## Two gear switching speed

We have a Andy Mark Super shifter with 9.6:1 and 24:1 gear ratios.  Question: What is the best speed to shift from 24:1 to 9.6:1?

The ratio of gears         n = 24/9.6 = 2.5

My thought is to switch when the power in first gear drops lower than the power available in second gear.

We can derive this point easily using the torque-speed curves for the two gears and setting them equal and solving for speed.

First gear torque speed   w1 = w1_max*(1- torq/torq1_max)

First gear pwr_1 = w1*torq

Second gear torque speed w2 = w2_max*(1 – torq/torq2_max)

Second gear power  pwr_2 = w2*torq

Setting pwr_1 = pwr_2 we can solve for torq and then for w_1  where the pwr curves cross and we should shift.

This leads to  w_1 = w_2  since torq cancels out.

w1_max*(1- torq/torq1_max)  =    w2_max*(1-torq/torq2_max)

Solving for torq

torq = (w2_max/w1_max-1)/(w2_max/w1_max/torq2_max -1/torq1_max)

And plugging into first gear speed equation gives

w1 _equal_pwr_point = w1_max*(1 – ( w2_max/w1_max-1)/(w2_max/w1_max*torq1_max/torq2max -1)

We can now make the following substitutions:

w2_wmax/ w1_max = n

torq1_max/torq2_max = n*eff1/eff2

where eff1 and eff2 are the efficiencies associated with each gear.

So  w1_equal_pwr_ point = w1_max*( 1 -(n-1)/(n^2*eff1/eff2 -1))

For spur gears we assume .95 for each 5:1 ratio so gear 25:1 for gear 1 would have eff1 = .95^2 and 9:1 ratio for gear 2 we assume just .95 .

Thats it:  if n= 2.5 then the best switching point is

w1_max*(1-(2.5-1)/(2.5^2*.95^2/.95-1)) = w1_max* .696

or   .7  w1_max

Also, the Torq_equal_pwr_point  = .3 *Torq1_max.

Using encoders to measure speed we would use a shift logic as follows:

If  (w_encoder > .7 w1_max)

then gear = gear2     // shift to high speed gear

else   gear = gear1    // shift to low gear

w1_max is calculated from w_free of Cim motors  = 5310 rpm * V_bat/12/gear1_ratio.  Since gear1_ratio = 24 then:

w1_max 12 v = 221 rpm (7.7  fps with 8 in wheels)

w1_max 10.5v = 194 rpm (6.8 fps with 8 in wheels)

So it would be best to read the bat voltage and take this into account when shifting.