Planetary gear systems are a type of gear train configuration where a central gear, called the sun, meshes with and is surrounded by planet gears, which in turn mesh with an outermost gear called the ring gear. Due to their advantages of high efficiency, compactness, and high power-to-weight ratios, they are widely used in various mechanical applications, including automotive transmissions, wind turbines, printing lathes, automation assembly, and semiconductor equipment.

Keep reading to understand the calculations necessary for designing planetary gear systems.

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## Planetary Gear Calculations

There are several different methods that can be used to perform planetary gear calculations, including analytical methods, graphical methods, and computer simulations.

Analytical methods involve solving mathematical equations to determine the system parameters. Graphical methods involve plotting the gear trains and calculating the parameters based on the geometry of the gears. Computer simulations use specialized software to model the system and simulate its behavior under different operating conditions.

In this article, the focus is only on the basic principles and parameters necessary in the design and analysis of planetary gear systems.

## Planetary Gear Systems

Before learning about the different calculations in detail, it is important to understand what planetary gear systems are and how they work.

In general, gears are combined into gear train systems in order to achieve the desired torque or velocity ratio in transmitting motion between rotating shafts. Gear train systems can be divided into two main classes: fixed-axis and epicyclic gear trains.

A fixed-axis gear train, also known as a parallel-axis gear train, consists of a series of gears mounted on fixed parallel shafts, while an epicyclic gear train consists of gears whose axes revolve around a fixed axis of another gear. Planetary gear systems are technically considered a subset of the epicyclic gear train family; however, these terms are often used interchangeably.

A planetary gear system consists of four main components: a central sun gear, planet gears, an outer ring gear, and an arm or carrier. The planet gears, which are mounted on a carrier, mesh with both the sun gear and the ring gear simultaneously, creating a compact and highly efficient gear arrangement, as shown in the diagram below.

Strictly speaking, a planetary gear train is a type of epicyclic gear train where the fixed member is the ring gear, the input member is the sun gear, and the output member is the carrier. If the fixed member is the sun gear, it is known as a solar epicyclic gear train, and if the fixed member is the carrier, it is known as a star epicyclic gear train.

However, for the sake of this article, all possible combinations of input, output, and fixed planetary gear train components are considered in the calculations.

## Velocity Ratio Calculation

The formula for the velocity ratio depends on which component is considered the input, output, or fixed. Suppose that α is the ratio between the number of teeth of the ring gear and the number of teeth of the sun gear.

If the sun gear is the input, the carrier is fixed, and the ring gear is the output, the formula for the velocity ratio is:

If the ring gear is the input, the carrier is fixed, and the sun gear is the output:

If the carrier is the input, the sun gear is fixed, and the ring gear is the output:

If the ring gear is the input, the sun gear is fixed, and the carrier is the output:

If the sun gear is the input, the ring gear is fixed, and the carrier is the output:

The same preceding formula applies if the carrier is the input, the ring gear is fixed, and the sun gear is the output.

## Overall Gear Ratio

Calculating the overall gear ratio also depends on whether the system is a planetary, star, or solar epicyclic gear train.

For a planetary epicyclic gear train:

Where:

- N
_{R}= number of teeth of the ring gear [unitless] - N
_{S}= number of teeth of the sun gear [unitless]

For a star epicyclic gear train:

For a solar epicyclic gear train:

In practice, planetary gear systems usually have a gear ratio ranging from 3:1 to 12:1, while star systems have a range from 2:1 to 11:1, and solar systems typically range from 1.2:1 to 1.7:1.

## Planetary Gear Torque Calculation

While it is not possible to develop a universal set of equations for determining component torques in every possible drive connection, there are several rules based on the fundamental law of gears that can be used to aid in torque analysis. For instance, in a basic planetary gear system, the ratio of ring torque to sun torque is equal to the ratio of the radii of the ring gear and the sun gear. Additionally, the torque ratio between the drive components and the external torque is equal to the reciprocal of the velocity ratio.

In the analysis of planetary systems, torque is typically correlated with speed for each component and then presented in tabular form to simplify the analysis process. Remember that the equation for the ratio of the torque and angular velocities in meshed gears is:

Where

- T
_{1}, T_{2}= torque values of the input and output shafts, respectively [N-m] - ω
_{1}, ω_{2}= corresponding angular velocities

The above equation is based on the fundamental law of gears, which states that the product of the gear’s pitch radius and its rotational speed is the same for all gears in a meshed system.

Moreover, note that torque can also be correlated to angular velocity and power using the following equation:

Where:

- T = torque on the gear [N-m]
- P = power [W]
- ω = angular velocity [rad/s]

## Planetary Gear Efficiency

In ordinary gear trains, the concept of power flow is normally used to estimate efficiency, with a constant percentage loss assumed for each gear mesh. For instance, a gear train with three meshes and a 1% loss for each mesh would result in an overall efficiency of 0.99^{3} or 97%. However, in actual practice, it is difficult to accurately estimate efficiency, especially for planetary gear systems.

To estimate the efficiency of a planetary gear system, one needs to calculate the power losses in the system due to various factors such as friction, oil churning, windage, and bearing losses. The efficiency of a gear system is defined as the ratio of the output power to the input power. Therefore, the efficiency can be estimated by measuring the input and output powers and calculating the ratio between them.

However, accurately measuring the power losses due to various factors can be challenging, and often requires complex modeling and simulation. Therefore, empirical tests are often used to estimate the efficiency, by measuring the input and output powers and comparing them to the theoretical values calculated from the gear ratios and torque values.

## Planetary Gear Systems vs Ordinary Gear Trains

Planetary gear systems offer several advantages over conventional gear trains due to their unique configuration. Firstly, because the load is shared between several gear meshes, they occupy less space and have a lighter weight. Secondly, the smaller and stiffer components used in epicyclic trains result in reduced noise and vibration, while increasing efficiency.

Additionally, because the input and output shafts are concentric, there are no bending moments or torques created by radial forces. This allows the driver and the driven equipment to be mounted in line, providing additional space savings.

However, there are also several drawbacks to planetary gear systems. These include its complexity, limited assembly of gear trains to specific gear ratios, and difficulty in calculating efficiency.