# Speed Reducer

Implement speed reducer

## Library

Electric Drives/Shafts and Speed Reducers

## Description

The high-level schematic shown below is built from three main blocks: a high-speed shaft, a reduction device, and a low-speed shaft. More details on the shaft model are included in the Mechanical Shaft reference pages.

## High-Level Schematic

The next figure shows the Simulink® schematic of the speed reducer model.

## Reduction Device

The reduction device dynamics are governed by the following equation:

${J}_{\text{rdh}}{\stackrel{¨}{\theta }}_{\text{rdh}}={T}_{h}-\frac{{T}_{l}}{\eta i},$

where Jrdh is the inertia of the reduction device with respect to the high-speed side, ${\stackrel{¨}{\theta }}_{\text{rdh}}$ is the acceleration of the high-speed side of the reduction device, Th is the torque transmitted by the high-speed shaft to the input of the reduction device, Tl is the torque transmitted by the low-speed shaft from the output of the reduction device, η is the efficiency of the reduction device, and i is the reduction ratio (i ≥ 1).

For reduction devices composed of gears, the efficiency varies according to the type of gears, the number of stages (thus the reduction ratio), the lubricant, etc. For small reduction ratios, the efficiency can climb up to 95%. For higher reduction ratios, the efficiency can be as low as 75%. However, most commercial speed reducers now have high efficiencies of 90% to 95%.

The output speed Nrdl (the speed of the driving side of the low-speed shaft) of the reduction device is given by the following equation:

Nrdl = Nrdh / i,

where Nrdh is the input speed of the reduction device (the speed of the loaded side of the high-speed shaft).

The following figure shows the reduction device schematic.

## Remarks

The stiffness of the shafts must be high enough to avoid large angular deflections that could cause misalignment inside the bearings and damage.

Keep in mind that the low-speed shaft will have a higher stiffness and a higher damping factor than the high-speed shaft, the torque on the low-speed shaft being a lot bigger. For proper simulation results, the damping factor of both shafts must be high enough to avoid undesired transient speed and torque oscillations.

Too high stiffness and damping factor values or too low gearbox inertias can cause simulation errors.

The model is discrete. Good simulation results have been obtained with a 1 µs time step.

## Dialog Box

Preset Model

This pop-up menu allows you to choose preset model parameters.

Reduction Ratio

The reduction ratio of the speed reducer (i ≥ 1).

Reduction Device Inertia

The inertia of the reduction device with respect to the high-speed side (kg.m2).

Efficiency

The efficiency of the reduction device.

High-Speed Shaft Stiffness

The stiffness of the high-speed shaft (N.m).

High-Speed Shaft Damping

The internal damping of the high-speed shaft (N.m.s).

Low-Speed Shaft Stiffness

The stiffness of the low-speed shaft (N.m).

Low-Speed Shaft Damping

The internal damping of the low-speed shaft (N.m.s).

## Block Inputs and Outputs

### Inputs

The block has two inputs: Nh and Nl.

The first input, Nh, is the speed (rpm) of the driving end of the high-speed shaft.

The second input, Nl, is the speed (rpm) of the loaded end of the low-speed shaft.

### Outputs

The block has two outputs: Th and Tl.

The Th output is the torque transmitted by the high-speed shaft to the reduction device.

The Tl output is the torque transmitted by the low-speed shaft to the load.

## Model Specifications

The library contains four preset models. The specifications of these speed reducer models are shown in the following table.

Preset Speed Reducer Models

1st

2nd

3rd

4th

Power (hp)

5

5

200

200

Reduction ratio

10

100

10

100

Max. output torque (N.m)

300

3000

12200

122000

The high-speed and low-speed shafts of the preset models have been designed in order to present 0.1 degrees of angular deflection at maximum torque.

## Example

The `reducer_example` example illustrates the speed reducer model.

The speed reducer is driven by a variable-speed source and is connected to a load. The load has an inertia of 30 kg.m2 and a viscous friction term of 0.5 N.m.s.

The speed reducer has a reduction ratio of 10, and the inertia of the reduction device with respect to the high-speed side is 0.0005 kg.m2. The reduction ratio being quite low, the efficiency is high and worth 0.95.

The high-speed shaft has a stiffness of 17190 N.m and an internal damping factor of 600 N.m.s. This shaft is designed to have 0.1 degree of angular deflection for a 30 N.m load torque. The low-speed shaft, having a higher torque to transmit, has a stiffness of 171900 N.m and an internal damping factor of 6000 N.m.s. This shaft is designed to have 0.1 degree of angular deflection for a 300 N.m load torque.

At t = 0 s, the driving speed starts climbing to 1750 rpm with a 500 rpm/s acceleration ramp. This causes the transmitted torque of the high-speed shaft to jump to about 18 N.m. Because of the reduction device, the torque transmitted to the load by the low-speed shaft is a lot bigger and is worth about 170 N.m.

During the accelerating phase, both torques keep increasing in order to compensate the viscous friction of the load. Notice that the load accelerates with a ramp of +50 rpm/s because of the reduction ratio of the gearbox.

At t = 3.5 s, the driving speed settles at 1750 rpm. Since no more accelerating torque is needed, the input and output torques decrease and stabilize respectively to 0.965 N.m and 9.16 N.m at t = 4 s. The load speed is now equal to 175 rpm.

The following figure shows the speed reducer input and output speeds and torques.

## References

[1] Norton, Robert L., Machine Design, Prentice Hall, 1998.

[2] Nise, Norman S., Control Systems Engineering, Addison-Wesley Publishing Company, 1995.