Find Transfer Function from Laplace transformation

Hi! I want to find the transfer function from a laplace transformation of a differential equation. Specifically this is the example I am working:

m = 1;

e = 0.06;

O = 25;

s = 25;

h = 1+e*s;

f = m*e*O^2;

P = e*f;

d = 0.1;

z = e*d;

%%%%%%%%%%%%%%%

syms t z u(t) e h P

f = diff(u(t),t,2) + 2*z*diff(u(t),t) + u(t) + e*(u(t))^3 - h^2 *P*cos(h*t);

F = laplace(f);

%transfer fuction

-------------------

The laplace transformation gives me this:

F = e*laplace(u(t)^3, t, s) - subs(diff(u(t), t), t, 0) - s*u(0) + s^2*laplace(u(t), t, s) - 2*z*(u(0) - s*laplace(u(t), t, s)) + laplace(u(t), t, s) - (P*h^2*s)/(h^2 + s^2).

I dont know how to convert it to a simplier form to find the transfer function. Please if you know something, comment below.

Thank you!!!

Answer 1

[Anurag Ojha]

Hi Christos

I have written MATLAB code to find a simpler form of the transfer function I have made use of some in built MATLAB functions. I have added those documentation at the end for your reference:

syms s U(s) u(t) e h P z

% Define the differential equation
f = diff(u(t), t, 2) + 2*z*diff(u(t), t) + u(t) + e*(u(t))^3 - h^2 *P*cos(h*t);

% Take the Laplace transform
F = laplace(f, t, s);

% Rearrange the terms to isolate the Laplace transform of the output variable
F = collect(F, laplace(u(t), t, s));

% Divide both sides by the Laplace transform of the input variable
H = F / U(s);

% Simplify the transfer function
H = simplify(H);

% Display the transfer function
disp(H);

• Simplify: https://www.mathworks.com/help/symbolic/simplify.html
• Laplace: https://www.mathworks.com/help/symbolic/sym.laplace.html

Answer 2

[Sam Chak]

The differential equation has a dependent variable with nonlinearity u(t)³. Since nonlinear systems cannot be analyzed by Laplace transform, MATLAB will return the result in that format.