Module 7   :  Free Undamped Vibration of Single Degree of Freedom Systems; Determination of Natural
                       Frequency ; Equivalent Inertia and Stiffness; Energy Method; Phase Plane Representation.
Lecture 2 : Determination of Natural Frequency
Torsional Spring-mass system
 
We have so far discussed the case of a translational spring and mass system. Another common case occurring in engineering systems is a shaft carrying rotors as shown in Fig. 7.2.1 For example, this could be the shaft of a compressor/turbine or crank shaft of an IC engine. The shaft represents a lumped torsional spring and the rotor's polar mass moment of inertia represents lumped mass/inertia. The d.o.f we will be dealing with is now the angular motion of the shaft rather than any translational motion. The governing equation can be readily shown to be
 

Fig. 7.2.1 Torsional system
7.2.6

Where represents the torsional stiffness of the shaft and represents the polar mass moment of inertia of the disk attached at the end of the shaft. is given by where G represents the modulus of rigidity of the shaft and is polar moment of area of the shaft cross-section and is the length of the shaft. The solution will be very similar to the case discussed earlier.
   
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