Lego motors, speed and physics question

[aol000xw]

Let me start saying that Philo technical pages are awesome.

There you can find all the useful data for any Lego motor and friction coeficients for different wheels. Also that 4x5292 prolific Sariel did got me thinking about Lego and speed. And what it takes to go fast with Lego.

Now I barely remember my physics lessons, but I think that with Philo charts it could be somehow predicted maximun theorical speed for a model based on weight, motors, voltage, wheel diameter, gear ratio....

Even transmission friction could be factored in based on number of gears and axles to get a better aproximation. but that goes beyond my main goal.

Is this possible? Could anyone point me in the right direction for this or give a hint with some formulas?

[Kelkschiz]

I am by no means a physicist but i do know that friction and energy loss are all important and it's difficult to calculate all friction and inefficiency in a system before hand, because one type of friction also interacts with other kinds of friction. On the flip side, a formula that ignores friction will be hugely inaccurate in predicting maximum vehicle speed. But i am sure some of the true engineers here will come up with a better answer.

[DrJB]

What makes friction difficult is that it often depends upon the load (torque) AND speed, and sometimes the friction mechanism is different. At low speed, friction is typically 'dry friction', where the friction force is constant and depends upon the load only. At higher speed, the friction often is proportional to the square of the speed, due to hydro-dynamics effects as an air-film tends to develop between rubbing parts. Want to make it complicated, friction is high at zero speed, drops a bit once you start moving, then icreases again as the speed increases further. This leads to the so-called 'stick-slip' mechanism ...

Possibly the best one can do is to calculate efficiency/losses (greek letter Eta) of a compound transmission under various load/speed combinations ... but then you'd need both speed and torque sensors. The former is easy, the latter is not... then there is Mr Heisenberg who wants to join the party as well ...

[Hrafn]

In principle, there are two factors to consider initially, both of which relate to getting maximum power out of your creation.

First, the Lego batteries have thermal protection circuits that kick in at a given point, so there's a certain current level you don't want to exceed if you want to have your motors run for more than a few seconds. According to Philo, the 5292 motor can reach its maximum power at 1.4 Amps, if I recall correctly - but (for example) the rechargeable battery pack will shut off if you draw more than 1 A. 2 5292s powered by 3 LiPo batteries would probably give the best power to weight ratio while using about all of the available current, though you'd have to do some funny things with the wiring (PF to 9V, then use a small (PF-M or mini-motor) to control a switch that fed the 9V to the motors.)

Second, any electric motor reaches maximum power at 1/2 of maximum torque and 1/2 of maximum speed.

The main equations to keep in mind - which you may already know - are these.

For motors:

P = ω * T (ω is rotational rate, in rotations per second; T is torque, in Newton-meters; P is power, in Watts)

ω = ω_max - (ω_max/T_max)*T for electric motors - in other words, there's a linear tradeoff between speed and torque

P_max = ω_max/2 * T_max/2 (restating point 2, above; this might not be obvious but is derived from the first two equations)

For wheels:

v = ω * r (v is velocity in meters per second, r is wheel radius in meters)

For batteries:

P = V * I (V is voltage in Volts, I is current in Amps; fresh non-rechargeable batteries give 9V, rechargeable ones give ~7.2V, and the LiPo gives ~7.4V)

As DrJB said, though, friction is important, tough to calculate in practice, and even harder to predict ahead of time. Probably the best thing to do is just observe best practices - keep your structure light and very rigid, keep axles short (and use axle joiners for longer ones, since they're more rigid), limit the number of gears you use, and use firm tires on flat, smooth, hard ground. Build a vehicle with no gearing, measure its speed, and calculate what the motor's rotational speed must be. If it's not close to half the theoretical maximum rotational speed for the motor, adjust the gearing and try again.

Edited June 22, 2014 by Hrafn

[Zerobricks]

And electricl motors are most efficient when they are rotating at 3/4 of unloaded speed.

[ALittleSlow]

The coefficients of friction Philo provides for tyres won't help you. What you need for the calculation you're interested in is the rolling friction, or rolling resistance. The relevant equation is Torque to overcome rolling resistance (T) = Vehicle speed / rotational speed (ω) * rolling resistance. Estimation of the rolling resistance, as you might guess from the other posts, can be a complicated affair but I expect at LEGO scales it mostly depends on the tire and the vehicle weight. -Brian A.

Thanks @jodawill for pointing out that the rolling resistance in the above equation is resistive force, not the coefficient of rolling resistance. For a rigid tire on a rigid surface, this equation reduces to T = r * F, where r is the tire radius and F is the force at the axle pushing the vehicle forward. The coefficient of rolling resistance, Crr, is defined from F = Crr * N, where N is the normal force the axle exerts on the tire (i.e. the weight on the wheel). If you make these two substitutions (the first one being dubious), the original equation becomes T = Crr * r * N.

That gives us an easy way to determine the Crr, with a caveat I'll give later. I'll call it a "coast-up test" because it's a coast-down test in the opposite direction. Build a simple test vehicle with all the same tires and no drive train. Put it on an inclined plane (i.e. a board). Measure the height of the vehicle off the ground level. Let the vehicle go and time how long it takes to reach the bottom. Using the standard acceleration equation s = 1/2 * a * t^2, where s is the length of the plane, a is all the accelerations experienced by the vehicle (assumed to be constant) and t is the time to cover distance s, it can be shown that the definition of Crr, Newton Laws and this last equation can be combined to h / s - (2 * s) / (g * t^2) = Crr, where h is the height and g is the acceleration of gravity. Now you have a coefficient of resistance, which is dominated by the rolling resistance. You can then apply this using T = Crr * r * N to any vehicle.

Now for the caveat. Crr is going to increase with torque, especially for softer tires, and in this coast-up test the torque is minimal. There are more caveats. It only applies to the surface you tested on. Carpet, wood, and gravel will give you different Crr values. Crr may also increase with speed. Finally, the equation neglects the momentum gained by the tires which will decrease the value of t (increasing the apparent Crr), so it would be good if the body of the test vehicle is much heavier than the tires. Probably at least by a factor of 10.

Edited July 12, 2014 by ALittleSlow

[aol000xw]

Thanks everyone for your answers.

I did not expect it to be easy and not even a close estimation -by now -, Indeed I had some notions about work, force and friction, and those equations Hrafn put in are bringing back some memories.

Now at least I got a start point. I remeber using two distinct formulas one to calculate initial rolling resistance and a second one for ongoing resistance, both taking different coefficients of friction, so there are a lot of factors to consider.

But again thanks guys. Physics + Lego? Double the fun

[Heppeng]

And electricl motors are most efficient when they are rotating at 3/4 of unloaded speed.

And maximum power when running at approx 1/2 of unloaded speed, and maximum torque at zero speed.