Internal Damping Estimation

This example teaches how to determine a rotor’s internal damping. Knowing the internal damping can help understand under which speeds machinery can operate safely.

A Jeffcott rotor is used in the example. It consists of a disk and a shaft. The rotor is excited by a load until the shaft is deflected and settled down.The applied load is then removed, and the oscillations are used to calculate the damping ratio.

Figure 1 shows the system with an external load, and the force is represented by an arrow. The disk is transparent to provide a better view of the setup.

Figure 1: Jeffcott Rotor

The response from the system can be seen in Figure 2, where the vertical deflection of the rotor can be seen over time. 

Figure 2: Applied force and vertical disk deflections 

The system's internal damping can be calculated using the logarithmic decrement, [1], of the deflections. For this model, the equivalent internal damping will be:

 c_i = ln( y_n / y_n+5 ) / (5 * 2* π) * 2√mk_y,

where m=100kg, k_y = F_y/y₀, F_y = 1000 N, y₀ = y_n= 0.1001 m, and y_n+5 = 0.021 m. By substituting the given values, c_i is calculated as: 

c_i = ln( 0.1001 / 0.021 ) / (5 * 2 * π) * 2 √100*(1000/0.1001) c_i = 99.36 Ns/m.

More information on internal damping can be found in this blog.

References

[1]  Inman, D. J. Engineering Vibration (3rd ed.). Pearson Education, Inc., 2008: 43-48.