Question

The mass of a single degree damped vibrating system is 7.5 kg and makes 24 free oscillations in 14 seconds when disturbed from its equilibrium position. The amplitude of vibration reduces to 0.25 of its initial value after five oscillations.

Determine: 

1.  stiffness of the spring, 

2.  logarithmic decrement, and 

3.  damping factor, i.e. the ratio of the system damping to critical damping.

Answer 1

Given : m = 7.5 kg

Since 24 oscillations are made in 14 seconds, therefore frequency of free vibrations,

f_n = 24/14 = 1.7

1.  stiffness of the spring,

Let s = stiffness of the spring in N/m.

2.  logarithmic decrement, 

Let x₁ = Initial amplitude,

x₆ = Final amplitude after five oscillations = 0.25 x₁ (Given)

We know that  logarithmic decrement, 

3.  damping factor

Let c = damping coefficient for the actual system, and

c_c = Damping coefficient for the critical damped system.

We know that logarithmic decrement (δ)