How to solve nonlinear equation?

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GUANGHE HUO
GUANGHE HUO 2022-12-20
评论： GUANGHE HUO 2022-12-23

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Sam Chak
Sam Chak 2022-12-21
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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
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Sam Chak
Sam Chak 2022-12-23
Ran in:
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
GUANGHE HUO 2022-12-23
Thanks for your help.@Sam Chak, You give me a lot of help, because i am new for MATLAB

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Torsten
Torsten 2022-12-20
Write as
xdot = y
ydot = inv(M)*(F-c*y-K*x)
and use ode45 to solve.

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