Modeling Impacts Between a Continuous System and a Rigid Obstacle Using Coefficient of Restitution

2009 ◽  
Vol 77 (2) ◽  
Author(s):  
Chandrika P. Vyasarayani ◽  
John McPhee ◽  
Stephen Birkett
Keyword(s):  
Initial Conditions ◽  
Equations Of Motion ◽  
Continuous System ◽  
Coefficient Of Restitution ◽  
Continuous Systems ◽  
Rigid Obstacle ◽  
Unit Impulse ◽  
Before And After ◽  
System Equations ◽  
The Impact

In this work, we discuss the limitations of the existing collocation-based coefficient of restitution method for simulating impacts in continuous systems. We propose a new method for modeling the impact dynamics of continuous systems based on the unit impulse response. The developed method allows one to relate modal velocity initial conditions before and after impact without requiring the integration of the system equations of motion during impact. The proposed method has been used to model the impact of a pinned-pinned beam with a rigid obstacle. Numerical simulations are presented to illustrate the inability of the collocation-based coefficient of restitution method to predict an accurate and energy-consistent response. We also compare the results obtained by unit impulse-based coefficient of restitution method with a penalty approach.

Differential Formulation for the Coefficient of Restitution of a Rigid Link

2015 ◽  
Vol 762 ◽  
pp. 175-182 ◽  
Author(s):  
Dorian Cojocaru ◽  
Dan B. Marghitu
Keyword(s):  
Mobile Robots ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
Coefficient Of Restitution ◽  
Numerical Techniques ◽  
Differential Impact ◽  
Differential Formulation ◽  
Nonlinear Contact ◽  
Nonlinear Equations Of Motion ◽  
The Impact

The differential impact equations of motion are developed using an nonlinear contact force. The nonlinear equations of motion are written using symbolical MATLAB and are solved using numerical techniques. The impact equations are based on the Kogut-Etsion model. The numerical results are obtained for different geometries of the link, different coefficients of friction, and different initial conditions. The coefficient of restitution (COR) is discussed for specific cases. The results can be used for the impact of mobile robots with different type of surfaces.

scholarly journals Dynamics of Space Particles and Spacecrafts Passing by the Atmosphere of the Earth

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Vivian Martins Gomes ◽  
Antonio Fernando Bertachini de Almeida Prado ◽  
Justyna Golebiewska
Keyword(s):  
Initial Conditions ◽  
Equations Of Motion ◽  
Center Of Mass ◽  
The Sun ◽  
Three Body Problem ◽  
The Earth ◽  
Before And After ◽  
Three Body ◽  
Body Energy ◽  
Spacecraft Position

The present research studies the motion of a particle or a spacecraft that comes from an orbit around the Sun, which can be elliptic or hyperbolic, and that makes a passage close enough to the Earth such that it crosses its atmosphere. The idea is to measure the Sun-particle two-body energy before and after this passage in order to verify its variation as a function of the periapsis distance, angle of approach, and velocity at the periapsis of the particle. The full system is formed by the Sun, the Earth, and the particle or the spacecraft. The Sun and the Earth are in circular orbits around their center of mass and the motion is planar for all the bodies involved. The equations of motion consider the restricted circular planar three-body problem with the addition of the atmospheric drag. The initial conditions of the particle or spacecraft (position and velocity) are given at the periapsis of its trajectory around the Earth.

Analytical Study of the Oblique Impact of a Rod with a Flat Using an Elasto-Plastic Contact Model

2015 ◽  
Vol 801 ◽  
pp. 25-32
Author(s):  
Ozdes Cermik ◽  
Hamid Ghaednia ◽  
Dan B. Marghitu
Keyword(s):  
Impact Velocity ◽  
Contact Force ◽  
Analytical Study ◽  
Initial Conditions ◽  
Permanent Deformation ◽  
Oblique Impact ◽  
Coefficient Of Restitution ◽  
Contact Model ◽  
Impact Angles ◽  
The Impact

In the current study a flattening contact model, combined with a permanent deformation expression, has been analyzed for the oblique impact case. The model has been simulated for different initial conditions using MATLAB. The initial impact velocity used for the simulations ranges from 0.5 to 3 m/s. The results are compared theoretically for four different impact angles including 20, 45, 70, and 90 degrees. The contact force, the linear and the angular motion, the permanent deformation, and the coefficient of restitution have been analyzed. It is assumed that sliding occurs throughout the impact.

Identification and Defect Detection of Continuous Dynamic Systems

2006 ◽  
Author(s):  
A. Siami ◽  
M. Farid
Keyword(s):  
Finite Difference ◽  
Differential Equations ◽  
Defect Detection ◽  
Equations Of Motion ◽  
Continuous System ◽  
Least Square Method ◽  
Least Square ◽  
Continuous Systems ◽  
Algebraic Equations ◽  
Continuous Dynamic

This paper presents a systematic and efficient algorithm using a coupled finite element - finite difference - least square method for identification and defect detection of continuous system using dynamic response of such systems. First the governing partial differential equations of motion of continuous systems such as beams are reduced to a set of ordinary differential equations in time domain using finite elements. Then finite difference method is used to convert these equations into a set of algebraic equations. This set of equations is considered as a set of equality constraints of an optimization problem in which the objective function is the summation of the squares of differences between measured data at specific points and the predicted data obtained by the solution of the governing system of differential of equations. This method has been successfully applied to find mechanical properties of aforementioned systems in an iterative procedure.

scholarly journals Energy dissipation of a particle colliding on a flat surface

2021 ◽  
Vol 249 ◽  
pp. 06007
Author(s):  
Fabricio Éric Fernández ◽  
Marcelo Fabián Piva ◽  
Román Gustavo Martino ◽  
María Alejandra Aguirre
Keyword(s):  
Kinetic Energy ◽  
Degrees Of Freedom ◽  
Flat Surface ◽  
Coefficient Of Restitution ◽  
Normal Direction ◽  
Normal Velocity ◽  
Factors Affecting ◽  
Different Types ◽  
Before And After ◽  
The Impact

To gain an understanding of the factors affecting the interaction of one grain with its environment as it reaches equilibrium, we study a particle bouncing off a flat surface. The bouncing of the particle leads to dissipation that is usually characterized with t, the coefficient of restitution, defined as the ratio between the velocity component that is normal to the contact surface just before impact (Vn) and the same component, but immediately after the collision (Vn’), i.e. related to a kinetic energy corresponding to motion in the normal direction. We will show how d is affected by energy stored in other degrees of freedom and transferred to kinetic energy that leads to an increase in normal velocity after the impact Vn’, and therefore to, ɛ >1. For this purpose, the evolution of potential, translational kinetic energy and rotational kinetic energy is analysed during the whole relaxation process and just before and after each collision for two different types of particle, a disk and a faceted particle.

scholarly journals Experiments on rebounding slow impacts under asteroid conditions

2021 ◽  
Author(s):  
Kolja Joeris ◽  
Laurent Schönau ◽  
Matthias Keulen ◽  
Philip Born ◽  
Jonathan E. Kollmer
Keyword(s):  
Size Distribution ◽  
High Speed ◽  
Coefficient Of Restitution ◽  
Size Segregation ◽  
Low Gravity ◽  
Before And After ◽  
Conservation Of Momentum ◽  
The Impact ◽  
Kinetics Of ◽  
Rubble Pile

<p class="p1"><span class="s1">The<span class="Apple-converted-space">  </span>surfaces<span class="Apple-converted-space">  </span>of<span class="Apple-converted-space">  </span>rubble-pile<span class="Apple-converted-space">  </span>asteroids<span class="Apple-converted-space">  </span>are<span class="Apple-converted-space">  </span>covered<span class="Apple-converted-space">  </span>in<span class="Apple-converted-space">  </span>regolith of a variety of sizes.<span class="Apple-converted-space">  </span>In some cases like for the asteroid Itokawa, the size distribution of regolith is not uniform across the surface [1]. Some areas are dominated by finer grains, while other areas are covered by larger rocks.<span class="Apple-converted-space">  </span>There are a number of competing explanations for this observed size segregation [2–4]. One approach is the so called ballistic-sorting-effect [2], where impacting particles sort themselves through different rebound behavior.</span></p> <p class="p1"><span class="s1">In our work we want to set practical limits on the role ballistic sorting can play in shaping an asteroids surface. To this end we conduct a series of drop tower experiments examining the impact kinetics of slow (cm/s)<span class="Apple-converted-space">  </span>3 mm sized projectiles into a regolith surface under conditions realistic for asteroid surfaces, i.e. vacuum and low gravity. We track the impactor with high-speed cameras and determine its velocity in 3 dimensions before and after the impact. From these velocities, we can then compute a coefficient of restitution (COR).<span class="Apple-converted-space">  </span>We then repeat the experiment for surfaces composed of differently sized material.<span class="Apple-converted-space">  </span>We find that for a regolith bed made from particles of similar size as the impactor we get a lower COR (0,1) than for beds made up of significantly larger (0,5) or smaller particles (0,8). The more elastic collisions for larger sized targets follows from conservation of momentum. For the finer material we suggest that the higher COR is a function of interparticle adhesion.</span></p> <p class="p1"><span class="s1">[1] A. Fujiwara, J. Kawaguchi, D.K. Yeomans, M. Abe, T. Mukai, T. Okada, J. Saito, H. Yano, M. Yoshikawa, D.J. Scheeres et al., Science 312, 1330 (2006)</span></p> <p class="p1"><span class="s1">[2] T. Shinbrot, T. Sabuwala, T. Siu, M.V. Lazo, P. Chakraborty, Phys. Rev. Lett. 118, 111101 (2017)</span></p> <p class="p1"><span class="s1">[3] S. Matsumura, D.C. Richardson, P. Michel, S.R. Schwartz, R.L. Ballouz, Mon. Not. the R. Astron. Soc. 443, 3368 (2014)</span></p> <p class="p1"><span class="s1">[4] A.J. Dombard, O.S. Barnouin, L.M. Prockter, P.C. Thomas, Icarus 210, 713 (2010)</span></p>

Spatial Impact of a Slender Beam

2002 ◽  
Author(s):  
Horatiu Barbulescu ◽  
Dan B. Marghitu ◽  
Uday Vaidya
Keyword(s):  
Composite Materials ◽  
Incident Angle ◽  
Contact Point ◽  
Equations Of Motion ◽  
Coefficient Of Restitution ◽  
Tangential Component ◽  
Sliding Direction ◽  
Generalized Momentum ◽  
Separation Force ◽  
The Impact

In this paper, the dynamics of the spatial impact of a slender beam is analyzed. The equations of motion are calculated using Kane’s impact method. The generalized momentum and generalized impulse of the beam are considered to find the equations of motion of the beam. The frictional phenomenon at the contact point is analyzed. For the case of impact without slipping, it is used the assumption that the tangential component of the velocity of separation is null. In the case with slipping, the tangential impulse (at the plane of impact) is computed. The sliding direction after impact is calculated. A simulation of the impact of beam with a surface is developed and the velocity of separation, force of impact and kinetic energy of the beam after impact are studied for different incident angles of the beam. The incident angle is varied from 0° to 57°. The results are function of the incident angle of impact. The results can be used to calculate the coefficient of restitution and friction for composite materials.

A Methodology to Detect the Precise Instant of Contact in Multibody Dynamics

2011 ◽  
Author(s):  
Paulo Flores
Keyword(s):  
Multibody Dynamics ◽  
Equations Of Motion ◽  
Contact Forces ◽  
Time Step ◽  
Dynamic Simulations ◽  
Integration Algorithm ◽  
Current Time ◽  
Impact Time ◽  
System Equations ◽  
The Impact

The main purpose of this work is to present a general and comprehensive approach to automatically adjust the time step for the contact and non contact periods in multibody dynamics. The basic idea of the described methodology is to ensure that the first impact within a multibody system does not occur with a large value for relative bodies’ penetration in order to avoid the artificially large contact forces associated. The detection of the instant of contact takes place when the distance between two bodies change the sign between two discrete moments in time. In fact, in theory, the contact starts when this distance is zero, or a very small value to prevent the round-off errors. Thus, during the numerical solution of the system equations of motion if the first penetration is below this small value previously specified, then the current time is taken as the impact time. On the other hand, if the first penetration is larger than the specified tolerance, then the current time step is beyond the impact time. In this case, integration algorithm is forced to go back and take a smaller time step until a step can be taken within the acceptable tolerance. The main features of this approach are the easiness to implement and the good computational efficiency. In addition, it can easily deal with the transitions between non contact and contact cases in multibody dynamics. Finally, results obtained from dynamic simulations are presented and discussed to study the validity of the methodology proposed in this work.

scholarly journals Statistical Analysis of Table-Tennis Ball Trajectories

2018 ◽  
Vol 8 (12) ◽  
pp. 2595
Author(s):  
Ralf Schneider ◽  
Lars Lewerentz ◽  
Karl Lüskow ◽  
Marc Marschall ◽  
Stefan Kemnitz
Keyword(s):  
Statistical Analysis ◽  
Graphics Processing Units ◽  
Initial Conditions ◽  
Statistical Significance ◽  
Equations Of Motion ◽  
Distribution Functions ◽  
Tennis Ball ◽  
Table Tennis ◽  
Ball Size ◽  
The Impact

In this work, the equations of motion for table-tennis balls were numerically solved on graphics processing units (GPUs) using Compute Unified Device Architecture (CUDA) for systematical statistical studies of the impact of ball size and weight, as well as of net height, on the distribution functions of successful strokes. Half a billion different initial conditions involving hitting location, initial spin, and velocities were analyzed to reach sufficient statistical significance for the different cases. In this paper, an advanced statistical analysis of the database generated by the simulation is presented.

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.

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