
The spring-mass-damper (SMD) model, Fig. 1, is the generalised representation of a single-degree-of-freedom system. It is a simple mechanical model whose motion we can easily develop equations for. The properties of our SDoF structure can be mapped directly onto the components of the model, making it a great analogue to base our analysis on.

Recall the water tower from the previous lecture and its lumped-mass approximation. The lumped-mass of the approximate model is represented by the mass in our SMD model. The lateral stiffness provided by the tower structure is represented by a simple spring with constant $k$.

All structures exhibit some degree of inherent damping that will dissipate energy. Inherent structural damping will eventually bring a freely vibrating system to rest. This characteristic is modelled with an oil damper and characterised through the damping coefficient, $c$ in the SMD model.

<Image
	src="/images/courses/fundamentals-of-structural-dynamics-with-python/the-spring-mass-damper-model/img1.jpg"
	alt="Spring-mass-damper SDoF model. | EngineeringSkills.com"
	width="600"
	height="299"
	caption="Fig 1. Spring-mass-damper SDoF model."
/>

We can visually link the lateral oscillation of the water tower lumped-mass with the lateral oscillation of the SMD system, Fig. 2.

<Image
	src="/images/courses/fundamentals-of-structural-dynamics-with-python/the-spring-mass-damper-model/img2.jpg"
	alt="Water tower lateral oscillations simulated by oscillations of the spring-mass-damper model. | EngineeringSkills.com"
	width="550"
	height="693"
	caption="Fig 2. Water tower lateral oscillations simulated by oscillations of the spring-mass-damper model."
/>

The important point to recognise here is that when we're solving for the dynamic behaviour of the SMD model, we're actually solving for the lateral oscillations of the water tower. Remember, the characteristics of the SMD model, mass $m$, spring stiffness, $k$ and damping coefficient $c$, are all directly derived from the original water tower structure.

Now that we understand that the SMD model is really just a proxy for the dynamic behaviour of the water tower, in the following lecture, we'll start modelling the behaviour of the SMD model.


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