Vibration Issues (Continued)

The vibration in my mower deck is caused by unbalanced mower blades. The distance between the mower blade’s center of mass and the rotation axis of the motor shaft is often referred to as eccentricity, and this eccentricity results in a centrifugal force as the mower blade’s center of mass is swung around the axis of rotation.

A cheap prop balancer from Amazon with a 0.5in diameter keyed shaft from McMaster Carr. The two head press fit bushings ensures the keyway doesn’t interfere with the shaft’s rotation.

At high rotation speeds, even a small amount of eccentricity can result in a big centrifugal force. To see how far out of balance my mower blades were, I purchased a small RC plane prop balancer off the internet. With a few pieces of hardware from McMaster Carr I was able to get it to work with my mower blades.

One side of the blade was heavy, so I started putting scotch tape on the opposite side to see how much mass it would take to balance the blade. It soon became apparent I was going to need more mass than a few strips of tape, so I stuck a few #8-32 hex nuts on there. Once I got it pretty well balanced, I removed the tape and hex nuts and weighed them.

It took almost 3 grams of mass to balance one of my mower blades.

Because the mass of the blade assembly is not coincident with the rotation axis of the prop balancer, there exists a moment about the rotation axis when the center of mass is not directly below it.

By putting tape and hex nuts on the opposite end of the blade, we created an additional balancing moment. The sum of these two moments must be zero for a balanced blade at equilibrium.

We can measure the mass of the blade, which is 798.1g. We also know the mass of the balance weight, 2.8g. And we know the position of the balance weight: it’s half the length of the blade, 6in.

Using this information, we can solve for the eccentricity:

Calculations for determining the mower blade’s eccentricity, or the distance away from the rotation axis the mower blade’s center of mass lies.

And the resulting centrifugal force is:

The resulting centrifugal force if a mass of 798.1g is 0.021in away from the center of rotation at 4800RPM. I find some calculations are easier to do in SI units, making the conversion back to imperial units at the end.

To be fair, I didn’t get my motors up to 4800RPM. But at speeds near the natural frequency of a structure you get a lot of force magnification, so it’s entirely possible the mower deck was experiencing a force at least that large or larger.

I know what you’re thinking: that’s a lot of words to say “balance the blades, dummy.” But there’s a few reasons this exercise is valuable.

Firstly, it gives me a rough idea of how much material needs to be removed from the mower blade. It needs to be 2.8g lighter on the side opposite where we placed the balance mass. The volume of material that needs removed is:

The volume of material that needs to be removed from this specific mower blade assembly to align the center of mass with the rotation axis.

If the blade is 0.203in thick and about 2in wide, that means we need to grind the end of the blade down by about:

How much material that needs to be ground off the heavy end of the mower blade to achieve balance.

That estimation will hopefully save me a few trips between the blade balancer and the grinder.

Secondly, these calculations shed some light on how significant the imbalance is between the various blade assembly components. It’s not just the mower blade that’s unbalanced: the adapter, screws and even the motor shaft (it’s keyed on one side, after all) also contribute to the unbalance.

However, their combined mass is smaller than the blade’s mass, and their locations from the axis of rotation are also relatively small. For example, you’d have to add nine 1/4 washers to one of the mounting bolts on the blade assembly to offset the unbalance that exists in the blade:

The number of 1/4 washers you’d have to add to one of the mounting bolts to offset the imbalance inherent in the mower blade.

That stack of washers would be almost 5/8 of an inch tall and weigh 20.7g. So the lowest hanging fruit in our attempt to balance the blade assembly is going to be working on getting the blades themselves better balanced.1

And lastly, these calculations reveal a major flaw with my mower blade assembly design: nothing holds the blade fixed relative to the adapter. The screws are inserted through clearance holes in the blade, and if you loosen them, you can wiggle the blade around relative to the adapter by about 1/16 of an inch.

The bottom of the mower blade assembly. The two screws are the only thing that sets the position of the blade relative to the shaft adapter.

If 0.021in eccentricity already exists in the mower blade, and that creates enough vibration to peel open an eyebolt on my mower deck mounts at resonance, then any slippage of the blade during operation will easily re-introduce that vibration.

It might be worthwhile to redesign the mower blade adapter to locate off the center hole in the blade to prevent this from happening. And to also redesign how the mower deck connects to the robot chassis. But for now we’ll focus on balancing the blades and see what that does for us.

  1. Adding washers like this would likely change the orientation of the inertial axes and introduce couple unbalance. I’m unsure how significant this couple unbalance would be. While one or two washer shims might work for fine tuning the balance, attempting to eliminate the unbalance like this may create a situation where the blade is balanced statically but still vibrates when rotating.

Vibration Issues

Before using the robot mower, I always power the blade motors on in my garage for a few minutes to make sure everything is working right. I pulled our vehicles out of the garage, told the missus to expect some loud noises, and made sure that Mr. Mower Jr. was safe inside the house.

The nice thing about the new motor controller I’m using on the blade motors is that I can vary the rotation speed of the blades. So instead of commanding full throttle from the get-go like I had been doing with a relay, I decided to slowly ramp up the speed. It’s pretty incredible listening to the crescendo of the blades as they spool up.

About halfway to full throttle, I started to get a ridiculous amount of vibration. It was extraordinarily loud. The vibration was so intense that it peeled open the eye of one of my deck support eyebolts.

The eyebolt on the turnbuckle supporting the mower deck. The vibration was so intense it peeled it open.

At first, I thought this was just a fluke, or that perhaps the eyebolt was already close to failure, so I replaced it with a spare and repeated the process. I’m not very smart. The result was identical.

When I was in engineering school, I had a choice to take Design of Heat Exchangers or Vibrations and Acoustics for technical electives. As you can probably guess by looking at my mower deck design, I chose Design of Heat Exchangers. It was a fun class, but pretty useless for anything mower related.

So to work through this issue, I’ve had to do some homework. Lucky for me, I managed to stumble upon a copy of Timoshenko’s Vibration Problems in Engineering at the Goodwill a few years ago, which has proven a valuable resource in my vibrations education.

Timoshenko says it better than I can, so I will quote him directly:

By an impulse or sudden application and removal of an external force, vibrations of the [spring mass] system can be produced. Such vibrations are maintained by the elastic force in the spring alone and are called free or natural vibrations.

Vibration Problems in Engineering, page 1

The mower deck is a kind of spring mass system, though geometrically more complex. There is a frequency at which it will vibrate if you were hit it with a hammer, just like a tuning fork will vibrate at 440Hz if you strike it against a hard surface. That unique frequency is called the system’s natural frequency, or frequency of free vibration.

Unfortunately, we’re not just hitting the mower deck with a hammer once. We’re hitting it with a hammer 4800 times per second. That oscillating force does not allow the mower deck to vibrate freely. That’s not free vibration. It’s forced vibration.

In the case of the mower deck, the frequency of that forced vibration is whatever speed the motors are spinning at. Because the motors go from zero to 4800RPM, the mower deck experiences vibration at every frequency along the way. Chances are good that the motors will create forced vibration at the natural frequency of the mower deck at some point as they spool up to operating speed.

A very curious thing happens when you force a structure to vibrate at it’s natural frequency. You get resonance. And the amplitude of vibration becomes very large at resonance. Absent damping it will become infinite. Timoshenko includes a chart that illustrates this phenomenon very well.

Resonance, and the Magnification Factor, from Timoshenko’s Vibration Problems in Engineering, page 42.

When I was spooling up the mower deck motors, I consciously decided to do it slowly to minimize electrical current spikes. This created a problem that didn’t previously exist with the relay, where we rapidly turned on the motors. Timoshenko explains:

The amplitude of vibration [at resonance] increases indefinitely with time… This shows that, while we theoretically obtain infinite amplitude of forced vibration at resonance in the absence of damping, it also takes time to build up these large amplitudes.

Vibration Problems in Engineering, page 48

In other words, if we maintain forced vibration at the natural frequency of a structure, the amplitude of vibration will increase continuously to the extent allowed by any damping present in the system.

By slowly increasing the speed of the motors, I was allowing the mower deck to experience resonance and the accompanying large vibration amplitudes for a period of time long enough to damage the turnbuckle eyes. Who’d have thought?

Something else I noticed was that I had trouble getting the motors to speed up once they were at speeds close to the mower deck’s natural frequency, when it started vibrating really bad. Timoshenko has something interesting to say about that, too:

Experiments show that if any vibrating system is once allowed to reach a steady state just below resonance, it then becomes difficult to accelerate the machine through the resonance condition. Additional power supplied for this purpose is simply used up in increasing the amplitude of vibration rather than the running speed of the machine.

Vibration Problems in Engineering, page 48

I think I have a good grip on what the problem is. Next post I’ll discuss some things that can be done to mitigate this surprising issue.