Explicit Equations Of Motion Research Articles

Quasi-velocities definition in Lagrangian multibody dynamics

Based on Lagrangian mechanics, use of velocity constraints as a special set of quasi-velocities helps derive explicit equations of motion. The equations are applicable to holonomic and nonholonomic constrained multibody systems. It is proved that in proposed quasi-spaces, the Lagrange multipliers are eliminated from equations of motion; however, it is possible to compute these multipliers once the equations of motion have been solved. The novelty of this research is employing block matrix inversion to find the analytical relations between the parameters of quasi-velocities and equations of motion. In other words, this research identifies arbitrary submatrices and their effects on equations of motion. Also, the present study aimed to provide appropriate criteria to select arbitrary parameters to avoid singularity, reduce constraints violations, and improve computational efficiency. In order to illustrate the advantage of this approach, the simulation results of a 3-link snake-like robot with nonholonomic constraints and a four-bar mechanism with holonomic constraints are presented. The effectiveness of the proposed approach is demonstrated by comparing the constraints violation at the position and velocity levels, conservation of the total energy, and computational efficiency with those obtained via the traditional methods.

Dynamic Modeling of Planar Multi-Link Flexible Manipulators

A closed-form dynamic model of the planar multi-link flexible manipulator is presented. The assumed modes method is used with the Lagrangian formulation to obtain the dynamic equations of motion. Explicit equations of motion are derived for a three-link case assuming two modes of vibration for each link. The eigenvalue problem associated with the mass boundary conditions, which changes with the robot configuration and payload, is discussed. The time-domain simulation results and frequency-domain analysis of the dynamic model are presented to show the validity of the theoretical derivation.

Geometrical model of massive spinning particle in four-dimensional Minkowski space

We propose the model of a massive spinning particle traveling in four-dimensional Minkowski space. The equations of motion of the particle are obtained from the requirement that its classical paths lie on a cylinder with the time-like axis in Minkowski space. All the paths on one and the same cylinder are gauge equivalent. The equations of motion are found in implicit form for general time-like paths, and they are non-Lagrangian. The explicit equations of motion are derived for trajectories with small curvature and helices. The momentum and total angular momentum are expressed in terms of characteristics of the path in all the cases. The constructed model of the spinning particle has a geometrical character, with no additional variables in the space of spin states being introduced.

Quantum corrections and the de Sitter swampland conjecture

Recently a swampland criterion has been proposed that rules out de Sitter vacua in string theory. Such a criterion should hold at all points in the field space and especially at points where the system is on-shell. However there has not been any attempt to examine the swampland criterion against explicit equations of motion. In this paper we study four-dimensional de Sitter and quasi-de Sitter solutions using dimensionally reduced M-theory that includes quantum corrections. These quantum corrections are found to allow for de Sitter solutions provided certain constraints are satisfied. A careful study however shows that generically such a constrained system does not allow for an effective field theory description in four-dimensions. Nevertheless, if some hierarchies between the various quantum pieces could be found, certain solutions with an effective field theory description might exist. Such hierarchies appear once some mild time dependence is switched on, in which case certain quasi-de Sitter solutions may be found without a violation of the swampland criterion.

Dynamic modeling of constrained planar multibody systems: A case of lower limbs rehabilitative robot

An explicit equation of motion for constrained systems, called the Udwadia-Kalaba equation, opens up a new way of dynamic modeling of constrained multibody systems, which is accomplished in a straightforward and novel three-step process. However, constraint violation arises from violation of the lower-order constraint equations because the aforementioned equations incorporate only the acceleration level constraints. There are two methods, Baumgarte stabilization method (BSM) and modified method, are implemented to realize the constraint violation suppression. The comparison of the BSM and the modified method is provided, based on numerical simulations for a sample constrained system that represents a lower limbs rehabilitative robot with continuous passive motion for therapy and rehabilitation. The results of numerical simulation obtained by the modified method are superior to that obtained by BSM, which confirms the numerical effectiveness and accuracy of the explicit solution based on the modified method to the constrained planar multibody systems.

Trajectory Tracking Control of Parallel Manipulator Based on Udwadia-Kalaba Approach

There have been many approaches for achieving the trajectory tracking control of parallel manipulator. However, these approaches are complex for calculating Lagrangian multipliers. In this paper, unlike the former approaches, a new approach of trajectory tracking control which is based on Udwadia-Kalaba approach is presented. Using this methodology, we can obtain a concise and explicit equation of motion and consider holonomic and nonholonomic constraint whether it is ideal or nonideal simultaneously. The most important difference is that we divide constraints into structural constraints and performance constraints in this paper. Structural constraints are used to establish dynamic model without regard for trajectory control. And performance constraints are used to represent the desired trajectory. For the parallel manipulator, a nonlinear dynamics system, its constraint forces are obtained by second-order constraints. And the numerical simulation in MATLAB shows the parallel manipulator’s movement meets the requirement; tracking trajectory is exact and perfect. Through this paper, we can see that the method can simplify calculation availably.

Coarse-grained equation of motion for many particle system containing internal degrees of freedom

In order to numerically investigate dynamics in large and complicated systems, it is important to derive the explicit equation of motion of an extracted tagged degree of freedom under the statistical influence of the other parts. This corresponds to adopting the procedure usually called “coarse-graining”. Up-to-now, many kinds of coarse grained (CG) simulation method have been proposed based on phenomenological equations of motion and investigated about their numerical efficiencies for practical applications. We can now investigate a dynamics of appropriate degrees of freedom, i.e., the center of mass, the orientation, the internal vibration modes, etc. of CG particles, for suitably chosen characteristics in a system. Although each of the phenomenological CG methods is well established for practical usages, it is still not sufficient to construct a multiscale simulation procedure for large and complicated systems. Some explicit relation between the coarser and finer descriptions should be grasped for deducing the CG expression based on atomistic pictures. Although Langevin equation, a typical CG equation of motion, has been derived from an atomistic equation of motion in previous literatures, most of these descriptions are limited within the equation of center-of-mass motions. In the present study, a microscopic description for arbitrarily extracted degree(s) of freedom was derived by restricting the system of CG particles constructed by tightly bounded finer particles, like polyatomic molecules. The derived equation of motion was shown to be applicable to dynamics simulation of reorientation and/or internal vibration motions of CG-particle systems by analyses of the equation for realistic conditions.

Analogue simulation of commonly encountered nonlinearity for dynamic structures

All structures are practically nonlinear to some degree. The study of nonlinear structures/systems is considerably more complicated than that of linear ones. The prediction of the dynamic responses of a nonlinear structure is considerably difficult unless the nature and extent of the nonlinearity of the structure is fully recognized. Although many types of nonlinearity have been studied extensively, based on the mathematics and analysis used in control engineering, the theory is often not directly applicable to real structures because of the absence of explicit equations of motion for practical vibration situations. The major difficulty in these situations is the detection and identification of the nonlinearity when what is available is the response of a nolinear structure to excitation by external forces rather than an explicit analytical description. This paper presents the results of simulating different types of nonlinearity using an analogue computer. Although mainly single degrees of freedom (SDOF) systems are used, the resultant analogue circuits can easily be combined or expanded to form multi-degrees of freedom nonlinear systems. The constructed systems represent various nonlinear structures whose dynamic response can be studied conveniently by inputting various time domain signals to the analogue systems and measuring the output of the systems.

On constrained motion

The general explicit equations of motion for constrained discrete dynamical systems are obtained. These new equations lead to a simple and new fundamental view of constrained motion where the forces of constraint may be ideal and/or non-ideal.

Closed-form dynamic model of planar multilink lightweight robots

Closed-form equations of motion are presented for planar lightweight robot arms with multiple flexible links. The kinematic model is based on standard frame transformation matrices describing both rigid rotation and flexible displacement, under small deflection assumption. The Lagrangian approach is used to derive the dynamic model of the structure. Links are modeled as Euler-Bernoulli beams with proper clamped-mass boundary conditions. The assumed modes method is adopted in order to obtain a finite-dimensional model. Explicit equations of motion are detailed for two-link case assuming two modes of vibration for each link. The associated eigenvalue problem is discussed in relation with the problem of time-varying mass boundary conditions for the first link. The model is cast in a compact form that is linear with respect to a suitable set of constant parameters. Extensive simulation results that validate the theoretical derivation are included. >

Theory of Longitudinal Relaxation of an AX Spin System under Double Resonance. I. Density Matrix Formulation of Saturation Recovery

Concrete equations of motion describing longitudinal relaxation of an AX system under irradiation of a selective rf field are worked out on the basis of Bloch's density matrix theory. These equations are composed of thermal relaxation and coherent motion of the stirred spin, which determine dynamical characteristics of the spin system. Explicit equations of motion are obtained also for the magnetizations of another spin observed in the saturation-recovery experiment under double resonance. When the stirring field is not too strong to cause any appreciable splitting of the observed line, the recovery of the observed line can be described in terms of several exponential decay terms together with a pair of oscillating terms corresponding to the coherent nutation.

Connection Force Analysis of Mechanisms Described by Explicit Equations of Motion in Generalized Coordinates

Connection forces acting at the joints of a kinematic mechanism are derived from the generalized variables of the mechanism equation of motion using an equivalent-force analysis. The connection force calculation is numerically efficient and is independent of the formulation and solution of the equations of motion. A numerical example illustrates an application to a planar four-bar linkage.

Derivation of the equations of motion for complex structures by symbolic manipulation

This paper outlines a computer program especially tailored to the task of deriving explicit equations of motion for structures with point-connected substructures. The special purpose program is written in FORTRAN and is designed for performing the specific algebraic operations encountered in the derivation of explicit equations of motion. The derivation is by the Lagrangian approach. Using an orderly kinematical procedure and a discretization and/or truncation scheme, it is possible to write the kinetic and potential energy of each substructure in a compact vector-matrix form. Then, if each element of the matrices and vectors encountered in the kinetic and potential energy is a known algebraic expression, the computer program performs the necessary operations to evaluate the kinetic and potential energy of the system explicitly. Lagrange's equations for small motions about equilibrium can be deduced directly from the explicit form of the system kinetic and potential energy.

An Improved Symbolic Manipulation Technique for the Simulation of Nonlinear Dynamic Systems With Mixed Time-Varying and Constant Terms

The time domain behavior of nonlinear dynamic systems often is obtained by numerical integration on the digital computer. These solutions are usually expensive and limit the scope of the dynamic study. The proposed improved technique results in a substantial increase in the computational efficiency by using automatic symbolic manipulation to generate explicit equations of motion algebraically prior to numerical integration. A model is presented which considers the effects of the presence of time-invariant and time-varying symbolic terms and of the sparsity of system elements to provide analytical guidelines for the use of this technique. A number of case studies including typical computational costs are presented.

Revision of the relativistic dynamics with variable rest mass and application to relativistic thermodynamics

For point bodies with variable rest mass it is shown that the definition of forcem0dUi/dτ=Fi is better than the usual d(m0Ui))/dτ=Fi and the explicit equation of motion is given bym0dUi/dτ=Fiext+(z−1UiB−Ui)dm0/dτ, where the external forceFiext has been separated from the reaction force due to the ejected mass,UiB is the four-velocity of the mass centre of the ejected mass andz=UsBU2c−2. For extended bodies it is shown that the von Laue mechanical-energy flux (in the usual «synchronous» formulation) is not plausible. Moreover a new formulation, called «asynchronous» is given. By this formulation many problems receive an immediate solution. The resultant «asynchronous» force density is always orthogonal to the four-velocity, and the equations of motion for continuous media turn out to be formally equal to the classical equations. Expanding the new equations to the first order when the pressure is a regular function of space and time, one obtains the usual «synchronous» equations. In the asynchronous formulation, all quantities relevant to extended bodies transform in a covariant way. In particular, momentum and energy turn out to have the expression proposed by Rohrlich. The above concepts are applied to relativistic thermodynamics. It is shown that, for point bodies, heat transforms as in the recent Ott formulation. For extended bodies, in the usual «synchronous» formulation, heat transforms as pointed out by von Laue. In the «asynchronous» formulation we always haveQ=γQ0. TemperatureT is considered invariant and the relation dQ=TdS valid in the rest system only.