Vibration and Forced Vibration Equations

1. Angular velocity (ω)
ω=2 πf
ω=2 π × 10=20 π rad /s
Time period of one cycle ( T )
T = 1
f
T = 1
10 =0.1 s
2. Maximum acceleration
amax=ω2 A
amax= ( 20 π ) 2 × 180
1000 =72 π2 m/s2Maximum velocity
v=ωA
v=20 π × 180
1000 =3.6 π m/s
3. Velocity at 100 mm from centre
v=± ω √ A2−x2
v=± 20 π √ ( 180
1000 )
2
− ( 100
1000 )
2
=± 2.99 π m/s
4. Acceleration at 50 mm from centre
a=−ω2 x
a=− ( 20 π ) 2
( 50
1000 )=−20 π2 m/s2
5. Kinetic energy at 150 mm from centre
KE= 1
2 m ω2 ( A2−x2 )
KE= 1
2 ( 30
1000 ) (20 π )2
( 180
1000
2
− 150
1000
2
)=5.94 π2 ×10−2 J