Inertia

A 1D-rotational component with inertia, subject to torques from two splines.

This component represents a rigid body with a moment of inertia J that can rotate in one dimension. It connects to other components via two rotational splines, spline_a and spline_b (inherited from PartialTwoSplines). The absolute rotation angle of the inertia is phi, its angular velocity is w, and its angular acceleration is a. The core dynamic behavior is defined by Newton's second law for rotation:

$$\text{J cdot a = text{spline\_a}.tau + text{spline\_b}.tau}$$

The kinematic relationships defining angular velocity and acceleration are:

$$\begin{gathered} w=\frac{d \\ phi}{dt} \end{gathered}$$$$a=\frac{dw}{dt}$$

Both splines are rigidly attached to the inertia, meaning phi = spline_a.phi and phi = spline_b.phi.

This component extends from PartialTwoSplines

Usage

Inertia(J)

Parameters:

Name	Description	Units	Default value
J	Moment of inertia of the rotating body	kg.m2

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)

Variables

Name	Description	Units
phi	Absolute rotation angle of the inertia	rad
w	Absolute angular velocity of the inertia	rad/s
a	Absolute angular acceleration of the inertia	rad/s2

Behavior

$$\begin{array}{rl}\mathtt{phi}\left(t\right)& =\mathtt{spline}\mathtt{\_}\mathtt{a}\mathtt{.}\mathtt{phi}\left(t\right)\\ \mathtt{phi}\left(t\right)& =\mathtt{spline}\mathtt{\_}\mathtt{b}\mathtt{.}\mathtt{phi}\left(t\right)\\ \frac{\mathrm{d}\mathtt{phi}\left(t\right)}{\mathrm{d}t}& =w\left(t\right)\\ \frac{\mathrm{d}w\left(t\right)}{\mathrm{d}t}& =a\left(t\right)\\ Ja\left(t\right)& =\mathtt{spline}\mathtt{\_}\mathtt{b}\mathtt{.}\mathtt{tau}\left(t\right)+\mathtt{spline}\mathtt{\_}\mathtt{a}\mathtt{.}\mathtt{tau}\left(t\right)\end{array}$$

Source

# A 1D-rotational component with inertia, subject to torques from two splines.
#
# This component represents a rigid body with a moment of inertia `J` that can rotate in one dimension.
# It connects to other components via two rotational splines, `spline_a` and `spline_b` (inherited from `PartialTwoSplines`).
# The absolute rotation angle of the inertia is `phi`, its angular velocity is `w`,
# and its angular acceleration is `a`.
# The core dynamic behavior is defined by Newton's second law for rotation:
# ```math
# J \\cdot a = \text{spline\\_a}.\\tau + \text{spline\\_b}.\\tau
# ```
# The kinematic relationships defining angular velocity and acceleration are:
# ```math
# w = \frac{d\\phi}{dt}
# ```
# ```math
# a = \frac{dw}{dt}
# ```
# Both splines are rigidly attached to the inertia, meaning `phi = spline_a.phi` and `phi = spline_b.phi`.
component Inertia
  extends PartialTwoSplines
  # Moment of inertia of the rotating body
  parameter J::MomentOfInertia
  # Absolute rotation angle of the inertia
  variable phi::Angle
  # Absolute angular velocity of the inertia
  variable w::AngularVelocity
  # Absolute angular acceleration of the inertia
  variable a::AngularAcceleration
relations
  phi = spline_a.phi
  phi = spline_b.phi
  der(phi) = w
  der(w) = a
  J * a = spline_a.tau + spline_b.tau
metadata {"Dyad": {"icons": {"default": "dyad://RotationalComponents/Inertia.svg"}}}
end

Flattened Source

# A 1D-rotational component with inertia, subject to torques from two splines.
#
# This component represents a rigid body with a moment of inertia `J` that can rotate in one dimension.
# It connects to other components via two rotational splines, `spline_a` and `spline_b` (inherited from `PartialTwoSplines`).
# The absolute rotation angle of the inertia is `phi`, its angular velocity is `w`,
# and its angular acceleration is `a`.
# The core dynamic behavior is defined by Newton's second law for rotation:
# ```math
# J \\cdot a = \text{spline\\_a}.\\tau + \text{spline\\_b}.\\tau
# ```
# The kinematic relationships defining angular velocity and acceleration are:
# ```math
# w = \frac{d\\phi}{dt}
# ```
# ```math
# a = \frac{dw}{dt}
# ```
# Both splines are rigidly attached to the inertia, meaning `phi = spline_a.phi` and `phi = spline_b.phi`.
component Inertia
  # First spline
  spline_a = Spline() [{"Dyad": {"placement": {"icon": {"x1": -50, "y1": 450, "x2": 50, "y2": 550}}}}]
  # Second spline
  spline_b = Spline() [{"Dyad": {"placement": {"icon": {"x1": 950, "y1": 450, "x2": 1050, "y2": 550}}}}]
  # Moment of inertia of the rotating body
  parameter J::MomentOfInertia
  # Absolute rotation angle of the inertia
  variable phi::Angle
  # Absolute angular velocity of the inertia
  variable w::AngularVelocity
  # Absolute angular acceleration of the inertia
  variable a::AngularAcceleration
relations
  phi = spline_a.phi
  phi = spline_b.phi
  der(phi) = w
  der(w) = a
  J * a = spline_a.tau + spline_b.tau
metadata {"Dyad": {"icons": {"default": "dyad://RotationalComponents/Inertia.svg"}}}
end

Test Cases

No test cases defined.

Related

• Examples

• Experiments

• Analyses

• Tests

  • TwoInertiasWithDrivingTorque