How is the fundamental frequency of a large rotating body determined?

Question

Okay, I understand that "large" is relative. I'm referring to the 60" diameter 5000 rpm counterrotating flywheels I've postulated in this question. When I was taking flying lessons in an 8KCAB Decathlon there was an RPM band (2000-2250 rpm) which we were told to avoid due to vibration resonance. It occurs to me that there should be a similar critical frequency band for my time travelers which they should either avoid or otherwise try to pass through as expeditiously as possible. Is this determined experimentally, or is there a way to craft a decent estimate of it given the postulated design parameters?

Answer 1

How is the fundamental frequency of a large rotating body determined?

5000 RPM / 60 = 83.33 Hz is just the fundamental resonance. Bearing rotation ratios create harmonics per fundamental revolution unless it ia maglev airbearing. In a spacecraft for storing excess energy, they might spin up to 60 kRPM. This is accentuated by static imbalance and dynamic imbalance that must be offset with calibrated counterweights sensed by an accelerometer just like car wheels. Structural and alias effects will occur as in all spring-mass systems. Standing waves occur when there is a discontinuity in impedance coupling waves from one member to another. Its wave frequency is determined by twice the path length and velocity. This is true for acoustic and electromagnetic waves and is measured by a reflection coefficient. The end-stop to a wave, if rigid acts as a short circuit to reflect waves inverted.

A recording is necessary with audio+video for any critical analysis. Please advise if possible, and find loudest vibration on the frame.

I don't know of any solid structural frame that does not have resonances unless it has an active overdamped design.

The key factor to minimize damaging effects with sufficiently high resonant frequency is such that the 2nd order pole effects of displacement vs acceleration do not strain the material and the amplification or Q is sufficiently low.

2100 RPM indicates the main structural resonance is around 35 Hz.

Anecdotal experiences

When I started in Aerospace R&D in '75 every product was tested in-house for shock and vibe. **The best damping material such as Solothane or Lord Mounts still had a Q of 5 even with the lossy or plasticity of an elastomer at the natural resonance with its compliance. The stiffness must be chosen to dampen the required looseness at the resonant frequency of acceleration. I had to find a method to dampen my design of an OCXO that was ultrastable and ruggedized yet subject to drift after resonance during a vibration sweep at 15g excitation for rockets. I was only allotted 5 mm of surrounding space to dampen the vibration. Oddly enough, I ended up using carbon ESD foam used for CMOS to pass my criteria for damping the resonance. The error criteria was frequency stability of 1e-10 or 0.01 part per million used for Doppler ranging.

In control system theory, I learned early that any 2nd order system with added latency in feedback or time delay or mechanical slack could resonate and require damping to prevent. Rotating bearings are commonly resonant and vibration analysis will indicate when maintenance is required. Harmonic effects occur from the primary rotation and gearing effects of a number of balls per revolution with aliasing effects that create difference frequencies. Harmonics indicated something was wearing- out by being out of round. These are common tests for turbines with 3-axis accelerometers added in critical locations for testing and correlations with acoustic spectrum and repair results. Any loose parts tend to resonate with fundamental and harmonic frequencies due to the available displacement and the latency in the structure to support itself from control theory.

When I was a senior Test Engineer for a disk drive manufacturer, I used 3D Modal Analysis tools with plastic hammer and 3D accelerometer to measure the 3D transfer function and resonances. The spring-mass structure with rotating and linear motion reveals all the many faults in the structure. The controlled high actuator had to be resonant-free to avoid overshoot and writing data off-track. The early 5.25" drives all had shock mounts to reduce the spectrum from an impulse or mechanical shock. The Q and resonances were unavoidable, but it was a tradeoff. As drives got smaller, the precision balance, stiffness of structure and air-bearing effects improved such that adding shock mounts only made it worse, so the solution was to make them as light as possible in the moving parts so they were tight and friction-free in the moving direction and very stiff in the orthogonal directions. Resonances were always deadly for the servo effects on data errors, with the high-speed noise from ultra-fast seeks so the designs required the most sophisticated complex electro-mechanical servo solutions (yet appear ultra simple), I have ever seen to become error-free with lots of margin.

The best stepper motor solution in the 80's, was from Hitachi-NPL for the first stepper-motor 5.25" hard disk drives. They used a thin plastic rotary disk that contained oil to surround around a brass disk inside to provide the fastest low mass seek speed with damping and a tight stainless steel foil for coupling the motor to moving mass or rails and a digital-controlled acceleration, velocity and position. I have yet to see any stepper-servo that was faster and so quiet. Their quality control system for documentation was state of the art.

Comments on counter-rotating masses

My primitive understanding is that imperfections in balance result in low-frequency resonances that challenge the natural cancellation of the moments of inertia. Each flywheel is a structural spring-mass system with resonant frequencies that will sound like a bell when struck. Any slight differences between the two flywheels will create lower and upper sideband resonances from intermodulation of the airwaves and eddy current frequencies unless it's in a vacuum on axial magnetic bearings.

there a way to craft a decent estimate of it given the postulated design parameters?

Design parameters depend on the degrees of freedom in each member, the mass and stress-strain stiffness. The parameters require modeling with test correlation with scale models for accuracy. The transfer function must be simplified just as it is done with Operation Amplifiers with many transistor stages by adding a capacitor to swamp the phase shift of each stage with a very low-frequency pass filter (10 Hz) and very high gain (10k to 10G).