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 $\tau \ast a$ and $\tau \ast b$ 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

RotationalComponents.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 - This connector represents a rotational spline with angle and torque as the potential and flow variables, respectively. (Spline)

• spline_b - This connector represents a rotational spline with angle and torque as the potential and flow variables, respectively. (Spline)

• support - This connector represents a rotational spline with angle and torque as the potential and flow variables, respectively. (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

math \phi_b = \text{spline}_b.\phi - \phi_\text

"""
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

math \phi_b = \text{spline}_b.\phi - \phi_\text

"""
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

No test cases defined.

Related

• Examples

• Experiments

• Analyses