Page 88 - Dynamics of Mechanical Systems

0593_C03_fm Page 69 Monday, May 6, 2002 2:03 PM

Kinematics of a Particle 69

By differentiating, the velocity of p in a reference frame R containing O is:

V = d p dt = ( dr dt) n + ( n dt)
P
r d
r r
= ( dr dt) n + × n d dt)
r( θ
r n r z (3.8.3)
r d dt)
= ( dr dt) n + ( θ n
r θ
where, as before, dn /dt is evaluated using Eq. (3.5.7).
r
Similarly, by differentiating again we obtain the acceleration of P as:
( [
r d dt) ]
a = d V dt = d dt dr dt) n + ( θ n θ
P
P
r
= ( d r dt 2 n ) r ( dr dt d )( n dt) +( dr dt d dt) n θ
+
)( θ
2
r
r d dt d )(
+ rd θ dt 2 n + ( θ n dt)
2
θ
θ
d dt)
θ
= d r dt 2 n +( dr dt d dt) n × n +( dr d )( θ n
)( θ
2
r z r θ θ
)
2
+ ( r d 2 θ dt 2 θ )n + θ ( r d dt n z × n θ
)
)(
)(
2
2
= (d r dt n r +(dr dt d θ dt θ )n +(dr dt d θ dt θ )n
)
2
+ ( rd 2 θ dt 2 θ )n − θ ( rd dt n r
or
p [ 2 2 r d dt) 2 r ] [ 2 2 dr dt d dt) ]
a = d r dt − ( θ n + rd θ dt + ( 2 )( θ n θ (3.8.4)
Suppose, as before, that we define ω and α to be:
θ
ω = d dt and α = d ω dt = d 2 θ dt 2 (3.8.5)
Thus, in terms of α and ω V and a are:
P
P
V =ω θ (3.8.6)
P
rn
and
a = [ d r dt − rω 2 n ] r [ r + ( dr dt)] n θ (3.8.7)
+ α
2ω
2
2
P
or, alternatively,
a = [ rrω 2 n ] r [ α 2 r] n θ (3.8.8)
−
+
r + ω
P
˙˙
˙
where the overdot represents differentiation with respect to time t.