We generate a multisine using the PNLSS toolbox[1], specifying number of N samples(related to fs), P periods(number of time the same signal is repeated), R realizations(where the phases are randomly drawn again) and kind=str('full') meaning that all frequency lines up till nyquist freq is excited.
[u,lines,non_exc_odd,non_exc_even] = fMultisine(N, , M, R); u = u/rms(u(:,1))*RMSu; % Scale multisine to the correct rms level u = repmat(u,[1 1 P]); % N x R x P u = permute(u,[1 3 2]); % N x P x R u_ext = u(:); % Concatenate the data: N*P*R
Note that the magnitude of the 0-frequency term is zero.
To use ode45, we need to interpolate the multisine. But do we destroy the frequency content? Yes! fex = @(t) interp1(time, u_ext, t, ‘linear’);
So we cannot use ode45 with u_ext, as it is not possible to force fixed time steps. We now have two options.
- Use a fixed time-step ode solver
- Use a time-domain formulation of multisine with ode45
- is somewhat straight forward. ode45 is using dopri[2], so from the Butcher tableau we find a 5-order accurate solution[3]. Why - or why not - we should use a higher order method, I do not know. Higher order methods requires move function evaluations per time step, so unless needed we should not go for them. I know that for celestial mechanics, eq. oscillations over large time scales, they need higher order methods. A former post.doc in our group used his own Prince-Dormand method of order 8, justified in [4].
- The time formulation looks like this:
y(t) = 1/sqrt(A) * ∑n=1^N A_n sin(2π n f_0 t + φ_n) with φ_n ~ U[0, 2π[ f_0 is the frequency resolution.
In matlab: % define harmonic content and draw the random phases har = ones(f2/f0,1); % excite all lines phase = 2*pi*rand(length(har),1); fex = @(t) har’*A*cos(2*pi*(1:N)’*f0*t + phase) / sqrt(sum(har));
and a file: https://github.com/pawsen/tmd/blob/master/ClampedClampedBeams/01_flat_beam/ode45_multisine.m
[1] http://homepages.vub.ac.be/~ktiels/pnlss.html [2] https://en.wikipedia.org/wiki/List_of_Runge%E2%80%93Kutta_methods#Dormand%E2%80%93Prince [4] https://link.springer.com/article/10.1007/BF00049361