Non-Uniform Gear Moment of Inertia

[Luke Smith]

I'm working on a project involving analyzing the motion of a reciprocating saw. This saw converts rotational motion from an electric motor to linear motion by means of a slider crank mechanism. The power transmission goes from motor and attached driving gear, to crank, to connecting rod, to slider (which acts as a blade clamp). The crank has non-uniform geometry. It has extrusions on one half and two holes on the other. My immediate assumption was that the geometry was intended to create an offset center of gravity. This would result in a varying moment of inertia when driving the crank with the motor. When the center of gravity is closer to the motor, it is easier to rotate. When it is further from the motor, it is harder to rotate. This essentially means the crank acts as a fast return mechanism that causing the cutting phase to accelerate faster than the extension phase. Here's my question: are non-uniform holes (and extrusions in this case) in gears intended to offset the center of gravity? I assume uniform holes are just made to lighten the gear and allow for better cooling. I haven't been able to dig up any information on the purpose of gears with non-uniform geometry. It would be great if anyone on the forums could confirm my hunch.

I've attached an image of crank solid model.

 

[Dr.D]

If I understand correctly what you are describing, you are proposing an acceleration due to gravity acting on the crank as an effective agent to return the stroke more quickly. This is most unlikely.

Gravity torques are almost always negligible in slider-crank mechanisms. This is not to say they are not there; most certainly they do exist. But they are almost always extremely small compared to the other torques in the machines.

I cannot imagine why the crank of your machine is made as you show it.

 

[Luke Smith]

I agree, the torque due to gravity is minuscule compared to the torque supplied by the motor. The question applies to Euler's second law for rotating bodies and the parallel axis theorem. Euler's law states ∑M = Iα for bodies rotating about an axis through their center (or wherever the moment of inertia was calculated from). For a body that rotates about an axis parallel to the original axis and radius r from the center of mass, Euler's law becomes ∑M = Iα + r_cm×ma_cm. If the crank was geometrically symmetric, the center of mass would be directly in the center. But, because the centroid is offset from the center of the gear, the radius to the center of mass r_cm varies as the crank rotates. At one point, the center of mass is close to the driving gear. At another point half a rotation later, the center of mass is further from the driving gear. This means, holding the torque constant, the angular acceleration will vary.

I've attached another image. This time of the entire gearbox assembly.

 

[Dr.D]

With the assembly drawing, the whole system is more transparent.

The "extrusions" as you have called them are simply a means to add mass to one side of the gear, while the holes add "negative" mass to the other side (equivalent to more mass added to the first side without increasing the total mass). This is all about attempting to obtain a first order balance of the shaking forces. Shaking forces are those forces transmitted the frame (ground) by the moving mechanism.

I am certain that this gear/crank turns about a fixed point on the frame, so its effective moment of inertia is a constant; it does not vary with position. It is whatever value the parallel axis theorem gives.

 

[Luke Smith]

Interesting. I'm not very familiar with vibrations on systems. Is there a simple way to understand how the offset mass helps buffer the vibrations felt by the frame?

My problem was that I visualized the crank as rotating about the driving gear. This would create a varying torque, but it would require a completely different system. Take a look at the attached diagrams if you don't mind. I try to explain my original thinking. Once I actually put it on paper I realize the problem. I still think they're is something to this though, just not my first assumption.

 

[SteamKing]

Since this gear is for a reciprocating saw, it would be reasonable to assume that the raised portion of the gear, since it is concentrated to one side of the center of rotation of the gear, is intended to act as a counterweight of some sort, much like the counterweights attached to the crankshaft of an internal combustion engine. The counterweight can be used to reduce or cancel out vibration which is set up by the reciprocating motion or to make sure that once set in motion, the gear keeps moving when force is not being applied to the saw, like in a piston engine after the power stroke has been applied.

The purpose of the other holes in the gear could be several:
1. They could serve to lighten the gear assembly and increase the effect of the counterweight.
2. They could be used to balance the assembly to reduce undesirable vibration when the gear is turning.
3. They could be used to provide access to bolts or other fasteners in the reciprocating mechanism, to which access must be gained thru the gear assembly.

 

[Luke Smith]

I do want to point out that during normal operation the crank gear is horizontal. I'd think any flywheel-like behavior would be negligible because gravity would create a moment perpendicular to the axis of rotation and, like Dr. D said, the torque due to gravity is negligible compared to the applied torque of the motor.

I can certainly get on board with idea that raised portion and holes reduce vibration in some way. I'm just not very familiar with that process.

I will be taking measurements with a high-speed camera on Tuesday to see whether or not the acceleration of the blade is non-uniform. It's obvious the blade will have a cosine-like acceleration curve, but I'm not sure if one peak will be higher than the other.

The saw is hand-held: http://imgur.com/oPFD8Ug

 

[Dr.D]

It appears that you have been focused on the rotational motion, but I would like to suggest that you take a different point of view.

Looking at the photograph you posted, it appears (from the outside) that the saw is a single body, but in fact, we know that it has several bodies inside, each executing a different motion -- the slider-crank mechanism. Imagine adding to that photograph a force aligned with the blade motion and applied to the handle. The purpose of this force is to hold the saw body still - to establish equilibrium in the axial direction.

If you do a careful and complete kinematic analysis, you can express the location of the overall system CM location as a function of the crank angle. This needs to include terms for the body (motor, drive pinion, frame), the crank gear, the connecting rod, the slider block/blade combination. The only concern is with the axial position of the CM. (The transverse position is another problem.)

The objective is to reduce the magnitude of the oscillatory part of the hand force to the minimum value. If the CM did not move at all, no force would be required; by making the motion small, only a small force is required. A larger oscillatory force promotes damage to the user's hand (carpal tunnel syndrome).

The arrangement given with the weighted crank gear is designed to minimize the motion of the overall system CM. It would be useful for you to work through the analysis just described to see for yourself how this works.