# How to calculate acceleration?

I remember reading a few threads on how smaller wheels accelerate faster than larger wheels due to mass. However, i’ve never actually seen anyone calculate it. What formula would you use to calculate something like this?

-I know friction, the weight of robot, motor differences, and other factors will affect calculations, but an estimate would be close enough.

Torque = Moment of Inertia X Radial Acceleration

Moment of Inertia (for a disk) = (mass X radius^2)/2

The wheels of the robot are roughly a disk. This number wont be axact, but it will be close enough for the most part.

## Basically a 2.75in wheel is the same as a 4in wheel with a 1.45 to 1 gear reduction. So while your acceleration is going to increase by a bit, your top speed is going to be reduced. So its like anything there are pros and cons, just be aware of what they are before making a decision.

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I remember one of the aura people doing calculations that showed that the difference in rotational inertia was almost irrelevant compared to the mass of the robot, so the difference is negligible, if you are planning to drive over sacks, then big wheels will help you climb better, but if you are planning to push them, small wheels will let you get your chassis closer to the ground, so it really depends on how you want to interact with the sacks, and how you want to gear your drive. You can find the original AURA post here https://vexforum.com/t/mecanums-lets-settle-this/21598/1&highlight=inertia&page=3(proving me wrong :o)

To be honest my calculation wasn’t the best one possible in the circumstances - what’s more relevant is the rotational kinetic energy of the wheel as a proportion of the kinetic energy of the robot.

As dontworryaboutit pointed out, if you assume that a wheel is like a ring (rather than a disk as discussed above - neither is completely accurate but the ring assumption leads to a worst-case estimate) then its translational and rotational kinetic energies are the same if it is rolling.

Ring assumption: I = mr^2*

E_rotational = 1/2(Iω^2)
E_rotational = 1/2(mr^2ω^2)

ω^2 = v^2/r^2

E_rotational = 1/2(mv^2) = E_translational

The total kinetic energy of the robot is
*E_k(robot) = E_t(robot) + E_r(wheels) *
where the robot includes the wheels.

*E_k(robot) = E_t(robot) + E_t(wheels)
E_r(wheels)/E_k(robot) = E_t(wheels)/(E_t(robot) + E_t(wheels))

E_r(wheels) / E_k(robot) = mass(wheels) / (mass(robot) + mass(wheels))*

So, if the mass of your wheels is a significant proportion of the mass of your robot then yes, they will make up a significant proportion of its kinetic energy. Their radius, however, doesn’t make a difference.

speed = distance X time

My physics class didnt go that deep into these calculations… i think ill just wait 1 more year to figure this out

There are a bunch of Moment of Inertia formulas here: http://en.wikipedia.org/wiki/List_of_moments_of_inertia
I like to write mr^2 or mrr as rmr:
A hoop-ring is rmr/2, a cylinder-ring is rmr, a uniform disk is rmr/4,
so a vex wheel is rmr/X where 4>X>1

To the extent that wheel inertia is form rmr/1,
vmv = rmr (translational energy = rotational energy),
E = vv (m of robot without wheels + 2*m of wheels).

A simple sound bite summary of your post is that,
regarding acceleration inertia, “mass of wheels counts (almost) twice”

Given the idea: “I want heavy wheels to lower my robot CG, so it doesnt tip”
the application of this principle is :
you’d have better acceleration with the same CG if you move mass from the wheels to the framework below the axles; eg lighter wheels and make up the difference in mass with low (at or below axle) ballast.

Another application:
So Rick TYler’s concept of speedy robots by reducing wheel size helps 2x as much as reduced mass alone would account for, but it is only the reduced wheel mass that helps, not so much the smaller diameter.

Is anyone interested in making an excel table model of robot acceleration?
Use variables like Number of motors, wheel gear ratio, robot mass, wheel mass, wheel diameter, motor speed-torque models, wheel inertia model.
Each row represents 25ms time increment or something. I can help review it.