How to solve nonlinear equation?
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Sam Chak
am 21 Dez. 2022
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
4 Kommentare
Sam Chak
am 23 Dez. 2022
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.
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
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