Showing papers on "Equations of motion published in 1969"
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TL;DR: The fundamental equation of classical plate theory can be found in this article, where anisotropic and variable-thickness versions of the classical plates are considered, as well as other considerations.
Abstract: : Contents: Fundamental Equations of Classical Plate Theory; Circular Plates; Elliptical Plates; Rectangular Plates; Parallelogram Plates; Other Quadrilateral Plates; Triangular Plates; Plates of Other Shapes; Anisotropic Plates; Plates With Inplane Forces; Plates With Variable Thickness; and Other Considerations.
2,137 citations
TL;DR: In this paper, the authors show that the particle number is an adiabatic invariant, but not a strict constant of the motion, and show that particle creation occurs in pairs.
Abstract: Spin-0 fields of arbitrary mass and massless fields of arbitrary spin are considered. The equations governing the fields are the covariant generalizations of the special-relativistic free-field equations. The metric, which is not quantized, is that of a universe with an expanding (or contracting) Euclidean 3-space. The spin-0 field of arbitrary mass is quantized in the expanding universe by the canonical procedure. The quantization is consistent with the time development dictated by the equation of motion only when the boson commutation relations are imposed. This consistency requirement provides a new proof of the connection between spin and statistics. We show that the particle number is an adiabatic invariant, but not a strict constant of the motion. We obtain an expression for the average particle density as a function of the time, and show that particle creation occurs in pairs. The canonical creation and annihilation operators corresponding to physical particles during the expansion are specified. Thus, we do not use an $S$-matrix approach. We show that in a universe with flat 3-space containing only massless particles in equilibrium, there will be precisely no creation of massless particles as a result of the expansion, provided the Einstein field equations without the cosmological term are correct. Furthermore, in a dust-filled universe with flat 3-space there will be precisely no creation of massive spin-0 particles in the limit of infinite mass, again provided that the Einstein field equations are correct. Conversely, without assuming any particular equations, such as the Einstein equations, as governing the expansion of the universe, we obtain the familiar Friedmann expansions for the radiation-filled and the dust-filled universes with flat 3-space. We only make a very general and natural hypothesis connecting the particle creation rate with the macroscopic expansion of the universe. In one derivation, we assume that in an expansion of the universe in which a particular type of particle is predominant, the type of expansion approached after a long time will be such as to minimize the average creation rate of that particle. In another derivation, we use the assumption that the reaction of the particle creation back on the gravitational field will modify the expansion in such a way as to reduce, if possible, the creation rate. This connection between the particle creation and the Einstein equations is surprising because the Einstein equations themselves played no part at all in the derivation of the equations governing the particle creation. Finally, on the basis of a so-called infinite-mass approximation, we argue that in the present predominantly dust-filled universe, only massless particles of zero spin might possibly be produced in significant amounts by the present expansion. In this connection, we show that massless particles of arbitrary nonzero spin, such as photons or gravitons, are not created by the expansion, regardless of its form.
996 citations
TL;DR: In this paper, it was shown that in spatially homogeneous two-dimensional turbulence, the mean square vorticity is unaffected by convection and can only decrease under the action of viscosity.
Abstract: Two‐dimensional and three‐dimensional turbulence have different properties, but both contain the two basic ingredients of randomness and convective nonlinearity, and some of the statistical hypotheses which have been proposed for three‐dimensional turbulence should be applicable to two‐dimensional motion. This justifies a numerical integration of the unaveraged equations of motion in two dimensions with random initial conditions as a means of testing the soundness of ideas such as those leading to the Kolmogoroff equilibrium theory. In spatially homogeneous two‐dimensional turbulence, the mean‐square vorticity is unaffected by convection and can only decrease under the action of viscosity. Consequently the rate of dissipation of energy tends to zero with the viscosity (v). On the other hand, the mean‐square vorticity gradient is increased by convective mixing, and it seems likely that the rate of decrease of mean‐square vorticity tends to a nonzero limit κ as ν → 0. This suggests the existence of a “casca...
901 citations
TL;DR: In this article, a modification of the Cowell method for the integration of orbits is proposed, which is characterized by the property that it will integrate unperturbed Kepler motion exactly (without truncation errors), thus the slight instability of the cowell method is avoided.
Abstract: This paper offers a modification of the Cowell method for the integration of orbits The modification is characterized by the property that it will integrate unperturbed Kepler motion exactly (excluding truncation errors), thus the slight instability of the Cowell Method is avoided Furthermore, the modification takes into account the most important secular effects of orbit motion As an example of the applicability of the modified method to perturbed motion, the equations of motion of an artificial earth satellite are integrated In the case of elliptic initial conditions regularization by a Levi-Civita transformation was used
381 citations
TL;DR: In this paper, a formal scale analysis of the equations of motion in a plane parallel atmosphere is made, assuming conditions to be such that relative fluctuations in density and temperature are small.
Abstract: A formal scale analysis of the equations of motion in a plane parallel atmosphere is made, assuming conditions to be such that relative fluctuations in density and temperature are small. It is found that an energetically consistent set of approximate equations can be derived which preclude the existence of acoustic motions. Such equations can be used to describe subsonic convection or internal gravity waves. Under certain conditions the analysis can be generalized to include vertical pulsations of the atmosphere.
301 citations
TL;DR: Light pulses propagation in nonlinear laser medium, obtaining equations of motion for density matrix was studied in this paper, where it was shown that light pulses propagate in a nonlinear nonlinear medium.
Abstract: Light pulses propagation in nonlinear laser medium, obtaining equations of motion for density matrix
241 citations
TL;DR: In this paper, an expression for the dynamic elastic field of a crack when one of its tips moves arbitrarily in the plane of the crack, starting from rest, was found with the help of a theorem of Bateman's.
Abstract: With the help of a theorem of Bateman's an expression is found for the dynamic elastic field of a crack when one of its tips moves arbitrarily in the plane of the crack, starting from rest. The linear isotropic theory of elasticity is used, and only states of anti-plane strain (mode III deformation) are considered. The crack is initially of finite length and subject to any static anti-plane loading. The solution obtained becomes inaccurate in regions into which disturbances reflected at the other tip have penetrated. The error is estimated for some special cases. The results are used to discuss the equation of motion of a crack tip.
191 citations
TL;DR: In this article, the force per unit volume which a flowing fluid exerts on a porous structure, and the divergence of the local volume-averaged extra stress tensor are examined with the aid of the principle of material indifference.
Abstract: The local volume average of the equation of motion is taken for an incompressible fluid flowing through a porous structure under conditions such that inertial effects may be neglected. The result has two terms beyond a pressure gradient: g, the force per unit volume which a flowing fluid exerts on a porous structure, and the divergence of the local volume-averaged extra stress tensor (viscous portion of the stress tensor).
Constitutive equations for g are examined with the aid of the principle of material indifference. When g is assumed to be a function of the velocity of the fluid relative to the solid as well as various scalars, the usual results for a nonoriented (isotropic) porous structure are obtained. When g is assumed to be a function of the local porosity gradient as well, we derive a new expression for g applicable to oriented (anisotropic) porous structures.
For a Newtonian fluid with a constant viscosity, the divergence of the local volume-averaged extra stress tensor is proportional to the Laplacian of the averaged velocity vector. Boundary conditions for the averaged velocity vector are discussed. Three problems are solved for the flow of an incompressible Newtonian fluid in a nonoriented permeable medium. These solutions, as well as an order-of-magnitude analysis, suggest that we may often neglect both the Laplacian of average velocity and the boundary conditions for the tangential components of averaged velocity at an impermeable wall.
Two specific constitutive equations for g are proposed for the flow of incompressible Noll simple fluids in nonoriented porous structures. Flow through a porous medium bounded by an impermeable cylindrical surface is solved for these two constitutive equations, and the results are compared with previously available experimental data.
160 citations
TL;DR: In this paper, the authors considered the traction boundary value problem in the general case where O(U,, ti) may be nonlinear in both u,, zi,.
137 citations
TL;DR: Aerodynamic coefficients from observed motion of body in flight, eliminating need for closed form solutions by employing numerical solutions to equations of motion were derived by as mentioned in this paper, where they were used to eliminate the need for a closed form solution.
Abstract: Aerodynamic coefficients from observed motion of body in flight, eliminating need for closed form solutions by employing numerical solutions to equations of motion
TL;DR: In this paper, the effect of an external magnetic field on the viscosity of a dilute suspension of magnetized particles is calculated to first order in the volume fraction of solution occupied by the particles.
Abstract: The effect of an external magnetic field on the viscosity of a dilute suspension of magnetized particles is calculated to first order in the volume fraction of solution occupied by the particles. The steady‐state motion of the particles under the influence of the external field is derived through an unusual application of Stokes' theorem to the equations of motion.
TL;DR: In this paper, an equations of motion, constitutive equations, and boundary conditions for a class of micropolar fluids which can stretch or contract are derived for a set of microelements of such fluids.
TL;DR: In this article, an electric charge moving in the field of a magnetic charge is discussed and solved by algebraic methods which exhibit the direct relation between charge quantization and angular momentum.
TL;DR: In this paper, the motion of a fluid in the closed annular cavity formed by two concentric vertical cylinders and two horizontal planes has been analyzed by a numerical solution of the equations of motion and energy using a high speed digital computer.
Abstract: The motion of a fluid in the closed annular cavity formed by two concentric vertical cylinders and two horizontal planes has been analyzed by a numerical solution of the equations of motion and energy using a high‐speed digital computer. The motion is generated by a radial density gradient caused by the thermal boundary conditions which are, typically: inner cylinder at a (dimensionless) temperature of unity; outer cylinder at a temperature of zero; horizontal boundaries adiabatic. The fluid is assumed to have constant thermodynamic and transport properties except for the density, which is temperature‐dependent in the buoyancy term of the equation of vertical motion (the Boussinesq approximation); the flow is assumed to be axisymmetric. The equations of time‐dependent motion have been solved, so that both transient and steady‐state solutions are obtained. The parameters of the problem, and the respective ranges of values which have been considered, are: Rayleigh number (based on gap width) up to 2 × 105; Prandtl number 0.5 to 5; radius ratio 1 to 4; aspect ratio (cavity height/gap) 1 to 20. At moderate Rayleigh numbers the motion consists of a single cell (i.e., torus), while at higher Rayleigh numbers the onset of a multicellular motion is observed. The local and average Nusselt numbers, of interest in determining the insulating value of an annular cavity, have been obtained.
TL;DR: In this paper, a nonlinear Galerkin analysis of curved plate flutter was performed using shallow shell/von Karman equations and quasi-steady aerodynamic theory, and the results showed that the flutter flutter can be represented by a curved plate.
Abstract: Nonlinear Galerkin analysis of curved plate flutter, using shallow shell/von Karman equations and quasi-steady aerodynamic theory
TL;DR: In this paper, the equations of motion of a vibrating string are derived, and it is shown that a coupling exists between the longitudinal and transverse modes of vibration, under sinusoidal initial conditions, and the equations are separable.
Abstract: The equations of motion of a vibrating string are derive, and it is shown that a coupling exists between the longitudinal and transverse modes of vibration. Free transverse vibration of small order, under sinusoidal initial conditions, is analyzed. Under these conditions, the equations are separable. The time‐dependent parts of the equations are solved by the method of variable amplitude and phase. It is seen that, when the vibration is nonplanar, part of the energy oscillates between the mutually perpendicular transverse components with a frequency proportional to the nonlinearity parameter α. The path of any point on the string is shown to be an ellipse with slowly rotating and shrinking axes.
TL;DR: In this article, it was shown that the von Laue mechanical-energy flux (in the usual ''synchronous'' formulation) is not plausible for point bodies with variable rest mass.
Abstract: For point bodies with variable rest mass it is shown that the definition of forcem
0dU
i
/dτ=F
i is better than the usual d(m
0
U
i))/dτ=F
i and the explicit equation of motion is given bym
0dU
i/dτ=F
ext
+(z
−1
U
i
B
−U
i)dm
0/dτ, where the external forceF
ext
has been separated from the reaction force due to the ejected mass,U
i
B
is the four-velocity of the mass centre of the ejected mass andz=U
s
B
U2c−2. For extended bodies it is shown that the von Laue mechanical-energy flux (in the usual «synchronous» formulation) is not plausible. Moreover a new formulation, called «asynchronous» is given. By this formulation many problems receive an immediate solution. The resultant «asynchronous» force density is always orthogonal to the four-velocity, and the equations of motion for continuous media turn out to be formally equal to the classical equations. Expanding the new equations to the first order when the pressure is a regular function of space and time, one obtains the usual «synchronous» equations. In the asynchronous formulation, all quantities relevant to extended bodies transform in a covariant way. In particular, momentum and energy turn out to have the expression proposed by Rohrlich. The above concepts are applied to relativistic thermodynamics. It is shown that, for point bodies, heat transforms as in the recent Ott formulation. For extended bodies, in the usual «synchronous» formulation, heat transforms as pointed out by von Laue. In the «asynchronous» formulation we always haveQ=γQ
0. TemperatureT is considered invariant and the relation dQ=TdS valid in the rest system only.
TL;DR: In this paper, a mixture of any number of materials with elastic and viscous properties in which the constituents may have different temperatures is discussed, and constitutive equations are discussed for a mixture.
Abstract: : Constitutive equations are discussed for a mixture of any number of materials with elastic and viscous properties in which the constituents may have different temperatures.
TL;DR: In this article, the effects of a superimposed radial outflow upon the motion of the fluid between a rotating and a stationary plane disc are dealt with, and the boundary layer approximations to the equations of fluid motion are integrated by an extension of the numerical technique proposed in Ref. 11.
Abstract: This paper deals with the effects of a superimposed radial outflow upon the motion of the fluid between a rotating and a stationary plane disc. This is an idealisation of the situation found in advanced turbo-machinery in which cooling of the rotor faces is necessary. The boundary layer approximations to the equations of fluid motion are integrated by an extension of the numerical technique proposed in Ref. 11. Predictions of drag torque and radial pressure distributions are compared with experimental results from a 30 in (762 mm) diameter system, using a range of speeds up to 4000 rev/min, while varying the axial spacing and rate of imposed radial flow. Qualitative agreement was satisfactory throughout the range, but quantitative discrepancies in the estimates of drag and in the pressure distributions at certain flow conditions suggest that the simple mixing length hypothesis used in this analysis for the turbulent shear terms in the equations of motion is not universally adequate.
TL;DR: In this article, the authors describe theoretical and experimental investigations of the large-amplitude vibrations of a flexible beam simply supported on a nearly rigid base, which is designed to minimize most secondary effects, such as transverse shear flexibility, rotatory inertia and nonlinearities in curvature and in the stress-strain curve.
Abstract: This paper describes theoretical and experimental investigations of the large-amplitude vibrations of a flexible beam simply supported on a nearly rigid base. The beam is designed to minimize most secondary effects, such as transverse shear flexibility, rotatory inertia, and nonlinearities in curvature and in the stress-strain curve. Detailed attention is given to quantitative verification of the assumptions made in deriving the equation of motion. Three different approaches are used to solve the equation: assumed-space mode, assumed-time mode, and Ritz-Galerkin solutions. In the experiments, the beam was base excited. The resonant frequencies and associated strain distributions, modal shapes, and waveforms were measured for various values of initial tension and amplitude ratios (ratio of maximum amplitude to beam thickness) up to 16. Within experimental accuracy, the experimental results verified all of the assumptions made in the analyses. However, no one method of solution resulted in the best predictions of all of the experimentally observed phenomena.
TL;DR: The linearized hydrodynamic equations of motion for a thin, flat, superfluid helium film are derived in some detail from standard two-fluid hydrodynamics.
Abstract: The linearized hydrodynamic equations of motion for a thin, flat, superfluid helium film are derived in some detail from standard two-fluid hydrodynamics. Interactions of the film with both the He vapor and the substrate which are in contract with it are included and discussed in detail. Boundary conditions for both the film-substrate and film-gas interfaces are derived. It is indicated how one may construct the equations of motion for the entire coupled system (gas-film-substrate). The equations are actually constructed and solved for the case when a certain parameter is small, which includes all the third-sound experiments on unsaturated films. A dispersion equation is found which is exact in the limit of vanishing frequency, and which is eminently suited to describe both the velocity and the attenuation of third sound in the regime of unsaturated films. No hydrodynamic instability is found. Results for the attenuation are shown to be in good agreement with preliminary experiments on unsaturated He films.
TL;DR: In this article, the authors apply the existence theorem for solutions of the equations of motion for infinite systems to study the time evolution of measures on the set of locally finite configurations of particles.
Abstract: We apply the existence theorem for solutions of the equations of motion for infinite systems to study the time evolution of measures on the set of locally finite configurations of particles. The set of allowed initial configurations and the time evolution mappings are shown to be measurable. It is shown that infinite volume limit states of thermodynamic ensembles at low activity or for positive potentials are concentrated on the set of allowed initial configurations and are invariant under the time evolution. The total entropy per unit volume is shown to be constant in time for a large class of states, if the potential satisfies a stability condition.
Journal Article•
TL;DR: Theorems on collision singularities in many body problem, and bibliography on equations of motion in celestial mechanics are presented in this article, where the collision singularity is defined as a singularity singularity in the problem of many body problems.
Abstract: Theorems on collision singularities in many body problem, and bibliography on equations of motion in celestial mechanics
TL;DR: In this article, a method for investigating the seismic response of multiple-story shear buildings on flexible elastic foundation media which are represented by an elastic half space is presented, where the frequency dependent dynamic properties of the foundation medium may be assumed to be constant with respect to the frequency.
Abstract: A method is presented for investigating the seismic response of multiple-story shear buildings on flexible elastic foundation media which are represented by an elastic half space. It is shown that the frequency dependent dynamic properties of the foundation medium may be assumed to be constant with respect to the frequency. This assumption leads to equations of motion with constant coefficients which are solved numerically. A study of several structure-foundation systems indicates that the flexibility of the foundation medium can increase or decrease the flexural response of the structure compared to the case of the rigid foundation.
TL;DR: In this article, the quantum theory of the stimulated Raman and Brillouin effects, the parametric amplifier, and the frequency converter is developed, starting with a simple Hamiltonian for a system of three coupled field modes.
Abstract: The quantum theory of the stimulated Raman and Brillouin effects, the parametric amplifier, and the frequency converter is developed, starting with a simple Hamiltonian for a system of three coupled field modes. The intense incident beam is treated classically. The evolution in time of the statistical properties of the quantized Stokes and anti-Stokes waves are analyzed by means of appropriate time-dependent phasespace distributions under various assumptions for the initial fields. Expansions of the field variables in terms of complete sets of orthogonal operators are discussed. Several regimes of operation are considered. It is shown that an initially coherent state develops, in general, into a superposition of a coherent state and a chaotic state. The amplitudes of the coherent components follow the same equations of motion as the mode operators. The chaotic components stem from the amplification of the vacuum fluctuations.
TL;DR: In this article, an alternative average force algorithm is described, which is shown to have both theoretical and practical advantages when compared to the central-difference method, and it is shown that the new method is more efficient than the central difference method.
Abstract: The equations of motion for N‐body systems are usually integrated by means of a central‐difference algorithm. An alternative average force algorithm is described in this paper. The new method is shown to have both theoretical and practical advantages when compared to the central‐difference method.