Equations of Motion

A method is proposed for deriving dynamical equations for systems with both rigid and flexible components. During the derivation, each flexible component of the system is represented by a ``surrogate element'''' which captures the response characteristics of that component and is easy to mathematically manipulate. The derivation proceeds essentially as if each surrogate element were a rigid body. Application of an extended form of Lagrange''s equation yields a set of simultaneous differential equations which can then be transformed to be the exact, partial differential equations for the original flexible system. This method''s use facilitates equation generation either by an analyst or through application of software-based symbolic manipulation.

A new class of linear multistep methods is proposed for the solution of the equations of motion of certain dynamical systems encountered in celestial mechanics and astrodynamics. These methods are distinguished from the classical predictor-corrector methods in that they permit ‘back-corrections’ of the solution to be made. As the integration advances in time, the numerical solution is corrected or improved at certain points in the past. The enhanced numerical stability of these methods allows the meaningful application of high-order algorithms. Consequently, stepsizes larger than those attainable with the classical methods may be adopted and thus greater over-all efficiency may be realized. The application of these methods to the problem of determining the orbit of an artificial satellite is accomplished and the results are compared with those obtained using classical methods.

Abstract: This report documents our study of active balance in dynamic legged systems. The purpose of this research is to build a foundation of knowledge that can lead both to the construction of useful legged vehicles and to a better understanding of animal locomotion. ...

Numerical orbit integrations have been conducted to characterize the types of tra-jectories in the one-dimensional Newtonian three-body problem with equal masses and negative energy. Essentially three different types of motions were found to exist. They may be ...

The equations of motion for straight fluid filled pipes are greatly simplified. It is found, for frequencies below a third of the ring frequency, that the radial-axial waves in cylinders are as if the circumferential motion were inextensional. This is the fundamental assumption for the analysis. The derivation is also based on the assumption of long axial wavelengths, resulting in the axial inertia of the fluid and the axial flexural stiffness of the pipe wall being negligible. The formulation is restricted to frequencies well below the cut-on of higher order fluid modes. For such frequencies, the compressibility of the fluid is neglected and the internal fluid loading, on the pipe, is approximated as an increase in the radial inertia. Upon this basis, the equations of motion, for each circumferential mode, are similar to those for a Timoshenko beam on a Winkler foundation. Numerical experiments are made, comparing the approximate theory with results from calculations from the Helmholtz equation for the fluid and accurate thin-walled cylinder theory. Criteria of the application of the simplified theory are formulated.

The results of a numerical investigation of Rayleigh-Taylor instability in an inviscid, incompressible flow by vortex methods was presented. The numerical procedure was briefly described. The effect of surface-tension on the smoothing of irregular motion of vortices in the vortex simulation of Reyleigh-Taylor instability is shown. The irregular motion appears as an effect of short-wave disturbances, the source of which is a round-off error. Inclusion of surface tension allows for the observation of the formation of singularities. The singularities make an infinite jump discontinuity in the curvature of the vortex sheet. It is observed that, for a sufficiently small Atwood number and for initial amplitude perturbation large enough, two singularities appear in a half period of the vortex sheet and only one appears for greater Atwood numbers. The results for desingularized equations of motion were also presented. Desingularization was made by introducing the delta-square parameter to t...

The 2-dimensional BF theory is both a gauge theory and a topological Poisson $\sigma$-model corresponding to a linear Poisson bracket. In \cite{To1}, Torossian discovered a connection which governs correlation functions of the BF theory with sources for the $B$-field. This connection is flat, and it is a close relative of the KZ connection in the WZW model. In this paper, we show that flatness of the Torossian connection follows from (properly regularized) quantum equations of motion of the BF theory.

This study analyzes the effects of link flexibility on the dynamic stability of a force-controlled flexible manipulator. The closed-loop dynamic equation for a l-link manipulator is first derived explicitly using the modal representation technique. Stability analysis is then carried out by computing the system eigenvalues. Results show that the link flexibility is one of the causes of dynamic instability of motion of the flexible robotic arm.

A two-dimensional, seven link, nine degrees of freedom biped model was developed to investigate the dynamic characteristics of normal and transfemoral amputee locomotion during the entire gait cycle. The equations of motion were derived using the Lagrange method and the stance foot-ground contact was simulated using a five-point penetration model. The joint driving torques were obtained using forward dynamic optimization of the normal human gait and applied to the intact joints of the amputee. Three types of motion controllers; frictional, elastic and hydraulic were considered for the prosthetic joints of the amputee and their design parameters were optimized to achieve the closest kinematics to that of the normal gait. It was found that, if optimally designed, all three passive controllers could reasonably reproduce a normal kinematical pattern in the swing phase. However, the stance phase kinematics could only be replicated by the hydraulic and elastic controllers; the performance of the latter was highly sensitive to the design parameters. It was concluded that an appropriately designed hydraulic motion controller can provide reasonably normal kinematics and reliable stability for stance knee flexion prostheses. © 2013 Shandiz et al.; licensee InTech.

cdn.intechopen.com/pdfs/43698/InTech-A_robotic_model_of_transfemoral_amputee_locomotion_for_design_optimization_of_knee_controllers.pdf

Inherent to any heavy-duty hydraulic machine operation with a large number of interconnected components are nonidealities such as gear backlash, friction and leakage. The swing motion of the operator’s cabin in an excavator is a typical example. In this paper we conduct a study comprising experimental, mathematical and simulation components to determine the degree to which these nonlinearities affect the performance of such machines. The inclusion of the conventional model of backlash in the simulation of the excavator swing motion is shown to be inefficient and unnecessary in terms of computation time and the final results. A new model which combines the fluid-flow and the gear train dynamics is developed. The study of contact and non-contact cases brings about proper sets of static and dynamic equations which efficiently simulate this phenomenon for the class of excavator machines under consideration. The inclusion of stick-slip friction model in the simulation shows two effects. ...

A dynamic model for flexible manipulators with prismatic joints is presented in Part I of this study. Floating frames following a nominal rigid body motion are introduced to describe the kinematics of the flexible links. A Lagrangian approach is used in deriving the equations of motion. The work done by the rigid body axial force through the axial shortening of the link due to transverse deformations is included in the Lagrangian function. Kinematic constraint equations are used to describe the compatibility conditions associated with revolute joints and prismatic joints, and incorporated into the equations of motion by Lagrange multipliers. The small displacements due to the flexibility of the links are then discretized by a displacement based finite element method. Equations of motion are derived for the cases of prescribed rigid body motion as well as prescribed joint torques/forces through application of Lagrange’s equations. The equations of motion and the constraint equations ...

The floating frame of reference formulation is currently the most widely used approach in flexible multibody simulations. The use of this approach, however, has been limited to small deformation problems. In this investigation, the computer implementation of the newabsolute nodal co-ordinate formulationand its use in the small and large deformation analysis of flexible multibody systems that consist of interconnected bodies are discussed. While in the floating frame of reference formulation a mixed set of absolute reference and local elastic co-ordinates are used, in the absolute nodal co-ordinate formulation only absolute co-ordinates are used. In the absolute nodal co-ordinate formulation, new interpretation of the nodal co-ordinates of the finite elements is used. No infinitesimal or finite rotations are used as nodal co-ordinates from beams and plates, instead, global slopes are used to define the element nodal co-ordinates. Using this interpretation of the element co-ordinates, beams and plates can be considered as isoparametric elements, and as a result, exact modelling of the rigid body dynamics can be obtained using the element shape function and the absolute nodal co-ordinates. Unlike the floating frame of reference approach, no co-ordinate transformation is required in order to determine the element inertia. The mass matrix of the finite elements is a constant matrix, and therefore, the centrifugal and Coriolis forces are equal to zero when the absolute nodal co-ordinate formulation is used. Another advantage of using the absolute nodal co-ordinate formulation in the dynamic simulation of multibody systems is its simplicity in imposing some of the joint constraints and also its simplicity in formulating the generalized forces due to spring-damper elements. The results obtained in this investigation show an excellent agreement with the results obtained using the floating frame of reference formulation when large rotation–small deformation problems are considered.