Page 438 - Marks Calculation for Machine Design

P2: Sanjay
P1: Shibu/Rakesh
January 4, 2005
Brown.cls
Brown˙C10
420
U.S. Customary 15:34 APPLICATION TO MACHINES SI/Metric
Example 2. Using the angle (β), the angular Example 2. Using the angle (β), the angular
velocity (ω rod ), and the velocity (v slider ) found velocity (ω rod ), and the velocity (v slider ) found
in Example 1, determine the angular acceler- in Example 1, determine the angular acceler-
ation (α rod ) and the acceleration of the slider ation (α rod ) and the acceleration of the slider
(a slider ), where (a slider ), where
ω crank = 2,000 rpm = 209 rad/s ω crank = 2,000 rpm = 209 rad/s
α crank = 0 (ω crank = constant) α crank = 0 (ω crank = constant)
φ = 50 ◦ φ = 50 ◦
L AB = 3in L AB = 7.5 cm
L BC = 8in L BC = 20 cm
and determined from Example 1: and determined from Example 1:
β = 14 ◦ β = 14 ◦
ω rod = 592 rpm = 62 rad/s ω rod = 592 rpm = 62 rad/s
v slider = 523 in/s = 43.6 ft/s v slider = 1,307 cm/s = 13.1 m/s
solution solution
Step 1. Using the given angular velocity Step 1. Using the given angular velocity
(ω crank ), angular acceleration (α crank ), the angle (ω crank ), angular acceleration (α crank ), the angle
(φ), the lengths (L AB ) and (L BC ), the angle (β), (φ), the lengths (L AB ) and (L BC ), the angle (β),
and angular velocity (ω rod ), calculate the angu- and angular velocity (ω rod ), calculate the angu-
lar acceleration (α rod ) from Eq. (10.22) as lar acceleration (α rod ) from Eq. (10.22) as
2 2
L AB ω crank cos φ + α crank sin φ L AB ω crank cos φ + α crank sin φ
α rod = α rod =
L BC cos β L BC cos β
−ω 2 rod tan β −ω 2 rod tan β
3in 7.5cm

= × = ×
8in 20 cm
2   2 
rad rad

 209 ◦ ◦  209 ◦ ◦
s cos 50 + (0) sin 50  s cos 50 + (0) sin 50 
   
   
cos 14 ◦ cos 14 ◦
   
2 2
rad rad

− 62 tan 14 ◦ − 62 tan 14 ◦
s s
2 2
28,078 rad/s 28,078 rad/s
= (0.375) = (0.375)
0.9703 0.9703
rad rad

− 3,844 (0.249) − 3,844 (0.249)
s 2 s 2

rad rad rad rad
= 10,851 − 958 = 10,851 − 958
s 2 s 2 s 2 s 2
rad 1rev 60 s rad 1rev 60 s
= 9,893 × × = 9,893 × ×
s 2 2π rad 1 min s 2 2π rad 1 min
rpm rpm
= 94,471 = 94,471
s s