How do you calculate the transmission ratio for a cycloidal needle wheel gearbox?

Cycloidal needle wheel gearboxes stand out in the realm of power transmission mechanisms for their efficiency, versatility, and robustness in various industrial applications. At the heart of understanding and effectively utilizing these gearboxes lies the calculation of their transmission ratio. So, how exactly do you calculate the transmission ratio for a cycloidal needle wheel gearbox? 
Understanding the Components:
Before diving into the calculation, it's crucial to grasp the key components of a cycloidal needle wheel gearbox. These gearboxes typically consist of an input shaft, an output shaft, and a cycloidal disc. The input shaft drives the cycloidal disc, which in turn drives the output shaft. Both the input and output shafts are characterized by a specific number of lobes, denoted as S1 and S2 respectively. Meanwhile, the cycloidal disc itself possesses a certain number of lobes, denoted as C. The Formula:
The transmission ratio (TR) of a cycloidal needle wheel gearbox is determined by the relationship between the lobes on the input and output shafts, as well as the lobes on the cycloidal disc. The calculation is succinctly captured by the following formula:
TR = (S2 / S1) * (C / S2)
Deciphering the Formula:
Let's break down this formula to better understand its components:
S1: Number of lobes on the input shaft
S2: Number of lobes on the output shaft
C: Number of lobes on the cycloidal disc
Practical Application:
To illustrate this calculation in practice, consider a hypothetical scenario where the input shaft has 10 lobes, the output shaft has 20 lobes, and the cycloidal disc boasts 80 lobes. Plugging these values into our formula, we get:
TR = (20 / 10) * (80 / 20) = 2 * 4 = 8
This means that for every revolution of the input shaft, the output shaft will complete 8 revolutions. Such clarity in understanding the transmission ratio empowers engineers and technicians to make informed decisions when designing or operating machinery employing cycloidal needle wheel gearboxes.
Significance of Transmission Ratio:
The transmission ratio serves as a fundamental parameter in assessing the performance and capabilities of a Cycloidal needle wheel gearbox. It directly influences factors such as speed, torque, and power output, thereby impacting the efficiency and effectiveness of various industrial processes.