Analytical dynamic model

Idealizing a stationary behavior, ignoring mechanical losses and considering only locked wheel, the principal gear induces a tangential force to lateral gear that is:

with RP primitive radius of principal gear. FP-L produces a torque on the wheel, about its principal axis:

with RL primitive radius of lateral gear. The unilateral bearing, fixed to support, stops lateral shaft relative rotation. TL is balanced by reaction that input shaft exerts on support FS at a distance RP+RL from lateral shaft axis:

Respecting force equilibrium for the lateral shaft, force FD which lateral shaft exerts on disk slope is:

Producing a torque on output shaft TOUT:

with Rc distance between output shaft axis and lateral shaft axis. Rc could be easily found analytically using analytical relationships collected in the paragraph “Analytical Cinematic model”.

Tout has its minimum value for Rc minimum value, i.e.  Rc min = RL + RP - TS, with TS input-output shaft offset.

Even if Tin is set constant, Tout and velocity ratio vary in one output shaft revolution because Rc varies.

For example, for an input torque Tin = 100 Nm and a shafts offset TS = 20 mm, one can calculate:

Analogous results can be obtained with numerical simulations where is set the same constant Tin and TS as shown in the figure below. In the diagram are reported the plots of resistance torque with time derived by the output shaft, and torque transmitted by the contact between one lateral shaft and its relative disk slope. Single lateral shaft contributes to transmit positive torque to output shaft only for a quarter of revolution period of the disk for reasons described before.

Figure A: Torque comparative

Minimum numerical value of output torque is similar to Tout min above calculated using the analytical model. This means that ideal hypothesis made for formalizing the model are not so far from reality.

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