In the two-rotor system shown in the figure, (I₁ < I₂), a node of vibration is situated

Concept:

Two-Rotor System:

• If a shaft held in bearing carries a rotor at each end, it can vibrate torsionally such that the two rotors move in the opposite directions.
• Thus some length of the shaft is twisted in one direction while the rest is twisted in the other.
• The section which does not undergo any twist is called the nodal section.
• The shaft behaves as if clamped at the nodal section and the two sections vibrate as two separate shafts with equal frequencies.

L₁ and L₂ = lengths of two portions of the shaft.

I₁ and I₂ = MOI of rotor 1 and 2 respectively.

q₁ and q₂ = torsional stiffness of lengths L1 and L2 of the shaft respectively.

f₁ and f₂ = natural frequencies of torsional vibrations of rotors 1 and 2 respectively.

\(f_1=\frac{1}{2\pi}\sqrt{\frac{q_1}{I_1}}\;\;\;and\;\;\;f_2=\frac{1}{2\pi}\sqrt{\frac{q_2}{I_2}}\)

At node point:

f₁ = f₂

\(\frac{q_1}{I_1}=\frac{q_2}{I_2}\)

Calculation:

Given:

I₁ < I₂

According to the torsional formula:

\(\frac{T}{J}=\frac{\tau}{r}=\frac{G\theta}{L}\)

\(q=\frac{T}{\theta}=\frac{GJ}{L}\)

At node point f₁ = f₂

\(\frac{q_1}{I_1}=\frac{q_2}{I_2}\)

\(\frac{GJ}{L_1I_1}=\frac{GJ}{L_2I_2}\)

∴ L₁I₁ = L₂I₂

\(\frac{I_1}{I_2}=\frac{L_2}{L_1}\)

∵ I₁ < I₂, ∴ L₁ > L₂

∴ node point shifts towards the rotor of higher MOI.