I know that, at steady state, the frequency response can be calculated relatively easily from the transfer function and the frequency of the input.
So if we have a system described by the transfer function \$ G(s) \$ then we have (again, at steady state): $$ u(t) = u_0 \sin(\omega t) \\ Y(s) = G(s) \cdot U(s) \\ |G(j\omega)|=\frac{y_0}{u_0} \\ \angle G(j\omega) = \phi \\ \implies y(t)=|G(j\omega)|u_0 \sin(\omega t+\phi)\\ $$
I never saw anywhere any comments in the many text books I read about a signal with a phase as an input.
Say I have this \$ u(t) \$ (the excitation) instead of the sine with a zero phase: $$ u(t)=u_0 \sin(\omega t + \theta) $$
I wonder what will happen. Does the relationships mentioned above still hold true, or not. How can I calculate \$ \phi \$ and \$ y_0/u_0 \$ while accounting for the input phase \$ \theta \$?
I would have difficulty to believe the presence of \$ \theta \$ would not have any effect on the steady state output...