Moment of Inertia of a Rotor System

Question

I need to settle a discussion on the following question:

Part A:

• Find the equivalent spring constant for springs in series
• Use equations of motion, sum of torques to find natural frequency of the system

where:

$$\begin{gather} \dfrac{1}{k_{eq}} = \dfrac{1}{k_1} + \dfrac{1}{k_2} + \dfrac{1}{k_3} \\ I_p\ddot\Theta = -(k_{eq} + k_4 + k_5 r^2 + k_6 r^2)\Theta \end{gather}$$

Part B:

System in torsion therefore:

$$w_{nf}=\sqrt{\dfrac{k}{J}}$$

where:

$$J = \dfrac{mr^2}{2}$$

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Issues

Solutions from my lecturer suggest adding the three strings in series and not using an equivalent

$$w_{nf}^2=\dfrac{k_1 + k_2 + k_3 + k_4 + k_5 r^2 + k_6 r^2}{J}$$

I believe you have to use the following, taking in to account the equivalent spring constant for the springs in series:

$$w_{nf}^2 = \dfrac{k_eq{} + k_4 + k_5 r^2 + k_6 r^2}{J}$$

Could someone please clarify whether the use of K123 being the equivalent is the right method. All literature suggests equating spring constants in series yet my lecturer appears to think differently.

Answer 1

You are correct. The schematic shows that k1,k2,k3 work in series with k4 and k5 and k6 being parallel at the upper side of the disk.

Your lecturer has made an error, and next time most likely will add an errata to his notes.