Current File : //lib/python3/dist-packages/sympy/physics/mechanics/tests/test_lagrange.py
from sympy.physics.mechanics import (dynamicsymbols, ReferenceFrame, Point,
RigidBody, LagrangesMethod, Particle,
inertia, Lagrangian)
from sympy import symbols, pi, sin, cos, tan, simplify, Function, \
Derivative, Matrix
def test_disc_on_an_incline_plane():
# Disc rolling on an inclined plane
# First the generalized coordinates are created. The mass center of the
# disc is located from top vertex of the inclined plane by the generalized
# coordinate 'y'. The orientation of the disc is defined by the angle
# 'theta'. The mass of the disc is 'm' and its radius is 'R'. The length of
# the inclined path is 'l', the angle of inclination is 'alpha'. 'g' is the
# gravitational constant.
y, theta = dynamicsymbols('y theta')
yd, thetad = dynamicsymbols('y theta', 1)
m, g, R, l, alpha = symbols('m g R l alpha')
# Next, we create the inertial reference frame 'N'. A reference frame 'A'
# is attached to the inclined plane. Finally a frame is created which is attached to the disk.
N = ReferenceFrame('N')
A = N.orientnew('A', 'Axis', [pi/2 - alpha, N.z])
B = A.orientnew('B', 'Axis', [-theta, A.z])
# Creating the disc 'D'; we create the point that represents the mass
# center of the disc and set its velocity. The inertia dyadic of the disc
# is created. Finally, we create the disc.
Do = Point('Do')
Do.set_vel(N, yd * A.x)
I = m * R**2/2 * B.z | B.z
D = RigidBody('D', Do, B, m, (I, Do))
# To construct the Lagrangian, 'L', of the disc, we determine its kinetic
# and potential energies, T and U, respectively. L is defined as the
# difference between T and U.
D.potential_energy = m * g * (l - y) * sin(alpha)
L = Lagrangian(N, D)
# We then create the list of generalized coordinates and constraint
# equations. The constraint arises due to the disc rolling without slip on
# on the inclined path. We then invoke the 'LagrangesMethod' class and
# supply it the necessary arguments and generate the equations of motion.
# The'rhs' method solves for the q_double_dots (i.e. the second derivative
# with respect to time of the generalized coordinates and the lagrange
# multipliers.
q = [y, theta]
hol_coneqs = [y - R * theta]
m = LagrangesMethod(L, q, hol_coneqs=hol_coneqs)
m.form_lagranges_equations()
rhs = m.rhs()
rhs.simplify()
assert rhs[2] == 2*g*sin(alpha)/3
def test_simp_pen():
# This tests that the equations generated by LagrangesMethod are identical
# to those obtained by hand calculations. The system under consideration is
# the simple pendulum.
# We begin by creating the generalized coordinates as per the requirements
# of LagrangesMethod. Also we created the associate symbols
# that characterize the system: 'm' is the mass of the bob, l is the length
# of the massless rigid rod connecting the bob to a point O fixed in the
# inertial frame.
q, u = dynamicsymbols('q u')
qd, ud = dynamicsymbols('q u ', 1)
l, m, g = symbols('l m g')
# We then create the inertial frame and a frame attached to the massless
# string following which we define the inertial angular velocity of the
# string.
N = ReferenceFrame('N')
A = N.orientnew('A', 'Axis', [q, N.z])
A.set_ang_vel(N, qd * N.z)
# Next, we create the point O and fix it in the inertial frame. We then
# locate the point P to which the bob is attached. Its corresponding
# velocity is then determined by the 'two point formula'.
O = Point('O')
O.set_vel(N, 0)
P = O.locatenew('P', l * A.x)
P.v2pt_theory(O, N, A)
# The 'Particle' which represents the bob is then created and its
# Lagrangian generated.
Pa = Particle('Pa', P, m)
Pa.potential_energy = - m * g * l * cos(q)
L = Lagrangian(N, Pa)
# The 'LagrangesMethod' class is invoked to obtain equations of motion.
lm = LagrangesMethod(L, [q])
lm.form_lagranges_equations()
RHS = lm.rhs()
assert RHS[1] == -g*sin(q)/l
def test_nonminimal_pendulum():
q1, q2 = dynamicsymbols('q1:3')
q1d, q2d = dynamicsymbols('q1:3', level=1)
L, m, t = symbols('L, m, t')
g = 9.8
# Compose World Frame
N = ReferenceFrame('N')
pN = Point('N*')
pN.set_vel(N, 0)
# Create point P, the pendulum mass
P = pN.locatenew('P1', q1*N.x + q2*N.y)
P.set_vel(N, P.pos_from(pN).dt(N))
pP = Particle('pP', P, m)
# Constraint Equations
f_c = Matrix([q1**2 + q2**2 - L**2])
# Calculate the lagrangian, and form the equations of motion
Lag = Lagrangian(N, pP)
LM = LagrangesMethod(Lag, [q1, q2], hol_coneqs=f_c,
forcelist=[(P, m*g*N.x)], frame=N)
LM.form_lagranges_equations()
# Check solution
lam1 = LM.lam_vec[0, 0]
eom_sol = Matrix([[m*Derivative(q1, t, t) - 9.8*m + 2*lam1*q1],
[m*Derivative(q2, t, t) + 2*lam1*q2]])
assert LM.eom == eom_sol
# Check multiplier solution
lam_sol = Matrix([(19.6*q1 + 2*q1d**2 + 2*q2d**2)/(4*q1**2/m + 4*q2**2/m)])
assert simplify(LM.solve_multipliers(sol_type='Matrix')) == simplify(lam_sol)
def test_dub_pen():
# The system considered is the double pendulum. Like in the
# test of the simple pendulum above, we begin by creating the generalized
# coordinates and the simple generalized speeds and accelerations which
# will be used later. Following this we create frames and points necessary
# for the kinematics. The procedure isn't explicitly explained as this is
# similar to the simple pendulum. Also this is documented on the pydy.org
# website.
q1, q2 = dynamicsymbols('q1 q2')
q1d, q2d = dynamicsymbols('q1 q2', 1)
q1dd, q2dd = dynamicsymbols('q1 q2', 2)
u1, u2 = dynamicsymbols('u1 u2')
u1d, u2d = dynamicsymbols('u1 u2', 1)
l, m, g = symbols('l m g')
N = ReferenceFrame('N')
A = N.orientnew('A', 'Axis', [q1, N.z])
B = N.orientnew('B', 'Axis', [q2, N.z])
A.set_ang_vel(N, q1d * A.z)
B.set_ang_vel(N, q2d * A.z)
O = Point('O')
P = O.locatenew('P', l * A.x)
R = P.locatenew('R', l * B.x)
O.set_vel(N, 0)
P.v2pt_theory(O, N, A)
R.v2pt_theory(P, N, B)
ParP = Particle('ParP', P, m)
ParR = Particle('ParR', R, m)
ParP.potential_energy = - m * g * l * cos(q1)
ParR.potential_energy = - m * g * l * cos(q1) - m * g * l * cos(q2)
L = Lagrangian(N, ParP, ParR)
lm = LagrangesMethod(L, [q1, q2], bodies=[ParP, ParR])
lm.form_lagranges_equations()
assert simplify(l*m*(2*g*sin(q1) + l*sin(q1)*sin(q2)*q2dd
+ l*sin(q1)*cos(q2)*q2d**2 - l*sin(q2)*cos(q1)*q2d**2
+ l*cos(q1)*cos(q2)*q2dd + 2*l*q1dd) - lm.eom[0]) == 0
assert simplify(l*m*(g*sin(q2) + l*sin(q1)*sin(q2)*q1dd
- l*sin(q1)*cos(q2)*q1d**2 + l*sin(q2)*cos(q1)*q1d**2
+ l*cos(q1)*cos(q2)*q1dd + l*q2dd) - lm.eom[1]) == 0
assert lm.bodies == [ParP, ParR]
def test_rolling_disc():
# Rolling Disc Example
# Here the rolling disc is formed from the contact point up, removing the
# need to introduce generalized speeds. Only 3 configuration and 3
# speed variables are need to describe this system, along with the
# disc's mass and radius, and the local gravity.
q1, q2, q3 = dynamicsymbols('q1 q2 q3')
q1d, q2d, q3d = dynamicsymbols('q1 q2 q3', 1)
r, m, g = symbols('r m g')
# The kinematics are formed by a series of simple rotations. Each simple
# rotation creates a new frame, and the next rotation is defined by the new
# frame's basis vectors. This example uses a 3-1-2 series of rotations, or
# Z, X, Y series of rotations. Angular velocity for this is defined using
# the second frame's basis (the lean frame).
N = ReferenceFrame('N')
Y = N.orientnew('Y', 'Axis', [q1, N.z])
L = Y.orientnew('L', 'Axis', [q2, Y.x])
R = L.orientnew('R', 'Axis', [q3, L.y])
# This is the translational kinematics. We create a point with no velocity
# in N; this is the contact point between the disc and ground. Next we form
# the position vector from the contact point to the disc's center of mass.
# Finally we form the velocity and acceleration of the disc.
C = Point('C')
C.set_vel(N, 0)
Dmc = C.locatenew('Dmc', r * L.z)
Dmc.v2pt_theory(C, N, R)
# Forming the inertia dyadic.
I = inertia(L, m/4 * r**2, m/2 * r**2, m/4 * r**2)
BodyD = RigidBody('BodyD', Dmc, R, m, (I, Dmc))
# Finally we form the equations of motion, using the same steps we did
# before. Supply the Lagrangian, the generalized speeds.
BodyD.potential_energy = - m * g * r * cos(q2)
Lag = Lagrangian(N, BodyD)
q = [q1, q2, q3]
q1 = Function('q1')
q2 = Function('q2')
q3 = Function('q3')
l = LagrangesMethod(Lag, q)
l.form_lagranges_equations()
RHS = l.rhs()
RHS.simplify()
t = symbols('t')
assert (l.mass_matrix[3:6] == [0, 5*m*r**2/4, 0])
assert RHS[4].simplify() == (
(-8*g*sin(q2(t)) + r*(5*sin(2*q2(t))*Derivative(q1(t), t) +
12*cos(q2(t))*Derivative(q3(t), t))*Derivative(q1(t), t))/(10*r))
assert RHS[5] == (-5*cos(q2(t))*Derivative(q1(t), t) + 6*tan(q2(t)
)*Derivative(q3(t), t) + 4*Derivative(q1(t), t)/cos(q2(t))
)*Derivative(q2(t), t)
Mini Shell