Page 77 - Dynamics of Mechanical Systems

0593_C03_fm Page 58 Tuesday, May 14, 2002 10:28 AM

58 Dynamics of Mechanical Systems

R
the derivative is calculated. Thus, instead of simply writing dV/dt we will write dV/dt,
unless the reference frame is clearly understood.
To explore the properties of vector differentiation, let n , n , and n be mutually perpen-
2
1
3
dicular unit vectors fixed in a reference frame R. Let V be a vector function of t and let V
be expressed in terms of n , n , and n as:
3
1
2
v
v
v
V = v n = () t n + () t n + n + () t n (3.2.3)
i i 1 1 2 2 2 3
By substituting into Eq. (3.2.2), the derivative of V with respect to t in R is then:
t n + (
t n + (
(
+
+
+
R dt = Lim [ { v t ∆ ) v t ∆ ) v t ∆ ) ]
dV 1 1 2 2 3 t n 3
∆ t→0 (3.2.4)
v t n + ()
− () 1 vt n + () ]} / t ∆
vt n
2
[ 1
2
3
3
Because the n (i = 1, 2, 3) are fixed in R, they are independent of t in R and thus are
i
constant in R. From elementary calculus, we recall that if the limits exist, the limit of a
sum is equal to the sum of the limits. Also, if the limits exist, the limit of a constant
multiplied by a function is equal to the constant multiplied by the limit of the function.
Hence, we can rewrite Eq. (3.2.4) as:
+
+
R t) ( t)
v t
v t
dV dt = v t ∆ − ()] / ∆ + v t ∆ − ()] / t ∆
1 ∆ t→0 1 t n [ 2 2
2
n Lim ( [ 1
t)
+ vt ∆ − ()] / t ∆ (3.2.5)
+
v t
n Lim ( [ 3
3 ∆ t→0 3
= n dv dt n dv dt n dv dt
+
+
1 1 2 2 3 3
This expression shows that vector derivatives are simply linear combinations of scalar
derivatives. Specifically, to calculate the derivative of a vector function in a reference frame
R we simply need to express V in terms of unit vectors fixed in R and then differentiate the
scalar components. Although this procedure always works, it may not be the most conve-
nient in all circumstances. We will explore and develop other procedures in Section 3.5.
Example 3.2.1: Differentiation of a Position Vector
To illustrate these ideas, consider the following example: Let the position vector p locating
a moving point P relative to the origin of a Cartesian reference frame R be expressed as:
p = x n + y n + z n (3.2.6)
1 2 3
where n , n , and n are mutually perpendicular unit vectors fixed in R. Let x, y, and z be
1
3
2
functions of time t given by:
4
3
x = (2 t − ) 2 , y = t − 3 t, z = − + 4 t (3.2.7)
t
where x, y, and z are measured in meters, and t is measured in seconds. Then, the derivative
of p with respect to t in R is:
2
3
dp dt = (2 t − ) 2 n +(3 t − ) 3 n +− ( 4 t + ) 4 n m sec (3.2.8)
1 2 3