9.2. Equation of Motion for Single Degree of Freedom (SDOF) System

For a linear SDOF system subjected to a force F(t), the equation of motion is:

mx¨(t) + cx˙(t) + kx(t) = F(t)

Where: • m: Mass • c: Damping coefficient • k: Stiffness • x(t): Displacement • F(t): External force (here, impulse)

In this equation, we describe how a single degree of freedom system behaves when a force is applied. The left side of the equation consists of three terms that represent: the inertia (mx¨(t)), damping (cx˙(t)), and stiffness (kx(t)). The right side shows the external force that is acting on the system, which can vary over time. Essentially, this equation serves as a balance between the forces acting on the mass and the forces exerted by the system due to its characteristics.

Imagine a swing (which lets us think of a mass hanging by a rope) as our SDOF system. When you push the swing (applying an external force), it starts moving (displacement). The swing also has resistance to motion due to air and friction at the pivot (damping) and will want to return to its resting position if displaced (stiffness), which can be thought of like a spring pulling back.

For unit impulse input: F(t) = δ(t) So, the governing equation becomes: mx¨(t) + cx˙(t) + kx(t) = δ(t)

Here, we introduce the concept of a unit impulse, represented by δ(t). This is a mathematical construct useful for analyzing responses of systems. When we substitute this impulse into our equation, it transforms our system’s behavior into one that can reveal essential dynamic characteristics. The idea here is that a unit impulse suddenly applies a force to the system, and the response we observe helps us understand how the system reacts to rapid applications of force.

Think about hitting a punching bag lightly. That sudden force is like our unit impulse. The bag suddenly compresses (the force is applied), and then it swings back due to its structure and materials (the system's reaction). By studying how far and fast it moves, we understand the dynamics of the punching bag — which parallels how engineers study other systems under impulsive forces.

Solving this equation gives the impulse response function of the system.

The impulse response function is a critical aspect of this analysis. It essentially tells us how the system will respond over time to the initial impulse. After applying the mathematical methods to solve the governing equation with the unit impulse, we can gain insights into how the system behaves in the short and long term. This response gives engineers vital information for ensuring safety and performance in real structures.

Consider how a pebble thrown into a pond creates ripples. The first ripple represents the initial impact (the impulse), while the subsequent ripples show how the water (our system) responds to that single input. The pattern of those ripples—including how quickly they fade—can tell us everything about the pond's shape and depth, much like how the impulse response function informs engineers about the structural behavior.

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