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International Journal of Impact Engineering 37 (2010) 995e998

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International Journal of Impact Engineering

journal homepage: www.elsevier .com/locate/ i j impeng

Short Communication

A correction on the calculation of frictional dissipation in planar impactof rough compliant bodies by W.J. Stronge

Tran Hien*

Department of Civil Engineering, National University of Singapore, Engineering Drive 2, Singapore 117576, Singapore

a r t i c l e i n f o

Article history:Received 14 March 2010Received in revised form3 April 2010Accepted 7 April 2010Available online 4 May 2010

Keywords:Planar impactFrictional dissipationSlip processLumped parameter model

* Tel.: þ65 90551905.E-mail address: tranhien@nus.edu.sg

0734-743X/$ e see front matter � 2010 Elsevier Ltd.doi:10.1016/j.ijimpeng.2010.04.002

a b s t r a c t

In the paper published by W.J. Stronge (Int. J. Impact Eng. 1994;15(4):435e50), a lumped parametermodel of contact between colliding bodies has been employed to calculate the tangential contact forceand energy dissipated by friction in planar impact of rough compliant bodies. The formulation of fric-tional energy loss was based on the work done by tangential force on the tangential motion of the bodyduring slip process. However, this formulation seems to be incorrect since it results in an inappropriateexplanation for the conservation of energy during the whole process of collision when the angles ofincidence are small or intermediate. In this paper, the cause of inaccuracy and the corrected formulationand calculation of the frictional dissipation in collinear collisions are presented.

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1. Introduction

In rigid body dynamics, a collision between two bodies isusually treated as an instantaneous process with contact at a singlepoint. An impulsive contact force which results in the change inrelative velocity of the colliding bodies is assumed to be applied atthe point of contact. For frictional collision, the tangential forceopposes the relative tangential velocity of the colliding bodies.Assuming that the collision has a finite duration that is smallcompared to a typical time scale of the motion of the bodies beforeand after collision, one can resolve the impact process as a functionof the normal component of impulse [1].

In order to determine the effect of tangential compliance duringthe collision process, Stronge [2] introduced a lumped parametermodel of contact (see Fig.1). In this figure, unit vectors n1 and n2 areoriented in direction tangent and normal to the common contactplane. At incidence (time t¼ 0), Stronge [2] defined that bothnormal and tangential components of relative velocities at contactpoint C are negative, v1ð0Þ < 0 and v2ð0Þ < 0. Thus, the angle ofincidence is q ¼ tan�1ðv1ð0Þ=v2ð0ÞÞ. The motion of the collidingbodies at the contact point C is represented by an inertia matrixmij.Notice that if the collision configuration is collinear, mij only hasnon-zero terms on principle diagonal which are m1 and m2 corre-sponding to tangential and normal direction respectively. Hence,

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the effects of normal and tangential impulse on changes in thecomponents of velocity at C are decoupled.

In Stronge’s model of collinear collision, the colliding body isdivided into 2 regions: a rigid body region (which has equivalentmasses m1 and m2 and relative velocities v1 and v2 correspondingto tangential and normal direction respectively at the contactpoint), and an infinitesimal deformable region at the point ofcontact C with negligible mass which is represented by a masslessparticle C connected to the rigid body by the normal and shearsprings. Contact forces Fi is applied on the particle C and trans-ferred to the rigid body region through these springs. This forceapplies a differential impulse dpi ¼ Fi dt in an incremental time dt.Thus, the equation of planar motion for the rigid body can beexpressed as [2]

dvi ¼ m�1ij dpj (1)

To solve the equation of motion above, periods of slip and stickat C during collision are considered so that tangential force andtangential compliance can be calculated. Stronge [2] found thatdepending on the angles of incidence, the contact point can initiallystick, slide before sticking or slide throughout the contact period. Ifthe angles of incidence is small, sticking initiates at initial contactand continues until time t1 when slip begins and goes on untilseparation at time tf. If the angle of incidence is intermediate, thecollision process includes three phases: initial sliding phase fromthe beginning to time t2, sticking phase from time t2 to time t3 andfinal sliding phase from time t3 to time tf. For large angle of

k, m2

2

k

ζ,m1

v(0)

v1(0) < 0

v2(0) < 0

n1

n2

μ C

u1

F1

F2

θ

Fig. 1. Lumped parameter model for impact of rough compliant body on half-space [2].

T. Hien / International Journal of Impact Engineering 37 (2010) 995e998996

incidence, slip continues throughout the entire contact period. Theslip process with respect to the angle of incidence can be summa-rized as follows.

e Small angle of incidence: initial sticking occurs if at incidence,the ratio of the tangential to normal components ofrelative velocity at contact point C is within a range0 < v1ð0Þ=v2ð0Þ < mz2 that is bounded by friction m and ratio ofnormal to tangential stiffness z2.

e Intermediate angle of incidence: when v1ð0Þ=v2ð0Þ > mz2, initialsliding occurs. This initial sliding phase may halt at time t2 < tfif v1ð0Þ=v2ð0Þ < mm2=m1ð1þ e�U2

=u2Þ where e is the coeffi-cient of restitution. Thus, the following sticking phase whichbegins at time t2 terminates at time t3 and a second slidingphase takes place from time t3 to time of separation tf.

e Large angle of incidence: if v1ð0Þ=v2ð0Þ > mm2=m1ð1þ e� U2

=u2Þ or t2 > tf , initial sliding phase continuesthroughout the entire contact period until time tf.

It could be noted that the upper bound for the range of inter-mediate angle of incidence v1ð0Þ=v2ð0Þ ¼ mm2=m1ð1þ e�U2

=u2Þis obtained from the condition that the sliding speed s ¼ v1 þ _u1 ininitial sliding phase will vanish at the time of separation tf, whichmeans that the initial sliding phase continues throughout the entirecontact period.

2. Calculation of frictional dissipation

During collision [2], the reaction forces on the colliding bodiesdecrease their kinetic energies in an initial phase of compression.Then elastic energy from the deformed region pushes the bodiesapart and restores some kinetic energy in the succeeding period ofrestitution. The energy dissipated in collision is the differencebetween the initial and final kinetic energy [3]. Thus, Stronge [3]defined that the kinetic energy dissipated at any time duringcollision is equal to the negative of work done by reaction force Fion the motion of the colliding body:

DðtÞ ¼ �WðtÞ ¼Zt

0

Fivi dt (2)

Later on, when calculating frictional dissipation of planar colli-sion using his lumped parameter model, Stronge [2] claimed thatwhereas the total work done on the body by tangential forcedepends on the sliding speed v1 þ _u1 (u1 e relative tangentialdisplacement of the particle C to the body), the frictional lossdepends only on the tangential speed of the body v1:

Ztb

D1 ¼ �W1 ¼ �

ta

F1v1 dt (3)

where [ta, tb] is period of a unidirectional sliding phase in collision.Thence, Stronge determined the total frictional dissipation

during a collision as the summation of thework done during slidingphases of the collision.

D1

�tf�

¼X

sliding phases

�Z

F1v1 dt (4)

In collinear collision, since mij only has non-zero terms onprinciple diagonal, the work done by tangential force F1 overa period [ta, tb] in view of Eq. (1) can be expressed as

Ztbta

F1v1 dt ¼Ztbta

v1 dp1 ¼Ztbta

v1m1 dv1 ¼ 12m1

hv21ðtbÞ � v21ðtaÞ

i

(5)

Thus, the calculation of frictional dissipation in Eq. (4) withrespect to different angles of incidence results in

e Small angle of incidence:

D1

�tf�

¼ �Ztf

F1v1 dt ¼ �12m1

hv21

�tf�� v21ðt1Þ

i(6)

t1

eIntermediate angle of incidence:

D1

�tf�

¼ �

264Zt2

F1v1 dt þZtf

F1v1 dt

375

0 t3

¼ �12m1

nhv21ðt2Þ � v21ð0Þ

iþhv21

�tf�� v21ðt3Þ

io(7)

e Large angle of incidence:

D1

�tf�

¼ �Ztf

F1v1 dt ¼ �1m1

hv21

�tf�� v21ð0Þ

i(8)

02

Normalizing the frictional dissipation in Eqs. (6)e(8) by the termð1=2Þm2v

22ð0Þ, which might be denoted as initial partial kinetic

energy for normal relative motion, gives the Eqs. (39), (40) and (41)in [2], respectively.

However, Stronge in his later book [4] has modified his state-ment to say that the frictional energy loss depends on the slidingspeed v1 þ _u1 at the contact point C during periods of slip. Though,no modification in formulation of frictional dissipation has beencarried out yet in this text.

3. Discussion and correction on the calculation of frictionaldissipation

In fact, Stronge was correct when he stated that total work doneon the body by tangential force depends on the sliding speedv1 þ _u1. However, the frictional dissipation should exactly equal tothe negative of this work done but not a part of it as Stronge hasclaimed in [2].

It is known that the negative of total work done by non-conservative forces on a mechanical system is the energy loss ofthis system. If we consider the whole colliding body as one system,

Table 1Upper bound for small and intermediate angles of incidence of sphere.

q1 (degree) q2 (degree)

Friction coefficient m¼ 0.1 6.9 30.1Friction coefficient m¼ 0.5 31.2 70.9

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

Non-dimensional contact time

Ener

gy c

ompo

nent

nor

mal

ised

by

initi

al k

inet

ic e

nerg

y

Frictional work doneChange of partial kinetic energy for tangential relative motion

Elastic energy stored in tangential spring

Fig. 2. Evolution of energy components during period of collision e Small angle ofincidence (m¼ 0.5, q¼ 20�). Small circles denote the transition between initial stickingphase and final sliding phase.

T. Hien / International Journal of Impact Engineering 37 (2010) 995e998 997

the negative of total work done by tangential force, which is alsofriction force or non-conservative force, applied at particle C insliding phases � R

F1ðv1 þ _u1Þdt should be the frictional loss of thesystem. However, the negative of work done by tangential forceapplied on the rigid body � R

F1v1 dt should not be considered asenergy loss since part of this work is required to deform thetangential spring.

Thus, if we calculate the frictional dissipation in sliding phasesof the collisions using sliding speed, the equation should be:

D1

�tf�

¼ �Z

sliding phases

F1�v1 þ _u1

�dt (9)

Since sliding speed vanishes v1 þ _u1 ¼ 0 in sticking phase, thisdissipation can be integrated over the whole period of collision asfollows:

D1

�tf�¼�

Zsliding phases

F1�v1þ _u1

�dt�

Zsticking phases

F1�v1þ _u1

�dt

¼�Ztf0

F1�v1þ _u1

�dt

or

D1

�tf�

¼ �

0B@Ztf

0

F1v1 dt þZtf0

F1 _u1 dt

1CA (10)

It is noted that the relative tangential displacement u1 onlydepends on the tangential force F1 at C and the stiffness oftangential spring k=z2. The integration term

RF1 _u1 dt over the

entire period of collision vanishes as the tangential spring is relaxedat the initial time of collision t(0) and at time of separation tf

Ztf0

F1ðtÞ _u1 dt¼Ztf0

F1ðtÞdu1 ¼Ztf0

k

z2u1ðtÞdu1

¼ 12k

z2

hu21

�tf��u21ð0Þ

i¼ 0 (11)

Hence, the frictional dissipation of collision in Eq. (9) can beexpressed as

D1

�tf�

¼ �Ztf0

F1v1 dt (12)

Though the negative of work done by tangential force applied onthe rigid body � R

F1v1 dt is not the frictional energy loss, this termwill be the same as the total frictional dissipation of the collisionwhen the integration is taken over the entire period of collision asin Eq. (12).

According to Eq. (5), the calculation of friction dissipation in Eq.(12) can be expressed as

D1

�tf�

¼ �12m1

hv21

�tf�� v21ð0Þ

i(13)

If we denote the term ð1=2Þm1v21 as partial kinetic energy for

tangential relative motion, the frictional dissipation calculated fromEq. (13) is also the negative of the change of this term during thewhole period of collision whereas the frictional dissipation calcu-lated by Stronge in Eqs. (6)e(8) is the summation of the changes ofthis term during sliding phases of the collision.

Normalizing the frictional dissipation D1ðtf Þ in Eq. (13) by theinitial partial kinetic energy for normal relative motion gives therevised version of Eqs. (39), (40) and (41) in [2]

2D1

�tf�

m2v22ð0Þ

¼ m2m1

m2

"v21ð0Þ

m2v22ð0Þ�

v21

�tf�

m2v22ð0Þ

#(14)

4. Modified results

In this section, impacts of a sphere on a plane surface withfriction coefficient of m at an angle of incidence of q are simulatedusing the lumped parameter model proposed by Stronge in [2]. Thecoefficient of restitution e is assumed to be unity as this study focuson the investigation of frictional dissipation. The upper bound q1 forthe range of small angle of incidence and the upper bound q2 for therange of intermediate angle of incidence are presented in Table 1.

Fig. 2 shows the evolution of the three terms in Eq. (10) duringthe period of a collision with friction coefficient m¼ 0.5 at a smallangle of incidence q¼ 20� where the motion is initial stick andterminal slip. The middle curve that initially is constant is thefrictional work done

R t0 F1ðv1 þ _u1Þdt from the initial time to the

current time t. The lower curveR t0 F1v1 dt represents the changes of

partial kinetic energy for tangential relative motion in this period,according to Eq. (5). The upper curve

R t0 F1 _u1 dt, in view of Eq. (11),

represents the elastic energy stored in the tangential spring. Allthese terms are normalized by the initial kinetic energy of thesphere ð1=2Þm2v

2ð0Þ where v2ð0Þ ¼ v21ð0Þ þ v22ð0Þ. Fig. 3 exhibitsthe evolution of these terms when the angle of incidence is inter-mediate (m ¼ 0:5 and q¼ 45�).

It can be seen from Fig. 2 that the frictional work done is zeroduring the initial sticking phase as the sliding speed vanish in thisperiod. Since the frictional work done is the summation of thechanges of partial kinetic energy for tangential relative motion andthe elastic energy stored in the tangential spring, there is only the

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.2

-0.15

-0.1

-0.05

0.05

0.1

0.15

Non-dimensional contact time

Ener

gy c

ompo

nent

nor

mal

ised

by

initi

al k

inet

ic e

nerg

y

Frictional work doneChange of partial kinetic energy for tangential relative motion

Elastic energy stored in tangential spring

Fig. 3. Evolution of energy components during period of collision e Intermediateangle of incidence (m¼ 0.5, q¼ 45�). Small circles denote the transitions among initialsliding phase, sticking phase and final sliding phase.

0 30 60 900

0.1

0.2

0.3

Angle of incidence (°)

μ=0.1μ=0.1

μ=0.5μ=0.5

2*D

1(t f)/m2v2 (0

)

Fig. 4. Frictional dissipation normalized by initial kinetic energy during impact ofsphere as a function of the angle of incidence for normal coefficient of restitution e¼ 1.The solid lines are Stronge’s results in [2] and the dash lines are the modified results.

T. Hien / International Journal of Impact Engineering 37 (2010) 995e998998

exchange between these two terms during the initial stickingphase. Besides, although the evolution of the middle curve isdifferent from the one of the lower curve, these two curves have thesame value at the end of the collision. It means that the total fric-tional work done is the same as the change of partial kinetic energyfor tangential relative motion for the whole period of collision. Thus,the frictional dissipation of the collision can be simply defined asthe negative of the difference between the initial and final partialkinetic energy for tangential relative motion. However, looking at thelower curve, if we consider the change of partial kinetic energy fortangential relative motion only in sliding phase like what Strongedid in [2], it can be easily seen from the figure that this value mustbe different from the real frictional energy in the entire period ofthe collision.

Similarly, for an intermediate angle of incidence where thecontact process is initial slip, stick, terminal slip, Fig. 3 shows thatduring the sticking phase, the frictional work done is unchangedand there is only the exchange between the lower curve and theupper curve. Again, the total frictional work done is the same aschange of partial kinetic energy for tangential relative motion at theend of the collision as it is expected fromEq. (13). The summation ofthe changes of partial kinetic energy for tangential relative motion ininitial sliding phase and final sliding phase can not be considered asthe frictional energy loss of the collision.

In both Figs. 2 and 3, it is also obvious that the upper curve,which is elastic energy stored in the tangential spring, goes to zeroat the end of the collision as the tangential spring becomes relaxedat the time of separation.

In the following Fig. 4, the plot of total frictional dissipationversus angle of incidence in Fig. 9 in [2] is revised by usingmodifiedformulation in this work. The dissipation is computed in collisionsof a sphere on plane surface at different angles of incidence forfriction coefficients m¼ 0.1 and m¼ 0.5 with normal coefficient of

restitution e¼ 1. The dissipation is also normalized by the initialkinetic energy of the sphere before impact ð1=2Þm2v

2ð0Þ. Thecurves in Fig. 4 show significant differences between Stronge’sresults and the modified results in the range of small and inter-mediate angles of incidence. It should be noted that this rangecovers qz0�e30:1� when m¼ 0.1 whereas it covers qz0�e70:9�

when m¼ 0.5. However, when the angles of incidence are large, thetwo results are the same since the sliding phase continuesthroughout the entire period of collision. For continuous (gross)slip, the modified calculation of frictional dissipation based on� R tf

0 F1v1 dt is the same as the frictional dissipation term� R

sliding phasesF1v1 dt which was used in [2].

5. Conclusion

Using the lumped parameter model of contact proposed byStronge in [2], one should calculate the frictional dissipation incollision of rough compliant bodies based on the work done bytangential force on the sliding motion at contact point instead ofthe tangential motion of the body. In other words, this dissipationcan be simply referred as the integration term� R tf

0 F1v1 dtwhich isalso the difference between the initial and final partial kineticenergy for tangential relative motion during the entire period ofcollision.

References

[1] Keller JB. Impact with friction. ASME J Appl Mech 1986;53:1e4.[2] Stronge WJ. Planar impact of rough compliant bodies. Int J Impact Eng 1994;15

(4):435e50.[3] Stronge WJ. Energy dissipated in planar collision. ASME J Appl Mech

1992;59:681e2.[4] Stronge WJ. Impact mechanics. New York: Cambridge University Press; 2000.

p. 110e111.