In a damped free vibration of a mass supported on a spring and a damper, where the damping force is proportional to the velocity, the ratio of two successive amplitudes

Explanation:

Vibration:  Vibration is a periodic motion of small magnitude. But for simplicity, we can assume it as a simple harmonic motion with a small amplitude.

Damped Vibration:

• When there is a reduction in amplitude over every cycle of vibration, the motion is said to be damped vibration.
  This is due to the fact that a certain amount of energy possessed by the vibrating system is always dissipated in overcoming frictional resistances to the motion.

Un-damped vibrations:

• When there is no friction or resistance present in the system to contract vibration then the body executes un-damped or damped free vibration.

Damped Free Vibration:

• Damped free vibration describes the mobility of an object without the action of any externally applied force.

​\(m\frac{{{d^2}x}}{{d{t^2}}} + c\frac{{dx}}{{dt}} + kx = 0\)

Where m is the mass suspended from the spring, s is the stiffness of the spring, x is the displacement of the mass from the mean position at time t and c is the damping coefficient. Since excitation force is absent in the equation hence it is the equation of free-damped vibration.

Damped Free Response x = Xe^-ζω_nt. sin(ω_d.t + ϕ₁)

where: x = displacement of mass from the mean position, X = Amplitude, ζ = damping factor or damping ratio, ω_d = Damped frequency, ϕ₁ = phase angle.

Decrement ratio or successive amplitudes ratio:

\(\frac{{{x_0}}}{{{x_1}}} =\frac{{{x_1}}}{{{x_2}}} =\frac{{{x_2}}}{{{x_3}}} =\frac{{{x_3}}}{{{x_4}}} ....=e^{ζω_nt}=Constant\)