Can't calculate g(x) as Derivative(f(x))

Question

Hello!

Problem: I can calculate xp(t) and xh(t) for any given t value, but not for ġ(t).

Context:

I'm calculating a general solution for a vibration system, using the Duhamel's integral. The result is a sum of 2 functions.

xg(t) = xp(t) + xh(t)

I'm also using Jet Brain's Datalore to quickly evaluate the results.

Lib:

import numpy as np
from numpy import pi, linspace
import sympy as smp
from sympy import sin, cos, tan, exp, symbols, Derivative, integrate, Integral, Function
from sympy.solvers.solveset import linsolve
import matplotlib.pyplot as plt

My constants:

k = 100*10**3   #[N/m] - Rigidity
m = 10000       #[Kg] - Mass
c = 20*10**3    #[N.s/m] - Damping
f0 = 5000       #[N] - Initial force
x0 = -0.05      #[m] - Initial position
v0 = 0.2        #[m/s] - Initial velocity

Main variables:

wn = float(np.sqrt(k/m))   #[rad/s]
zeta = float(c/(2*m*wn))    
wd = wn*np.sqrt(1-zeta**2)
t0 = 2*pi/wn/2             #[s]
t1 = 5*2*pi/wn/2           #[s]
sloap = f0/t0

What I can do:

I can create a function xh(t), insert a t value and obtain a valid result (using my ti-nspire cx CAS to check):

A, phi = symbols('A phi', real=True)
t = symbols('t', real=True, positive=True)

def xh(t):
    return A * exp(-zeta*wn*t)*sin(wd*t+phi)

I can do the same for xp(t):

x = symbols('x', real=True, positive=True)
xp_integral = (f0/t0) * x * exp(-zeta*wn*(t-x)) * sin(wd*(t-x))

def xp(t):
    return 1/(m*wd) * integrate(xp_integral, (x, 0, t)).simplify()

And even for xg(t):

def xg(t):
    return xh(t) + xp(t)

Where the trouble begins:

Now I need to solve a system of 2 equations, where I can get the values for A and phi.

xg(t=0) = x0 ← Normal function, for t=0 the result is the initial position [m].

ẋg(t=0) = v0 ← Derivative of xg(t), for t=0 the result is the initial velocity [m/s].

But I can't get the derivative of xg(t) to produce any value:

a, phi = symbols('a phi', real=True)

def d_xg(t):
    return Derivative(xg(t),t, evaluate=True)

No mater the t value I throw in d_xg(t), it will always give me the same expression.

Any help is welcome, Thanks!

#Edit_01: changed def d_xg(t) to the current state.

Answer 1

When you call d_xg(t) you are computing the symbolic derivative of xg with respect to t. If you call d_xg(t) multiple times you will always get the same expression because you are doing the same computation over and over again. Similarly, if you call d_xg(A), you will compute the derivative of xg with respect to A.

Once you have computed d_xg(t), you want to substitute t=0 into that expression:

d_xg_at_t_0 = d_xg(t).subs(t, 0)