DV-SOB-ligger-met-veer

import sympy as sp

w = sp.symbols('w', cls=sp.Function)
q0, x, r = sp.symbols('q0 x r')
L, EI = sp.symbols('L EI')
C1, C2, C3, C4 = sp.symbols('C1 C2 C3 C4')

DV = sp.Eq(EI*sp.diff(w(x),x,4),q0) 
display(DV)

\[\displaystyle EI \frac{d^{4}}{d x^{4}} w{\left(x \right)} = q_{0}\]

w = sp.dsolve(DV, w(x)) 
w = w.rhs 
display(w)

\[\displaystyle C_{1} + C_{2} x + C_{3} x^{2} + C_{4} x^{3} + \frac{q_{0} x^{4}}{24 EI}\]

phi = -sp.diff(w, x)
kappa = sp.diff(phi, x)
M = EI * kappa
V = sp.diff(M, x)

Eq1 = sp.Eq(w.subs(x, 0), 0) 
Eq2 = sp.Eq(phi.subs(x, 0), 0)
Eq3 = sp.Eq(w.subs(x, L), 0)
Eq4 = sp.Eq(M.subs(x, L), -r*phi.subs(x,L))

sol = sp.solve((Eq1,Eq2,Eq3,Eq4),(C1,C2,C3,C4))
display(sol)

{C1: 0,
 C2: 0,
 C3: (6*EI*L**2*q0 + L**3*q0*r)/(96*EI**2 + 24*EI*L*r),
 C4: (-5*EI*L*q0 - L**2*q0*r)/(48*EI**2 + 12*EI*L*r)}

w_sol = w.subs(sol)
phi_sol = phi.subs(sol)
M_sol = M.subs(sol)
V_sol = V.subs(sol)
display(phi_sol.subs(x,0))
display(phi_sol.subs(x,L))
display(w_sol.subs(x,L/2))

\[\displaystyle 0\]

\[\displaystyle - \frac{3 L^{2} \left(- 5 EI L q_{0} - L^{2} q_{0} r\right)}{48 EI^{2} + 12 EI L r} - \frac{2 L \left(6 EI L^{2} q_{0} + L^{3} q_{0} r\right)}{96 EI^{2} + 24 EI L r} - \frac{L^{3} q_{0}}{6 EI}\]

\[\displaystyle \frac{L^{3} \left(- 5 EI L q_{0} - L^{2} q_{0} r\right)}{8 \cdot \left(48 EI^{2} + 12 EI L r\right)} + \frac{L^{2} \cdot \left(6 EI L^{2} q_{0} + L^{3} q_{0} r\right)}{4 \cdot \left(96 EI^{2} + 24 EI L r\right)} + \frac{L^{4} q_{0}}{384 EI}\]

w_subs = w_sol.subs([(EI,10000),(q0,5),(L,8),(r,2000)])
phi_subs = phi_sol.subs([(EI,10000),(q0,5),(L,8),(r,2000)])
M_subs = M_sol.subs([(EI,10000),(q0,5),(L,8),(r,2000)])
V_subs = V_sol.subs([(EI,10000),(q0,5),(L,8),(r,2000)])

sp.plot(-w_subs,(x,0,8),title='$w$');
sp.plot(-phi_subs,(x,0,8),title='$\phi$');
sp.plot(-M_subs,(x,0,8),title='$M$');
sp.plot(-V_subs,(x,0,8),title='$V$');