# How to find the steady state response of a Multi-Degree of Freedom (MDOF) system?

The Problem

I currently have a Multi-Degree of Freedom (MDOF) system with the following equation:

$$\mathbf{M\ddot{X}}+ \mathbf{D}(t)\mathbf{\dot{X}}^2 + \mathbf{C\dot{X}} + \mathbf{KX} = \mathbf{F}(t) \tag{1}$$

where $$\mathbf{M, D}$$ are diagonal matrices, $$\mathbf{C,X}$$ are tri-diagonal matrices, and $$\mathbf{F}(t)$$ is a vector containing a complicated sinusoid function.

I proceed to solve the system numerically, by treating it as Linear Time Invariant (LTI), which due to its week nonlinearity found in the $$\mathbf{D\dot{X}}^2$$ term, works well in terms of accuracy (i.e. validated with measured data).

However, it takes a really long time for the system to reach its steady state; meaning to reach a state where I get a repeating oscillatory pattern every 360 degrees. To demonstrate, if we extract the displacement at the $$n^{\text{th}}$$ DOF (Degree of Freedom) you can tell that steady state is reached roughly after the point I have marked below with a vertical line, and you can see the the zoomed in version of the last few cycles to the right graph on the figure below. Solving the problem this way wastes time in computations which I would like to avoid since I am only interested on the last few cycles.

$n^{\text{th}}$ DOF" />

Therefore, I would like to find a way to calculate the 'steady state' response straight away.

My reasoning behind a possible solution

I think that what I am asking is feasible, especially after looking at this open source paper of the equations listed on page 3. However I am unable to fully grasp the concept. As I understand it, I have to solve Eq.(1) in the frequency domain, and then by bringing it back to the time domain via an inverse Fourier transform, I'll get the steady state response.

1. Is my above reasoning correct?
2. What is the kind of formulation in terms of an LTI system that I would have to use in order to achieve the above? The LTI would have a form such as $$\begin{bmatrix}\mathbf{\dot{X}} \\ \mathbf{\ddot{X}} \end{bmatrix} = \begin{bmatrix}\mathbf{0} & \mathbf{I} \\ \mathbf{-M^{-1}K} & \mathbf{-M^{-1}C} \end{bmatrix}\begin{bmatrix}\mathbf{X} \\ \mathbf{\dot{X}} \end{bmatrix}+\begin{bmatrix}\mathbf{0} \\ \mathbf{-M^{-1}}(\mathbf{F}(t)-\mathbf{D}(t)\dot{\mathbf{X}}^2) \end{bmatrix}$$
• I find it odd that you identify an obviously oscillating system as being at steady state. Oct 7, 2021 at 17:03
• It sounds more like you are asking for the asymptotic behavior of the system as $t\to\infty$. Oct 7, 2021 at 17:51