DYNAMIC FORMULATION OF COEFFICIENT OF RESTITUTION

1964 ◽  
Author(s):  
E. H. Jakubowski
Keyword(s):  
Coefficient Of Restitution ◽  
Dynamic Formulation

Use of the Generalized Impulse Momentum Equations in Analysis of Wave Propagation

1991 ◽  
Vol 113 (4) ◽  
pp. 532-542 ◽  
Author(s):  
Wei-Hsin Gau ◽  
A. A. Shabana
Keyword(s):  
Wave Equations ◽  
Initial Conditions ◽  
Fourier Method ◽  
Coefficient Of Restitution ◽  
Theory Of Elasticity ◽  
Jump Discontinuity ◽  
Mechanical Joints ◽  
Dynamic Formulation ◽  
Propagation Of Elastic Waves ◽  
Algebraic Constraint

A procedure is developed in this paper to study the propagation of impact-induced axial waves in constrained beams that undergo large rigid body displacements. The solution of the wave equations is obtained using the Fourier method. Kinematic conditions that describe mechanical joints in the system are formulated using a set of nonlinear algebraic constraint equations that are introduced to the dynamic formulation using the vector of Lagrange multipliers. The initial conditions which represent the jump discontinuity in the elastic coordinates as the result of impact are predicted using the generalized impulse momentum equations that involve the coefficient of restitution as well as the Jacobian matrix of the kinematic constraints. The convergence of the series solutions presented in this paper is examined and the analytical and numerical results are found to be consistent with the solutions obtained by the use of the classical theory of elasticity in the case of plastic impact. The cases in which the coefficient of restitution is different from zero are also examined and it is shown that the generalized impulse momentum equations can be used with confidence to study the propagation of elastic waves in applications related to multibody dynamics.

Comparative Test on Performance of Golf Ball with Different Coefficient of Restitution and Ball Compression

2018 ◽  
Vol 27 (4) ◽  
pp. 1049-1057
Author(s):  
Kwang-Hyeuk Kim ◽  
Jae-Won Choi ◽  
In-Cheol Kang ◽  
Jae-Kil Han
Keyword(s):  
Coefficient Of Restitution ◽  
Comparative Test ◽  
Golf Ball

scholarly journals An Analytical Solution for Non-Linear Viscoelastic Impact

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1849
Author(s):  
Stelian Alaci ◽  
Constantin Filote ◽  
Florina-Carmen Ciornei ◽  
Oana Vasilica Grosu ◽  
Maria Simona Raboaca
Keyword(s):  
Analytical Solution ◽  
Hysteresis Loop ◽  
Viscoelastic Model ◽  
First Integral ◽  
Coefficient Of Restitution ◽  
Contact Period ◽  
Maximum Deformation ◽  
Impact Parameters ◽  
Simplified Algorithm ◽  
The Moment

The paper presents an analytical solution for the centric viscoelastic impact of two smooth balls. The contact period has two phases, compression and restitution, delimited by the moment corresponding to maximum deformation. The motion of the system is described by a nonlinear Hunt–Crossley equation that, when compared to the linear model, presents the advantage of a hysteresis loop closing in origin. There is only a single available equation obtained from the theorem of momentum. In order to solve the problem, in the literature, there are accepted different supplementary hypotheses based on energy considerations. In the present paper, the differential equation is written under a convenient form; it is shown that it can be integrated and a first integral is found—this being the main asset of the work. Then, all impact parameters can be calculated. The effect of coefficient of restitution upon all collision characteristics is emphasized, presenting importance for the compliant materials, in the domain of small coefficients of restitution. The results (variations of approach, velocity, force vs. time and hysteresis loop) are compared to two models due to Lankarani and Flores. For quasi-elastic collisions, the results are practically the same for the three models. For smaller values of the coefficient of restitution, the results of the present paper are in good agreement only to the Flores model. The simplified algorithm for the calculus of viscoelastic impact parameters is also presented. This algorithm avoids the large calculus volume required by solving the transcendental equations and definite integrals present in the mathematical model. The method proposed, based on the viscoelastic model given by Hunt and Crossley, can be extended to the elasto–visco–plastic nonlinear impact model.

An Improved Mathematical Model of Galton Board With Velocity-Dependent Restitution

2017 ◽  
Vol 12 (6) ◽  
Author(s):  
Auni Aslah Mat Daud
Keyword(s):  
Mathematical Model ◽  
Random Walk ◽  
Symbolic Dynamics ◽  
Coefficient Of Restitution ◽  
Dynamics Analysis ◽  
The Novel ◽  
Francis Galton ◽  
The Way

A Galton board is an instrument invented in 1873 by Francis Galton (1822–1911). It is a box with a glass front and many horizontal nails or pins embedded in the back and a funnel. Galton and many modern statisticians claimed that a lead ball descending to the bottom of the Galton board would display random walk. In this study, a new mathematical model of Galton board is developed, to further improve three very recently proposed models. The novel contribution of this paper is the introduction of the velocity-dependent coefficient of restitution. The developed model is then analyzed using symbolic dynamics. The results of the symbolic dynamics analysis prove that the developed Galton board model does not behave the way Galton envisaged.

Dynamic Formulation and Performance Evaluation of the 6-d.o.f. Parallel Structure Seismic Simulator

2009 ◽  
Vol 23 (12-13) ◽  
pp. 1617-1640
Author(s):  
Yongjie Zhao ◽  
Feng Gao ◽  
Xianchao Zhao ◽  
Nengsheng Bao
Keyword(s):  
Performance Evaluation ◽  
Parallel Structure ◽  
Dynamic Formulation ◽  
And Performance

Single Particle Interaction Properties: Investigations on the Coefficient of Restitution and Coefficient of Friction

2012 ◽  
Author(s):  
Martin C. Marinack ◽  
Patrick S. M. Dougherty ◽  
C. Fred Higgs
Keyword(s):  
Coefficient Of Friction ◽  
Particle Interaction ◽  
Low Carbon Steel ◽  
Granular Flows ◽  
Coefficient Of Restitution ◽  
Natural Phenomena ◽  
Low Carbon ◽  
Solids Processing ◽  
Particle Level ◽  
The Individual

Understanding granular flows has always been important for predicting natural phenomena such as rockslides and soil erosion, as well as industrial processes such as coal-based fossil fuel systems and solids processing. As such, it becomes important to understand granular flows from both a classical granular flow and tribological perspective. Inherently important in the study of granular flows is the study of the individual particle level interactions, which define the global behavior of the flow. The current work examines both the coefficient of restitution (COR) and coefficient of friction (COF) for various material combinations. COR and tribological experiments are performed on various sphere and plate (disk) materials, such as low carbon steel, tungsten carbide (WC), and NITINOL 60.

Dynamics and Performance of a Harmonically Excited Vertical Impact Damper

2008 ◽  
Vol 130 (2) ◽  
Author(s):  
Sanjiv Ramachandran ◽  
George Lesieutre
Keyword(s):  
Loss Factor ◽  
Vertical Direction ◽  
Theoretical Models ◽  
Coefficient Of Restitution ◽  
Numerical Continuation ◽  
Particle Impact ◽  
Impact Damper ◽  
High Loss ◽  
Wide Range ◽  
And Performance

Particle impact dampers (PIDs) have been shown to be effective in vibration damping. However, our understanding of such dampers is still limited, based on the theoretical models existing today. Predicting the performance of the PID is an important problem, which needs to be investigated more thoroughly. This research seeks to understand the dynamics of a PID as well as those parameters which govern its behavior. The system investigated is a particle impact damper with a ceiling, under the influence of gravity. The base is harmonically excited in the vertical direction. A two-dimensional discrete map is obtained, wherein the variables at one impact uniquely dictate the variables at the next impact. This map is solved using a numerical continuation procedure. Periodic impact motions and “irregular” motions are observed. The effects of various parameters such as the gap clearance, coefficient of restitution, and the base acceleration are analyzed. The dependence of the effective damping loss factor on these parameters is also studied. The loss factor results indicate peak damping for certain combinations of parameters. These combinations of parameters correspond to a region in parameter space where two-impacts-per-cycle motions are observed over a wide range of nondimensional base accelerations. The value of the nondimensional acceleration at which the onset of two-impacts-per-cycle solutions occurs depends on the nondimensional gap clearance and the coefficient of restitution. The range of nondimensional gap clearances over which two-impacts-per-cycle solutions are observed increases as the coefficient of restitution increases. In the regime of two-impacts-per-cycle solutions, the value of nondimensional base acceleration corresponding to onset of these solutions initially decreases and then increases with increasing nondimensional gap clearance. As the two-impacts-per-cycle solutions are associated with high loss factors that are relatively insensitive to changing conditions, they are of great interest to the designer.

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