Simple Gear Design

Introduction

            The animation produced here is an attempt to educate the user about gear ratios and is not intended to provide design information for durable and low noise gear boxes.

 

Design of Simple Gears

            The simple design shown here makes each gear tooth into a trapezoid as I attempt to show in the Figure below:

 

where the cutout (notch) and the actual tooth have the same shape.

Gears that mesh have the requirement that their circumferential spacing has to be the same regardless of the average radius or the number of teeth of the gear.  Therefore the actual meshing circumference, c, of the gear is given by the equation:

 

                                                                                                                           (1)

where n is the number of teeth and ts is the tooth spacing.  It is reasonable to choose the radius at which this circumference exists as the radius at half of the tooth height as shown above so that the equation for the radius becomes:

                                                                                                               (2)

Another parameter is the tooth depth, d.  This has to be large enough to keep the gears meshed even after there has been some wear.

To design a single tooth and notch (a single period on the gear) we define 6 angles and 6 radii.  The angles are then obviously each 1/6 of the total angle subtended by the period:

                                                                                                                        (3)

The radii and angles are therefore:

           

                          (4)

We then use the following equations to define the (x,y) coordinates of the points on the single tooth period:

                                                                                                                      (5)

where i ranges from 1 through 6.

To provide the (x,y) points on the remaining n-1 teeth, we add kqperiod to each of the angles in equation 5 and re-compute:

           

                                                                                                     (6)

where k ranges from 1 to n.

 

Meshing gears

            It is obvious from equations 4 that the center of the first tooth will always exist at f=0 which means there will always be a tooth centered on the positive x axis.  If the number of teeth, n, is an even number then there will also be a tooth centered on the negative x axis.  By the same reasoning, if the number of teeth is odd, there will always be a tooth notch centered on the negative x axis.   If we want a gear whose axis is to the left of another gear to mesh, then, if the gear to the right has an odd number of teeth, no rotation is required.  By the same reasoning, if we want a gear to the left of a gear with an even number of teeth to mesh, we can rotate the left hand gear by half of its tooth period and meshing can be accomplished.  This takes care of the angles between the two gears.  To get the distance between the axes, recall that we defined the meshing radii as r(n) in equation 2.  Therefore we can just separate the axes by

                                                                                                (7)

where d is the distance between the axes of gears 1 and 2.  For example, if the left hand gear is centered at (x,y)=(0,0) we would set the axis of the right hand gear at (x,y)=(r1+r2,0).

 

Internal Gears

These are gears where the teeth are internal to the ring which contains them.  The design for internal gears is most simply done by following the above prescription for the internal teeth.  This means that centered on the positive x axis we have a notch rather than a tooth like we had on external gears.   So, if we want to mesh an external gear of radius re on an internal gear of radius ri, we can place its center at distance ri-re from the center of the internal gear and we would not have to rotate either gear to get them to mesh.

 

Hybrid Gears

These are gears where the teeth are both internal and external to the ring which contains them.  The design for the internal teeth is done exactly the same as for Internal Gears. The prescription for meshing an inside External gear is again exactly the same as above. The external teeth also follow the same prescription as for any other external gear which means that a tooth is centered on the positive x axis.  Then, if the hybrid’s number of external teeth is even, there will also be a tooth centered on the negative x axis so we would need to rotate a standard external gear by ½ tooth in order to mesh it on the left side of a hybrid gear. To mesh a standard external gear on the right side (positive x axis) of a hybrid’s exterior, we would need to rotate it by ½ tooth if its number of teeth is even.