Rigidity versus deformability hypothesis in impact dynamics

The complexity of the study concerning an impact phenomenon depends strongly on the accepted simplifying hypothesis for the model. One of the most important assumptions in the study of mechanical impact refers to the absolutely rigid or compliant character of colliding bodies. It is anticipated that the employment of hypothesis of deformable body should lead to a better modelling of the process. The present paper presents a qualitative comparison between the results obtained via the two methods, considering one of the simplest impact models, namely the collision with dry friction between a metallic ball and an immobile flat body. 1 Study methods of impact phenomenon Two main directions in approaching the impact phenomena have been outlined in technical literature. The first method considers the impact between any two bodies as an instantaneous event. The second approach regards bodies as deformable ones and the impact happens during a finite time. So, every impact phenomenon is characterized by the existence of two distinctive phases: compression and restitution. The compression phase lasts the moment the first points of the two bodies are contacting until the instance the normal approach between the two bodies reaches the maximum. The restitution phase begins from the moment corresponding to the maximum approach and lasts until the last points of the two bodies separate from contact. No matter the considered hypothesis, the coefficient of restitution (COR) is a parameter characteristic to both methods. Newton [1] describes cinematically the coefficient of restitution e as the ratio with changed sign between the normal components of relative velocities of initial contact points: Newton 1 2 1 2 e ( " " ) ( ' ' ) v v n v v n       (1) * Corresponding author: florina@fim.usv.ro DOI: 10.1051/ , 07005 (2017) 711207005 112 MATEC Web of Conferences matecconf/201 IManE&E 2017 © The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution License 4.0 (http://creativecommons.org/licenses/by/4.0/). In (1), n represents the unit vector of the normal to the bodies’ surfaces. For the present work, the symbol ) (' is used for the parameters characteristic to initial collision time and the symbol ) (" is used for the parameters corresponding to the final moment. The definition (1) for the COR was useful until Kane [2], analyzing the plane impact with friction for a double pendulum reaches the conclusion that accepting the definition (1) for the COR leads, when the geometry and initial kinematical state are conveniently chosen, to disobedience of the law of conservation of energy. To surpass this intricacy another definition for the COR is required. To this end, the hypothesis of finite continuous variation of impact force must be accepted. The new definition of the COR is due to Poisson [6]. Thus: f c c i t t Poisson r c t t e dt dt P P       F n F n (2)


Study methods of impact phenomenon
Two main directions in approaching the impact phenomena have been outlined in technical literature.The first method considers the impact between any two bodies as an instantaneous event.The second approach regards bodies as deformable ones and the impact happens during a finite time.So, every impact phenomenon is characterized by the existence of two distinctive phases: compression and restitution.The compression phase lasts the moment the first points of the two bodies are contacting until the instance the normal approach between the two bodies reaches the maximum.The restitution phase begins from the moment corresponding to the maximum approach and lasts until the last points of the two bodies separate from contact.No matter the considered hypothesis, the coefficient of restitution (COR) is a parameter characteristic to both methods.Newton [1] describes cinematically the coefficient of restitution e as the ratio with changed sign between the normal components of relative velocities of initial contact points: In (1), n represents the unit vector of the normal to the bodies' surfaces.For the present work, the symbol ) (' is used for the parameters characteristic to initial collision time and the symbol ) (" is used for the parameters corresponding to the final moment.The definition (1) for the COR was useful until Kane [2], analyzing the plane impact with friction for a double pendulum reaches the conclusion that accepting the definition (1) for the COR leads, when the geometry and initial kinematical state are conveniently chosen, to disobedience of the law of conservation of energy.To surpass this intricacy another definition for the COR is required.To this end, the hypothesis of finite continuous variation of impact force must be accepted.The new definition of the COR is due to Poisson [6].Thus: Considering that: represents the percussion, according to Hibbler [3] the definition states that the COR is the ratio between normal percussions corresponding to the restitution (r index) and compression phases (c index), respectively.There were denoted i t , c t f t the moments corresponding to contact initiation, maximum approach and impact ending, respectively.

The plane of percussions method (Routh)
The models mentioned above are applicable where the friction force presents a continuous variation with velocity.In the case of dry Coulomb friction, the condition is not obeyed due to the fact that the friction force is characterized via inequalities, the dry friction forces being unilateral constraints [4].A reference work in the impact with friction domain is owing to Wang and Mason [1].For the study of two-dimensional impact with dry friction, they apply the plane of percussions method, an extremely intuitive method, proposed by Routh [5].As a principle, as shown in Figure 1, for the two bodies denoted 1 and 2, contacting in point O , an axis system is defined, with the axes directed along the normal and tangent to the surfaces of the bodies, and the centers of mass of the two bodies, 1 C and 2 C are established according to it.For both bodies, the Newton-Euler dynamic equations [3] are written and the final kinematical parameters are found, as function of tangential t P and normal n P components of interaction percussion, Figure 2. Finding the relative velocity of impact points, in the plane of percussions [5], the geometrical locus of the points where the normal component of the velocity 21 v (relative velocity of the two contacting points [1]) is zero is represented by the line of maximum compression ) C ( , Figure 3, and the geometrical locus of the points from percussions plane where the tangential component of the velocity 21 v is zero is represented by the stiction line ) S ( ; in addition to these two straight lines, the straight line of limit friction is traced, defined as: and the ending straight line , on which the impact finishes according to Newton's hypothesis.For the study of a plane impact with friction, the hypothesis is made that during the entire impact process the normal percussion increases monotonically.Thus, at the beginning of impact, the characteristic point starts from origin and moves along the limit friction line LF .When the compression line is reached, the approaching phase ends.The impact ends when (knowing that f c r P P P   ) according to equation ( 2), the next relation is fulfilled: The motion state of the system modifies only once, when the characteristic point reaches the stiction line ) S ( .The characteristic point will move along the steepest of the stiction lines (rolling relative motion) or along the reverse friction line ) RF ( when the sliding reverses its sign.At the intersections between the stiction line and compression line, and stiction line and terminal line, two points are defined.These points, together with the origin, describe two half-lines crossing the plane in three regions, 1, 2 and 3, Figure 3.Only in the domain 1 the same result is obtained with both definitions of COR.

Percussions plane for the impact between a dropping ball and a tilted immobile plane -rigid bodies hypothesis
For the impact between a ball in free fall and an inclined immobile plane tilted at  angle with respect to the horizontal, the percussions plane is presented in Figure 3. Applying the methodology described by Goldsmith [6] concerning the manner proposed by Routh for the impact between a mobile sphere and a fixed plane, the following conclusions become known: the stiction line has the equation of a vertical line, while the maximum compression line, as well the termination line, become two horizontal lines.From Figure 4 one can notice two domains, depending on the relation between the friction angle   atan


, and the angle  , namely smaller or greater than it.The angle  is defined [6] by:  The certainty of two different behaviours of the ball stated by the tilting angle of the plane was proved in a recent work [7] where, considering the impact between a ball and a metallic disc that rotates about a vertical axis, Figure 5, it is shown that the mentioned impact case can be assimilated as the impact between a ball and an inclined plane; the space run by the ball after impact is found and there is an obvious rotation velocity value delimiting the linear increase of the space from the space decrease, Figure 6.

Percussions plane for the impact between a dropping ball and a tilted immobile plane -deformable bodies hypothesis
The problem of experimental study impact assumes finding both the variation of normal approach and impact force with time.For the case of elastic centric impact, Timoshenko [8]  IManE&E 2017 determines analytical expressions for maximum impact force, maximum normal approach and impact time, based on the assumption that during the whole impact period, the impact force has the same relation as in Hertzian case.The equations show that the period of a collision is quite short, of the order of milliseconds, while the accelerations are of the order g ) 000 . 10 1000 (  .The significant values of impact force and reduced time of collision are the main impediments in the study of collisions.In a recent paper, Garland and Rogers [9] present the experimental study of the plane impact with friction between a ball and an inclined plane.The schematic of the principle [9] is shown in Figure 7, using a mathematic pendulum consisting in a wire and steel ball.A body 2 having great mass is placed eccentrically with respect to the plane of oscillation of the pendulum.On the surface of the body a tri-axial shock sensor 3 is mounted, the axes of the sensor being oriented on the following directions: Oz normal to the plane surface, Oy vertical in the impact point and Ox tangent in the impact point.The ball is launched from ' B position, strikes the sensor 3 and reaches the post-impact position " B .The launching amplitude is small enough as to consider the trajectory of the centre of the ball as a plane one.This approximation is confirmed by the experimental values of vertical force Fy which are negligible compared to normal force Fz .In [9] there are presented graphs of the variations of normal and tangential forces, for different incident angles  of the ball.For a tilt of  With known dependencies for Fn,t Fn,t( t )  , using the relation 3, the variations in time of tangential and normal percussions were established, Pn( t ),Pt( t ) , and the plane of percussions was traced using these percussions, as presented in Figure 9; also represented is a broken line consisting in an inclined straight line and a vertical line, according to Figure 4. To accept the hypothesis of rigid bodies and to compare the results with the ones under the hypothesis of deformable bodies is equivalent to accepting that the broken line form Figure 9 (obtained as in explained in Figure 4) is an approximation of the continuous curve from the same figure.But, as one can remark, there is a significant difference between the two graphs.It can be considered that, for the first stage of impact, the broken tilted line is a good approximation of the continuous curve, but for the restitution phase this equivalence is hardly plausible.One of the most significant differences between the two methods consists in the fact that while the rigid body model estimates that tangential percussion increases or at least is constant during the last phase of collision, the deformable body model assumed the occurrence of a maximum for tangential percussion during collision, and afterwards a decrease of it.Given the variety of applications in the domain of multibody dynamics, [10][11][12][13], the results could be a base for future research.

Conclusions
The paper analyses one of the simplest collision cases, namely the dry friction impact of a ball with a tilted plane.There are two main methods of approaching the problem: the rigid bodies assumption and the compliant bodies method.For the ball-inclined plane impact, the authors used both data from research literature and experimental results from own research work, and a qualitative comparison between the two methods is presented.Differences between the two methods are noticeable even from qualitative analysis.The results of the paper are a starting phase for future work aiming to correlate the results of the two methods, considering that the broad fields of applications, from granular and powder materials phenomena, grasping in robotics, landing trains in aircraft to percussions in sports.The infrastructure used for this work was partially supported by the project "Integrated Center for research, development and innovation in Advanced Materials, Nanotechnologies, and Distributed Systems for fabrication and control", Contract No. 671/09.04.2015,Sectoral Program for Increase of the Economic Competitiveness co-funded from the European Regional Development Fund.

Fig. 4 .
Fig. 4. Percussion plne for impact between a ball and a tilted plane.

P
Postimpact horizontal velocity [m/s]Velocity of contact point from the disc [m/s]