Centroid Position Vector and Engaging Force of Gear Pair

The dynamic center distance of a gear pair at any time is the modulus of the difference between the vector of the centroid position of each gear:

Where Δx and Δy are respectively:

The engagement angle at any moment, i.e. the dynamic engagement angle, is obtained from the dynamic center distance of the gear pair:

Gear pair relative position angle at any time obtained from geometric relation:

Use the angle ψ between the engagement line and the coordinate axis OY to determine the direction of action of the engagement force. As shown in the figure, the angle ψ is:

The engagement line direction of tooth back engagement state is different from that of tooth surface engagement state. As shown in the figure, M2N2 is the engagement line of tooth back engagement state.Tooth back engagement, sharp angle ψ between engagement line and coordinate axis OY is: