
All structures are continuous, meaning their mass and stiffness is continuously distributed throughout the structure. To model the dynamic behaviour of a continuous structure, we need to understand how the structure's response varies as a function of time and position (in the structure). This is certainly achievable through the use of partial differential equations, however when our structural models become large and complex, a far more practical approach is to discretise the structure into lumped masses. In this way, the dynamic behaviour of a complex continuous structure with distributed mass and stiffness can be modelled as a collection of discrete connected lumped masses.

As an example, the multi-storey shear building in Fig. 1 below could be approximated by _lumping_ the mass from each floor level into a single _lumped mass_ constrained to move laterally. As there is 4 floors and therefore, 4 lumped masses and 4 possible displacements, this system is said to have 4 degrees of freedom. When you've completed this course, you can investigate the behavour of these multi-degree of freedom systems in [this course](/course/multi-degree-of-freedom-dynamics-modal-analysis-and-seismic-response-simulation-in-python).

<Image
	src="/images/courses/fundamentals-of-structural-dynamics-with-python/lumped-mass-analysis/img1.jpg"
	alt="Multi-storey shear building approximated as a series of connected lumped masses. | EngineeringSkills.com"
	width="600"
	height="591"
/>
<Caption>
	Fig 1. Multi-storey shear building approximated as a series of connected lumped
	masses. The analysis of these structures is covered in [this
	course](/course/multi-degree-of-freedom-dynamics-modal-analysis-and-seismic-response-simulation-in-python).
</Caption>

For now, we will focus on the behaviour of single-degree-of-freedom (SDoF) systems. In such a system, we approximate the behaviour of the structure as that of a single lumped mass free to move along a single axis. The diagram below shows a water tower approximated by a lumped mass on top of a massless rod. The lumped mass can oscillate laterally and thus the system has a single-degree-of-freedom. The sketched time-displacement trace indicates the lateral oscillations that would be observed if the system was subject to a harmonic (sinusoid) load.

<Image
	src="/images/courses/fundamentals-of-structural-dynamics-with-python/lumped-mass-analysis/img2.jpg"
	alt="Water tower approximated as a single-degree-of-freedom system. | EngineeringSkills.com"
	width="700"
	height="605"
	caption="Fig 2. Water tower approximated as a single-degree-of-freedom system."
/>

The analysis of SDoOF systems acts as our entry point into the study of dynamic behaviour. A solid grasp of the concepts as they relate to SDoF systems can carry us a **very** long way and give us the tools to understand and analyse quite complex structures.

Let's develop our water tower example further. As we've stated above, the lateral oscillation of the water tower could be approximated by a SDoF system. We can abstract the system dynamics away from the water tower structure and represent the behaviour by a **generalised SDoF model**, also known as a **spring-mass-damper model**. We will use this abstraction to develop our mathematical models of the behaviour. But remember, we're still talking about the lateral oscillation of the water tower (or any other structure whose dynamic behaviour can be modelled as a SDoF system). We'll explore the spring-mass-damper model further in the next lecture.


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