Showing papers on "Equations of motion published in 1988"

Algorithms Based on Hamilton-Jacobi Formulations

Abstract: We devise new numerical algorithms, called PSC algorithms, for following fronts propagating with curvature-dependent speed. The speed may be an arbitrary function of curvature, and the front can also be passively advected by an underlying flow. These algorithms approximate the equations of motion, which resemble Hamilton-Jacobi equations with parabolic right-hand-sides, by using techniques from the hyperbolic conservation laws. Non-oscillatory schemes of various orders of accuracy are used to solve the equations, providing methods that accurately capture the formation of sharp gradients and cusps in the moving fronts. The algorithms handle topological merging and breaking naturally, work in any number of space dimensions, and do not require that the moving surface be written as a function. The methods can be also used for more general Hamilton-Jacobitype problems. We demonstrate our algorithms by computing the solution to a variety of surface motion problems.

Fourth-order finite-difference P-SV seismograms

Abstract: I describe the properties of a fourth-order accurate space, second-order accurate time two-dimensional P-Sk’ finite-difference scheme based on the MadariagaVirieux staggered-grid formulation. The numerical scheme is developed from the first-order system of hyperbolic elastic equations of motion and constitutive laws expressed in particle velocities and stresses. The Madariaga-Virieux staggered-grid scheme has the desirable quality that it can correctly model any variation in material properties, including both large and small Poisson’s ratio materials, with minimal numerical dispersion and numerical anisotropy. Dispersion analysis indicates that the shortest wavelengths in the model need to be sampled at 5 gridpoints/wavelength. The scheme can be used to accurately simulate wave propagation in mixed acoustic-elastic media, making it ideal for modeling marine problems. Explicitly calculating both velocities and stresses makes it relatively simple to initiate a source at the free-surface or within a layer and to satisfy free-surface boundary conditions. Benchmark comparisons of finite-difference and analytical solutions to Lamb’s problem are almost identical, as are comparisons of finite-difference and reflectivity solutions for elastic-elastic and acoustic-elastic layered models.

Wave propagation simulation in a linear viscoacoustic medium

Abstract: SUMMARY A new approach for viscoacoustic wave propagation is developed. The Boltzmann’s superposition principle based on the general standard linear solid rheology is implemented in the equation of motion by the introduction of memory variables. This approach replaces the conventional convolutional rheological relation, and thus the complete time history of the material is no longer required, and the equations of motion become a coupled first-order linear system in time. The propagation in time is done by a direct expansion of the evolution operator by a Chebycheff polynomial series. The resulting method is highly accurate and effects such as the numerical dispersion often encountered in time-stepping methods are avoided. The numerical algorithm is tested in the problem of wave propagation in a homogeneous viscoacoustic medium. For this purpose the l-D and 2-D viscoacoustic analytical solutions were derived using the correspondence principle.

Flight dynamics of aeroelastic vehicles

Abstract: The nonlinear equations of motion for an elastic airplane are developed from first principles. Lagrange's equation and the Principle of Virtual Work are used to generate the equations of motion, and aerodynamic strip theory is then employed to obtain closed-form integral expressions for the generalized forces. The inertial coupling is minimized by appropriate choice of the body-reference axes and by making use of free vibration modes of the body. The mean axes conditions are discussed, a form that is useful for direct application is developed, and the rigid-body degrees of freedom governed by these equations are defined relative to this body-reference axis. In addition, particular attention is paid to the simplifying assumptions used during the development of the equations of motion. Since closed-form, analytic expressions are obtained for the generalized aerodynamic forces, insight can be gained into the effects of parameter variations that is not easily obtained from numerical models. An example is also presented in which the modeling method is applied to a generic elastic aircraft, and the model is used to parametrically address the effects of flexibility. The importance of residualizing elastic modes in forming an equivalent rigid model is illustrated, but as vehicle flexibility is increased, even modal residualization is shown to yield a poor model.

Soil-structure-interaction analysis in time domain

Abstract: The procedures to perform nonlinear soil-structure-interaction analysis in the time domain are summarized. The nonlinearity is restricted to the structure and possibly an adjacent irregular soil region. The unbounded soil (far field) must remain linear in this formulation. Besides the direct method where local frequency-independent boundary conditions are enforced on the artificial boundary, various formulations based on the substructure method are addressed, ranging from a discrete model with springs, dashpots and masses to boundary-element methods with convolution integrals involving either the dynamic-stiffness coefficients or the Green's functions in the time domain via the iterative hybrid-frequency-time-domain analysis procedure with the nonlinearities affecting only the right-hand side of the equations of motion.

Nonlinear gyrokinetic theory for finite‐beta plasmas

Abstract: A self‐consistent and energy‐conserving set of nonlinear gyrokinetic equations, consisting of the averaged Vlasov and Maxwell’s equations for finite‐beta plasmas, is derived. The method utilized in the present investigation is based on the Hamiltonian formalism and Lie transformation. The resulting formulation is valid for arbitrary values of k⊥ρi and, therefore, is most suitable for studying linear and nonlinear evolution of microinstabilities in tokamak plasmas as well as other areas of plasma physics where the finite Larmor radius effects are important. Because the underlying Hamiltonian structure is preserved in the present formalism, these equations are directly applicable to numerical studies based on the existing gyrokinetic particle simulation techniques.

A modified Lagrangian formulation for the dynamic analysis of constrained mechanical systems

Abstract: A modified Lagrangian formulation is presented for the dynamic analysis of constraint mechanisms. The proposed method is based on a Hamiltonian description of the dynamics which leads to the Lagrange's equations. However, the constraint conditions are not appended to the Lagrange's equations in the form of algebraic or differential constraints, but instead inserted in them by means of a penalty formulation, and therefore the number of equations of the system does not increase. In addition, this approach directly leads to a system of ordinary differential equations, as opposed to the classical Lagrange's formulation which results in differential algebraic equations. The resulting set of equations is of the form dot y =g(y,t) , which can be integrated by standard numerical algorithms. Finally, the proposed method is very systematic and general, and can model any type of constraint conditions, either holonomic or nonholonomic. A series of illustrative examples are analyzed. The results demonstrate the capabilities of the proposed method for simulation analysis.

Entropy and 2-D velocity distribution in open channels

Abstract: Equations based on the entropy concept have been derived for describing the two‐dimensional velocity distribution in an openchannel cross section. The velocity equation derived is capable of describing the variation of velocity in both the vertical and transverse directions, with the maximum velocity occurring on or below the water surface. Equations for determining the location of mean velocity have also been derived along with those, such as the entropy function, that can be used as measures of the homogeneity of velocity distribution in a channel cross section. A dimensionless parameter of the entropy function named the M number has been found useful as an index for characterizing and comparing various patterns of velocity distribution and states of open‐channel flow systems. The definition and demonstrated usefulness of this parameter indicate the importance and value of the information given by the location and magnitude of maximum velocity in a cross section, and suggest the need for future experime...

The mechanics of vibrations of cylindrical shells

Abstract: 1. The Membrane Theory of Cylindrical Shells. Equations of static equilibrium. Equations of dynamic equilibrium. Equations of deformation of the cylindrical shell. Solution of the natural vibration problem. Constraints of the membrane theory of cylindrical shells. References. 2. The Bending Theory of Cylindrical Shells. Axisymmetrically loaded cylindrical shells. The general deformation theory of thin cylindrical shells. 3. Accurate Equations of Motion of Cylindrical Shell Vibration. Solution of the equations of motion. The frequency equation. Comparison of the precise theory with Flugge's approximate theory. References. 4. The Theory of Damped, Two-Layered Cylindrical Shells. Fundamental assumptions. Derivation of the equations of motion. Characteristics of damped vibrations of two-layered shells. References. Additional bibliography on shell vibration. Index.

Inverse dynamic analysis and simulation of a platform type of robot

Abstract: This article presents an algorithm to solve the inverse dynamics for platform type of manipulators using Newton-Euler equations of motion. We found that the inverse dynamics of the system is governed by thirty-six linear equations. The number of these simultaneous equations can be reduced to six, if a proper sequence is taken. The relationships between the actuating forces and the shape of the structure are analyzed. Based on the algorithm, computer code for simulation was developed. Three cases were studied. As a result, configurations which minimize the actuating forces are suggested. It is also found that the fluctuation of the driving forces is lesser when the path is closer to the center of the base. These results are believed to be useful in the design and control of this type of manipulating devices.

Numerically stable finite element techniques for viscoelastic calculations in smooth and singular geometries

Abstract: Finite element calculations for viscoelastic flows are reported that use a restructured form of the equation of motion that makes explicit the elliptic character of this equation. We call this restructured equation the Explicity Elliptic Momentum Equation, and its use is illustrated for flow of an upper convected Maxwell (UCM) model between eccentric and concentric rotating cylinders and also for a modified upper convected Maxwell (MUCM) model in the stick-slip problem. Sets of mixed-order approximations for velocity, stress, and a modified pressure are used to test the algorithm in both problems. Both sets of calculations are shown to converge with mesh refinement and are limited at high values of Deborah number by the formation of elastic boundary layers that are identified in the momentum equation by the growth of low-order derivative terms that involve the local velocity gradient and divergence of stress. Similar convergence properties are observed for bilinear and biquadratic Lagrangian approximations to the stress components. However, calculations with the more accurate basis for stress converge to higher values of De and are sensitive to the weighted residual method used for the constitutive equation, particularly for the eccentric cylinder problem. Streamline-upwind Petroy-Galerkin (SUPG) and artificial diffusivity (AD) formulations of the constitutive equation are tested for solution of both problems by calculations of the stress fields with fixed kinematics and by solution of the coupled problem. The SUPG method improves the performance of the calculations with the biquadratic basis set for the eccentric cylinder problem. For the UCM model, adding artificial diffusion to the constitutive equation in the stick-slip problem changes the dominant balance for the stress field near the singularity, making it appear as an integrable stress approximation for fixed mesh. For the MUCM model the Newtonian-like behavior of the stress near this point is unaffected by the AD method and calculations converge to moderate De .

Bulk elastic wave propagation in partially saturated porous solids

Abstract: The linear equations of motion that describe the behavior of small disturbances in a porous solid containing both liquid and gas are solved for bulk wave propagation. The equations have been simplified by neglecting effects due to changes in capillary pressure. With this simplifying assumption, the equations reduce to two coupled (vector) equations of the form found in Biot’s equations (for full saturation) but with more complicated coefficients. As in fully saturated solids, two shear waves with the same speed but different polarizations exist as do two compressional waves with distinct speeds. Attenuation effects can be enhanced in the partially saturated solid, depending on the distribution of gas in the pore space. Two models of the liquid/gas spatial distribution are considered: a segregated‐fluids model and a mixed‐fluids model. The two models predict comparable attentuation when the gas saturation is low, but the segregated‐fluids model predicts a more rapid roll‐off of attenuation as the gas saturation increases.

Geometric non‐linear substructuring for dynamics of flexible mechanical systems

Abstract: A substructure synthesis formulation is presented that permits use of established flexible multibody dynamic analysis computer codes to account for structural geometric non-linear effects. Large relative displacement is permitted between points within bodies that undergo small strain elastic deformation. Components are divided into substructures, on each of which the theory of linear elasticity relative to a body reference frame is adequate to describe deformation and its coupling with system motion. Normal vibration and static correction deformation modes are used to account for elastic deformation within each substructure. Compatibility conditions are derived and imposed as constraint equations at boundary points between substructures. System equations of motion that include geometric non-linear effects of large rotation, in terms of generalized co-ordinates of a reference frame for each substructure and a set of deformation modes that are defined within the substructure, are assembled. The method is implemented in an industry standard flexible multibody dynamics code, with minimal modification. Use of the formulation is illustrated on the classical problem of a spinning beam with geometric stiffening and on a space structure that experiences large deformation.

High-Speed Forming of Metal Sheets by Electromagnetic Force

Abstract: A numerical method to simulate the high-speed free forming of a clamped circular disk in an electromagnetic forming system using a flat spiral coil is presented. The method combines field penetration into the disk and dynamic elasto-plastic deformation of the disk. The spiral coil is approximated by coaxial circular loops carrying the discharge current from a capacitor bank. The penetration of the magnetic field of the coil into the disk is formulated as a boundary-value problem for the diffusion equation. Magnetic field, eddy currents and electromagnetic force density in the disk are calculated. An equation of motion for the disk loaded by the magnetic force is combined with the field equation as well as an equivalent circuit equation. Plane stress condition is assumed. Strain-rate effect on the work-hardening law of the disk material is taken into account. An experiment on the free bulging of annealed aluminum disks is also presented, showing fairly good agreement with the numerical solution.

Lyapunov instability of dense Lennard-Jones fluids

Abstract: We present calculations of the full spectra of Lyapunov exponents for 8- and 32-particle systems in three dimensions with periodic boundary conditions and interacting with the repulsive part of a Lennard-Jones potential. A new algorithm is discussed which incorporates ideas from control theory and constrained nonequilibrium molecular dynamics. Equilibrium and nonequilibrium steady states are examined. The latter are generated by the application of an external field F/sub e/ through which equal numbers of particles are accelerated in opposite directions, and by thermostatting the system using Nose-Hoover or Gauss mechanics. In equilibrium (F/sub e/ = 0) the Lyapunov spectra are symmetrical and may be understood in terms of a simple Debye model for vibrational modes in solids. For nonequilibrium steady states (F/sub e/not =0) the Lyapunov spectra are not symmetrical and indicate a collapse of the phase-space density onto an attracting fractal subspace with an associated loss in dimensionality proportional to the square of the applied field. Because of this attractor's vanishing volume in phase space and the instability of the corresponding repellor it is not possible to observe trajectories violating the second law of thermodynamics in spite of the time-reversal invariance of the equations of motion. Thus Nose-Hoover mechanics, ofmore » which Gauss's isokinetic mechanics is a special case, resolves the reversibility paradox first stated by J. Loschmidt (Sitzungsber. kais. Akad. Wiss. Wien 2. Abt. 73, 128 (1876)) for nonequilibrium steady-state systems.« less

A recursive formulation for flexible multibody dynamics, Part I: open-loop systems

Abstract: This paper presents a recursive formulation for dynamics of flexible multibody systems. Deformation modes are used to represent elastic deformation of each body, relative to a body reference frame that is permitted to undergo large displacement and rotation in space. Kinematic relations between contiguous bodies that are connected by articulated joints are defined by relative joint coordinates. A recursive variational vector calculus method is presented for efficient formulation and solution of the equations of motion on parallel processors. A manipulator and a rotating blade with geometric nonlinear effects are studied to illustrate computational efficiency of the recursive formulation.

Calculation procedure for viscous incompressible flows in complex geometries

Abstract: A general calculation procedure for computing fluid flow and related phenomena in arbitrary-shaped domains is presented. The scheme has been developed for a generalized nonorthogonal coordinate system and is based on a control-volume approach with a staggered grid arrangement. The physical covariant velocity components are selected as the dependent variables in the momentum equations. The discretization equations for these velocity components are obtained by an algebraic manipulation of the discretization equations for the Cartesian velocity components. Such a practice avoids any reference to the differential form of the governing equations for the covariant velocity components. The coupling between the continuity and momentum equations is effected using the SIMPLER algorithm.

A Lagrangian Formulation of the Dynamic Model for Flexible Manipulator Systems

Abstract: This paper presents a procedure for deriving dynamic equations for manipulators containing both rigid and flexible links. The equations are derived using Hamilton's principle, and are nonlinear integro-differential equations. The formulation is based on expressing the kinetic and potential energies of the manipulator system in terms of generalized coordinates. In the case of flexible links, the mass distribution and flexibility are taken into account. The approach is a natural extension of the well-known Lagrangian method for rigid manipulators. Properties of the dynamic matrices, which lead to a less computation, are shown. Boundary-value problems of continuous systems are briefly described. A two-link manipulator with one rigid link and one flexible link is analyzed to illustrate the procedure.

Computed torque for the position control of open-chain flexible robots

Abstract: A technique is presented for the solution of the inverse dynamics of open-chain flexible robots. The proposed method finds the joint torques necessary to produce a specified end-effector motion. The formulation includes all the nonlinear terms due to the large rotation of the links, together with Timoshenko beam theory to model their elastic characteristics. The finite-element method is used to discretize the equations of motion. The performance and capabilities of this technique are tested through a simulation analysis. Results show the potential of the method not only for feedforward control, but also for incorporation in feedback control strategies. >

Numerical modelling of nonlinear effects in laminar flow through a porous medium

Abstract: This paper deals with the introduction of a nonlinear term into Darcy's equation to describe inertial effects in a porous medium The method chosen is the numerical resolution of flow equations at a pore scale The medium is modelled by cylinders of either equal or unequal diameters arranged in a regular pattern with a square or triangular base For a given flow through this medium the pressure drop is evaluated numericallyThe Navier-Stokes equations are discretized by the mixed finite-element method The numerical solution is based on operator-splitting methods whose purpose is to separate the difficulties due to the nonlinear operator in the equation of motion and the necessity of taking into account the continuity equation The associated Stokes problems are solved by a mixed formulation proposed by Glowinski & PironneauFor Reynolds numbers lower than 1, the relationship between the global pressure gradient and the filtration velocity is linear as predicted by Darcy's law For higher values of the Reynolds number the pressure drop is influenced by inertial effects which can be interpreted by the addition of a quadratic term in Darcy's lawOn the one hand this study confirms the presence of a nonlinear term in the motion equation as experimentally predicted by several authors, and on the other hand analyses the fluid behaviour in simple media In addition to the detailed numerical solutions, an estimation of the hydrodynamical constants in the Forchheimer equation is given in terms of porosity and the geometrical characteristics of the models studied

Computer simulation of ion clouds in a Penning trap.

Abstract: In a recent series of experiments, ' a collection of N ions (where N=100-1000) were stored in a Penning trap and cooled to very low temperature (T= 1-10 mK). One expects these ions to be strongly correlated since the coupling parameter I is larger than unity. The relatively small number of ions used in the experiments makes the system ideal for molecular-dynamics (MD) simulations with realistic boundary conditions. We have performed such simulations for N 100 and N 256 in the large-I regime and have seen behavior quite unlike that observed in simulations of an unbounded homogeneous system of ions. These latter simulations predict a liquid phase for I =2 and a transition to a body-centered cubic lattice for I =170. In contrast, we observe that at large I the system of ions arranges itself into concentric spheroidal shells. However, the ions wander randomly over the surface of the shells. The system might therefore be characterized as a crystal in the direction perpendicular to the shells and as a liquid on the shells; similar behavior is observed in smectic-liquid crystals. As I is further increased, diffusion decreases and a 2D hexagonal lattice forms on the outer shells. However, the lattice is imperfect and diffusion persists even for I =300-400. A 2D hexagonal lattice on cylindrical shells was observed previously by Rahman and Schiffer in a simulation designed to model a system of ions in a storage ring. These authors also considered spherically symmetric potentials as a test of the effect of boundary conditions on their model. While space does not allow for a detailed comparison, their results are consistent with those presented here. Our MD code is novel in that it is based on guidingcenter equations of motion. We will 6rst discuss the advantages and range of applicability of the code, and then we present the results of the simulations. In the experiments, a strong magnetic 6eld is applied to con6ne the ion cloud, and this 6eld makes a straightforward simulation difficult by introducing a small time scale and a small length scale: the cyclotron period and the cyclotron radius. To overcome this difficulty, we average out the cyclotron dynamics, replacing the exact equations of motion by guiding-center equations of motion. The idea here is that for a sufficiently strong magnetic field the cyclotron motion decouples from the motion associated with the collision dynamics. For I ) 1, this decoupling requires the cyclotron frequency to be large compared to the plasma frequency. This strong-magnetic-field limit is often achieved in the experiments, and in this case the guiding-center equations of motion provide a good approximation to the exact dynamics of the system. Furthermore, we will see that for N large, the spatial properties of the guiding-center system in equilibrium are the same as those of an equivalent system undergoing exact dynamics. In the guiding-center approximation the state of each ion is specified by its guiding-center position x and its velocity U parallel to the magnetic field B. We take B to be uniform and directed along the z axis of a cylindrical-coordinate system (p, p, z). The equations of motion are then

An Equation of State for Steam for Turbomachinery and Other Flow Calculations

Abstract: Large-scale equations of state for steam used for generating tables are unsuitable for inclusion infinite difference flow calculation computer codes. Such codes, which are in common use in the turbomachinery industry, may require 106 property evaluations before convergence is achieved. This paper describes a simple equation of state for superheated and two-phase property calculations for use in these circumstances. Computational efficiency is excellent and accuracy over the range of application is comparable to that of the large-scale equations. Further advantages are that complete thermodynamic consistency is maintained and the equation can be differentiated analytically for direct substitution into the gas dynamic equations where required. A truncated virial form is used to represent superheated properties and a new empirical correlation for the third virial coefficient of steam is presented.

Molecular dynamics with internal coordinate constraints

Abstract: The method of Ryckaert, Ciccotti, and Berendsen [J. Comp. Phys. 23, 327 (1977)] for integrating the Cartesian equations of motion of a system with holonomic constraints, has been extended to allow the independent constraint of arbitrary internal coordinates. To illustrate this new methodology, and to investigate the effects of dihedral angle constraints on the equilibrium and dynamical properties of macromolecules, we have carried out parallel sets of molecular dynamics simulations and normal mode analyses of a small dipeptide: one without constraints, and one with a single backbone dihedral angle constrained. We find that the averages and the fluctuations of the energies, and of the internal degrees of freedom are not significantly modified by the constraint. However, in the region between 100 and 1400 cm−1 of the normal mode spectrum, the constraint shifts the frequencies of the modes, and modifies their contributions to the spectra of the internal coordinates. Except for the lowest frequency torsional ...