Rotor dynamics: a case study

A rotor-dynamic analysis for evaluation of the critical speeds of an axial flux electric motor is carried out both numerically by simulation and experimentally on a test bench dyno.

A Computer Aided Design (CAD) 3D model is created for Finite Elements Analysis (FEA) according to the following picture depicting all the rotating parts, except shaft and rolling bearings whose modelling for our purpose doesn’t require a solid model.

Despite not axial-symmetric, overall rotor geometry is pretty “flat” allowing 2D surfaces extraction from original 3D geometry getting shell-mesh elements with thicknesses and offsets accurately matching actual solid geometry. This solution provides both a more precise analytical model and basically a much faster runtime than the equivalent solid alternative.

Glue connection is applied at every mating interface between rotor discs, magnets and the intermediate spiral-coil disc whose mechanical orthotropic properties are defined according to the following compliance matrix and constrain relations:

Both a static bending test on one coil specimen and the actual test on bench provided useful information for FEA correlation of unknown parameters, thus acceptably modelling the low shear stiffness along axial direction.

Central supporting shaft between discs is modeled by 1D elements and grounded with bearing elements with linear stiffness properties.

Numerical results

The main goal is finding the critical speeds which clearly need to be identified as one possible failure condition of the motor. A complex modal analysis output is summarized by a Campbell diagram with modes visualization as depicted in next figure. This typically is a great visualization tool for identification of critical speeds by intersection of orders (dash dotted lines multiples of the first one being the rotor speed itself) with vibration modes (continuous and dotted lines each one relative to a particular deformation mode at a certain frequency range varying with speed).

Numerical analysis also provides a way to identify the most relevant modes by looking at participation factors, a kind of “significance” parameter to distinguish the main vibrations among the negligible ones.

In this report we’re focusing on the first one, indicating 2000rpm as a possible critical speed by the intersection of a saddle-like vibration mode close to 600Hz and the excitation by stator cores as multiple of the rotation speed.

Figure below reports all computed vibration modes in order of appearance.

Test results

Let’s have a look now at some measurements.

One motor is placed on a dyno test bench with a set of 3 laser sensors sampling at 200kHz positioned in a circular array with phasing 0° - 60° - 90° to possibly match nodes and anti-nodes of the expected disc’s modal shapes as previously computed in simulation.

more in detail, for each rotor speed acquisition data appears like:

the first window on left depicts the axial profile oscillation of the rotor at 50rpm, basically a static condition with no dynamic effects and thus a measure of the actual disc axial runout error. At 500rpm dynamics effect appears, but raw data provides misleading information. By subtracting the 50rpm "baseline" you get the cleaned expected oscillation, yet still being in 0.01mm range.

It's time to look at the 2000rpm condition.

this diagram is showing the cyclic deformation you saw in the previous animation above, where the 3 moving dots are plotting each phase reading.

For each rotor’s revolution a double oscillation per period (1 disc rotation) appears and confirm this as the first critical speed we were looking for.

Ref. mode function overlaying signals is a simple trigonometric function defined as sum of a main sinusoidal of 66.7Hz (1/9th order sub-harmonic of 1st saddle mode @600Hz) and a secondary of 600Hz identifying the critical speed of 2000rpm by the intersection with 18th order related to the number of stator cores as source of excitation.

The overall speed range mapping for a single rotor revolution assumes the following shape:

Key aspects:

• Quadratic gradient of displacement by centrifugal effect
• 1st saddle mode at 2000rpm: 1/9th order sub-harmonic + 18th order vibration (1st critical speed)
• Effective geometric error compensation only for lower rpm range, a better quality component is needed for significant vibration measurements at higher speeds

Conclusions

I tried to summarize a recent engineering activity which cost me a lot of effort, yet enjoying it very much. It's not the end.. it's actually just started. Much more to be done!

Thanks for reading.