Velocity and Acceleration Machinery's Handbook, 31st Edition
185
Table 3. Rotary Motion with Constant Acceleration
To Find Known
To Find Known
Formula
Formula
Motion Uniformly Accelerated from Rest ( ω o = 0)
θ , ω f θ , α α , ω θ , ω f ω f , t θ , t
θ = 1 ∕
α , t ω f , t α , t θ , t α , θ ω f , α
t = 2 θ ÷ ω f
2
2 α t
2 ' i = α
t
θ =
1 ∕
2 ω f t
t
θ
θ = ω f ω f = α t
2 ÷ 2 α
t = ω f ÷ α α = 2 θ ÷ t 2
f
ω f = 2 θ ÷ t
α = ω f
2 ÷ 2 θ
ω f
α
2
f ~ α = ω f ÷ t Motion Uniformly Accelerated from Initial Velocity ω o α i =
θ = ω o t +
ω o , ω f , θ α = ( ω f
2 ) ÷ 2 θ
α , t , ω o ω o , ω f , t ω f , α , t ω o , α , t ω o , θ , t ω o , α , θ
1 ∕
2
2 α t
2 − ω
o
θ = ( ω f + ω o ) t ÷ 2
ω o , ω f , t
α = ( ω f − ω o ) ÷ t α = 2( θ − ω o t ) ÷ t 2 α = 2( ω f t − θ ) ÷ t 2
θ
α
ω o , ω f , α θ = ( ω f
2 ) ÷ 2 α
o , θ , t f , θ , t
2 − ω
ω ω
o
θ = ω f t − ω f = ω o + α t 1 ∕
2
2 α t
Meanings of Symbols
ω f = (2 θ ÷ t ) − ω o
ω f
2
f o 2 ~ ~
α i = +
θ = angle of rotation, radians ω f = final angular velocity, rad/s ω o = initial angular velocity, rad/s α = angular acceleration, rad/s 2 t = time in seconds
θ , α , t
ω f = ( θ ÷ t ) +
1 ∕
2 α t
ω f , α , θ ω f , θ , t ω f , α , t θ , α , t ω o , ω f , α ω o , ω f , θ
2
o f 2 ~ ~
α i = −
ω o = (2 θ ÷ t ) − ω f
ω o
ω o = ω f − α t ω o = ( θ ÷ t ) −
1 ∕
2 α t
t = ( ω f − ω o ) ÷ α t = 2 θ ÷ ( ω f + ω o )
1 degree = 0.01745 radian (See degree-radian conversion table on page 103 )
t
The acceleration of point P is the resultant of r ω 2 and r α and is given by the formula a r r 2 2 2 ~ α = + ^ h ^ h When α = 0, this formula reduces to: a = r ω 2 Example: A flywheel on a press rotating at 120 rpm is slowed to 102 rpm during a punch ing operation that requires 3 ∕ 4 second for the punching portion of the cycle. What angular deceleration does the flywheel experience? Solution: From the table on page 184 , angular velocities corresponding to 120 and 102 rpm, respectively, are 12.57 and 10.68 radians per second. Therefore, using the formula: α ω f ω o – ( ) t ÷ = 10.68 – 12.57 ( ) 3 ⁄ 4 ÷ –1.89 3 ⁄ 4 ÷ = = –2.52 rad/s 2 = which is, from the table on page 184 , − 24 rpm per second. The minus sign in the answer indicates that acceleration α acts to slow the flywheel, that is, the flywheel is decelerating.
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