
The dynamic behaviour of the spring-mass-damper SDoF system is described by a differential equation. This is really just a force equation that has differential terms due to the presence of velocity and acceleration in the equation; this will become clearer below. This equation can be constructed by first considering a free-body diagram of the system, Fig. 1.

<Image
	src="/images/courses/fundamentals-of-structural-dynamics-with-python/the-equation-of-motion/img1.jpg"
	alt="Free-body diagram of the SMD model showing all forces acting on the mass. | EngineeringSkills.com"
	width="600"
	height="299"
	caption="Fig 1. Free-body diagram of the SMD model showing all forces acting on the mass."
/>

The forces acting on the mass can be summarised as:

- $ku =$ stiffness force, proportional to displacement

- $c\dot{u} = $ damping force, proportional to velocity

- $m\ddot{u} = $ inertia force, proportional to acceleration

Note that both $ku$ and $c\dot{u}$ model structural properties, i.e. stiffness (a function of $EI$) and inherent damping. However, $m\ddot{u}$ models the inertia of the mass. As we discussed in an earlier lecture, this is a _fictitious_ force that we apply to model the mass resistance to a change in its velocity.

The external forces that induce a change in the velocity of the system are,

$$
p(t) - ku - c\dot{u}
\tag{1}
$$

These can be equated to the fictitious inertia force, $m\ddot{u}$ resisting a change in the velocity of the mass,

$$
p(t) - ku - c\dot{u} = m\ddot{u}
\tag{2}
$$

Rearranging, we have,

$$
m\ddot{u} + c\dot{u} + ku  = p(t)
\tag{3}
$$

or using alternative notation,

$$
\underbrace{m\frac{\mathrm{d}^2u}{\mathrm{d}t^2} + c\frac{\mathrm{d}u}{\mathrm{d}t}}_\text{dynamic component} + \underbrace{ku}_\text{static component} = p(t)
\tag{4}
$$

Note that the inertia and damping forces are only present when there is non-zero acceleration and velocity respectively. Thus they can be considered the dynamic components of the response. Without them, the equation reduces to the familiar static equation,

$$
ku = p(t)
\tag{5}
$$

The equation describing the full system response is a second-order differential equation. The solution of this equation allows us to determine the state of the mass (position, velocity and acceleration) at any time provided we know the loading history, $p(t)$ and the system's **initial conditions** (position and velocity), $u(0)$ and $\dot{u}(0)$.

If we set the forcing term to zero, the equation becomes homogeneous and represents the free (non-forced) vibration of the system. In such a case, motion of the system occurs if the system is displaced and allowed to freely oscillate. In the next lecture we'll examine this free vibration behaviour.


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