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The centre of percussionCOP is not a unique point on
+ N1 C+ ]% z+ f) a4 B$ Bthe barrel since it is defined in terms of a conjugate point in2 }- K. Q7 Q2 [6 o% i, G' z
2,6,7
7 d4 g" ~4 _3 e0 dthe handle. Each point in the handle is associated with a
1 {+ H! O9 {$ u1 Hdifferent COP in the barrel. When a ball strikes a freely w1 r0 B; f, O8 ]
suspended bat, the bat will translate in the direction of the
& r5 `& s1 g# G! p/ ]9 Q2 L! jincident ball and it will also rotate about the centre of mass.3 `- z& C- w' x$ u/ V- [3 a+ l
The combined effects of translation and rotation result in a
+ z: v' D) ~8 b1 l' kshift of the actual axis of rotation, away from the centre of
$ ]4 J! Z2 P# j$ {& ^mass toward the handle. The axis passes through the conju-; {- L: E$ g! [1 [$ p
gate point for that particular impact. If b is the distance from
' ?8 v0 s; T+ S4 z6 d; M3 Dthe impact point to the centre of mass, and x is the distance
0 q( w/ e5 R3 O" r; h& J2 J+ cfrom the centre of mass to the conjugate point, then xIcm/(Mb) where Icm is the moment of inertiaof the bat) H; u) B$ w% u( U
about its centre of mass and M is the mass of the bat. Con-
- K2 X% O- a. I# [3 p* |' N* xsequently, if the bat is pivoted about an axis through the& Q7 p# Q7 s9 A! E
conjugate point, and if the ball strikes the corresponding) x" P8 x6 f3 B. j' s1 Q9 k. t" Q
COP, then there is no reaction force at the pivot point. It is' _* O" \; z: i1 ]- h7 K% i) T
therefore reasonable to expect that every point under the H& \3 R$ y, f& D) w$ X
hands will have its own COP and that an impact at any given9 u% x: n/ L8 s! v* _5 e
COP will generate zero reaction force at the corresponding
1 b4 `0 \* n; N- h# L) w+ Y8- S+ e* P0 K0 b
conjugate point under the hands. Brancazio has suggested0 D8 R* I; y+ z5 {. Z" ^
that the sweet spot zone might therefore encompass all cor-
. F" y3 |% n+ C8 Wresponding COP points along the barrel. There is, however,( I* _' i$ n: B W9 C
an uncertainty in this argument since a hand-held bat is not5 q2 K; o3 q. G6 ~* D ?6 n2 D, Z0 x
freely suspended, nor is it pivoted about a single axis, and
! r* ?% z6 I Y- B" cmay behave differently as a result of the reaction forces ex-
7 w Q% Y& Q0 zerted by the two hands. Furthermore, the axis of rotation may
$ t( Q& @8 s4 A8 Q1 |even shift as a function of time since the ball remains in
% b1 t+ V; m" j- ycontact with the bat for about 1.5 ms, the transmission time
; {2 a* k; \- fof the pulse from the impact point to the hands is about 1 ms,
# w2 F% j+ s0 U$ U- uand the impact forces on the hands last for about 20 ms as a7 D0 S! o6 D, D8 m9 V; Q
result of the induced vibrations and the induced motion of9 m5 ]2 I. Q; ?; x
the bat. |
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