
Now, returning to our beam example; the displacement of the beam depends on two factors:

- The time-varying external force $F(t)$.

- The inertia forces which oppose the accelerations producing them.

As the external load is applied quickly or dynamically in this case, the rate of change of velocity along the beam will be high or at the very least non-negligible. We now know that this means the beam will exhibit correspondingly high resistance to a change in its velocity. We model this by applying inertia forces along the length of the beam, Fig 1.

<Image
	src="/images/courses/fundamentals-of-structural-dynamics-with-python/dynamic-response-and-inertia/img1.jpg"
	alt="Cantilever beam with dynamically applied load exhibiting an exponentially decaying dynamic response. | EngineeringSkills.com"
	width="571"
	height="468"
	caption="Fig 1. Cantilever beam with dynamically applied load exhibiting an exponentially decaying dynamic response."
/>

Remember the beam deflection now depends on the external force and the relatively large inertia forces induced by the beam's acceleration. This means that the internal moments and shears in the beam must resist not only the externally applied force but also the inertia forces resulting from accelerations of the beam. **It's the presence of inertia forces that fundamentally differentiate a dynamic problem from a static problem.**

If inertia forces represent a significant portion of the total loading resisted by a structure, a dynamic analysis should be performed as the dynamic characteristics of the problem cannot be ignored and the structural response will vary with time.

Conversely, if all external loads are applied slowly such that they do not induce significant inertia forces, the problem can be considered static with a single response corresponding to the externally applied load(s). In fact, as we progress it will become apparent that static structural analyses are a sub-set of the more general case of dynamic analyses; a static analysis is simply a dynamic analysis with zero or negligible inertia forces.


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