2.4.2.A.sagews

from sympy import *
from sympy import N as Num
import matplotlib.pyplot as plt
from numpy import linspace, full_like, array, arange

# solve for these variables:
C1, C2 = var('C1, C2')

# use constants:
l, q = var('l, q')

print("--- 1-3: Compute C1, C2 -------------------")
eqns = [
    Eq(0, q/24*l**4 + C1/6*l**3 + C2/2*l**2),
    Eq(0, q/2*l**2 + C1*l + C2),
    ]

sol = solve(eqns, [C1, C2])

print sol[C1]
print sol[C2]
# C1:  -5*l*q/8
# C2:  q*l**2/8

print("--- e: find xi with w'(xi)=0 --------------")
# w(xi):
xi = var('xi')

def w(xi):
    t1 =   xi**2/16
    t2 =-5*xi**3/48
    t3 =   xi**4/24
    return t1+t2+t3

# w':
w_prime = diff(w(xi),xi)

# Solve w' = 0:
xiw = solve(w_prime)

# xi und w(xi):
for Xi in xiw:
    print "xi, w(xi): ", Num(Xi,4), Num(w(Xi),3)

print("--- f: ------------------------------------")
q, a, EI = var("q, a, EI")
W = q*a*a*a*a/EI * w(xi)
print "W:"
pprint(W)
# xi = x/a
# dxi_dx = 1/a
# dW_dx = dW_dxi dxi_dx
W_p = diff(W,xi)/a
W_pp = diff(W_p,xi)/a
W_ppp = diff(W_pp,xi)/a
# print(latex(W))
# print(latex(W_p))
# print(latex(W_pp))
# print(latex(W_ppp))

MA = -EI*W_pp.subs(xi,0)
Av = -EI*W_ppp.subs(xi,0)
Bv =  EI*W_ppp.subs(xi,1)
# pprint(latex(MA))
# pprint(latex(Av))
# pprint(latex(Bv))

print("--- e: plot -------------------------------")

# Plotting:
xi = linspace(float(0),float(1))

tmp = plt.plot(xi,w(xi), label="$w EI / qa^4$")
tmp = plt.xlabel("$x / a$")
tmp = plt.legend()
plt.gca().invert_yaxis()

# plt.savefig("2.4.D_w.svg", transparent=True)
plt.show()

--- 1-3: Compute C1, C2 ------------------- -5*l*q/8 l**2*q/8 --- e: find xi with w'(xi)=0 -------------- xi, w(xi): 0 0 xi, w(xi): 0.5785 0.00542 xi, w(xi): 1.297 -0.00423 --- f: ------------------------------------ W: / 4 3 2\ 4 |xi 5*xi xi | a *q*|--- - ----- + ---| \ 24 48 16/ ------------------------ EI --- e: plot -------------------------------

/ext/sage/sage-8.0/local/lib/python2.7/site-packages/matplotlib/font_manager.py:273: UserWarning: Matplotlib is building the font cache using fc-list. This may take a moment. warnings.warn('Matplotlib is building the font cache using fc-list. This may take a moment.')