How to solve nonlinear equation?

Answer 1

[Sam Chak]

Hi @GUANGHE HUO

The nonlinear matrix ODE with time-varying stiffness matrix K can be transformed into a nonlinear state-space model. See example below.

tspan  = [0 40];
x0     = [1 0.5 0 0];
[t, x] = ode45(@odefcn, tspan, x0);
plot(t, x), grid on, xlabel('t')

function xdot = odefcn(t, x)
    xdot = zeros(4, 1);	
    M = diag([3 5]);
    C = 2*eye(2);
    K = [1+0.5*sin(2*pi/40*t) 0; 0 1+0.5*sin(2*pi/40*t)];   % time-varying K
    A = [zeros(2) eye(2); -M\K -M\C];
    B = [zeros(2); eye(2)];
    F = [0; 0];     % Requires your input
    u = M\F;
    xdot = A*x + B*u;
end

[GUANGHE HUO]

thanks for your answer@Sam Chak, I will try

[GUANGHE HUO]

Hi, @Sam Chak, for every time point, I have stored the Force vector and Stiffness Matrix by using cell, like F{i,1} and K{i,1}. I want to use ode45 in a LOOP, as following:

for kk=1:1:Length(T) % T is time point vector

end

but now I do not know how to use the template that you give me? Do you have some advice?

[Sam Chak]

Hi @GUANGHE HUO

I use ordinary numeric array in my simulations. Perhaps you can try using the cell2mat() command to convert the selected cell array into the desired numeric array.

https://www.mathworks.com/help/matlab/ref/cell2mat.html

If your Force vector and the Stiffness matrix are time series data (cannot be expressed in any fundamental mathematical form), then you need to use the interp1() function to interpolate and to obtain the value of the time-dependent terms at the specified time.

Here is an example of using a data-driven Force to stabilize the Double Integrator system:

% Force data set recorded over some intervals of time
ft = linspace(0, 20, 2001);
f  = 2*exp(-ft).*ft - exp(-ft).*(1 + ft);   % made-up to generate the data

tspan  = [0 20];
y0     = [1 0];
opts   = odeset('RelTol', 1e-4, 'AbsTol', 1e-8);
[t, y] = ode45(@(t, y) doubleInt(t, y, ft, f), tspan, y0, opts);

plot(t, y), grid on, xlabel('t'), ylabel('Y(t)')
legend('y_{1}(t)', 'y_{2}(t)')

% Double Integrator system
function dydt = doubleInt(t, y, ft, f)
    dydt    = zeros(2, 1);
    f       = interp1(ft, f, t);    % Interpolate the data set (ft, f) at time t
    dydt(1) = y(2);
    dydt(2) = f;
end

[GUANGHE HUO]

Thanks for your help.@Sam Chak, You give me a lot of help, because i am new for MATLAB

Answer 2

[Torsten]

Write as

xdot = y
ydot = inv(M)*(F-c*y-K*x)

and use ode45 to solve.

[GUANGHE HUO]

thanks