SCIENCE CHINA Physics, Mechanics & Astronomy • Article •
May 2013 Vol.56 No.5: 875–881
Progress of Projects Supported by NSFC
doi: 10.1007/s11433-013-5061-1
Energy evolution in complex impacts with friction ZHANG HongJian1, LIU CaiShan1*, ZHAO Zhen2 & BROGLIATO Bernard3 1
State Key Laboratory for Turbulence and Complex Systems, College of Engineering, Peking University, Beijing 100871, China; 2 School of Aeronautic Science and Engineering, Beihang University, Beijing 100191, China; 3 INRIA, BIPOP project-team, ZIRST Montbonnot, 655 avenue de l'Europe, Saint Ismier 38334, France Received December 12, 2012; accepted February 26, 2013; published online April 15, 2013
In this paper, we base a theory established in an impulse-energy level to solve a problem of a disk-ball system, in which a moving ball collides perpendicularly against an disk staying on a horizontal surface. The impact process is an ensemble consisting of a point impact coupled with a line contact between bodies of the disk, the ball and the fixed horizontal surface. We experimentally and theoretically show that the post-impact states of the disk dramatically vary with the impacting position of the ball. Explanations are given by investigating the evolutions of the potential energies resided in the points involved in the complex frictional impacts. Good agreements between numerical and experimental results strongly suggest that the evolution of energy together with the dissipation must be reflected in mathematical models if a precise description for the post-impact state of systems is expected. complex impact, friction, non-smooth mechanics, disk-ball system, experiments PACS number(s): 45.20.dg, 46.55.+d, 45.20.D-, 45.20.Jj Citation:
Zhang H J, Liu C S, Zhao Z, et al. Energy evolution in complex impacts with friction. Sci China-Phys Mech Astron, 2013, 56: 875881, doi: 10.1007/s11433-013-5061-1
1 Introduction Impacts are common phenomena occurring in nature and many mechanical systems, such as kinematic chains in mechanisms [1–3] and granular materials [4–6]. Methods in dealing with those phenomena depend on the requirements of practical problems, and in general are classified within the following categories: (i) the finite-element methods focusing on the shock stress and deformation in structural dynamics [7–10]; (ii) the compliant models to mimic contact/impact process in multibody system dynamics [11–13]; (iii) the various impact laws with an algebraic form in dealing with the impacts between rigid bodies [14–16]. Although different methods can be applied to the same problem to obtain useful information in understanding impact *Corresponding author (email:
[email protected]) © Science China Press and Springer-Verlag Berlin Heidelberg 2013
process, the decision in choosing a method adopted in simulations not only depends on the available computational resources, but also on the physical pictures to be expected for purpose. Therefore, the research activities of developing various simple methods to deal with impact problems are still vivid in various academic fields, including physics, classical mechanics, and a variety of engineering projects. Among different methods, gaps exist that mainly concern the complex impact problem, in which collisions occur simultaneously among different contact points. In particular, some inconsistent phenomena, which violate the basic conservation law of energy, are often found in the theoretical predictions by choosing simple impacting laws, such as the restitution of coefficients, or by selecting a compliant model with inappropriate parameters. It can be understood that those incorrect predictions originate from excessive simplifications in modeling. Nevertheless, the question is how we could find a technical routine to reasonably balance the phys.scichina.com
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complexity and accuracy of a mathematical model. Generally complex impacts, in which the transition between different forms of energy finishes during an extremely short time scale, vividly exhibit peculiar properties that challenge conventional concepts adopted in classical mechanics. At present, it has been well known that the limitations of classical impact laws come from the ignorance of couplings among different contact points together with or without frictional effects. To reflect those effects on a proper model, a comprehensive framework has been established in [17, 18], and has been successfully applied in the problems such as Newton’s cradle [19], the Bernoulli problem [20], a bouncing dimer on a vibrating plate [21,22] and a rocking Block [23]. The primary idea of the theory is that the evolution of energy during an impact process can be governed by a set of impulsive differential equations with respect to a “time-like” independent primary impulse. Different from the Darboux-Keller impact dynamics [24–29], which only concerns a single impact point coupled with friction, the couplings among the simultaneously impact points are reflected in our theory by modeling the transitions of energy among contacting bodies. In particular, those transitions are implemented, without resorting to the local deformation in small size and time scales, but effectively using the macroscopic physical properties of materials on the contact interfaces, such as the friction coefficients [30] and the Stronge's coefficient of restitution [26]. In terms of the recent developments of impact dynamics in [17,18], this paper will deal with a specific problem relating a disk-ball system, in which a moving ball collides against a disk standing vertically on a rough fixed horizontal surface. The impact happens in the system that contains a point impact triggered by a ball, and a line contact between the disk and the fixed rough surface. The complexity of the problem is because of the presence of the line contact between the disk and the horizontal rough surface, which significantly influence the impact between the ball and disk. To get a simple model of accounting the effects from the line contact, we adopt the conventional techniques in numerical discretization by substituting the distribution interaction on the line contact for three discrete contact points, such that a discrete system involving multiple impacts with friction is established. In order to verify theoretical predictions, an experimental setup is built with measurements by using two layer-Doppler vibrometers to capture the postimpact velocities of the disk. Nine cases of experiments are carried out by assigning the ball colliding against the disk at the different positions along a vertical line passing through the center of the disk surface. We emphasize that numerical simulations are implemented by using a set of physical parameters (the coefficients of friction and restitution) identified from certain specific experiments, so no artificial effects are inserted in the numerical procedure. By fixing those physical parameters for all experiments, we get good agreement between numerical and experimental results.
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Furthermore, detailed inspection of the numerical results validated by experiments demonstrate that the evolution of energy during the very short impacted-time scale has a significant role in determining the post-impact state of the disk. First, we give a description for the disk-ball system and present the governing equations of the system. Together with the common assumptions in an impulsive process, the impact dynamics of the disk-ball system is established by setting a local constitutive relationship to reflect the physical property of materials on the contact interface. Lastly, after giving experimental results, we present the pictures for the evolution of the potential energies resided in different contact points, which varies with the impacting position of the ball.
2 The governing equations for a disk-ball dynamical system The disk-ball system is shown in Figure 1, in which a homogeneous disk with radius r, thickness 2h and mass md stands on a rough horizontal surface, while a ball with radius rb, mass mb, initially takes a velocity of vb along the normal direction of the disk surface colliding against the stationary disk at point D, which is located in the vertical symmetry axis of the disk surface. Since rb r, the ball is thought of a particle, such that the rigid body model of the disk-ball system belongs to a planar system with five degrees of freedom. Let (xd, yd, zd) and (xb, yb, zb) be the Cartesian coordinates, with respect to a Galilean frame (O′, i, j, k), of the center of mass of the disk O and the position of the particle, respectively. b is the vertical distance from the impacting point D to the center of mass O. For the planar case concerned in in the configurathis paper, we have xd=xb=0, and 2 tion of the system shown in Figure 1, where is defined as the angle of the vertical symmetry axis of the disk relative to the Cartesian space axis j. The set (O, e1, e2, e3) is a Cartesian set of coordinates fixed in the disk and located at the
Figure 1 System of a moving ball impacting an stationary disk placed on a rough surface. The measurement for its 2D motion by using two laser sensors.
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center of mass. Let us select q=(yb, zb, yd, zd, )T as the generalized coordinates for the disk-ball system. Defining a Lagrangian function and using Euler-Lagrange equations, the governing equations for the disk-ball system take the following form: Mq G W (q ) F n N (q) F ,
(1)
where M R 55 is a mass matrix, which can be expressed as M=diag{mb, mb, md, md, I}, G=[0, mbg, 0, mdg, 0]T is the generalized forces obtained from the potential energy, F n [ FAn , FBn , FCn , FDn ]T , F [ FA , FB , FC , FD ]T are the normal and tangential components of the contact forces at contact points A, B, C, D, respectively. WT(q) is a transformation matrix for the relative contact point normal velocities i with respect to the generalized velocities q : W T (q)q , NT(q) is a transformation matrix for the relative
tangential velocities v with respect to the generalized velocities q : v N T (q )q. Symbols of s and c are the abbreviations for sin() and cos().
3 The impact dynamics in the disk-ball system For the disk-ball system, the presence of the interaction between the disk and the horizontal surface brings simultaneously multiple impacts when the ball collides against the stationary disk. The assumptions commonly adopted in shock dynamics are introduced as follows: the configuration of the system is constant during impact; except the contact force, the effects from other forces on impact are ignorable; the dissipation of energy induced by the normal motion is not influenced by the tangential motion, and the time rate of the dissipation is uniform during impact. These assumptions together with the definitions of dPn=Fndt, dP=Fdt, eq. (1) are simplified into the following form: Mdq W (q )dP n N (q )dP .
(6)
Let us specify that the relationship between elastic force Fi n and small elastic deformation i at contact point i satisfies a constitutive relationship Fi n ki i , where ki is a contact stiffness, and exponent =3/2 for a Hertz contact, =1 for linear elastic contact. The variation of the elastic potential energy dEi induced by the work dwin done by the
s 0 0 0 0 0 0 c W (q) 0 0 0 s , c 1 1 1 hs rc rc hs rc b
elastic force Fi n through a small elastic deformation i is expressed as dEi Fi n d i , which can be rewritten as dPi n d i i dPi n , with a form of mapping the dt potential energy in an impulse-velocity level. Together with the constitutive relationship, Liu et al. [17] indicates that the ratios of normal impulses at different points can be exdEi
and c 0 0 0 0 s 0 0 N (q) 1 1 1 c . 0 0 s 0 (hc rs ) rs (hc rs ) h
dPi n
The second-order differentials relating the normal and tangential relative displacements can be expressed as:
1
k 1 Ei 1 pressed as, i Rin, j , where dPjn is n E dPj k j j thought of “time-like” independent normal impulse at contact point j, whose potential energy is a maximum in comparison with other points. The tangential impulse dPi is connected with the nor-
W T (q )q S n (q, q ),
(2)
mal impulse dPi n by Coulomb's friction law. If vi 0,
v N T (q )q S (q, q ).
(3)
the relationship between dPi n and dPi is simple, and is
where S (q, q ) W q , S (q, q ) N q. Combining eq. (1) with eqs. (4) and (5), it can be determined that the relative motions of the various contact points are governed by the following local dynamic equations: n
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T
T
(4)
v N T M 1WF n N T M 1 NF T
1
+N M G S .
tionship between dPi n and dPi should be established by using an additional condition of dvi 0 [18]. Taking into consideration that dPi n has been connected with the primary impulse dPjn , the tangential impulsive dPi can
W T M 1WF n W T M 1 NF +W T M 1G S n ,
expressed as: dPi i dPi n sign(vi ). If vi 0, the rela-
(5)
always be expressed as a function with respect of the primary impulse dPjn . It is worth noting that the stick-slip transition at the instant of vi 0, is limited by an upper bound s defined by Coulomb's friction law for the ratio of
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tangential to normal impulses. Therefore, a transition from a stick state to a slip state at contact point i can occur only if dPi dPi n
is . Similarly, a slip motion at point i can be trans-
ferred into a stick state at the instant of vi 0 only if the ratio of dPi and dPi n , which is obtained by setting dvi 0, satisfies the condition of
dPi dPi n
is , where
i>0 and is i are coefficients of the dynamic and static friction at contact point i. In order to reflect the energy dissipation occurring in impact, let us consider the definition of the energetic coefficient of restitution ei, given by Stronge [26], to characterize the loss of the energy when a full compression-expansion cycle is finished at point i. Together with the assumption that the dissipated energy is uniformly distributed in the expansion phase, the potential energy at contact i with an initial value Ei ( Pjn* ) at the impulsive instant Pjn* of the impact can be generally expressed as: n
Ei ( Pjn ) Ei ( Pjn* )
Pj 1 R n dPjn 0, n* i i , j P i ( i ) j
(7)
where if 0, 2 ei , if 0. 1,
i (i )
Clearly, the value of the potential energy Ei ( Pjn ) increases in a compressional phase ( i 0 ) because of the negative work done by the normal contact force, but decreases in an expansion phase ( i 0 ) due to the positive work of the normal contact force. For a fully elastic impact, in which ei=1, the energy absorbed in the contact point can be completely released through an expansion phase without dissipation. If 0
