
In this lecture we're going to derive the equations of motion for a MDoF system. We'll start by considering a simple free-standing low-rise multi-storey structure with 3 floors above ground level, Fig 1 (left). We'll also assume a solid concrete floor slab on each floor level. In order to model the dynamic response of this structure we will extract a sub-frame and model this as a 2D planar frame, Fig 1 (right).

<Image
	src="/images/courses/multi-degree-of-freedom-dynamics-modal-analysis-and-seismic-response-simulation-in-python/mdof-equations-of-motion/img1.jpg"
	alt="Free-standing 3-storey structure steel frame structure (left) and sub-frame extracted for modelling as a 2D planar frame (right). | EngineeringSkills.com"
	width="500"
	height="384"
	caption="Fig 1. Free-standing 3-storey structure steel frame structure (left) and sub-frame extracted for modelling as a 2D planar frame (right)."
/>

It would be quite reasonable to model this structure as a shear building with its mass concentrated at each floor level, not least because of the mass concentration provided by the floor slab at each level. This will allow us to abstract the dynamic behaviour of the structure into a relatively simple mechanical model on which to base our dynamic equations.

<Image
	src="/images/courses/multi-degree-of-freedom-dynamics-modal-analysis-and-seismic-response-simulation-in-python/mdof-equations-of-motion/img2.jpg"
	alt="Typical 2D plane frame (left), shear building dynamic model that approximates the dynamic behaviour of a typical frame within the structure. | EngineeringSkills.com"
	width="500"
	height="369"
	caption="Fig 2. Typical 2D plane frame (left), shear building dynamic model that approximates the dynamic behaviour of a typical frame within the structure."
/>

For now, we'll ignore ground excitation and simply say that the top floor of the structure is subject to a harmonic excitation,

$$
p_3(t) = F\sin(\omega t)
\tag{1}
$$

At this point we're also going to conveniently skip over the topic of structural damping. We'll need to discuss this in a little more detail later, so for now we'll simply apply some arbitrary values for the damping coefficients, $c_1$, $c_2$ and $c_3$. At this point, it's sufficient to simply represent damping forces within our system of equations.

In order to determine the governing differential equations of motion we need to consider each mass as a free body diagram and explicitly identify the forces acting on them, in much the same way that we did for a SDoF system, Fig. 3.

<Image
	src="/images/courses/multi-degree-of-freedom-dynamics-modal-analysis-and-seismic-response-simulation-in-python/mdof-equations-of-motion/img3.jpg"
	alt="Free-body diagram showing the forces applied to each mass. | EngineeringSkills.com"
	width="300"
	height="334"
	caption="Fig 3. Free-body diagram showing the forces applied to each mass."
/>

Considering mass $m_1$ and evaluating the sum of the horizontal forces gives us the differential equation describing the motion of $m_1$,

$$
m_1\ddot{u}_1 + \dot{u}_1(c_1+c_2)-\dot{u}_2 c_2 + u_1(k_1+k_2) - u_2 k_2 = 0
\tag{2}
$$

Repeating this for $m_2$ we get,

$$
m_2\ddot{u}_2 -\dot{u}_1 c_2 +  \dot{u}_2(c_2+c_3) - \dot{u}_3 c_3 - u_1 k_2 + u_2(k_2+k_3)-u_3 k_3 = 0
\tag{3}
$$

And finally for $m_3$,

$$
m_3\ddot{u}_3 -\dot{u}_2 c_3 + \dot{u}_3 c_3-u_2k_3+u_3k_3 = p_3(t)
\tag{4}
$$

Note that these three equations are no more complex than the differential equation that describes a SDoF system; we have an inertial, damping and stiffness force and an optional external excitation force,

$$
m\ddot{u} + c\dot{u} + k u = p(t)
\tag{5}
$$

However one slight complicating factor is that **the three equations are coupled** i.e. we now have a system of coupled second order differential equations. This means they cannot be solved in isolation as the behaviour of each mass influences its neighbours. This means that we must solve the equations in parallel (simultaneously). We'll discuss how to do this in the next lecture, for now, let's tidy up our system of equations.

We can more conveniently represent this system of three simultaneous equations as a matrix equation,

$$
\begin{bmatrix}
m_1 & 0 & 0\\
0 & m_2 & 0\\
0 & 0& m_3
\end{bmatrix}
\begin{Bmatrix}
\ddot{u}_1 \\
\ddot{u}_2 \\
\ddot{u}_3
\end{Bmatrix}+
\begin{bmatrix}
c_1+c_2 & -c_2 & 0\\
-c_2 & c_2 +c_3 & -c_3\\
0 &-c_3 & c_3
\end{bmatrix}
\begin{Bmatrix}
\dot{u}_1\\ \dot{u}_2\\ \dot{u}_3
\end{Bmatrix} +
\begin{bmatrix}
k_1+k_2 & -k_2 & 0\\
-k_2 & k_2 +k_3 & -k_3\\
0 &-k_3 & k_3
\end{bmatrix}
\begin{Bmatrix}
u_1\\ u_2\\ u_3
\end{Bmatrix} =
\begin{Bmatrix}
0\\
0\\
p_3(t)
\end{Bmatrix}
\tag{6}
$$

Not the presence of coupling terms in the square damping and stiffness matrices. By contrast note that the mass matrix as an example of an **uncoupled matrix**. If it wasn't already apparent, this tells us that we have coupling between our equations. For convenience, let's now define the following,

$$
\begin{bmatrix}
M
\end{bmatrix} =
\begin{bmatrix}
m_1 & 0 & 0\\
0 & m_2 & 0\\
0 & 0 & m_3
\end{bmatrix}
\tag{7}
$$

$$
\begin{bmatrix}
C
\end{bmatrix} =
\begin{bmatrix}
c_1+c_2 & -c_2 & 0\\
-c_2 & c_2+c_3 & -c_3\\
0 & -c_3 & c_3
\end{bmatrix}
\tag{8}
$$

$$
\begin{bmatrix}
K
\end{bmatrix} =
\begin{bmatrix}
k_1+k_2 & -k_2 & 0\\
-k_2 & k_2+k_3 & -k_3\\
0 & -k_3 & k_3
\end{bmatrix}
\tag{9}
$$

$$
\begin{Bmatrix}
P
\end{Bmatrix} =
\begin{Bmatrix}
0\\
0\\
p_3(t)
\end{Bmatrix}
\tag{10}
$$

$$
\begin{Bmatrix}
u
\end{Bmatrix} =
\begin{Bmatrix}
u_1\\
u_2\\
u_3
\end{Bmatrix} \: \text{,}\:
\begin{Bmatrix}
\dot{u}
\end{Bmatrix} =
\begin{Bmatrix}
\dot{u}_1\\
\dot{u}_2\\
\dot{u}_3
\end{Bmatrix}\: \text{and}\:
\begin{Bmatrix}
\ddot{u}
\end{Bmatrix} =
\begin{Bmatrix}
\ddot{u}_1\\
\ddot{u}_2\\
\ddot{u}_3
\end{Bmatrix}
\tag{11}
$$

Therefore we can represent the system of equations in a simpler form that resembles the SDoF system equation from above,

$$
\begin{bmatrix}
M
\end{bmatrix}
\begin{Bmatrix}
\ddot{u}
\end{Bmatrix} +
\begin{bmatrix}
C
\end{bmatrix}
\begin{Bmatrix}
\dot{u}
\end{Bmatrix} +
\begin{bmatrix}
K
\end{bmatrix}
\begin{Bmatrix}
u
\end{Bmatrix}=
\begin{Bmatrix}
P
\end{Bmatrix}
\tag{12}
$$

where $\begin{bmatrix}M\end{bmatrix}$, $\begin{bmatrix}C\end{bmatrix}$ and $\begin{bmatrix}K\end{bmatrix}$ are referred to as the mass, damping and stiffness matrices respectively.

It's important to appreciate the generality of what we've done so far. We started this lecture basing the discussion around a three storey shear building. This allowed us to easily present free body diagrams and eventually to construct mass, damping and stiffness matrices from first principles. But remember, the preceding matrix equation can be solved for any dynamic system, _provided the appropriate mass, damping and stiffness matrices are determined_. This means that the methods of solving these equations that we're about to explore, will work for any dynamic system. The _trick_ is to determine $\begin{bmatrix}M\end{bmatrix}$, $\begin{bmatrix}C\end{bmatrix}$ and $\begin{bmatrix}K\end{bmatrix}$ matrices that adequately model the structure.


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