Introduction:
The cycloidal tooth profile is a type of tooth profile used in gears. It is characterized by its smooth, curved shape and is commonly used in applications where low noise and high efficiency are desired. When analyzing the behavior of the cycloidal tooth profile, the pressure angle at different points in the engagement process is an important parameter to consider.

Pressure Angle:
The pressure angle is the angle between the direction of the force exerted by one gear tooth on the mating gear tooth and the line perpendicular to the tangent of the pitch circle. It is an important parameter that determines the ability of the gear teeth to transmit power efficiently and smoothly.

Pressure Angle at Different Points in Engagement:
I. Commencement of Engagement:
At the commencement of engagement, when the gear teeth start to come into contact, the pressure angle is at its maximum value. This is because the involute curve, which is the profile of the cycloidal tooth, is steepest at this point. The maximum pressure angle ensures that the contact between the gear teeth is established smoothly and efficiently.

II. Pitch Point:
At the pitch point, which is the point of maximum contact between the gear teeth, the pressure angle is zero. This means that the force exerted by one gear tooth on the mating gear tooth is purely tangential to the pitch circle. The zero pressure angle at the pitch point results in smooth and efficient power transmission without any axial forces.

III. End of Engagement:
At the end of engagement, when the gear teeth are about to disengage, the pressure angle is again at its maximum value. This is because the involute curve is steepest at this point, similar to the commencement of engagement. The maximum pressure angle ensures that the disengagement of the gear teeth occurs smoothly and efficiently.

Conclusion:
In summary, the pressure angle for a cycloidal tooth profile is maximum at the commencement and end of engagement, and it is zero at the pitch point. This characteristic of the cycloidal tooth profile allows for smooth and efficient power transmission without any axial forces at the pitch point, resulting in low noise and high efficiency.