I Seemingly a contradiction of conservation of energy?

[jrmichler]

Where did the potential energy in scenario 2 go?

To clarify: I am not arguing that conservation of energy is actually being broken here. I just want to find out how my reasoning is incorrect/incomplete.

This is about servomotors and perceptions. To oversimplify, there are two types of servomotors:
1) The little low voltage motors that the Arduino people play with,
and
2) Industrial servomotors.

Small low voltage motors tend to have low efficiency, and tend to be used in applications where friction is a large percentage of the total load. In these systems, it is possible that friction force could be larger than gravity force. There will be little, if any, power regenerated on a down move. In this case, the perception that electricity is used for both up and down moves is correct.

Industrial servomotors are different. They are more powerful, more efficient, and are used in applications where friction force is small compared to acceleration and gravity forces. When there is deceleration torque where the load is driving the servomotor, the motor acts as a generator and pushes energy back to the drive. Depending on the drive, that energy is either pumped back into the powerline, or absorbed by capacitors on the drive DC buss. This is well known to industrial servomotor manufacturers, and calculation of the maximum regeneration energy is an important part of system design. This is a case where perception is misleading, and it is critically important to do the calculations and trust the results.

I was part of an application that required moving a 900 lbs mass vertically about 36 inches in slightly over 500 milliseconds. The acceleration for both the up and down moves was 500 in/sec$^2$. The motion profile was a simple triangular velocity profile, where the acceleration was a constant 500 in/sec$^2$ for halfway, followed by equal acceleration in the opposite direction to bring it to a stop. Yes, it was being pushed down faster than gravity for the first half of the down move. The deceleration torque during the last half of the down move regenerated power into the drive. Since the regeneration energy was a known amount of energy, and would be followed by hard acceleration, it was only necessary to put enough capacitance on the drive DC buss to absorb that energy without exceeding the maximum DC buss voltage.

 

[kered rettop]

This is about servomotors and perceptions. ...

Yes to all of that. But the OP went to extreme lengths to describe an ideal (frictionless) case, not realising that they had overlooked a very fundamental part of the system. Which sent them off on a wild goose chase looking for the missing energy in the dynamics - acceleration etc - and unaccounted-for kinetic energy dissipated when the mass comes to rest. So I'd like to be sure they understand that motor losses and friction are actually just improvements on the basic model, after the fundamental flaw has been corrected. That's not a criticism of your post, of course, just putting it into the context of fine-tuning the model. I did like the examples, especially the reference to Arduino people. I immediately saw them as little elves posing on their microchips. That probably says more about me than the physics.

addendum:
I just looked up the meaning of "Arduino". Apparently, it means "little helper". My subconscious must be smarter than I thought!