Page 123 - Dynamics of Mechanical Systems

0593_C04*_fm Page 104 Monday, May 6, 2002 2:06 PM

104 Dynamics of Mechanical Systems

By differentiating in Eq. (4.10.4), we can obtain a relation determining the acceleration
of P. That is,

a = a + d V dt+ αα × r+ ωω × d dt (4.10.7)
R P RQ RB P R B R B R
r
R Q
R
R P
where, as before, a and a are the acceleration of P and Q in R, and α is the acceleration
B
P
R B
of B in R. By using Eq. (4.6.6), we can express d V /dt as:
a + ωω
R
R B P dt = B B P dt+ ωω B × B V = B P R B × B P
P
d V d V V (4.10.8)
Then, by using Eq. (4.10.3), a may be written as:
R P
a +
a + ωω
P
R a = R Q B P R B + B V + αα B × r+ ωω B × [ B V ×( ωω B × r)]
P
R
R
R
P
or
a +2 ωω
a +
R
R
P
R
R a = B P R Q R B × B V + αα B × r+ ωω B ×( ωω B × r) (4.10.9)
P
Suppose again that at an instant of interest P happens to coincide with Q. Then, r is
zero and a is:
R P
a +
a +2 ωω
P
R a = B P R Q R B × B V P (4.10.10)
*
Hence, in general, if P is the point of B coinciding with P we have:
a = a + a +2 ωω × V (4.10.11)
R P B P R P* R B B P
The term 2 ωω ωω × V is called the Coriolis acceleration, after the French mechanician de
B
B
R
P
Coriolis (1792–1843) who is credited with being the first to discover it. We have already
seen this term in our analysis of the movement of a point in a plane in Chapter 3 (see Eq.
(3.8.7)). This term is not generally intuitive, and it often gives rise to surprising and
unexpected effects.
Example 4.10.1: Movement of Sports Car Operator’s Hands
Equations (4.10.6) and (4.10.11) may also be used to determine the velocity and acceleration
of the sports car operator’s left hand of Example 4.9.2. Recall that the sports car is making
a turn to the right at 15 mph with a turn radius of 25 feet, and that the operator’s left
hand is at 10 o’clock on a 12-in.-diameter vertical steering wheel, as in Figures 4.10.2 and
4.10.3. Recall further that the operator is turning the wheel clockwise at one revolution
per second, as in Figure 4.10.3.
Solution: From Eq. (4.10.6), the velocity of the left hand L may be expressed as:
V = V + V (4.10.12)
S L C L S L*