LIBRARY

PlanarMechanics.Cable

Planar cable / rope of total length l and mass m, modeled as n rigid BodyShape segments connected by Revolute joints. This is the planar (PlanarMechanics) analog of the 3D Cable, using 1-DOF revolute joints (rotation about the z-axis) in place of 3-DOF Spherical joints.

When end_joints = true (default), the topology uses n + 1 joints: frame_a → R₁ → link₁ → R₂ → … → Rₙ → linkₙ → Rₙ₊₁ → frame_b

When end_joints = false, only the n − 1 internal joints are kept and both ends attach rigidly to the adjacent frame: frame_a → link₁ → R₁ → link₂ → … → Rₙ₋₁ → linkₙ → frame_b

By default each Revolute joint transmits no torque (its rotational flange is left unconnected), so the cable swings freely. Setting hasdamping = true adds a viscous Damper of constant d_joint to every joint, dissipating energy as the cable flexes (the planar analog of the Rope's d_joint in Multibody.jl). Setting hasstiffness = true adds a rotational Spring of constant c_joint to every joint, giving the cable flexural stiffness (a restoring torque toward the straight configuration). The two are independent and may be combined.

The cable starts straight (along dir); seed an initial bend by setting a joint's angle as an initial condition, e.g. a 0.9 rad bend on the first joint: cable.joints[1].phi = 0.9.

This component extends from PartialTwoFrames This component extends from MultibodyComponents.Renderable

Usage

MultibodyComponents.PlanarMechanics.Cable(render=true, color=[0.5, 0.4, 0.4, 1], specular_coefficient=1.5, d_joint=0, c_joint=0, l=1, m=1, dir=[0, -1], radius=0.05, z_position=0, r_segment=(l / n) * (dir / norm(dir)), m_segment=m / n, I_segment=m_segment * (l / n) ^ 2 / 12)

Parameters:

Name	Description	Units	Default value
n	Number of rigid segments	–	5
end_joints	Include Revolute joints at both ends (frame_a and frame_b)	–	true
hasdamping	Add viscous damping in the joints between links (dissipates energy while the cable flexes)	–	false
hasstiffness	Add rotational stiffness in the joints between links (the cable resists bending toward straight)	–	false
nj	Number of joints	–	ifelse(end_...+ 1, n - 1)
render		–	true
color		–	[0.5, 0.4, 0.4, 1]
specular_coefficient		–	1.5
d_joint	Viscous damping constant of each joint (used only if hasdamping = true)	N.m.s/rad	0
c_joint	Rotational spring constant of each joint (used only if hasstiffness = true)	N.m/rad	0
l	Total unstretched length of the rope	m	1
m	Total mass of the rope (each segment gets m/n)	kg	1
dir	Hanging direction in the plane; unit-normalized internally	–	[0, -1]
radius	Rendering radius of each segment	–	0.05
z_position	z-position of the cable segments in animations	–	0

Connectors

• frame_a - Coordinate system (2-dim.) fixed to the component with one cut-force and cut-torque.

All variables are resolved in the planar world frame. (Frame2D)

• frame_b - Coordinate system (2-dim.) fixed to the component with one cut-force and cut-torque.

All variables are resolved in the planar world frame. (Frame2D)

Behavior

Source

"""
Planar cable / rope of total length `l` and mass `m`, modeled as `n` rigid
`BodyShape` segments connected by `Revolute` joints. This is the planar
(PlanarMechanics) analog of the 3D `Cable`, using 1-DOF revolute joints
(rotation about the z-axis) in place of 3-DOF `Spherical` joints.

When `end_joints = true` (default), the topology uses `n + 1` joints:
`frame_a → R₁ → link₁ → R₂ → … → Rₙ → linkₙ → Rₙ₊₁ → frame_b`

When `end_joints = false`, only the `n − 1` internal joints are kept and
both ends attach rigidly to the adjacent frame:
`frame_a → link₁ → R₁ → link₂ → … → Rₙ₋₁ → linkₙ → frame_b`

By default each `Revolute` joint transmits no torque (its rotational flange is
left unconnected), so the cable swings freely. Setting `hasdamping = true` adds
a viscous `Damper` of constant `d_joint` to every joint, dissipating energy as
the cable flexes (the planar analog of the `Rope`'s `d_joint` in Multibody.jl).
Setting `hasstiffness = true` adds a rotational `Spring` of constant `c_joint`
to every joint, giving the cable flexural stiffness (a restoring torque toward
the straight configuration). The two are independent and may be combined.

The cable starts straight (along `dir`); seed an initial bend by setting a
joint's angle as an initial condition, e.g. a 0.9 rad bend on the first joint:
`cable.joints[1].phi = 0.9`.
"""
component Cable
  extends PartialTwoFrames
  extends MultibodyComponents.Renderable(color = [0.5, 0.4, 0.4, 1])
  joints = [Revolute(render = false, phi(initial = 0), w(initial = 0)) for i in 1:nj]
  links = [BodyShape(r = r_segment, m = m_segment, I = I_segment, color = color, radius = radius, z_position = z_position) for i in 1:n]
  dampers = [RotationalComponents.Components.Damper(d = d_joint) for i in 1:nj] if hasdamping
  damper_ground = RotationalComponents.Components.Fixed() if hasdamping
  springs = [RotationalComponents.Components.Spring(c = c_joint) for i in 1:nj] if hasstiffness
  spring_ground = RotationalComponents.Components.Fixed() if hasstiffness
  "Number of rigid segments"
  structural parameter n::Integer = 5
  "Include Revolute joints at both ends (frame_a and frame_b)"
  structural parameter end_joints::Boolean = true
  "Add viscous damping in the joints between links (dissipates energy while the cable flexes)"
  structural parameter hasdamping::Boolean = false
  "Viscous damping constant of each joint (used only if hasdamping = true)"
  parameter d_joint::RotationalDampingConstant = 0
  "Add rotational stiffness in the joints between links (the cable resists bending toward straight)"
  structural parameter hasstiffness::Boolean = false
  "Rotational spring constant of each joint (used only if hasstiffness = true)"
  parameter c_joint::RotationalSpringConstant = 0
  "Number of joints"
  final structural parameter nj::Integer = ifelse(end_joints, n + 1, n - 1)
  "Total unstretched length of the rope"
  parameter l::Length = 1
  "Total mass of the rope (each segment gets m/n)"
  parameter m::Mass = 1
  "Hanging direction in the plane; unit-normalized internally"
  parameter dir::Real[2] = [0, -1]
  "Rendering radius of each segment"
  parameter radius::Real = 0.05
  "z-position of the cable segments in animations"
  parameter z_position::Real = 0
  "Vector from frame_a to frame_b of one segment, resolved in segment frame_a"
  final parameter r_segment::Length[2] = (l / n) * (dir / norm(dir))
  "Mass of one segment"
  final parameter m_segment::Mass = m / n
  "Moment of inertia of one segment about its center of mass (thin rod)"
  final parameter I_segment::MomentOfInertia = m_segment * (l / n) ^ 2 / 12
relations
  if end_joints
    connect(frame_a, joints[1].frame_a)
    for i in 1:n
      connect(joints[i].frame_b, links[i].frame_a)
      connect(links[i].frame_b, joints[i + 1].frame_a)
    end
    connect(joints[nj].frame_b, frame_b)
  else
    connect(frame_a, links[1].frame_a)
    for i in 1:nj
      connect(links[i].frame_b, joints[i].frame_a)
      connect(joints[i].frame_b, links[i + 1].frame_a)
    end
    connect(links[n].frame_b, frame_b)
  end
  # viscous joint damping: each joint flange is damped against a fixed reference,
  # so the damper dissipates energy proportional to the joint's angular velocity
  if hasdamping
    for i in 1:nj
      connect(joints[i].flange_a, dampers[i].spline_a)
      connect(dampers[i].spline_b, damper_ground.spline)
    end
  end
  # rotational joint stiffness: each joint flange is sprung against a fixed
  # reference, so the spring applies a restoring torque toward the straight cable
  if hasstiffness
    for i in 1:nj
      connect(joints[i].flange_a, springs[i].spline_a)
      connect(springs[i].spline_b, spring_ground.spline)
    end
  end
metadata {
  "Dyad": {
    "icons": {"default": "dyad://MultibodyComponents/Cable.svg"},
    "labels": [
      {
        "label": "$(instance)",
        "x": 500,
        "y": 150,
        "rot": 0,
        "attrs": {"font-size": "160"}
      }
    ]
  }
}
end

Flattened Source

"""
Planar cable / rope of total length `l` and mass `m`, modeled as `n` rigid
`BodyShape` segments connected by `Revolute` joints. This is the planar
(PlanarMechanics) analog of the 3D `Cable`, using 1-DOF revolute joints
(rotation about the z-axis) in place of 3-DOF `Spherical` joints.

When `end_joints = true` (default), the topology uses `n + 1` joints:
`frame_a → R₁ → link₁ → R₂ → … → Rₙ → linkₙ → Rₙ₊₁ → frame_b`

When `end_joints = false`, only the `n − 1` internal joints are kept and
both ends attach rigidly to the adjacent frame:
`frame_a → link₁ → R₁ → link₂ → … → Rₙ₋₁ → linkₙ → frame_b`

By default each `Revolute` joint transmits no torque (its rotational flange is
left unconnected), so the cable swings freely. Setting `hasdamping = true` adds
a viscous `Damper` of constant `d_joint` to every joint, dissipating energy as
the cable flexes (the planar analog of the `Rope`'s `d_joint` in Multibody.jl).
Setting `hasstiffness = true` adds a rotational `Spring` of constant `c_joint`
to every joint, giving the cable flexural stiffness (a restoring torque toward
the straight configuration). The two are independent and may be combined.

The cable starts straight (along `dir`); seed an initial bend by setting a
joint's angle as an initial condition, e.g. a 0.9 rad bend on the first joint:
`cable.joints[1].phi = 0.9`.
"""
component Cable
  frame_a = Frame2D() {
    "Dyad": {
      "placement": {
        "diagram": {"iconName": "default", "x1": 50, "y1": 450, "x2": 150, "y2": 550, "rot": 0}
      },
      "tags": []
    }
  }
  frame_b = Frame2D() {
    "Dyad": {
      "placement": {
        "diagram": {"iconName": "default", "x1": 850, "y1": 450, "x2": 950, "y2": 550, "rot": 0}
      },
      "tags": []
    }
  }
  parameter render::Boolean = true
  parameter color::Real[4] = [0.5, 0.5, 0.5, 1.0]
  parameter specular_coefficient::Real = 1.5
  joints = [Revolute(render = false, phi(initial = 0), w(initial = 0)) for i in 1:nj]
  links = [BodyShape(r = r_segment, m = m_segment, I = I_segment, color = color, radius = radius, z_position = z_position) for i in 1:n]
  dampers = [RotationalComponents.Components.Damper(d = d_joint) for i in 1:nj] if hasdamping
  damper_ground = RotationalComponents.Components.Fixed() if hasdamping
  springs = [RotationalComponents.Components.Spring(c = c_joint) for i in 1:nj] if hasstiffness
  spring_ground = RotationalComponents.Components.Fixed() if hasstiffness
  "Number of rigid segments"
  structural parameter n::Integer = 5
  "Include Revolute joints at both ends (frame_a and frame_b)"
  structural parameter end_joints::Boolean = true
  "Add viscous damping in the joints between links (dissipates energy while the cable flexes)"
  structural parameter hasdamping::Boolean = false
  "Viscous damping constant of each joint (used only if hasdamping = true)"
  parameter d_joint::RotationalDampingConstant = 0
  "Add rotational stiffness in the joints between links (the cable resists bending toward straight)"
  structural parameter hasstiffness::Boolean = false
  "Rotational spring constant of each joint (used only if hasstiffness = true)"
  parameter c_joint::RotationalSpringConstant = 0
  "Number of joints"
  final structural parameter nj::Integer = ifelse(end_joints, n + 1, n - 1)
  "Total unstretched length of the rope"
  parameter l::Length = 1
  "Total mass of the rope (each segment gets m/n)"
  parameter m::Mass = 1
  "Hanging direction in the plane; unit-normalized internally"
  parameter dir::Real[2] = [0, -1]
  "Rendering radius of each segment"
  parameter radius::Real = 0.05
  "z-position of the cable segments in animations"
  parameter z_position::Real = 0
  "Vector from frame_a to frame_b of one segment, resolved in segment frame_a"
  final parameter r_segment::Length[2] = (l / n) * (dir / norm(dir))
  "Mass of one segment"
  final parameter m_segment::Mass = m / n
  "Moment of inertia of one segment about its center of mass (thin rod)"
  final parameter I_segment::MomentOfInertia = m_segment * (l / n) ^ 2 / 12
relations
  if end_joints
    connect(frame_a, joints[1].frame_a)
    for i in 1:n
      connect(joints[i].frame_b, links[i].frame_a)
      connect(links[i].frame_b, joints[i + 1].frame_a)
    end
    connect(joints[nj].frame_b, frame_b)
  else
    connect(frame_a, links[1].frame_a)
    for i in 1:nj
      connect(links[i].frame_b, joints[i].frame_a)
      connect(joints[i].frame_b, links[i + 1].frame_a)
    end
    connect(links[n].frame_b, frame_b)
  end
  # viscous joint damping: each joint flange is damped against a fixed reference,
  # so the damper dissipates energy proportional to the joint's angular velocity
  if hasdamping
    for i in 1:nj
      connect(joints[i].flange_a, dampers[i].spline_a)
      connect(dampers[i].spline_b, damper_ground.spline)
    end
  end
  # rotational joint stiffness: each joint flange is sprung against a fixed
  # reference, so the spring applies a restoring torque toward the straight cable
  if hasstiffness
    for i in 1:nj
      connect(joints[i].flange_a, springs[i].spline_a)
      connect(springs[i].spline_b, spring_ground.spline)
    end
  end
metadata {
  "Dyad": {
    "icons": {"default": "dyad://MultibodyComponents/Cable.svg"},
    "labels": [
      {
        "label": "$(instance)",
        "x": 500,
        "y": 150,
        "rot": 0,
        "attrs": {"font-size": "160"}
      }
    ]
  }
}
end

Test Cases

No test cases defined.

Related

• Examples

• Experiments

• Analyses

• Tests

  • TestCableEndJoints

  • TestCableNoEndJoints

  • TestPlanarCableEndJoints

  • TestPlanarCableNoEndJoints