I am trying to simulate an electronical device that can be modeled by a mass-spring-damper system with an additional non-linear force. The equation at the equilibrium for the system is the following :

frac{epsilon_0 a V^2}{2 x^2} – Bx’ + Kx = 0

The goal of my MATLAB code is to solve this equation for x and find the equilibrium. For this purpose, I use the ode45 function like this (all coefficient are defined in my code but not shown here) :

x0 = 0;
[t, x] = ode45(@(t, x) odefun(t, x, eps0, a, V0, B, d0, K), tspan, x0);

function dxdt = odefun(t, x, eps, a, V, B, d0, K)
    dxdt = ((eps * a * V^2) ./ (2 * B * (d0 - x).^2)) + ((K / B) .* x);
end

The equation is good according to my teachers and several papers but the solver never converges and I can’t see why. All coefficient are defined according to the dimensions of a capacitive micromachined ultrasound transducer. The solution of ode45 shows an infinite displacement of the membrane. This is obviously wrong because the dimensions of a cmut are bellow millimeter.
Can you see any mistakes in the way I use the ode45 solver ?

Full code :

% DIMENSIONS DE LA MEMBRANE
e = 500e-9;                 % epaisseur [m]
r = 20e-6;                  % rayon [m]
d0 = 550e-9;                % epaisseur de cavite [m]
a = pi * r^2;               % surface de la membrane [m^2]

% PARAMETRES MECANIQUES
E = 200e9;                  % module d'Young du SiN [Pa]
nu = 0.25;                  % coefficient de poisson du SiN
eta = 18.5e-6;              % viscosité dynamique de l'air [Pa.s]
p0 = 1e5;                   % pression exterieure [Pa]
rhoSiN = 3170;              % densite du SiN [Kg/m^3]
m = rhoSiN * a * e;         % masse membrane [Kg]
K = (16 * E * e^3) / (3 * (1 - nu^2) * r^2);   % raideur [N/m]
B = (eta * pi * r^2) / e;                      % amortissement [N.s/m]

% PARAMETRES ELECTRIQUES
eps0 = 8.85e-12;            % permittivité diélectrique du vide [F/m]
V0 = 10;                    % tension de polarisation [V]

% Conditions initiales et plage de temps
x0 = 0;
ti = 0; tf = 1e-3;
dt = 1e-6;
tspan = linspace(ti, tf, 1/dt);

% Résolution par RK4
[t, x] = ode45(@(t, x) odefun(t, x, eps0, a, V0, B, d0, K), tspan, x0);

plot(t,x,'-')

function dxdt = odefun(t, x, eps, a, V, B, d0, K)
    dxdt = ((eps * a * V^2) ./ (2 * B * (d0 - x).^2)) + ((K / B) .* x);
end

References :

• Y. Wang, L. -M. He, Z. Li, W. Xu and J. Ren, “A computationally efficient nonlinear dynamic model for cMUT based on COMSOL and MATLAB/Simulink”
• T. Merrien, A. Boulmé and D. Certon, “Lumped-Parameter Equivalent Circuit Modeling of CMUT Array Elements”

I tried several solver from MATLAB and implemented my own Runge-Kutta algorithm based on the Wikipedia example. I also verified the coefficients according to my references. Finally I tried using ode15s stiff equations solver with additional options and got the same results.