Gear Design - Force and Stress Analysis

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

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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 (F_t)
2. Radial components (F_r)

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

F_t = 2 M_t/d

Where, M_t = 60*106(kW)/(2πn)

The radial force or separating force is given by

F_r = F_t tanα

The resultant force normal to the surface is given by

F_N = F_t /cosα

Where,

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

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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 (F_t) 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:

F_t = m b σ_b Y

Y= t²/6 h m

S_b ≥ m b σ_b Y

S_b ≥ F_t …………… (For the safe design of gears)

Where,

• S_b = Beam strength of gear tooth (N)
• σ_b = 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.