11. Multiple Degree of Freedom (MDOF) System

In real-world structures, motion due to earthquakes cannot be accurately modeled with Single Degree of Freedom (SDOF) systems alone. Most structures such as buildings, bridges, and towers possess multiple masses distributed in space and can vibrate in several modes simultaneously. These types of systems are best represented as Multiple Degree of Freedom (MDOF) systems.

MDOF systems are crucial in accurately representing the behavior of structures during events like earthquakes. Unlike SDOF systems, which simplify motion to a single point of movement, MDOF systems account for multiple points of motion. This means that things like skyscrapers or long bridges, which have many parts that can vibrate in combination, need to be analyzed as MDOF systems for better understanding and safety.

Think of a musical orchestra where each musician plays their own instrument. If one musician plays alone, it's like an SDOF system—simple and straightforward. However, when the entire orchestra plays together, similar to an MDOF system, the sound is complex because each instrument adds its own unique tone and rhythm, creating a rich musical experience.

• Definition: A mechanical or structural system that requires two or more independent coordinates (degrees of freedom) to describe its motion. • Examples: o A shear building model with multiple floors. o Multi-span bridges. o Towers with mass concentrated at various levels. • Key Properties: o Each DOF has an associated mass and stiffness. o Coupled differential equations govern motion. o System responds in multiple vibration modes.

MDOF systems require more than one degree of freedom because they involve complex interactions between different parts of a structure. For instance, a multi-floor building sways not just as one unit but each floor moves independently to some extent, which must be accounted for to understand the overall response during vibrations. These systems typically lead to coupled equations, meaning that the motion of one part affects others, creating various vibration modes.

Imagine a multi-level parking garage. When cars move on one floor, the other floors experience movement too, affecting the whole structure. If we just looked at one floor in isolation (like SDOF), we wouldn’t capture all the dynamics and potential safety issues that arise during an earthquake.

Consider an n-DOF linear system with lumped masses and linear springs. The general form of the equations of motion without damping and external forces is: [M]{u¨(t)}+[K]{u(t)}={0} Where: • [M] = Mass matrix (n×n) • [K] = Stiffness matrix (n×n) • {u(t)} = Displacement vector • {u¨(t)} = Acceleration vector Key Concepts: • Mass matrix is usually diagonal in lumped-mass systems. • Stiffness matrix is symmetric and positive-definite. • This leads to a system of n coupled second-order differential equations.

The equation presented describes how the forces acting on an MDOF system relate to its motion. The mass matrix ([M]) represents how much 'weight' is concentrated in different parts of the structure, while the stiffness matrix ([K]) indicates how resistant those parts are to deformation. Together, they create a set of coupled equations that need to be solved to understand how the structure moves when impacted by forces like earthquakes.

Think of a team of people standing in a row holding onto a rope. When one person pulls on the rope, their movement also affects the others due to the tension in the rope. Just like that team, the different parts of a structure are interconnected, and understanding their collective movement requires analyzing these coupled equations.

To determine the natural behavior of the system, we assume a harmonic solution: {u(t)}={ϕ}sin(ωt) Substituting into the equation of motion: ([K]−ω2 [M]){ϕ}={0} This is a generalized eigenvalue problem, where: • ω2 are the eigenvalues (square of natural frequencies) • {ϕ} are the corresponding eigenvectors (mode shapes) Properties: • There are n natural frequencies and n mode shapes. • Mode shapes are orthogonal with respect to both [M] and [K].

In this section, we explore how to predict how structures behave naturally without any external influence. By assuming that displacements follow a sinusoidal pattern, we can derive 'natural frequencies'—specific speeds at which a structure likes to oscillate—and 'mode shapes'—the patterns of how it sways or vibrates at those frequencies. This leads us to a system where each mode is independent of the others, making calculations much simpler.

Imagine pushing a child on a swing. The swing has a natural rhythm—if you push it just right (at its natural frequency), it goes higher with less effort. Similarly, understanding a structure's natural frequency helps engineers predict how it will respond to external forces, such as the winds or earthquakes, avoiding potential structural failures.

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