Page 107 - Dynamics of Mechanical Systems

0593_C04*_fm Page 88 Monday, May 6, 2002 2:06 PM

88 Dynamics of Mechanical Systems

n ˆ
3

ˆ
V n
2
ˆ
R
FIGURE 4.6.1 R
Vector V and reference frames R n ˆ
ˆ
and R. 1
ˆ
ˆ
ˆ n
Because the are fixed in , the derivative of V in is obtained by simply differen-
R
R
i
tiating the scalar components in Eq. (4.6.1). That is,
ˆ n +
ˆ n +
ˆ R ( ˆ ) ( ˆ ) ( ˆ )
dV dt = dV dt 1 dV dt 2 dV dt ˆ n 3 (4.6.2)
1
3
2
Next, relative to reference frame R, the derivative of V is:
)
ˆ
R dt = ( dV dt n + ˆ R dt + ( ˆ )
dV 1 ˆ 1 V dn ˆ 1 dV dt n ˆ 2
2
1
)
ˆ
ˆ
ˆ
+ V dn ˆ 3 dt + ( dV dt n + V dn ˆ 3 dt (4.6.3)
ˆ
R
R
3
3
3
2
ˆ
ˆ
ˆ
+
+
+
= R ˆ dV dtV dn ˆ 1 dtV dn 2 dt V dn ˆ 3 dt
R
R
R
2
1
3
ˆ n
where the second equality is determined from Eq. (4.6.2). Because the are fixed in , R ˆ
i
the derivatives in R are:
R R R ˆ ˆ ( ,,3)
d ˆ n dt= ωω × n i = 12 (4.6.4)
1 i
R
Hence, dV/dt becomes:
+
R
R dt = R ˆ dt V ωω R ˆ n + V ωω R ˆ ×
R
dV dV 1 × ˆ 1 2 n ˆ 2
+ V ωω R ˆ × n ˆ 3
R
3
= R ˆ dV dt+ ωω R ˆ × ( 1 1 V n + V n ˆ 3 )
ˆ
ˆ
ˆ
V n +
R
ˆ
ˆ
3
2
2
or
R R ˆ R R ˆ
dV dt = dV dt+ ωω × V (4.6.5)
Because there were no restrictions on V, Eq. (4.6.5) may be written as:
R
R d() dt = R ˆ d() dt+ ωω R ˆ ×() (4.6.6)
where any vector quantity may be inserted in the parentheses.