11.8. Response of MDOF Systems to Dynamic Loading

When subjected to external forces or ground acceleration (e.g., earthquakes), the equation becomes: [M]{u¨(t)}+[C]{u˙(t)}+[K]{u(t)}={f(t)}

This equation represents how a Multiple Degree of Freedom (MDOF) system responds when external forces, like those caused by an earthquake, act upon it. Here, [M] denotes the mass matrix of the system, {u¨(t)} is the acceleration of the displacement vector, [C] is the damping matrix, {u˙(t)} signifies the velocity of the displacement, and [K] represents the stiffness matrix. The term {f(t)} symbolizes the external force applied to the system. In essence, these elements come together to form a dynamic relationship that helps us understand how structures react to sudden forces.

Imagine a car that is driving on a highway when suddenly it hits a pothole (the external force). The car's suspension system (comparable to the MDOF system's damping and stiffness) works to absorb the shock. Just like the car's movement is affected by the force from the pothole, a structure’s movement under seismic activity is defined by the interplay of forces, mass, and its ability to absorb and dissipate energy.

Or, for base excitation due to earthquake: [M]{u¨(t)}+[C]{u˙(t)}+[K]{u(t)}=−[M]{u¨ (t)} g

In this equation, we are discussing a scenario where the base of the structure is subjected to ground acceleration during an earthquake, denoted as {u¨ g(t)}. The right side, which equals - [M]{u¨(t)}, illustrates how the mass of the system experiences this base excitation caused by the ground shaking. The negative sign indicates that the acceleration of the ground affects the structure in an opposite manner, challenging its stability and causing it to respond dynamically to prevent failure.

Think of a building on a shaking ground as a person standing on a moving train. As the train suddenly lurches, the person must adjust their balance to avoid falling. Similarly, the structure must react to the shifting ground to maintain its integrity. This highlights the importance of understanding the ground's effects on the structure's accelerations.

Using modal superposition, the solution is: {u(t)}=∑ϕ q (t) i i i=1

Modal superposition is a method used to solve the equations of motion in MDOF systems. It involves expressing the total displacement {u(t)} as a sum of contributions from individual modes of vibration, represented by the mode shapes ϕ and their respective modal coordinates q_i(t). This technique allows us to simplify complex dynamic problems into manageable parts, focusing on each mode separately while still capturing the overall behavior of the system.

Think of a choir singing a complex piece of music. Each singer (mode) contributes to the overall harmony (total response). If we analyze each singer's part separately and then bring them together, we can better understand how the choir’s performance (structural response) comes together, making it easier to identify and fine-tune each part for a beautiful sound.

Each modal coordinate q i(t) is found by solving the uncoupled modal equation with forcing.

After expressing the displacement as a sum of modal contributions, each modal coordinate q_i(t) is solved independently using specific modal equations that consider the forces acting on each mode. This uncoupling allows engineers to analyze the response of each vibration mode under the external forces one at a time. Each of these equations reveals how the corresponding mode of vibration responds to the seismic loading, leading to a more comprehensive understanding of the MDOF system's dynamics.

Consider a multi-layer cake where each layer represents a different mode of vibration. When you push down on one layer (applying external force), each layer will respond in its way, possibly compressing or bouncing back. By examining each layer's response independently, we can understand how the entire cake (the MDOF system) reacts when subjected to external forces.

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