xplot = np.empty((nsteps, len(state)))
def wilber_derivs(s, t, params):
''' Returns the derivative of the state vector '''
m, I, k, delta, eps = params
derivs = np.empty(len(s))
derivs[0] = s[2]
derivs[1] = s[3]
derivs[2] = -k/m*s[0] - eps/(2*m)*s[1]
derivs[3] = -delta/I*s[1] - eps/(2*I)*s[0]
return derivs
for i in range(nsteps):
state = rk4(state, time, tau, wilber_derivs, params)
time += tau
xplot[i, :] = np.copy(state)
tplot[i] = time
plt.figure()
plt.plot(tplot, xplot[:, 0], '-', label='Longitudinal Displacement')
plt.xlabel('Time')
plt.ylabel('Longitudinal Displacement (m)')
plt.figure()
plt.plot(tplot, xplot[:, 1]*180/(np.pi), '-', label='Torsional Displacement')
plt.xlabel('Time')
plt.ylabel('Torsional Displacement (degrees)')
f = np.arange(nsteps)/(tau*nsteps)
x1 = xplot[:, 0]
def main(pos, param_r):
state = pos
r = param_r
sigma = 10.
b = 8. / 3.
param = np.array([r, sigma, b])
tau = 0.04
err = 1e-3
# loop over desired number of steps
time = 0.
nsteps = 500
tplot = np.empty(nsteps)
tauplot = np.empty(nsteps)
diffplot = np.empty((3, nsteps))
x1plot, y1plot, z1plot = np.empty(nsteps), np.empty(nsteps), np.empty(nsteps)
x2plot, y2plot, z2plot = np.empty(nsteps), np.empty(nsteps), np.empty(nsteps)
for istep in range(nsteps):
# Record values for plotting
x1, y1, z1 = state[0], state[1], state[2]
x2, y2, z2 = state[3], state[4], state[5]
tplot[istep] = time
x1plot[istep] = x1
y1plot[istep] = y1
z1plot[istep] = z1
x2plot[istep] = x2
y2plot[istep] = y2
z2plot[istep] = z2
diffplot[0, istep] = np.abs(x2 - x1)
diffplot[1, istep] = np.abs(y2 - y1)
diffplot[2, istep] = np.abs(z2 - z1)
total_diff = np.linalg.norm(diffplot, axis=0)
if (istep + 1) % 50 == 0:
print(f'Completed {istep} steps out of {nsteps}')
# Find new state using adaptive Runge-Kutta
state = rk4(state, time, tau, lorzrk, param)
time += tau
# Graph the time series x(t)
fig = plt.figure()
plt.plot(tplot, x1plot, 'g-')
plt.plot(tplot, x2plot, 'r--')
plt.xlabel('Time')
plt.ylabel('x(t)')
plt.title(f'Lorenz model time series, r = {r}')
fig = plt.figure()
plt.semilogy(tplot, diffplot[0, :])
plt.xlabel('time')
plt.ylabel('Absolute difference in x-coordinates')
plt.grid(True, which='both')
fig=plt.figure()
plt.semilogy(tplot, total_diff)
plt.xlabel('time')
plt.ylabel('Norm difference')
plt.grid(True, which='both')
# Graph the x, y, z, phase space trajectory
# Mark the location of the three stredy states
x_ss, y_ss, z_ss = np.empty(3), np.empty(3), np.empty(3)
x_ss[0] = 0.
y_ss[0] = 0.
z_ss[0] = 0.
x_ss[1] = np.sqrt(b*(r-1))
y_ss[1] = x_ss[1]
z_ss[1] = r - 1
x_ss[2] = -np.sqrt(b*(r-1))
y_ss[2] = x_ss[2]
z_ss[2] = r - 1
fig = plt.figure()
ax = fig.gca(projection='3d')
ax.plot(x1plot, y1plot, z1plot, 'g-')
ax.plot(x2plot, y2plot, z2plot, 'r--')
ax.plot(x_ss, y_ss, z_ss, '*')
ax.set_xlabel('x')
ax.set_ylabel('y')
ax.set_zlabel('z')
ax.set_title(f'Lorenz model phase space, r = {r}')
print(state)
plt.show()
x = np.array([0.3, 0., 0.]) # Initial positions (m)
v = np.array([0., 0., 0.]) # Initial velocities (m/s)
state = np.array([x[0], x[1], x[2], v[0], v[1], v[2]])
tau = 0.5 # Time step (sec)
k_over_m = 1. # Spring constant divided by mass (N/(kg-m))
time = 0.
nsteps = 256
nprint = nsteps/8 # Number of steps between printing process
tplot = np.empty(nsteps)
xplot = np.empty((nsteps, 3))
### MAIN LOOP ###
for i in range(nsteps):
# Use Runge-Kutta to find new mass displacements
state = rk4(state, time, tau, sprrk, k_over_m)
time += tau
# Record positions for plotting
tplot[i] = time
xplot[i, :] = np.copy(state[:3])
if i % nprint < 1:
print(f'Finished {i} out of {nsteps} steps')
# Graph displacements
plt.figure()
plt.plot(tplot, xplot[:, 0], '-', label='Mass 1')
plt.plot(tplot, xplot[:, 1], '-', label='Mass 2')
plt.plot(tplot, xplot[:, 2], '-', label='Mass 3')
plt.title('Displacement of masses relative to rest position')
plt.xlabel('Time')
nsteps = 100
params = np.array([k1, k2, k3, k4, L1, L2, L3, L4, Lw, m1, m2, m3])
state = np.array([x1_0, x2_0, x3_0, v1_0, v2_0, v3_0])
xplot = np.empty((3, nsteps))
vplot = np.empty((3, nsteps))
tplot = np.empty(nsteps)
for i in range(nsteps):
xplot[0, i] = state[0]
xplot[1, i] = state[1]
xplot[2, i] = state[2]
vplot[0, i] = state[3]
vplot[1, i] = state[4]
vplot[2, i] = state[5]
tplot[i] = time
state = rk4(state, time, tau, params)
time += tau
plt.figure(1)
plt.plot(tplot, xplot[0, :], 'b-', label='mass 1 position')
plt.plot(tplot, xplot[1, :], 'g--', label='mass 2 position')
plt.plot(tplot, xplot[2, :], 'r-.', label='mass 3 position')
plt.hlines(disp[0], tplot[0], tplot[-1], 'k', '--')
plt.hlines(disp[1], tplot[0], tplot[-1], 'k', '--')
plt.hlines(disp[2], tplot[0], tplot[-1], 'k', '--')
plt.xlabel('time (sec)')
plt.ylabel('mass position (m)')
plt.legend()
plt.figure(2)