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Planar impact of rough compliant bodies
~ )
Int. J. Impact Engng Vol. 15, No. 4, pp. 435-450, 1994 Elsevier Science Ltd Printed in Great Britain 0734-743X(94)E0013-L 0734-743X/94 $7.00 + 0.00
Pergamon
PLANAR
IMPACT
OF
ROUGH
COMPLIANT
BODIES
W. J. STRONGE Department of Engineering, University of Cambridge, Cambridge CB2 1PZ, U.K. (Received 23 November 1993; and in revised form 31 January 1994)
Summary--A lumped parameter model of contact between colliding bodies is used to obtain the tangential force and energy dissipated by friction during oblique impact of rough compliant bodies in collinear configurations. For Coulomb's law of friction and an energetic coefficient of restitution, this analysis distinguishes between angles of incidence where the contact point initially sticks, slides before sticking or slides throughout the contact period. During part of the contact period tangential compliance significantly alters friction; this affects changes in relative velocity unless the angle of incidence is so large that there is continuous sliding in the initial direction. For oblique impact, tangential compliance reduces the largest friction force in comparison with friction generated if the contact region has negligible tangential compliance. While the part of the initial kinetic energy dissipated by friction is limited by the coefficient of restitution, this coefficient has no effect on the largest friction force generated during collision.
INTRODUCTION
Collisions between hard bodies result in changes in relative velocity in a very brief period of time. These changes in velocity are caused by large contact forces; e.g. a typical hammer blow applies a force of several tons. Nevertheless, if the bodies are hard and the contacting surfaces are nonconforming, the area of contact between the bodies remains small in comparison with the size of either body. For hard bodies, the region of significant deformations is limited to the immediate vicinity of the contact area despite the large contact force; this is because stresses (and strains) decrease rapidly with increasing radius from the initial contact point (roughly as r -2 in an elastic solid if the cross-section is much larger than the contact area). Consequently, collisions between hard and compact bodies ordinarily can be assumed to occur impulsively; i.e. we assume the changes in velocity occur instantaneously so there is no movement of colliding bodies while they are in contact. For oblique impact between rough bodies there are both normal and tangential components of contact force; the tangential force (friction) opposes the tangential relative velocity which is termed sliding or slip. Effects of dry friction on changes in velocity during impact can be obtained from Coulomb's law only by summation of changes during successive stages of unidirectional slip 1-1-3]. If the direction of slip is continually changing a dynamic analysis that incorporates effects of dry friction during sliding can be achieved if the impact process is resolved as a function of the normal component of impulse [4, 5]. In the past such analyses have assumed however that tangential compliance of the bodies is negligible in the contact region. While this assumption is an asymptotic limit that represents large initial slip, it may not be accurate for small initial slip where the direction of slip can change during contact. Maw et al. 1-6, 7"] performed a dynamic analysis of oblique impact between rough elastic spheres using Hertz contact theory to obtain the normal tractions in the contact area. They found that at small sliding speeds the contact area had an outer annulus which was sliding. The annulus surrounded a central area where there was no tangential component of relative velocity; i.e. the central area was sticking. This combined state of slip and stick was termed microslip. Both the analysis and experiments of Maw et al. showed that if initial slip was small, the direction of slip could be reversed during collision by tangential compliance. With negligible tangential compliance such reversals are not possible if the impact configuration is collinear (i.e. if the centres of mass of the colliding bodies are on the common normal line passing through the contact point). 435
436
W.J. STRONGE
There have been several attempts to develop approximations which produce the effect of slip reversal for small angles of incidence in collinear as well as noncollinear collisions. Bilbao et al. I-8] define a kinematic tangential coefficient of restitution which varies exponentially with the coefficient of friction and the ratio of normal to tangential components of incident velocity. Smith [9] defined a kinetic coefficient of restitution that relates the tangential impulse to the coefficient of friction, normal impulse and an average velocity of sliding during collision. Brach [I0] used two linear relations for changes in tangential velocity; these employed a kinematic coefficient of restitution (negative) at very small angles of incidence and a kinetic coefficient of restitution at larger angles. All of these approaches were designed to produce at large angles of incidence a ratio of tangential to normal impulse equal to the coefficient of friction; they represent Coulomb's law of friction only in the limit as sliding becomes continuous in the initial direction. In contrast to the elastic continuum approach of Maw et al. [6] and the approximations above, the present paper uses a lumped parameter representation for compliance of the contact region. Equations of motion are developed from a few physical laws while system characteristics are expressed in terms of coefficients that are independent of the angle of incidence. This model yields either slip or stick at the contact point depending on whether the ratio of tangential to normal contact f o r c e is as large as the coefficient of friction. During stick the tangential force depends on relative displacement so a time dependent analysis is required to resolve the changes in velocity that occur in a collision; nevertheless we assume that the total period of contact is so brief that there is no change in configuration during collision. Comparison reveals that the present lumped parameter modelling gives velocity changes that are almost the same as those in experiments by Maw et al. (1981) and contact forces that are similar to measurements by Lewis and Rogers rl 1]. In addition the present analysis yields results for a model of inelastic collisions with tangential compliance. DYNAMICS OF PLANAR COLLISION FOR HARD BODIES To focus on the effects of tangential compliance during collision, consider a body with mass M that collides against a half-space at contact point C as shown in Fig. 1. At C the body and the half-space have a common tangent plane. Let unit vectors n~ and n2 be oriented in directions tangent and normal to this plane respectively. At incidence (time t =0) a point on each body comes into contact; at incidence these points have a relative velocity v~(0) with tangential and normal components, v~(0) and 02(0), respectively. The orientation of the coordinate system is defined such that at incidence both normal and tangential components of relative velocity are negative, o1(0)< 0 and ~2(0)< 0. The collision period is separated into an initial period of normal compression and a subsequent period
x--~t~:
~
x. . . . . ~:
/
,-,4.;., "~ I I ~.'~~b.?.i:ii:~i:~.:!.~:i~:~i:i~::.'~;"
n
~i:~ii~!~::.'.:i~ii~.~' , i..':~" ~
,~,-~
•
t
"-' $.~:.:.-~'i'~::"
FIG. 1. Discreteparameter model for impact of rough, compliant body on half-space.
Planar impact of rough compliant bodies
437
of separation. The compression period terminates at time t c when the normal component of relative velocity vanishes, v2(tc)=0. The contact points separate at time tt when the final relative velocity has a normal component v2(tf), where v2(tf)>0. Hence during compression, kinetic energy is absorbed by deformation of the bodies while during restitution, elastic strain energy generates the force that drives the bodies apart and restores some of the kinetic energy that was absorbed during compression. To simplify the dynamic analysis we assume that both bodies are rigid except for an infinitesimally small deformable region that separates the bodies at the contact point. Let the body with mass M have a radius of gyration k about the centre of mass. From the centre of mass of the colliding body the coordinates of C are xl in directions ni, i = 1, 2. Thus the body has an inertia matrix for C with an inverse mi~ 1 where
m-Z1 L_xlx2/Mk 2
mi~l=[
ml = Mk2/(k2 + x2),
-xxx2/Mk21 m2 1
d
m2= Mk2/(k2-1- x2).
(1) (2)
Notice that this matrix has terms off the principal diagonal that all vanish if the collision configuration is collinear. We model the infinitesimal deforming region around C by assuming that the colliding body is connected to a massless particle located at C as illustrated in Fig. 1. The connection is via two independent compliant elements--one tangential and one normal to the tangent plane. This particle has a tangential component of displacement u~(t) relative to the body that depends on the tangential force at C and the tangential compliance. Contact force F~ acts at the contact point; it has components in directions n~. This force applies a differential impulse dpl = Fi dt in an increment of time dr. Thus the equations of planar motion for the rigid body can be expressed as (see Stronge, 1994) do i = mi-~ 1 dp r
(3)
To progress further we must be specific about the compliance in the contact region so that contact force can be calculated. The components of this force are necessary in order to distinguish between periods of slip and stick at C. In order to calculate the changes in relative velocity that occur while the contact sticks, it is necessary first to obtain the components of contact force.
Linear compliance model Here we assume that both normal and tangential compliant elements are piecewise linear. During compression let the normal element have an arbitrary stiffness x while the tangential element has stiffness x/( 2 as shown in Fig. 2. The parameter (2 is a ratio of normal to tangential stiffness at the contact point; this ratio depends on the structure of the colliding
F1
0
(1-e2lu2(lc)
u2(t e)
u,
FIG. 2. Stiffnessof normal and tangential compliant elementsduring loading and unloading.The energydissipated by irreversibleinternal deformationsis proportional to the ratio of the area under the curve during unloading to this area during loading.
438
W.J. STRONGE
body in the deforming region. These spring constants and the inertia matrix mij give two natural frequencies, co and ~ where
co2,f,~2= m~1 {( 2
(2
m1~2~ ./(1
ml(2~ 2
1+ m2 / +_ ~1\ - - - m2 / +
4m~(4x2x2~ x 2 M2k4 j.
For a collinear collision and linear stiffness, the colliding body undergoes independent simple harmonic motions (s.h.m.) in the normal and tangential directions while the contact point sticks. These motions have frequencies that depend on the stiffness and the components mi of the inverse of mass matrix that were given in (2). The normal frequency f~ and tangential frequency co are defined as f~-~2
n =-f~,
co=
~/ x _ n__n___ m/~2 ~2ml 2 ( t ~ m l
(4,
where time t c is the instant when the compression period terminates. At this instant the normal compliant element has a maximum compression u2(tc).
Coefficient of restitution The coefficient of restitution, e, can be defined as the square root of the ratio of elastic strain energy released at the contact point during restitution to the kinetic energy absorbed by internal deformation during compression. For negligible tangential compliance, the loss of kinetic energy due to irreversible internal deformations in the contact region can be obtained from the work done on the bodies by the normal component of contact force. Stronge (1990) has defined the energetic coefficient of restitution, e, according to this work. For negligible tangential compliance, this is the only part of the work done on the bodies that goes into deformation. Consequently, it is the only part that can be stored as elastic strain energy if tangential compliance is negligible--the work done by the tangential component of force is all dissipated by friction. If tangential compliance is not negligible, however, there is also energy absorbed by tangential deformations u 1. Here I assume that this deformation is entirely elastic. This assumption is based on considering the coefficient of restitution as representing nonfrictional energy losses that are due primarily to plastic deformation. In an initial range of elastic-plastic deformation the region of plasticity is contained beneath the surface of a deforming body; this contained or subsurface plasticity has very little on tangential compliance. Hence for elastic-plastic bodies which collide at low speeds, Stronge's definition of the energetic coefficient of restitution still applies, i.e. -- ~t tt g2v 2 dt e: =
°
(5)
0cF2v2 dt With linear compliance the normal component of force does work W2(tc) on the bodies during compression; this is simply the area under the normal compression line in Fig. 2, W2(tc)=F2(tc)u2(tc)/2. In the present model the effect of coefficient of restitution (5) is obtained by changing the stiffness of the normal compliant element at the transition time t~ when compression terminates. Hence at the instant of maximum compression t~ the stiffness of the normal element increases from x to x/e2~ For changes in the normal component of relative velocity, this change in stiffness makes the frequency of s.h.m, larger during restitution than it was during compression. The collision terminates and separation occurs at a final time t6 for collinear collisions tf=(1 +e)tc. At separation the normal component of relative displacement has a terminal value u2(tf) ----(1 --e2)u2(tc).
Normal components of velocity and force in collinear collision Henceforth in this paper the analysis is limited to coUinear collisions so that normal and tangential equations of motion (3) are decoupled. For a collinear collision the normal
P l a n a r i m p a c t o f r o u g h c o m p l i a n t bodies
439
force (and hence any change in the normal component of relative velocity) is independent of the process of slip or stick at the contact point. For a linear compliant element, the colliding bodies undergo separate stages of s.h.m, during compression and restitution periods of the collision. Thus at any time during the collision, the normal component of relative velocity is as follows:
v2(t) = v2(0) cos f~t v2(t) = evz(0) cos
+ 2\
eJJ
0 < t < tc
(6a)
t¢ < t < tr
(6b)
This normal component of velocity is continuous at the time of maximum compression t c when the frequency of s.h.m, increases from ~ to f~/e. The normal component of impulse that causes these changes in velocity can be obtained directly, and this impulse can be differentiated to obtain the normal component of contact force. The expressions for these variables are listed in Table 1.
Tangential velocity and force during stick During the period in which the contact point sticks, s.h.m, applies also to tangential changes in velocity if the tangential compliant element is linear. The tangential oscillations during stick occur with frequency o9. It is convenient to express the velocity and force during stick in terms of an initial displacement ua(T) and an initial velocity v~(z) at time t = r when stick begins. While the contact point sticks there is no sliding, v~(t)+ ill(t)= 0 where ti 1-dua/dt; thus tangential displacement, velocity and the contact force on C can be written as
ul(t) = ul(z) cos w ( t - r ) - l v l ( z ) sin og(t- z)
(7a)
O9
vl(0 = ogul(z) sin co(t- r) + vl(r) cos co(t- z)
(7b)
Fl(t)=mlo92ul(r) COS og(t--z)--miogvi(z) sin co(t--z),
t>z.
(7c)
The state of stick persists while the ratio between tangential and normal components of contact force satisfies IFll/F2 <1~.
Sliding contact with Coulomb friction For dry friction that can be represented by Coulomb's law, sliding occurs if the ratio between the tangential and normal components of force is equal to the coefficient of friction #. Thus sliding depends on the tangential compliance and relative displacement u l(t ) between the particle at C and the body; if there is sliding contact, the particle is sliding at velocity vl(t)+ til(t ). On the other hand, if IFiI/F 2 < # then the contact is sticking so vl(t)= -til(t).
TABLE 1.
NORMAL DISPLACEMENT, VELOCITY, FORCE AND IMPULSE DURING COLLISION Compression, 0 < t < tc
Disp.
u~(t)= (v2(O)/[l)sin fit
Velocity
v2(t)= v~(O)cos fit
Force
F2(t) = -- m2flv2(O) sin fit > 0
Impulse
p(t)= - m2v2(O)[1--cos lit] _>0
R e s t i t u t i o n , t e < t < tf
e2sin( + (, !))] v2,,ev2,0,cos<+(1
440
W.J. STRONGE
For simplicity, coefficients of static and dynamic friction are assumed to be equal. While the contact slides in direction s = sgn(vl + riO, the contact force applies a differential impulse
dpj= { - i S } dp
(8)
where d p - @2. Consequently, for sliding the differential Eqn (3) can be expressed in terms of a new independent variable--the normal component of impulse p rather than time,
~dv,/dp~=[m;' [dv2/dpJ
m°~ ' l [
1 J"
(9)
Initial stick or slip? With the present model, the contact point C either sticks or slips depending on the ratio between the components of contact force; i.e. whether or not the contact force is inside the cone of friction.* First suppose that stick begins at the initial instant of contact and then test whether this satisfies the limiting force ratio. An initial period of stick terminates at a time tl when the ratio of tangential to normal force first becomes as large as/~; i.e. time tl is obtained from IF1[ 1 v1(0) f~sin cot 1 . . . . /.tF 2 (2 #V2(0) COsin t2t~
IFll /'tF2
_ 1 o1(0) f~
1,
0_
sin cot I
(2 kw2(0) co s i n ( ~
= 1,
(10a)
to
(10b)
+ 2(1 _ ~) )
The process of initial stick takes place if t 1 >0; i.e. if in the limit a s t I ---~0 the force ratio is inside the cone of friction. This requires an angle of incidence such that vl(0)
(11)
02(0) Thus initial stick occurs if the angle of incidence is small; i.e. if at incidence, the ratio of the tangential to normal component of relative velocity at C is within a range 0 < v 1(0)/v2(0) < p~2 that is bounded by friction and the ratio of normal to tangential stiffness. For collisions with an angle of incidence at C which is larger in magnitude than t a n - 1(p¢2), the contact point begins sliding when contact initiates. SLIP PROCESS
Small anole of incidence v l(0)/v2(0)
vl(tl) = vl(0) costotl.
(12)
Thus from (11) the terminal velocity can be obtained vt(tf) =
vx(tx)--I~S[p(te)--p(t,)] ml
(13)
* This model givesa period of initial stick if the incident tangential velocityis small--initial stick occurs because the particle at C has negligiblemass. Mindlin and Deresiewicz1-12]have shown that for elastic bodies, any finite tangential velocity results in central stick plus a peripheral annulus of slip around the small contact patch during an early stage of collision.
Planar impact of rough compliant bodies
441
slip (e :1.0) I = F "1
st,ck 1.0-/f~ m2v2(tc)/ / ~/,/ 0.5
/
/
X
\
~.--_
F~/tl~m2v2(t~)
\\
(al "
1.o ~
'/o
\
-o.s
*1.0
/
\
-1.0
t/t~
\
~1 i
\
\
"/
~I
\
/
,,,~.i i
CI
/
/
compression-.,P---~ resti'fulion _
v1 (t~,, IJ'v2(O)
(b)
1.0
1.5
2.0 ~/i=
-1.0
FIG. 3. (a) Normal and tangential forces and (b) tangential velocity during collision for small angle of incidence vl(O)/v2(O) <#(2 and frequency ratio o•/D= 1.7. Forces are illustrated for both e=0.5 and 1.0. For each coefficient of restitution, the final velocity at separation is indicated by an X. where the change in normal impulse during the final period of slip is given by p(tf) - p(t 1) = - # I m 202(0) •
(14)
The impulse ratio during slip gl is the ratio of the normal impulse applied during period t c - t t to the normal impulse during compression, gl =e{1 + c o s ( e - l f 2 t 1 +(n/2)(l - e - 1 ) ) } ; this impulse ratio depends on initial conditions. In the subsequent analysis we consider co/fl > 1 since this applies to most elastic bodies*. F o r this case slip begins during restitution at a time tl > t c when u l ( q ) > 0 so that s = + 1 and the terminal velocity can be expressed as Vl(tf) -- 01(tl) Jr m2g 1 #xo2(0) #xo2(0) m~
(15)
Intermediate angle of incidence #~2 < vl(0)/v2(0 ) < Ix(1 + e)m2/ml If the angle of incidence is larger than t a n - ~(#x~2), initially there is sliding at the contact point as illustrated in Fig. 4. D u r i n g the initial period of sliding the b o d y has a tangential c o m p o n e n t of velocity at C that is given by
vl(t ) = vl(O ) - lZSp(t). ml
(16)
* For elastic spheres with Poisson's ratio v=0.3 the ratio of stiffnesses (2 = 1.21 and elements of the inverse of mass matrix give m2/ml =3.5 so that w/D= 1.7 (e.g. see Johnson [15]) whereas for incompressible elastic spheres, (2= 1.5 and oJ/f~= 1.53. Furthermore, in the asymptotic limit of negligible tangential compliance (2--*0 so that O.)'--+ 0 0 .
442
W.J. STRONGE
slip ,
stick
~'"
1.0
_ IS}ip(e=l'O)l_ -II
"~"-- ~'-- "~" ~-"1 I"-slip (e=0.5,
- -
F'21/~m2v2 { 0 ) /
0-5 •
''
/~
/
',
',., "¢.o
\
/
\t~(e=0.5) \ \" tc ~' t~le=l} \
/
0
.,l
,~/)J~m2v2 (0 } \
1'0
t~'l¢
V1"5
X
-0.5
--/2"0
,\
~t c
/
~,~//
\ \ ..
-1.0
i1
compression :
',
= restitution
Ibl
-1.0
LL t2/t¢
I
~
' I '
1.0
I
1-5
2.0 ~"tc
-1.0 FIG. 4. (a) Normal and tangential forces and (b) tangential velocity during collision for intermediate angle of incidence ,u(2
This initial sliding terminates and slip begins at time t 2 when subsequent sliding or stick have the same rate of change for the tangential force; i.e. the transition to stick occurs when l i m r dFl(t2-I-z) dF2(t2-t-z!1 T-,o El dz =# dz "
(17)
For the period t > t2, Eqn (7c) gives dFl(t) = dt
mla~aUl(t2)sin ~ ( t -
where the dynamics for sliding during t < ul(t2) =
(2Fl(t2) K
t2
give transition values at the beginning of stick,
m2f~02(0)
= # - mlo) 2
(18a)
sin ['It 2
vl(t2)=vl(O)--Ilm2V2(O)[1--COS ~)t21, ml x
cos og(t-- t2)
t2)--mlo92ol(t2)
=# ~
ml~
Vl(t2)=vl(O)--#--v2(O)
sin/--+ \ e
(18b)
t2 <_tc
2\
1--ecos __2+~\
(19a)
e,/J
t2>te.
(19b)
Planar impact of rough compliant bodies
443
The normal force during restitution is given in Table 1; hence by differentiation we obtain dF2(t-~)-dt
m2['~21)2(0)cos ~')t,
t 2 <_t <_t~
dF2(t) m2f1202(0) ( - ~ -n(1 -- 1'~'~ - = cos + t > t 2 > t c. (20) dt e 2\ e},/ Thus after equating the rates of change for components of force, in the limit as t--~t 2 we obtain the expression to be solved for f~t2, 01(0) -- m2 I-1 --COS ~t2] + (2 COS~t2, /~v2(O) ml
v 1(0) _
_1)1(0) _ > p _ _ .m2 1)2(0) ml (21) The limits of applicability for these equations have been expressed in terms of the ratio of velocity components by noting that the transition time t2 = tc occurs ifvl(O)/v2(O) = lam2/ml. For intermediate angles of incidence the contact point sticks at time t 2 and then the tangential compliant element begins a period of s.h.m. During the period of sticking the tangential components of velocity and force are given by Eqn (7); i.e. Vl(t) = f.OU1(t2) sin og(t-- t2) q- vl(t2) cos og(t-- t2) (22a) --01(0) -/~1)2(0)
m2~l --cos(-~-t---n(1--1~x~l ~ (_~2 _ n ( l _ l ' ~ ' mlL 2\ eJJd + cos + 2\ e}}
Fl(t) = mlco2ul(t2) cos og(t-- t2)- mlcovl(t2) sin ~ ( t - t2).
(22b)
This period of stick terminates and slip begins again at time t a when the ratio of components of force next becomes as large as the coefficient of friction, [FI[/FE=lt. This gives a final phase of slip which begins at time ta--a time that can be obtained from
f~ul(t2)c°sc°(ta-t2)#o2(0) co/~v2(0)f~vl(t2----~) sin c o ( t 3 - t 2 ) = ( 2 s i n [ - ~ + 2(1 - ~) 1.
(23)
At time t 3 when sticking ceases, a second phase of slip begins. For this second phase of slip, the initial tangential components of velocity and force, vx(t3) and Fl(t3), are given by (22). The second phase of slip terminates at separation. During this final phase of slip the particle at the contact point is sliding as the elastic strain energy in the compliant elements is decreasing to zero. Thus changes in velocity during this phase are given by (9). The final tangential velocity of the contact point at time tf=(1 +e)to is given by
vl(tf) = vl(t3)--/zm~- 1[p(tf) -- p(t3) ]
(24)
where s = + 1 since the final direction of slip is opposite to the initial direction as illustrated in Fig. 4. The non-dimensional tangential velocity at separation is vl(tf) - v l ( t 3 ) ~-m ~ [ - I1- e - p(t3)l" /zv2(0) Pv2(0) p(to)l
(25)
Large angle of incidence v 1(0)/v2(0) > #(1 + e)m2/m 1 If initial slip does not cease before separation, the transition time t 2 > ( 1 +e)tc; i.e. slip continues in the initial direction throughout the entire contact period. This contact process is sometimes termed gross slip. For this case, at separation Eqn (9) gives the tangential v e l o c i t y vl(tf) as follows, vl(tf) : Vl(0 ) "4-/~mi- 1(1 +
e)p(tc)
(26)
where s = - 1 . This can be rewritten as vl(tf) _ Vx(0) m2(1 +e). /~v2(0) #v2(0) ml
(27)
For gross slip, the direction of sliding is constant--friction merely slows the speed of sliding.
444
W.J. STRONGE
FRICTION DURING OBLIQUE IMPACT OF A SPHERE Combined effects of friction and tangential compliance have been evaluated for oblique impact of a sphere on a half-space. For a solid sphere composed of material with Poisson's ratio v=0.3, the ratio of stiffness ( 2 = 1.21 while the mass ratio m2/mt=3.5 so that the ratio of frequencies og/D = 1.7. These values are used in the following examples of oblique impact of a sphere against a massive half-space.
Tangential oelocity at separation For a sphere striking a half-space at angle of incidence 0 where 0=tan-l(Vl(0)/v2(0)), Eqns (15), (25) and (27) were used to calculate the tangential relative velocity for contact point C at separation. In Fig. 5, the results from the present discrete parameter model are compared with the elasticity solution given by Maw et al. [6] and experimental measurements of Johnson [ 14] for a rubber sphere (Poisson's ratio v = 0.5) striking a heavy steel plate at a small speed. The elastic solution and the discrete parameter model each have similar processes that develop at the contact point in three parts of the range of angle of incidence. The predictions of these two models are most different for small and intermediate angles of incidence where the discrete parameter model has a final period of slip that is prolonged by elastic strain energy stored in the tangential compliant element. Throughout most of the range of small to intermediate angles of incidence, both the elastic continuum and the discrete parameter models of sphere impact have a tangential relative velocity at separation that is in the opposite direction to the incident tangential velocity. For a collinear collision this velocity reversal at C is entirely due to tangential compliance. In almost all respects the results of these models are practically identical. Figure 5 also illustrates final velocities calculated for negligible tangential compliance; in this case there is a nonzero terminal tangential velocity only if the angle of incidence is large; i.e. vl(O)/v2(O) > p(1 + e)m2/ml. The effect of coefficient of restitution e on the change in the tangential component of relative velocity at C is shown in Fig. 6. The angle of incidence for gross slip decreases with increasing internal dissipation. The coefficient of restitution e effects only the impulse imparted during restitution so that changes in velocity during restitution decrease with e. This causes the shift in the curve for separation velocity that is apparent in Fig. 6.
Angle of incidence for maximum friction Experiments using repeated impacts at oblique angles of incidence were performed by Ko [15] who showed that for relatively small normal impact speeds, o2 < 1 ms -1, wear rate of steel tubes is closely correlated with maximum tangential force Flma~ and that this force varies with angle of obliquity. For the present model, the tangential force Fl(t) can
initiol
stick
|erminol
slip
slip - stick - r e v e r s e slip
'
A
~
continuous tsp
D 4
•
1
~ _ . ~ .
. . . . 2 . . . . _~ . . . .
oo-\
-1
,o slip. ~
7,,,-
ye:10
~
?
v2'0' '
o
•
-2
~ . . . . 9,_ . . . .
r~,,~
ill
sphere
continuous
v = 0.31
-3
FIG. 5. Tangentialvelocityof contact point on elastic sphere at instant of separation as a function of the angle of incidence. Lumped parameter model - - , elastic continuum analysis - - - , analysis for negligibletangential compliance. . . . . , experimentswith rubber ball o o o.
Planar impact of rough compliant bodies
445
/
>~
S
.I_+.23
)/?
/
/
>
?
v~(O)/~v2101
/ -3 FIG. 6. Tangential velocity of contact point at separation as function of the angle of incidence of sphere for coefficients of restitution, e =0, 0.5 and 1.0.
be calculated as a function of the angle of incidence, tan- l(vl(O)/v2(O)). Irrespective of the angle of incidence, the largest value of friction Fxmax o c c u r s during the compression period if w/f2 > 1; consequently, the maximum tangential force is independent of the coefficient of restitution. For any impact speed S(0)-[v2(0)+ v2(0)] 1/2 the tangential component of force can be compared with the largest normal force F2m,x = F2(t¢) where F2max
f~m2S(0)
1 F1 +v~(0)/v~(0)] ~/2"
(28)
Expressions for the maximum tangential force are given below for different ranges of the angle of incidence. Small angles of incidence v~(0)/v2(0)_<#~ 2. For small angles of obliquity the contact point initially sticks and only begins to slide during the restitution period. The maximum tangential force occurs during compression when the tangential velocity reverses in direction at time z =-~t¢/o). At this time the contact is sticking so the maximum tangential force an be obtained from Eqn (7c).
F l m.x = -- wm 1v 1(0).
(29)
This maximum tangential component of force can be expressed as a nondimensional ratio defined as IFlma~l/m2f~S(O). Thus for small angles of obliquity Flm._____.~_ wmllv~(0)l _ 1 m/~
va(0)/v2(0)
(30)
nm2S(0) nm~S(0) ~/m2 [l+v~(0)/v~,(0)]"2" Intermediate angles of incidence #~2 < v l(0)/Vz(0) _<#(1 + e)mz/m 1. At intermediate angles of obliquity there is initial sliding but then stick begins at time tz during the compression period. When stick begins the contact point is still moving in the initial direction; i.e. vx(tz)>0. Maximum friction develops shortly after the period of stick begins and before the instant of maximum compression. The friction force during sticking can be expressed as F~(t) . ~"ul(t2)cosw(t_t2)+Q m ~ ~ , s i n o ) ( t _ t 2 ) } . f~m2S(0) ~2S(0)/./xv2(0) V mz/xv2tu)
(31)
At the transition from sliding to stick the displacement ul(tz) and velocity vl(tz) depend on the coefficient of friction. The transition velocity vl(t2) is obtained from (9) while the displacement ux(t2) is calculated from the friction law F t = -IxsF 2 and the force Fz given in Table 1. Thus ff]Ul(t2) = ( 2 sin ~ t 2 ,
/w2(O)
Vl(t2) -- vl(O)
m2(1 --COS ~t2).
(32)
/xv2(O) #v2(O) mi
Large angles of incidence vl(O)/v2(O)>/x(1 + e)m2/mi.
If the direction of slip is constant
446
W.J.
STRONGE
throughout the collision period then the maximum friction force is directly proportional to the normal force and the coefficient of friction,
Flmax /.t f~mzS(O ) - [ l + v~(O)/vZ~(O)]'/2"
(33)
For gross sliding this maximum tangential force occurs simultaneously with the largest normal force; i.e. tangential force is a maximum at time t c when the compression period terminates. These expressions for the maximum values of components of contact force have been used to calculate the largest normal and tangential forces which occur during oblique collisions. These largest values vary with the angle of incidence and the coefficient of friction as shown in Fig. 7. The largest values for the peak force occur for collisions at intermediate angles of incidence. The angle of incidence where the peak force is largest increases from about 20 ° for a coefficient of friction p = 0.1 to almost 60 ° for # = 1.0. Maximum friction for negligible tangential compliance. If tangential compliance is negligible, oblique collision always results in an initial period of sliding; this sliding is halted before separation and there is a successive final period of stick unless the angle of incidence is large enough to cause gross slip. Since we are considering central or collinear collisions, slip reversal does not occur. Consequently, if initial slip comes to a halt during compression, the peak tangential force occurs at the instant t s when sliding terminates. /2 sin f~ts [1 + v~(O)/v~(O)] 1/2
Flmax
f~m2S(O)
fh, =cos-
where
'F,L -
(34)
~\~v--~0)/l"
On the other hand, if the bodies are still sliding when compression terminates at time tc then the largest friction force occurs at this instant simultaneously with the largest normal force.
Flmax f~m2S(0--~)- [1
+
/2 v~(O)/v~(O)] ,/z"
(35)
In Fig. 7 the dashed lines show the maximum friction force for a compliant solid sphere whereas the light extensions to the left of these curves are results for similar collisions between spheres with negligible tangential compliance. For solid spheres, the largest
0 1.0
0.2
0.4
I
I
v~(0)/v2(0) 0.6 1.0 i
i
2.0
4.0
I
I
1.0
~o.8 O.6 i
0.8 ~
i 0 I~
•
0.t.
E o 0
~)
0 E 30* 60 ° angle of incidence, 0
9~
FIG.7. Maximumnormal force~ and tangential force----- during oblique impact of sphere. The light curves - - are max. tangential force if tangential compliance is negligible. In the crosshatched area there is gross slip.
Planar impact of rough compliant bodies
80
447 t,0
n ~
g =0.18
,,. 60-
o
z
E 30 C w"
~,~
,9.o -~ 4 0 -
20 c o
~ ~o
10 ox E
0 ~"
0
t
i
i
t
30* 60* ongle of incidence. O
t
~,,lO
90 °
FIG. 8. Comparison of the m a x i m u m normal and tangential components of contact force with experimental measurements by Lewis and Rogers (1988). At each angle of incidence, the normal and friction forces on a 101 g sphere colliding against a half-space were calculated for an incident speed 0.048 ms-~ and coefficient of friction/~=0.18.
tangential force for a compliant body is substantially less than the largest force calculated with the assumption of negligible tangential compliance. In either case the largest tangential force occurs in the range of small to intermediate angles of incidence. At large angles of incidence there is gross slip; in this case the maximum tangential force is independent of tangential compliance. Comparison with measurement of peak force during oblique impact. Lewis and Rogers [11] performed impact experiments at which a 25.4 mm diameter steel sphere collided against a heavy steel plate at angles of incidence that varied between 0 ° and 85 ° from normal. The impact speeds were small, being in the range 0.01-0.05 ms- ~. The sphere was attached at the free end of a 1.8 m long pendulum by a steel "ball holder". Piezoelectric force transducers were used to make separate measurements of normal and tangential components of contact force during impact. For gross slip or continuous sliding of this ball on the plate, Lewis and Rogers reported a coefficient of dynamic friction,/a=0.179. In Fig. 8 experimental data taken from collisions at an impact speed of 0.048 ms-1 are compared with normal and tangential components of force calculated by the present theory (using a coefficient of friction # = 0.18). The calculations depend on an estimate of the mass of the "ball holder". The agreement between experiment and theory shown in Fig, 8 was achieved by increasing the mass of the ball by 50% in order to account for inertia of the support system. In addition the calculations used a relative compliance ratio ~2 = 1.21 and mass ratio m2/m~=3.5 that are representative of solid spheres. This resulted in both qualitative and quantitative agreement between the calculations and the experiments for the full range of possible angles of incidence. Four series of tests using different impact speeds each gave a largest measurement of tangential force at an angle of incidence of about 40 °. For p=0.18 the present lumped parameter model gives a largest value of peak tangential force at about 35 ° irrespective of impact speed. (For any angle of incidence, the ratio between peak tangential and normal components of force can only be as large as the coefficient of friction if these peak values occur simultaneously; i.e. if Ol(O)/l)2(O)>_llm2/m 1 or the angle of incidence 0 > 32 ° for a rough solid sphere with/~ = 0.18.)
Dissipation of energy Internal dissipationfrom hysteresis of normalforce. During partly elastic collisions (e < 1) there is always irreversible internal deformation that dissipates a part of the initial kinetic energy K o where K o = M[v~(0) + o~(0)- 2x2v~(O)¢,(O)+ (x~ +
K2)I~/2(0)-]
and ~b(0) is the rate of rotation of the body at incidence. In the present model this internal
448
W.J. STRONGE
d i s s i p a t i o n D 2 is entirely due to hysteresis of the normal component of force. It can be
obtained as the negative of work done by this component of force, D2(tf)=-
f~'F2v2
(36)
d t = ( 1 - - e 2 ) m2v2(O)
since in collinear collisions m 2 = M, the part of the initial kinetic energy that is dissipated internally by irreversible deformations can be expressed as 2D2(t¢___~)=(1 -e2).
(37)
MrS(O) Frictional dissipation. Friction dissipates energy only during periods of slip. During these periods the tangential force does some work that changes the tangential strain energy and also some D1 that is dissipated by friction. Whereas the total work done on the body by the tangential force depends on the sliding speed vl + t~l, the frictional energy loss depends only on the tangential speed of the body yr. The remainder of the work done by the tangential force is stored as elastic strain energy in the tangential compliant element and later recovered before separation. Thus for small angles of incidence where the contact point slides only after an initial period of sticking, the dissipation due to friction is calculated from*
Dt(tf)= -
Fly 1 dt= - /
~Vl(tl)+
[p(t)-p(q)]
IJ dp
(38)
Using Eqns (12) and (14) this gives 2o,(t,) _
v,(0)
m2v22(0)
2p2gt (/~v2(0) cos coq
+
2-m-T1J"
(39)
On the other hand, if the angle of incidence is moderate, the contact point slides prior to t i m e t 2 and again slides after time t3. Thus for t2 < t:,
2D,(tf) _/~2m 1f v~(0)
m2v~(0)
v2(t2)
v~(t3)
m2 (~2v~2(0) 1,2v~(0) ~ ~2v~(0)
v12(t4)r"(
~2v,~(0)J"
(40)
Finally, if the angle of incidence is large so that there is gross sliding, the part of the energy dissipation due to friction can be expressed as 2D 1 (tf) + e)'~. rn2v22(0)-2/~2(l+e)~"v'(0)(pv2(0) 2 ~ ( 1 j
(41)
Equations (39)-(41) were used to evaluate the part of the initial kinetic energy that is dissipated by friction 2Dl(tt)/m2S2(O). In Fig. 9 this frictional dissipation is illustrated as a function of angle of incidence for coefficients of friction/.t=0.1 and 0.5. In contact regions with tangential compliance there is almost no frictional dissipation if the angle of incidence is small. If there is gross slip due to a large angle of incidence, tangential compliance has no effect on frictional dissipation. The fraction of the initial kinetic energy which is dissipated by friction is maximum at an angle of incidence slightly larger than the smallest angle giving gross slip. These results are based on the supposition that coefficient 'of friction is a parameter that is constant during impact. For impact speeds S(0)> 50 ms-1 where indentation of metals results from uncontained plastic deformation, Sundararajan [16] has pointed out that tangential force can be increased by indentation and decreased by frictional heating. * During any period t b- t. with a constant direction of sliding, the part of the total energydissipation Di(tb)--D,(t.) due to a component of contact force F t can be calculated from a theorem of Stronge [17]. Di(tb)--Di(ta)=[vi(tb)+v,(t.)][.pl(tb)--pj(t.)]/2. This gives Dl(tb)--Di(t.)= rat[v~(tb)--u~(t.)]/2 for a collinear impact configuration if the direction of sliding is constant.
Planar impact of rough compliant bodies
0
I
03-
vl(O) / v2(O) 0.6 1.0 2.0
0.20..t. I
I
I
449
t..O I ,,.,,P ~ I'fl 1
to/£ =1.7
o
% 0.2
* / d 0.1
0
.......
.
b.~
I I 30* 60* angle of inctdence. O
go °
FIG. 9. Dissipation of initial kinetic energy by friction during ,mpact of sphere as a function of the angle of incidence for coefficients of restitution e = 1 - - and e =0.5 - - - . If tangential compliance is negligible.., and friction is large enough to halt initial slip before separation, frictional dissipation is independent of coefficients of friction and restitution. For gross slip however, frictional dissipation does not depend on tangential compliance.
CONCLUSIONS
For almost all angles of incidence, the response of this simple lumped parameter model is identical with that of the quasistatic (Hertz) elastic analysis. The microslip present in the continuum analysis has no significant effect on changes in velocity of the colliding bodies. Both the elastic continuum analysis and the lumped parameter model show that if slip is brought to a halt during collision, tangential compliance can subsequently reverse the direction of slip. Slip can be brought to a halt during collision only if the tangential component of incident velocity is not too large; i.e. if o~(0)/Oz(0)
450
W.J. STRONGE REFERENCES
1. W. GOLDSMITH,Impact: the Theory and Physical Behaviour of Colliding Solids. Edward Arnold (1960). 2. W. J. STRONGE,Rigid body collisions with friction. Proceedings of The Royal Society, London A431, 169-181 (1990). 3. R. M. BRACH,Classical planar impact theory and the tip impact of a slender rod. Int. J. Impact Engng 13, 21-33 (1993). 4. J. B. KELLER, Impact with friction. ASME J. Appl. Mech. 53, 1-4 (1986). 5. W. J. STRONGE,Swerve during 3-dimensional impact of rough bodies. ASME J. Appl. Mech. 61, accepted for publication (1994). 6. N. MAW, J. R. BARBERand J. N. FAWCETT,The oblique impact of elastic spheres. Wear 38, 101-114 (1976). 7. N. MAW, J. R. BARBERand J. N. FAWCE'rr, The role of elastic tangential compliance in oblique impact. ASME J. Lub. Technol. 103(74), 74-80 (1981). 8. A. BILBAO,J. CAMPOSand C. BASTERO,On the planar impact of an elastic body with a rough surface. Int. J. Mech. Engng Education 17, 205-210 (1989). 9. C. E. SMITH, Predicting rebounds using rigid-body dynamics. ASME J. Appl. Mech. 58, 754-758 (1991). 10. R. M. BRACH,Tangential restitution in collisions. Computational Techniques for Contact Impact, Penetration and Perforation of Solids. ASME AMD 103 (eds L. E. Schwer, N. J. Salamon and W. K. Liu) pp. 1-7 (1989). 11. A. D. LEWISand R. J. ROGERS,Experimental and numerical study of forces during oblique impact. J. Sound Vibration 125(3), 403-412 0988). 12. R. D. MINDLIN and H. DERESIEWlCZ,Elastic spheres in contact under varying oblique forces. ASME J. Applied Mechanics 75, 27-344 (1953). 13. K. L. JOHNSON, Contact Mechanics, Cambridge University Press, Cambridge (1985). 14. K. L. JOHNSON,The bounce of'superball'. Int. J. Mech. Engng Education 111, 57-63 (1983). 15. P. L. Ko, The significance of shear and normal force components on tube wear due to fretting and periodic impacting. Wear 106, 261-281 (1985). 16. G. SUNDARARAJAN,The energy absorbed during the oblique impact of a hard ball against ductile target materials. Int. J. Impact Engng 9, 343-358 (1990). 17. W. J. STRONGE, Energy dissipated in planar collision. ASME J. Appl. Mech. 59, 681-682 (1992).
Int. J. Impact Engng Vol. 15, No. 4, pp. 435-450, 1994 Elsevier Science Ltd Printed in Great Britain 0734-743X(94)E0013-L 0734-743X/94 $7.00 + 0.00
Pergamon
PLANAR
IMPACT
OF
ROUGH
COMPLIANT
BODIES
W. J. STRONGE Department of Engineering, University of Cambridge, Cambridge CB2 1PZ, U.K. (Received 23 November 1993; and in revised form 31 January 1994)
Summary--A lumped parameter model of contact between colliding bodies is used to obtain the tangential force and energy dissipated by friction during oblique impact of rough compliant bodies in collinear configurations. For Coulomb's law of friction and an energetic coefficient of restitution, this analysis distinguishes between angles of incidence where the contact point initially sticks, slides before sticking or slides throughout the contact period. During part of the contact period tangential compliance significantly alters friction; this affects changes in relative velocity unless the angle of incidence is so large that there is continuous sliding in the initial direction. For oblique impact, tangential compliance reduces the largest friction force in comparison with friction generated if the contact region has negligible tangential compliance. While the part of the initial kinetic energy dissipated by friction is limited by the coefficient of restitution, this coefficient has no effect on the largest friction force generated during collision.
INTRODUCTION
Collisions between hard bodies result in changes in relative velocity in a very brief period of time. These changes in velocity are caused by large contact forces; e.g. a typical hammer blow applies a force of several tons. Nevertheless, if the bodies are hard and the contacting surfaces are nonconforming, the area of contact between the bodies remains small in comparison with the size of either body. For hard bodies, the region of significant deformations is limited to the immediate vicinity of the contact area despite the large contact force; this is because stresses (and strains) decrease rapidly with increasing radius from the initial contact point (roughly as r -2 in an elastic solid if the cross-section is much larger than the contact area). Consequently, collisions between hard and compact bodies ordinarily can be assumed to occur impulsively; i.e. we assume the changes in velocity occur instantaneously so there is no movement of colliding bodies while they are in contact. For oblique impact between rough bodies there are both normal and tangential components of contact force; the tangential force (friction) opposes the tangential relative velocity which is termed sliding or slip. Effects of dry friction on changes in velocity during impact can be obtained from Coulomb's law only by summation of changes during successive stages of unidirectional slip 1-1-3]. If the direction of slip is continually changing a dynamic analysis that incorporates effects of dry friction during sliding can be achieved if the impact process is resolved as a function of the normal component of impulse [4, 5]. In the past such analyses have assumed however that tangential compliance of the bodies is negligible in the contact region. While this assumption is an asymptotic limit that represents large initial slip, it may not be accurate for small initial slip where the direction of slip can change during contact. Maw et al. 1-6, 7"] performed a dynamic analysis of oblique impact between rough elastic spheres using Hertz contact theory to obtain the normal tractions in the contact area. They found that at small sliding speeds the contact area had an outer annulus which was sliding. The annulus surrounded a central area where there was no tangential component of relative velocity; i.e. the central area was sticking. This combined state of slip and stick was termed microslip. Both the analysis and experiments of Maw et al. showed that if initial slip was small, the direction of slip could be reversed during collision by tangential compliance. With negligible tangential compliance such reversals are not possible if the impact configuration is collinear (i.e. if the centres of mass of the colliding bodies are on the common normal line passing through the contact point). 435
436
W.J. STRONGE
There have been several attempts to develop approximations which produce the effect of slip reversal for small angles of incidence in collinear as well as noncollinear collisions. Bilbao et al. I-8] define a kinematic tangential coefficient of restitution which varies exponentially with the coefficient of friction and the ratio of normal to tangential components of incident velocity. Smith [9] defined a kinetic coefficient of restitution that relates the tangential impulse to the coefficient of friction, normal impulse and an average velocity of sliding during collision. Brach [I0] used two linear relations for changes in tangential velocity; these employed a kinematic coefficient of restitution (negative) at very small angles of incidence and a kinetic coefficient of restitution at larger angles. All of these approaches were designed to produce at large angles of incidence a ratio of tangential to normal impulse equal to the coefficient of friction; they represent Coulomb's law of friction only in the limit as sliding becomes continuous in the initial direction. In contrast to the elastic continuum approach of Maw et al. [6] and the approximations above, the present paper uses a lumped parameter representation for compliance of the contact region. Equations of motion are developed from a few physical laws while system characteristics are expressed in terms of coefficients that are independent of the angle of incidence. This model yields either slip or stick at the contact point depending on whether the ratio of tangential to normal contact f o r c e is as large as the coefficient of friction. During stick the tangential force depends on relative displacement so a time dependent analysis is required to resolve the changes in velocity that occur in a collision; nevertheless we assume that the total period of contact is so brief that there is no change in configuration during collision. Comparison reveals that the present lumped parameter modelling gives velocity changes that are almost the same as those in experiments by Maw et al. (1981) and contact forces that are similar to measurements by Lewis and Rogers rl 1]. In addition the present analysis yields results for a model of inelastic collisions with tangential compliance. DYNAMICS OF PLANAR COLLISION FOR HARD BODIES To focus on the effects of tangential compliance during collision, consider a body with mass M that collides against a half-space at contact point C as shown in Fig. 1. At C the body and the half-space have a common tangent plane. Let unit vectors n~ and n2 be oriented in directions tangent and normal to this plane respectively. At incidence (time t =0) a point on each body comes into contact; at incidence these points have a relative velocity v~(0) with tangential and normal components, v~(0) and 02(0), respectively. The orientation of the coordinate system is defined such that at incidence both normal and tangential components of relative velocity are negative, o1(0)< 0 and ~2(0)< 0. The collision period is separated into an initial period of normal compression and a subsequent period
x--~t~:
~
x. . . . . ~:
/
,-,4.;., "~ I I ~.'~~b.?.i:ii:~i:~.:!.~:i~:~i:i~::.'~;"
n
~i:~ii~!~::.'.:i~ii~.~' , i..':~" ~
,~,-~
•
t
"-' $.~:.:.-~'i'~::"
FIG. 1. Discreteparameter model for impact of rough, compliant body on half-space.
Planar impact of rough compliant bodies
437
of separation. The compression period terminates at time t c when the normal component of relative velocity vanishes, v2(tc)=0. The contact points separate at time tt when the final relative velocity has a normal component v2(tf), where v2(tf)>0. Hence during compression, kinetic energy is absorbed by deformation of the bodies while during restitution, elastic strain energy generates the force that drives the bodies apart and restores some of the kinetic energy that was absorbed during compression. To simplify the dynamic analysis we assume that both bodies are rigid except for an infinitesimally small deformable region that separates the bodies at the contact point. Let the body with mass M have a radius of gyration k about the centre of mass. From the centre of mass of the colliding body the coordinates of C are xl in directions ni, i = 1, 2. Thus the body has an inertia matrix for C with an inverse mi~ 1 where
m-Z1 L_xlx2/Mk 2
mi~l=[
ml = Mk2/(k2 + x2),
-xxx2/Mk21 m2 1
d
m2= Mk2/(k2-1- x2).
(1) (2)
Notice that this matrix has terms off the principal diagonal that all vanish if the collision configuration is collinear. We model the infinitesimal deforming region around C by assuming that the colliding body is connected to a massless particle located at C as illustrated in Fig. 1. The connection is via two independent compliant elements--one tangential and one normal to the tangent plane. This particle has a tangential component of displacement u~(t) relative to the body that depends on the tangential force at C and the tangential compliance. Contact force F~ acts at the contact point; it has components in directions n~. This force applies a differential impulse dpl = Fi dt in an increment of time dr. Thus the equations of planar motion for the rigid body can be expressed as (see Stronge, 1994) do i = mi-~ 1 dp r
(3)
To progress further we must be specific about the compliance in the contact region so that contact force can be calculated. The components of this force are necessary in order to distinguish between periods of slip and stick at C. In order to calculate the changes in relative velocity that occur while the contact sticks, it is necessary first to obtain the components of contact force.
Linear compliance model Here we assume that both normal and tangential compliant elements are piecewise linear. During compression let the normal element have an arbitrary stiffness x while the tangential element has stiffness x/( 2 as shown in Fig. 2. The parameter (2 is a ratio of normal to tangential stiffness at the contact point; this ratio depends on the structure of the colliding
F1
0
(1-e2lu2(lc)
u2(t e)
u,
FIG. 2. Stiffnessof normal and tangential compliant elementsduring loading and unloading.The energydissipated by irreversibleinternal deformationsis proportional to the ratio of the area under the curve during unloading to this area during loading.
438
W.J. STRONGE
body in the deforming region. These spring constants and the inertia matrix mij give two natural frequencies, co and ~ where
co2,f,~2= m~1 {( 2
(2
m1~2~ ./(1
ml(2~ 2
1+ m2 / +_ ~1\ - - - m2 / +
4m~(4x2x2~ x 2 M2k4 j.
For a collinear collision and linear stiffness, the colliding body undergoes independent simple harmonic motions (s.h.m.) in the normal and tangential directions while the contact point sticks. These motions have frequencies that depend on the stiffness and the components mi of the inverse of mass matrix that were given in (2). The normal frequency f~ and tangential frequency co are defined as f~-~2
n =-f~,
co=
~/ x _ n__n___ m/~2 ~2ml 2 ( t ~ m l
(4,
where time t c is the instant when the compression period terminates. At this instant the normal compliant element has a maximum compression u2(tc).
Coefficient of restitution The coefficient of restitution, e, can be defined as the square root of the ratio of elastic strain energy released at the contact point during restitution to the kinetic energy absorbed by internal deformation during compression. For negligible tangential compliance, the loss of kinetic energy due to irreversible internal deformations in the contact region can be obtained from the work done on the bodies by the normal component of contact force. Stronge (1990) has defined the energetic coefficient of restitution, e, according to this work. For negligible tangential compliance, this is the only part of the work done on the bodies that goes into deformation. Consequently, it is the only part that can be stored as elastic strain energy if tangential compliance is negligible--the work done by the tangential component of force is all dissipated by friction. If tangential compliance is not negligible, however, there is also energy absorbed by tangential deformations u 1. Here I assume that this deformation is entirely elastic. This assumption is based on considering the coefficient of restitution as representing nonfrictional energy losses that are due primarily to plastic deformation. In an initial range of elastic-plastic deformation the region of plasticity is contained beneath the surface of a deforming body; this contained or subsurface plasticity has very little on tangential compliance. Hence for elastic-plastic bodies which collide at low speeds, Stronge's definition of the energetic coefficient of restitution still applies, i.e. -- ~t tt g2v 2 dt e: =
°
(5)
0cF2v2 dt With linear compliance the normal component of force does work W2(tc) on the bodies during compression; this is simply the area under the normal compression line in Fig. 2, W2(tc)=F2(tc)u2(tc)/2. In the present model the effect of coefficient of restitution (5) is obtained by changing the stiffness of the normal compliant element at the transition time t~ when compression terminates. Hence at the instant of maximum compression t~ the stiffness of the normal element increases from x to x/e2~ For changes in the normal component of relative velocity, this change in stiffness makes the frequency of s.h.m, larger during restitution than it was during compression. The collision terminates and separation occurs at a final time t6 for collinear collisions tf=(1 +e)tc. At separation the normal component of relative displacement has a terminal value u2(tf) ----(1 --e2)u2(tc).
Normal components of velocity and force in collinear collision Henceforth in this paper the analysis is limited to coUinear collisions so that normal and tangential equations of motion (3) are decoupled. For a collinear collision the normal
P l a n a r i m p a c t o f r o u g h c o m p l i a n t bodies
439
force (and hence any change in the normal component of relative velocity) is independent of the process of slip or stick at the contact point. For a linear compliant element, the colliding bodies undergo separate stages of s.h.m, during compression and restitution periods of the collision. Thus at any time during the collision, the normal component of relative velocity is as follows:
v2(t) = v2(0) cos f~t v2(t) = evz(0) cos
+ 2\
eJJ
0 < t < tc
(6a)
t¢ < t < tr
(6b)
This normal component of velocity is continuous at the time of maximum compression t c when the frequency of s.h.m, increases from ~ to f~/e. The normal component of impulse that causes these changes in velocity can be obtained directly, and this impulse can be differentiated to obtain the normal component of contact force. The expressions for these variables are listed in Table 1.
Tangential velocity and force during stick During the period in which the contact point sticks, s.h.m, applies also to tangential changes in velocity if the tangential compliant element is linear. The tangential oscillations during stick occur with frequency o9. It is convenient to express the velocity and force during stick in terms of an initial displacement ua(T) and an initial velocity v~(z) at time t = r when stick begins. While the contact point sticks there is no sliding, v~(t)+ ill(t)= 0 where ti 1-dua/dt; thus tangential displacement, velocity and the contact force on C can be written as
ul(t) = ul(z) cos w ( t - r ) - l v l ( z ) sin og(t- z)
(7a)
O9
vl(0 = ogul(z) sin co(t- r) + vl(r) cos co(t- z)
(7b)
Fl(t)=mlo92ul(r) COS og(t--z)--miogvi(z) sin co(t--z),
t>z.
(7c)
The state of stick persists while the ratio between tangential and normal components of contact force satisfies IFll/F2 <1~.
Sliding contact with Coulomb friction For dry friction that can be represented by Coulomb's law, sliding occurs if the ratio between the tangential and normal components of force is equal to the coefficient of friction #. Thus sliding depends on the tangential compliance and relative displacement u l(t ) between the particle at C and the body; if there is sliding contact, the particle is sliding at velocity vl(t)+ til(t ). On the other hand, if IFiI/F 2 < # then the contact is sticking so vl(t)= -til(t).
TABLE 1.
NORMAL DISPLACEMENT, VELOCITY, FORCE AND IMPULSE DURING COLLISION Compression, 0 < t < tc
Disp.
u~(t)= (v2(O)/[l)sin fit
Velocity
v2(t)= v~(O)cos fit
Force
F2(t) = -- m2flv2(O) sin fit > 0
Impulse
p(t)= - m2v2(O)[1--cos lit] _>0
R e s t i t u t i o n , t e < t < tf
e2sin( + (, !))] v2,,ev2,0,cos<+(1
440
W.J. STRONGE
For simplicity, coefficients of static and dynamic friction are assumed to be equal. While the contact slides in direction s = sgn(vl + riO, the contact force applies a differential impulse
dpj= { - i S } dp
(8)
where d p - @2. Consequently, for sliding the differential Eqn (3) can be expressed in terms of a new independent variable--the normal component of impulse p rather than time,
~dv,/dp~=[m;' [dv2/dpJ
m°~ ' l [
1 J"
(9)
Initial stick or slip? With the present model, the contact point C either sticks or slips depending on the ratio between the components of contact force; i.e. whether or not the contact force is inside the cone of friction.* First suppose that stick begins at the initial instant of contact and then test whether this satisfies the limiting force ratio. An initial period of stick terminates at a time tl when the ratio of tangential to normal force first becomes as large as/~; i.e. time tl is obtained from IF1[ 1 v1(0) f~sin cot 1 . . . . /.tF 2 (2 #V2(0) COsin t2t~
IFll /'tF2
_ 1 o1(0) f~
1,
0_
sin cot I
(2 kw2(0) co s i n ( ~
= 1,
(10a)
to
(10b)
+ 2(1 _ ~) )
The process of initial stick takes place if t 1 >0; i.e. if in the limit a s t I ---~0 the force ratio is inside the cone of friction. This requires an angle of incidence such that vl(0)
(11)
02(0) Thus initial stick occurs if the angle of incidence is small; i.e. if at incidence, the ratio of the tangential to normal component of relative velocity at C is within a range 0 < v 1(0)/v2(0) < p~2 that is bounded by friction and the ratio of normal to tangential stiffness. For collisions with an angle of incidence at C which is larger in magnitude than t a n - 1(p¢2), the contact point begins sliding when contact initiates. SLIP PROCESS
Small anole of incidence v l(0)/v2(0)
vl(tl) = vl(0) costotl.
(12)
Thus from (11) the terminal velocity can be obtained vt(tf) =
vx(tx)--I~S[p(te)--p(t,)] ml
(13)
* This model givesa period of initial stick if the incident tangential velocityis small--initial stick occurs because the particle at C has negligiblemass. Mindlin and Deresiewicz1-12]have shown that for elastic bodies, any finite tangential velocity results in central stick plus a peripheral annulus of slip around the small contact patch during an early stage of collision.
Planar impact of rough compliant bodies
441
slip (e :1.0) I = F "1
st,ck 1.0-/f~ m2v2(tc)/ / ~/,/ 0.5
/
/
X
\
~.--_
F~/tl~m2v2(t~)
\\
(al "
1.o ~
'/o
\
-o.s
*1.0
/
\
-1.0
t/t~
\
~1 i
\
\
"/
~I
\
/
,,,~.i i
CI
/
/
compression-.,P---~ resti'fulion _
v1 (t~,, IJ'v2(O)
(b)
1.0
1.5
2.0 ~/i=
-1.0
FIG. 3. (a) Normal and tangential forces and (b) tangential velocity during collision for small angle of incidence vl(O)/v2(O) <#(2 and frequency ratio o•/D= 1.7. Forces are illustrated for both e=0.5 and 1.0. For each coefficient of restitution, the final velocity at separation is indicated by an X. where the change in normal impulse during the final period of slip is given by p(tf) - p(t 1) = - # I m 202(0) •
(14)
The impulse ratio during slip gl is the ratio of the normal impulse applied during period t c - t t to the normal impulse during compression, gl =e{1 + c o s ( e - l f 2 t 1 +(n/2)(l - e - 1 ) ) } ; this impulse ratio depends on initial conditions. In the subsequent analysis we consider co/fl > 1 since this applies to most elastic bodies*. F o r this case slip begins during restitution at a time tl > t c when u l ( q ) > 0 so that s = + 1 and the terminal velocity can be expressed as Vl(tf) -- 01(tl) Jr m2g 1 #xo2(0) #xo2(0) m~
(15)
Intermediate angle of incidence #~2 < vl(0)/v2(0 ) < Ix(1 + e)m2/ml If the angle of incidence is larger than t a n - ~(#x~2), initially there is sliding at the contact point as illustrated in Fig. 4. D u r i n g the initial period of sliding the b o d y has a tangential c o m p o n e n t of velocity at C that is given by
vl(t ) = vl(O ) - lZSp(t). ml
(16)
* For elastic spheres with Poisson's ratio v=0.3 the ratio of stiffnesses (2 = 1.21 and elements of the inverse of mass matrix give m2/ml =3.5 so that w/D= 1.7 (e.g. see Johnson [15]) whereas for incompressible elastic spheres, (2= 1.5 and oJ/f~= 1.53. Furthermore, in the asymptotic limit of negligible tangential compliance (2--*0 so that O.)'--+ 0 0 .
442
W.J. STRONGE
slip ,
stick
~'"
1.0
_ IS}ip(e=l'O)l_ -II
"~"-- ~'-- "~" ~-"1 I"-slip (e=0.5,
- -
F'21/~m2v2 { 0 ) /
0-5 •
''
/~
/
',
',., "¢.o
\
/
\t~(e=0.5) \ \" tc ~' t~le=l} \
/
0
.,l
,~/)J~m2v2 (0 } \
1'0
t~'l¢
V1"5
X
-0.5
--/2"0
,\
~t c
/
~,~//
\ \ ..
-1.0
i1
compression :
',
= restitution
Ibl
-1.0
LL t2/t¢
I
~
' I '
1.0
I
1-5
2.0 ~"tc
-1.0 FIG. 4. (a) Normal and tangential forces and (b) tangential velocity during collision for intermediate angle of incidence ,u(2
This initial sliding terminates and slip begins at time t 2 when subsequent sliding or stick have the same rate of change for the tangential force; i.e. the transition to stick occurs when l i m r dFl(t2-I-z) dF2(t2-t-z!1 T-,o El dz =# dz "
(17)
For the period t > t2, Eqn (7c) gives dFl(t) = dt
mla~aUl(t2)sin ~ ( t -
where the dynamics for sliding during t < ul(t2) =
(2Fl(t2) K
t2
give transition values at the beginning of stick,
m2f~02(0)
= # - mlo) 2
(18a)
sin ['It 2
vl(t2)=vl(O)--Ilm2V2(O)[1--COS ~)t21, ml x
cos og(t-- t2)
t2)--mlo92ol(t2)
=# ~
ml~
Vl(t2)=vl(O)--#--v2(O)
sin/--+ \ e
(18b)
t2 <_tc
2\
1--ecos __2+~\
(19a)
e,/J
t2>te.
(19b)
Planar impact of rough compliant bodies
443
The normal force during restitution is given in Table 1; hence by differentiation we obtain dF2(t-~)-dt
m2['~21)2(0)cos ~')t,
t 2 <_t <_t~
dF2(t) m2f1202(0) ( - ~ -n(1 -- 1'~'~ - = cos + t > t 2 > t c. (20) dt e 2\ e},/ Thus after equating the rates of change for components of force, in the limit as t--~t 2 we obtain the expression to be solved for f~t2, 01(0) -- m2 I-1 --COS ~t2] + (2 COS~t2, /~v2(O) ml
v 1(0) _
_1)1(0) _ > p _ _ .m2 1)2(0) ml (21) The limits of applicability for these equations have been expressed in terms of the ratio of velocity components by noting that the transition time t2 = tc occurs ifvl(O)/v2(O) = lam2/ml. For intermediate angles of incidence the contact point sticks at time t 2 and then the tangential compliant element begins a period of s.h.m. During the period of sticking the tangential components of velocity and force are given by Eqn (7); i.e. Vl(t) = f.OU1(t2) sin og(t-- t2) q- vl(t2) cos og(t-- t2) (22a) --01(0) -/~1)2(0)
m2~l --cos(-~-t---n(1--1~x~l ~ (_~2 _ n ( l _ l ' ~ ' mlL 2\ eJJd + cos + 2\ e}}
Fl(t) = mlco2ul(t2) cos og(t-- t2)- mlcovl(t2) sin ~ ( t - t2).
(22b)
This period of stick terminates and slip begins again at time t a when the ratio of components of force next becomes as large as the coefficient of friction, [FI[/FE=lt. This gives a final phase of slip which begins at time ta--a time that can be obtained from
f~ul(t2)c°sc°(ta-t2)#o2(0) co/~v2(0)f~vl(t2----~) sin c o ( t 3 - t 2 ) = ( 2 s i n [ - ~ + 2(1 - ~) 1.
(23)
At time t 3 when sticking ceases, a second phase of slip begins. For this second phase of slip, the initial tangential components of velocity and force, vx(t3) and Fl(t3), are given by (22). The second phase of slip terminates at separation. During this final phase of slip the particle at the contact point is sliding as the elastic strain energy in the compliant elements is decreasing to zero. Thus changes in velocity during this phase are given by (9). The final tangential velocity of the contact point at time tf=(1 +e)to is given by
vl(tf) = vl(t3)--/zm~- 1[p(tf) -- p(t3) ]
(24)
where s = + 1 since the final direction of slip is opposite to the initial direction as illustrated in Fig. 4. The non-dimensional tangential velocity at separation is vl(tf) - v l ( t 3 ) ~-m ~ [ - I1- e - p(t3)l" /zv2(0) Pv2(0) p(to)l
(25)
Large angle of incidence v 1(0)/v2(0) > #(1 + e)m2/m 1 If initial slip does not cease before separation, the transition time t 2 > ( 1 +e)tc; i.e. slip continues in the initial direction throughout the entire contact period. This contact process is sometimes termed gross slip. For this case, at separation Eqn (9) gives the tangential v e l o c i t y vl(tf) as follows, vl(tf) : Vl(0 ) "4-/~mi- 1(1 +
e)p(tc)
(26)
where s = - 1 . This can be rewritten as vl(tf) _ Vx(0) m2(1 +e). /~v2(0) #v2(0) ml
(27)
For gross slip, the direction of sliding is constant--friction merely slows the speed of sliding.
444
W.J. STRONGE
FRICTION DURING OBLIQUE IMPACT OF A SPHERE Combined effects of friction and tangential compliance have been evaluated for oblique impact of a sphere on a half-space. For a solid sphere composed of material with Poisson's ratio v=0.3, the ratio of stiffness ( 2 = 1.21 while the mass ratio m2/mt=3.5 so that the ratio of frequencies og/D = 1.7. These values are used in the following examples of oblique impact of a sphere against a massive half-space.
Tangential oelocity at separation For a sphere striking a half-space at angle of incidence 0 where 0=tan-l(Vl(0)/v2(0)), Eqns (15), (25) and (27) were used to calculate the tangential relative velocity for contact point C at separation. In Fig. 5, the results from the present discrete parameter model are compared with the elasticity solution given by Maw et al. [6] and experimental measurements of Johnson [ 14] for a rubber sphere (Poisson's ratio v = 0.5) striking a heavy steel plate at a small speed. The elastic solution and the discrete parameter model each have similar processes that develop at the contact point in three parts of the range of angle of incidence. The predictions of these two models are most different for small and intermediate angles of incidence where the discrete parameter model has a final period of slip that is prolonged by elastic strain energy stored in the tangential compliant element. Throughout most of the range of small to intermediate angles of incidence, both the elastic continuum and the discrete parameter models of sphere impact have a tangential relative velocity at separation that is in the opposite direction to the incident tangential velocity. For a collinear collision this velocity reversal at C is entirely due to tangential compliance. In almost all respects the results of these models are practically identical. Figure 5 also illustrates final velocities calculated for negligible tangential compliance; in this case there is a nonzero terminal tangential velocity only if the angle of incidence is large; i.e. vl(O)/v2(O) > p(1 + e)m2/ml. The effect of coefficient of restitution e on the change in the tangential component of relative velocity at C is shown in Fig. 6. The angle of incidence for gross slip decreases with increasing internal dissipation. The coefficient of restitution e effects only the impulse imparted during restitution so that changes in velocity during restitution decrease with e. This causes the shift in the curve for separation velocity that is apparent in Fig. 6.
Angle of incidence for maximum friction Experiments using repeated impacts at oblique angles of incidence were performed by Ko [15] who showed that for relatively small normal impact speeds, o2 < 1 ms -1, wear rate of steel tubes is closely correlated with maximum tangential force Flma~ and that this force varies with angle of obliquity. For the present model, the tangential force Fl(t) can
initiol
stick
|erminol
slip
slip - stick - r e v e r s e slip
'
A
~
continuous tsp
D 4
•
1
~ _ . ~ .
. . . . 2 . . . . _~ . . . .
oo-\
-1
,o slip. ~
7,,,-
ye:10
~
?
v2'0' '
o
•
-2
~ . . . . 9,_ . . . .
r~,,~
ill
sphere
continuous
v = 0.31
-3
FIG. 5. Tangentialvelocityof contact point on elastic sphere at instant of separation as a function of the angle of incidence. Lumped parameter model - - , elastic continuum analysis - - - , analysis for negligibletangential compliance. . . . . , experimentswith rubber ball o o o.
Planar impact of rough compliant bodies
445
/
>~
S
.I_+.23
)/?
/
/
>
?
v~(O)/~v2101
/ -3 FIG. 6. Tangential velocity of contact point at separation as function of the angle of incidence of sphere for coefficients of restitution, e =0, 0.5 and 1.0.
be calculated as a function of the angle of incidence, tan- l(vl(O)/v2(O)). Irrespective of the angle of incidence, the largest value of friction Fxmax o c c u r s during the compression period if w/f2 > 1; consequently, the maximum tangential force is independent of the coefficient of restitution. For any impact speed S(0)-[v2(0)+ v2(0)] 1/2 the tangential component of force can be compared with the largest normal force F2m,x = F2(t¢) where F2max
f~m2S(0)
1 F1 +v~(0)/v~(0)] ~/2"
(28)
Expressions for the maximum tangential force are given below for different ranges of the angle of incidence. Small angles of incidence v~(0)/v2(0)_<#~ 2. For small angles of obliquity the contact point initially sticks and only begins to slide during the restitution period. The maximum tangential force occurs during compression when the tangential velocity reverses in direction at time z =-~t¢/o). At this time the contact is sticking so the maximum tangential force an be obtained from Eqn (7c).
F l m.x = -- wm 1v 1(0).
(29)
This maximum tangential component of force can be expressed as a nondimensional ratio defined as IFlma~l/m2f~S(O). Thus for small angles of obliquity Flm._____.~_ wmllv~(0)l _ 1 m/~
va(0)/v2(0)
(30)
nm2S(0) nm~S(0) ~/m2 [l+v~(0)/v~,(0)]"2" Intermediate angles of incidence #~2 < v l(0)/Vz(0) _<#(1 + e)mz/m 1. At intermediate angles of obliquity there is initial sliding but then stick begins at time tz during the compression period. When stick begins the contact point is still moving in the initial direction; i.e. vx(tz)>0. Maximum friction develops shortly after the period of stick begins and before the instant of maximum compression. The friction force during sticking can be expressed as F~(t) . ~"ul(t2)cosw(t_t2)+Q m ~ ~ , s i n o ) ( t _ t 2 ) } . f~m2S(0) ~2S(0)/./xv2(0) V mz/xv2tu)
(31)
At the transition from sliding to stick the displacement ul(tz) and velocity vl(tz) depend on the coefficient of friction. The transition velocity vl(t2) is obtained from (9) while the displacement ux(t2) is calculated from the friction law F t = -IxsF 2 and the force Fz given in Table 1. Thus ff]Ul(t2) = ( 2 sin ~ t 2 ,
/w2(O)
Vl(t2) -- vl(O)
m2(1 --COS ~t2).
(32)
/xv2(O) #v2(O) mi
Large angles of incidence vl(O)/v2(O)>/x(1 + e)m2/mi.
If the direction of slip is constant
446
W.J.
STRONGE
throughout the collision period then the maximum friction force is directly proportional to the normal force and the coefficient of friction,
Flmax /.t f~mzS(O ) - [ l + v~(O)/vZ~(O)]'/2"
(33)
For gross sliding this maximum tangential force occurs simultaneously with the largest normal force; i.e. tangential force is a maximum at time t c when the compression period terminates. These expressions for the maximum values of components of contact force have been used to calculate the largest normal and tangential forces which occur during oblique collisions. These largest values vary with the angle of incidence and the coefficient of friction as shown in Fig. 7. The largest values for the peak force occur for collisions at intermediate angles of incidence. The angle of incidence where the peak force is largest increases from about 20 ° for a coefficient of friction p = 0.1 to almost 60 ° for # = 1.0. Maximum friction for negligible tangential compliance. If tangential compliance is negligible, oblique collision always results in an initial period of sliding; this sliding is halted before separation and there is a successive final period of stick unless the angle of incidence is large enough to cause gross slip. Since we are considering central or collinear collisions, slip reversal does not occur. Consequently, if initial slip comes to a halt during compression, the peak tangential force occurs at the instant t s when sliding terminates. /2 sin f~ts [1 + v~(O)/v~(O)] 1/2
Flmax
f~m2S(O)
fh, =cos-
where
'F,L -
(34)
~\~v--~0)/l"
On the other hand, if the bodies are still sliding when compression terminates at time tc then the largest friction force occurs at this instant simultaneously with the largest normal force.
Flmax f~m2S(0--~)- [1
+
/2 v~(O)/v~(O)] ,/z"
(35)
In Fig. 7 the dashed lines show the maximum friction force for a compliant solid sphere whereas the light extensions to the left of these curves are results for similar collisions between spheres with negligible tangential compliance. For solid spheres, the largest
0 1.0
0.2
0.4
I
I
v~(0)/v2(0) 0.6 1.0 i
i
2.0
4.0
I
I
1.0
~o.8 O.6 i
0.8 ~
i 0 I~
•
0.t.
E o 0
~)
0 E 30* 60 ° angle of incidence, 0
9~
FIG.7. Maximumnormal force~ and tangential force----- during oblique impact of sphere. The light curves - - are max. tangential force if tangential compliance is negligible. In the crosshatched area there is gross slip.
Planar impact of rough compliant bodies
80
447 t,0
n ~
g =0.18
,,. 60-
o
z
E 30 C w"
~,~
,9.o -~ 4 0 -
20 c o
~ ~o
10 ox E
0 ~"
0
t
i
i
t
30* 60* ongle of incidence. O
t
~,,lO
90 °
FIG. 8. Comparison of the m a x i m u m normal and tangential components of contact force with experimental measurements by Lewis and Rogers (1988). At each angle of incidence, the normal and friction forces on a 101 g sphere colliding against a half-space were calculated for an incident speed 0.048 ms-~ and coefficient of friction/~=0.18.
tangential force for a compliant body is substantially less than the largest force calculated with the assumption of negligible tangential compliance. In either case the largest tangential force occurs in the range of small to intermediate angles of incidence. At large angles of incidence there is gross slip; in this case the maximum tangential force is independent of tangential compliance. Comparison with measurement of peak force during oblique impact. Lewis and Rogers [11] performed impact experiments at which a 25.4 mm diameter steel sphere collided against a heavy steel plate at angles of incidence that varied between 0 ° and 85 ° from normal. The impact speeds were small, being in the range 0.01-0.05 ms- ~. The sphere was attached at the free end of a 1.8 m long pendulum by a steel "ball holder". Piezoelectric force transducers were used to make separate measurements of normal and tangential components of contact force during impact. For gross slip or continuous sliding of this ball on the plate, Lewis and Rogers reported a coefficient of dynamic friction,/a=0.179. In Fig. 8 experimental data taken from collisions at an impact speed of 0.048 ms-1 are compared with normal and tangential components of force calculated by the present theory (using a coefficient of friction # = 0.18). The calculations depend on an estimate of the mass of the "ball holder". The agreement between experiment and theory shown in Fig, 8 was achieved by increasing the mass of the ball by 50% in order to account for inertia of the support system. In addition the calculations used a relative compliance ratio ~2 = 1.21 and mass ratio m2/m~=3.5 that are representative of solid spheres. This resulted in both qualitative and quantitative agreement between the calculations and the experiments for the full range of possible angles of incidence. Four series of tests using different impact speeds each gave a largest measurement of tangential force at an angle of incidence of about 40 °. For p=0.18 the present lumped parameter model gives a largest value of peak tangential force at about 35 ° irrespective of impact speed. (For any angle of incidence, the ratio between peak tangential and normal components of force can only be as large as the coefficient of friction if these peak values occur simultaneously; i.e. if Ol(O)/l)2(O)>_llm2/m 1 or the angle of incidence 0 > 32 ° for a rough solid sphere with/~ = 0.18.)
Dissipation of energy Internal dissipationfrom hysteresis of normalforce. During partly elastic collisions (e < 1) there is always irreversible internal deformation that dissipates a part of the initial kinetic energy K o where K o = M[v~(0) + o~(0)- 2x2v~(O)¢,(O)+ (x~ +
K2)I~/2(0)-]
and ~b(0) is the rate of rotation of the body at incidence. In the present model this internal
448
W.J. STRONGE
d i s s i p a t i o n D 2 is entirely due to hysteresis of the normal component of force. It can be
obtained as the negative of work done by this component of force, D2(tf)=-
f~'F2v2
(36)
d t = ( 1 - - e 2 ) m2v2(O)
since in collinear collisions m 2 = M, the part of the initial kinetic energy that is dissipated internally by irreversible deformations can be expressed as 2D2(t¢___~)=(1 -e2).
(37)
MrS(O) Frictional dissipation. Friction dissipates energy only during periods of slip. During these periods the tangential force does some work that changes the tangential strain energy and also some D1 that is dissipated by friction. Whereas the total work done on the body by the tangential force depends on the sliding speed vl + t~l, the frictional energy loss depends only on the tangential speed of the body yr. The remainder of the work done by the tangential force is stored as elastic strain energy in the tangential compliant element and later recovered before separation. Thus for small angles of incidence where the contact point slides only after an initial period of sticking, the dissipation due to friction is calculated from*
Dt(tf)= -
Fly 1 dt= - /
~Vl(tl)+
[p(t)-p(q)]
IJ dp
(38)
Using Eqns (12) and (14) this gives 2o,(t,) _
v,(0)
m2v22(0)
2p2gt (/~v2(0) cos coq
+
2-m-T1J"
(39)
On the other hand, if the angle of incidence is moderate, the contact point slides prior to t i m e t 2 and again slides after time t3. Thus for t2 < t:,
2D,(tf) _/~2m 1f v~(0)
m2v~(0)
v2(t2)
v~(t3)
m2 (~2v~2(0) 1,2v~(0) ~ ~2v~(0)
v12(t4)r"(
~2v,~(0)J"
(40)
Finally, if the angle of incidence is large so that there is gross sliding, the part of the energy dissipation due to friction can be expressed as 2D 1 (tf) + e)'~. rn2v22(0)-2/~2(l+e)~"v'(0)(pv2(0) 2 ~ ( 1 j
(41)
Equations (39)-(41) were used to evaluate the part of the initial kinetic energy that is dissipated by friction 2Dl(tt)/m2S2(O). In Fig. 9 this frictional dissipation is illustrated as a function of angle of incidence for coefficients of friction/.t=0.1 and 0.5. In contact regions with tangential compliance there is almost no frictional dissipation if the angle of incidence is small. If there is gross slip due to a large angle of incidence, tangential compliance has no effect on frictional dissipation. The fraction of the initial kinetic energy which is dissipated by friction is maximum at an angle of incidence slightly larger than the smallest angle giving gross slip. These results are based on the supposition that coefficient 'of friction is a parameter that is constant during impact. For impact speeds S(0)> 50 ms-1 where indentation of metals results from uncontained plastic deformation, Sundararajan [16] has pointed out that tangential force can be increased by indentation and decreased by frictional heating. * During any period t b- t. with a constant direction of sliding, the part of the total energydissipation Di(tb)--D,(t.) due to a component of contact force F t can be calculated from a theorem of Stronge [17]. Di(tb)--Di(ta)=[vi(tb)+v,(t.)][.pl(tb)--pj(t.)]/2. This gives Dl(tb)--Di(t.)= rat[v~(tb)--u~(t.)]/2 for a collinear impact configuration if the direction of sliding is constant.
Planar impact of rough compliant bodies
0
I
03-
vl(O) / v2(O) 0.6 1.0 2.0
0.20..t. I
I
I
449
t..O I ,,.,,P ~ I'fl 1
to/£ =1.7
o
% 0.2
* / d 0.1
0
.......
.
b.~
I I 30* 60* angle of inctdence. O
go °
FIG. 9. Dissipation of initial kinetic energy by friction during ,mpact of sphere as a function of the angle of incidence for coefficients of restitution e = 1 - - and e =0.5 - - - . If tangential compliance is negligible.., and friction is large enough to halt initial slip before separation, frictional dissipation is independent of coefficients of friction and restitution. For gross slip however, frictional dissipation does not depend on tangential compliance.
CONCLUSIONS
For almost all angles of incidence, the response of this simple lumped parameter model is identical with that of the quasistatic (Hertz) elastic analysis. The microslip present in the continuum analysis has no significant effect on changes in velocity of the colliding bodies. Both the elastic continuum analysis and the lumped parameter model show that if slip is brought to a halt during collision, tangential compliance can subsequently reverse the direction of slip. Slip can be brought to a halt during collision only if the tangential component of incident velocity is not too large; i.e. if o~(0)/Oz(0)
450
W.J. STRONGE REFERENCES
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