# Necessary conditions for energy harvesting

Summary:
Is there a general expression for the mechanical power that can be extracted from a moving vehicle by means of an inertial device?
Assume that a contraption with loosely suspended internal weights is fixed to the body of a vehicle that is moving in free 3D space without gravitational sources nearby. Given the position and orientation of the vehicle as functions of time, how can one tell whether it is possible for the contraption to extract power from the movement? Is there a general expression for this? I am particularly interested in continuous extraction of power from repetitive movement.

anorlunda
Staff Emeritus
Such as in a windmill? A hydro dam? A vehicle brake? ... can you be more specific?

Also, energy harvesting can be done at micro watt scale and at mega watt scale. Which do you want?

phyzguy
Are you talking about a body moving freely in empty space? In the reference frame of the body , it is at rest, so there is no energy to harvest. It takes the interaction of the body with something else to transfer energy.

tech99 and russ_watters
Take, for example, the rotation of the earth. One would think that it cannot be converted to useful work (excluding tides from the consideration), but how do we know for sure? On the other hand, we know that the movement of a cocktail shaker performs work on the ingredients because we can see that they get mixed. What is the characteristic of the shaking movement that is missing from the rotational movement? I am interested in a mathematical expression that tells the two types of movement apart.

russ_watters
Mentor
Take, for example, the rotation of the earth. One would think that it cannot be converted to useful work (excluding tides from the consideration), but how do we know for sure? On the other hand, we know that the movement of a cocktail shaker performs work on the ingredients because we can see that they get mixed. What is the characteristic of the shaking movement that is missing from the rotational movement? I am interested in a mathematical expression that tells the two types of movement apart.
You could extract work from Earth's rotation, slowing it. Your cocktail shaker example makes no sense - maybe like a shake weight? You're going to need to be more specific and detailed.

jbriggs444
Homework Helper
As has been pointed out, a moving center of mass is indistinguishable from a stationary center of mass. Without a place to stand, there is no way to harvest the associated energy.

Relative motion, including rotational motion, can be harvested completely. As long as we are ignoring gravity anyway.

If one has two component objects moving apart, one can imagine firing ideal threads with barbed stickers and harvesting the energy as the string pays out and the objects are no longer moving apart. If there is residual rotation, the thread can be paid out further -- as far as is needed so that the residual relative velocity is arbitrarily slow even though angular momentum is still conserved.

If one has component objects moving together, one harvests energy from the collision, if any, and then sends them apart. The problem is thus reduced the case of component objects moving apart.

If you want a formula, first shift to coordinates where the center of mass is at rest. Then add up the kinetic energies (including rotational energy) of all of the pieces. That's it. You're done.

It is not a formula for power. It is a formula for energy.

Seppo Turunen
I want to attach to a push rod of an engine a sensor box that uses the movement of the rod to power its circuitry. For this purpose, I plan to use a solenoid, a magnet and whatever mechanical parts that are needed to allow inertial forces to cause the magnet to move inside the solenoid. The movement pattern of the rod is somewhat complicated but it is known. How can I decide whether the task is possible?

anorlunda
Staff Emeritus
You are describing a shake flashlight. If that is the amount of power you need, you could cannibalize a shake flashlight and connect it to the rod.

Seppo Turunen and russ_watters
A.T.
I want to attach to a push rod of an engine a sensor box that uses the movement of the rod to power its circuitry. For this purpose, I plan to use a solenoid, a magnet and whatever mechanical parts that are needed to allow inertial forces to cause the magnet to move inside the solenoid. The movement pattern of the rod is somewhat complicated but it is known. How can I decide whether the task is possible?
If you search the web for "Vibration Energy Harvesting" or "Vibration Power Harvesting", you will find various products. Have you checked if some of them would work for you, in terms of frequency range and power output?

Seppo Turunen
My question was and still is: " Is there a general expression for the mechanical power that can be extracted from a moving vehicle by means of an inertial device?". In my subsequent postings I only try to keep the discussion from diverging by describing what a practical application could look like.

phyzguy
My question was and still is: " Is there a general expression for the mechanical power that can be extracted from a moving vehicle by means of an inertial device?". In my subsequent postings I only try to keep the discussion from diverging by describing what a practical application could look like.

If you want a formula, first shift to coordinates where the center of mass is at rest. Then add up the kinetic energies (including rotational energy) of all of the pieces. That's it. You're done.

It is not a formula for power. It is a formula for energy.

The idea is not that the contraption purported to harvest power from a vehicle should absorb all of its kinetic energy. The idea is that the vehicle has an engine or a motor that keeps it performing a pre-known movement no matter what the contraption is or does.

A.T.
The idea is not that the contraption purported to harvest power from a vehicle should absorb all of its kinetic energy.
Then you have to specify how much power you want to extract.

The idea is that the vehicle has an engine or a motor that keeps it performing a pre-known movement no matter what the contraption is or does.
You cannot extract energy, without having any effect. If you mean that the extracted power should be negligible compared to the power supplied by the engine, then you have to specify how much that is.

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russ_watters
Mentor
My question was and still is: " Is there a general expression for the mechanical power that can be extracted from a moving vehicle by means of an inertial device?". In my subsequent postings I only try to keep the discussion from diverging by describing what a practical application could look like.
This question is very generic, so I'll try the generic answer:

Ph=Pe-Pm

Where:
Ph = harvested power
Pe = engine power
Pm = power to sustain movement

That's the basic equation for this sort of thing. Does it meet your needs?

anorlunda
Staff Emeritus
There are countless ways to convert rotary motion to linear reciprocal motion, and visa versa. You can even make the electric power via rotation, then transmit it to the sensors by wire. Think of the alternator and the headlights in your car. There are many factors other than power to consider, including lifetime and reliability. You should not be fixed on a particular choice such as harvesting vibrations.

You need to begin with this:
1. What is the power source?
2. What is the power need for the sensors?

One more important question. Do you have the mechanical and electrical skills to design this yourself?

russ_watters
What you say is obviously true, but does not yet answer my problem. Let me try to make myself more clear: In my problem setting the engine power is not known. What are known are the trajectory and the speed of the movement. I then assume that both of these can be maintained by adjusting the engine power as necessary. There are two trivial cases: the steady circular motion and the linear shaking motion. In the previous case, no energy can be extracted, while in the latter case, energy can be extracted using, for example, the commercial products referred to above. However, many machine parts move in a more complex fashion. What if, for example, their trajectory is ellipsoidal and their speed is varying, akin to that of a planet circulating the sun? The x, y and z coordinates and the orientation of the motion are usually easy to express as functions of time. Having those functions, can I plug them into some expression and learn whether power is available, like in the case of the shaking movement, or not available, like in the case of the steady circular motion?

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russ_watters
Mentor
Having those functions, can I plug them into some expression and learn whether power is available like in the case of the shaking movement or not available like in the case of the steady circular motion?
No. You are approaching the problem backwards. There is no power available to extract until your device demands it. You need a function to describe the required power, not just the motion. There is no power just "there" to be harvested. Once you know the demand of your device, you can design a mechanism to extract it from the car by the appropriate means (if excess engine capacity exists).

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There are two trivial cases: the steady circular motion and the linear shaking motion. In the previous case, no energy can be extracted, while in the latter case, energy can be extracted using, for example, the commercial products referred to above. However, many machine parts move in a more complex fashion. What if, for example, their trajectory is ellipsoidal and their speed is varying, akin to that of a planet circulating the sun?
This implies you do actually think energy is just "there" to be extracted. If that's what you think, it is wrong. The amount of energy available in both the steady circular and linear shaking motion (and ellipsoidal motion) cases is zero. Once you attach and turn on your device, it applies a load to the engine and if the engine has enough spare capacity, it increases its power output to compensate.

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sophiecentaur
A.T.
The x, y and z coordinates and the orientation of the motion are usually easy to express as functions of time. Having those functions, can I plug them into some expression and learn whether power is available, like in the case of the shaking movement, or not available, like in the case of the steady circular motion?

For linear accelerations:

I think it boils down to proper acceleration in the local reference frame of the extractor device. If the motion of the machine part and attachment of the device to it are such, that in the local frame of the device you have a constant proper acceleration, then you cannot extract energy using some suspended mass, because your device experiences a conservative force field.

Note that proper acceleration is relative to free fall. So in horizontal steady circular motion it is constant (you cannot extract energy), but in vertical steady circular motion it varies (you can extract energy).

To compute how much energy you could potentially extract (upper limit), you would have to compute the difference between the achievable maximal and minimal potential energy of the suspended mass over a cycle, based on the varying acceleration field in the local reference frame of the device, and on the limits of the suspended mass' displacement.

For angular accelerations:

It is similar to the above, but instead of the mass of the suspended object, you would use its moment of inertia. And instead of proper acceleration you would you the extreme values of the angular accelerations and angular displacements.

The total extractable energy would be the difference of the maximal and minimal combined potential energies from linear and angular accelerations.

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Seppo Turunen
sophiecentaur
Gold Member
2020 Award
The idea is not that the contraption purported to harvest power from a vehicle should absorb all of its kinetic energy. The idea is that the vehicle has an engine or a motor that keeps it performing a pre-known movement no matter what the contraption is or does.
You seem to have no idea just how vague you are being. In a generic sense, there are a lot of what could be called "Energy harvesting" systems but I have a serious issue with the term "Harvesting" because it seems to imply using Energy that's free. If you have a system that's supplying energy for some need (like propelling a motor car) then you are paying for absolutely all the Joules that your Harvester is extracting. The energy from an internal combustion engine is usually electrical and that supplies the ignition, dashboard meters, windscreen wiper motors etc.. There would be no point in using anything other than the existing alternator.
In any system that involves rotary motion at any stage then a rotary electrical dynamo / alternator is your best bet. Reciprocating parts are not convenient sources of energy so why use them?

You seem to be very selective about this Harvesting idea. The only systems that are truly harvesting energy are using wind, sunlight and water power. The energy comes from large sources like the Earth's rotation or the Sun. That is truly sustainable.

PS If I am being to 'practical' in my comments then the OP should not be couched in terms of Harvesting. Energy or Power transfer in moving systems should be the sort of title for the thread. That's nicely 'ideal'.

A.T.
You seem to have no idea just how vague you are being.
I think post #16 clarifies the question. Here is how I understand it:

You have a closed box that undergoes some given cyclic motion. Inside you have a suspended mass. What is the general approach to get the upper limit for the energy per cycle that you can extract continuously from the relative motion of the box and suspended mass?

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sophiecentaur
sophiecentaur
Gold Member
2020 Award
I think post #16 clarifies the question. Here is how I understand it:

You have a closed box that undergoes some given cyclic motion. Inside you have a suspended mass. What is the general approach to get the upper limit for the energy per cycle that you can extract continuously from the relative motion of the box and suspended mass?
That's a good summary. But what's the power for". Put the box on the piston of that locomotive in the picture and how could you use it? A magnet on the con rod and a nearby coil would at least give you a feasible take-off for the energy from the stationary coil.

A.T.
The x, y and z coordinates and the orientation of the motion are usually easy to express as functions of time. Having those functions, can I plug them into some expression and learn whether power is available, like in the case of the shaking movement, or not available, like in the case of the steady circular motion?
A simpler (than post #18) approach to estimate the upper bound for the harvestable energy:

Assume your suspended mass remains fixed to the device, and express its kinetic energy in the inertial frame as function of time. Now consider only the time intervals where this KE is decreasing, and sum these ΔKE. This is the energy that you could at best extract from the suspended mass, if you allow it to move a bit. In the periods where the KE is increasing the suspended mass is kept fixed to the device, so it draws all the KE from the external work done on the device.

Seppo Turunen
For linear accelerations:

I think it boils down to proper acceleration in the local reference frame of the extractor device. If the motion of the machine part and attachment of the device to it are such, that in the local frame of the device you have a constant proper acceleration, then you cannot extract energy using some suspended mass, because your device experiences a conservative force field.

Note that proper acceleration is relative to free fall. So in horizontal steady circular motion it is constant (you cannot extract energy), but in vertical steady circular motion it varies (you can extract energy).

To compute how much energy you could potentially extract (upper limit), you would have to compute the difference between the achievable maximal and minimal potential energy of the suspended mass over a cycle, based on the varying acceleration field in the local reference frame of the device, and on the limits of the suspended mass' displacement.

For angular accelerations:

It is similar to the above, but instead of the mass of the suspended object, you would use its moment of inertia. And instead of proper acceleration you would you the extreme values of the angular accelerations and angular displacements.

The total extractable energy would be the difference of the maximal and minimal combined potential energies from linear and angular accelerations.
Makes sense to me. To calculate the energy, I then obviously need to multiply the acceleration with the mass and with the distance available for movement, and to do the same for the angular quantities. To maximize power output, the movement should perhaps take place when the acceleration is at its highest?

A.T.
To maximize power output, the movement should perhaps take place when the acceleration is at its highest?
Acceleration is a vector, so you have to consider the direction too. Ideally the suspended mass would move all the time, when there is any acceleration, such that the local force field is doing positive work on it.

Finding the optimal path for a given acceleration profile within given position boundaries could be a complex optimization problem. Then realizing that path practically a difficult engineering problem. If you need just a little power, from a machine that has plenty of it, you should go with a simple reliable mechanism, even if it is very inefficient in terms of the theoretical upper bounds discussed above.

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Seppo Turunen
epenguin
Homework Helper
Gold Member
On the other hand, we know that the movement of a cocktail shaker performs work on the ingredients because we can see that they get mixed.
Just to say it takes no work to mix them. They will mix over time without you shaking. From this spontaneous tendency to mix, work can be extracted – osmotic work. You are into thermodynamics here rather than dynamics. I'm afraid I can't give you a quick account and you frankly have some plodding study before you. For simple 'conservative' system with a small number of components perhaps somebody can illustrate convincingly to you the mechanics, maybe with a simple example.

(just thinking aloud now) in a conservative system total energy is conserved - that's what 'conservative' means. To say that there really are such systems (at least to good approximation in the real world) is not trivial or tautological. Such a system is one in which the forces acting on bodies depend on only the spatial positions of the bodies. It can then be proved that as such a system evolves under the effect of all the forces, the total of a certain quantity, which gets called 'energy', remains constant all the time despite the positions of the bodies changing. This concept, which turns out to be very useful for thinking, was first formulated (deduced from Newton's laws and definitions) as far as I know by one Md. du Châtelet. What you call extracting work is really transferring energy from one part of a system to another, the total remaining constant. (It could be said that the answer to your question is that the condition for extracting work is that the forces be not in equilibrium.)

Well there is another thing called extracting work – that is when you transfer the energy to a non-conservative part of the system. What happens there is that the energy is so to speak used up in e.g. generating heat, or in changing the shape of a non-springy piece of metal or plastic etc. You then get into a deeper principle which states that despite appearances all systems are conservative.You just don't see at a gross macroscopic level, it is there in the movement of molecules etc.

Could illustrate with example if you can't work one out yourself but no time at moment.

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