10.5. Application to Base Excitation (Earthquake Ground Motion)

In earthquake engineering, instead of an external force F(t), the system is subjected to ground acceleration $u¨ g(t)$, which induces relative motion in the structure.

In the context of earthquake engineering, the primary concern is how structures respond to the motion of the ground caused by an earthquake. Unlike situations where we apply forces directly to a structure, during an earthquake, the ground itself moves. This movement is represented as ground acceleration, noted mathematically as $u¨ g(t)$. This acceleration affects how the structure reacts, making it crucial to consider this change in force application.

Imagine standing in a car that suddenly accelerates forward. You feel pushed back against your seat, not because something is pushing you directly, but because the car (the ground in a structural context) is moving. When an earthquake occurs, the ground shifts beneath buildings and bridges, causing them to sway or even collapse, much like the sudden motion of that car would affect your stance.

The equation of motion in this case becomes: $mx¨(t) + cx˙(t) + kx(t) = -mu¨ g(t)$

This equation describes how the system behaves when subjected to base excitation. Here, $m$ represents the mass of the structure, $c$ is the damping coefficient which affects the energy dissipation of the structure, and $k$ symbolizes the stiffness of the structure. The term on the right, $-mu¨ g(t)$, indicates that the relative displacement of the structure is affected by the negative of the ground acceleration. This means that as the ground shakes, it induces movement that needs to be accounted for in the dynamics of the system.

Think of a tall building on a shaky foundation during an earthquake. The ground motion is like a child shaking a toy building back and forth. The building's response (how much it sways) depends on its weight, how well it can absorb the shaking (damping), and its sturdiness (stiffness). This equation helps engineers predict how much the building will sway when the ground shakes, ensuring that it doesn't collapse.

Then using Duhamel’s integral: $x(t) = -\int_{0}^{t} h(t−τ) mu¨ g(τ) dτ$

This formulation uses Duhamel's integral to calculate the relative displacement of the structure over time due to the ground acceleration. Here, $x(t)$ is the displacement response at a given time, while the integral sums the effects of the ground acceleration at previous times. By evaluating this integral, engineers can understand how the structure will behave (sway, deform) during an earthquake, based on the specific way the ground has moved up to that point.

Imagine you're at a concert where each song triggers waves of movement through the crowd. Each wave creates a ripple effect; those at the front feel the movement first, and it travels backward. Similarly, Duhamel’s integral captures how past ground movements affect the current response of a structure. Each previous movement contributes to the overall sway of the building, helping us predict how it will move in response to ongoing shaking.

Or: $x(t) = -\int_{0}^{t} \frac{1}{\omega_d} e^{-ζω_n(t−τ)} \sin[ω_d(t−τ)] u¨ g(τ) dτ$

This alternative formulation of the relative displacement response again uses Duhamel's Integral but expresses the impulse response in a sinusoidal form, taking damping into account. Here, $ω_d$ represents the damped natural frequency, and $ζ$ is the damping ratio, which represents how fast the oscillations decay. This equation is particularly insightful because it encapsulates how the structural response is influenced not only by sheer acceleration but also by how the system naturally reacts over time, considering its intrinsic properties.

Consider a swing: when you push it, it oscillates back and forth. If you push it too hard, it might swing wildly (high response), but if you do it gently, the motion feels smoother and eventually dampens. This equation mathematically describes how a building sways in response to shaking—high-frequency ground movements can cause it to sway significantly, but if the building is well-designed (damping), it will settle down faster, just like the swing returning to a calm state after being pushed.

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