Replace Gear Segments with a Mechanism

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Replace Gear Segments with a Mechanism

Replace Gear Segments with a Four-bar Mechanism

It is often the case in machine design, that a motion needs to be transmitted between shafts for small angles of rotation, say an oscillation of 30º. The shafts may rotate in the same and the opposite direction to give 'positive' and 'negative' gear ratios.

Frequently, Gear Segments are used. As an alternative, you can use a four-bar mechanism to replace the gear segments.

This topic gives the calculations to find the near optimum four-bar mechanism that can replace the gear segments.

At the design position, the gear ratio is EXACT. The accuracy is within 0.2% over 30º of input shaft rotation.

Design Methods

There are three design methods:

Method 1: Perpendiculars

Method 2: Inflection Circle and Cubic of Stationary Circle

Method 3: Collineation Axis

With all three methods, you have some design choices to make:

Methods 1 & 2 :  One Design Variable

Method 3 : Two Design Variables

Methods 1 & 3: use only the Part-Editor.

Method 2: One or two simple calculations.


Cylinder- External and Internal Gear Segments with Instant Centre, I

Kinematically, the required gearing motion - one shaft rotates at a speed relative to a different shaft - is obtained with two pitch cylinders rolling on each other without slipping. Where these two cylinders touch, the corresponding points on each cylinder have identical velocities. It is the same with gears.

Such a Point, I, is called the Instantaneous center of Rotation for the relative motion of the two cylinders. or gears.

Linkages and Instantaneous centers

Diagram for Instant Centres of Linkage with Perpendicular Links

In the Image, A0B0 is the Frame, B0B and A0A are the input and output links, AB is the coupler that transfers the motion from the input to the output link.

The lines drawn to I are construction lines.

We can find the ratio of the links in the four-bar linkage so that links A0A and B0B move with a relative velocity equal to the Gear Ratio of the gear segments we want to replace.

The input and output links will rotate with the relative speeds at the 'design position', and also for a number of degrees to each side of the design position.

Equation : locate Point 'I'

Every four-bar linkages has an Instantaneous center, I.

The first step in all three methods is to find the Instantaneous center, I, that gives the Gear Ratio at the design position.

Find length a to locate Point I.

a = GD/(1-G)....Equation 1


G = Gear Ratio,

D = distance between the Point Ao and Bo

G = Input Speed / Output Speed. You should choose the Input to be the faster gear. Usually the shorter Link.

G is positive when the gears segments rotate in the same direction, and negative when in the opposite direction.

G should be between –1 and +1. Swap the input and output shafts if G lies outside this range.

D is a scaling factor. For convenience, it is easy to make the length A0B0 = 100mm


If G = 0, then B.B0 become zero length. If G = 1, then you get a parallelogram.

tog_minusMethod 1: Perpendiculars
tog_minusMethod 2: Inflection Circle and Cubic of Stationary Curvature
tog_minusMethod 3: Use the Collineation Axis.