16.3. Mathematical Modeling of MDOF Systems

In seismic analysis, mass is usually lumped at each floor level (nodes), and the stiffness is represented by springs connecting these nodes. The model simplifies:

• Mass matrix [M]: Diagonal matrix with masses at each DOF.
• Stiffness matrix [K]: Symmetric matrix representing inter-storey stiffness.
• Damping matrix [C]: Often assumed as proportional damping for simplicity.

In order to analyze the behavior of multi-degree-of-freedom (MDOF) systems during seismic events, we use a simplified model where we focus on mass, stiffness, and damping characteristics.

1. Lumped Mass: Instead of considering every part of the building’s mass, we represent the mass concentrated at each floor level. This makes the analysis more manageable without losing essential dynamic characteristics.
2. Mass Matrix: The mass matrix [M] is a diagonal matrix, meaning that it lists the masses of each floor along the diagonal and zeros elsewhere. This shows the system has 'n' degrees of freedom, corresponding to the number of floors.
3. Stiffness Matrix: The stiffness matrix [K] is symmetric, capturing how the floors are connected and their ability to resist deformation under loads. The off-diagonal elements may represent interaction effects if floors affect each other’s stiffness.
4. Damping Matrix: Damping is a mechanism that dissipates energy and is often simplified by assuming it is proportional to the mass and stiffness matrices. This means we can calculate how the system will respond to dynamic loading more easily.

Think of a multi-story building like a stack of blocks. Each block represents a floor with a certain weight (mass). In this analogy, the way the blocks are connected (structure stiffness) affects how they sway when pushed (dynamic loading), and putting sponge under each block (damping) can help absorb and slow down the motion during such disturbances, mimicking how real buildings behave during an earthquake.

For an undamped, linear elastic MDOF system subjected to external forces: [M]{u¨(t)}+[K]{u(t)}={f(t)} For a damped system: [M]{u¨(t)}+[C]{u˙(t)}+[K]{u(t)}={f(t)} Where:

• {u(t)} = displacement vector
• {u˙(t)} = velocity vector
• {u¨(t)} = acceleration vector
• {f(t)} = external force vector

To understand how MDOF systems react to external forces, we utilize mathematical equations that depict their motion:

1. Undamped System: The equation [M]{u¨(t)} + [K]{u(t)} = {f(t)} links the mass matrix with the acceleration of the structure and the stiffness matrix with the displacement, thereby showing how the movement of the structure depends on the forces acting upon it.
2. Damped System: In practice, we often account for damping since it plays a key role in energy dissipation. The equation for a damped system [M]{u¨(t)} + [C]{u˙(t)} + [K]{u(t)} = {f(t)} adds a term for the damping matrix [C] which is related to the velocity of the structure.
3. Vector Definitions: In these equations, {u(t)} is the displacement vector describing where each mass is located, {u˙(t)} is the velocity vector showing how fast each mass is moving, {u¨(t)} indicates the acceleration, and {f(t)} is the external force vector indicating the forces acting on the system.

Imagine you are trying to push a swing (which represents our MDOF system). The motion of the swing is influenced by how hard you push it (external forces), how fast it moves (velocity), and how much it slows down when you stop pushing (damping). The equations we use help us predict how the swing will move over time, just like those modeling a building's response to seismic activity.

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