# Power flow analysis

In the numerical model simulations, resistance torque is set proportional to angular velocity of output shaft through a damping coefficient: Tout = C ωout.

Because of this constraint between output torque and output velocity, setting a constant entering torque, input shaft velocity results variable in time as well as setting a constant velocity of the input shaft, entering torque to move it results variable. In fact, as shown in previous paragraphs, velocity ratio τ is not constant during an output shaft revolution. So, for example, fixing a constant input torque and a constant shafts offset, resistance torque Tout = Tin τ η varies with time because η but especially τ varies in time.

In the same way ωout = Tout / C varies and so ωin = τ ωout = τ (Tin τ η) / C = (Tin τ2 η) / C.

Typical trends of gearbox efficiency with entering power, absorbed by input shaft loaded by a constant torque, Pin = Tin ωin and output  power, released by output shaft to external damper, Pout = Tout ωout = C ωout2 are reported in the image below. These plots are obtained setting a constant Tin = 100 Nm, C = 3 m2 Kg/(rad s), and a shafts offset of 10 mm. The maximum power transfer develops when Tout (and so ωout) has is maximum, when locked wheel is farthest from disk axis. In these moments η has its minimum values for the locking wheel transitory.

Figure A: Efficiency, input and output power with slide displacement magnitude equal to 10 mm

### Working with lower transmission ratio

Increasing input-output shafts offset, revolution multiplier velocity ratio τ decreases. In constant input torque hypothesis, resistant torque Tout = (Tin τ η) decreases as τ decreases (η also decreases) and similarly ωout (= Tout / C) and ωin (= τ ωout). So at constant input torque, lower transmission ratios involve lower power exchanges. Increasing input-output shafts offset, unstationary characteristics of motion become also stronger, dynamic and cinematic parameters (like velocity ratio) oscillates with higher amplitude. This is due to greater angular velocity oscillations of lateral shafts in the rotations about their axis and about disk axis. Mainly for this reason efficiency medium value decreases with decreasing τ and has greater drops in time.

Figure B shows a comparison of numerical results of efficiency (red lines) and input power (blue lines) calculated for two different configuration of the gearbox. Plots on the left are about a shafts offset of 10 mm, plots on the right are about a shafts offset of 20 mm. In simulations related to these results a constant torque of 100 Nm and C = 3 m2 Kg/(rad s) are set. In reference to what described above, for 10 mm shafts offset the medium value of mechanical efficiency is 0.981, its minimum value is -2.5 % and its maximum value is +1.0 % the mean one. For 20 mm shafts offset the mean efficiency is 0.972, its minimum value is -4.4 % and its maximum value is +2.0 % the mean one. For bigger velocity ratio input power is greater and its oscillations less wide. Smaller τ involves smaller ωout and longer revolution periods.

Figure B: Comparison between efficiency and input power at two different slide positions

Reducing the transmission ratio, with fixed value of angular velocity of input shaft ωin, output shaft velocity ωout (= ωin / τ) and resistant torque Tout (= C ωout) increases. Entering torque Tin (=Tout / (τ η)= C ωout / (τ η)= C ωin / (τ2 η) ) increase with lower τ, and so entering power does it, but efficiency decreases.

### Working with higher damping

Results showed above are about a fixed damping coefficient C of resistant torque.

Increasing C with constant input torque Tin and transmission ratio τ, output torque Tout = (Tin τ η) rests about unchanged, but output shaft velocity ωout (= Tout / C) and input shaft velocity ωin (= τ ωout) decrease. Power exchange reduces because the same torques are transmitted with low velocities.

The diagram below represents efficiency trends calculated with numerical simulations setting the same input torque Tin = 100 Nm, the same slide position with a shafts offset of 20 mm but a damping coefficient C1 = 3 m2 Kg/(rad s) for red line results and C2 = 6 m2 Kg/(rad s) for green line results. Setting higher damping coefficient, mean output velocity is lower and so trend periodicity and fluctuations amplitude are lower. However lateral wheel locking transitory is slower and produces superior mechanical losses: low damping mean efficiency is 0.972 while high damping mean efficiency is 0.968.

Figure C: Working with higher damping

### Working with higher input torque

Instead, leaving the same damping coefficient of the initial case but raising input torque at constant shafts offset, output torque, input shaft velocity and output shaft velocity increase.

Results illustrated below are related to simulations where C = 3 m2 Kg/(rad s) and 20 mm of shafts offset are imposed in both a cases, but red line is about an input torque Tin 1 = 100 Nm while green line is about an input torque Tin 2 = 2 Tin 1 = 200 Nm.

In this latter case Tout 2 = Tin 2 τ η ≈ 2 Tout 1 ; ωout 2 = Tout 2 / C 2 ωout 1ωin 2 = τ ωout 2 2 ωin 1.

Power exchange with the double torque is about 4 times greater than the base one. In the case of higher torques and higher velocities, efficiency presents higher fluctuations. In the way that was defined efficiency, it reaches also values higher than 1 for short time intervals after it has touched its minimum values. Anyway high torque mean efficiency is equal to low torque mean efficiency of 0.972.

Figure D: Working with higher input torque

### Working with higher damping and higher input torque

Doubling both damping coefficient and entering torque with respect to the initial case, one obtains about a double value of output torque but constant values of input and output shaft velocities. Power exchange is multiplied by a factor 2. Efficiency plots for the base configuration (red line) and for double damping coefficient and double entering torque set (green line) are reported in the diagrams below, shafts offset is maintained constant and equal to 20 mm. Green line efficiency trend is similar to the base one, even if its drops are wider because of higher power losses due to higher torques transmitted. Efficiency medium value of base configuration is 0.972, this value decreases to 0.969 doubling C and Tin.

Figure E: Working with higher damping and input torque