IdealPlanetaryGear

Ideal planetary gear set with three rotational flanges (sun, ring, carrier).

This component models the kinematic and torque relationships of an ideal planetary gear system. It connects three shafts: the sun gear (sun), the ring gear (ring), and the planet carrier (carrier). The gear ratio ratio is defined as the number of teeth on the ring gear divided by the number of teeth on the sun gear. The fundamental kinematic relationship, also known as the Willis equation, links the angles of the three shafts:

$$(1+\text{ratio}) \\cdot \\phi_{\text{carrier}} = \\phi_{\text{sun}} + \text{ratio} \\cdot \\phi_{\text{ring}}$$

For an ideal, massless gear, the torques are related as follows:

$$\tau_{\text{ring}} = \text{ratio} \\cdot \tau_{\text{sun}}$$

$$\tau_{\text{carrier}} = -(1+\text{ratio}) \\cdot \tau_{\text{sun}}$$

Ideal planetary gear box

Usage

IdealPlanetaryGear(ratio)

Parameters:

Connectors

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

Behavior

$$\begin{align} \left( 1 + \mathtt{ratio} \right) \mathtt{carrier.phi}\left( t \right) &= \mathtt{sun.phi}\left( t \right) + \mathtt{ratio} \mathtt{ring.phi}\left( t \right) \\ \mathtt{ring.tau}\left( t \right) &= \mathtt{ratio} \mathtt{sun.tau}\left( t \right) \\ \mathtt{carrier.tau}\left( t \right) &= \left( -1 - \mathtt{ratio} \right) \mathtt{sun.tau}\left( t \right) \end{align}$$

Source

# Ideal planetary gear set with three rotational flanges (sun, ring, carrier).
#
# This component models the kinematic and torque relationships of an ideal planetary gear system.
# It connects three shafts: the sun gear (`sun`), the ring gear (`ring`), and the planet carrier (`carrier`).
# The gear ratio `ratio` is defined as the number of teeth on the ring gear divided by the number of teeth on the sun gear.
# The fundamental kinematic relationship, also known as the Willis equation, links the angles of the three shafts:
# ```math
# (1+\text{ratio}) \\cdot \\phi_{\text{carrier}} = \\phi_{\text{sun}} + \text{ratio} \\cdot \\phi_{\text{ring}}
# ```
# For an ideal, massless gear, the torques are related as follows:
# ```math
# \tau_{\text{ring}} = \text{ratio} \\cdot \tau_{\text{sun}}
# ```
# ```math
# \tau_{\text{carrier}} = -(1+\text{ratio}) \\cdot \tau_{\text{sun}}
# ```
# Ideal planetary gear box
component IdealPlanetaryGear
  # Flange of sun shaft, representing the sun gear's connection point.
  sun = Spline()
  # Flange of carrier shaft, representing the planet carrier's connection point.
  carrier = Spline()
  # Flange of ring shaft, representing the ring gear's connection point.
  ring = Spline()
  # Gear ratio defined as (number of ring_teeth) / (number of sun_teeth).
  parameter ratio::Real
relations
  # Kinematic relationship between shaft angles (Willis equation).
  (1+ratio)*carrier.phi = sun.phi+ratio*ring.phi
  # Torque balance equation: ring torque in terms of sun torque.
  ring.tau = ratio*sun.tau
  # Torque balance equation: carrier torque in terms of sun torque.
  carrier.tau = -(1+ratio)*sun.tau
metadata {
  "Dyad": {"icons": {"default": "dyad://RotationalComponents/IdealPlanetary.svg"}}
}
end

# Ideal planetary gear set with three rotational flanges (sun, ring, carrier).
#
# This component models the kinematic and torque relationships of an ideal planetary gear system.
# It connects three shafts: the sun gear (`sun`), the ring gear (`ring`), and the planet carrier (`carrier`).
# The gear ratio `ratio` is defined as the number of teeth on the ring gear divided by the number of teeth on the sun gear.
# The fundamental kinematic relationship, also known as the Willis equation, links the angles of the three shafts:
# ```math
# (1+\text{ratio}) \\cdot \\phi_{\text{carrier}} = \\phi_{\text{sun}} + \text{ratio} \\cdot \\phi_{\text{ring}}
# ```
# For an ideal, massless gear, the torques are related as follows:
# ```math
# \tau_{\text{ring}} = \text{ratio} \\cdot \tau_{\text{sun}}
# ```
# ```math
# \tau_{\text{carrier}} = -(1+\text{ratio}) \\cdot \tau_{\text{sun}}
# ```
# Ideal planetary gear box
component IdealPlanetaryGear
  # Flange of sun shaft, representing the sun gear's connection point.
  sun = Spline()
  # Flange of carrier shaft, representing the planet carrier's connection point.
  carrier = Spline()
  # Flange of ring shaft, representing the ring gear's connection point.
  ring = Spline()
  # Gear ratio defined as (number of ring_teeth) / (number of sun_teeth).
  parameter ratio::Real
relations
  # Kinematic relationship between shaft angles (Willis equation).
  (1+ratio)*carrier.phi = sun.phi+ratio*ring.phi
  # Torque balance equation: ring torque in terms of sun torque.
  ring.tau = ratio*sun.tau
  # Torque balance equation: carrier torque in terms of sun torque.
  carrier.tau = -(1+ratio)*sun.tau
metadata {
  "Dyad": {"icons": {"default": "dyad://RotationalComponents/IdealPlanetary.svg"}}
}
end

Test Cases

No test cases defined.

Related

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