Page 83 - Dynamics of Mechanical Systems

0593_C03_fm Page 64 Monday, May 6, 2002 2:03 PM

64 Dynamics of Mechanical Systems

FIGURE 3.5.1
A point P moving on a curve C in a
reference frame R.
we can calculate the derivative of n by expressing it in terms of unit vectors fixed in R.
As we will see presently, however, there are more convenient procedures.
We focus our attention on unit vectors because they define the direction characteristics
of vectors, which distinguish vectors from scalars. It is quickly seen that the derivative of
a unit vector is a vector perpendicular to the unit vector. To see this, recall that the
magnitude of a unit vector is 1. Therefore,

nn⋅= 1 (3.5.3)

Then, by differentiating, we have:

)
n (
ddt n n⋅ d dt = 0 (3.5.4)
⋅+
n
The dot product, however, is commutative (see Eq. (2.6.2)). That is,
ddt n )⋅= ⋅( d dt) (3.5.5)
n (
n
n
Hence, Eq. (3.5.4) becomes:

n ) =
2n⋅(ddt 0 or n⋅(d n ) = 0 (3.5.6)
t
d
This shows that if dn/dt is not zero, it is perpendicular to n.
Next, consider the case of a unit vector that moves in such a way that it always remains
perpendicular to a fixed line. Specifically, let Z be a line fixed in a reference frame R, and
let L be a line that intersects Z and is perpendicular to Z. Let the unit vector n be parallel
to L. Finally, let X and Y be lines fixed in R such that X, Y, and Z form a mutually
perpendicular set. Let L rotate in the X–Y plane (see Figure 3.5.2). Let θ be the angle
between L and X as shown. As L rotates, θ changes. Thus, θ is a function of time t and is
a measure of the rotation of L and of n, as well. Because n rotates with L, it is called a
rotating unit vector.
Let n , n , and n be unit vectors parallel to X, Y, and Z, as shown in Figure 3.5.2. Then,
z
y
x
the derivative of n in R is simply:
θ
ddt = n × n d dt (3.5.7)
n
z