How to determine the energy recovered by regenerative braking on a roller coaster

Question

I am a high school science research student and am investigating the implementation of regenerative braking on roller coasters. Ive read the papers that were available on this topic and its said that usually around 70% of the coasters original kinetic energy is up for recapture.

My first problem is the formula, ive seen a couple motors using specific formulas that calculate the energy recovered but I don't have a specific motor to draw an equation from.

My second problem is the data required to calculate the energy recovered. What general stats do I need to find out the energy recovered such as velocity, mass, incline etc.?

There aren't many papers on this topic so anything pointing me in the right direction would help me significantly, thank you!

Answer 1

To simplify things, let's say that a DC motor is being used as a generator.

In order to compute some physical quantities (like braking power), you'll need to create a mathematical model of this system. You don't need to create a perfect model. A rough approximation should suffice.

The description of the model:

• The speed of the rollercoaster is known.
• The system has no mechanical frictions (otherwise, you cannot compute everything without computational methods or advanced mathematics).
• A DC motor is coupled to the roller coaster's main shaft.
• The internal resistance of the DC motor is zero.
• The inductance of the DC motor is neglected.
• A DC motor's terminals are connected with an electrical resistance so that the roller coaster's kinetic energy is converted into heat caused by the current generated by the generator. If you don't connect anything to the DC motor, it won't generate braking.

Mathematics of the model:

Find a relationship between the roller coasters' speed ($v$) and the DC motor's rotation speed ($\omega$):

$$ v = N\omega $$

Assume N is a constant number (it depends on your mechanical design).

The rotation speed of a DC motor is always linearly proportional to the voltage difference between its terminals ($V$):

$$ V = K_V \omega $$

where $K_V$ is a constant that depends on the DC motor.

The braking torque generated by a DC motor ($T$) is always linearly proportional to the current flow through the DC motor ($i$):

$$ T=K_Ti $$

where $K_T$ is a constant that depends on the DC motor.

Since the DC motor's terminals are connected with a resistance ($R$):

$$ V=iR $$

Then, the braking power exerted by the motor is:

$$ \begin{align} \text{Braking Power} &= T \omega \\ &= K_T i \omega \\ &=K_T \frac{V}{R} \omega \\ &=K_T \frac{K_V \omega}{R} \omega \\ & =\frac{K_T K_V v^2}{R N^2} \end{align} $$

So the braking power depends on the DC motor selection ($K_T$, $K_V$), the speed of the rollercoaster ($v$), the mechanical design to couple the DC motor to the rollercoaster's shaft ($N$), the electrical resistance connected to the DC motor ($R$).

With no mechanical friction assumption, you will find that 100% of the kinetic energy of the roller coaster can be recovered. So, calculating the recovered energy is straightforward (the difference between the kinetic energies of the roller coaster at two different times). Accounting for friction in these types of models is an undergraduate-level topic, and I don't think you should dive into this.

For references to all the equations above, check out this post: sinaatalay.com/posts/design-and-construction-of-a-dynamometer/

Answer 2

My starting point would be to consider the conversion of energy between forms. If we model the rollercoaster car as a mass starting at its highest point, initially with zero speed but teetering on the edge of its first descent, then in an idealised system with no energy loss externally it would be be able to roll all the way around its track, returning to the exact same position and speed. This would involve a conversion of gravitational potential energy (which we can calculate, based on the mass of the car and the height difference in the track) into kinetic energy and then back again.

Now suppose that we want to stop and let passengers in and out at a point that isn't the highest, we need some way of taking energy out of our simple system (in order to deccelerate at the bottom) and then putting it back in (in order to accelerate and get back to the top).

Next we can think about the reasons this model is not realistic, these would include: friction at the wheels, air resistance, resistance in the wiring, losses in the motor, losses in the battery, etc. All these represent energy lost from the system (typically as heat). It also may not be feasible to recover all the kinetic energy into the electrical system (e.g. for capacity or safety reasons) and we may need to add mechanical brakes (which will convert all their kinetic energy into heat).

With the energy conversions quantified and the losses understood we can next start to think about how far from perfectly efficient each conversion might be. You may be able to find some typical % efficiency numbers for each step or loss mechanism. Ultimately, for a closed loop like a rollercoaster there will be an amount of energy lost per lap, and this will be equal to the amount you will need to add in order to complete the lap.