Design of spur gear

Mechanical engineers working in transmission field would often have to decide upon kind of gears they have to use. Although this task has become a matter of selection of gear based on standards, it is also important to know what goes behind this. In this article you will learn how to design a pair of spur gears for mechanical strength, surface resistance and fluctuating load.

AGMA standard of gear design

A designed gear should meet following design criteria conforming to AGMA standards. It should have

1. Enough mechanical strength to withstand force transmitted.

2. Enough surface resistance to overcome pitting failure.

3. Enough dynamic resistance to carry fluctuating loads.

Design inputs and outputs in gear design

Following figure shows design inputs and outputs of a gear design.

Fig 1 : Input and output parameters for a gear design
Fig 2 : A general spur gear nomenclatures

Design for space constrains

The designed gear system should fit within a space limit. So the designer could say if he sums pitch diameters of the mating gears, it should be less than or equal to allowable space limit as shown in Fig 3.

Fig 3 : Space constrain of gear design

The blue rectangle represents space on which gear should get fit. One can take 80% of width of this space as allowable width for gear design. So following is the relation obtained by this condition.

Dp1 + Dp2 = Allowable Width

We also know speed ratio of gears, this will lead to one more relation in terms of pitch circle diameters.

By solving above 2 equations simultaneously we can obtain pitch circle diameters of both the gears.

Determination of number of teeth – Interference

Here we will understand how to determine number of teeth on both the gears. To do this we have to assume number of teeth on one gear(T1), say the smaller gear. Now using the relation given below we can determine number of teeth on other gear,T2.

So we got number of teeth on both the gears, but one should also check for a phenomenon called interference if gear system has to have a smooth operation. Interference happens when gear teeth has got profile below base circle. This will result high noise and material removal problem. This phenomenon is shown in Fig 4.

Fig 4 : A pair of gear teeth under interference

If one has to remove interference , the pinion should have a minimum number of teeth specified by following relation.

Where aw represents addendum of tooth. For 20 degree pressure angle(which is normally taken by designers) aw = 1 m and bw = 1.2 m. Module m, and pitch circle diameter Pd are defined as follows.

If this relation does not hold for a given case, then one has to increase number of teeth T1, and redo the calculation.

The algorithm for deciding number of teeth T1 and T2 is shown below.

Fig 5 : Flow chart to determine number of teeth on each gears

Design for mechanical strength – Lewis equation

Now the major parameter remaining in gear design is width of the gear teeth, b. This is determined by checking whether maximum bending stress induced by tangential component of transmitted load, Ft at the root of gear is greater than allowable stress. As we know power transmitted,P and pitch line velocity V of the gear Ft can be determined using following relation.

FtV = Power Transmitted

Here we consider gear tooth like a cantilever which is under static equilibrium. Gear forces and detailed geometry of the tooth is shown in figure below.

Fig 6 : Gear tooth under load

One can easily find out maximum value of bending stress induced if all geometrical parameters shown in above figure are known. But the quantities t and l are not easy to determine, so we use an alternate approach to find out maximum bending stress value using Lewis approach. Maximum bending stress induced is given by Lewis bending equation as follows.

σ = Ft Pd / bY

Where Y is Lewis form factor, which is a function of pressure angle, number of teeth and addendum and dedendum. Value of Y is available as in form of table or graph. Using above relation one can determine value of b, by substituting maximum allowable stress value of material in LHS of equation. But a gear design obtained so will be so unrealistic, because in this design we are considering gear tooth like a cantilever which is under static equilibrium. But that’s not the actual case. In next session we will incorporate many other parameters which will affect mechanical strength of the gear in order to get more realistic design.

AGMA strength equation

When a pair of gear rotates we often hear noise from this, this is due to collision happening between gear teeth due to small clearance in between them. Such collisions will raise load on the gear more than the previously calculated value. This effect is incorporated in dynamic loading loading factor, Kv value of which is a function of pitch line velocity. At root of the gear there could be fatigue failure due to stress concentration effect. Effect of which is incorporated in a factor called Kf value of which is more than 1. There will be factors to check for overload (Ko) and load distribution on gear tooth (Km). While incorporating all these factors Lewis strength equation will be modified like this

The above equation can also be represented in an alternating form (AGMA Strength equation) like shown below.

Where J is

Using above equation we can solve for value of b, so we have obtained all the output parameters required for gear design. But such a gear does not guarantee a peacefull operation unless it does not a have enough surface resistance.

Design for surface resistance

Usually failure happens in gears due to lack of surface resistance, this is also known as pitting failure. Here when 2 mating surfaces come in contact under a specified load a contact stress is developed at contact area and surfaces get deformed. A simple case of contact stress development is depicted below, where 2 cylinders come in contact under a load F.

Fig 7 : Surface deformation and development of surface stress due to load applied

For a gear tooth problem one can determine contact stress as function of following parameters.

σc = ƒ (Ft,ɸ ,r1 ,r2, Kv ,E1,E2 ,v1 ,v2)

If contact stress developed in a gear interface is more than a critical value(specified by AGMA standard), then pitting failure occurs. So designer has to make sure that this condition does not arise.


Sabin Mathew

This article is written by Sabin Mathew, an IIT Delhi postgraduate in mechanical engineering. Sabin is passionate about understanding the physics behind complex technologies and explaining them in simple words. He is the founder of YouTube channel 'LESICS', Engineering educational platform. To know more about the author check this link.