Question 7 of 7: First Torsional Natural Frequency by Holzer's Method

Question 7: First Torsional Natural Frequency by Holzer's Method (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. A free–free, six-inertia torsional chain: propeller inertia $J_P=0.14$ kg·m² and gearbox/flywheel inertia $J_0=1.6$ kg·m² coupled by stiffness $K_1=2.6\times10^5$ N·m/rad, then four equal engine-cylinder inertias $J=0.75$ kg·m² each, each pair coupled by stiffness $K=4.2\times10^5$ N·m/rad.

Find. The first (lowest non-zero) natural torsional frequency of the chain, expected between 300 and 400 rad/s.

Approach. Assume unit angular amplitude at J_P, march station-by-station through Holzer's recursion – at each inertia accumulate the running inertia torque and use it to find the next station's amplitude across the connecting stiffness – and search for the trial frequency at which the residual torque left over after the last (free) inertia vanishes; that frequency is a natural frequency of the free–free system.

1. Holzer recursion. With $\theta_1=1$ (at $J_P$) and running torque sum $\Sigma J_k\omega^2\theta_k$, each subsequent amplitude is $$\theta_{i+1}=\theta_i-\frac{\sum_{k\le i}J_k\omega^2\theta_k}{K_i}$$ where $K_i$ is the stiffness connecting station $i$ to $i+1$ ($K_1$ first, then $K$ four times). A trial $\omega$ is a natural frequency of the free–free chain when the FULL running torque sum (after including the last, free-end inertia $J_4$) comes out to zero – that end has no stiffness left to react a residual torque against.
2. Scanning the stated range. Evaluating the residual torque at trial frequencies swept from 300 to 400 rad/s (the range the question itself suggests) shows exactly one sign change, between 388.0 and 388.5 rad/s.
3. Bisecting to the root. Refining by bisection inside that bracket converges to $$\boxed{\omega_1 \approx 388.4\ \text{rad/s}}$$ with residual torque $<10^{-9}$ N·m at convergence, comfortably inside the suspected 300–400 rad/s band. The corresponding (unnormalised) mode shape is $\theta = [1.000,\ 0.919,\ 0.340,\ -0.330,\ -0.911,\ -1.247]$ for $[J_P, J_0, J_1, J_2, J_3, J_4]$ – the sign change partway along the engine's four cylinders (between $J_2$ and $J_3$) marks the node of this first torsional mode.