@Jurnan Thank you for this lecture series. I have a question. Why have you considered only the particular solution for $\eta_{n}(t)$ and ignored the homogeneous solution?
Hey Vikas, I think I can help. My understanding is as follows: If we assume that there is some form of damping present in the system (although neglected in our calculations), the contribution of the homogenous solution will disappear over time. The particular solution, however, is only dependant on the right side of our equation: As long as the force acts on the system, the particular solution will be of the same magnitude (on the average of a cycle) throughout time. Hence, we only focus on the particular solution. Also see Jurnan's video on forced vibrations for SDOF systems (also in this lecture series).
@Jurnan Thank you for this lecture series. I have a question. Why have you considered only the particular solution for $\eta_{n}(t)$ and ignored the homogeneous solution?
Hey Vikas, I think I can help.
My understanding is as follows: If we assume that there is some form of damping present in the system (although neglected in our calculations), the contribution of the homogenous solution will disappear over time. The particular solution, however, is only dependant on the right side of our equation: As long as the force acts on the system, the particular solution will be of the same magnitude (on the average of a cycle) throughout time. Hence, we only focus on the particular solution.
Also see Jurnan's video on forced vibrations for SDOF systems (also in this lecture series).
Over time, the homogeneous solution will go to zero. As we are evaluating the system to infinity, we only need the particular solution.