Loopout driven by rotation parameter.

I would approach the problem a little differently. I don't have time to generate screenshots or show the expressions now, but the workflow I would follow goes like this.

Create the wheel layer and add an expression and a slider. The expression comes from Dan Ebberts's mothinscript.com Speed control and looks like this:

spd = effect("Speed Control")("Slider");
n = spd.numKeys;
if (n > 0 && spd.key(1).time < time){
  accum = spd.key(1).value*(spd.key(1).time - inPoint);
  for (i = 2; i <= n; i++){
    if (spd.key(i).time > time) break;
    k1 = spd.key(i-1);
    k2 = spd.key(i);
    accum += (k1.value + k2.value)*(k2.time - k1.time)/2;
  }
  accum += (spd.value + spd.key(i-1).value)*(time - spd.key(i-1).time)/2;
}else{
  accum = spd.value*(time - inPoint);
}
value + accum

Another way to solve the problem is to create an expression that uses the diameter of the wheel and the movement of the wheel layer or another layer to control speed. You'll find both approaches easier to control than fiddling with time-remapping.

Hey there, you're really close! There are a handful of ways to do this, but using most of the expression you already have, you just need to change the last argument (parameter) in the linear() function. In expressions, After Effects measures time as whole seconds, so you're telling the Time Remapping parameter to go from 0 to 500 seconds. The easiest fix here is to add a new function called framesToTime() which will convert the number of frames inside the parenthesis to the proper amount of time based on your comp's frame rate.

You don't have to create the new endDuration variable, I just did it to improve readability:

var backrotation=thisComp.layer("Rotation - BackWheel").transform.rotation;

var endDuration = framesToTime(500);
linear(backrotation, 0, 360, 0, endDuration);

Now when you keyframe your wheel's rotation from 0 > 360 and put loopOut("cycle") on that property, you should get exactly what you're after.