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Implement speed reducer

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.

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

The reduction device dynamics are governed by the following equation:

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

where *J*_{rdh} is the inertia of the reduction
device with respect to the high-speed side, $${\ddot{\theta}}_{\text{rdh}}$$ is the acceleration of the high-speed side of the reduction device,
*T _{h}* is the torque transmitted by the
high-speed shaft to the input of the reduction device,

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 *N*_{rdl} (the speed of the
driving side of the low-speed shaft) of the reduction device is given by the following
equation:

*N*_{rdl} =
*N*_{rdh} /
*i*,

where *N*_{rdh} 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.

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.

**Preset model**This pop-up menu allows you to choose preset model parameters. When you select a preset model, the other block parameters become inaccessible. Default is

`01: 5 HP — i = 10 — Tlmax = 300 N.m`

.**Reduction ratio**The reduction ratio of the speed reducer (

*i*≥ 1). Default is`10`

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

`0.0005`

.**Efficiency**The efficiency of the reduction device. Default is

`0.95`

.**High-speed shaft stiffness**The stiffness of the high-speed shaft (N.m). Default is

`17190`

.**High-speed shaft damping**The internal damping of the high-speed shaft (N.m.s). Default is

`600`

.**Low-speed shaft stiffness**The stiffness of the low-speed shaft (N.m). Default is

`171900`

.**Low-speed shaft damping**The internal damping of the low-speed shaft (N.m.s). Default is

`6000`

.

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.

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.

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.

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

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