How to convolution integration(Duhamel Integration) by python?

Question

Hi~ I’m studying about structural dynamics.
I want to make a code about Duhamel Integration which is kind of Convoution Integration.

If the initial conditions are y(0)=0 and y'(0)=0,
Duhamel Integration is like this.
enter image description here

Using Ti Nspire
I solved this problem with my Ti Npire softwere. The result is like that.
enter image description here

Its response(y) of t=1 is -0.006238

Using python(sympy)
I tried to solve this problem using by Python(Jupyter Notebook).
But I couldn’t solve the problem.

I wrote the code like this.

from sympy import *

t, tau=symbols('t, tau')

m=6938.78
k=379259
wn=sqrt(k/m)
wd=wn*sqrt(1-0.05**2)

eq1=(900*sin(5.3*tau))
eq2=exp(-0.05*wn*(t-tau))
eq3=sin(wd*(t-tau))

y0=1/(m*wd)*integrate(eq1*eq2*eq3,(tau,0,t))
y0

But I couldn’t get the result.
enter image description here

Is there other way to solve this problem?

enter image description here

Asked By: SM KIM

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Source

Answer 1

Use the unevaluated Integral and then substitute in a value for t and use the doit method:

...
>>> y0=1/(m*wd)*Integral(eq1*eq2*eq3,(tau,0,t))
>>> y0.subs(t,1).doit()
-0.00623772329557205

To see the symbolic result before substituting t=1 you need to help SymPy a little by expanding your integrand; I also evaluate floats to show only 3 digits from sake of simplicity:

>>> integrate((eq1*eq2*eq3).expand(),(tau,0,t)).simplify().replace(
... lambda x:x.is_Float, lambda x: x.n(3))
245.0*sin(5.3*t) - 36.1*cos(5.3*t) - 174.0*exp(-0.37*t)*sin(7.38*t) + 36.1*exp(-0.37*t)*cos(7.38*t)

Answered By: smichr

How to convolution integration(Duhamel Integration) by python?

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