Discretized Equations Of Motion Research Articles

Explicit/implicit partitioning and a new explicit form of the generalized alpha method

Abstractabstract Most finite element packages use the Newmark algorithm for time integration of structural dynamics. Various algorithms have been proposed to better optimize the high frequency dissipation of this algorithm. Hulbert and Chung proposed both implicit and explicit forms of the generalized alpha method. The algorithms optimize high frequency dissipation effectively, and despite recent work on algorithms that possess momentum conserving/energy dissipative properties in a non‐linear context, the generalized alpha method remains an efficient way to solve many problems, especially with adaptive timestep control. However, the implicit and explicit algorithms use incompatible parameter sets and cannot be used together in a spatial partition, whereas this can be done for the Newmark algorithm, as Hughes and Liu demonstrated, and for the HHT‐αalgorithm developed from it. The present paper shows that the explicit generalized alpha method can be rewritten so that it becomes compatible with the implicit form. All four algorithmic parameters can be matched between the explicit and implicit forms. An element interface between implicit and explicit partitions can then be used, analogous to that devised by Hughes and Liu to extend the Newmark method. The stability of the explicit/implicit algorithm is examined in a linear context and found to exceed that of the explicit partition. The element partition is significantly less dissipative of intermediate frequencies than one using the HHT‐αmethod. The explicit algorithm can also be rewritten so that the discrete equation of motion evaluates forces from displacements and velocities found at the predicted mid‐point of a cycle. Copyright © 2003 John Wiley & Sons, Ltd.

Thermal diffusion of supersonic solitons in an anharmonic chain of atoms.

We study the nonequilibrium diffusion dynamics of supersonic lattice solitons in a classical chain of atoms with nearest-neighbor interactions coupled to a heat bath. As a specific example we choose an interaction with cubic anharmonicity. The coupling between the system and a thermal bath with a given temperature is made by adding noise, delta correlated in time and space, and damping to the set of discrete equations of motion. Working in the continuum limit and changing to the sound velocity frame we derive a Korteweg-de Vries equation with noise and damping. We apply a collective coordinate approach which yields two stochastic ODEs which are solved approximately by a perturbation analysis. This finally yields analytical expressions for the variances of the soliton position and velocity. We perform Langevin dynamics simulations for the original discrete system which confirm the predictions of our analytical calculations, namely, noise-induced superdiffusive behavior which scales with the temperature and depends strongly on the initial soliton velocity. A normal diffusion behavior is observed for solitons with very low energy, where the noise-induced phonons also make a significant contribution to the soliton diffusion.

Finite‐difference modeling of SH‐wave propagation in nonwelded contact media

Many geological structures of interest are known to exhibit fracturing. Fracturing directly affects seismic wave propagation because, depending on its scale, fracturing may give rise to scattering and/or anisotropy. A fracture may be described mathematically as an interface in nonwelded contact (i.e., as a displacement discontinuity). This poses a difficulty for finite‐difference modeling of seismic wave propagation in fractured media, because the standard heterogeneous approach assumes welded contact. In the past, this difficulty has been circumvented by incorporating nonwelded contact into the medium parameters using equivalent medium theory. We present an alternate method based on the homogeneous approach to finite differencing, whereby nonwelded contact boundary conditions are imposed explicitly. For simplicity, we develop the method in the SH‐wave case.In the homogeneous approach, nonwelded contact boundary conditions are discretized by introducing auxiliary, so‐called fictitious, grid points. Wavefield values at fictitious grid points are then used in the discrete equation of motion, so that the time‐evolved wavefield satisfies the correct boundary conditions. Although not as general as the heterogeneous approach, the homogeneous approach has the advantage of being relatively simple and manifestly satisfying nonwelded contact boundary conditions. For fractures aligned with the numerical grid, the homogeneous and heterogeneous approaches yield identical results. In particular, in both approaches nonwelded contact results in a larger maximum stable time step size than in the welded contact case.

How to find the conserved quantities of nonlinear discrete equations

A method of finding conserved quantities of nonlinear ordinary difference equations is briefly discussed. The method is based on the hypothesis that the conserved quantities are expressed by a ratio of two polynomials of dependent variables. We obtain the conserved quantities of the `discrete equation of motion of an anharmonic oscillator' and `a nonlinear difference equation of third order'.

Fokker-Planck equation for lattice deposition models.

An asymptotically exact Fokker-Planck equation for the height fluctuations of lattice deposition models is derived from a Van Kampen expansion of the master equation. Using an Edwards-Wilkinson-type model as an example, the solution of the equivalent Langevin equation reproduces the surface roughness and lateral height correlations obtained with kinetic Monte Carlo (KMC) simulations. Our discrete equations of motion thereby provide an exact analytic and computational alternative to KMC simulations of these models.

Equivalent circuit representation of a vibrating structure withpiezoelectric transducers and the stability consideration in the activedamping control

Active damping control with piezoelectric transducers is presented.Based on the finite element expression for a vibrationsystem with electromechanical coupling included, the discretizedequation of motion is derived. Modal analysis is then made to developthe corresponding electrical equivalent circuit, the transfer function of which isconsidered within a loop of the active control system. The limitationsand conditions for stable control are discussed.

Equations of motion and conserved quantities in non-Abelian discrete integrable models

Conserved quantities for the Hirota bilinear difference equation, which is satisfied by eigenvalues of the transfer matrix, are studied. The transfer-matrix eigenvalue combinations that are integrals of motion for discrete integrable models, which correspond to Ak−1 algebras and satisfy zero or quasi-periodic boundary conditions, are found. Discrete equations of motion for a non-Abelian generalization of the Liouville model and the discrete analogue of the Tsitseiko equation are obtained.

Adaptive FEM computation of geometric and material nonlinearities with application to brittle failure

A model is presented for the dynamic finite element analysis of large-strain, high strain rate deformation behavior of materials. A total Lagrangian formulation is used in the derivation of discrete equations of motion. Both an isochoric finite deformation plasticity model, including rate and temperature effects, for metals, and a multiple-plane microcracking model for ceramics are introduced. In addition, algorithms are presented for correcting finite element mesh distortion through mesh rezoning, optimization, and refinement. A surface-defined multibody contact algorithm designed to handle large relative displacements between bodies, with addition for friction, is included. Extensions of the mechanical contact to account for heat fluxes between sliding bodies and the treatment of body interfaces with cohesive strength are presented within a unified framework. Two test examples are examined, simulating a modified Taylor rod impact experiment, in which an aluminum anvil strikes a confined or unconfined ceramic rod. Axial and radial velocities are computed at the free end of the ceramic and at 30 mm from the impact surface, respectively. Comparisons with experimental traces reveal that the simulations produce the same overall features observed in the experimental data.

Vibration and crack detection of a rotor with speed-dependent bearings

The vibration of a shaft-disk rotor containing a transverse crack and supported by speed-dependent bearings is investigated. With crack released energy, the flexibility due to crack is first evaluated. An energy principle in conjunction with the assumed-mode method follow to yield the discrete equations of motion of periodic, time-varying coefficients From the FFT analysis of the displacement responses, the two-times shaft speed component (2Ω) is extracted and serves as a good index to detect the crack location and depth. Response amplitudes to the variations of crack depth and crack location are then discussed and a technique of crack identification is introduced with the aid of response contour maps of two sensing probes.

On The Vibration of a Cracked Rotating Blade

The dynamic behavior of a rotating blade containing a transverse crack was investigated. First, the local flexibility of the cracked blade was obtained by using the method of the released energy. An energy principle, in conjunction with a weighted residual method, was then applied to yield the discrete equations of motion. The equations of motion were further utilized to study the influences of the crack depth and location on the bending natural frequencies under various of rotation speeds. The numerical calculation showed that the crack effects the natural frequencies and the response appreciably only if it is relatively deep and locates near the root of the blade. However, the effects increase exponentially with the depth increases. In addition to the natural frequencies, the displacement responses of the blade with a crack under a constant lateral forces were discussed as well. This was done by calculating the deflections at the tip of the blade for various crack depths and locations. Similar to the rotation speed of the blade frequency, the deflection was offset by the increase of the rotation. However, the centrifugal effects increased significantly such that the crack’s effects became relatively insignificant. Nevertheless, the study showed that the changes on the natural frequency and the tip-deflection of the blade due to a crack may be used as indices for on-line detection of cracks.

Influence of static disorder on the dynamics of Fermi resonance solitary waves at the interface between two molecular layers

We consider the nonlinear dynamics of the energy transport along the interface between two contacting molecular layers A and B. Between neighboring molecules of different layers we assume a Fermi resonance coupling, where the excitation energy ω B of a localized optical phonon at molecule B is approximately twice the excitation energy ω A of a localized phonon at molecule A. For the classical equations of motion in continuum approximation, we find exact analytical solutions in the form of solitary waves propagating along the interface. By numerically integrating the discrete equations of motion, we (a) confirm the stability of this solitary wave solutions on discrete ideal systems and (b) consider the dynamics of the solitary waves on discrete systems with static disorder.

Dispersion and wave propagation in discrete and continuous models for granular materials

A generalised continuum model for granular media is derived by direct homogenisation of the discrete equations of motion. In contrast to previous works on this topic, continuum concepts such as stress and moment stress are introduced after homogenisation. First, a very simple one-dimensional model is considered and the continuum version for this model is derived by replacing the difference quotients of the discrete model by differential quotients. The dispersion relations of the discrete and the continuous model are derived and compared. Variational boundary conditions for the continuous model are deduced from the stationarity of the corresponding Lagrangian. The three-dimensional case is treated in an essentially similar fashion. The resulting continuum theory is a combination of a Cosserat Continuum and a higher-order deformation gradient continuum. The salient features of the theory are illustrated by means of the dispersion relations for planar wave propagation.

Whirl speeds and stability of rotating shaft subjected to end loads

The variation of natural whirl speeds and instability of a rotating shaft subjected to end loads (i.e., axial com- pressive forces, moments, and torques) are studied. A finite element model of a Timoshenko beam is adopted to approximate the continuous shaft, and important factors such as the rotatory inertia, shear deformations, and gyroscopic moments are taken into account. The results show that the natural whirl speeds of the shaft do not always decrease when the loads increase and the types of system instability resulting from different end loads are not the same. moments, and torques are studied in a kinetic approach. To obtain discrete equations of motion of the system and to acquire accurate numerical results, a finite element model of a Timoshenko beam proposed by Thomas and Abbas13 is adopted. This model is suc- cessfully used to investigate vibration problems of rotating blades in Refs. 14 and 15. By using the model, all kinds of boundary con- ditions of beam-type structures can be satisfied, and only a few elements are needed to obtain good results. II. Finite Element Formulation Energy of the Shaft A cantilever rotating shaft with a rigid disk at its end and a fixed reference XYZ: J- triad, with the X axis being coincident with the undeformed rotor centerline, is illustrated in Fig. 1. In a deformed state, an element of the shaft of length d;c, located at a distance x from the left end, can be described by the translations V (x, t) and W(x, t) of the center of the element in the Y and Z directions, as well as the small rotations (/>Y(x, t) and <pz(x, t) and the twisting angle fi(x,t) about Y, Z, and X. For a circular shaft, the kinetic energy K Es and the strain energy Us, taking account of the effects of the gyroscopic moments, rotatory inertia, and shears, are given by

Solitary excitations in two-dimensional molecular crystals

Intramolecular excitations nonlinearly coupled with lattice vibrations in two-dimensional molecular systems are studied. The dynamical discrete equations of motion are presented. In the quasi-continuum approximation, we derive a spatial soliton solution with a sech-profile. It is shown that the solitary bound state formation is due to the anharmonicity of the intramolecular vibrations (excitons).

Introduction of the digital signal processing methodology (DSPM) for time integration

Although the existence of a close relationship between the areas of digital signal processing and time integration methodology is known, a systematic application of the concepts and methods of the first area to the second is missing. Such an approach is followed in this paper, arising from the fact that any time integration formula can be viewed as a digital filter of the applied excitation force, approximating as close as possible to the behaviour of a ‘prototype analogue filter’, which is in fact the semi discrete equations of motion of the system. This approach provides a universal framework for handling and analysing all various aspects of time integration formulae, such as analysis in the frequency domain, algebraic operations, accuracy and stability, aliasing, spurious oscillations generation, introduction of digital filters within the time integration formula, initial conditions handling and overshooting. Additionally it is shown that digital signal processing methods, such as pre‐ or post‐processing, time delays, etc. can be in certain cases a quite effective complement of the time integration scheme.

Morphological equilibration of a corrugated crystalline surface.

We study the flattening of a corrugated crystal surface for the case when mass transport is driven by surface diffusion and the temperature is well below the roughening temperature of the equilibrium facet. For the considered geometry, the problem is one dimensional and the surface morphology consists of a collection of terraces separated by straight, parallel steps. In this situation, the kinetics is driven by step-step interactions alone. We obtain separated variable (shape-preserving) solutions to both the differential equation of motion for the surface profile z(x,t) and the discrete equations of motion for individual terraces' widths ${\mathit{l}}_{\mathit{n}}$(t). Numerical solutions of these equations demonstrate that the analytic solution well describes the kinetics in the latter case. As morphological equilibration proceeds, we find that z(x,t)\ensuremath{\sim}${\mathit{t}}^{\mathrm{\ensuremath{-}}1}$ and that ${\mathit{l}}_{\mathit{n}}$(t)\ensuremath{\sim}${\mathit{t}}^{1/5}$ or ${\mathit{l}}_{\mathit{n}}$(t)\ensuremath{\sim}${\mathit{t}}^{1/4}$ for the cases of diffusion-limited or step-attachment-limited step propagation, respectively. These predictions are amenable to direct experimental tests.

A numerical approach to vibrations of orthotropic plates using Gureen's function for a static bending problem. (2nd Report, Application to transient vibrations).

An approach is proposed for the estimation of free, forced and transient vibrations of an orthotropic plate by using Green's function for a static bending problem. In the previous paper, free and forced vibrations of a simply supported orthotropic rectangular plate were discussed to check the accuracy of the approach. As numerical examples, a free vibration of a rectangular plate with a mass attached in the center was discussed. In the present paper, a transient vibration of an orthotropic plate is discussed by applying Wilson's θ method to a set of discrete equations of motion. First, to discuss the accuracy, numerical computations are carried out for a simply supported rectangular plate subject to a uniformly distributing stepping load. Next, the approach is applied to a simply supported rectangular plate with an attached mass.

Discrete gradients in discrete classical mechanics

A simple model of discrete classical mechanics is given where, starting from the continuous Hamilton equations, discrete equations of motion are established together with a proper discrete gradient definition. The conservation laws of the total discrete momentum, angular momentum, and energy are demonstrated.

A hybrid approach to modeling the dynamics of macromolecules

A new approach to modeling macromolecular systems is presented which can be used to study both the low frequency collective motion of the entire macromolecule and the high frequency motion of local regions within the molecule. The approach combines the classical atomistic model of the molecule with a continuum model to form an overall classical hybird model. This hybrid model is developed by using a long wavelength limit to the discrete equations of motion to obtain a field of local elastic modulus tensors which depend on the intramolecular potential and molecular configuration. These tensors can then be used in a qualitative manner to correlate the elasticity of different regions in the macromolecule with the dynamics of the molecule as a whole. In addition, the tensors obey specific symmetry conditions which can be utilized to identify regions of the macromolecule that are displaced from equilibrium. The field of tensors can also be used in a quantitative manner to decrease the number of degrees of freedom for the macromolecule by the lumping of regions of uniform elasticity as dictated by the field. The symmetry properties may also be used to obtain equations of motion corresponding to an equilibrium configuration and an intramolecular potential which are consistent with each other. This latter result is important in the cases where the molecular configuration in the atomistic model does not properly correspond to the equilibrium configuration for the intramolecular potential. The applicability of the tensor as an analytic tool for investigating macromolecular systems is illustrated using several molecules and molecular analogs.

NUMERICAL SOLUTION OF PROBLEMS IN INCOMPRESSIBLE FLUID FLOW: TREATMENT OF THE TEMPERATURE-VELOCITY COUPLING

The cost of obtaining solutions to fluid flows depends strongly on how well the solution procedure treats the interequation couplings. For natural convection flows, treatment of the temperature-velocity coupling is often of the utmost importance. The first purpose of the present paper is to report the conditions under which the temperature-velocity coupling is important in natural convection flows, and the improvement in convergence rate that may be achieved by implicitly accounting for this coupling within the discrete equations of motion. It is shown that the strength of the coupling depends on the Prandtl number, and that substantial acceleration of the convergence rate can be achieved by appropriate treatment of the coupling. To translate improved convergence rate into computational savings requires that an efficient method be found for solution of the coupled linear set of equations. The second purpose of the paper is to present such a method, and to report the computation effort required to obtain s...