L2

L2 computes the root-mean-square (RMS) value of the response over the supplied time samples. It is a compact time-domain magnitude: useful when you want one number per signal but do not care about phase or frequency.

For each mode \(i\),

\[ A_i = \sqrt{ \frac{1}{N} \sum_{k=1}^{N} x_i(t_k)^2 } \]

with \(x_i(t_k)\) the modal displacement at sample \(t_k\).

If \(\text{modal_contributions} = (\phi_1, \dots, \phi_{n_{\mathrm{modes}}})\) is provided, the total signal is

\[ x_{\mathrm{total}}(t_k) = \sum_{i=1}^{n_{\mathrm{modes}}} \phi_i \, x_i(t_k) \]

and the total RMS is

\[ A_{\mathrm{total}} = \sqrt{ \frac{1}{N} \sum_{k=1}^{N} x_{\mathrm{total}}(t_k)^2 } \]

If modal_contributions is omitted, unit weights are used. When those weights come from a mode shape at a measurement point, the total entry becomes the physical RMS response at that point.