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.

GEOMETRY OF CONTACT BETWEEN SELF-AFFINE SURFACES

Fractals ◽  
1995 ◽  
Vol 03 (01) ◽  
pp. 113-122 ◽  
Author(s):  
FRANCK PLOURABOUÉ ◽  
STÉPHANE ROUX ◽  
JEAN SCHMITTBUHL ◽  
JEAN-PIERRE VILOTTE
Keyword(s):  
Probability Distribution ◽  
Contact Point ◽  
Relative Shift ◽  
Lévy Walk ◽  
Contact Points ◽  
Roughness Exponent ◽  
Mean Shape ◽  
Before And After ◽  
Levy Walk ◽  
The Mean

We study the geometry of the contact between two rigid self-affine surfaces. We investigate the mean shape of the surface in the vicinity of the contact point as well as the probability distribution of apertures a as a function of the distance to the contact point. The latter reveals two distinct scaling regimes which are characterized. We show that as the two surfaces are shifted with respect to each other, the contact point on one surface undergoes a “Levy walk”. If u is the relative shift of the surfaces, the distance d between the two contact points (before and after the shift), scales as d ∝ uα where the exponent α is shown to be equal to the roughness exponent of the surfaces.

scholarly journals Experimental Investigation of the Painlevé Paradox in a Robotic System

2008 ◽  
Vol 75 (4) ◽  
Author(s):  
Zhen Zhao ◽  
Caishan Liu ◽  
Wei Ma ◽  
Bin Chen
Keyword(s):  
Contact Point ◽  
Tangential Velocity ◽  
Rigid Bodies ◽  
Robotic System ◽  
Experimental Results ◽  
Painlevé Paradox ◽  
Dynamical Behaviors ◽  
Tangential Impact ◽  
System Of Rigid Bodies ◽  
Approach Zero

This paper aims at experimentally investigating the dynamical behaviors when a system of rigid bodies undergoes so-called paradoxical situations. An experimental setup corresponding to the analytical model presented in our prior work Liu et al. [2007, “The Bouncing Motion Appearing in a Robotic System With Unilateral Constraint,” Nonlinear Dyn., 49(1–2), 217–232] is developed, in which a two-link robotic system comes into contact with a moving rail. The experimental results show that a tangential impact exists at the contact point and takes a peculiar property that well coincides with the maximum dissipation principle stated in the work of Moreau [1988, “Unilateral Contact and Dry Friction in Finite Freedom Dynamics,” Nonsmooth Mechanics and Applications, Springer-Verlag, Vienna, pp. 1–82] the relative tangential velocity of the contact point must immediately approach zero once a Painlevé paradox occurs. After the tangential impact, a bouncing motion may be excited and is influenced by the speed of the moving rail. We adopt the tangential impact rule presented by Liu et al. to determine the postimpact velocities of the system, and use an event-driven algorithm to perform numerical simulations. The qualitative comparisons between the numerical and experimental results are carried out and show good agreements. This study not only presents an experimental support for the shock assumption related to the problem of the Painlevé paradox, but can also find its applications in better understanding the instability phenomena appearing in robotic systems.

Neighboring Degree-of-Approximation Equating Method for Computing Contact Points between Robot and Convex Object

1993 ◽  
Vol 5 (1) ◽  
pp. 19-25
Author(s):  
Makoto Kaneko ◽  
◽  
Kazuo Tanie ◽  
Keyword(s):  
Contact Point ◽  
Degree Of Approximation ◽  
Mathematical Framework ◽  
Object Surface ◽  
Convex Object ◽  
Contact Points ◽  
Nearest Point ◽  
Before And After ◽  
Compliant Joint ◽  
Computation Scheme

With a proper combination of compliant joint and position- controlled joint, a link system changes its posture with keeping contact between link system and environment. This is so-called Self-Posture Changing Motion (SPCM) and conveniently used to detect a contact point between robot and unknown object. In conventional approach, a contact point has been computed as an intersecting point between two different link postures before and after SPCM. In general, this computation scheme gives us a contact point with unsatisfactory approximation, while it furnishes us with an exact contact point for a sharp-edged object. In this paper, we first introduces a concept of degree-of-approximation, which is defined as the distance between a computed point and the nearest point over the object surface, and then propers the Neighboring Degree-Of Approximation Equating Method which improves the degree-of-approximation drastically and provides exact contact points not only for a sharp· edged object but also for a constant curved object. Mathematical framework is provided for the proposed computation scheme. Finally, the effectiveness of the proposed scheme is verified by simulations.

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.

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.

Peristaltic flow of a Jeffery fluid over a porous conduit in the presence of variable liquid properties and convective boundary conditions

2019 ◽  
Vol 6 (2) ◽  
Author(s):  
G. Manjunatha ◽  
C. Rajashekhar ◽  
K. V. Prasad ◽  
Hanumesh Vaidya ◽  
Saraswati
Keyword(s):  
Boundary Conditions ◽  
Frictional Force ◽  
Variable Viscosity ◽  
Pressure Rise ◽  
Velocity Slip ◽  
Peristaltic Flow ◽  
Physical Parameters ◽  
Convective Boundary Conditions ◽  
Convective Boundary ◽  
Closed Form Solutions

The present article addresses the peristaltic flow of a Jeffery fluid over an inclined axisymmetric porous tube with varying viscosity and thermal conductivity. Velocity slip and convective boundary conditions are considered. Resulting governing equations are solved using long wavelength and small Reynolds number approximations. The closed-form solutions are obtained for velocity, streamline, pressure gradient, temperature, pressure rise, and frictional force. The MATLAB numerical simulations are utilized to compute pressure rise and frictional force. The impacts of various physical parameters in the interims for time-averaged flow rate with pressure rise and is examined. The consequences of sinusoidal, multi-sinusoidal, triangular, trapezoidal, and square waveforms on physiological parameters are analyzed and discussed through graphs. The analysis reveals that the presence of variable viscosity helps in controlling the pumping performance of the fluid.

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.

scholarly journals Cavitation between cylinder-liner and piston-ring in a new designed optical IC engine

2021 ◽  
pp. 146808742110080
Author(s):  
Jamshid Malekmohammadi Nouri ◽  
Ioannis Vasilakos ◽  
Youyou Yan
Keyword(s):  
High Speed ◽  
Contact Point ◽  
Cylinder Liner ◽  
Optical Methods ◽  
Engine Management System ◽  
Great Success ◽  
Ic Engine ◽  
Contact Points ◽  
Lubricant Flow ◽  
Cavitation Intensity

A new engine block with optical access has been designed and manufactured capable of running up to 3000 r/min with the same specification as the unmodified engine. The optical window allowed access to the full length of the liner over a width of 25 mm to investigate the lubricant flow and cavitation at contact point between the rings and cylinder-liner. In addition, it allowed good access into the combustion chamber to allow charged flow, spray and combustion visualisation and measurements using different optical methods. New custom engine management system with build in LabView allowed for the precise full control of the engine. The design of the new optical engine was a great success in producing high quality images of lubricant flow, cavitation formation and development at contact point at different engine speeds ranging from 208 to 3000 r/min and lubricant temperatures (30°C–70°C) using a high-speed camera. The results under motorised operation confirmed that there was no cavitation at contact points during the intake/exhaust strokes due to low in-cylinder presure, while during compression/expansion strokes, with high in-cylinder pressure, considerable cavities were observed, in particular, during the compression stroke. Lubricant temperatures had the effect of promoting cavities both in their intensity and covered ring area up to 50°C as expected. Beyond that, although the cavitation intensity increases further with temperature, its area reduces due to possible collapse of the cavitating bubbles at higher temperature. The change of engine speed from 208 to 800 r/min increased cavitating area considerably by 52% of the ring area and was further increased by 19% at 1000 r/min. After that, the results showed very small increase in cavitation area (1.3% at 2000 r/min) with similar intensity and distribution across the ring.

scholarly journals The macroscopic shape of assemblies formed from microparticles based on host–guest interaction dependent on the guest content

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Takahiro Itami ◽  
Akihito Hashidzume ◽  
Yuri Kamon ◽  
Hiroyasu Yamaguchi ◽  
Akira Harada
Keyword(s):  
Molecular Recognition ◽  
Smart Materials ◽  
Self Assembly ◽  
Early Stage ◽  
Contact Point ◽  
Sodium Acrylate ◽  
Contact Points ◽  
Assembly Behavior ◽  
Small Clusters ◽  
Macroscopic Shape

AbstractBiological macroscopic assemblies have inspired researchers to utilize molecular recognition to develop smart materials in these decades. Recently, macroscopic self-assemblies based on molecular recognition have been realized using millimeter-scale hydrogel pieces possessing molecular recognition moieties. During the study on macroscopic self-assembly based on molecular recognition, we noticed that the shape of assemblies might be dependent on the host–guest pair. In this study, we were thus motivated to study the macroscopic shape of assemblies formed through host–guest interaction. We modified crosslinked poly(sodium acrylate) microparticles, i.e., superabsorbent polymer (SAP) microparticles, with β-cyclodextrin (βCD) and adamantyl (Ad) residues (βCD(x)-SAP and Ad(y)-SAP microparticles, respectively, where x and y denote the mol% contents of βCD and Ad residues). Then, we studied the self-assembly behavior of βCD(x)-SAP and Ad(y)-SAP microparticles through the complexation of βCD with Ad residues. There was a threshold of the βCD content in βCD(x)-SAP microparticles for assembly formation between x = 22.3 and 26.7. On the other hand, the shape of assemblies was dependent on the Ad content, y; More elongated assemblies were formed at a higher y. This may be because, at a higher y, small clusters formed in an early stage can stick together even upon collisions at a single contact point to form elongated aggregates, whereas, at a smaller y, small clusters stick together only upon collisions at multiple contact points to give rather circular assemblies. On the basis of these observations, the shape of assembly formed from microparticles can be controlled by varying y.

Sign in / Sign up

Export Citation Format

Share Document