Page 427 - The Mechatronics Handbook

0066_Frame_C19 Page 49 Wednesday, January 9, 2002 5:27 PM

Fundamental Concepts
Angular Displacement, Velocity, and Acceleration
The concept of rotational motion is readily formalized: all points within a rotating rigid body move in
parallel or coincident planes while remaining at fixed distances from a line called the axis. In a perfectly
rigid body, all points also remain at fixed distances from each other. Rotation is perceived as a change
in the angular position of a reference point on the body, i.e., as its angular displacement, ∆θ, over some
time interval, ∆t. The motion of that point, and therefore of the whole body, is characterized by its
clockwise (CW) or counterclockwise (CCW) direction and by its angular velocity, ω = ∆θ/∆t. If during
a time interval ∆t, the velocity changes by ∆ω, the body is undergoing an angular acceleration, α = ∆ω/∆t.
–1
With angles measured in radians, and time in seconds, units of ω become radians per second (rad s )
–2
and of α, radians per second per second (rad s ). Angular velocity is often referred to as rotational speed
and measured in numbers of complete revolutions per minute (rpm) or per second (rps).
Force, Torque, and Equilibrium
Rotational motion, as with motion in general, is controlled by forces in accordance with Newton’s laws.
Because a force directly affects only that component of motion in its line of action, forces or components of
forces acting in any plane that includes the axis produce no tendency for rotation about that axis. Rotation
can be initiated, altered in velocity, or terminated only by a tangential force F t acting at a finite radial
distance l from the axis. The effectiveness of such forces increases with both F t and l; hence, their product,
called a moment, is the activating quantity for rotational motion. A moment about the rotational axis
constitutes a torque. Figure 19.45(a) shows a force F acting at an angle β to the tangent at a point P,
distant l (the moment arm) from the axis. The torque T is found from the tangential component of F as

T = F t l = ( F cos b)l (19.58)

The combined effect, known as the resultant, of any number of torques acting at different locations along
a body is found from their algebraic sum, wherein torques tending to cause rotation in CW and CCW
directions are assigned opposite signs. Forces, hence torques, arise from physical contact with other solid
bodies, motional interaction with fluids, or via gravitational (including inertial), electric, or magnetic
force fields. The source of each such torque is subjected to an equal, but oppositely directed, reaction
torque. With force measured in newtons and distance in meters, Eq. (19.58) shows the unit of torque to
be a Newton meter (Nm).

P F L
I β
F T
t b' b' φ
d
+ O
CCW a b b
Axis
CW T

(a) (b)

FIGURE 19.45 (a) The off-axis force F at P produces a torque T = (F cos β)l tending to rotate the body in the CW
direction. (b) Transmitting torque T over length L twists the shaft through angle φ.

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