Page 148 - Dynamics of Mechanical Systems

0593_C05_fm Page 129 Monday, May 6, 2002 2:15 PM

Planar Motion of Rigid Bodies — Methods of Analysis 129

derivatives are thus parallel to the motion planes and therefore perpendicular to n (see
3
Section 4.5). Hence, (dn /dt) · n is zero. Equation (5.3.1) thus becomes:
2
3
+ ( [ ] D
ωω= 0n 1 + 0n 2 d n dt )⋅n n 3 = nω 3 (5.3.2)
1
2
where by inspection the angular speed ω is defined as (dn /dt) • n .
2
1
Equation (5.3.2) shows that the angular velocity ω of B is always parallel to n , a fixed
3
unit vector. Therefore, ωω ωω is characterized by the angular speed ω. Hence, in planar motion,
ω is sometimes called the angular velocity of B.
By differentiating in Eq. (5.3.2), the angular acceleration α of B takes the simple form:

αα = d ωω dt = (dω dt )n 3 D 3 (5.3.3)
= αn
where the scalar α is called the angular acceleration of B and is often written as . ˙ ω
Because the angular velocity and angular acceleration of a body in planar motion are
defined by the two scalars ω and α, respectively, the objectives of kinematic analyses are
often reduced to finding expressions for velocities and accelerations of points on the body.
In such analyses, it is often convenient to categorize the motion of the body as being
translation, rotation, or general motion.
Translation occurs when all particles of the body have equal velocities. If, in addition to
this, all the particles have straight-line motion, the body is said to have rectilinear motion.
Rotation occurs when the particles of the body move in concentric circles (this is some-
times called pure rotation).
General plane motion occurs when a body has planar motion that is neither translation
nor rotation. General plane motion may be considered as a superposition of translation
and rotation. (Translation and rotation are thus special cases of general plane motion.)

5.3.1 Translation
The kinematic analysis of a body in translation is relatively simple. The angular velocity
ω and the angular acceleration α are zero:

ω = α = 0 (5.3.4)

P
If P is a typical particle of the body B with velocity v , and if Q is any other particle, then:
v = v P (5.3.5)
Q

Then, by differentiation, we have:

a = a P (5.3.6)
Q
If the translating body also has rectilinear motion, the direction of the velocity and
acceleration of the particles is constant. The particle velocity and acceleration can then be
defined in terms of scalars v and a as: