3. Kinematic Analysis (Calculations)

The setup for the kinematic analysis and example calculations are found in the attached file below.

In summary, each speed setting relies on two gear sets. The first is between the input and the intermediate shaft, and the second is between the intermediate shaft and the output. In the table below, D1 is on the input, D2 and D3 are on the intermediate shaft, and D4 is on the output. The gear ratio is calculated as the product of diameters of driver gears divided by the product of the diameters of driven gears. D1 and D3 are driver gears; D2 and D4 are driven gears. An example calculation is as follows for the first speed setting (R is the gear ratio):

R = (D1*D3) / (D2*D4) = (6*4)/(6*8) = 0.5

Speed Setting	D₁	D₂	D₃	D4	Gear Ratio
#1	6cm	6cm	4cm	8cm	0.5
#2	6cm	6cm	5cm	7cm	0.7143
#3	6cm	6cm	6cm	6cm	1
#4	7cm	5cm	6cm	6cm	1.4

The gear ratio will amplify the input speed by the given amount. For example, at speed setting #2, the output speed will be 0.7143 times the input speed. Conversely, the input torque will be amplified by the inverse of the gear ratio. Thus, at speed setting #2, the output torque will be (1/0.7143)=1.4 times the input torque. This demonstrates the speed-torque trade-off available through different gear box settings.

After fabrication, the number of teeth for each gear was counted and the results are presented in the table below. As expected, the gear ratios calculated using the number of gear teeth are the same as the gear ratios calculated above with gear diameters. This holds true despite potential inconsistencies in the internal cardboard mesh used in fabrication of the gear teeth.

Speed Setting	_N1	_N2	_N3	N4	Gear Ratio
#1	24	24	16	32	0.5
#2	24	24	20	28	0.7143
#3	24	24	24	24	1
#4	28	20	24	24	1.4

The images below show the gear box in its 4 speed settings (first row, left to right is speeds #1 and #2; second row, left to right is speeds #3 and #4). The speed setting can be seen on the panel above the gear box. The closer shaft in the image is the output, and the further shaft (in the very back) is the input shaft, connected to a motor visible to the right. The intermediate shaft has an attached dowel used for shifting gears.

Note that for speed setting #3, the same gear on the intermediate shaft is driven by the input and also acts as the driver for the output.