Short Answer:

The line of action is an imaginary straight line that passes through the point of contact between the teeth of two meshing gears and is tangent to both base circles. It represents the direction along which the force is transmitted from one gear tooth to another during meshing. The smooth transfer of motion and power between gears occurs along this line.

The angle between the line of action and the common tangent to the pitch circles of the gears is known as the pressure angle. The line of action ensures constant velocity ratio and proper rolling contact between the gears without slipping.

Detailed Explanation :

Line of Action

The line of action is one of the most important concepts in the study of gears and their motion. It is a geometrical line that defines how power and motion are transmitted from the driving gear to the driven gear. This line lies tangential to the base circles of both gears and passes through the point where their teeth come into contact. It is also the path along which the force between the two teeth acts during the engagement of gears.

In simple terms, when two gears mesh, the pressure between their teeth is transmitted along a straight line. This line is the line of action, and it determines how smoothly the gears will rotate and transmit power. The shape of the teeth, the angle of contact, and the efficiency of the gear system all depend on this line.

Definition and Geometry of Line of Action

When two involute gears mesh, each gear has a base circle, and the involute profile of each tooth is generated from this base circle. The point where the two involute profiles touch at any instant lies on a common tangent to their base circles. This tangent line is known as the line of action.

The line of action makes an angle with the tangent to the pitch circles of the gears. This angle is called the pressure angle (usually 20° or 25° in standard gears). During rotation, the contact point between the two teeth moves along the line of action—from the point of approach, through the pitch point, and then to the point of recess.

The line of action is therefore the locus of all contact points during a single engagement between two teeth.

Function of Line of Action

The line of action plays a vital role in the working of gears. Its main functions include:

Transmission of Force:
The driving gear applies a force on the driven gear along the line of action. This ensures that the force acts tangentially to the base circles, leading to smooth and efficient torque transfer.
Maintaining Constant Velocity Ratio:
Since the line of action is tangent to both base circles, the velocity ratio between the gears remains constant throughout their rotation. This is one of the key advantages of the involute gear profile.
Reducing Slippage:
The line of action ensures that the teeth roll on each other with minimal sliding, reducing friction and wear.
Defining Pressure Angle:
The line of action also helps define the pressure angle, which affects the strength and efficiency of the gear set.
Ensuring Smooth Meshing:
By maintaining continuous contact along this line, gears can transfer motion smoothly without shocks or jerks.

Points on the Line of Action

There are three main points associated with the line of action during gear contact:

Point of Approach:
This is the point where the teeth first come into contact at the beginning of engagement. It lies on the line of action before the pitch point.
Pitch Point:
This is the point on the line of action where the line of centers of both gears intersects the line of action. At this point, the velocity of sliding between the teeth is zero.
Point of Recess:
This is the point where the teeth disengage after contact. It lies on the line of action after the pitch point.

The segment of the line of action between the points of approach and recess is called the path of contact, and its length determines how smoothly the gears mesh.

Relation with Pressure Angle and Base Circles

The angle between the line of action and the tangent to the pitch circles of the gears is the pressure angle. A smaller pressure angle results in smoother operation but lower strength, while a larger pressure angle gives stronger teeth but higher radial forces.

Both gears have base circles smaller than their pitch circles, and the line of action always touches these base circles tangentially. The tangency ensures that the contact between the teeth follows the involute law, maintaining a constant velocity ratio.

Mathematical Relation

The geometry of the line of action can be expressed using trigonometric relations:

= pitch circle radius
= base circle radius
= pressure angle

Then,

This shows that the line of action is always tangent to the base circles of both gears, and its direction is defined by the pressure angle. The motion of contact along this line from approach to recess determines the contact ratio, which represents the average number of teeth in contact during meshing.

Importance of Line of Action

The line of action is crucial in gear design because it ensures:

Constant velocity ratio between gears.
Minimum vibration and noise.
Uniform load distribution across teeth.
Longer life of gears due to reduced wear.
Smooth engagement and disengagement of teeth.

If the line of action is not correctly maintained (for example, if the teeth do not follow the involute profile), the gear motion will not be uniform, leading to vibration, wear, and loss of efficiency.

Example

Consider two spur gears with 20° pressure angle in mesh. The line of action will be tangent to both base circles and inclined at 20° to the common tangent at the pitch circles. During operation, the contact between teeth starts at the point of approach, passes through the pitch point, and ends at the point of recess—all along the same line of action. This ensures constant speed ratio and smooth transmission of torque.

Conclusion

The line of action is the straight line that passes through the point of contact between gear teeth and is tangent to the base circles of both gears. It represents the direction along which the force is transmitted between the gears. The line of action plays a crucial role in maintaining a constant velocity ratio, ensuring smooth motion, and minimizing wear and noise. Proper design of gears ensures that the contact between teeth always takes place along this line, resulting in efficient and reliable gear operation.