Hello, I am trying to calculate the profile of a double circular arc helical gear for a gear pump that I am considering getting made. I am working off of a couple papers, and I have three of the four sections of the gear tooth profile calculated and graphed, but the last section, the conjugate curve, just won't fit with the rest.

Papers:

Article Profile design and displacement analysis of the low pulsatin...

(The original paper is in Chinese and is available at

https://doi.org/10.3969/j.issn.1004-132X.2018.02.009)

Article Study on the Design of the Gear Pair and Flow Characteristic...

The math for the sine curve and the tooth root and tip all seem to work fine and lineup more or less as expected. Something in the math for the conjugate curve doesn't seem to be working though, and I wanted to see if anyone can help me figure out why my math isn't working. It's kind of hard to explain without uploading the whole Excel file and going through it sheet by sheet, so I will try to explain as best I can.

The 2025 paper says that for calculating the curve of AA' (the conjugate curve), the equation is:

x2 = −tR cos(2φ + π/Z ) + 2R cos(φ + π/Z ) + aR sin N(t − t0) sin(2φ + π/Z )

y2 = −tR sin(2φ + π/Z ) + 2R sin(φ + π/Z ) − aR sin N(t − t0) cos(2φ + π/Z )

(tE ≤ t ≤ tA)

(The x and y coordinates of the universal frame are calculated by x = t*R and y = aR sin N(t − t0), so they appear in the equations.)

Where φ satisfies the following equations:

tan γ = dx/dy

cos ψ = (y cos γ+x sin γ)/R

φ = π/2 − (γ + ψ)

The 2018 paper says that the AA' curve can be calculated with:

x1 = -tr cos (2θ + β)+Ar sin n(t - t0) sin (2θ+β) + 2r cos (θ + β)

y1 = -tr sin (2θ + β)  Ar sin n(t - t0) cos (2θ+β) + 2r sin (θ + β)

As near as I can determine, β = 15.1° (α = 14.5°, ∠ADP = α + β, and I think ∠ADP = 29.6° based on the tangent at A). However, I am not sure what θ is.

My values:

tE = 0.947529994

tA = 0.965925826

t0 = 0.577251834

N = 1.702302301

R = 12

(I don't know why, but my values of a 6-tooth gear are different than those published in the 2018 paper, but I have calculated/approximated them multiple different ways with the same answer coming back that does not match the paper.)

As near as I can interpret from the paper, if I plug in the values of x and y as calculated based on t, the equations should provide the appropriate x2 and y2 to allow me to graph the results. The line is supposed to go from (xA=11.59,yA=3.11) to the start of the tooth tip at

(xA'C'=12.43,yA'C'=3.73), but instead, it's going from (13.45,4.93) to (13.86,5.38), and it's not even long enough to fill the gap between A and A' if it was in the right place.

I have tried reaching out to the corresponding authors of the two papers, but I haven't heard back yet.

Can anyone tell me what I am doing wrong with my math, where I am misinterpreting the paper, or how to properly calculate this curve? I've attached a picture of what I have been able to graph so far and the problematic section. Thank you.