Showing papers on "Equations of motion published in 1974"
TL;DR: In this article, it is shown that if the phase space of general relativity is defined so as to contain the trajectories representing solutions of the equations of motion then, for asymptotically flat spaces, the Hamiltonian does not vanish but its value is given rather by a nonzero surface integral.
1,365 citations
TL;DR: In this paper, the equations of motion of a 180° domain wall in an infinite uniaxially anisotropic medium which is exposed to an instantaneously applied uniform dc magnetic field H0 have been integrated numerically.
Abstract: The equations of motion of a 180° domain wall in an infinite uniaxially anisotropic medium which is exposed to an instantaneously applied uniform dc magnetic field H0 have been integrated numerically. Below the critical field Hc =2παM0 (α is the Gilbert loss parameter and M0 the saturation magnetization), where a steady‐state solution is known to exist, it is shown that the wall motion tends smoothly to this solution. Above Hc, the magnetization precesses about the field and a periodic component appears in the forward motion of the wall. Analytic solutions for the wall motion have been found based upon approximations suggested by the computed behavior; these reproduce the computer results very accurately.
901 citations
TL;DR: In this paper, the existence of integrals of Toda's exponential lattice is proved by a different method, which shows the Toda lattice to be a finite-dimensional analog of the Korteweg-de Vries partial differential equation.
Abstract: Following recent computer studies which suggested that the equations of motion of Toda's exponential lattice should be completely H\'enon discovered analytical expressions for the constants of the motion. In the present paper, the existence of integrals is proved by a different method. Our approach shows the Toda lattice to be a finite-dimensional analog of the Korteweg-de Vries partial differential equation. Certain integrals of the Toda equations are the counterparts of the conserved quantities of the Korteweg-de Vries equation, and the theory initiated here has been used elsewhere to obtain solutions of the infinite lattice by inverse-scattering methods.
670 citations
01 Dec 1974
TL;DR: In this article, the Hamilton's principle and the Newtonian method are used to derive the equations of motion for long, straight, slender, homogeneous, isotropic beams undergoing moderate displacements.
Abstract: 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.
576 citations
TL;DR: In this paper, it was shown how to construct a set of reduced multipole moments of the charge-current vector for an extended body in arbitrary gravitational and electromagnetic fields, and the uniqueness of the chosen set of moments was investigated, leading to the discovery of properties which are sufficient to characterize them uniquely.
Abstract: A study is made of the motion of an extended body in arbitrary gravitational and electromagnetic fields In a previous paper it was shown how to construct a set of reduced multipole moments of the charge-current vector for such a body This is now extended to a corresponding treatment of the energy-momentum tensor It is shown that, taken together, these two sets of moments have the following three properties First, they provide a full description of the body, in that they determine completely the energy-momentum tensor and charge-current vector from which they are constructed Secondly, they include the total charge, total momentum vector and total angular momentum (spin) tensor of the body Thirdly, the only restrictions on the moments, apart from certain symmetry and orthogonality conditions, are the equations of motion for the total momentum and spin, and the conservation of total charge The time dependence of the higher moments is arbitrary, since the process of reduction used to construct the moments has eliminated those contributions to these moments whose behaviour is determinate The uniqueness of the chosen set of moments is investigated, leading to the discovery of a set of properties which is sufficient to characterize them uniquely The equations of motion are first obtained in an exact form Under certain conditions, the contributions from the moments of sufficiently high order are seen to be negligible It is then convenient to make the multipole approximation, in which these high order terms are omitted When this is done, further simplifications can be made to the equations of motion It is shown that they take an especially simple form if use is made of the extension operator of Veblen & Thomas This is closely related to repeated covariant differentiation, but is more useful than that for present purposes By its use, an explicit form is given for the equations of motion to any desired multipole order It is shown that they agree with the corresponding Newtonian equations in the appropriate limit
254 citations
TL;DR: In this article, the authors recast Zwanzig's approach to inelastic atom/harmonic chain scattering in a form which suggests generalization to atomic collisions with real solid surfaces.
Abstract: Zwanzig's approach to inelastic atom/harmonic chain scattering is recast in a form which suggests generalization to atomic collisions with real solid surfaces. We reduce the infinite set of equations of motion for the impinging atom and harmonic chain to a pair of equations of motion. These equations describe the collision of the incident particle with a generalized Langevin oscillator. The first and second fluctuation‐dissipation theorems for the Langevin oscillator are verified and their implications for the scattering process are discussed.
215 citations
TL;DR: In this article, the equations of motion of a beam whose length changes with time are derived in the form of four nonlinear partial differential equations and one algebraic relation, assuming that the deflection gradients of the beam are small, and that the beam is axially rigid.
Abstract: The equations of motion of a beam whose length changes with time are derived. These take the form of four nonlinear partial differential equations and one algebraic relation. By assuming that the deflection gradients of the beam are small, and that the beam is axially rigid, the equations of motion are linearized. One closed-form similarity solution and a semi-analytic solution is obtained for specific axial velocities. Also, a procedure is described for obtaining approximate solutions for various axial velocities of the beam.
192 citations
TL;DR: In this paper, a mathematical model was developed to describe the mechanism of normal perforation of projectiles in metallic targets, which was considered to be divided into three interconnected stages, and the analysis enabled the residual velocity to be calculated as a function of the target thickness and its mechanical and physical properties, and of the mass, geometry and impact velocity of the projectile.
180 citations
TL;DR: In this paper, the problem of the harmonic forced vibrations of a massless rigid disc supported on an elastic layered medium is studied and the resulting mixed boundary value problems are reduced to sets of Fredholm integral equations that are solved numerically for a wide range of frequencies.
162 citations
TL;DR: In this article, the results of an investigation of these three mechanisms used simultaneously, and the development of a computational procedure to predict efficiencies of a fibrous filter for both the continuum and the slip-flow regimes are presented.
147 citations
TL;DR: In this paper, a hydromagnetic dynamo model is considered, where a Boussinesq, electrically conducting fluid is confined between two horizontal planes and is heated from below.
Abstract: A hydromagnetic dynamo model is considered. A Boussinesq, electrically conducting fluid is confined between two horizontal planes and is heated from below. The system rotates rapidly about the vertical axis with constant angular velocity. It is supposed that instability first sets in as stationary convection characterized by a small horizontal length scale. In preliminary calculations the Lorentz force is neglected so that the magnetic induction equation and the equation of motion are decoupled. The possibility that motions occurring at the onset of instability may sustain magnetic fields is thus reduced to a kinematic dynamo problem. Moreover, the existence of two length scales introduces simplifications which enable the problem to be studied by well-known techniques. The effect of the Lorentz force on the finite amplitude dynamics of the system is investigated also. Since only weak magnetic fields are considered the kinetic energy of the motion is fixed by other considerations and it is only the fine structure of the flow that is influenced by the magnetic field. A set of nonlinear equations, which govern the evolution of the hydromagnetic dynamo, are derived from an asymptotic analysis. The equations are investigated in detail both analytically and numerically. In spite of serious doubts concerning the existence of sufficiently complex stable motions, stable periodic dynamos are shown to exist. An interesting analytic solution of these equations, which may be pertinent to other problems arising from finite-amplitude Benard convection, is presented in the final section.
TL;DR: In this paper, a mathematical theory of chemical reaction systems is proposed, which separate equilibrium, nonequilibrium and stoichiometric aspects of a chemical reaction system, and the role of various constitutive assumptions is investigated.
Abstract: A mathematical theory of chemical reaction systems is proposed. Generic equations of motion are developed which separate equilibrium, nonequilibrium and stoichiometric aspects. The role of various constitutive assumptions is investigated.
TL;DR: In this article, a general theory of the spin dynamics of anisotropic Fermi superfluids of the generalized BCS type was developed, under conditions which should be realistic for any such phase of liquid 3 He occurring below 3 mK.
TL;DR: In this paper, a transversely isotropic plate in a general state of nonuniform initial stress where the effects of transverse shear and rotary inertia are included are derived by two methods.
Abstract: Equations of motion for a transversely isotropic plate in a general state of nonuniform initial stress where the effects of transverse shear and rotary inertia are included are derived by two methods. The first method is to perturb the nonlinear equations of elasticity by an incremental deformation. The resulting equations are linearized and integrated through the thickness of the plate to obtain the plate equations. The second method is to derive nonlinear equations of motion for a thick plate variationally by Hamilton's principle. These equations are then perturbed and suitably linearized to obtain the same equations as were obtained by the first method. A reduced set of equations for a thin plate are also given. Finally, the thick plate equations are solved for a simply supported rectangular plate in a state of uniform compressive stress plus a uniform bending stress both acting in the same direction.
TL;DR: In this paper, the envelope model of fluid/cilia interaction was employed to construct equations of motion for the oscillating boundary layer of a self-propelling body. But the results were compared with the relatively sparse observations for a number of micro-organisms.
Abstract: This paper analyses the locomotion of a finite body propelling itself through a viscous fluid by means of travelling harmonic motions of its surface. The methods are developed with application to the propulsion of ciliated micro-organisms in mind. Provided that the metachronal wavelength (of the surface motions) is much smaller than the overall dimensions of the body, the flow can be divided into an oscillating-boundary-layer flow to which is matched an external complementary Stokes flow. The present paper employs the envelope model of fluid/cilia interaction to construct equations of motion for the oscillating boundary layer. The final solution for the propulsive velocity is obtained by application of the condition of zero total force on the self-propelling body; alternatively, if the organism is held at rest, the thrust it generates can be computed. Various optimum propulsive velocities for self-propelling bodies and optimum thrusts for restrained bodies are analysed in some simple examples. The results are compared with the relatively sparse observations for a number of micro-organisms.
TL;DR: In this article, the critical dynamical behavior of the pseudospin-phonon coupled system has been investigated in connection with the structural phase transition of molecular crystals based on Onsager's equation of motion.
Abstract: The critical dynamical behavior of the pseudospin-phonon coupled system has been investigated in connection with the structural phase transition of molecular crystals. Based on Onsager's equation of motion, explicit expressions for three correlation functions, spin-spin correlation, phonon-phonon correlation and spin-phonon correlation, are obtained. It has been shown that the characteristic of the spectra of these correlation functions shows wide variation depending on the values of two parameters: The ratio of the relaxation rate of reorientational motion of an isolated spin and the frequency of the phonon, and the ratio of the interaction parameter of direct spin-spin coupling and that of spin-phonon coupling. Especially, in `fast relaxation case', the spectrum of phonon-phonon correlation shows critical softening, whereas, in `show relaxation case' the spectrum gives `triple peak' structure and only the central (relaxational) mode shows critical slowing down leaving phonon side peaks unchanged.
TL;DR: In this article, the interaction between charged particles and sharply localized fields is investigated and the random particle scattering is described by a Fokker-Planck equation whose time-dependent solution exhibits the formation of a highly populated superthermal tail.
Abstract: The interaction between charged particles and sharply localized fields is investigated. The random particle scattering is described by a Fokker-Planck equation whose time-dependent solution exhibits the formation of a highly populated superthermal tail.
TL;DR: The work of Aarseth and Zare (1974) is extended to provide aglobal regularisation of the classical gravitational three-body problem by transformation of the variables in a way that does not depend on the particular configuration as discussed by the authors.
Abstract: The work of Aarseth and Zare (1974) is extended to provide aglobal regularisation of the classical gravitational three-body problem: by transformation of the variables in a way that does not depend on the particular configuration, we obtain equations of motion which are regular with respect to collisions between any pair of particles. The only cases excepted are those in which collisions between more than one pair occur simultaneously and those in which at least one of the masses vanishes. However, by means of the same principles the restricted problem is regularised globally if collisions between the two primaries are excluded. Results of numerical tests are summarised, and the theory is generalised to provide global regularisations, first, for perturbed three-body motion and, second, for theN-body problem. A way of increasing the number of degrees of freedom of a dynamical system is central to the method, and is the subject of an Appendix.
TL;DR: In this article, a condensation method for large discrete parameter vibration analysis of complex structures is presented, which greatly reduces truncation errors and provides accurate definition of modes in a selected frequency range.
Abstract: This paper presents a condensation method for large discrete parameter vibration analysis of complex structures that greatly reduces truncation errors and provides accurate definition of modes in a selected frequency range. A dynamic transformation is obtained from the partitioned equations of motion that relates modes not explicitly in the condensed solution to the retained modes at a selected system frequency. The generalized mass and stiffness matrices, obtained with existing modal synthesis methods, are reduced using this transformation and solved. Revised solutions are then obtained using new transformations at the calculated eigenvalues and are also used to assess the accuracy of the results. If all the modes of interest have not been obtained, the results are used to select a new set of retained coordinates and a new transformation frequency and the procedure repeated for another group of modes. Computations are made tractable by simplified forms of the transformation that result with various modal synthesis methods. Three examples using the dynamic transformation in conjunction with a General Electric stiffness coupling method and the method of Craig and Bampton indicate large reductions in truncation errors and demonstrate the method for sequential groups of modes. Comparisons with truncated results using current methods indicate that two to three times as many accurate modes are obtained from solutions keeping less than half the component modes. Nomenclature [/c] = stiffness matrix for total structure in {x} physical coordinates [X] = generalized stiffness matrix for total structure in {q} modal coordinates [/CCPL] = stiffness matrix in {x} coordinates for coupling substructures [m] = mass matrix for total structure in {x} physical coordinates [M] = generalized mass matrix for total structure in {q} modal coordinates [Am] = incremental mass matrix of coupling structures in {x} coordinates
TL;DR: In this article, a computational method was developed for the finite-amplitude three-dimensional motion of inextensible elastic rods with equal principal stiffnesses, and a semi-explicit method of solution was devised.
Abstract: A computational method is developed for the finite-amplitude three-dimensional motion of inextensible elastic rods with equal principal stiffnesses. The method also applies to the two-dimensional motion of such rods with unequal principal stiffnesses. For these two classes of problems the equations of the classical theory of rods are reduced to a non-linear vector equation of motion together with the inextensibility condition and appropriate boundary and initial conditions. Consistent finite-difference approximations are introduced and a semi-explicit method of solution is devised. The approximate limitation for numerical stability of the method is shown to be the same as for the usual explicit method in linear beam dynamics. By way of example the method is applied to the free fall of a circular pipe through water onto a rigid plane from a suspended initial configuration.
TL;DR: In this article, a new numerical method for use in the solution of classical equations of motion is described, accurate to third order in the coordinates and second-order in the velocities, which has the unique property of preserving the energy and total linear and angular momenta at their initial values in the computation.
TL;DR: In this paper, the authors derive macroscopic equations for the vibrations of nuclei from the RPA equations of motion and derive an energy for the quadrupole vibration 30% lower than the empirical.
TL;DR: In this paper, it was shown that physically reasonable solutions of the field equations based on an R2 lagrangian are possible, however, experimental predictions of such a theory are at variance with observations.
Abstract: It is shown that physically reasonable solutions of the field equations based on an R2 lagrangian are possible. (R is the scalar curvature.) However, it is shown that experimental predictions of such a theory are at variance with observations. The most general quadratic lagrangian is also considered and it is shown that the R2 term must dominate thus invalidating gravitational equations based on a general quadratic lagrangian.
TL;DR: In this paper, the equations of motion of a spinning body in the gravitational field of a much larger mass are found using both the Corinaldesi-Papapetrou spin supplementary condition (SSC) and the Pirani SSC.
Abstract: The equations of motion of a spinning body in the gravitational field of a much larger mass are found using both the Corinaldesi-Papapetrou spin supplementary condition (SSC) and the Pirani SSC. These equations of motion are compared with our previous result derived from Gupta's quantum theory of Gravitation. It is found that the spin-dependent terms differ in each of the above three results due to a different location of the center of mass of the spinning body. As expected, these terms are not affected by the choice of either Schwarzschild or isotropic coordinates. Finally, for the presently planned Stanford gyroscope experiment, we find the maximum secular displacement of the orbit of the gyro with respect to the orbit of its non-rotating housing to be of the order of (10−7 cm/year)t, a result much smaller than Schiff's result which is proportional to time squared.
TL;DR: In this paper, an approximation method for solving the equations of motion that describe the vibration of an elastic structure immersed in an infinite acoustic fluid medium is presented, which avoids the difficulties encountered at eigenvalues of the interior problem by determining exactly the one degree of freedom needed to overdetermine the surface equations.
Abstract: An approximation method is presented for solving the equations of motion that describe the vibration of an elastic structure immersed in an infinite acoustic fluid medium. The mathematical model that is developed uses the finite element method to calculate the vibrational characteristics of the elastic body and the acoustic pressure field of that portion of the fluid which closely surrounds the vibrating body. Analytical methods are used to obtain the boundary conditions for this mathematical model. This technique can be used to predict the response of an elastic structure over a wide range of frequencies and the acoustic pressure at a large number of field points in both the nearfield and farfields. It avoids the difficulties encountered at eigenvalues of the interior problem by determining exactly the one degree of freedom needed to overdetermine the surface equations. Experimental validation of theoretical predictions is given for a piezoelectric cylinder, and a computer‐generated contour plot of a predicted nearfield pressure distribution is shown.
Abstract: A number of alternative solution schemes for problems of dynamic structural analysis involving large displacements and plastic deformations are compared. The equations of motion, derived from the finite element approximation, are integrated directly using a number of well-known time integration operators; namely, the Houbolt, Newmark, Wilson, and central difference operators. This latter operator serves as a basis for a comparison of accuracy and economy with all methods. Numerical results for nonlinear systems are used to indicate the true worth of integration operators whose properties have been determined theoretically by studying their behavior with linear systems only.
IBM1
TL;DR: A new approach to the analysis of wide foil bearings is investigated, where the equation of motion for a finite length of tape is coupled to the transient lubrication equation for the air film between the tape and the recording head.
Abstract: A new approach to the analysis of wide foil bearings is investigated. The equation of motion for a finite length of tape is coupled to the transient lubrication equation for the air film between the tape and the recording head. Compressibility and slip flow are retained in the fluid mechanics equation; flexural rigidity and high-speed dynamic effects are retained in the tape equation. The steady-state solution to the coupled equations is obtained as the limiting case of the transient initial value problem. Describing the system equations relative to the undeflected tape (as opposed to conventional foil-bearing theory, which uses the head as the reference surface) permits investigation of noncircular head geometries. In addition, wave propagation effects in the tape and the interaction of waves in the tape with the air-bearing region may be studied.
Journal Article•
TL;DR: In this article, a unified description of systems with a condensed phase in terms of hydrodynamic equations of motion is given, and the eigenmodes of the linearized form of these equations, typically first and second sound, are discussed in particular with respect to their manifestation in inelastic light and neutron scattering.
Abstract: A unified description of systems with a condensed phase in terms of hydrodynamic equations of motion is given. These equations are of two kinds: First those equations obeyed by the thermal excitations (the "first fluid") typically are local conservation equations of mass, energy and momentum. Second the equation obeyed by the condensed phase (the "second fluid") is an equation of motion related to the order parameter of the broken symmetry. These equations are established on phenomenological grounds making use of irreversible thermodynamics. The eigenmodes of the linearized form of these equations, typically first and second sound, are discussed in particular with respect to their manifestation in inelastic light and neutron scattering. The systems considered are homogenous superfluids, superconductors, dielectric crystals and magnetic systems. Except for superfluid $^{4}\mathrm{He}$ the critical behavior at the phase transition to the ordered state is not systematically discussed.
01 Oct 1974
TL;DR: In this paper, the equations of motion for a system of coupled flexible bodies, rigid bodies, point masses, and symmetric wheels were derived, and the equations were cast into a partitioned matrix form in which certain partitions became nontrivial when the effects of flexibility were treated.
Abstract: The equations of motion for a system of coupled flexible bodies, rigid bodies, point masses, and symmetric wheels were derived. The equations were cast into a partitioned matrix form in which certain partitions became nontrivial when the effects of flexibility were treated. The equations are shown to contract to the coupled rigid body equations or expand to the coupled flexible body equations all within the same basic framework. Furthermore, the coefficient matrix always has the computationally desirable property of symmetry. Making use of the derived equations, a comparison was made between the equations which described a flexible body model and those which described a rigid body model of the same elastic appendage attached to an arbitrary coupled body system. From the comparison, equivalence relations were developed which defined how the two modeling approaches described identical dynamic effects.