Tie rack calculations

From DDL Wiki

Gear Train

Speed of Output

The gear train creates a decrease of speed and increase of torque from the actual motor output. The final speed is slow enough to measure with a stopwatch. This was done by marking a point on the rotating belt, and measuring the time that it took for one rotation (the point returns to where it started).

Setup:

Time for one rotation of belt gear = 5s : 38ms Circumference of belt gear = 0.126 m

Speed = length/time (m/s)

Angular speed = ω = 2π/T (radians/s)

ω10 = length / [r10 * time] = (0.126 m) / [(0.02m)(5.38 s)] = 1.17 rotations/sec

ω10 = 70.26 RPM

Gear Ratios/Calculations (Relating input to output)

ω1 r1 = ω2 r2 and ω2 = ω3

ω2 = ω4 = [r1 / r2] ω1 (1)

ω3 r3 = ω7 r7 and ω7 = ω4

ω7 = ω3 = [r3 / r7] ω3 (2)

substitute (1) into (2)

ω7 = ω4 = [r3 r1 / r7 r2] ω1 (3)

ω4 r4 = ω8 r8 and ω8 = ω5

ω8 = ω5 = [r4 / r8] ω34 (4)

substitute (3) into (4)

ω3 = ω5 = [r4 r3 r1 / r8 r7 r2] ω1 (5)

ω5 r5 = ω9 r9 and ω9 = ω6

ω9 = ω6 = [r5 / r9] ω5 (6)

substitute (5) into (6)

ω9 = ω6 = [r5 r4 r3 r1 / r9 r8 r7 r2] ω1 (7)

ω6 r6 = ω10 r10 and ω10 = ω6 = ω9

ω10 = ω6 = ω9 = [r6 / r10] ω6 (8)

substitute (7) into (8)

ω10 = [r6 r5 r4 r3 r1 / r10 r9 r8 r7 r2] ω1 (9)

ω10/ ω1 = [r5 r4 r3 r1 / r9 r8 r7 r2] (10)

ω1/ ω10 = [r9 r8 r7 r2 / r5 r4 r3 r1] (11)

ω1 = [(0.019m)( 0.019m)(0.019m)(0.0115m) / (0.003m)(0.003m) (0.003m)(0.004m)] * 70.26 RPM

ω1 = 51314.84 RPM = motor speed

Stall Torque

The stall torque of the motor was calculated by finding an equilibrium point between an opposing torque and motor output torque. This was done by turning the motor sideways, powering it, and hanging weights on the pulley attached to the motor shaft until weight suspension was reached (τ1 = τ2). The string is of negligible mass and was tightly affixed to the pulley to prevent slipping by both knotting it with a rubber band.

Setup:

Result:

Motor stalled when 0.8 lb weight was creating a torque. Thus:

0.8 lbs = 0.363 kg

Radius of motor pulley = 0.004 m

τs = Fr = (0.363 kg) (9.81 m/s2) (0.004 m)

τs = 0.0142 Nm = 14.24 mN-m