Gear Design - Force and Stress Analysis

By Mohit Uniyal|Updated : August 3rd, 2022

The Gear design is one of the greatest and oldest inventions in history. It was invented in the 17th century B.C. at that time, it was used by one of the Greek Inventors in clocks and water wheels. The various types of sketches and gear designs of this time have been found in history. At that time, gear was defined as the opposite direction of rotation when one gear wheel drives the other.

There are various mechanical drives like chain drive, belt drive, and gear drive. In the list of mechanical drives, gear is one of the most suitable for transmitting mechanical power from one shaft to another. In this article, we have seen the detail about gears, their purpose and requirement, and the design of gears.

What is the Design of Gear?

Gear designing is the process of defining the dimensions and shapes of the gears. Various factors such as gear size, tooth shape, number of teeth, amount of profile shift, etc., are considered while designing the gears. Gear is a toothed element that transmits rotary motion from one shaft to another with a constant velocity ratio. The concept of gear comes from the friction wheels; gear design is done in a way to increase the friction and avoid slipping between the wheels; proper teeth are cut over it. The tooth profile should follow the law of gearing; then, only we can say it as gear. General involute and cycloidal profiles are most used and suitable for gear.

A combination of two or more gears, which are arranged so that power is transmitted from a driving shaft to the driven shaft, is known as a gear train. The gear train consists of the main driver known as the pinion, the main drive known as the gear, intermediate gear, and some, in some cases, arms. As the gear has the advantage of no-slip condition, it is used in applications where accurate and precise motion is required, like watches, lathe machines, etc.

Various types of analysis are done to create efficient gear design which will sustain various loads and stress over time. The two major types of analysis done for gear design are:

  • Force Analysis
  • Stress Analysis

Forces Analysis in Gear Design

Regarding the gear design, the first thing that comes into the picture is the forces acting on the gear tooth. According to the gear tooth profile type, gears have a different tooth profile. As the mashing gear starts rotating and transferring, the power and torque force are normally applied to the gear's face and shank; according to their tooth profile, the resultant forces are resolved into two components at a pitch point.

  1. Tangetial components (Ft)
  2. Radial components (Fr)

In the design of gears, we have to consider the following assumptions:

  • As the point of contact moves, the magnitude of the resultant force Fn changes. This effect is neglected in the analysis.
  • It is assumed that only one pair of teeth takes the entire load. At times there are two pairs which are simultaneously in contact and share the load. This aspect is neglected in the analysis.
  • The analysis is valid under static conditions, i.e., when the gears run at very low velocities. In practice, power transmission has a dynamic force and force. The effect of this dynamic force is neglected in the analysis.

The tangential force responsible for transmitting the torque is given by

Ft = 2 Mt/d

Where, Mt = 60*106(kW)/(2πn)

The radial force or separating force is given by

Fr = Ft tanα

The resultant force normal to the surface is given by

FN = Ft /cosα

Where,

  • Mt = Torque transmitted by gears
  • α = Pressure angle
  • d = Dimeter of gear
  • n = Speed of rotation (rpm)

Stress Analysis in Gear Design

While doing design analysis of spur gear, we consider the teeth of gear as a cantilever beam. As we see that two forces acting on the tooth, the radial component results in the compressive stress and tangential components result in the bending stress in the gear tooth, as the magnitude of radial force is less than that of the tangential component. Therefore we considered only bending moment while doing the stress analysis in gear design.

In stress analysis of the design of gear following assumptions have to be considered:

  • The radial component (Fr) effect, which induces compressive stresses, is neglected.
  • It is assumed that the tangential component (Ft) is uniformly distributed over the face width of the gear. This is possible when the gears are rigid and accurately machined.
  • The effect of stress concentration is neglected.
  • It is assumed that at any time, only one pair of teeth is in contact and takes the total load.

The beam strength of the gear tooth is given by the equation:

Ft = m b σb Y

Y= t2/6 h m

Sb ≥ m b σb Y

Sb ≥ Ft …………… (For the safe design of gears)

Where,

  • Sb = Beam strength of gear tooth (N)
  • σb = Permissible bending stress (N/mm2)
  • m = Module of gear
  • b = Width of gear tooth
  • Y = Lewis form factor
  • h = Effective length of gear tooth
  • t = Thickness of gear tooth

For the safe gear design, the beam strength of the gear should be greater than or equal to the tangential force applied to the gear.

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FAQs on Gear Design

  • Gear design is the process of designing the gears used in mechanical processes based on force and stress analysis. Gears are designed in order to transfer maximum power with least wastage of the energy.

  • Different terminology in the design of gear are as follow:

    • Pitch circle diameter​
    • Pitch circle​
    • Pitch point​
    • Circular pitch (𝑝𝑐)​
    • The diametrical pitch (𝑝𝑑)​
    • Module (m)​
    • Addendum (a) and Addendum circle​
    • Dedendum and Dedendum circle​
    • Full depth of teeth​
    • Working depth of teeth​
    • Space width​
    • Face width
    • Tooth thickness​
    • Fillet ​
    • Backlash​
    • Face of the tooth​
    • Flank of the tooth​
    • Top and bottom land​
    • Clearance​
  • Consider factors like size, tooth shape, pitch, number of teeth, degree of profile shift, material, required heat treatment and/or grinding, type of tooth surface finish, degree of backlash, mounting method, precision class, etc., as well as the manufacturing process and lubrication technique when designing gears.

  • The following five design factors are taken into account: 

    • The number of teeth,
    • The pressure angle, 
    • The reduction ratio, 
    • The profile shifting factor, 
    • The addendum factor, and
    • The damping ratio.
  • The gears can be classified as follow:

    a. Parallel shaft

    1. Spur gear.
    2. Spur rack and pinion gear.
    3. Helical spur gear.
    4. Double helical gear.
    5. Herringbone gear.

    b. Intersecting shaft

    1. Zero bevel gear.
    2. Spiral bevel gear.

    c. Skew shaft (non-parallel, Non-intersecting shaft)

    1. Cross helical gear.
    2. Worm gear.
    3. Hypoid gear.
  • Apart from bending stress, gear can be failed due to failure of its surface

    • Abrasive wear - It is due to the impurities present in the lubricating oil such as dirt, rust, weld spatter or metallic debris roughening the gear surface.
    • Corrosive wear - Due to improper lubrication and moisture, corrosive element, and foreign material are present in the lubrication.
    • Scoring - ​Scoring is a stick-slip phenomenon, in which alternate welding and shearing take place rapidly at the high spots.
    • Pitting - It is a type of fatigue failure, when the load on the gear tooth exceeds the limits of surface endurance strength of the material, pitting failure will occur.

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