How fast was our January club robot going?
An image showing the actual robot drive system is attached to this post. The datasheet for the motor can be found here (Our motor model number is 455A116-2)
In the CAD model and the actual build, a large pulley (P1, with radius r1= 1.5 inches) was attached to the motor shaft and a small pulley (P2, with radius r2= 0.5 inches) was attached to the drive wheel. Becuase each of the pulleys and the belt have teeth, there is not supposed to be any slip between them. In each case, it is as if the pulley is "rolling" on the belt without slip as previously discussed. So, in each case the linear speed of the belt (v) relates to rotational speed of the pulley (w) as v = r * w. You might call this a "wheel equation"...Because the belt doesn't stretch, every part of the belt moves at the same speed. So, v = r1*w1 AND v=r2*w2. Conveniently, the rotational speeds of the two pulleys can then be related by r1*w1 = r2*w2. We'll call this our "belt equation." This relationship holds regardless of the units we use for rotational speed. This relationship can be seen in the detailed bike example from the original post.
If our motor is supplied 12Volts, the datasheet says that it would have a no-load speed of 5200rpm...which we divide by the "reduction ratio" of our gearbox to find the actual output shaft speed = 5200/65.53 = 79rpm. However, our robot batteries are only rated at 7.2 volts instead of 12V. So, we can estimate our actual no-load speed proportionally as shaft speed = 79 *7.2/12 = 47 rpm. (this is close to what is actually observed).
Our large pulley, being connected to the motor output shaft is then spinning at 47 rpm (no-load speed with 7.2 volts supplied) (i.e. w1 = 47rpm). This drives the belt which, in turn, drives the second pulley. Rearranging our "belt equation," makes it easy to calculate w2=w1*r1/r2 = 47 *1.5/0.5 = 141 rpm.
However, P2 isn't the end of the story. The second pulley is attached to the robot drive wheel. The radius of the wheel is approximately 1.5inches. Again, we use our "wheel equation" to relate linear speed to rotational speed, v = r*w. Now, we must be careful that the units are balanced in the equation. As described in the previous post, the units of rotational speed need to be radians/time in order to work. So, we'll multiply by 2pi.
Then v = 1.5(inches)*141(rev/min)*2pi (rad/rev) = 1328 inch/minute...which is about 1.8 feet/second.
But there are two really big differences between this theory and our reality. First, our motors are not operating under a no-load condition. There is friction, slipping, and masses in the real thing. Second, we cheated by changing the gearbox. Yes, we actually opened up the motor and rearranged the gears. [don't EVER try this with a VEX kit!!!] Now, our no-load speed is way more than 47 rpm...it is more like 180rpm. But that means we lost a lot of torque in exchange. In the end, maybe it balances out some. Because, if I observe the rotational speed of the large pulley on the actual robot, I get approximately 112rpm (which suggests a ground speed of about 4.4 feet/second). But, if I give the robot an actual speed test on the ground, I get a little less than 3 feet/second. Would fully charging the battery improve the performance? Probably...