Page 452 - Marks Calculation for Machine Design

P2: Sanjay
P1: Shibu/Rakesh
15:34
January 4, 2005
Brown˙C10
Brown.cls
APPLICATION TO MACHINES
434
Fig. 10.23 shows the velocities of four additional points E, F, G, and H.
rw v E
w
F
E v A
v A
v F
r A rw
45∞
v = rw
A
rw v H
v A G v
H rw A
v G
FIGURE 10.23 Velocity of four additional points on a rolling wheel.
Choosing point F in Fig. 10.23, its velocity has a magnitude given by the Pythagorean
theorem as

◦ 2
◦ 2
v F = (v A + rω cos 45 ) + (−rω sin 45 )

◦ 2
◦ 2
= (v A + v A cos 45 ) + (−v A sin 45 )
(10.55)

2 ◦ 2 ◦ 2 2
= v [(1 + cos 45 ) + (− sin 45 ) ] = v [3.414]
A A
= (1.85) v A
and its direction is downward from the horizontal, a negative angle (θ) given by the
expression
−rω sin 45 ◦ −v A sin 45 ◦ − sin 45 ◦
tan θ = = = =−0.414
v A + rω cos 45 ◦ v A + v A cos 45 ◦ 1 + cos 45 ◦
(10.56)
θ =−22.5 ◦
The magnitude of the other three velocities is the same as that given by Eq. (10.55);
however, each velocity is at a different angle relative to the horizontal.
U.S. Customary SI/Metric
Example 2. Determine the velocity, both its Example 2. Determine the velocity, both its
magnitude and direction, of point H on the magnitude and direction, of point H on the
rolling wheel shown in Fig. 10.23, where rolling wheel shown in Fig. 10.23, where
v A = 60 mph = 88 ft/s v A = 96.5 kph = 26.8 m/s
solution solution
Step1. Substitutethegivenvelocity(v A )ofthe Step1. Substitutethegivenvelocity(v A )ofthe
center of the wheel in Eq. (10.55) to determine center of the wheel in Eq. (10.55) to determine
the magnitude of the velocity (v H ) as the magnitude of the velocity (v H ) as
v H = (1.85)v A = (1.85)(88 ft/s) v H = (1.85)v A = (1.85)(26.8 m/s)
= 162.8 ft/s = 49.6 m/s