Question

 A machine of mass 75 kg is mounted on springs and is fitted with a dash pot to damp out vibrations there are three springs each of stiffness 10 n/mm and it is found that the amplitude of vibration diminishes from 38.4 mm to 6.4 mm in two complete oscillations. Assuming that the damping force varies as the velocity, determine 

1. The resistance of dash pot at unit velocity

2. The ratio of frequency of the damped vibration to the un damped vibration

3. The periodic time of the damped vibration.

Answer 1

m = 75 kg; s = 10 N/mm = 10 x 10³ N/m; x₁ = 38.4 mm = 0.0384 m;  x₃ = 6.4 mm = 0.00064 m

Since the stiffness of each spring is 10 x 10³ N/m and there are 3 springs, therefore total stiffness, 

s = 3 x 10 x 10³ = 30 x 10³ N/m

We know that natural circular frequency of motion,

 Resistance of dash pot at unit velocity

Let c =  resistance of dash pot in newtons at unit velocity i.e. in N/m/s,

x₂ = Amplitude after one complete oscillation in metres, and

x₃ = Amplitude after two complete oscillation in metres.

We know that x₁/x₂ = x₂/x₃

2. Ratio of frequency of the damped vibration to the un damped vibration

Let f_n1 =  frequency of the damped vibration  = ω_d/2π

f_n2 =  frequency of the undamped vibration = ω_n/2π

3. Periodic time of the damped vibration.

We know that periodic time of the damped vibration.