Damping Factor or Damping Ratio

          The ratio of the actual damping coefficient (C) to the critical damping coefficient (C_C) is known as Damping factor or Damping Ratio. Mathematically it is expressed as, 

The Damping Factor is the measure of the relative amount of damping in the existing system with necessary for the critical system.

Logarithmic Decrement

          It is used to find the damping ratio of an underdamped system in the time domain. It is defined as the natural logarithm of the amplitude reduction factor. The amplitude reduction factor is the ratio of any two successive amplitudes(Peaks) on the same  side of the mean position. 

          If x₁ and x₂ are successive values of the amplitude on the same side of the mean position, as shown in Fig.,

Then the amplitude reduction factor, 

Where t_p is the time period of forced oscillation (or) the time difference between two consecutive amplitudes. As per definition, logarithmic decrement,

In general, amplitude reduction factor,

Logarithmic decrement,

Magnification Factor (or) Dynamic Magnifier

          It is the ratio of maximum displacement of the forced vibration  (x_max) to the deflection due to the static force F(x₀). We have proved in the previous article that the maximum displacement or the amplitude of forced vibration. 

Relation between magnification factor and phase angle for different values of ω/ω_n

Magnification factor or dynamic magnifier,

The magnification factor or dynamic magnifier gives the factor by which the static deflection produced by a force F (i.e x₀) must be multiplied in-order to obtain the maximum amplitude of the forced vibration (i.e x_max) by harmonic force F Cos ωt. 

                                                 x_max = x₀ X D

Above Fig shows the relation between the magnification factor (D) and phase angle Φ for different value of and for  values of damping factor  C/C_C =0.1, 0.2, and 0.5.

Note: 1. If there is no damping (i.e if the vibration is undamped), then c=0. In that case, magnification factor, 

2. At resonance, ω/ω_n . Therefore magnification factor,