Dynamics of Structures

Chapter 2. Damped Single Degree of Freedom System

2.3 Estimating Damping Using Logarithmic Decrement

Figure 2.3.1 Estimating Damping Using Logarithmic Decrement

The damping ratio ζ can be estimated by measuring two successive peak displacement from the system response to a free vibration. In the above Figure, u₁ and u₂ are the first and second peaks for a one DOF system, these two peak displacements are separated by TD,

Eq. 2.3.1

And,

Eq. 2.3.2

the logarithmic decrement δ can be calculated as,

Eq. 2.3.3

For small values of the damping ratio ζ, this equation can be approximated to,

Eq. 2.3.4

Here is an interesting interactive graph practice!

By selecting the system parameters, you can check the values of the first and second peaks, calculate the logarithmic devrement and confirm that the calculated damping ratio marches the value used to create the graph!

uo

vo

ω

ζ

Observations

The first positive peak is at  t=0.040 \space t = 0.040 \space sec and is equal to  u1=1.020 \space u_1 = 1.020\space length unit
The second positive peak is at  t=1.300 \space t = 1.300 \space sec and is equal to  u2=0.899 \space u_2 = 0.899\space length unit
The Logarithmic Decrement can be calculated as δ=ln⁡(u1u2)=ln⁡(1.0200.899)=0.126  \delta = \ln\left(\frac{u_1}{u_2}\right) = \ln(\frac{1.020}{0.899}) = 0.126 \space
We can now calculate the damping ratio  ζ=δ2π=0.1262π=0.02 \space \zeta = \frac{\delta}{2 \pi} = \frac{0.126}{2 \pi} = 0.02