Introductie verplaatsingenmethode

import sympy as sp

a, b, c, k1, k2, k3 = sp.symbols('a b c k1 k2 k3')

Bg = sp.Matrix([[0,-1,-a],[0,-1,b],[-1,0,c]])
BgT = sp.transpose(Bg)
Dg = sp.Matrix([[k1,0,0],[0,k2,0],[0,0,k3]])
K = BgT*Dg*Bg
display(K)
F = K.inv()
display(F)
load = sp.Matrix([50,150,-5])
disp = F*load
disp_sol = disp.subs([(a,3),(b,2),(c,1),(k1,1000),(k2,2000),(k3,3000)])
display(disp_sol)
display(disp_sol.evalf())

\[\begin{split}\displaystyle \left[\begin{matrix}k_{3} & 0 & - c k_{3}\\0 & k_{1} + k_{2} & a k_{1} - b k_{2}\\- c k_{3} & a k_{1} - b k_{2} & a^{2} k_{1} + b^{2} k_{2} + c^{2} k_{3}\end{matrix}\right]\end{split}\]

\[\begin{split}\displaystyle \left[\begin{matrix}\frac{a^{2} k_{1} k_{2} + 2 a b k_{1} k_{2} + b^{2} k_{1} k_{2} + c^{2} k_{1} k_{3} + c^{2} k_{2} k_{3}}{a^{2} k_{1} k_{2} k_{3} + 2 a b k_{1} k_{2} k_{3} + b^{2} k_{1} k_{2} k_{3}} & \frac{- a c k_{1} + b c k_{2}}{a^{2} k_{1} k_{2} + 2 a b k_{1} k_{2} + b^{2} k_{1} k_{2}} & \frac{c k_{1} + c k_{2}}{a^{2} k_{1} k_{2} + 2 a b k_{1} k_{2} + b^{2} k_{1} k_{2}}\\\frac{- a c k_{1} + b c k_{2}}{a^{2} k_{1} k_{2} + 2 a b k_{1} k_{2} + b^{2} k_{1} k_{2}} & \frac{a^{2} k_{1} + b^{2} k_{2}}{a^{2} k_{1} k_{2} + 2 a b k_{1} k_{2} + b^{2} k_{1} k_{2}} & \frac{- a k_{1} + b k_{2}}{a^{2} k_{1} k_{2} + 2 a b k_{1} k_{2} + b^{2} k_{1} k_{2}}\\\frac{c k_{1} + c k_{2}}{a^{2} k_{1} k_{2} + 2 a b k_{1} k_{2} + b^{2} k_{1} k_{2}} & \frac{- a k_{1} + b k_{2}}{a^{2} k_{1} k_{2} + 2 a b k_{1} k_{2} + b^{2} k_{1} k_{2}} & \frac{k_{1} + k_{2}}{a^{2} k_{1} k_{2} + 2 a b k_{1} k_{2} + b^{2} k_{1} k_{2}}\end{matrix}\right]\end{split}\]

\[\begin{split}\displaystyle \left[\begin{matrix}\frac{671}{30000}\\\frac{519}{10000}\\\frac{57}{10000}\end{matrix}\right]\end{split}\]

\[\begin{split}\displaystyle \left[\begin{matrix}0.0223666666666667\\0.0519\\0.0057\end{matrix}\right]\end{split}\]