CIE4140_Lecture_13_Part_1_Python#
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,real=True)
x,t = sp.symbols('x t',real=True)
nd, c, Omega= sp.symbols('nd c Omega',real=True)
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(x)*sp.exp(sp.I*Omega*t)
display(EQM)
\[\displaystyle - c^{2} \frac{\partial^{2}}{\partial x^{2}} w{\left(x,t \right)} + 2 nd \frac{\partial}{\partial t} w{\left(x,t \right)} - \operatorname{q_{1}}{\left(x \right)} e^{i \Omega t} + \frac{\partial^{2}}{\partial t^{2}} w{\left(x,t \right)}\]
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)
\[\displaystyle - \Omega^{2} W{\left(x \right)} e^{i \Omega t} + 2 i \Omega nd W{\left(x \right)} e^{i \Omega t} - c^{2} e^{i \Omega t} \frac{d^{2}}{d x^{2}} W{\left(x \right)} - \operatorname{q_{1}}{\left(x \right)} e^{i \Omega t}\]
\(e^{i \Omega t}\) is not dropped, so can dropped manually, not necessary:
display(EQM_fr.subs(sp.exp(sp.I * Omega * t),1))
\[\displaystyle - \Omega^{2} W{\left(x \right)} + 2 i \Omega nd W{\left(x \right)} - c^{2} \frac{d^{2}}{d x^{2}} W{\left(x \right)} - \operatorname{q_{1}}{\left(x \right)}\]
Let us find the soluton to this equation that satisfies the boundary conditions of a fixed-fixed string
L = sp.symbols('L')
W_sol = sp.dsolve(EQM_fr,W(x),ics={W(0):0,W(L):0}).rhs
display(W_sol)
\[\displaystyle \left(- \frac{e^{\frac{2 L \sqrt{- \Omega^{2} + 2 i \Omega nd}}{c}} \int \operatorname{q_{1}}{\left(L \right)} e^{- \frac{L \sqrt{- \Omega^{2} + 2 i \Omega nd}}{c}}\, dL}{2 c \sqrt{- \Omega^{2} + 2 i \Omega nd} e^{\frac{2 L \sqrt{- \Omega^{2} + 2 i \Omega nd}}{c}} - 2 c \sqrt{- \Omega^{2} + 2 i \Omega nd}} + \frac{e^{\frac{2 L \sqrt{- \Omega^{2} + 2 i \Omega nd}}{c}} \int\limits^{0} \operatorname{q_{1}}{\left(x \right)} e^{- \frac{x \sqrt{- \Omega^{2} + 2 i \Omega nd}}{c}}\, dx}{2 c \sqrt{- \Omega^{2} + 2 i \Omega nd} e^{\frac{2 L \sqrt{- \Omega^{2} + 2 i \Omega nd}}{c}} - 2 c \sqrt{- \Omega^{2} + 2 i \Omega nd}} - \frac{e^{\frac{2 L \sqrt{- \Omega^{2} + 2 i \Omega nd}}{c}} \int\limits^{0} \operatorname{q_{1}}{\left(x \right)} e^{\frac{x \sqrt{- \Omega^{2} + 2 i \Omega nd}}{c}}\, dx}{2 c \sqrt{- \Omega^{2} + 2 i \Omega nd} e^{\frac{2 L \sqrt{- \Omega^{2} + 2 i \Omega nd}}{c}} - 2 c \sqrt{- \Omega^{2} + 2 i \Omega nd}} + \frac{\int \operatorname{q_{1}}{\left(L \right)} e^{\frac{L \sqrt{- \Omega^{2} + 2 i \Omega nd}}{c}}\, dL}{2 c \sqrt{- \Omega^{2} + 2 i \Omega nd} e^{\frac{2 L \sqrt{- \Omega^{2} + 2 i \Omega nd}}{c}} - 2 c \sqrt{- \Omega^{2} + 2 i \Omega nd}}\right) e^{- \frac{x \sqrt{\Omega \left(- \Omega + 2 i nd\right)}}{c}} + \left(\frac{e^{\frac{2 L \sqrt{- \Omega^{2} + 2 i \Omega nd}}{c}} \int \operatorname{q_{1}}{\left(L \right)} e^{- \frac{L \sqrt{- \Omega^{2} + 2 i \Omega nd}}{c}}\, dL}{2 c \sqrt{- \Omega^{2} + 2 i \Omega nd} e^{\frac{2 L \sqrt{- \Omega^{2} + 2 i \Omega nd}}{c}} - 2 c \sqrt{- \Omega^{2} + 2 i \Omega nd}} - \frac{\int \operatorname{q_{1}}{\left(L \right)} e^{\frac{L \sqrt{- \Omega^{2} + 2 i \Omega nd}}{c}}\, dL}{2 c \sqrt{- \Omega^{2} + 2 i \Omega nd} e^{\frac{2 L \sqrt{- \Omega^{2} + 2 i \Omega nd}}{c}} - 2 c \sqrt{- \Omega^{2} + 2 i \Omega nd}} - \frac{\int\limits^{0} \operatorname{q_{1}}{\left(x \right)} e^{- \frac{x \sqrt{- \Omega^{2} + 2 i \Omega nd}}{c}}\, dx}{2 c \sqrt{- \Omega^{2} + 2 i \Omega nd} e^{\frac{2 L \sqrt{- \Omega^{2} + 2 i \Omega nd}}{c}} - 2 c \sqrt{- \Omega^{2} + 2 i \Omega nd}} + \frac{\int\limits^{0} \operatorname{q_{1}}{\left(x \right)} e^{\frac{x \sqrt{- \Omega^{2} + 2 i \Omega nd}}{c}}\, dx}{2 c \sqrt{- \Omega^{2} + 2 i \Omega nd} e^{\frac{2 L \sqrt{- \Omega^{2} + 2 i \Omega nd}}{c}} - 2 c \sqrt{- \Omega^{2} + 2 i \Omega nd}}\right) e^{\frac{x \sqrt{\Omega \left(- \Omega + 2 i nd\right)}}{c}} - \frac{e^{\frac{x \sqrt{\Omega \left(- \Omega + 2 i nd\right)}}{c}} \int \operatorname{q_{1}}{\left(x \right)} e^{- \frac{x \sqrt{\Omega \left(- \Omega + 2 i nd\right)}}{c}}\, dx}{2 c \sqrt{\Omega \left(- \Omega + 2 i nd\right)}} + \frac{e^{- \frac{x \sqrt{\Omega \left(- \Omega + 2 i nd\right)}}{c}} \int \operatorname{q_{1}}{\left(x \right)} e^{\frac{x \sqrt{\Omega \left(- \Omega + 2 i nd\right)}}{c}}\, dx}{2 c \sqrt{\Omega \left(- \Omega + 2 i nd\right)}}\]
We introduce numerical values for the parameters
W_sol = W_sol.subs([(L,1),(c,2),(nd,1/2)])
display(W_sol)
\[\displaystyle \left(- \frac{e^{\sqrt{- \Omega^{2} + 1.0 i \Omega}} \int\limits^{1} \operatorname{q_{1}}{\left(L \right)} e^{- \frac{L \sqrt{- \Omega^{2} + 1.0 i \Omega}}{2}}\, dL}{4 \sqrt{- \Omega^{2} + 1.0 i \Omega} e^{\sqrt{- \Omega^{2} + 1.0 i \Omega}} - 4 \sqrt{- \Omega^{2} + 1.0 i \Omega}} + \frac{e^{\sqrt{- \Omega^{2} + 1.0 i \Omega}} \int\limits^{0} \operatorname{q_{1}}{\left(x \right)} e^{- \frac{x \sqrt{- \Omega^{2} + 1.0 i \Omega}}{2}}\, dx}{4 \sqrt{- \Omega^{2} + 1.0 i \Omega} e^{\sqrt{- \Omega^{2} + 1.0 i \Omega}} - 4 \sqrt{- \Omega^{2} + 1.0 i \Omega}} - \frac{e^{\sqrt{- \Omega^{2} + 1.0 i \Omega}} \int\limits^{0} \operatorname{q_{1}}{\left(x \right)} e^{\frac{x \sqrt{- \Omega^{2} + 1.0 i \Omega}}{2}}\, dx}{4 \sqrt{- \Omega^{2} + 1.0 i \Omega} e^{\sqrt{- \Omega^{2} + 1.0 i \Omega}} - 4 \sqrt{- \Omega^{2} + 1.0 i \Omega}} + \frac{\int\limits^{1} \operatorname{q_{1}}{\left(L \right)} e^{\frac{L \sqrt{- \Omega^{2} + 1.0 i \Omega}}{2}}\, dL}{4 \sqrt{- \Omega^{2} + 1.0 i \Omega} e^{\sqrt{- \Omega^{2} + 1.0 i \Omega}} - 4 \sqrt{- \Omega^{2} + 1.0 i \Omega}}\right) e^{- \frac{x \sqrt{\Omega \left(- \Omega + 1.0 i\right)}}{2}} + \left(\frac{e^{\sqrt{- \Omega^{2} + 1.0 i \Omega}} \int\limits^{1} \operatorname{q_{1}}{\left(L \right)} e^{- \frac{L \sqrt{- \Omega^{2} + 1.0 i \Omega}}{2}}\, dL}{4 \sqrt{- \Omega^{2} + 1.0 i \Omega} e^{\sqrt{- \Omega^{2} + 1.0 i \Omega}} - 4 \sqrt{- \Omega^{2} + 1.0 i \Omega}} - \frac{\int\limits^{1} \operatorname{q_{1}}{\left(L \right)} e^{\frac{L \sqrt{- \Omega^{2} + 1.0 i \Omega}}{2}}\, dL}{4 \sqrt{- \Omega^{2} + 1.0 i \Omega} e^{\sqrt{- \Omega^{2} + 1.0 i \Omega}} - 4 \sqrt{- \Omega^{2} + 1.0 i \Omega}} - \frac{\int\limits^{0} \operatorname{q_{1}}{\left(x \right)} e^{- \frac{x \sqrt{- \Omega^{2} + 1.0 i \Omega}}{2}}\, dx}{4 \sqrt{- \Omega^{2} + 1.0 i \Omega} e^{\sqrt{- \Omega^{2} + 1.0 i \Omega}} - 4 \sqrt{- \Omega^{2} + 1.0 i \Omega}} + \frac{\int\limits^{0} \operatorname{q_{1}}{\left(x \right)} e^{\frac{x \sqrt{- \Omega^{2} + 1.0 i \Omega}}{2}}\, dx}{4 \sqrt{- \Omega^{2} + 1.0 i \Omega} e^{\sqrt{- \Omega^{2} + 1.0 i \Omega}} - 4 \sqrt{- \Omega^{2} + 1.0 i \Omega}}\right) e^{\frac{x \sqrt{\Omega \left(- \Omega + 1.0 i\right)}}{2}} - \frac{e^{\frac{x \sqrt{\Omega \left(- \Omega + 1.0 i\right)}}{2}} \int \operatorname{q_{1}}{\left(x \right)} e^{- \frac{x \sqrt{\Omega \left(- \Omega + 1.0 i\right)}}{2}}\, dx}{4 \sqrt{\Omega \left(- \Omega + 1.0 i\right)}} + \frac{e^{- \frac{x \sqrt{\Omega \left(- \Omega + 1.0 i\right)}}{2}} \int \operatorname{q_{1}}{\left(x \right)} e^{\frac{x \sqrt{\Omega \left(- \Omega + 1.0 i\right)}}{2}}\, dx}{4 \sqrt{\Omega \left(- \Omega + 1.0 i\right)}}\]
Let us consider two different shapes of the exernal force:
q1_second_normal_mode = sp.sin(2 * sp.pi * x / 1)
q1_arbitrary = 14.39 * x **5 * (x - 1)
q1_constant = 1
p0 = sp.plotting.plot(q1_second_normal_mode, (x, 0, 1),label='$q1_{second\ normal\ mode}$' ,legend=True,show=False)
p1 = sp.plotting.plot(q1_arbitrary , (x, 0, 1),label='$q1_{arbitrary}$' ,show=False)
p2 = sp.plotting.plot(q1_constant , (x, 0, 1),label='$q1_{constant}$' ,show=False)
p0.append(p1[0])
p0.append(p2[0])
p0.show()
W_to_second_normal_mode = W_sol.subs(q1(x),q1_second_normal_mode).subs(q1(L),q1_second_normal_mode.subs(x,L)).subs(q1(0),q1_second_normal_mode.subs(x,0))
display(W_to_second_normal_mode)
W_to_arbitrary = W_sol.subs(q1(x),q1_arbitrary).subs(q1(L),q1_second_normal_mode.subs(x,L)).subs(q1(0),q1_second_normal_mode.subs(x,0))
display(W_to_arbitrary)
W_to_constant = W_sol.subs(q1(x),q1_constant).subs(q1(L),q1_second_normal_mode.subs(x,L)).subs(q1(0),q1_second_normal_mode.subs(x,0))
display(W_to_constant)
\[\displaystyle \left(- \frac{e^{\sqrt{- \Omega^{2} + 1.0 i \Omega}} \int\limits^{1} e^{- \frac{L \sqrt{- \Omega^{2} + 1.0 i \Omega}}{2}} \sin{\left(2 \pi L \right)}\, dL}{4 \sqrt{- \Omega^{2} + 1.0 i \Omega} e^{\sqrt{- \Omega^{2} + 1.0 i \Omega}} - 4 \sqrt{- \Omega^{2} + 1.0 i \Omega}} + \frac{e^{\sqrt{- \Omega^{2} + 1.0 i \Omega}} \int\limits^{0} e^{- \frac{x \sqrt{- \Omega^{2} + 1.0 i \Omega}}{2}} \sin{\left(2 \pi x \right)}\, dx}{4 \sqrt{- \Omega^{2} + 1.0 i \Omega} e^{\sqrt{- \Omega^{2} + 1.0 i \Omega}} - 4 \sqrt{- \Omega^{2} + 1.0 i \Omega}} - \frac{e^{\sqrt{- \Omega^{2} + 1.0 i \Omega}} \int\limits^{0} e^{\frac{x \sqrt{- \Omega^{2} + 1.0 i \Omega}}{2}} \sin{\left(2 \pi x \right)}\, dx}{4 \sqrt{- \Omega^{2} + 1.0 i \Omega} e^{\sqrt{- \Omega^{2} + 1.0 i \Omega}} - 4 \sqrt{- \Omega^{2} + 1.0 i \Omega}} + \frac{\int\limits^{1} e^{\frac{L \sqrt{- \Omega^{2} + 1.0 i \Omega}}{2}} \sin{\left(2 \pi L \right)}\, dL}{4 \sqrt{- \Omega^{2} + 1.0 i \Omega} e^{\sqrt{- \Omega^{2} + 1.0 i \Omega}} - 4 \sqrt{- \Omega^{2} + 1.0 i \Omega}}\right) e^{- \frac{x \sqrt{\Omega \left(- \Omega + 1.0 i\right)}}{2}} + \left(\frac{e^{\sqrt{- \Omega^{2} + 1.0 i \Omega}} \int\limits^{1} e^{- \frac{L \sqrt{- \Omega^{2} + 1.0 i \Omega}}{2}} \sin{\left(2 \pi L \right)}\, dL}{4 \sqrt{- \Omega^{2} + 1.0 i \Omega} e^{\sqrt{- \Omega^{2} + 1.0 i \Omega}} - 4 \sqrt{- \Omega^{2} + 1.0 i \Omega}} - \frac{\int\limits^{1} e^{\frac{L \sqrt{- \Omega^{2} + 1.0 i \Omega}}{2}} \sin{\left(2 \pi L \right)}\, dL}{4 \sqrt{- \Omega^{2} + 1.0 i \Omega} e^{\sqrt{- \Omega^{2} + 1.0 i \Omega}} - 4 \sqrt{- \Omega^{2} + 1.0 i \Omega}} - \frac{\int\limits^{0} e^{- \frac{x \sqrt{- \Omega^{2} + 1.0 i \Omega}}{2}} \sin{\left(2 \pi x \right)}\, dx}{4 \sqrt{- \Omega^{2} + 1.0 i \Omega} e^{\sqrt{- \Omega^{2} + 1.0 i \Omega}} - 4 \sqrt{- \Omega^{2} + 1.0 i \Omega}} + \frac{\int\limits^{0} e^{\frac{x \sqrt{- \Omega^{2} + 1.0 i \Omega}}{2}} \sin{\left(2 \pi x \right)}\, dx}{4 \sqrt{- \Omega^{2} + 1.0 i \Omega} e^{\sqrt{- \Omega^{2} + 1.0 i \Omega}} - 4 \sqrt{- \Omega^{2} + 1.0 i \Omega}}\right) e^{\frac{x \sqrt{\Omega \left(- \Omega + 1.0 i\right)}}{2}} - \frac{e^{\frac{x \sqrt{\Omega \left(- \Omega + 1.0 i\right)}}{2}} \int e^{- \frac{x \sqrt{\Omega \left(- \Omega + 1.0 i\right)}}{2}} \sin{\left(2 \pi x \right)}\, dx}{4 \sqrt{\Omega \left(- \Omega + 1.0 i\right)}} + \frac{e^{- \frac{x \sqrt{\Omega \left(- \Omega + 1.0 i\right)}}{2}} \int e^{\frac{x \sqrt{\Omega \left(- \Omega + 1.0 i\right)}}{2}} \sin{\left(2 \pi x \right)}\, dx}{4 \sqrt{\Omega \left(- \Omega + 1.0 i\right)}}\]
\[\displaystyle \left(- \frac{e^{\sqrt{- \Omega^{2} + 1.0 i \Omega}} \int\limits^{1} e^{- \frac{L \sqrt{- \Omega^{2} + 1.0 i \Omega}}{2}} \sin{\left(2 \pi L \right)}\, dL}{4 \sqrt{- \Omega^{2} + 1.0 i \Omega} e^{\sqrt{- \Omega^{2} + 1.0 i \Omega}} - 4 \sqrt{- \Omega^{2} + 1.0 i \Omega}} + \frac{e^{\sqrt{- \Omega^{2} + 1.0 i \Omega}} \int\limits^{0} 14.39 x^{5} \left(x - 1\right) e^{- \frac{x \sqrt{- \Omega^{2} + 1.0 i \Omega}}{2}}\, dx}{4 \sqrt{- \Omega^{2} + 1.0 i \Omega} e^{\sqrt{- \Omega^{2} + 1.0 i \Omega}} - 4 \sqrt{- \Omega^{2} + 1.0 i \Omega}} - \frac{e^{\sqrt{- \Omega^{2} + 1.0 i \Omega}} \int\limits^{0} 14.39 x^{5} \left(x - 1\right) e^{\frac{x \sqrt{- \Omega^{2} + 1.0 i \Omega}}{2}}\, dx}{4 \sqrt{- \Omega^{2} + 1.0 i \Omega} e^{\sqrt{- \Omega^{2} + 1.0 i \Omega}} - 4 \sqrt{- \Omega^{2} + 1.0 i \Omega}} + \frac{\int\limits^{1} e^{\frac{L \sqrt{- \Omega^{2} + 1.0 i \Omega}}{2}} \sin{\left(2 \pi L \right)}\, dL}{4 \sqrt{- \Omega^{2} + 1.0 i \Omega} e^{\sqrt{- \Omega^{2} + 1.0 i \Omega}} - 4 \sqrt{- \Omega^{2} + 1.0 i \Omega}}\right) e^{- \frac{x \sqrt{\Omega \left(- \Omega + 1.0 i\right)}}{2}} + \left(\frac{e^{\sqrt{- \Omega^{2} + 1.0 i \Omega}} \int\limits^{1} e^{- \frac{L \sqrt{- \Omega^{2} + 1.0 i \Omega}}{2}} \sin{\left(2 \pi L \right)}\, dL}{4 \sqrt{- \Omega^{2} + 1.0 i \Omega} e^{\sqrt{- \Omega^{2} + 1.0 i \Omega}} - 4 \sqrt{- \Omega^{2} + 1.0 i \Omega}} - \frac{\int\limits^{1} e^{\frac{L \sqrt{- \Omega^{2} + 1.0 i \Omega}}{2}} \sin{\left(2 \pi L \right)}\, dL}{4 \sqrt{- \Omega^{2} + 1.0 i \Omega} e^{\sqrt{- \Omega^{2} + 1.0 i \Omega}} - 4 \sqrt{- \Omega^{2} + 1.0 i \Omega}} - \frac{\int\limits^{0} 14.39 x^{5} \left(x - 1\right) e^{- \frac{x \sqrt{- \Omega^{2} + 1.0 i \Omega}}{2}}\, dx}{4 \sqrt{- \Omega^{2} + 1.0 i \Omega} e^{\sqrt{- \Omega^{2} + 1.0 i \Omega}} - 4 \sqrt{- \Omega^{2} + 1.0 i \Omega}} + \frac{\int\limits^{0} 14.39 x^{5} \left(x - 1\right) e^{\frac{x \sqrt{- \Omega^{2} + 1.0 i \Omega}}{2}}\, dx}{4 \sqrt{- \Omega^{2} + 1.0 i \Omega} e^{\sqrt{- \Omega^{2} + 1.0 i \Omega}} - 4 \sqrt{- \Omega^{2} + 1.0 i \Omega}}\right) e^{\frac{x \sqrt{\Omega \left(- \Omega + 1.0 i\right)}}{2}} - \frac{e^{\frac{x \sqrt{\Omega \left(- \Omega + 1.0 i\right)}}{2}} \int 14.39 x^{5} \left(x - 1\right) e^{- \frac{x \sqrt{\Omega \left(- \Omega + 1.0 i\right)}}{2}}\, dx}{4 \sqrt{\Omega \left(- \Omega + 1.0 i\right)}} + \frac{e^{- \frac{x \sqrt{\Omega \left(- \Omega + 1.0 i\right)}}{2}} \int 14.39 x^{5} \left(x - 1\right) e^{\frac{x \sqrt{\Omega \left(- \Omega + 1.0 i\right)}}{2}}\, dx}{4 \sqrt{\Omega \left(- \Omega + 1.0 i\right)}}\]
\[\displaystyle \left(- \frac{e^{\sqrt{- \Omega^{2} + 1.0 i \Omega}} \int\limits^{1} e^{- \frac{L \sqrt{- \Omega^{2} + 1.0 i \Omega}}{2}} \sin{\left(2 \pi L \right)}\, dL}{4 \sqrt{- \Omega^{2} + 1.0 i \Omega} e^{\sqrt{- \Omega^{2} + 1.0 i \Omega}} - 4 \sqrt{- \Omega^{2} + 1.0 i \Omega}} + \frac{e^{\sqrt{- \Omega^{2} + 1.0 i \Omega}} \int\limits^{0} e^{- \frac{x \sqrt{- \Omega^{2} + 1.0 i \Omega}}{2}}\, dx}{4 \sqrt{- \Omega^{2} + 1.0 i \Omega} e^{\sqrt{- \Omega^{2} + 1.0 i \Omega}} - 4 \sqrt{- \Omega^{2} + 1.0 i \Omega}} - \frac{e^{\sqrt{- \Omega^{2} + 1.0 i \Omega}} \int\limits^{0} e^{\frac{x \sqrt{- \Omega^{2} + 1.0 i \Omega}}{2}}\, dx}{4 \sqrt{- \Omega^{2} + 1.0 i \Omega} e^{\sqrt{- \Omega^{2} + 1.0 i \Omega}} - 4 \sqrt{- \Omega^{2} + 1.0 i \Omega}} + \frac{\int\limits^{1} e^{\frac{L \sqrt{- \Omega^{2} + 1.0 i \Omega}}{2}} \sin{\left(2 \pi L \right)}\, dL}{4 \sqrt{- \Omega^{2} + 1.0 i \Omega} e^{\sqrt{- \Omega^{2} + 1.0 i \Omega}} - 4 \sqrt{- \Omega^{2} + 1.0 i \Omega}}\right) e^{- \frac{x \sqrt{\Omega \left(- \Omega + 1.0 i\right)}}{2}} + \left(\frac{e^{\sqrt{- \Omega^{2} + 1.0 i \Omega}} \int\limits^{1} e^{- \frac{L \sqrt{- \Omega^{2} + 1.0 i \Omega}}{2}} \sin{\left(2 \pi L \right)}\, dL}{4 \sqrt{- \Omega^{2} + 1.0 i \Omega} e^{\sqrt{- \Omega^{2} + 1.0 i \Omega}} - 4 \sqrt{- \Omega^{2} + 1.0 i \Omega}} - \frac{\int\limits^{1} e^{\frac{L \sqrt{- \Omega^{2} + 1.0 i \Omega}}{2}} \sin{\left(2 \pi L \right)}\, dL}{4 \sqrt{- \Omega^{2} + 1.0 i \Omega} e^{\sqrt{- \Omega^{2} + 1.0 i \Omega}} - 4 \sqrt{- \Omega^{2} + 1.0 i \Omega}} - \frac{\int\limits^{0} e^{- \frac{x \sqrt{- \Omega^{2} + 1.0 i \Omega}}{2}}\, dx}{4 \sqrt{- \Omega^{2} + 1.0 i \Omega} e^{\sqrt{- \Omega^{2} + 1.0 i \Omega}} - 4 \sqrt{- \Omega^{2} + 1.0 i \Omega}} + \frac{\int\limits^{0} e^{\frac{x \sqrt{- \Omega^{2} + 1.0 i \Omega}}{2}}\, dx}{4 \sqrt{- \Omega^{2} + 1.0 i \Omega} e^{\sqrt{- \Omega^{2} + 1.0 i \Omega}} - 4 \sqrt{- \Omega^{2} + 1.0 i \Omega}}\right) e^{\frac{x \sqrt{\Omega \left(- \Omega + 1.0 i\right)}}{2}} - \frac{e^{\frac{x \sqrt{\Omega \left(- \Omega + 1.0 i\right)}}{2}} \int e^{- \frac{x \sqrt{\Omega \left(- \Omega + 1.0 i\right)}}{2}}\, dx}{4 \sqrt{\Omega \left(- \Omega + 1.0 i\right)}} + \frac{e^{- \frac{x \sqrt{\Omega \left(- \Omega + 1.0 i\right)}}{2}} \int e^{\frac{x \sqrt{\Omega \left(- \Omega + 1.0 i\right)}}{2}}\, dx}{4 \sqrt{\Omega \left(- \Omega + 1.0 i\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_to_second_normal_mode_sin = sp.im(W_to_second_normal_mode*sp.exp(sp.I*Omega*t)).subs(Omega,5)
W_to_arbitrary_sin = sp.im(W_to_arbitrary*sp.exp(sp.I*Omega*t)).subs(Omega,5)
W_to_constant_sin = sp.im(W_to_constant*sp.exp(sp.I*Omega*t)).subs(Omega,5)
display(W_to_second_normal_mode_sin.subs([(x,0.25),(t,3)]).evalf())
\[\displaystyle 0.829017160665234 \operatorname{re}{\left(\frac{\int\limits^{1} e^{\frac{L \sqrt{-25 + 5.0 i}}{2}} \sin{\left(2 \pi L \right)}\, dL}{- 4 \sqrt{-25 + 5.0 i} + 4 \sqrt{-25 + 5.0 i} e^{\sqrt{-25 + 5.0 i}}}\right)} - 0.0849116508438914 \operatorname{re}{\left(\frac{\int\limits^{0} e^{- \frac{x \sqrt{-25 + 5.0 i}}{2}} \sin{\left(2 \pi x \right)}\, dx}{- 4 \sqrt{-25 + 5.0 i} + 4 \sqrt{-25 + 5.0 i} e^{\sqrt{-25 + 5.0 i}}}\right)} + 0.0849116508438914 \operatorname{re}{\left(\frac{\int\limits^{0} e^{\frac{x \sqrt{-25 + 5.0 i}}{2}} \sin{\left(2 \pi x \right)}\, dx}{- 4 \sqrt{-25 + 5.0 i} + 4 \sqrt{-25 + 5.0 i} e^{\sqrt{-25 + 5.0 i}}}\right)} + 0.162571960039279 \operatorname{re}{\left(\frac{e^{- 0.125 \sqrt{-25 + 5.0 i}} \int\limits^{0.25} e^{\frac{x \sqrt{-25 + 5.0 i}}{2}} \sin{\left(2 \pi x \right)}\, dx}{\sqrt{-25 + 5.0 i}}\right)} - 0.162571960039279 \operatorname{re}{\left(\frac{e^{0.125 \sqrt{-25 + 5.0 i}} \int\limits^{0.25} e^{- \frac{x \sqrt{-25 + 5.0 i}}{2}} \sin{\left(2 \pi x \right)}\, dx}{\sqrt{-25 + 5.0 i}}\right)} - 0.829017160665234 \operatorname{re}{\left(\frac{e^{\sqrt{-25 + 5.0 i}} \int\limits^{1} e^{- \frac{L \sqrt{-25 + 5.0 i}}{2}} \sin{\left(2 \pi L \right)}\, dL}{- 4 \sqrt{-25 + 5.0 i} + 4 \sqrt{-25 + 5.0 i} e^{\sqrt{-25 + 5.0 i}}}\right)} + 0.913928811509126 \operatorname{re}{\left(\frac{e^{\sqrt{-25 + 5.0 i}} \int\limits^{0} e^{- \frac{x \sqrt{-25 + 5.0 i}}{2}} \sin{\left(2 \pi x \right)}\, dx}{- 4 \sqrt{-25 + 5.0 i} + 4 \sqrt{-25 + 5.0 i} e^{\sqrt{-25 + 5.0 i}}}\right)} - 0.913928811509126 \operatorname{re}{\left(\frac{e^{\sqrt{-25 + 5.0 i}} \int\limits^{0} e^{\frac{x \sqrt{-25 + 5.0 i}}{2}} \sin{\left(2 \pi x \right)}\, dx}{- 4 \sqrt{-25 + 5.0 i} + 4 \sqrt{-25 + 5.0 i} e^{\sqrt{-25 + 5.0 i}}}\right)} + 0.842203412678742 \operatorname{im}{\left(\frac{\int\limits^{1} e^{\frac{L \sqrt{-25 + 5.0 i}}{2}} \sin{\left(2 \pi L \right)}\, dL}{- 4 \sqrt{-25 + 5.0 i} + 4 \sqrt{-25 + 5.0 i} e^{\sqrt{-25 + 5.0 i}}}\right)} + 1.06077451419394 \operatorname{im}{\left(\frac{\int\limits^{0} e^{- \frac{x \sqrt{-25 + 5.0 i}}{2}} \sin{\left(2 \pi x \right)}\, dx}{- 4 \sqrt{-25 + 5.0 i} + 4 \sqrt{-25 + 5.0 i} e^{\sqrt{-25 + 5.0 i}}}\right)} - 1.06077451419394 \operatorname{im}{\left(\frac{\int\limits^{0} e^{\frac{x \sqrt{-25 + 5.0 i}}{2}} \sin{\left(2 \pi x \right)}\, dx}{- 4 \sqrt{-25 + 5.0 i} + 4 \sqrt{-25 + 5.0 i} e^{\sqrt{-25 + 5.0 i}}}\right)} - 0.189921978214705 \operatorname{im}{\left(\frac{e^{- 0.125 \sqrt{-25 + 5.0 i}} \int\limits^{0.25} e^{\frac{x \sqrt{-25 + 5.0 i}}{2}} \sin{\left(2 \pi x \right)}\, dx}{\sqrt{-25 + 5.0 i}}\right)} + 0.189921978214705 \operatorname{im}{\left(\frac{e^{0.125 \sqrt{-25 + 5.0 i}} \int\limits^{0.25} e^{- \frac{x \sqrt{-25 + 5.0 i}}{2}} \sin{\left(2 \pi x \right)}\, dx}{\sqrt{-25 + 5.0 i}}\right)} - 0.842203412678742 \operatorname{im}{\left(\frac{e^{\sqrt{-25 + 5.0 i}} \int\limits^{1} e^{- \frac{L \sqrt{-25 + 5.0 i}}{2}} \sin{\left(2 \pi L \right)}\, dL}{- 4 \sqrt{-25 + 5.0 i} + 4 \sqrt{-25 + 5.0 i} e^{\sqrt{-25 + 5.0 i}}}\right)} - 0.218571101515199 \operatorname{im}{\left(\frac{e^{\sqrt{-25 + 5.0 i}} \int\limits^{0} e^{- \frac{x \sqrt{-25 + 5.0 i}}{2}} \sin{\left(2 \pi x \right)}\, dx}{- 4 \sqrt{-25 + 5.0 i} + 4 \sqrt{-25 + 5.0 i} e^{\sqrt{-25 + 5.0 i}}}\right)} + 0.218571101515199 \operatorname{im}{\left(\frac{e^{\sqrt{-25 + 5.0 i}} \int\limits^{0} e^{\frac{x \sqrt{-25 + 5.0 i}}{2}} \sin{\left(2 \pi x \right)}\, dx}{- 4 \sqrt{-25 + 5.0 i} + 4 \sqrt{-25 + 5.0 i} e^{\sqrt{-25 + 5.0 i}}}\right)}\]
W_to_second_normal_mode_sin_func = sp.lambdify((x,t),W_to_second_normal_mode_sin)
W_to_arbitrary_sin_func = sp.lambdify((x,t),W_to_arbitrary_sin)
W_to_constant_sin_func = sp.lambdify((x,t),W_to_constant_sin)
---------------------------------------------------------------------------
NotImplementedError Traceback (most recent call last)
~\AppData\Local\Temp\ipykernel_2968\3484483245.py in <module>
----> 1 W_to_second_normal_mode_sin_func = sp.lambdify((x,t),W_to_second_normal_mode_sin)
2 W_to_arbitrary_sin_func = sp.lambdify((x,t),W_to_arbitrary_sin)
3 W_to_constant_sin_func = sp.lambdify((x,t),W_to_constant_sin)
~\Anaconda3\lib\site-packages\sympy\utilities\lambdify.py in lambdify(args, expr, modules, printer, use_imps, dummify, cse)
873 else:
874 cses, _expr = (), expr
--> 875 funcstr = funcprinter.doprint(funcname, iterable_args, _expr, cses=cses)
876
877 # Collect the module imports from the code printers.
~\Anaconda3\lib\site-packages\sympy\utilities\lambdify.py in doprint(self, funcname, args, expr, cses)
1150 funcbody.append('{} = {}'.format(s, self._exprrepr(e)))
1151
-> 1152 str_expr = _recursive_to_string(self._exprrepr, expr)
1153
1154
~\Anaconda3\lib\site-packages\sympy\utilities\lambdify.py in _recursive_to_string(doprint, arg)
954
955 if isinstance(arg, (Basic, MatrixOperations)):
--> 956 return doprint(arg)
957 elif iterable(arg):
958 if isinstance(arg, list):
~\Anaconda3\lib\site-packages\sympy\printing\codeprinter.py in doprint(self, expr, assign_to)
148 self._number_symbols = set() # type: tSet[tTuple[Expr, Float]]
149
--> 150 lines = self._print(expr).splitlines()
151
152 # format the output
~\Anaconda3\lib\site-packages\sympy\printing\printer.py in _print(self, expr, **kwargs)
329 printmethod = getattr(self, printmethodname, None)
330 if printmethod is not None:
--> 331 return printmethod(expr, **kwargs)
332 # Unknown object, fall back to the emptyPrinter.
333 return self.emptyPrinter(expr)
~\Anaconda3\lib\site-packages\sympy\printing\str.py in _print_Add(self, expr, order)
56 l = []
57 for term in terms:
---> 58 t = self._print(term)
59 if t.startswith('-'):
60 sign = "-"
~\Anaconda3\lib\site-packages\sympy\printing\printer.py in _print(self, expr, **kwargs)
329 printmethod = getattr(self, printmethodname, None)
330 if printmethod is not None:
--> 331 return printmethod(expr, **kwargs)
332 # Unknown object, fall back to the emptyPrinter.
333 return self.emptyPrinter(expr)
~\Anaconda3\lib\site-packages\sympy\printing\codeprinter.py in _print_Mul(self, expr)
543 a_str = [self.parenthesize(a[0], 0.5*(PRECEDENCE["Pow"]+PRECEDENCE["Mul"]))]
544 else:
--> 545 a_str = [self.parenthesize(x, prec) for x in a]
546 b_str = [self.parenthesize(x, prec) for x in b]
547
~\Anaconda3\lib\site-packages\sympy\printing\codeprinter.py in <listcomp>(.0)
543 a_str = [self.parenthesize(a[0], 0.5*(PRECEDENCE["Pow"]+PRECEDENCE["Mul"]))]
544 else:
--> 545 a_str = [self.parenthesize(x, prec) for x in a]
546 b_str = [self.parenthesize(x, prec) for x in b]
547
~\Anaconda3\lib\site-packages\sympy\printing\str.py in parenthesize(self, item, level, strict)
35 def parenthesize(self, item, level, strict=False):
36 if (precedence(item) < level) or ((not strict) and precedence(item) <= level):
---> 37 return "(%s)" % self._print(item)
38 else:
39 return self._print(item)
~\Anaconda3\lib\site-packages\sympy\printing\printer.py in _print(self, expr, **kwargs)
329 printmethod = getattr(self, printmethodname, None)
330 if printmethod is not None:
--> 331 return printmethod(expr, **kwargs)
332 # Unknown object, fall back to the emptyPrinter.
333 return self.emptyPrinter(expr)
~\Anaconda3\lib\site-packages\sympy\printing\str.py in _print_Add(self, expr, order)
56 l = []
57 for term in terms:
---> 58 t = self._print(term)
59 if t.startswith('-'):
60 sign = "-"
~\Anaconda3\lib\site-packages\sympy\printing\printer.py in _print(self, expr, **kwargs)
329 printmethod = getattr(self, printmethodname, None)
330 if printmethod is not None:
--> 331 return printmethod(expr, **kwargs)
332 # Unknown object, fall back to the emptyPrinter.
333 return self.emptyPrinter(expr)
~\Anaconda3\lib\site-packages\sympy\printing\codeprinter.py in _print_Mul(self, expr)
543 a_str = [self.parenthesize(a[0], 0.5*(PRECEDENCE["Pow"]+PRECEDENCE["Mul"]))]
544 else:
--> 545 a_str = [self.parenthesize(x, prec) for x in a]
546 b_str = [self.parenthesize(x, prec) for x in b]
547
~\Anaconda3\lib\site-packages\sympy\printing\codeprinter.py in <listcomp>(.0)
543 a_str = [self.parenthesize(a[0], 0.5*(PRECEDENCE["Pow"]+PRECEDENCE["Mul"]))]
544 else:
--> 545 a_str = [self.parenthesize(x, prec) for x in a]
546 b_str = [self.parenthesize(x, prec) for x in b]
547
~\Anaconda3\lib\site-packages\sympy\printing\str.py in parenthesize(self, item, level, strict)
35 def parenthesize(self, item, level, strict=False):
36 if (precedence(item) < level) or ((not strict) and precedence(item) <= level):
---> 37 return "(%s)" % self._print(item)
38 else:
39 return self._print(item)
~\Anaconda3\lib\site-packages\sympy\printing\printer.py in _print(self, expr, **kwargs)
329 printmethod = getattr(self, printmethodname, None)
330 if printmethod is not None:
--> 331 return printmethod(expr, **kwargs)
332 # Unknown object, fall back to the emptyPrinter.
333 return self.emptyPrinter(expr)
~\Anaconda3\lib\site-packages\sympy\printing\str.py in _print_Add(self, expr, order)
56 l = []
57 for term in terms:
---> 58 t = self._print(term)
59 if t.startswith('-'):
60 sign = "-"
~\Anaconda3\lib\site-packages\sympy\printing\printer.py in _print(self, expr, **kwargs)
329 printmethod = getattr(self, printmethodname, None)
330 if printmethod is not None:
--> 331 return printmethod(expr, **kwargs)
332 # Unknown object, fall back to the emptyPrinter.
333 return self.emptyPrinter(expr)
~\Anaconda3\lib\site-packages\sympy\printing\codeprinter.py in _print_Mul(self, expr)
541 # an operator precedence between multiplication and exponentiation,
542 # so we use this to compute a weight.
--> 543 a_str = [self.parenthesize(a[0], 0.5*(PRECEDENCE["Pow"]+PRECEDENCE["Mul"]))]
544 else:
545 a_str = [self.parenthesize(x, prec) for x in a]
~\Anaconda3\lib\site-packages\sympy\printing\str.py in parenthesize(self, item, level, strict)
37 return "(%s)" % self._print(item)
38 else:
---> 39 return self._print(item)
40
41 def stringify(self, args, sep, level=0):
~\Anaconda3\lib\site-packages\sympy\printing\printer.py in _print(self, expr, **kwargs)
329 printmethod = getattr(self, printmethodname, None)
330 if printmethod is not None:
--> 331 return printmethod(expr, **kwargs)
332 # Unknown object, fall back to the emptyPrinter.
333 return self.emptyPrinter(expr)
~\Anaconda3\lib\site-packages\sympy\printing\numpy.py in _print_re(self, expr)
229
230 def _print_re(self, expr):
--> 231 return "%s(%s)" % (self._module_format(self._module + '.real'), self._print(expr.args[0]))
232
233 def _print_sinc(self, expr):
~\Anaconda3\lib\site-packages\sympy\printing\printer.py in _print(self, expr, **kwargs)
329 printmethod = getattr(self, printmethodname, None)
330 if printmethod is not None:
--> 331 return printmethod(expr, **kwargs)
332 # Unknown object, fall back to the emptyPrinter.
333 return self.emptyPrinter(expr)
~\Anaconda3\lib\site-packages\sympy\printing\codeprinter.py in _print_Mul(self, expr)
543 a_str = [self.parenthesize(a[0], 0.5*(PRECEDENCE["Pow"]+PRECEDENCE["Mul"]))]
544 else:
--> 545 a_str = [self.parenthesize(x, prec) for x in a]
546 b_str = [self.parenthesize(x, prec) for x in b]
547
~\Anaconda3\lib\site-packages\sympy\printing\codeprinter.py in <listcomp>(.0)
543 a_str = [self.parenthesize(a[0], 0.5*(PRECEDENCE["Pow"]+PRECEDENCE["Mul"]))]
544 else:
--> 545 a_str = [self.parenthesize(x, prec) for x in a]
546 b_str = [self.parenthesize(x, prec) for x in b]
547
~\Anaconda3\lib\site-packages\sympy\printing\str.py in parenthesize(self, item, level, strict)
37 return "(%s)" % self._print(item)
38 else:
---> 39 return self._print(item)
40
41 def stringify(self, args, sep, level=0):
~\Anaconda3\lib\site-packages\sympy\printing\printer.py in _print(self, expr, **kwargs)
329 printmethod = getattr(self, printmethodname, None)
330 if printmethod is not None:
--> 331 return printmethod(expr, **kwargs)
332 # Unknown object, fall back to the emptyPrinter.
333 return self.emptyPrinter(expr)
~\Anaconda3\lib\site-packages\sympy\printing\numpy.py in _print_Integral(self, e)
452
453 def _print_Integral(self, e):
--> 454 integration_vars, limits = _unpack_integral_limits(e)
455
456 if len(limits) == 1:
~\Anaconda3\lib\site-packages\sympy\printing\pycode.py in _unpack_integral_limits(integral_expr)
546 integration_var, lower_limit, upper_limit = integration_range
547 else:
--> 548 raise NotImplementedError("Only definite integrals are supported")
549 integration_vars.append(integration_var)
550 limits.append((lower_limit, upper_limit))
NotImplementedError: Only definite integrals are supported
Integral cannot be evaluated numerically, see CIE4140_Lecture_13_Part_1_Python_seperate_q
Now we plot the amplitude-frequency response functions at two locations along the string for the two load shapes
AFRF_second_normal_mode = sp.Abs(W_to_second_normal_mode.subs(x,3/4))
display(AFRF_second_normal_mode.evalf())
\[\displaystyle \left|{\left(- \frac{e^{\sqrt{- \Omega^{2} + 1.0 i \Omega}} \int\limits^{1} e^{- \frac{L \sqrt{- \Omega^{2} + 1.0 i \Omega}}{2}} \sin{\left(2 \pi L \right)}\, dL}{4 \sqrt{- \Omega^{2} + 1.0 i \Omega} e^{\sqrt{- \Omega^{2} + 1.0 i \Omega}} - 4 \sqrt{- \Omega^{2} + 1.0 i \Omega}} + \frac{e^{\sqrt{- \Omega^{2} + 1.0 i \Omega}} \int\limits^{0} e^{- \frac{x \sqrt{- \Omega^{2} + 1.0 i \Omega}}{2}} \sin{\left(2 \pi x \right)}\, dx}{4 \sqrt{- \Omega^{2} + 1.0 i \Omega} e^{\sqrt{- \Omega^{2} + 1.0 i \Omega}} - 4 \sqrt{- \Omega^{2} + 1.0 i \Omega}} - \frac{e^{\sqrt{- \Omega^{2} + 1.0 i \Omega}} \int\limits^{0} e^{\frac{x \sqrt{- \Omega^{2} + 1.0 i \Omega}}{2}} \sin{\left(2 \pi x \right)}\, dx}{4 \sqrt{- \Omega^{2} + 1.0 i \Omega} e^{\sqrt{- \Omega^{2} + 1.0 i \Omega}} - 4 \sqrt{- \Omega^{2} + 1.0 i \Omega}} + \frac{\int\limits^{1} e^{\frac{L \sqrt{- \Omega^{2} + 1.0 i \Omega}}{2}} \sin{\left(2 \pi L \right)}\, dL}{4 \sqrt{- \Omega^{2} + 1.0 i \Omega} e^{\sqrt{- \Omega^{2} + 1.0 i \Omega}} - 4 \sqrt{- \Omega^{2} + 1.0 i \Omega}}\right) e^{- 0.375 \sqrt{\Omega \left(- \Omega + 1.0 i\right)}} + \left(\frac{e^{\sqrt{- \Omega^{2} + 1.0 i \Omega}} \int\limits^{1} e^{- \frac{L \sqrt{- \Omega^{2} + 1.0 i \Omega}}{2}} \sin{\left(2 \pi L \right)}\, dL}{4 \sqrt{- \Omega^{2} + 1.0 i \Omega} e^{\sqrt{- \Omega^{2} + 1.0 i \Omega}} - 4 \sqrt{- \Omega^{2} + 1.0 i \Omega}} - \frac{\int\limits^{1} e^{\frac{L \sqrt{- \Omega^{2} + 1.0 i \Omega}}{2}} \sin{\left(2 \pi L \right)}\, dL}{4 \sqrt{- \Omega^{2} + 1.0 i \Omega} e^{\sqrt{- \Omega^{2} + 1.0 i \Omega}} - 4 \sqrt{- \Omega^{2} + 1.0 i \Omega}} - \frac{\int\limits^{0} e^{- \frac{x \sqrt{- \Omega^{2} + 1.0 i \Omega}}{2}} \sin{\left(2 \pi x \right)}\, dx}{4 \sqrt{- \Omega^{2} + 1.0 i \Omega} e^{\sqrt{- \Omega^{2} + 1.0 i \Omega}} - 4 \sqrt{- \Omega^{2} + 1.0 i \Omega}} + \frac{\int\limits^{0} e^{\frac{x \sqrt{- \Omega^{2} + 1.0 i \Omega}}{2}} \sin{\left(2 \pi x \right)}\, dx}{4 \sqrt{- \Omega^{2} + 1.0 i \Omega} e^{\sqrt{- \Omega^{2} + 1.0 i \Omega}} - 4 \sqrt{- \Omega^{2} + 1.0 i \Omega}}\right) e^{0.375 \sqrt{\Omega \left(- \Omega + 1.0 i\right)}} - \frac{e^{0.375 \sqrt{\Omega \left(- \Omega + 1.0 i\right)}} \int\limits^{0.75} e^{- \frac{x \sqrt{\Omega \left(- \Omega + 1.0 i\right)}}{2}} \sin{\left(2 \pi x \right)}\, dx}{4 \sqrt{\Omega \left(- \Omega + 1.0 i\right)}} + \frac{e^{- 0.375 \sqrt{\Omega \left(- \Omega + 1.0 i\right)}} \int\limits^{0.75} e^{\frac{x \sqrt{\Omega \left(- \Omega + 1.0 i\right)}}{2}} \sin{\left(2 \pi x \right)}\, dx}{4 \sqrt{\Omega \left(- \Omega + 1.0 i\right)}}}\right|\]
AFRF can neither be evaluated