Question

A machine of mass 75 kg is mounted on springs and is fitted with a dash *** 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 *** 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; * = 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,

• = 3 x 10 x 103 = 30 x 103 N/m

We know that natural circular frequency of motion,

Resistance of dash *** at unit velocity

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

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 fn1 = frequency of the damped vibration = ωd/2π

fn2 = frequency of the undamped vibration = ωn/2π

3. Periodic time of the damped vibration.

We know thatperiodic time of the damped vibration.