## Modelling Flywheels on a Shaft

Three flywheels, 1, 2, 3, with moment of inertia
$I_1, \: I_2, \: I_3$
respectively, are on a shaft with torsional stiffness
$C$
. What are the natural frequencies of oscillations of the flywheels?

Let the angular displacements of the flywheels 1, 2, 3 be
$\theta_1, \: \theta_2, \: \theta_3$
respectively. The rotation of each flywheel exer_1ts a torque on the shaft and hence on the other flywheels.
For flywheel 1
$\Gamma_1 = I_1 \ddot{\theta}_1= C(\theta_2 - \theta_1)$

For flywheel 2
$\Gamma_1 = I_2 \ddot{\theta}_2= C(\theta_1 - \theta_2)+C(\theta_3 - \theta_2)=C(\theta_1-2 \theta_2+ \theta_3)$

For flywheel 3
$\Gamma_3 = I_3 \ddot{\theta}_3= C(\theta_2 - \theta_3)$

Divevide the equations by
$I_1, \: I_2, \: I_3$
resctively and write in matrix form as


$\begin{pmatrix} \ddot{\theta}_1 \\ \ddot{\theta}_2 \\ \ddot{\theta}_3 \end{pmatrix} \left( \begin{array}{ccc} - \frac{C}{\Gamma_1} & \frac{C}{\Gamma_1} & 0 \\ \frac{C}{\Gamma_2} & -2 \frac{C}{\Gamma_2} & \frac{C}{\Gamma_2} \\ 0 & \frac{C}{\Gamma_3} & - \frac{C}{\Gamma_3} \end{array} \right) \begin{pmatrix}\theta_1 \\ \theta_2 \\ \theta_3 \end{pmatrix}$

The amgular frequencies of the vibration are the square roots of the negative eigenvalues,
$\omega_i = \sqrt{- \lambda_i}$
and the frequenceies are then
$f_= =2 \pi \sqrt{- \lambda_i}$
.

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