def exercise3(clargs):
""" Exercise 3 """
parameters = PendulumParameters() # Checkout pendulum.py for more info
pylog.info(parameters)
# Simulation parameters
time = np.arange(0, 30, 0.01) # Simulation time
x0 = [1, 0.0] # Initial state
# To use/modify pendulum parameters (See PendulumParameters documentation):
# parameters.g = 9.81 # Gravity constant
# parameters.m = 1. # Mass
# parameters.L = 1. # Length
# parameters.I = 1. # Inertia (Automatically computed!)
# parameters.d = 0.3 # damping
# parameters.sin = np.sin # Sine function
# parameters.dry = False # Use dry friction (True or False)
# ---------------------------------------------------------------------------
# "Evolution of pendulum in normal conditions must be implemented"
# ---------------------------------------------------------------------------
res = integrate(pendulum_system, x0, time, args=(parameters,))
res.plot_state("State - normal")
res.plot_phase("Phase - normal")
# ---------------------------------------------------------------------------
# "Evolution of pendulum without damping must be implemented"
# ---------------------------------------------------------------------------
res = integrate(pendulum_system, x0, time, args=(PendulumParameters(d=0),))
res.plot_state("State - d=0")
res.plot_phase("Phase - d=0")
# ---------------------------------------------------------------------------
# "Evolution of pendulum with perturbations must be implemented"
# ---------------------------------------------------------------------------
res = integrate(pendulum_system, x0, time, args=(PendulumParameters(), -2.0))
res.plot_state("State - torque")
res.plot_phase("Phase - torque")
# ---------------------------------------------------------------------------
# "Evolution of pendulum with dry friction must be implemented"
# ---------------------------------------------------------------------------
res = integrate(pendulum_system, x0, time, args=(PendulumParameters(dry=True), 0.0))
res.plot_state("State - dry")
res.plot_phase("Phase - dry")
# Show plots of all results
if not clargs.save_figures:
plt.show()
def evolution_dry(x0, time):
""" Dry friction simulation """
pylog.info("Evolution with dry friction")
parameters = PendulumParameters(d=0.03, dry=True)
pylog.info(parameters)
title = "{} with dry friction (x0={})"
res = integrate(pendulum_system, x0, time, args=(parameters, ))
res.plot_state(title.format("State", x0))
res.plot_phase(title.format("Phase", x0))
def evolution_perturbation(x0, time):
""" Perturbation and no damping simulation """
pylog.info("Evolution with perturbations")
parameters = PendulumParameters(d=0.0)
pylog.info(parameters)
title = "{} with perturbation (x0={})"
res = integrate(pendulum_perturbation, x0, time, args=(parameters, ))
res.plot_state(title.format("State", x0))
res.plot_phase(title.format("Phase", x0))
def evolution_no_damping(x0, time):
""" No damping simulation """
pylog.info("Evolution with no damping")
parameters = PendulumParameters(d=0.0)
pylog.info(parameters)
title = "{} without damping (x0={})"
res = integrate(pendulum_system, x0, time, args=(parameters, ))
res.plot_state(title.format("State", x0))
res.plot_phase(title.format("Phase", x0))
def evolution_cases(time):
""" Normal simulation """
pylog.info("Evolution with basic paramater")
x0_cases = [["Normal", [0.1, 0]], ["Stable", [0.0, 0.0]],
["Unstable", [np.pi, 0.0]], ["Multiple loops", [0.1, 10.0]]]
title = "{} case {} (x0={})"
parameters = PendulumParameters()
for name, x0 in x0_cases:
res = integrate(pendulum_system, x0, time, args=(parameters, ))
res.plot_state(title.format(name, "state", x0))
res.plot_phase(title.format(name, "phase", x0))
def fixed_points_analysis():
""" Exercise 3a - Compute and analyse fixed points """
g, L, d, time = sp.symbols("g, L, d, t")
params = PendulumParameters(g=g, L=L, d=d, sin=sp.sin)
theta = sp.Function('theta')(time)
dt_sym = theta.diff(time)
d2t_sym = theta.diff(time, 2)
# System
sys_matrix = sp.Matrix(pendulum_system([theta, dt_sym], parameters=params))
sys = sp.Equality(sp.Matrix(np.array([[dt_sym], [d2t_sym]])), sys_matrix)
pylog.info(u"Pendulum system:\n{}".format(sp.pretty(sys)))
# Solve when dtheta = 0 and d2theta=0
dt = 0
d2t = 0
t_sol = sp.solveset(
sp.Equality(d2t, pendulum_equation(theta, dt, parameters=params)),
theta # Theta, the variable to be solved
)
sys_fixed = sp.Equality(sp.Matrix(np.array([[0], [0]])), sys_matrix)
pylog.info(u"Solution for fixed points:\n{}\n{}\n{}\n{}".format(
sp.pretty(sys_fixed), sp.pretty(sp.Equality(theta, t_sol)),
sp.pretty(sp.Equality(dt_sym, dt)),
sp.pretty(sp.Equality(d2t_sym, d2t))))
# Alternative way of solving
sol = sp.solve(sys_fixed, [theta, dt_sym])
pylog.info(u"Solution using an alternative way of solving:\n{}".format(
sp.pretty(sol)))
# Jacobian, eigenvalues and eigenvectors
jac = sys_matrix.jacobian(sp.Matrix([theta, dt_sym]))
eigenvals = jac.eigenvals()
eigenvects = jac.eigenvects()
pylog.info(u"Jacobian:\n{}\nEigenvalues:\n{}\nEigenvectors:\n{}".format(
sp.pretty(jac), sp.pretty(eigenvals), sp.pretty(eigenvects)))
# Stability analysis
for i, s in enumerate(sol):
pylog.info(u"Case {}:\n{}".format(
i + 1,
sp.pretty(sp.Equality(sp.Matrix([theta, dt_sym]), sp.Matrix(s)))))
res = [None for _ in eigenvals.keys()]
for j, e in enumerate(eigenvals.keys()):
res[j] = e.subs({theta: s[0], dt_sym: s[1], g: 9.81, L: 1, d: 0.1})
pylog.info(u"{}\n{}".format(
sp.pretty(sp.Equality(e, res[j])),
"{} {}".format(sp.re(res[j]),
"> 0" if sp.re(res[j]) > 0 else "< 0")))
fixed_point_type = None
if all([sp.re(r) < 0 for r in res]):
fixed_point_type = " stable point"
elif all([sp.re(r) > 0 for r in res]):
fixed_point_type = "n unstable point"
else:
fixed_point_type = " saddle point"
pylog.info("Fixed point is a{}".format(fixed_point_type))
def exercise3(clargs):
""" Exercise 3 """
fixed_points() # Optional
parameters = PendulumParameters() # Checkout pendulum.py for more info
pylog.info(parameters)
# Simulation parameters
time = np.arange(0, 30, 0.01) # Simulation time
x0 = [0.1, 0.0] # Initial state
# Evolutions
evolution_cases(time)
x0 = [0.1, 0]
evolution_no_damping(x0, time)
evolution_perturbation(x0, time)
evolution_dry(x0, time)
# Show plots of all results
if not clargs.save_figures:
plt.show()
def exercise3(clargs):
""" Exercise 3 """
parameters = PendulumParameters() # Checkout pendulum.py for more info
pylog.info(parameters)
# Simulation parameters
time = np.arange(0, 30, 0.01) # Simulation time
x0 = [0.1, 0.0] # Initial state
# To use/modify pendulum parameters (See PendulumParameters documentation):
# parameters.g = 9.81 # Gravity constant
# parameters.m = 1. # Mass
# parameters.L = 1. # Length
# parameters.I = 1. # Inertia (Automatically computed!)
# parameters.d = 0.3 # damping
# parameters.sin = np.sin # Sine function
# parameters.dry = False # Use dry friction (True or False)
# Example of system integration (Similar to lab1)
# (NOTE: pendulum_equation must be imlpemented first)
pylog.debug("Running integration example")
res = integrate(pendulum_system, x0, time, args=(parameters,))
res.plot_state("State")
res.plot_phase("Phase")
# Evolutions
# Write code here (You can add functions for the different cases)
pylog.warning(
"Evolution of pendulum in normal conditions must be implemented"
)
pylog.warning(
"Evolution of pendulum without damping must be implemented"
)
pylog.warning(
"Evolution of pendulum with perturbations must be implemented"
)
pylog.warning(
"Evolution of pendulum with dry friction must be implemented"
)
# Show plots of all results
if not clargs.save_figures:
plt.show()