Kane’s Method in Physics/Mechanics#
sympy.physics.mechanics
provides functionality for deriving equations of motion
using Kane’s method [Kane1985]. This document will describe Kane’s method
as used in this module, but not how the equations are actually derived.
Structure of Equations#
In sympy.physics.mechanics
we are assuming there are 5 basic sets of equations needed
to describe a system. They are: holonomic constraints, non-holonomic
constraints, kinematic differential equations, dynamic equations, and
differentiated non-holonomic equations.
In sympy.physics.mechanics
holonomic constraints are only used for the linearization
process; it is assumed that they will be too complicated to solve for the
dependent coordinate(s). If you are able to easily solve a holonomic
constraint, you should consider redefining your problem in terms of a smaller
set of coordinates. Alternatively, the time-differentiated holonomic
constraints can be supplied.
Kane’s method forms two expressions, and
, whose sum
is zero. In this module, these expressions are rearranged into the following
form:
For a non-holonomic system with total speeds and
motion constraints, we
will get o - m equations. The mass-matrix/forcing equations are then augmented
in the following fashion:
Kane’s Method in Physics/Mechanics#
The formulation of the equations of motion in sympy.physics.mechanics
starts with
creation of a KanesMethod
object. Upon initialization of the
KanesMethod
object, an inertial reference frame needs to be supplied. along
with some basic system information, such as coordinates and speeds
>>> from sympy.physics.mechanics import *
>>> N = ReferenceFrame('N')
>>> q1, q2, u1, u2 = dynamicsymbols('q1 q2 u1 u2')
>>> q1d, q2d, u1d, u2d = dynamicsymbols('q1 q2 u1 u2', 1)
>>> KM = KanesMethod(N, [q1, q2], [u1, u2])
It is also important to supply the order of coordinates and speeds properly if there are dependent coordinates and speeds. They must be supplied after independent coordinates and speeds or as a keyword argument; this is shown later.
>>> q1, q2, q3, q4 = dynamicsymbols('q1 q2 q3 q4')
>>> u1, u2, u3, u4 = dynamicsymbols('u1 u2 u3 u4')
>>> # Here we will assume q2 is dependent, and u2 and u3 are dependent
>>> # We need the constraint equations to enter them though
>>> KM = KanesMethod(N, [q1, q3, q4], [u1, u4])
Additionally, if there are auxiliary speeds, they need to be identified here. See the examples for more information on this. In this example u4 is the auxiliary speed.
>>> KM = KanesMethod(N, [q1, q3, q4], [u1, u2, u3], u_auxiliary=[u4])
Kinematic differential equations must also be supplied; there are to be provided as a list of expressions which are each equal to zero. A trivial example follows:
>>> kd = [q1d - u1, q2d - u2]
Turning on mechanics_printing()
makes the expressions significantly
shorter and is recommended. Alternatively, the mprint
and mpprint
commands can be used.
If there are non-holonomic constraints, dependent speeds need to be specified (and so do dependent coordinates, but they only come into play when linearizing the system). The constraints need to be supplied in a list of expressions which are equal to zero, trivial motion and configuration constraints are shown below:
>>> N = ReferenceFrame('N')
>>> q1, q2, q3, q4 = dynamicsymbols('q1 q2 q3 q4')
>>> q1d, q2d, q3d, q4d = dynamicsymbols('q1 q2 q3 q4', 1)
>>> u1, u2, u3, u4 = dynamicsymbols('u1 u2 u3 u4')
>>> #Here we will assume q2 is dependent, and u2 and u3 are dependent
>>> speed_cons = [u2 - u1, u3 - u1 - u4]
>>> coord_cons = [q2 - q1]
>>> q_ind = [q1, q3, q4]
>>> q_dep = [q2]
>>> u_ind = [u1, u4]
>>> u_dep = [u2, u3]
>>> kd = [q1d - u1, q2d - u2, q3d - u3, q4d - u4]
>>> KM = KanesMethod(N, q_ind, u_ind, kd,
... q_dependent=q_dep,
... configuration_constraints=coord_cons,
... u_dependent=u_dep,
... velocity_constraints=speed_cons)
A dictionary returning the solved ’s can also be solved for:
>>> mechanics_printing(pretty_print=False)
>>> KM.kindiffdict()
{q1': u1, q2': u2, q3': u3, q4': u4}
The final step in forming the equations of motion is supplying a list of
bodies and particles, and a list of 2-tuples of the form (Point, Vector)
or (ReferenceFrame, Vector)
to represent applied forces and torques.
>>> N = ReferenceFrame('N')
>>> q, u = dynamicsymbols('q u')
>>> qd, ud = dynamicsymbols('q u', 1)
>>> P = Point('P')
>>> P.set_vel(N, u * N.x)
>>> Pa = Particle('Pa', P, 5)
>>> BL = [Pa]
>>> FL = [(P, 7 * N.x)]
>>> KM = KanesMethod(N, [q], [u], [qd - u])
>>> (fr, frstar) = KM.kanes_equations(BL, FL)
>>> KM.mass_matrix
Matrix([[5]])
>>> KM.forcing
Matrix([[7]])
When there are motion constraints, the mass matrix is augmented by the
matrix, and the forcing vector by the
vector.
There are also the “full” mass matrix and “full” forcing vector terms, these include the kinematic differential equations; the mass matrix is of size (n + o) x (n + o), or square and the size of all coordinates and speeds.
>>> KM.mass_matrix_full
Matrix([
[1, 0],
[0, 5]])
>>> KM.forcing_full
Matrix([
[u],
[7]])
Exploration of the provided examples is encouraged in order to gain more
understanding of the KanesMethod
object.