CIE4140_Lecture_13_Part_1_Python_seperate_q

Consider the steady-state vibrations of a string subjected to distributed viscous damping under a harmonic force

import sympy as sp

We represent the harmonic time-dependence by \(e^{i \Omega t}\). The real part of the response to this load will give the response to the corresponding cosinusoidal load , whereas the imaginary part will give the response to the corresponding sinusoidal load

Let us write the equation of motion

w = sp.symbols('w',cls=sp.Function)
q1 = sp.symbols('q1',cls=sp.Function)
x,t = sp.symbols('x t')
nd, c, Omega= sp.symbols('nd c Omega',real=True)

q1_second_normal_mode = sp.sin(2 * sp.pi * x / 1)
q1_arbitrary = 14.39 * x **5 * (x - 1)
q1_constant = 1
q1 = q1_arbitrary

EQM = sp.diff(w(x,t),t,2) + 2 * nd * sp.diff(w(x,t),t) - c**2 * sp.diff(w(x,t),x,2) - q1*sp.exp(sp.I*Omega*t)
display(EQM)

\[\displaystyle - c^{2} \frac{\partial^{2}}{\partial x^{2}} w{\left(x,t \right)} + 2 nd \frac{\partial}{\partial t} w{\left(x,t \right)} - 14.39 x^{5} \left(x - 1\right) e^{i \Omega t} + \frac{\partial^{2}}{\partial t^{2}} w{\left(x,t \right)}\]

sp.plot(q1, (x, 0, 1));

We search for the steady-state solution in the form:

W = sp.symbols('W',cls=sp.Function)
w_form = W(x) * sp.exp(sp.I * Omega * t)

Substitution of this equation in the equation of motion gives:

EQM_fr = sp.simplify(EQM.subs(w(x,t),w_form))
display(EQM_fr)

\[\displaystyle - \Omega^{2} W{\left(x \right)} e^{i \Omega t} + 2 i \Omega nd W{\left(x \right)} e^{i \Omega t} - c^{2} e^{i \Omega t} \frac{d^{2}}{d x^{2}} W{\left(x \right)} + 14.39 x^{5} \cdot \left(1 - x\right) e^{i \Omega t}\]

\(e^{i \Omega t}\) is not dropped, so can dropped manually, not necessary:

display(EQM_fr.subs(sp.exp(sp.I * Omega * t),1))

\[\displaystyle - \Omega^{2} W{\left(x \right)} + 2 i \Omega nd W{\left(x \right)} - c^{2} \frac{d^{2}}{d x^{2}} W{\left(x \right)} + 14.39 x^{5} \cdot \left(1 - x\right)\]

Let us find the soluton to this equation that satisfies the boundary conditions of a fixed-fixed string

L = sp.symbols('L',real=True)
W_sol = sp.dsolve(EQM_fr.subs(sp.exp(sp.I * Omega * t),1),W(x),ics={W(0):0,W(L):0}).rhs

\[\displaystyle \left(- \frac{1439.0 L^{6} \Omega^{6} e^{\frac{1.4142135623731 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}}}{100.0 \Omega^{8} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 100.0 \Omega^{8} - 800.0 i \Omega^{7} nd e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 800.0 i \Omega^{7} nd - 2400.0 \Omega^{6} nd^{2} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 2400.0 \Omega^{6} nd^{2} + 3200.0 i \Omega^{5} nd^{3} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 3200.0 i \Omega^{5} nd^{3} + 1600.0 \Omega^{4} nd^{4} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 1600.0 \Omega^{4} nd^{4}} + \frac{8634.0 i L^{6} \Omega^{5} nd e^{\frac{1.4142135623731 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}}}{100.0 \Omega^{8} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 100.0 \Omega^{8} - 800.0 i \Omega^{7} nd e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 800.0 i \Omega^{7} nd - 2400.0 \Omega^{6} nd^{2} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 2400.0 \Omega^{6} nd^{2} + 3200.0 i \Omega^{5} nd^{3} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 3200.0 i \Omega^{5} nd^{3} + 1600.0 \Omega^{4} nd^{4} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 1600.0 \Omega^{4} nd^{4}} + \frac{17268.0 L^{6} \Omega^{4} nd^{2} e^{\frac{1.4142135623731 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}}}{100.0 \Omega^{8} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 100.0 \Omega^{8} - 800.0 i \Omega^{7} nd e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 800.0 i \Omega^{7} nd - 2400.0 \Omega^{6} nd^{2} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 2400.0 \Omega^{6} nd^{2} + 3200.0 i \Omega^{5} nd^{3} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 3200.0 i \Omega^{5} nd^{3} + 1600.0 \Omega^{4} nd^{4} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 1600.0 \Omega^{4} nd^{4}} - \frac{11512.0 i L^{6} \Omega^{3} nd^{3} e^{\frac{1.4142135623731 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}}}{100.0 \Omega^{8} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 100.0 \Omega^{8} - 800.0 i \Omega^{7} nd e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 800.0 i \Omega^{7} nd - 2400.0 \Omega^{6} nd^{2} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 2400.0 \Omega^{6} nd^{2} + 3200.0 i \Omega^{5} nd^{3} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 3200.0 i \Omega^{5} nd^{3} + 1600.0 \Omega^{4} nd^{4} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 1600.0 \Omega^{4} nd^{4}} + \frac{1439.0 L^{5} \Omega^{6} e^{\frac{1.4142135623731 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}}}{100.0 \Omega^{8} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 100.0 \Omega^{8} - 800.0 i \Omega^{7} nd e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 800.0 i \Omega^{7} nd - 2400.0 \Omega^{6} nd^{2} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 2400.0 \Omega^{6} nd^{2} + 3200.0 i \Omega^{5} nd^{3} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 3200.0 i \Omega^{5} nd^{3} + 1600.0 \Omega^{4} nd^{4} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 1600.0 \Omega^{4} nd^{4}} - \frac{8634.0 i L^{5} \Omega^{5} nd e^{\frac{1.4142135623731 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}}}{100.0 \Omega^{8} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 100.0 \Omega^{8} - 800.0 i \Omega^{7} nd e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 800.0 i \Omega^{7} nd - 2400.0 \Omega^{6} nd^{2} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 2400.0 \Omega^{6} nd^{2} + 3200.0 i \Omega^{5} nd^{3} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 3200.0 i \Omega^{5} nd^{3} + 1600.0 \Omega^{4} nd^{4} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 1600.0 \Omega^{4} nd^{4}} - \frac{17268.0 L^{5} \Omega^{4} nd^{2} e^{\frac{1.4142135623731 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}}}{100.0 \Omega^{8} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 100.0 \Omega^{8} - 800.0 i \Omega^{7} nd e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 800.0 i \Omega^{7} nd - 2400.0 \Omega^{6} nd^{2} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 2400.0 \Omega^{6} nd^{2} + 3200.0 i \Omega^{5} nd^{3} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 3200.0 i \Omega^{5} nd^{3} + 1600.0 \Omega^{4} nd^{4} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 1600.0 \Omega^{4} nd^{4}} + \frac{11512.0 i L^{5} \Omega^{3} nd^{3} e^{\frac{1.4142135623731 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}}}{100.0 \Omega^{8} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 100.0 \Omega^{8} - 800.0 i \Omega^{7} nd e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 800.0 i \Omega^{7} nd - 2400.0 \Omega^{6} nd^{2} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 2400.0 \Omega^{6} nd^{2} + 3200.0 i \Omega^{5} nd^{3} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 3200.0 i \Omega^{5} nd^{3} + 1600.0 \Omega^{4} nd^{4} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 1600.0 \Omega^{4} nd^{4}} + \frac{43170.0 L^{4} \Omega^{4} c^{2} e^{\frac{1.4142135623731 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}}}{100.0 \Omega^{8} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 100.0 \Omega^{8} - 800.0 i \Omega^{7} nd e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 800.0 i \Omega^{7} nd - 2400.0 \Omega^{6} nd^{2} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 2400.0 \Omega^{6} nd^{2} + 3200.0 i \Omega^{5} nd^{3} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 3200.0 i \Omega^{5} nd^{3} + 1600.0 \Omega^{4} nd^{4} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 1600.0 \Omega^{4} nd^{4}} - \frac{172680.0 i L^{4} \Omega^{3} c^{2} nd e^{\frac{1.4142135623731 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}}}{100.0 \Omega^{8} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 100.0 \Omega^{8} - 800.0 i \Omega^{7} nd e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 800.0 i \Omega^{7} nd - 2400.0 \Omega^{6} nd^{2} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 2400.0 \Omega^{6} nd^{2} + 3200.0 i \Omega^{5} nd^{3} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 3200.0 i \Omega^{5} nd^{3} + 1600.0 \Omega^{4} nd^{4} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 1600.0 \Omega^{4} nd^{4}} - \frac{172680.0 L^{4} \Omega^{2} c^{2} nd^{2} e^{\frac{1.4142135623731 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}}}{100.0 \Omega^{8} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 100.0 \Omega^{8} - 800.0 i \Omega^{7} nd e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 800.0 i \Omega^{7} nd - 2400.0 \Omega^{6} nd^{2} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 2400.0 \Omega^{6} nd^{2} + 3200.0 i \Omega^{5} nd^{3} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 3200.0 i \Omega^{5} nd^{3} + 1600.0 \Omega^{4} nd^{4} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 1600.0 \Omega^{4} nd^{4}} - \frac{28780.0 L^{3} \Omega^{4} c^{2} e^{\frac{1.4142135623731 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}}}{100.0 \Omega^{8} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 100.0 \Omega^{8} - 800.0 i \Omega^{7} nd e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 800.0 i \Omega^{7} nd - 2400.0 \Omega^{6} nd^{2} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 2400.0 \Omega^{6} nd^{2} + 3200.0 i \Omega^{5} nd^{3} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 3200.0 i \Omega^{5} nd^{3} + 1600.0 \Omega^{4} nd^{4} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 1600.0 \Omega^{4} nd^{4}} + \frac{115120.0 i L^{3} \Omega^{3} c^{2} nd e^{\frac{1.4142135623731 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}}}{100.0 \Omega^{8} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 100.0 \Omega^{8} - 800.0 i \Omega^{7} nd e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 800.0 i \Omega^{7} nd - 2400.0 \Omega^{6} nd^{2} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 2400.0 \Omega^{6} nd^{2} + 3200.0 i \Omega^{5} nd^{3} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 3200.0 i \Omega^{5} nd^{3} + 1600.0 \Omega^{4} nd^{4} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 1600.0 \Omega^{4} nd^{4}} + \frac{115120.0 L^{3} \Omega^{2} c^{2} nd^{2} e^{\frac{1.4142135623731 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}}}{100.0 \Omega^{8} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 100.0 \Omega^{8} - 800.0 i \Omega^{7} nd e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 800.0 i \Omega^{7} nd - 2400.0 \Omega^{6} nd^{2} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 2400.0 \Omega^{6} nd^{2} + 3200.0 i \Omega^{5} nd^{3} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 3200.0 i \Omega^{5} nd^{3} + 1600.0 \Omega^{4} nd^{4} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 1600.0 \Omega^{4} nd^{4}} - \frac{518040.0 L^{2} \Omega^{2} c^{4} e^{\frac{1.4142135623731 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}}}{100.0 \Omega^{8} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 100.0 \Omega^{8} - 800.0 i \Omega^{7} nd e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 800.0 i \Omega^{7} nd - 2400.0 \Omega^{6} nd^{2} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 2400.0 \Omega^{6} nd^{2} + 3200.0 i \Omega^{5} nd^{3} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 3200.0 i \Omega^{5} nd^{3} + 1600.0 \Omega^{4} nd^{4} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 1600.0 \Omega^{4} nd^{4}} + \frac{1036080.0 i L^{2} \Omega c^{4} nd e^{\frac{1.4142135623731 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}}}{100.0 \Omega^{8} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 100.0 \Omega^{8} - 800.0 i \Omega^{7} nd e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 800.0 i \Omega^{7} nd - 2400.0 \Omega^{6} nd^{2} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 2400.0 \Omega^{6} nd^{2} + 3200.0 i \Omega^{5} nd^{3} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 3200.0 i \Omega^{5} nd^{3} + 1600.0 \Omega^{4} nd^{4} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 1600.0 \Omega^{4} nd^{4}} + \frac{172680.0 L \Omega^{2} c^{4} e^{\frac{1.4142135623731 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}}}{100.0 \Omega^{8} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 100.0 \Omega^{8} - 800.0 i \Omega^{7} nd e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 800.0 i \Omega^{7} nd - 2400.0 \Omega^{6} nd^{2} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 2400.0 \Omega^{6} nd^{2} + 3200.0 i \Omega^{5} nd^{3} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 3200.0 i \Omega^{5} nd^{3} + 1600.0 \Omega^{4} nd^{4} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 1600.0 \Omega^{4} nd^{4}} - \frac{345360.0 i L \Omega c^{4} nd e^{\frac{1.4142135623731 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}}}{100.0 \Omega^{8} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 100.0 \Omega^{8} - 800.0 i \Omega^{7} nd e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 800.0 i \Omega^{7} nd - 2400.0 \Omega^{6} nd^{2} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 2400.0 \Omega^{6} nd^{2} + 3200.0 i \Omega^{5} nd^{3} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 3200.0 i \Omega^{5} nd^{3} + 1600.0 \Omega^{4} nd^{4} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 1600.0 \Omega^{4} nd^{4}} + \frac{1036080.0 c^{6} e^{\frac{1.4142135623731 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}}}{100.0 \Omega^{8} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 100.0 \Omega^{8} - 800.0 i \Omega^{7} nd e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 800.0 i \Omega^{7} nd - 2400.0 \Omega^{6} nd^{2} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 2400.0 \Omega^{6} nd^{2} + 3200.0 i \Omega^{5} nd^{3} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 3200.0 i \Omega^{5} nd^{3} + 1600.0 \Omega^{4} nd^{4} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 1600.0 \Omega^{4} nd^{4}} - \frac{1036080.0 c^{6} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}}}{100.0 \Omega^{8} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 100.0 \Omega^{8} - 800.0 i \Omega^{7} nd e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 800.0 i \Omega^{7} nd - 2400.0 \Omega^{6} nd^{2} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 2400.0 \Omega^{6} nd^{2} + 3200.0 i \Omega^{5} nd^{3} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 3200.0 i \Omega^{5} nd^{3} + 1600.0 \Omega^{4} nd^{4} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 1600.0 \Omega^{4} nd^{4}}\right) e^{- \frac{x \sqrt{\Omega \left(- \Omega + 2 i nd\right)}}{c}} + \left(\frac{1439.0 L^{6} \Omega^{6} e^{\frac{1.4142135623731 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}}}{100.0 \Omega^{8} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 100.0 \Omega^{8} - 800.0 i \Omega^{7} nd e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 800.0 i \Omega^{7} nd - 2400.0 \Omega^{6} nd^{2} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 2400.0 \Omega^{6} nd^{2} + 3200.0 i \Omega^{5} nd^{3} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 3200.0 i \Omega^{5} nd^{3} + 1600.0 \Omega^{4} nd^{4} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 1600.0 \Omega^{4} nd^{4}} - \frac{8634.0 i L^{6} \Omega^{5} nd e^{\frac{1.4142135623731 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}}}{100.0 \Omega^{8} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 100.0 \Omega^{8} - 800.0 i \Omega^{7} nd e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 800.0 i \Omega^{7} nd - 2400.0 \Omega^{6} nd^{2} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 2400.0 \Omega^{6} nd^{2} + 3200.0 i \Omega^{5} nd^{3} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 3200.0 i \Omega^{5} nd^{3} + 1600.0 \Omega^{4} nd^{4} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 1600.0 \Omega^{4} nd^{4}} - \frac{17268.0 L^{6} \Omega^{4} nd^{2} e^{\frac{1.4142135623731 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}}}{100.0 \Omega^{8} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 100.0 \Omega^{8} - 800.0 i \Omega^{7} nd e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 800.0 i \Omega^{7} nd - 2400.0 \Omega^{6} nd^{2} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 2400.0 \Omega^{6} nd^{2} + 3200.0 i \Omega^{5} nd^{3} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 3200.0 i \Omega^{5} nd^{3} + 1600.0 \Omega^{4} nd^{4} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 1600.0 \Omega^{4} nd^{4}} + \frac{11512.0 i L^{6} \Omega^{3} nd^{3} e^{\frac{1.4142135623731 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}}}{100.0 \Omega^{8} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 100.0 \Omega^{8} - 800.0 i \Omega^{7} nd e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 800.0 i \Omega^{7} nd - 2400.0 \Omega^{6} nd^{2} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 2400.0 \Omega^{6} nd^{2} + 3200.0 i \Omega^{5} nd^{3} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 3200.0 i \Omega^{5} nd^{3} + 1600.0 \Omega^{4} nd^{4} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 1600.0 \Omega^{4} nd^{4}} - \frac{1439.0 L^{5} \Omega^{6} e^{\frac{1.4142135623731 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}}}{100.0 \Omega^{8} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 100.0 \Omega^{8} - 800.0 i \Omega^{7} nd e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 800.0 i \Omega^{7} nd - 2400.0 \Omega^{6} nd^{2} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 2400.0 \Omega^{6} nd^{2} + 3200.0 i \Omega^{5} nd^{3} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 3200.0 i \Omega^{5} nd^{3} + 1600.0 \Omega^{4} nd^{4} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 1600.0 \Omega^{4} nd^{4}} + \frac{8634.0 i L^{5} \Omega^{5} nd e^{\frac{1.4142135623731 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}}}{100.0 \Omega^{8} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 100.0 \Omega^{8} - 800.0 i \Omega^{7} nd e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 800.0 i \Omega^{7} nd - 2400.0 \Omega^{6} nd^{2} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 2400.0 \Omega^{6} nd^{2} + 3200.0 i \Omega^{5} nd^{3} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 3200.0 i \Omega^{5} nd^{3} + 1600.0 \Omega^{4} nd^{4} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 1600.0 \Omega^{4} nd^{4}} + \frac{17268.0 L^{5} \Omega^{4} nd^{2} e^{\frac{1.4142135623731 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}}}{100.0 \Omega^{8} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 100.0 \Omega^{8} - 800.0 i \Omega^{7} nd e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 800.0 i \Omega^{7} nd - 2400.0 \Omega^{6} nd^{2} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 2400.0 \Omega^{6} nd^{2} + 3200.0 i \Omega^{5} nd^{3} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 3200.0 i \Omega^{5} nd^{3} + 1600.0 \Omega^{4} nd^{4} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 1600.0 \Omega^{4} nd^{4}} - \frac{11512.0 i L^{5} \Omega^{3} nd^{3} e^{\frac{1.4142135623731 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}}}{100.0 \Omega^{8} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 100.0 \Omega^{8} - 800.0 i \Omega^{7} nd e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 800.0 i \Omega^{7} nd - 2400.0 \Omega^{6} nd^{2} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 2400.0 \Omega^{6} nd^{2} + 3200.0 i \Omega^{5} nd^{3} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 3200.0 i \Omega^{5} nd^{3} + 1600.0 \Omega^{4} nd^{4} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 1600.0 \Omega^{4} nd^{4}} - \frac{43170.0 L^{4} \Omega^{4} c^{2} e^{\frac{1.4142135623731 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}}}{100.0 \Omega^{8} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 100.0 \Omega^{8} - 800.0 i \Omega^{7} nd e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 800.0 i \Omega^{7} nd - 2400.0 \Omega^{6} nd^{2} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 2400.0 \Omega^{6} nd^{2} + 3200.0 i \Omega^{5} nd^{3} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 3200.0 i \Omega^{5} nd^{3} + 1600.0 \Omega^{4} nd^{4} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 1600.0 \Omega^{4} nd^{4}} + \frac{172680.0 i L^{4} \Omega^{3} c^{2} nd e^{\frac{1.4142135623731 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}}}{100.0 \Omega^{8} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 100.0 \Omega^{8} - 800.0 i \Omega^{7} nd e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 800.0 i \Omega^{7} nd - 2400.0 \Omega^{6} nd^{2} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 2400.0 \Omega^{6} nd^{2} + 3200.0 i \Omega^{5} nd^{3} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 3200.0 i \Omega^{5} nd^{3} + 1600.0 \Omega^{4} nd^{4} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 1600.0 \Omega^{4} nd^{4}} + \frac{172680.0 L^{4} \Omega^{2} c^{2} nd^{2} e^{\frac{1.4142135623731 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}}}{100.0 \Omega^{8} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 100.0 \Omega^{8} - 800.0 i \Omega^{7} nd e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 800.0 i \Omega^{7} nd - 2400.0 \Omega^{6} nd^{2} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 2400.0 \Omega^{6} nd^{2} + 3200.0 i \Omega^{5} nd^{3} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 3200.0 i \Omega^{5} nd^{3} + 1600.0 \Omega^{4} nd^{4} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 1600.0 \Omega^{4} nd^{4}} + \frac{28780.0 L^{3} \Omega^{4} c^{2} e^{\frac{1.4142135623731 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}}}{100.0 \Omega^{8} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 100.0 \Omega^{8} - 800.0 i \Omega^{7} nd e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 800.0 i \Omega^{7} nd - 2400.0 \Omega^{6} nd^{2} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 2400.0 \Omega^{6} nd^{2} + 3200.0 i \Omega^{5} nd^{3} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 3200.0 i \Omega^{5} nd^{3} + 1600.0 \Omega^{4} nd^{4} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 1600.0 \Omega^{4} nd^{4}} - \frac{115120.0 i L^{3} \Omega^{3} c^{2} nd e^{\frac{1.4142135623731 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}}}{100.0 \Omega^{8} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 100.0 \Omega^{8} - 800.0 i \Omega^{7} nd e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 800.0 i \Omega^{7} nd - 2400.0 \Omega^{6} nd^{2} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 2400.0 \Omega^{6} nd^{2} + 3200.0 i \Omega^{5} nd^{3} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 3200.0 i \Omega^{5} nd^{3} + 1600.0 \Omega^{4} nd^{4} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 1600.0 \Omega^{4} nd^{4}} - \frac{115120.0 L^{3} \Omega^{2} c^{2} nd^{2} e^{\frac{1.4142135623731 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}}}{100.0 \Omega^{8} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 100.0 \Omega^{8} - 800.0 i \Omega^{7} nd e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 800.0 i \Omega^{7} nd - 2400.0 \Omega^{6} nd^{2} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 2400.0 \Omega^{6} nd^{2} + 3200.0 i \Omega^{5} nd^{3} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 3200.0 i \Omega^{5} nd^{3} + 1600.0 \Omega^{4} nd^{4} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 1600.0 \Omega^{4} nd^{4}} + \frac{518040.0 L^{2} \Omega^{2} c^{4} e^{\frac{1.4142135623731 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}}}{100.0 \Omega^{8} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 100.0 \Omega^{8} - 800.0 i \Omega^{7} nd e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 800.0 i \Omega^{7} nd - 2400.0 \Omega^{6} nd^{2} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 2400.0 \Omega^{6} nd^{2} + 3200.0 i \Omega^{5} nd^{3} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 3200.0 i \Omega^{5} nd^{3} + 1600.0 \Omega^{4} nd^{4} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 1600.0 \Omega^{4} nd^{4}} - \frac{1036080.0 i L^{2} \Omega c^{4} nd e^{\frac{1.4142135623731 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}}}{100.0 \Omega^{8} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 100.0 \Omega^{8} - 800.0 i \Omega^{7} nd e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 800.0 i \Omega^{7} nd - 2400.0 \Omega^{6} nd^{2} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 2400.0 \Omega^{6} nd^{2} + 3200.0 i \Omega^{5} nd^{3} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 3200.0 i \Omega^{5} nd^{3} + 1600.0 \Omega^{4} nd^{4} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 1600.0 \Omega^{4} nd^{4}} - \frac{172680.0 L \Omega^{2} c^{4} e^{\frac{1.4142135623731 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}}}{100.0 \Omega^{8} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 100.0 \Omega^{8} - 800.0 i \Omega^{7} nd e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 800.0 i \Omega^{7} nd - 2400.0 \Omega^{6} nd^{2} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 2400.0 \Omega^{6} nd^{2} + 3200.0 i \Omega^{5} nd^{3} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 3200.0 i \Omega^{5} nd^{3} + 1600.0 \Omega^{4} nd^{4} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 1600.0 \Omega^{4} nd^{4}} + \frac{345360.0 i L \Omega c^{4} nd e^{\frac{1.4142135623731 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}}}{100.0 \Omega^{8} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 100.0 \Omega^{8} - 800.0 i \Omega^{7} nd e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 800.0 i \Omega^{7} nd - 2400.0 \Omega^{6} nd^{2} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 2400.0 \Omega^{6} nd^{2} + 3200.0 i \Omega^{5} nd^{3} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 3200.0 i \Omega^{5} nd^{3} + 1600.0 \Omega^{4} nd^{4} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 1600.0 \Omega^{4} nd^{4}} - \frac{1036080.0 c^{6} e^{\frac{1.4142135623731 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}}}{100.0 \Omega^{8} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 100.0 \Omega^{8} - 800.0 i \Omega^{7} nd e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 800.0 i \Omega^{7} nd - 2400.0 \Omega^{6} nd^{2} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 2400.0 \Omega^{6} nd^{2} + 3200.0 i \Omega^{5} nd^{3} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 3200.0 i \Omega^{5} nd^{3} + 1600.0 \Omega^{4} nd^{4} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 1600.0 \Omega^{4} nd^{4}} + \frac{1036080.0 c^{6}}{100.0 \Omega^{8} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 100.0 \Omega^{8} - 800.0 i \Omega^{7} nd e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 800.0 i \Omega^{7} nd - 2400.0 \Omega^{6} nd^{2} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} + 2400.0 \Omega^{6} nd^{2} + 3200.0 i \Omega^{5} nd^{3} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 3200.0 i \Omega^{5} nd^{3} + 1600.0 \Omega^{4} nd^{4} e^{\frac{2.82842712474619 L \sqrt{- 0.5 \Omega^{2} + i \Omega nd}}{c}} - 1600.0 \Omega^{4} nd^{4}}\right) e^{\frac{x \sqrt{\Omega \left(- \Omega + 2 i nd\right)}}{c}} - \frac{1439.0 x^{6}}{\Omega \left(100.0 \Omega - 200.0 i nd\right)} + \frac{1439.0 x^{5}}{\Omega \left(100.0 \Omega - 200.0 i nd\right)} + \frac{4317.0 c^{2} x^{4}}{\Omega^{2} \cdot \left(10.0 \Omega^{2} - 40.0 i \Omega nd - 40.0 nd^{2}\right)} - \frac{1439.0 c^{2} x^{3}}{\Omega^{2} \cdot \left(5.0 \Omega^{2} - 20.0 i \Omega nd - 20.0 nd^{2}\right)} - \frac{25902.0 c^{4} x^{2}}{\Omega^{3} \cdot \left(5.0 \Omega^{3} - 30.0 i \Omega^{2} nd - 60.0 \Omega nd^{2} + 40.0 i nd^{3}\right)} + \frac{8634.0 c^{4} x}{\Omega^{3} \cdot \left(5.0 \Omega^{3} - 30.0 i \Omega^{2} nd - 60.0 \Omega nd^{2} + 40.0 i nd^{3}\right)} + \frac{51804.0 c^{6}}{\Omega^{4} \cdot \left(5.0 \Omega^{4} - 40.0 i \Omega^{3} nd - 120.0 \Omega^{2} nd^{2} + 160.0 i \Omega nd^{3} + 80.0 nd^{4}\right)}\]

We introduce numerical values for the parameters

W_sol = W_sol.subs([(L,1),(c,2),(nd,1/2)])

\[\displaystyle \left(- \frac{57560.0 \Omega^{4} e^{0.5 \sqrt{- \Omega^{2} + i \Omega}}}{100.0 \Omega^{8} e^{1.0 \sqrt{- \Omega^{2} + i \Omega}} - 100.0 \Omega^{8} - 400.0 i \Omega^{7} e^{1.0 \sqrt{- \Omega^{2} + i \Omega}} + 400.0 i \Omega^{7} - 600.0 \Omega^{6} e^{1.0 \sqrt{- \Omega^{2} + i \Omega}} + 600.0 \Omega^{6} + 400.0 i \Omega^{5} e^{1.0 \sqrt{- \Omega^{2} + i \Omega}} - 400.0 i \Omega^{5} + 100.0 \Omega^{4} e^{1.0 \sqrt{- \Omega^{2} + i \Omega}} - 100.0 \Omega^{4}} + \frac{115120.0 i \Omega^{3} e^{0.5 \sqrt{- \Omega^{2} + i \Omega}}}{100.0 \Omega^{8} e^{1.0 \sqrt{- \Omega^{2} + i \Omega}} - 100.0 \Omega^{8} - 400.0 i \Omega^{7} e^{1.0 \sqrt{- \Omega^{2} + i \Omega}} + 400.0 i \Omega^{7} - 600.0 \Omega^{6} e^{1.0 \sqrt{- \Omega^{2} + i \Omega}} + 600.0 \Omega^{6} + 400.0 i \Omega^{5} e^{1.0 \sqrt{- \Omega^{2} + i \Omega}} - 400.0 i \Omega^{5} + 100.0 \Omega^{4} e^{1.0 \sqrt{- \Omega^{2} + i \Omega}} - 100.0 \Omega^{4}} + \frac{5583320.0 \Omega^{2} e^{0.5 \sqrt{- \Omega^{2} + i \Omega}}}{100.0 \Omega^{8} e^{1.0 \sqrt{- \Omega^{2} + i \Omega}} - 100.0 \Omega^{8} - 400.0 i \Omega^{7} e^{1.0 \sqrt{- \Omega^{2} + i \Omega}} + 400.0 i \Omega^{7} - 600.0 \Omega^{6} e^{1.0 \sqrt{- \Omega^{2} + i \Omega}} + 600.0 \Omega^{6} + 400.0 i \Omega^{5} e^{1.0 \sqrt{- \Omega^{2} + i \Omega}} - 400.0 i \Omega^{5} + 100.0 \Omega^{4} e^{1.0 \sqrt{- \Omega^{2} + i \Omega}} - 100.0 \Omega^{4}} - \frac{5525760.0 i \Omega e^{0.5 \sqrt{- \Omega^{2} + i \Omega}}}{100.0 \Omega^{8} e^{1.0 \sqrt{- \Omega^{2} + i \Omega}} - 100.0 \Omega^{8} - 400.0 i \Omega^{7} e^{1.0 \sqrt{- \Omega^{2} + i \Omega}} + 400.0 i \Omega^{7} - 600.0 \Omega^{6} e^{1.0 \sqrt{- \Omega^{2} + i \Omega}} + 600.0 \Omega^{6} + 400.0 i \Omega^{5} e^{1.0 \sqrt{- \Omega^{2} + i \Omega}} - 400.0 i \Omega^{5} + 100.0 \Omega^{4} e^{1.0 \sqrt{- \Omega^{2} + i \Omega}} - 100.0 \Omega^{4}} - \frac{66309120.0 e^{0.5 \sqrt{- \Omega^{2} + i \Omega}}}{100.0 \Omega^{8} e^{1.0 \sqrt{- \Omega^{2} + i \Omega}} - 100.0 \Omega^{8} - 400.0 i \Omega^{7} e^{1.0 \sqrt{- \Omega^{2} + i \Omega}} + 400.0 i \Omega^{7} - 600.0 \Omega^{6} e^{1.0 \sqrt{- \Omega^{2} + i \Omega}} + 600.0 \Omega^{6} + 400.0 i \Omega^{5} e^{1.0 \sqrt{- \Omega^{2} + i \Omega}} - 400.0 i \Omega^{5} + 100.0 \Omega^{4} e^{1.0 \sqrt{- \Omega^{2} + i \Omega}} - 100.0 \Omega^{4}} + \frac{66309120.0}{100.0 \Omega^{8} e^{1.0 \sqrt{- \Omega^{2} + i \Omega}} - 100.0 \Omega^{8} - 400.0 i \Omega^{7} e^{1.0 \sqrt{- \Omega^{2} + i \Omega}} + 400.0 i \Omega^{7} - 600.0 \Omega^{6} e^{1.0 \sqrt{- \Omega^{2} + i \Omega}} + 600.0 \Omega^{6} + 400.0 i \Omega^{5} e^{1.0 \sqrt{- \Omega^{2} + i \Omega}} - 400.0 i \Omega^{5} + 100.0 \Omega^{4} e^{1.0 \sqrt{- \Omega^{2} + i \Omega}} - 100.0 \Omega^{4}}\right) e^{\frac{x \sqrt{\Omega \left(- \Omega + 1.0 i\right)}}{2}} + \left(\frac{57560.0 \Omega^{4} e^{0.5 \sqrt{- \Omega^{2} + i \Omega}}}{100.0 \Omega^{8} e^{1.0 \sqrt{- \Omega^{2} + i \Omega}} - 100.0 \Omega^{8} - 400.0 i \Omega^{7} e^{1.0 \sqrt{- \Omega^{2} + i \Omega}} + 400.0 i \Omega^{7} - 600.0 \Omega^{6} e^{1.0 \sqrt{- \Omega^{2} + i \Omega}} + 600.0 \Omega^{6} + 400.0 i \Omega^{5} e^{1.0 \sqrt{- \Omega^{2} + i \Omega}} - 400.0 i \Omega^{5} + 100.0 \Omega^{4} e^{1.0 \sqrt{- \Omega^{2} + i \Omega}} - 100.0 \Omega^{4}} - \frac{115120.0 i \Omega^{3} e^{0.5 \sqrt{- \Omega^{2} + i \Omega}}}{100.0 \Omega^{8} e^{1.0 \sqrt{- \Omega^{2} + i \Omega}} - 100.0 \Omega^{8} - 400.0 i \Omega^{7} e^{1.0 \sqrt{- \Omega^{2} + i \Omega}} + 400.0 i \Omega^{7} - 600.0 \Omega^{6} e^{1.0 \sqrt{- \Omega^{2} + i \Omega}} + 600.0 \Omega^{6} + 400.0 i \Omega^{5} e^{1.0 \sqrt{- \Omega^{2} + i \Omega}} - 400.0 i \Omega^{5} + 100.0 \Omega^{4} e^{1.0 \sqrt{- \Omega^{2} + i \Omega}} - 100.0 \Omega^{4}} - \frac{5583320.0 \Omega^{2} e^{0.5 \sqrt{- \Omega^{2} + i \Omega}}}{100.0 \Omega^{8} e^{1.0 \sqrt{- \Omega^{2} + i \Omega}} - 100.0 \Omega^{8} - 400.0 i \Omega^{7} e^{1.0 \sqrt{- \Omega^{2} + i \Omega}} + 400.0 i \Omega^{7} - 600.0 \Omega^{6} e^{1.0 \sqrt{- \Omega^{2} + i \Omega}} + 600.0 \Omega^{6} + 400.0 i \Omega^{5} e^{1.0 \sqrt{- \Omega^{2} + i \Omega}} - 400.0 i \Omega^{5} + 100.0 \Omega^{4} e^{1.0 \sqrt{- \Omega^{2} + i \Omega}} - 100.0 \Omega^{4}} + \frac{5525760.0 i \Omega e^{0.5 \sqrt{- \Omega^{2} + i \Omega}}}{100.0 \Omega^{8} e^{1.0 \sqrt{- \Omega^{2} + i \Omega}} - 100.0 \Omega^{8} - 400.0 i \Omega^{7} e^{1.0 \sqrt{- \Omega^{2} + i \Omega}} + 400.0 i \Omega^{7} - 600.0 \Omega^{6} e^{1.0 \sqrt{- \Omega^{2} + i \Omega}} + 600.0 \Omega^{6} + 400.0 i \Omega^{5} e^{1.0 \sqrt{- \Omega^{2} + i \Omega}} - 400.0 i \Omega^{5} + 100.0 \Omega^{4} e^{1.0 \sqrt{- \Omega^{2} + i \Omega}} - 100.0 \Omega^{4}} + \frac{66309120.0 e^{0.5 \sqrt{- \Omega^{2} + i \Omega}}}{100.0 \Omega^{8} e^{1.0 \sqrt{- \Omega^{2} + i \Omega}} - 100.0 \Omega^{8} - 400.0 i \Omega^{7} e^{1.0 \sqrt{- \Omega^{2} + i \Omega}} + 400.0 i \Omega^{7} - 600.0 \Omega^{6} e^{1.0 \sqrt{- \Omega^{2} + i \Omega}} + 600.0 \Omega^{6} + 400.0 i \Omega^{5} e^{1.0 \sqrt{- \Omega^{2} + i \Omega}} - 400.0 i \Omega^{5} + 100.0 \Omega^{4} e^{1.0 \sqrt{- \Omega^{2} + i \Omega}} - 100.0 \Omega^{4}} - \frac{66309120.0 e^{1.0 \sqrt{- \Omega^{2} + i \Omega}}}{100.0 \Omega^{8} e^{1.0 \sqrt{- \Omega^{2} + i \Omega}} - 100.0 \Omega^{8} - 400.0 i \Omega^{7} e^{1.0 \sqrt{- \Omega^{2} + i \Omega}} + 400.0 i \Omega^{7} - 600.0 \Omega^{6} e^{1.0 \sqrt{- \Omega^{2} + i \Omega}} + 600.0 \Omega^{6} + 400.0 i \Omega^{5} e^{1.0 \sqrt{- \Omega^{2} + i \Omega}} - 400.0 i \Omega^{5} + 100.0 \Omega^{4} e^{1.0 \sqrt{- \Omega^{2} + i \Omega}} - 100.0 \Omega^{4}}\right) e^{- \frac{x \sqrt{\Omega \left(- \Omega + 1.0 i\right)}}{2}} - \frac{1439.0 x^{6}}{\Omega \left(100.0 \Omega - 100.0 i\right)} + \frac{1439.0 x^{5}}{\Omega \left(100.0 \Omega - 100.0 i\right)} + \frac{17268.0 x^{4}}{\Omega^{2} \cdot \left(10.0 \Omega^{2} - 20.0 i \Omega - 10.0\right)} - \frac{5756.0 x^{3}}{\Omega^{2} \cdot \left(5.0 \Omega^{2} - 10.0 i \Omega - 5.0\right)} - \frac{414432.0 x^{2}}{\Omega^{3} \cdot \left(5.0 \Omega^{3} - 15.0 i \Omega^{2} - 15.0 \Omega + 5.0 i\right)} + \frac{138144.0 x}{\Omega^{3} \cdot \left(5.0 \Omega^{3} - 15.0 i \Omega^{2} - 15.0 \Omega + 5.0 i\right)} + \frac{3315456.0}{\Omega^{4} \cdot \left(5.0 \Omega^{4} - 20.0 i \Omega^{3} - 30.0 \Omega^{2} + 20.0 i \Omega + 5.0\right)}\]

Let us animate the reponses to these two loads assuming that the time-depandence is sinusoidal and introducing a numerical value of the load frequency: \(\Omega=5\)

W_sol_sin = sp.im(W_sol*sp.exp(sp.I*Omega*t))

%matplotlib notebook

import numpy as np
import matplotlib.pyplot as plt
from matplotlib.animation import FuncAnimation

W_sol_func = sp.lambdify((x,t,Omega),W_sol_sin)

fig, ax = plt.subplots()
xdata = np.linspace(0,1,100)
line1, = ax.plot([], [])
ax.set_xlim(0, 1)
ax.set_ylim(-0.085,0.085)
frame = 0.1

def update(frame):
    ydata1 = W_sol_func(x=xdata,t=frame,Omega=5)
    ax.set_title("Displacement for t = "+str(np.round(frame,2)))
    line1.set_data(xdata, ydata1)

ani = FuncAnimation(fig, update, frames=np.linspace(0,2*np.pi/5,100),interval = 100)
plt.show()

Now we plot the amplitude-frequency response functions at two locations along the string for the two load shapes

AFRF_sol = sp.Abs(W_sol.subs(x,3/4))
sp.plot(AFRF_sol,(Omega,0.1,50),adaptive=False);

Let us animate the reponses to these two loads assuming that the time-depandence is sinusoidal and introducing a numerical value of the load frequency: \(\Omega=13\)

fig, ax = plt.subplots()
xdata = np.linspace(0,1,100)
line1, = ax.plot([], [])
ax.set_xlim(0, 1)
ax.set_ylim(-0.085,0.085)
frame = 0.1

def update(frame):
    ydata1 = W_sol_func(x=xdata,t=frame,Omega=13)
    ax.set_title("Displacement for t = "+str(np.round(frame,2)))
    line1.set_data(xdata, ydata1)

ani = FuncAnimation(fig, update, frames=np.linspace(0,2*np.pi/5,100),interval = 100)
plt.show()