IdealGear

An ideal mechanical gear unit with a fixed housing, connecting two rotational shafts.

This component models a lossless gearbox with a user-defined gear ratio, connecting two mechanical rotational splines, spline_a and spline_b. The housing of the gear is assumed to be rigidly connected to a support. The kinematic relationship between the relative angle of shaft 'a' and shaft 'b' with respect to the support is given by:

$$\begin{gathered} ph{i}_a=ratio \\ cdot \\ ph{i}_b \end{gathered}$$

The corresponding torque relationship, assuming an ideal (lossless) transmission, is:

$$\begin{gathered} \text{ratio} \\ cdot \\ ta{u}_a+ \\ ta{u}_b=0 \end{gathered}$$

where ${\tau }_a$ and ${\tau }_b$ are the torques at spline_a and spline_b respectively. The relative angles are defined as:

$$\text{phi\_a = text{spline\_a}.phi - phi\_text{support}}$$$$\begin{gathered} ph{i}_b=\text{spline}{ \\ }_b. \\ phi- \\ phi{ \\ }_{\text{support}} \end{gathered}$$

This component extends from PartialElementaryTwoSplinesAndSupport

Usage

IdealGear(ratio)

Parameters:

Name	Description	Units	Default value
ratio	The fixed transmission ratio between shaft 'a' and shaft 'b' (phi_a / phi_b)	–

Connectors

• spline_a - (Spline)

• spline_b - (Spline)

• support - (Spline)

Variables

Name	Description	Units
phi_support	Variable for the absolute angle of the support spline	rad
phi_a	Relative angle of shaft 'a' with respect to the support	rad
phi_b	Relative angle of shaft 'b' with respect to the support	rad

Behavior

$$\begin{array}{rl}\mathtt{support}\mathtt{.}\mathtt{phi}\left(t\right)& =\mathtt{phi}\mathtt{\_}\mathtt{support}\left(t\right)\\ \mathtt{support}\mathtt{.}\mathtt{tau}\left(t\right)& =-\mathtt{spline}\mathtt{\_}\mathtt{b}\mathtt{.}\mathtt{tau}\left(t\right)-\mathtt{spline}\mathtt{\_}\mathtt{a}\mathtt{.}\mathtt{tau}\left(t\right)\\ \mathtt{phi}\mathtt{\_}\mathtt{a}\left(t\right)& =-\mathtt{phi}\mathtt{\_}\mathtt{support}\left(t\right)+\mathtt{spline}\mathtt{\_}\mathtt{a}\mathtt{.}\mathtt{phi}\left(t\right)\\ \mathtt{phi}\mathtt{\_}\mathtt{b}\left(t\right)& =\mathtt{spline}\mathtt{\_}\mathtt{b}\mathtt{.}\mathtt{phi}\left(t\right)-\mathtt{phi}\mathtt{\_}\mathtt{support}\left(t\right)\\ \mathtt{phi}\mathtt{\_}\mathtt{a}\left(t\right)& =\mathtt{ratio}\mathtt{phi}\mathtt{\_}\mathtt{b}\left(t\right)\\ 0& =\mathtt{spline}\mathtt{\_}\mathtt{b}\mathtt{.}\mathtt{tau}\left(t\right)+\mathtt{ratio}\mathtt{spline}\mathtt{\_}\mathtt{a}\mathtt{.}\mathtt{tau}\left(t\right)\end{array}$$

Source

# An ideal mechanical gear unit with a fixed housing, connecting two rotational shafts.
#
# This component models a lossless gearbox with a user-defined gear ratio,
# connecting two mechanical rotational splines, `spline_a` and `spline_b`.
# The housing of the gear is assumed to be rigidly connected to a support.
# The kinematic relationship between the relative angle of shaft 'a' and shaft 'b'
# with respect to the support is given by:
# ```math
# \\phi_a = ratio \\cdot \\phi_b
# ```
# The corresponding torque relationship, assuming an ideal (lossless) transmission, is:
# ```math
# \text{ratio} \\cdot \\tau_a + \\tau_b = 0
# ```
# where \$\tau_a\$ and \$\tau_b\$ are the torques at `spline_a` and `spline_b` respectively.
# The relative angles are defined as:
# ```math
# \\phi_a = \text{spline\\_a}.\\phi - \\phi\\_\text{support}
# ```
# ```math
# \\phi_b = \text{spline}\\_b.\\phi - \\phi\\_\text{support}
# ```
component IdealGear
  extends PartialElementaryTwoSplinesAndSupport
  # The fixed transmission ratio between shaft 'a' and shaft 'b' (phi_a / phi_b)
  parameter ratio::Real
  # Relative angle of shaft 'a' with respect to the support
  variable phi_a::Angle
  # Relative angle of shaft 'b' with respect to the support
  variable phi_b::Angle
relations
  phi_a = spline_a.phi-phi_support
  phi_b = spline_b.phi-phi_support
  phi_a = ratio*phi_b
  0 = ratio*spline_a.tau+spline_b.tau
metadata {"Dyad": {"icons": {"default": "dyad://RotationalComponents/IdealGear.svg"}}}
end

Flattened Source

# An ideal mechanical gear unit with a fixed housing, connecting two rotational shafts.
#
# This component models a lossless gearbox with a user-defined gear ratio,
# connecting two mechanical rotational splines, `spline_a` and `spline_b`.
# The housing of the gear is assumed to be rigidly connected to a support.
# The kinematic relationship between the relative angle of shaft 'a' and shaft 'b'
# with respect to the support is given by:
# ```math
# \\phi_a = ratio \\cdot \\phi_b
# ```
# The corresponding torque relationship, assuming an ideal (lossless) transmission, is:
# ```math
# \text{ratio} \\cdot \\tau_a + \\tau_b = 0
# ```
# where \$\tau_a\$ and \$\tau_b\$ are the torques at `spline_a` and `spline_b` respectively.
# The relative angles are defined as:
# ```math
# \\phi_a = \text{spline\\_a}.\\phi - \\phi\\_\text{support}
# ```
# ```math
# \\phi_b = \text{spline}\\_b.\\phi - \\phi\\_\text{support}
# ```
component IdealGear
  # First rotational 1-dimensional shaft spline connector (flange_a)
  spline_a = Spline() [{"Dyad": {"placement": {"icon": {"x1": -50, "y1": 450, "x2": 50, "y2": 550}}}}]
  # Second rotational 1-dimensional shaft spline connector (flange_b)
  spline_b = Spline() [{"Dyad": {"placement": {"icon": {"x1": 950, "y1": 450, "x2": 1050, "y2": 550}}}}]
  # Support spline connector, typically for housing or reaction torques
  support = Spline() [{
    "Dyad": {
      "placement": {"icon": {"iconName": "support", "x1": 450, "y1": 950, "x2": 550, "y2": 1050}}
    }
  }]
  # Variable for the absolute angle of the support spline
  variable phi_support::Angle
  # The fixed transmission ratio between shaft 'a' and shaft 'b' (phi_a / phi_b)
  parameter ratio::Real
  # Relative angle of shaft 'a' with respect to the support
  variable phi_a::Angle
  # Relative angle of shaft 'b' with respect to the support
  variable phi_b::Angle
relations
  support.phi = phi_support
  support.tau = -spline_a.tau-spline_b.tau
  phi_a = spline_a.phi-phi_support
  phi_b = spline_b.phi-phi_support
  phi_a = ratio*phi_b
  0 = ratio*spline_a.tau+spline_b.tau
metadata {"Dyad": {"icons": {"default": "dyad://RotationalComponents/IdealGear.svg"}}}
end

Test Cases

This is setup code, that must be run before each test case.

using RotationalComponents
using ModelingToolkit, OrdinaryDiffEqDefault
using Plots
using CSV, DataFrames

snapshotsdir = joinpath(dirname(dirname(pathof(RotationalComponents))), "test", "snapshots")

"/home/actions-runner-10/.julia/packages/RotationalComponents/0VPxm/test/snapshots"

Related

• Examples

• Experiments

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