
So far in this course and in the Fundamentals of Structural Dynamics course that precedes this one, we have focused exclusively on the behaviour of SDoF systems. This has allowed us to establish many of the foundational concepts that we will continue to rely on.

We've also seen how the behaviour of SDoF systems can be modelled using differential equations and we've explored closed-form and numerical solutions to these equations. Indeed, the behaviour of SDoF system will have special relevance in our study of modal analysis in the next section.

While SDoF systems are an essential precursor to the study of multi-degree of freedom systems - we can readily acknowledge that all structures are indeed continuous in nature. This means that they have spatially distributed, mass, stiffness and damping. As a result, the governing equations that describe these systems will be partial differential equations. Consider the simple beam in Fig 1. below.

<Image
	src="/images/courses/multi-degree-of-freedom-dynamics-modal-analysis-and-seismic-response-simulation-in-python/introduction-to-mdof-systems/img1.jpg"
	alt="Simply supported beam subject to dynamic load p(t) (top) and beam
	response v(x,t) at location x and time t (bottom). | EngineeringSkills.com"
	width="450"
	height="333"
/>
<Caption>
	Fig 1. Simply supported beam subject to dynamic load $p(t)$ (top) and beam
	response $v(x,t)$ at location $x$ and time $t$ (bottom).
</Caption>

To fully describe the dynamic response of this beam we need to know how the displacement at every location along the beam varies with time. A SDoF system can only tell us how the position of a single point on the beam varies with time. Therefore we need to formulate the problem with two independent variables, position and time, hence the partial differential equations.

While we can pursue the solution of a partial differential equation for relatively simple structures, like the beam in Fig 1, there is far more practical utility in discretising the system, which leads us to the study of MDoF systems.

This is the same logic that leads us to discretise a structure in order to perform a finite element analysis. By breaking a structure into finite elements, we can obtain an approximation of the structural behaviour. The accuracy of this approximation is to some extent a function of the degree of discretisation. The same is true for a dynamic analysis. As the number of degrees of freedom increases, the accuracy of our approximate solution approaches that of a continuous solution.

Let's park the comparison with finite element analysis for now and simply consider a comparison between SDoF and MDoF systems. A SDoF system is capable of only describing motion in a single coordinate direction - this is literally what we mean by _single degree of freedom_.

However, a MDoF system is capable of describing motion in multiple coordinate directions simultaneously. Now, if each of these coordinates is associated with a different part of the structure, we have the ability to capture or approximate the spatial variation in dynamic behaviour, across the structure.

Consider again, the same beam from Fig 1, but now with its mass concentrated into three discrete locations or lumped masses. If we constrain these masses to move vertically, ignoring any rotation or longitudinal displacement along the beam, we can say that our lumped mass beam model has three **degrees of freedom**.

<Image
	src="/images/courses/multi-degree-of-freedom-dynamics-modal-analysis-and-seismic-response-simulation-in-python/introduction-to-mdof-systems/img2.jpg"
	alt="Lumped mass approximation of a simply supported beam. | EngineeringSkills.com"
	width="450"
	height="198"
	caption="Fig 2. Lumped mass approximation of a simply supported beam."
/>

By formulating three simultaneous equations of motion with time as the only independent variable, we can solve for the displacement of each mass. In doing so, we obtain an approximation of the dynamic behaviour of the full beam.

### Equations of motion

We've said that a MDoF system is a discretised approximation of a continuous system. In this course we'll focus on MDoF systems that have localised mass, stiffness and damping. This is completely analogous to the SDoF system we're already familiar with. So, the mass of the structure is distributed into a number of individual rigid lumped masses, the stiffness is modelled by discrete localised springs and the inherent structural damping is modelled as viscous dashpot damping (we explore damping in more detail later in the course).

The beauty of discretised MDoF analysis when compared with the analysis of continuous systems is its relative simplicity. We can deploy much of what we already know from the SDoF world. At a high level, for now, let's imagine a MDoF system consisting of three masses and three translational degrees of freedom.

<Image
	src="/images/courses/multi-degree-of-freedom-dynamics-modal-analysis-and-seismic-response-simulation-in-python/introduction-to-mdof-systems/img3.jpg"
	alt="Three degree of freedom spring-mass-damper system. | EngineeringSkills.com"
	width="550"
	height="144"
	caption="Fig 3. Three degree of freedom spring-mass-damper system."
/>

The system's three masses can each move relative to each other. The motion of each mass will be described by a differential equation of motion. In order to obtain the governing equation for each mass, we simply need to consider a free body diagram for each mass and evaluate the sum of the forces on the mass. In the next lecture we'll do exactly this.

Provided we solve these equation in parallel, we can determine the dynamic behaviour of the system. To achieve this we've done little more than was required for a SDoF system. This approach does have its limitations, it's not very scalable for one. As the number of degrees of freedom increase, this **direct solution method** gets very laborious.

More importantly, direct solution of the equations of motion also obscures some of the underlying dynamic characteristics of the system, it's so-called **modes of vibration** that are of immense practical value to the engineer. But we'll deal with that in the next section. The takeaway message here is that we can move from SDoF to MDoF dynamic analysis relatively easily. We'll prove this in the remaining lectures in this section.

### Shear buildings

Throughout the rest of this course we will typically represent our MDoF dynamic models as **shear buildings**. This conveniently fits the context of our discussion; structural vibration of multi-story buildings due to ground motion. But there is absolutely no loss of generality by _just_ modelling shear buildings; the theory and methods of dynamic analysis we discuss apply to any discretised dynamic system.

We've already seen an example of a single story shear building in the previous section. These models are completely dynamically equivalent to the spring-mass-damper cart models we've also seen previously, Fig 3. The flexural stiffness of the columns is equivalent to the spring stiffness and the structural damping is again modelled by a velocity proportional viscous damper.

In addition, we assume that all structural mass is concentrated in the floor plates and the each floor plate only experiences lateral motion. There are no rotational degrees of freedom and all motion is confined to the 2D plane of the structure.

<Image
	src="/images/courses/multi-degree-of-freedom-dynamics-modal-analysis-and-seismic-response-simulation-in-python/introduction-to-mdof-systems/img4.jpg"
	alt="Dynamic model of a shear building that is dynamically equivalent to the spring-mass-damper model in Fig 3. | EngineeringSkills.com"
	width="347"
	height="400"
	caption="Fig 4. Dynamic model of a shear building that is dynamically equivalent to the spring-mass-damper model in Fig 3."
/>


Please log in or enroll to continue