energy.- 6.1 Surface and Volume Forces in Vacuum.- 6.2 Maxwell's Stress Tensor.- 6.3 A Few Simple Force Calculations.- 6.4 Radiation Pressure on a Moving Mirror.- 6.5 Radiation Force on a Dielectric Cylinder.- 6.6 Static Electric Force on a Dielectric Body.- 6.7 Magnetic Levitation.- 6.8 Levitation on a Line Current.- 6.9 Electromagnetic Energy in an Inertial System.- 6.10 Four-Dimensional Formulation in Vacuum.- 6.11 The Electromagnetic Energy-Momentum Tensor in Material Media.- Problems.- 7. Accelerated Systems of Reference.- 7.1 Coordinate Transformations.- 7.2 The Metric Tensor.- 7.3 Examples of Transformations.- 7.4 Coordinates and Measurements.- 7.5 Time and Length.- 7.6 Four-Vectors and Tensors.- 7.7 Three-Vectors.- 7.8 Velocities and Volume Densities.- 7.9 Covariant Derivative.- Problems.- 8. Gravitation.- 8.1 Inertial and Gravitational Masses.- 8.2 The Principle of Equivalence.- 8.3 Curvature.- 8.4 Einstein's Equations.- 8.5 The Small-Field Approximation.- 8.6 Gravitational Frequency Shift.- 8.7 Time Measurement Problems.- 8.8 Some Important Solutions of Einstein's Equations.- 8.9 Point Dynamics.- 8.10 Motion in the Schwarzschild Metric.- 8.11 Motion of a Photon in the Schwarzschild Metric.- 8.12 Strongly Concentrated Masses.- 8.13 Static Cosmological Metrics.- 8.14 Nonstatic Cosmological Metrics.- 8.15 Recent Cosmological Observations.- Problems.- 9. Maxwell's Equations in a Gravitational Field.- 9.1 Field Tensors and Maxwell's Equations.- 9.2 Maxwell's Equations in Rotating Coordinates.- 9.3 Transformation Equations for Fields and Sources.- 9.4 Constitutive Equations in Vacuum.- 9.5 Constitutive Equations in a Time-Orthogonal Metric.- 9.6 Constitutive Equations in Material Media.- 9.7 The Co-Moving Frame Assumption.- 9.8 Boundary Conditions.- Problems.- 10. Electromagnetism of Accelerated Bodies.- 10.1 Conducting Body of Revolution Rotating in a Static Magnetic Field.- 10.2 Conducting Sphere Rotating in a Uniform Magnetic Field.- 10.3 Motional E.M.F.- 10.4 Generators with Contact Electrodes.- 10.5 Dielectric Body of Revolution Rotating in a Static Field.- 10.6 Rotating Permanent Magnets.- 10.7 Scattering by a Rotating Circular Dielectric Cylinder.- 10.8 Scattering by a Rotating Circular Conducting Cylinder.- 10.9 Scattering by a Rotating Dielectric Body of Revolution.- 10.10 Scattering by a Rotating Sphere.- 10.11 Reflection from a Mirror in Arbitrary Linear Motion.- 10.12 Reflection from an Oscillating Mirror, at Normal Incidence.- 10.13 Reflection from an Oscillating Mirror, at Oblique Incidence.- 10.14 Scattering by Other Moving Surfaces.- Problems.- 11. Field Problems in a Gravitational Field.- 11.1 Fields Associated with Rotating Charges.- 11.2 Schiff's Paradox.- 11.3 Kennard's Experiment.- 11.4 Optical Rotation Sensors.- 11.5 Scattering by a Rotating Body of Arbitrary Shape.- 11.6 Transformation of an Incident Wave to Rotating Coordinates.- 11.7 Scattered Field in Rotating Coordinates.- 11.8 Two Examples.- 11.9 Low Frequency Scattering by Rotating Cylinders.- 11.10 Quasistationary and Relativistic Fields.- 11.11 Axes in Hyperbolic Motion.- 11.12 The Induction Law.- 11.13 Maxwell's Equations in a Schwarzschild Metric.- 11.14 Light Deflection in a Gravitational Field.- Problems.- Appendix A. Complements of Kinematics and Dynamics.- A.1 Transformation Matrix for the "Parallel" Transformation.- A.2 Transformation with Rotation.- A.3 Transformation of Velocities.- A.4 Relationship Between Force and Acceleration.- A.5 Equations of Motion in Cylindrical Coordinates (r,?,z).- A.6 Equations of Motion in Spherical Coordinates (R,?,?).- Appendix B. Dyadics.- B.1 The Dyadic Notation.- B.2 Operators on Dyadics.- B.3 Green's Dyadic.- Appendix C. Basis Vectors.- Appendix D. Moving Open Circuits.- List of Symbols.- Some Useful Numerical Constants.- References.
TL;DR: In this article, the weakly nonlinear, resonant response of a damped, spherical pendulum (length l, damping ratio δ, natural frequency ω 0 ) to the planar displacement e l cos ω t (e ⪡ 1) of its point of suspension is examined in a four-dimensional phase space in which the coordinates are slowly varying amplitudes of a sinusoidal motion.
TL;DR: In this article, a principle of local definiteness is introduced which is needed beyond equations of motion and commutation relations to fix the theory uniquely, and it also allows us to formulate local stability.
Abstract: We discuss quantum fields on Riemannian space-time. A principle of local definiteness is introduced which is needed beyond equations of motion and commutation relations to fix the theory uniquely. It also allows us to formulate local stability. In application to a region with a time-like Killing vector field and horizons it yields the value of the Hawking temperature. The concept of vacuum and particles in a non-stationary metric is treated in the example of the Robertson-Walker metric and some remarks on detectors in non-inertial motion are added.
01 Mar 1984
TL;DR: In this article, the authors classified the equations governing the flow of viscoelastic liquids according to the symbol of their differential operators, and discussed propagation of singularities and conditions for a change of type.
Abstract: The equations governing the flow of viscoelastic liquids are classified according to the symbol of their differential operators. Propagation of singularities is discussed and conditions for a change of type are investigated. The vorticity equation for steady flow can change type when a critical condition involving speed and stresses is satisfied. This leads to a partitioning of the field of flow into subcritical and supercritical regions, as in the problem of transonic flow.
01 Mar 1984
TL;DR: In this article, the authors considered the Stokes equations for the flow of a viscous, incompressible fluid and the equations of linear plane-strain elasticity for the deformation of an isotropic, nearly-incoherent solid.
Abstract: : Interest here is in finite element discretizations of problems involving an incompressibility condition. As model problems we consider the Stokes equations for the flow of a viscous, incompressible fluid and the equations of linear plane-strain elasticity for the deformation of an isotropic, nearly incompressible solid. In both cases the incompressibility condition takes the form of a divergence constraint. Although this is the most simple formulation, the proper understanding of how an approximate method satisfies the constraint represents an important step towards the understanding of more complicated situations, involving e.g. the Navier-Stokes equations or the equations of nonlinear elasticity. The finite element methods we study have the property that the approximations to the velocities, respectively to the displacements, are continuous; such methods are generally referred to as conforming.
TL;DR: In this paper, an economical method of solving the equations of motion for two and three dimensional problems using non-orthogonal boundary-fitted meshes is described, which is intended for flows in which compressibility effects do not dominate.
Abstract: For three-dimensional fluid flows in complex geometries, it is convenient to make predictions using a non-orthogonal boundary-fitted mesh. The present paper describes an economical method of solving the equations of motion for two and three dimensional problems using such meshes. The locations on the mesh at which the depenent variables are calculated, and the methods used to solve the equations, are key issues in the development of a successful algorithm; these are discussed in the present paper. Results obtained when the proposed method is applied to several problems are also described. The method is intended for flows in which compressibility effects do not dominate.
TL;DR: In this paper, the one-dimensional imbricate nonlocal continuum is extended to two or three dimensions and a proper variational method is developed to derive the equations of motion from the principle of virtual work.
Abstract: The one-dimensional imbricate nonlocal continuum, which was developed in another paper in order to model strain-softening within zones of finite size, is extended here to two or three dimensions. The continuum represents a limit of a system of imbricated (overlapping) elements that have a fixed size and a diminishing cross section as the mesh is refined. The proper variational method for the imbricate continuum is developed, and the continuum equations of motion are derived from the principle of virtual work. They are of difference-differential type and involve not only strain averaging but also stress gradient averaging for the so-called broad-range stresses characterizing the forces within the characteristic volume of heterogeneous material. The gradient averaging may be defined by a difference operator, or an averaging integral, or by least-square fitting of a homogeneous strain field. A differential approximation with higher order displacement derivatives is also shown. The theory implies a boundary layer which requires special treatment. The blunt crack band model, previously used in finite element analysis of progressive fracturing, is extended by the present theory into the range of mesh sizes much smaller than the characteristic width of the crack band front. Thus, the crack band model is made part of a convergent discretization scheme. The nonlocal continuum aspects are captured by an imbricated arrangement of finite elements, which are of the usual type.
TL;DR: In this paper, it was shown that Sternberg's space is, in a natural way, a symplectic leaf of a reduced Poisson manifold and relations to a construction of Kummer's for dynamics on the cotangent bundle of a principle bundle are clarified.
Abstract: Wong [14] introduced equations of motion for a spin 0 particle in a Yang-Mills field which was widely accepted among physicists. It is shown that these are equivalent to the various mathematical formulations for the motion of such particles as given by the Kaluza-Klein formulation of Kerner [4], and those of Sternberg [11], and Weinstein [12]. In doing this, we show that Sternberg's space is, in a natural way, a symplectic leaf of a reduced Poisson manifold and relations to a construction of Kummer's [5] for dynamics on the cotangent bundle of a principle bundle are clarified.
TL;DR: In this article, the generalized equations of motion for elastic mechanism systems were developed by utilizing finite element theory, which provided the capability to model a general two-or three-dimensional complex elastic mechanism, to include the nonlinear rigid-body and elastic motion coupling terms in a general representation.
Abstract: Until recently, vibration effects have generally been neglected in the design of high-speed machines and mechanisms. This has been primarily due to the complexity of the mathematical analysis of mechanisms with elastic links. With the advent of high-speed computers and structural dynamics techniques, such as finite element analysis, this is no longer regarded as such a formidable task. To date, with few exceptions, the analysis of elastic mechanism systems have been limited to a single type of mechanism (i.e., a four-bar or slider-crank) modeled with a small number of simple finite elements (usually beam elements). This paper develops the generalized equations of motion for elastic mechanism systems by utilizing finite element theory. The derivation and final form of the equations of motion provide the capability to model a general two- or three-dimensional complex elastic mechanism, to include the nonlinear rigid-body and elastic motion coupling terms in a general representation, and to allow any finite element type to be utilized in the model. A discussion of a solution method, applications, as well as an experimental investigation of an elastic four-bar mechanism will be presented in subsequent publications.
TL;DR: In this paper, a two-dimensional melting of a solid phase change material in a rectangular enclosure heated from one side is simulated numerically by dividing the process in a large number of quasi-static steps.
Abstract: Two-dimensional melting of a solid phase change material in a rectangular enclosure heated from one side is simulated numerically. The simulations are carried out by dividing the process in a large number of quasi-static steps. In each quasi-static step, steady-state natural convection in the liquid phase is calculated by directly solving the governing equations of motion with a finite difference technique. This is used to predict the shape and motion of the solid-liquid boundary at the beginning of the next step. The predictions are found to be in good agreement with experiment. Influence of some of the governing parameters on the time development of the melting process is studied using the numerical simulation procedure.
TL;DR: In this paper, the free vibration problem of thin elastic cross-ply laminated circular cylindrical panels is considered and a theoretical unification as well as a numerical comparison of the thin shell theories most commonly used (in engineering applications) is presented.
TL;DR: In this article, an optimal guidance law for a vehicle pursuing a maneuvering evader is derived using the exact nonlinear equations of motion in the plane, using the complete knowledge of the evader's motion is available to the pursuer.
Abstract: Using the exact nonlinear equations of motion in the plane, an optimal guidance law for a vehicle pursuing a maneuvering evader is derived. It is assumed that a complete knowledge of the evader's motion is available to the pursuer. Both the pursuer and the evader move with constant velocities. The guidance law minimizes a weighted linear combination of the time of capture and the expended maneuvering energy. The equations of motion are solved in closed form in terms of elliptic integrals. Numerical results are presented in order to illustrate the advantages of the optimal guidance law as compared both with proportional navigation and a trajectory formed by a hard turn followed by a straight line. The extension of the approach to the three-dimensional case is also outlined.
TL;DR: In this article, the rectilinear motion and the conditions of reattachment and separation of a rigid body, in friction contact with another body are considered, and analytical expressions for the velocities and displacements are derived.
Abstract: The rectilinear motion and the conditions of reattachment and separation of a rigid body, in friction contact with another body are considered. A graphical representation of the motion is indicated, and analytical expressions for the velocities and displacements are derived. The existence of limiting values of velocity and displacement is shown for a special class of periodic ground motions which include harmonic motions. Also, the equations of motion and the conditions of reattachment and separation of a two degrees of freedom model of a sliding structure and foundation are derived. The numerical integration of the response of this system is carried out, as well as a parametric study showing the effect of different values of the mass ratio, coefficient of friction and amplitude of the ground acceleration. Use of results of the parametric study, concerning amount of slippage, resonance frequency ratios, minimum allowable frequency for sticked mode, etc. in the design for structural base isolation is indic...
TL;DR: In this article, a nonlinear equation of motion for quantum systems consisting of a single elementary constituent of matter is proposed, which is satisfied by pure states and by a special class of mixed states evolving unitarily.
Abstract: A novel nonlinear equation of motion is proposed for quantum systems consisting of a single elementary constituent of matter. It is satisfied by pure states and by a special class of mixed states evolving unitarily. But, in general, it generates a nonunitary evolution of the state operator. It keeps the energy invariant and causes the entropy to increase with time until the system reaches a state of equilibrium or a limit cycle.
Book•
01 Jan 1984
TL;DR: In this article, the authors present an analysis of the relationship between velocity gradient tensors and spin tensors, and show that the latter is a function of the acceleration of the tensors.
Abstract: 1 Kinematics of Flow.- 1.1 Introduction.- 1.2 Velocity Gradient Tensor.- 1.3 Rate of Deformation Tensor.- 1.4 Analysis of Strain Rates.- 1.5 Spin Tensor.- 1.6 Curvature-Twist Rate Tensor.- 1.7 Objective Tensors.- 1.8 Balance of Mass.- 1.9 Concluding Remarks.- 1.10 References.- 2 Field Equations.- 2.1 Introduction.- 2.2 Measures for Mechanical Interactions.- 2.3 Euler's Laws of Motion.- 2.4 Stress and Couple Stress Vectors.- 2.5 Stress and Couple Stress Tensors.- 2.6 Cauchy's Laws of Motion.- 2.7 Analysis of Stress.- 2.8 Energy Balance Equation.- 2.9 Entropy Inequality.- 2.10 Concluding Remarks.- 2.11 References.- 3 Couple Stresses in Fluids.- 3.1 Introduction.- 3.2 Constitutive Equations.- 3.3 Equations of Motion.- 3.4 Boundary Conditions.- 3.5 Steady Flow Between Parallel Plates.- 3.6 Steady Tangential Flow Between Two Coaxial Cylinders.- 3.7 Poiseuille Flow Through Circular Pipes.- 3.8 Creeping Flow Past a Sphere.- 3.9 Some Time-Dependent Flows.- 3.10 Stability of Plane Poiseuille Flow.- 3.11 Hydromagnetic Channel Flows.- 3.12 Some Effects on Heat Transfer.- 3.13 Concluding Remarks.- 3.14 References.- 4 Anisotropic Fluids.- 4.1 Introduction.- 4.2 Balance Laws.- 4.3 Microstructure of a Dumbbell-Shaped Particle.- 4.4 Field Equations.- 4.5 Constitutive Equations.- 4.6 Implications of the Second Law of Thermodynamics.- 4.7 Incompressible Fluids.- 4.8 Simple Shearing Motion.- 4.9 Orientation Induced by Flow.- 4.10 Poiseuille Flow Through Circular Pipes.- 4.11 Cylindrical Couette Flow.- 4.12 Concluding Remarks.- 4.13 References.- 5 Micro Fluids.- 5.1 Introduction.- 5.2 Description of Micromotion.- 5.3 Kinematics of Deformation.- 5.4 Conservation of Mass.- 5.5 Balance of Momenta.- 5.6 Microinertia Moments.- 5.7 Balance of Energy.- 5.8 Entropy Inequality.- 5.9 Constitutive Equations for Micro Fluids.- 5.10 Linear Theory of Micro Fluids.- 5.11 Equations of Motion.- 5.12 Concluding Remarks.- 5.13 References.- 6 Micropolar Fluids.- 6.1 Introduction.- 6.2 Skew-Symmetry of the Gyration Tensor and Microisotropy.- 6.3 Micropolar Fluids.- 6.4 Thermodynamics of Micropolar Fluids.- 6.5 Equations of Motion.- 6.6 Boundary and Initial Conditions.- 6.7 Two Limiting Cases.- 6.8 Steady Flow Between Parallel Plates.- 6.9 Steady Couette Flow Between Two Coaxial Cylinders.- 6.10 Pipe Poiseuille Flow.- 6.11 Micropolar Fluids with Stretch.- 6.12 Concluding Remarks.- 6.13 References.- Notation.
TL;DR: In this paper, the authors derived the 6 deg equations of motion for an aircraft incorporating variable wind terms and discussed the influence of wind shear on inputs to computing the aerodynamic coefficients, such as the effects of wind velocity vector rotation on relative angular rates of rotation and on the time rate of change of angles of attack and sideslip.
Abstract: Conventional analyses of aircraft motion in the atmosphere have neglected wind speed variability on the scales associated with many atmospheric phenomena such as thunderstorms, low level jets, etc These phenomena produce wind shears that have been determined as the probable cause in many recent commercial airline ac cidents This paper derives the 6 deg equations of motion for an aircraft incorporating the variable wind terms The equations are presented in several coordinate systems (i e , body coordinates, inertia! coordinates, etc ) The wind shear terms, including the temporal and spatial gradients of the wind, appear differently in the various coordinate systems; these terms are discussed. Also, the influence of wind shear on inputs to computing the aerodynamic coefficients, such as the effects of wind velocity vector rotation on relative angular rates of rotation and on the time rate of change of angles of attack and sideslip, are addressed
TL;DR: In this paper, the motion of suspended sand particles is studied by means of a new perturbation solution to the equation of motion, where the relative velocity between sand and water is everywhere equal to the still water settling velocity.
Abstract: The motion of suspended sand particles is studied by means of a new perturbation solution to the equation of motion. In the first approximation, fluid accelerations are neglected so that the relative velocity between sand and water is everywhere equal to the still water settling velocity . It turns out that some of the most important mechanisms of sediment suspension, such as trapping in vortices, can be derived from this “zero order solution” when the flow structure is given proper consideration. The next level of solutions takes into account the effect of fluid accelerations by including terms of order of magnitude . As a main result it is shown that pure wave motion does not cause a net reduction of settling velocity of this order of magnitude. Finally the effect of drag nonlinearity is studied for the case of an oscillatory flow. It is shown that this effect is of the order of magnitude e2 and without practical importance for sediment suspension by waves.
TL;DR: In this paper, the authors derived the equation of motion of the thermodynamical average of the Higgs field and the friction term in the equation was explicitly given in simple models.
Abstract: We derive the equation of motion of the thermodynamical average of the Higgs field. The friction term in the equation is explicitly given in simple models. The equation obtained may be relevant to the cosmological phase transitions.