Design and Analysis of Conjugated Straight-Line Internal Gear Pairs


In comparison to involute internal gear pumps, conjugated straight-line internal gear pumps have the advantages of smaller trapped volume, lower flow ripple, and higher reliability. Conjugated straight-line internal gear pumps were invented by Truninger in 1966; however, because of their unique line type, the main characteristics of these pumps are still not completely understood. Internal gear pairs are the key component of these pumps. To better understand the performance of these pumps, the present research focused on the design of an internal gear pair. A numerical model was developed to demonstrate the design method of the conjugated straight-line internal gear pair. To point out the effective engagement of the gear pair, three fundamental requirements were considered—a conjugated point existed in the tooth profile, no interference occurred during the meshing process, and the overlap coefficient was greater than one. Furthermore, according to the design of the gear pair, flow characteristics and some optimization suggestions for tooth parameters were proposed. The main characteristics of the gear pair were compared with those of an involute gear pair. Finally, a conjugated straight-line internal gear pair machined by the linear cutting method was tested, and the obtained results revealed that the design method was correct.

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\(M_{1} ,M_{2}\) :

Transformation matrices around the origins of \(S_{1}\) and \(S_{2}\), respectively

\(M_{21}\) :

Transformation matrix from \(S_{1}\) to \(S_{2}\)

\(M_{g1}\) :

Transformation matrix from \(S_{1}\) to \(S_{g}\)

\(\varphi_{1} ,\varphi_{2}\) :

Rotation angles of the gear and the ring gear, respectively

\(R\) :

Gear numerical matrix

\(M\) :

Meshing line matrix

\(R_{ab} ,R_{bc} ,R_{cd}\) :

Matrices of lines \(ab,bc,cd\), respectively

\(r_{1} ,r_{2}\) :

Pitch circle radii of the gear and the ring gear, respectively

\(k,b\) :

Slope and intercept of the gear profile, respectively

\(\theta\) :

Tooth thickness central angle

\(\beta\) :

Addendum pressure angle

\(w_{1} ,w_{2}\) :

Angular velocities of the gear and the ring gear, respectively

\(z_{1} ,z_{2}\) :

Tooth numbers of the gear and the ring gear, respectively

\(v_{12}\) :

Relative motion velocity between the gear and the ring gear

\(G\) :

Ring gear numerical matrix

\(B\) :

Tooth width

\(\psi\) :

Meshing point rotation angle

\(r_{a1} ,r_{a2}\) :

Addendum radii of the gear and the ring gear, respectively

\(r_{d1} ,r_{d2}\) :

Dedendum radii of the gear and the ring gear, respectively

\(r_{n1} ,r_{n2}\) :

Meshing radii of the gear and the ring gear, respectively

\(e\) :

Eccentricity of the internal gear pump

\(q,\overline{q}\) :

Instantaneous displacement and average displacement, respectively

\(\varepsilon\) :

Overlap ratio

\(\eta\) :

Flow pulsation coefficient


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This work was supported by the Natural Science Foundation of Hubei Province(2020CFB156).

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Correspondence to Liao Jian.

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Zongbin, C., Lin, H. & Jian, L. Design and Analysis of Conjugated Straight-Line Internal Gear Pairs. Int. J. Precis. Eng. Manuf. 22, 1425–1440 (2021).

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  • Conjugated straight-line
  • Gear pair
  • Internal gear pump