How to Calculate the Number of Gears in a Gearbox: A Simplified Approach

Introduction

Designing and understanding the number of gears in a gearbox can be a complex task, especially when dealing with planetary gear sets. This article simplifies the process by utilizing an analogy with levers, making it easier to understand and apply these principles in practical scenarios.

Counting Gears: A Basic Overview

Counting the number of gears in a gearbox can be approached in several ways:

Inspection: Use a shifter knob or dash markings to identify the gear positions. Manual Manipulation: Move the shifter by hand to see the number of gear positions. Owners Manual: Refer to the manual where the gear count is typically documented.

However, for a deeper understanding or customization, it is essential to know the precise number of teeth on each gear and how they are connected.

Planetary Gear Sets and Levers

Professor Henry Zhang from Purdue University introduced an innovative approach using levers to simplify gear calculations. This method reduces complex gear interactions to simpler mechanical principles, making it easier to analyze and design gear systems.

Step-by-Step Guide

The procedure involves the following steps:

Replace Each Gear Set with a Lever: Each planetary gear set is represented by a vertical lever. Rescale Levers: Adjust the lengths of the levers based on the number of teeth on the sun and ring gears. Identify Connections: Determine the connections between the levers, analogous to the gear sets' connections.

Mapping Gear Sets and Calculating Ratios

The tooth ratio of the sun and ring gears determines the lever proportions. Here is how you can manually calculate the gear ratio:

Example: Consider a gear set with a sun gear (NS) and a ring gear (NR).

Invert Proportions: To simulate the planetary gear set, adjust the lever arms such that Tr is the torque from the ring side, and the lever length for the sun side is Nr. Static Equilibrium: To achieve equilibrium, the torque from the ring side (Tr) must equal the torque from the sun side (Ts) multiplied by the ratio of their lengths: Mathematical Representation: Ts Tpc NS / NR

For a transmission with two planetary gear sets (often a six-speed), the process involves adding additional lever arms and adjusting their proportions.

Combining Gear Sets

Each gear set can be represented by a lever, and combining them requires proportionally adjusting the lengths of the levers:

Create Ratios: For the second gear set, divide the lengths between the ring and carrier and the carrier and sun to determine proportional adjustments. Final Length Calculation: Multiply the length between the carrier and sun to determine the length between new pivot points.

Finding Gear Ratios

Once the lever system is set up, run a statics analysis to determine the torque output for any linkage on the arm. The ratio of the lever lengths provides the gear ratio:

Example: If the length of the carrier and sun in set 1 is 65, and the length between the carrier and sun in the new set is calculated, the gear ratio can be determined.

Conclusion

The lever analogy significantly simplifies the calculations involved in gear set design. While the subject can be complex, understanding the basic principles can greatly enhance the effectiveness of gearbox design. For further clarification, it is recommended to read the linked document and practice with example problems.

Keywords: gear calculations, gearbox design, gear ratios