
Now, returning to our beam example. The displacement of the beam depends on three factors:

- the time-varying external force $F(t)$
- the beam’s stiffness, which we observe as a restoring force acting to return the beam to its equilibrium position
- the inertial response of the mass, which is proportional to the beam’s acceleration and resists changes in velocity.

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

Remember the beam deflection now depends on the external force, the beam’s stiffness 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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