Torsional vibrations is predominant whenever there is large discs on relatively thin shafts (e.g. flywheel of a punch press). Torsional vibrations may original from the following forcings (i) inertia forces of reciprocating mechanisms (such as pistons in IC engines) (ii) impulsive loads occurring during a normal machine cycle (e.g. during operations of a punch press) (iii) shock loads applied to electrical machinery (such as a generator line fault followed by fault removal and automatic closure) (iv) torques related to gear mesh frequencies, turbine blade passing frequencies, etc. For machines having massive rotors and flexible shafts (where the system natural frequencies of torsional vibrations may be close to, or within, the source frequency range during normal operation) torsional vibrations constitute a potential design problem area. In such cases designers should ensure the accurate prediction of machine torsional frequencies and frequencies of any torsional load fluctuations should not coincide with the torsional natural frequencies. Hence, the determination of torsional natural frequencies of the system is very important.
Simple System with Single Rotor Mass:
Consider a rotor system The shaft is considered as massless and it provides torsional stiffness only. The disc is considered as rigid and has no flexibility. If an initial disturbance is given to the disc in the torsional mode and allow it to oscillate its own, it will execute the free vibrations .It shows that rotor is spinning with a nominal speed of ω and excuting torsional vibrations, θ(t), due to this it has actual speed of (ω + θ(t)). It should be noted that the spinning speed remains same however angular velocity due to torsion have varying direction over a period. The oscillation will be simple harmonic motion with a unique frequency, which is called the torsional natural frequency of the rotor system.
From the theory of torsion of shaft, we have K = T/ θ = GJ/I
where, Kt is the torsional stiffness of shaft, Ip is the rotor polar moment of inertia, kg-m2 , J is the shaft polar second moment of area, l is the length of the shaft and θ is the angular displacement of the rotor. From the free body diagram of the disc as shown in Figure
∑ External torque of disc Ip θ ⇒ – K t θ = I pθ