Classification of Solutions to the Plane Extremal Distance Problem for Bodies With Smooth Boundaries

2016 ◽  
Vol 11 (6) ◽  
Author(s):  
Jochen Damerau ◽  
Robert J. Low
Keyword(s):  
Equations Of Motion ◽  
Time Integration ◽  
Contact Problems ◽  
Extremal Point ◽  
Point Problem ◽  
Transversality Conditions ◽  
Contact Points ◽  
Extremal Distance ◽  
Two Bodies

The determination of the contact points between two bodies with analytically described boundaries can be viewed as the limiting case of the extremal point problem, where the distance between the bodies is vanishing. The advantage of this approach is that the solutions can be computed efficiently along with the generalized state during time integration of a multibody system by augmenting the equations of motion with the corresponding extremal point conditions. Unfortunately, these solutions can degenerate when one boundary is concave or both boundaries are nonconvex. We present a novel method to derive degeneracy and nondegeneracy conditions that enable the determination of the type and codimension of all the degenerate solutions that can occur in plane contact problems involving two bodies with smooth boundaries. It is shown that only divergence bifurcations are relevant, and thus, we can simplify the analysis of the degeneracy by restricting the system to its one-dimensional center manifold. The resulting expressions are then decomposed by applying the multinomial theorem resulting in a computationally efficient method to compute explicit expressions for the Lyapunov coefficients and transversality conditions. Furthermore, a procedure to analyze the bifurcation behavior qualitatively at such solution points based on the Tschirnhaus transformation is given and demonstrated by examples. The application of these results enables in principle the continuation of all the solutions simultaneously beyond the degeneracy as long as their number is finite.

On Solvability of Multibody Contact Problems

2007 ◽  
Author(s):  
Alexander P. Ivanov
Keyword(s):  
Contact Problem ◽  
Equations Of Motion ◽  
Sufficient Conditions ◽  
Contact Problems ◽  
Point Of View ◽  
Contact Forces ◽  
Empty Intersection ◽  
Contact Points ◽  
Whole Space ◽  
Multibody Contact

The paper is devoted to dynamic multi-rigid-body contact problems with dry friction. It is known that such problem may have multiple solutions or none solution (so-called Painleve´ paradoxes). A great deal of works concerning overcoming of the paradoxes was published last century, but general conditions of existence and uniqueness were not derived yet. We consider systems with a finite numbers of contact points with well-defined contact directions and Coulomb friction. The equations of motion contain unknowns of two kinds: the accelerations and the contact forces. According to the friction law, some of these variables vanish, and remaining ones can be treated as a coordinate system in the space of the generalized forces. Thus, this space splits to a finite number of regions with different coordinates. From a geometrical point of view, the solvability of the multi-contact problem means that the union of these regions equals to the whole space. Furthemore, the solution is unique ⇔ any pair of regions has empty intersection, and the coordinates within any region are defined uniquely. We present some algebraic conditions, which are equivalent to these geometric properties. Therefore, necessary and sufficient conditions of correct solution to multibody contact problem are obtained for the first time. A number of mechanical examples are considered to illustrate new results.

The post-Newtonian approximation

2018 ◽  
Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan
Keyword(s):  
General Relativity ◽  
Gravitational Field ◽  
Body Problem ◽  
Equations Of Motion ◽  
Two Body Problem ◽  
Newtonian Approximation ◽  
Actual Motion ◽  
Law Of Attraction ◽  
Universal Law ◽  
Two Bodies

This chapter embarks on a study of the two-body problem in general relativity. In other words, it seeks to describe the motion of two compact, self-gravitating bodies which are far-separated and moving slowly. It limits the discussion to corrections proportional to v2 ~ m/R, the so-called post-Newtonian or 1PN corrections to Newton’s universal law of attraction. The chapter first examines the gravitational field, that is, the metric, created by the two bodies. It then derives the equations of motion, and finally the actual motion, that is, the post-Keplerian trajectories, which generalize the post-Keplerian geodesics obtained earlier in the chapter.

scholarly journals X-Ray Determination of the Black-Hole Mass in Cygnus X-1

1998 ◽  
Vol 188 ◽  
pp. 388-389
Author(s):  
A. Kubota ◽  
K. Makishima ◽  
T. Dotani ◽  
H. Inoue ◽  
K. Mitsuda ◽  
...  
Keyword(s):  
Black Hole ◽  
Compact Object ◽  
Optical Observations ◽  
Black Hole Mass ◽  
X Ray ◽  
Upper Limit ◽  
Supergiant Star ◽  
X Ray Binaries ◽  
Two Bodies

About 10 X-ray binaries in our Galaxy and LMC/SMC are considered to contain black hole candidates (BHCs). Among these objects, Cyg X-1 was identified as the first BHC, and it has led BHCs for more than 25 years(Oda 1977, Liang and Nolan 1984). It is a binary system composed of normal blue supergiant star and the X-ray emitting compact object. The orbital kinematics derived from optical observations indicates that the compact object is heavier than ~ 4.8 M⊙ (Herrero 1995), which well exceeds the upper limit mass for a neutron star(Kalogora 1996), where we assume the system consists of only two bodies. This has been the basis for BHC of Cyg X-1.

scholarly journals IV. On the dynamical theory of incompressible viscous fluids and the determination of the criterion

1895 ◽  
Vol 186 ◽  
pp. 123-164 ◽  
Keyword(s):  
Singular Solution ◽  
Physical State ◽  
Theoretical Calculations ◽  
Equations Of Motion ◽  
Dynamical Theory ◽  
Viscous Fluids ◽  
Linear Functions ◽  
Incompressible Viscous Fluids ◽  
Theoretical Results

1. The equations of motion of viscous fluid (obtained by grafting on certain terms to the abstract equations of the Eulerian form so as to adapt these equations to the case of fluids subject to stresses depending in some hypothetical manner on the rates of distortion, which equations Navier seems to have first introduced in 1822, and which were much studied by Cauchy and Poisson) were finally shown by St. Venant and Sir Gabriel Stokes, in 1845, to involve no other assumption than that the stresses, other than that of pressure uniform in all directions, are linear functions of the rates of distortion, with a co-efficient depending on the physical state of the fluid. By obtaining a singular solution of these equations as applied to the case of pendulums in steady periodic motion, Sir G. Stokes was able to compare the theoretical results with the numerous experiments that had been recorded, with the result that the theoretical calculations agreed so closely with the experimental determinations as seemingly to prove the truth of the assumption involved. This was also the result of comparing the flow of water through uniform tubes with the flow calculated from a singular solution of the equations so long as the tubes were small and the velocities slow. On the other hand, these results, both theoretical and practical, were directly at variance with common experience as to the resistance encountered by larger bodies moving with higher velocities through water, or by water moving with greater velocities through larger tubes. This discrepancy Sir G. Stokes considered as probably resulting from eddies which rendered the actual motion other than that to which the singular solution referred and not as disproving the assumption.

Time Integration Incorporated Sensitivity Analysis With Generalized-α Method for Multibody Dynamics Systems

2011 ◽  
Author(s):  
Guang Dong ◽  
Zheng-Dong Ma ◽  
Gregory Hulbert ◽  
Noboru Kikuchi
Keyword(s):  
Sensitivity Analysis ◽  
Topology Optimization ◽  
Dynamic Loading ◽  
Multibody Dynamics ◽  
Equations Of Motion ◽  
Time Integration ◽  
Optimization Method ◽  
Sensitivity Analysis Method ◽  
System Equations ◽  
Consecutive Time

The topology optimization method is extended for the optimization of geometrically nonlinear, time-dependent multibody dynamics systems undergoing nonlinear responses. In particular, this paper focuses on sensitivity analysis methods for topology optimization of general multibody dynamics systems, which include large displacements and rotations and dynamic loading. The generalized-α method is employed to solve the multibody dynamics system equations of motion. The developed time integration incorporated sensitivity analysis method is based on a linear approximation of two consecutive time steps, such that the generalized-α method is only applied once in the time integration of the equations of motion. This approach significantly reduces the computational costs associated with sensitivity analysis. To show the effectiveness of the developed procedures, topology optimization of a ground structure embedded in a planar multibody dynamics system under dynamic loading is presented.

Investigation of Train Dynamics in Passing Through Curves Using a Full Model

Joint Rail ◽  
2004 ◽  
Author(s):  
Mohammad Durali ◽  
Mohammad Mehdi Jalili Bahabadi
Keyword(s):  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Full Model ◽  
Bogie Frame ◽  
Nonlinear Characteristics ◽  
Train Derailment ◽  
Train Dynamics

In this article a train model is developed for studying train derailment in passing through bends. The model is three dimensional, nonlinear, and considers 43 degrees of freedom for each wagon. All nonlinear characteristics of suspension elements as well as flexibilities of wagon body and bogie frame, and the effect of coupler forces are included in the model. The equations of motion for the train are solved numerically for different train conditions. A neural network was constructed as an element in solution loop for determination of wheel-rail contact geometry. Derailment factor was calculated for each case. The results are presented and show the major role of coupler forces on possible train derailment.

scholarly journals Thermal impedance at the interface of contacting bodies: 1-D examples solved by semi-derivatives

2012 ◽  
Vol 16 (2) ◽  
pp. 623-627 ◽  
Author(s):  
Jordan Hristov
Keyword(s):  
Heat Conduction ◽  
Fractional Calculus ◽  
Half Time ◽  
Thermal Impedance ◽  
Entire Domain ◽  
Transient Thermal ◽  
Two Bodies

Simple 1-D semi-infinite heat conduction problems enable to demonstrate the potential of the fractional calculus in determination of transient thermal impedances of two bodies with different initial temperatures contacting at the interface ( x = 0 ) at t = 0 . The approach is purely analytic and uses only semi-derivatives (half-time) and semi-integrals in the Riemann-Liouville sense. The example solved clearly reveals that the fractional calculus is more effective in calculation the thermal resistances than the entire domain solutions.

A Hybrid Highly Parallelizable Algorithm for the Dynamics of Rigid Body Chain Systems

1999 ◽  
Author(s):  
Shanzhong Duan ◽  
Kurt S. Anderson
Keyword(s):  
Rigid Body ◽  
Degrees Of Freedom ◽  
High Efficiency ◽  
Equations Of Motion ◽  
Constraint Forces ◽  
Dynamics Of Rigid Body ◽  
A Chain ◽  
Explicit Determination ◽  
Order Algorithm

Abstract The paper presents a new hybrid parallelizable low order algorithm for modeling the dynamic behavior of multi-rigid-body chain systems. The method is based on cutting certain system interbody joints so that largely independent multibody subchain systems are formed. These subchains interact with one another through associated unknown constraint forces f¯c at the cut joints. The increased parallelism is obtainable through cutting the joints and the explicit determination of associated constraint loads combined with a sequential O(n) procedure. In other words, sequential O(n) procedures are performed to form and solve equations of motion within subchains and parallel strategies are used to form and solve constraint equations between subchains in parallel. The algorithm can easily accommodate the available number of processors while maintaining high efficiency. An O[(n+m)Np+m(1+γ)Np+mγlog2Np](0<γ<1) performance will be achieved with Np processors for a chain system with n degrees of freedom and m constraints due to cutting of interbody joints.

Determination of Instability Boundaries for a Highly Flexible Prismatic-Jointed Robot Performing Repetitive Maneuvers

1991 ◽  
Author(s):  
Keith W. Buffinton
Keyword(s):  
Floquet Theory ◽  
Equations Of Motion ◽  
Perturbation Techniques ◽  
Axial Motion ◽  
The Third ◽  
Straightforward Application ◽  
Method Yield ◽  
Robotic Devices ◽  
Axially Moving

Abstract Presented in this work are the equations of motion governing the behavior of a simple, highly flexible, prismatic-jointed robotic manipulator performing repetitive maneuvers. The robot is modeled as a uniform cantilever beam that is subject to harmonic axial motions over a single bilateral support. To conveniently and accurately predict motions that lead to unstable behavior, three methods are investigated for determining the boundaries of unstable regions in the parameter space defined by the amplitude and frequency of axial motion. The first method is based on a straightforward application of Floquet theory; the second makes use of the results of a perturbation analysis; and the third employs Bolotin’s infinite determinate method. Results indicate that both perturbation techniques and Bolotin’s method yield acceptably accurate results for only very small amplitudes of axial motion and that a direct application of Floquet theory, while computational expensive, is the most reliable way to ensure that all instability boundaries are correctly represented. These results are particularly relevant to the study of prismatic-jointed robotic devices that experience amplitudes of periodic motion that are a significant percentage of the length of the axially moving member.

A comparative study of the methods for calculation of surface elastic deformation

2003 ◽  
Vol 217 (2) ◽  
pp. 145-154 ◽  
Author(s):  
W-Z Wang ◽  
H Wang ◽  
Y-C Liu ◽  
Y-Z Hu ◽  
D Zhu
Keyword(s):  
Surface Deformation ◽  
Contact Problems ◽  
Piecewise Constant Function ◽  
Bilinear Interpolation ◽  
Numerical Accuracy ◽  
Piecewise Constant ◽  
Computation Efficiency ◽  
Influence Coefficients ◽  
Grid Points

A fundamental issue of lubrication analysis is the calculation of surface deformation, which includes two major steps: determination of influence coefficients and multiplication and summation. There are various interpolation schemes, such as the bilinear interpolation, the piecewise constant function or Green's function, available for determining the influence coefficients, while the summation operation may be performed by using one of the following methods: direct summation (DS), multilevel multi-integration (MLMI) or the discrete convolution and fast Fourier transform (DC-FFT) method. To limit the periodical errors, the proper way to implement the DC-FFT method is described in detail. The computation efficiency and numerical accuracy are compared by applying the different methods to typical contact problems. The results show that the three methods can achieve comparable numerical accuracy, but the DC-FFT method shows much higher computation efficiency than the others, especially when a great number of grid points are involved. It is concluded that the DC-FFT method has great potential in applications to the numerical analysis of, for example, surface deformations and temperature rises.

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