scholarly journals On a paradox in the impact dynamics of smooth rigid bodies

2018 ◽  
Vol 24 (3) ◽  
pp. 573-597 ◽  
Author(s):  
Peter Palffy-Muhoray ◽  
Epifanio G Virga ◽  
Mark Wilkinson ◽  
Xiaoyu Zheng
Keyword(s):  
Conservation Laws ◽  
Contact Point ◽  
Classical Mechanics ◽  
Rigid Bodies ◽  
Rigid Body Dynamics ◽  
Impact Dynamics ◽  
Contact Points ◽  
Impact Impulse ◽  
2D And 3D ◽  
The Impact

Paradoxes in the impact dynamics of rigid bodies are known to arise in the presence of friction. We show here that, on specific occasions, in the absence of friction, the conservation laws of classical mechanics are also incompatible with the collisions of smooth, strictly convex rigid bodies. Under the assumption that the impact impulse is along the normal direction to the surface at the contact point, two convex rigid bodies that are well separated can come into contact, and then interpenetrate each other. This paradox can be demonstrated in both 2D and 3D when the collisions are tangential, in which case no momentum or energy transfer between the two bodies is possible. The postcollisional interpenetration can be realized through the contact points or through neighboring points only. The penetration distance is shown to be [Formula: see text]. The conclusion is that rigid-body dynamics is not compatible with the conservation laws of classical mechanics.

scholarly journals Slipping–rolling transitions of a body with two contact points

2021 ◽  
Author(s):  
Mate Antali ◽  
Gabor Stepan
Keyword(s):  
Rigid Body ◽  
Contact Point ◽  
Static Friction ◽  
Rigid Body Dynamics ◽  
Contact Forces ◽  
Friction Forces ◽  
Contact Points ◽  
Body Dynamics ◽  
Coulomb Model ◽  
Rigid Surfaces

AbstractIn this paper, the general kinematics and dynamics of a rigid body is analysed, which is in contact with two rigid surfaces in the presence of dry friction. Due to the rolling or slipping state at each contact point, four kinematic scenarios occur. In the two-point rolling case, the contact forces are undetermined; consequently, the condition of the static friction forces cannot be checked from the Coulomb model to decide whether two-point rolling is possible. However, this issue can be resolved within the scope of rigid body dynamics by analysing the nonsmooth vector field of the system at the possible transitions between slipping and rolling. Based on the concept of limit directions of codimension-2 discontinuities, a method is presented to determine the conditions when the two-point rolling is realizable without slipping.

Unraveling Paradoxical Theories for Rigid Body Collisions

1991 ◽  
Vol 58 (4) ◽  
pp. 1049-1055 ◽  
Author(s):  
W. J. Stronge
Keyword(s):  
Frictional Force ◽  
Contact Point ◽  
Tangential Velocity ◽  
Velocity Slip ◽  
Rigid Bodies ◽  
Contact Points ◽  
Before And After ◽  
Limited Applicability ◽  
Work Done ◽  
Impulsive Reaction

A collision between two rigid bodies has a normal impulsive reaction at the contact point (CP). If the bodies are slightly rough and the contact points have a relative tangential velocity (slip), there is also a frictional force that opposes slip. Small initial slip can halt before contact terminates; when slip halts the frictional force changes and the collision process is separated into periods before and after halting. An energetically consistent theory for collisions with slip that halts is based on the work done by normal (nonfrictional) forces during restitution and compression phases. This theory clearly separates dissipation due to frictional forces from that due to internal irreversible deformation. With this theory, both normal and tangential components of the impulsive reaction always dissipate energy during collisions. In contrast, Newton’s impact law results in calculations of paradoxical increases in energy for collisions where slip reverses. This law relates normal components of relative velocity for the CP at separation and incidence by a constant (the coefficient of restitution e). Newton’s impact law is a kinematic definition for e that generally depends on the slip process and friction; consequently it has limited applicability.

scholarly journals Planar dynamics of a rigid body system with frictional impacts. II. Qualitative analysis and numerical simulations

2009 ◽  
Vol 465 (2107) ◽  
pp. 2267-2292 ◽  
Author(s):  
Zhen Zhao ◽  
Caishan Liu ◽  
Bernard Brogliato
Keyword(s):  
Qualitative Analysis ◽  
Rigid Body ◽  
Numerical Simulations ◽  
Contact Point ◽  
Impulsive Differential Equations ◽  
Companion Paper ◽  
Rigid Body Dynamics ◽  
Drift Motion ◽  
Contact Points ◽  
Static Coefficient Of Friction

The objective of this paper is to implement and test the theory presented in a companion paper for the non-smooth dynamics exhibited in a bouncing dimer. Our approach revolves around the use of rigid body dynamics theory combined with constraint equations from the Coulomb's frictional law and the complementarity condition to identify the contact status of each contacting point. A set of impulsive differential equations based on Darboux–Keller shock dynamics is established that can deal with the complex behaviours involved in multiple collisions, such as the frictional effects, the local dissipation of energy at each contact point, and the dispersion of energy among various contact points. The paper will revisit the experimental phenomena found in Dorbolo et al . ( Dorbolo et al . 2005 Phys. Rev. Lett. 95 , 044101), and then present a qualitative analysis based on the theory proposed in part I. The value of the static coefficient of friction between the plate and the dimer is successfully estimated, and found to be responsible for the formation of the drift motion of the bouncing dimer. Plenty of numerical simulations are carried out, and precise agreements are obtained by the comparisons with the experimental results.

Rigid body collisions with friction

1990 ◽  
Vol 431 (1881) ◽  
pp. 169-181 ◽  
Keyword(s):  
Energy Dissipation ◽  
Rigid Body ◽  
Rigid Bodies ◽  
Consistent Theory ◽  
Contact Points ◽  
Collision Processes ◽  
Inconsistent Theory ◽  
The Impact ◽  
Impact Hypothesis ◽  
Alternative Theories

An energetically consistent theory is presented for dynamics of partly elastic collisions between somewhat rough rigid bodies with friction that opposes slip. This theory is based on separately accounting for frictional and non-frictional sources of dissipation. Alternative theories derived from Newton’s impact law or Poisson’s impact hypothesis are shown to be valid only for central (collinear) or non-frictional collisions; generally the latter theories yield erroneous energy dissipation if small initial slip stops during collision between eccentric bodies. Collision processes are complex when small slip is stopped by friction; then either the direction of slip reverses or contact points roll without slip. An inconsistent theory based on Newton’s impact law can yield erroneous energy increases when slip stops during collision; the consistent theory always dissipates energy. The impact law that specifies a simple proportionality between normal components of contact velocity for incidence and rebound is not applicable in any range of incident velocities with small slip if the collision is non-collinear with friction. In Percussion the force or Impetus whereby one body is moved may cause another body against which it strikes to be put in motion, and withal lose some of its strength or swiftness. (J. Wallis, 1668)

Using an Anti-Relaxation Step to Improve the Accuracy of the Frictional Contact Solution in a DVI Framework for Rigid Body Dynamics

2016 ◽  
Author(s):  
Daniel Melanz ◽  
Hammad Mazhar ◽  
Dan Negrut
Keyword(s):  
Frictional Contact ◽  
Rigid Bodies ◽  
Rigid Body Dynamics ◽  
Contact Forces ◽  
Science And Engineering ◽  
Complementarity Condition ◽  
Body Dynamics ◽  
Engineering Problems ◽  
The Impact

Systems composed of rigid bodies interacting through frictional contact are manifest in several science and engineering problems. The number of contacts can be small, such as in robotics and geared machinery, or large, such as in terrame-chanics applications, additive manufacturing, farming, food industry, and pharmaceutical industry. Currently, there are two popular approaches for handling the frictional contact problem in dynamic systems. The penalty method calculates the frictional contact force based on the kinematics of the interaction, some representative parameters, and an empirical force law. Alternatively, the complementarity method, based on a differential variational inequality (DVI), enforces non-penetration of rigid bodies via a complementarity condition. This contribution concentrates on the latter approach and investigates the impact of an anti-relaxation step that improves the accuracy of the frictional contact solution. We show that the proposed anti-relaxation step incurs a relatively modest cost to improve the quality of a numerical solution strategy which poses the calculation of the frictional contact forces as a cone-complementarity problem.

scholarly journals A kinematics-based model for the settling of gravity-driven arbitrary-shaped particles on a surface

PLoS ONE ◽  
2021 ◽  
Vol 16 (2) ◽  
pp. e0243716
Author(s):  
Mohsen Daghooghi ◽  
Iman Borazjani
Keyword(s):  
Equilibrium State ◽  
Contact Point ◽  
Center Of Mass ◽  
Stable State ◽  
Rigid Body Dynamics ◽  
Contact Points ◽  
Shaped Particle ◽  
Instantaneous Center ◽  
Point Line ◽  
Discrete Element Methods

A discrete model is proposed for settling of an arbitrary-shaped particle onto a flat surface under the gravitational field. In this method, the particle dynamics is calculated such that (a) the particle does not create an overlap with the wall and (b) reaches a realistic equilibrium state, which are not guaranteed in the conventional discrete element methods that add a repulsive force (torque) based on the amount of overlap between the particle and the wall. Instead, upon the detection of collision, the particle’s kinematics is modified depending on the type of contact, i.e., point, line, and surface types, by assuming the contact point/line as the instantaneous center/line of rotation for calculating the rigid body dynamics. Two different stability conditions are implemented by comparing the location of the projection of the center of mass on the wall along gravity direction against the contact points to identify the equilibrium (stable) state on the wall for particles with multiple contact points. A variety of simulations are presented, including smooth surface particles (ellipsoids), regular particles with sharp edges (cylinders and pyramids) and irregular-shaped particles, to show that the method can provide the analytically-known equilibrium state.

Defining the Coefficient of Restitution for the Impact Between Free and Constrained Bodies

1999 ◽  
Author(s):  
J. L. Escalona ◽  
J. M. Mayo ◽  
J. Domínguez
Keyword(s):  
Mechanical Energy ◽  
Coefficient Of Restitution ◽  
Rigid Bodies ◽  
Momentum Balance ◽  
Balance Equations ◽  
Flexible Bodies ◽  
Local Loss ◽  
Contact Points ◽  
Before And After ◽  
The Impact

Abstract This paper revisits the coefficient of restitution involved in the impulse-momentum balance equations for colliding rigid bodies and examines its extension to impacts between flexible bodies. The analytical solution to axial impact on a flexible rod is used to demonstrate that the coefficient of restitution is not inherent in the underlying physical process. In fact, the type of coefficient to be used in each case depends on the particular model employed by the analyst to describe flexibility in the bodies concerned. It is demonstrated that the coefficient of restitution used in the generalized impulse-momentum balance for flexible bodies does not represent a physical magnitude. In any case, as shown in this paper, the ratio between the relative velocities at the contact points or surfaces of the flexible bodies before and after impact is no measure of the local loss of mechanical energy during the process.

scholarly journals Modeling and Control of 2D Grasping under Rolling Contact Constraints between Arbitrary Shapes: A Riemannian-Geometry Approach

2010 ◽  
Vol 2010 ◽  
pp. 1-13 ◽  
Author(s):  
Suguru Arimoto ◽  
Morio Yoshida
Keyword(s):  
Lagrange Equation ◽  
Contact Point ◽  
Equation Of Motion ◽  
Second Fundamental Form ◽  
Rigid Bodies ◽  
Rolling Contact ◽  
Modeling And Control ◽  
Contact Points ◽  
Dimensional Object ◽  
And Control

Modeling, control, and stabilization of dynamics of two-dimensional object grasping by using a pair of multijoint robot fingers are investigated under rolling contact constraints and arbitrariness of the geometry of the object and fingertips. First, modeling of rolling motion between 2D rigid bodies with arbitrary shape is treated under the assumption that the two contour curves coincide at the contact point and share the same tangent. The rolling constraints induce the Euler equation of motion that is parameterized by a pair of arclength parameters and constrained onto the kernel space as an orthogonal complement to the image space spanned from all the constraint gradients. Furthermore, it is shown that all the Pfaffian forms of the rolling constraints are integrable in the sense of Frobenius and therefore the rolling contacts are regarded as a holonomic constraint. The Euler-Lagrange equation of motion of the overall fingers/object system is rederived together with a couple of first-order differential equations that express evolution of contact points in terms of quantities of the second fundamental form. A control signal called “blind grasping” is defined and shown to be effective in maintenance or stabilization of grasping without using the details of object shape and parameters or external sensing. An extension of the Dirichlet-Lagrange stability theorem to a system of DOF-redundancy under constraints is discussed by introducing a Morse-Bott function and deriving its Hessian, in a special case that the object to be grasped is a parallelepiped.

scholarly journals A Simple Treatment of Constraint Forces and Constraint Moments in the Dynamics of Rigid Bodies

2014 ◽  
Vol 67 (1) ◽  
Author(s):  
Oliver M. O'Reilly ◽  
Arun R. Srinivasa
Keyword(s):  
Rigid Body ◽  
Classical Mechanics ◽  
Rigid Bodies ◽  
Rigid Body Dynamics ◽  
Constraint Forces ◽  
Simple Treatment ◽  
Body Dynamics ◽  
Expository Article

In this expository article, a simple concise treatment of Lagrange's prescription for constraint forces and constraint moments in the dynamics of rigid bodies is presented. The treatment is suited to both Newton–Euler and Lagrangian treatments of rigid body dynamics and is illuminated with a range of examples from classical mechanics and orthopedic biomechanics.

A Contact Force Solution for Non-Colliding Contact Dynamics Simulation

2006 ◽  
Author(s):  
Inna Sharf ◽  
Yuning Zhang
Keyword(s):  
Rigid Body ◽  
Contact Force ◽  
Contact Point ◽  
Closed Form Solution ◽  
Rigid Body Dynamics ◽  
Contact Forces ◽  
Contact Dynamics ◽  
Dynamics Simulation ◽  
Friction Forces ◽  
Contact Points

Rigid-body impact modeling remains an intensive area of research spurred on by new applications in robotics, biomechanics, and more generally multibody systems. By contrast, the modeling of non-colliding contact dynamics has attracted significantly less attention. The existing approaches to solve non-colliding contact problems include compliant approaches in which the contact force between objects is defined explicitly as a function of local deformation, and complementarity formulations in which unilateral constraints are employed to compute contact interactions (impulses or forces) to enforce the impenetrability of the contacting objects. In this article, the authors develop a novel approach to solve the non-colliding contact problem for objects of arbitrary geometry in contact at multiple points. Similarly to the complementarity formulation, the solution is based on rigid-body dynamics and enforces contact kinematics constraints at the acceleration level. Differently, it leads to an explicit closed-form solution for the normal forces at the contact points. Integral to the proposed formulation is the treatment of tangential contact forces, in particular the static friction. These friction forces must be calculated as a function of microslip velocity or displacement at the contact point. Numerical results are presented for three test cases: 1) a thin rod sliding down a stationary wedge; 2) a cube rotating off the stationary wedge under application of an external moment and 3) the cube and the wedge both moving under application of a moment. To ascertain validity and correctness, the solutions to frictionless and frictional scenarios obtained with the proposed formulation are compared to those generated by using a commercial simulation tool MSC ADAMS.

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