#!/usr/bin/env python
# -*- coding: utf-8 -*-
"""This module contains some sample symbolic models used for testing and
examples."""
# external libraries
import sympy as sm
import sympy.physics.mechanics as me
# local
from .system import System
from .utils import sympy_newer_than
[docs]def multi_mass_spring_damper(n=1, apply_gravity=False,
apply_external_forces=False):
"""Returns a system containing the symbolic equations of motion and
associated variables for a simple mutli-degree of freedom point mass,
spring, damper system with optional gravitational and external
specified forces. For example, a two mass system under the influence of
gravity and external forces looks like:
::
----------------
| | | | g
\ | | | V
k0 / --- c0 |
| | | x0, v0
--------- V
| m0 | -----
--------- |
| | | |
\ v | | |
k1 / f0 --- c1 |
| | | x1, v1
--------- V
| m1 | -----
---------
| f1
V
Parameters
----------
n : integer
The number of masses in the serial chain.
apply_gravity : boolean
If true, gravity will be applied to each mass.
apply_external_forces : boolean
If true, a time varying external force will be applied to each mass.
Returns
-------
system : pydy.system.System
A system constructed from the KanesMethod object.
"""
mass = sm.symbols('m:{}'.format(n))
stiffness = sm.symbols('k:{}'.format(n))
damping = sm.symbols('c:{}'.format(n))
acceleration_due_to_gravity = sm.symbols('g')
coordinates = me.dynamicsymbols('x:{}'.format(n))
speeds = me.dynamicsymbols('v:{}'.format(n))
specifieds = me.dynamicsymbols('f:{}'.format(n))
ceiling = me.ReferenceFrame('N')
origin = me.Point('origin')
origin.set_vel(ceiling, 0)
points = [origin]
kinematic_equations = []
particles = []
forces = []
for i in range(n):
center = points[-1].locatenew('center{}'.format(i),
coordinates[i] * ceiling.x)
center.set_vel(ceiling, points[-1].vel(ceiling) +
speeds[i] * ceiling.x)
points.append(center)
block = me.Particle('block{}'.format(i), center, mass[i])
kinematic_equations.append(speeds[i] - coordinates[i].diff())
total_force = (-stiffness[i] * coordinates[i] -
damping[i] * speeds[i])
try:
total_force += (stiffness[i + 1] * coordinates[i + 1] +
damping[i + 1] * speeds[i + 1])
except IndexError: # no force from below on last mass
pass
if apply_gravity:
total_force += mass[i] * acceleration_due_to_gravity
if apply_external_forces:
total_force += specifieds[i]
forces.append((center, total_force * ceiling.x))
particles.append(block)
kane = me.KanesMethod(ceiling, q_ind=coordinates, u_ind=speeds,
kd_eqs=kinematic_equations)
if sympy_newer_than('1.0'):
kane.kanes_equations(particles, forces)
else:
kane.kanes_equations(forces, particles)
return System(kane)
[docs]def n_link_pendulum_on_cart(n=1, cart_force=True, joint_torques=False):
"""Returns the system containing the symbolic first order equations of
motion for a 2D n-link pendulum on a sliding cart under the influence of
gravity.
::
|
v y o
g ^ /
| /
----|-/-- theta0
| |/ |
F-> | o --|---> x
| |
---------
o o
Parameters
----------
n : integer
The number of links in the pendulum.
cart_force : boolean, default=True
If true an external specified lateral force is applied to the cart.
joint_torques : boolean, default=False
If true joint torques will be added as specified inputs at each
joint.
Returns
-------
system : pydy.system.System
The system containing the symbolics.
Notes
-----
The degrees of freedom of the system are n + 1, i.e. one for each
pendulum link and one for the lateral motion of the cart.
M x' = F, where x = [u0, ..., un+1, q0, ..., qn+1]
The joint angles are all defined relative to the ground where the x axis
defines the ground line and the y axis points up. The joint torques are
applied between each adjacent link and the between the cart and the
lower link where a positive torque corresponds to positive angle.
"""
if n <= 0:
raise ValueError('The number of links must be a positive integer.')
q = me.dynamicsymbols('q:{}'.format(n + 1))
u = me.dynamicsymbols('u:{}'.format(n + 1))
if joint_torques is True:
T = me.dynamicsymbols('T1:{}'.format(n + 1))
m = sm.symbols('m:{}'.format(n + 1))
l = sm.symbols('l:{}'.format(n))
g, t = sm.symbols('g t')
I = me.ReferenceFrame('I')
O = me.Point('O')
O.set_vel(I, 0)
P0 = me.Point('P0')
P0.set_pos(O, q[0] * I.x)
P0.set_vel(I, u[0] * I.x)
Pa0 = me.Particle('Pa0', P0, m[0])
frames = [I]
points = [P0]
particles = [Pa0]
forces = [(P0, -m[0] * g * I.y)]
kindiffs = [q[0].diff(t) - u[0]]
if cart_force is True or joint_torques is True:
specified = []
else:
specified = None
for i in range(n):
Bi = I.orientnew('B{}'.format(i), 'Axis', [q[i + 1], I.z])
Bi.set_ang_vel(I, u[i + 1] * I.z)
frames.append(Bi)
Pi = points[-1].locatenew('P{}'.format(i + 1), l[i] * Bi.x)
Pi.v2pt_theory(points[-1], I, Bi)
points.append(Pi)
Pai = me.Particle('Pa' + str(i + 1), Pi, m[i + 1])
particles.append(Pai)
forces.append((Pi, -m[i + 1] * g * I.y))
if joint_torques is True:
specified.append(T[i])
if i == 0:
forces.append((I, -T[i] * I.z))
if i == n - 1:
forces.append((Bi, T[i] * I.z))
else:
forces.append((Bi, T[i] * I.z - T[i + 1] * I.z))
kindiffs.append(q[i + 1].diff(t) - u[i + 1])
if cart_force is True:
F = me.dynamicsymbols('F')
forces.append((P0, F * I.x))
specified.append(F)
kane = me.KanesMethod(I, q_ind=q, u_ind=u, kd_eqs=kindiffs)
if sympy_newer_than('1.0'):
kane.kanes_equations(particles, forces)
else:
kane.kanes_equations(forces, particles)
return System(kane)