Showing papers on "Equations of motion published in 1975"

Dynamics Of Structures

Abstract: Part 1 Single-degree-of-freedom systems: overview of structural dynamics analysis of free vibrations response to harmonic loading response to periodic loading response to impulse loading responses to general dynamic loading - step by step methods, superposition methods generalized single degree-of-freedom systems. Part 2 Multi-degree-of-freedom systems: formulation of the MDOF equations of motion evaluation of structural-property matrices undamped free vibrations analysis of dynamic response using superposition vibration analysis by matrix iteration selection of dynamic degrees of freedom analysis of MDOF dynamic response - step by step methods variational formulation of the equations of motion. Part 3 Distributed parameter systems: partial differential equations of motion analysis of undamped free vibrations analysis if dynamic response. Part 4 Random vibrations: probability theory random processes stochastic response of linear SDOF systems stochastic response of non-linear MDOF systems. Part 5 Earthquake engineering: seismological background free-field surface ground motions deterministic structural response - including soil-structure interaction stochastic structural response.

Time‐dependent approach to semiclassical dynamics

Abstract: In this paper we develop a new approach to semiclassical dynamics which exploits the fact that extended wavefunctions for heavy particles (or particles in harmonic potentials) may be decomposed into time−dependent wave packets, which spread minimally and which execute classical or nearly classical trajectories. A Gaussian form for the wave packets is assumed and equations of motion are derived for the parameters characterizing the Gaussians. If the potential (which may be nonseparable in many coordinates) is expanded in a Taylor series about the instantaneous center of the (many−particle) wave packet, and up to quadratic terms are kept, we find the classical parameters of the wave packet (positions, momenta) obey Hamilton’s equation of motion. Quantum parameters (wave packet spread, phase factor, correlation terms, etc.) obey similar first order quantum equations. The center of the wave packet is shown to acquire a phase equal to the action integral along the classical path. State−specific quantum information is obtained from the wave packet trajectories by use of the superposition principle and projection techniques. Successful numerical application is made to the collinear He + H2 system widely used as a test case. Classically forbidden transitions are accounted for and obtained in the same manner as the classically allowed transitions; turning points present no difficulties and flux is very nearly conserved.

Dynamics and statistical mechanics of a one-dimensional model Hamiltonian for structural phase transitions

Abstract: We have studied thermodynamic and some dynamic properties of a one-dimensional-model system whose displacement field Hamiltonian is strongly anharmonic, and is representative of those used to study displacive phase transitions. By studying the classical equations of motion, we find important solutions (domain walls) which cannot be represented effectively by the usual phonon perturbation expansions. The thermodynamic properties of this system can be calculated exactly by functional integral methods. No Hartree or decoupling approximations are made nor is a temperature dependence of the Hamiltonian introduced artificially. At low temperature, the thermodynamic behavior agrees with that found from a phenomenological model in which both phonons and domain walls are included as elementary excitations. We then show that equal-time correlation functions calculated by both functional-integral and phenomenological methods agree, and that the dynamic correlation functions (calculated only phenomenologically) exhibit a spectrum with both phonon peaks and a central peak due to domain-wall motion.

Calculation of nonlinear ground response in earthquakes

Abstract: A method is presented for calculating the seismic response of a system of horizontal soil layers. The essential element of the method is a rheological model suggested by Iwan which takes account of the nonlinear hysteretic behavior of soils and has considerable flexibility for incorporating laboratory results on the dynamic behavior of soils. Finite rigidity is allowed in the underlying elastic medium, permitting energy to be radiated back into the underlying medium. Three alternate ways of integrating the equations of motion are compared, an implicit technique, an explicit technique, and integration along characteristics. An example is set up for comparing the different methods of integration and for comparing the nonlinear solution with a solution based on the widely used equivalent linear assumption. The example consists of a 200-m section of firm alluvium excited at its base by the N21E component of the Taft accelerogram multiplied by four to produce a peak acceleration of 0.7 g and a peak velocity of 67 cm/sec. The three techniques of integration give very similar results, but integration along characteristics has the advantage of avoiding spurious high-frequency oscillations in the acceleration time history at the surface. For the chosen example, which has a thick soil column and a strong input motion, the equivalent linear solution underestimates the intensity of surface motion for periods between 0.1 and 0.6 sec by factors exceeding two. The discrepancies, however, would probably be less for input motion of lower intensity. At longer periods the equivalent linear solution is in essential agreement with the nonlinear solution. For the same example both solutions show that, compared to a site with rock at the surface, motion at the surface of the soil is amplified for periods longer than 1.5 sec by as much as a factor of two. At shorter periods the amplitude is reduced.

Quantization of nonlinear waves

Abstract: We suggest that certain nonlinear field theories possess a particle spectrum, richer than has been heretofore discussed. In addition to states associated with quantization of the free-field modes of oscillation---these are the conventional particles of the theory---there also appear heavy particles, which carry a new quantum number and are stable. An approximation scheme is developed in which the signal for these new particles is the existence of stable static solutions with finite energy to the classical equations of motion. We give a systematic expansion for the theory, with special emphasis on the translational motion.

On the Formation of Dissipative Structures in Reaction-Diffusion Systems

Abstract: A unified viewpoint on the dynamics of spatio-temporal organization in various reaction­ diffusion systems is presented. A dynamical similarity law attained near the instability points plays a decisive role in our whole theory. The method of reductive perturbation is used for extracting a scale-invariant part from original macroscopic equations of motion. It is shown that in many cases the dynamics near the instability point is governed by the time­ dependett Ginzburg-Landau equation with coefficients which are in general complex numbers. A~ important effeCt of the imaginary parts of these coefficients on the stability of a spatially uniform limit cycle against inhomogeneous perturbation is also discussed.

Noether's theory in classical nonconservative mechanics

Abstract: Noether's theorem and Noether's inverse theorem for mechanical systems with nonconservative forces are established. The existence of first integrals depends on the existence of solutions of the generalized Noether-Bessel-Hagen equation or, which is the same, on the existence of solutions of the Killing system of partial differential equations. The theory is based on the idea that the transformations of time and generalized coordinates together with dissipative forces determine the transformations of generalized velocities, as it is the case with variations in a variational principle of Hamilton's type for purely nonconservative mechanics [17], [18]. Using the theory a few new first integrals for nonconservative problems are obtained.

Lagrangian Dynamics of Spinning Particles and Polarized Media in General Relativity

Abstract: The general form of the Lagrangian equations of motion is derived for a spinning particle having arbitrary multipole structure in arbitrary external fields. It is then shown how these equations, together with the complete system of field equations can be recovered from a fourdimensional action integral representing a polarized dustlike medium interacting with an arbitrary set of fields. These general results are then specialized to the case of Einstein-Maxwell fields in order to obtain the general-relativistic extension of Lorentz's dielectric theory.

Energy and momentum conserving methods of arbitrary order for the numerical integration of equations of motion

Abstract: Conventional numerical methods, when applied to the ordinary differential equations of motion of classical mechanics, conserve the total energy and angular momentum only to the order of the truncation error. Since these constants of the motion play a central role in mechanics, it is a great advantage to be able to conserve them exactly. A new numerical method is developed, which is a generalization to arbitrary order of the "discrete mechanics" described in earlier work, and which conserves the energy and angular momentum to all orders. This new method can be applied much like a "corrector" as a modification to conventional numerical approximations, such as those obtained via Taylor series, Runge-Kutta, or predictor-corrector formulae. The theory is extended to a system of particles in Part II of this work.

The stability of periodic orbits in the three-body problem

Abstract: A method is developed to study the stability of periodic motions of the three-body problem in a rotating frame of reference, based on the notion of surface of section. The method is linear and involves the computation of a 4×4 variational matrix by integrating numerically the differential equations for time intervals of the order of a period. Several properties of this matrix are proved and also it is shown that for a symmetric periodic motion it can be computed by integrating for half the period only.

Quantum-statistical approach to gross properties of peripheral collisions between heavy nuclei

Abstract: Non-equilibrium quantum-statistical mechanics is applied to peripheral collisions between heavy nuclei (A≳40) where a large number of degrees of freedom are involved during the process. By eliminating the relative motion from the explicit consideration, the transitions between different channels are determined by a Liouville equation with timedependent coupling matrix elements. The introduction of subsets of channels (coarse graining) leads to the definition of macroscopic variables which correspond to observable quantities. The equation of motion for the macroscopic variables become irreversible by assuming the values of the coupling matrix elements to be randomly distributed. The validity and possible applications of the resulting master equations are discussed.

Relativistic Hydrodynamic Theory of Heavy Ion Collisions

Abstract: By use of finite-difference methods the classical relativistic equations of motion for the head-on collision of two heavy nuclei are solved. For $sup 16$O projectiles incident onto various targets at laboratory bombarding energies per nucleon less than or equal to2.1 GeV, curved shock waves develop. The target and projectile are deformed and compressed into crescents of revolution. This is followed by rarefaction waves and an overall expansion of the matter into a moderately wide distribution of angles.

Coupled-states approach for elastic and for rotationally and vibrationally inelastic atom-molecule collisions

Abstract: The elastic and inelastic collisions of an atom with a diatomic molecule are treated quantum mechanically in the body−fixed coordinate system. The body−fixed equations of motion are first compared with the usual spaced−fixed ’’close−coupling’’ equations and limiting cases are considered in which the two formalisms become equivalent. The recently developed ’’coupled−states’’ approximation in the body−fixed system is then described in which intermultiplet transitions are neglected and the eigenvalue of the orbital angular momentum operator l2 is approximated by h/l (l + 1). Numerically computed cross sections from this approximation are compared to those computed from the standard space−fixed close−coupling equations for the test system He−H2. Agreement to within a few percent is obtained for the integral as well as for the differential cross sections for elastic and for rotationally and vibrationally inelastic scattering in the energy range of 0.9 to 4.2 eV. A coupled−states large basis calculation (j = 0, 2, 4, 6, 8, 10 for n = 0 and j = 0, 2, 4, 6 for n = 1) at 4.2 eV is presented which demonstrates the enormous utility of the method.

Transport of Heat Across a Plane Turbulent Mixing Layer

Abstract: Publisher Summary A well-known characteristic of turbulent shear flows is that the spread of scalar quantities—that is, heat or matter, is faster than the spread of momentum This chapter describes a model for the transport mechanism of a scalar quantity in typical turbulent shear flows In the present state of the investigation, measurements of the temperature field alone have been obtained and the temperature field is mathematically represented by four equations; the equation of motion, the continuity equation for the mean velocities, the heat-transfer equation, and the equation of the balance of temperature fluctuations The chapter provides some important observations, such as based on the shape of the temperature fluctuations as well as the mean quantity distributions it can be assumed that the transport mechanism is largely due to a large scale vortical motion, the mean temperature distribution in the turbulent domain is largely homogeneous, the temperature across a single vortex is linear in the mean, and the temperature fluctuation variance in the turbulent domain is approximately constant

Generalized Ginzburg-Landau equations for phase transition-like phenomena in lasers, nonlinear optics, hydrodynamics and chemical reactions

Abstract: The recently found close analogies between the continuous mode laser, the Benard instability, and chemical instabilities with respect to their phase transition-like behaviour are shown to have a common root. We start from equations of motion containing fluctuations. We first assume external parameters permitting only stable solutions and linearize the equations, which define a set of modes. When the external parameters are changed the modes getting unstable are taken as order parameters. Since their relaxation time tends to infinity the damped modes can be eliminated adiabatically leaving us with a set of nonlinear coupled order parameter equations resembling the time dependent Ginzburg-Landau equations with fluctuating forces. In two and three dimensions additional terms occur which allow for e.g. hexagonal spatial structures. We also treat the hard mode instability and obtain the stationary distribution function as solution of the Fokker-Planck equation. Our procedure has immediate applications to the Taylor instability, to various chemical reaction models, to the parametric oscillator in nonlinear optics and to some biological models. Furthermore, it allows us to treat analytically the onset of laser pulses, higher instabilities in the Benard and Taylor problems and chemical oscillations including fluctuations.

Equations of motion for interconnected rigid and elastic bodies: A derivation independent of angular momentum

Abstract: Equations of motion are derived for systems of rotationally interconnected bodies in which the terminal bodies may be flexible and the remaining bodies are rigid. The bodies may have an arbitrary ‘topological tree’ arrangement; that is, there are no closed loops of bodies. This derivation extends earlier results for systems of interconnected rigid bodies only, and is much simpler than several other recent works on terminal flexible bodies. The model for a flexible body assumes that the elastic deformation is representable as a time-varying linear combination of given mode shapes. The paper also derives the appropriate form for gravitational terms, so that the equations can be used for flexible satellites. Also included are expressions for kinetic energy and angular momentum so that in case these are theoretically constant, they can be used to monitor the accuracy of the numerical integration. The paper concludes with a section showing how interbody constraint forces and torques (which do not appear in the equations of motion) can be recovered from quantities available in this formulation, and also how to treat state variables which are prescribed functions of time. A digital computer program based on the equations derived here has been used to simulate a spinning Skylab (with flexible booms) and also the interplanetary Viking (with flexible solar panels and thrust vector control).

Equations of hydrodynamics in general relativity in the slow motion approximation

Abstract: By means of a formal solution to the Einstein gravitational field equations a slow motion expansion in inverse powers of the speed of light is developed for the metric tensor. The formal solution, which satisfies the deDonder coordinate conditions and the Trautman outgoing radiation condition, is in the form of an integral equation which is solved iteratively. A stress-energy tensor appropriate to a perfect fluid is assumed and all orders of the metric needed to obtain the equations of motion and conserved quantities to the 21/2post-Newtonian approximation are found. The results are compared to those obtained in another gauge by S. Chandrasekhar. In addition, the relation of the fast motion approximation to the slow motion approximation is examined.

Exact localized solutions of two-dimensional field theories of massive fermions with Fermi interactions

Abstract: The classical equations of motion for field theories of massive fermions with Fermi interactions in one space and one time dimension are investigated. It is shown that they all possess exact stationary solutions which are localized, due to the vanishing of the stress tensor, a feature for this type of solutions only in two dimensions. The explicit forms of these solutions are presented. More importantly, exact solutions for the bound state of $N$ localized massive fermions with scalar or vector Fermi interactions are also found.