15.2. Mathematical Formulation of Mode Shapes

Consider an undamped linear MDOF system governed by:

[M]{u¨}+[K]{u}={0}

Where: • [M] = Mass matrix • [K] = Stiffness matrix • {u} = Displacement vector

This chunk introduces the basic equation governing a multi-degree-of-freedom (MDOF) system. The equation [M]{u¨}+[K]{u}={0} represents how the mass ([M]) and stiffness ([K]) of a structure relate to its motion ({u}). This is fundamental in the study of vibrations in structures, particularly how they respond to forces without any damping, meaning they continue to vibrate without losing energy.

Imagine a room full of pendulum clocks. Each clock's swinging (like the displacement vector {u}) is affected by its own mass (weight) and the way it's attached (stiffness). The room (the system) can be thought of as one big MDOF system where each clock's movement contributes to the overall behavior of the room's atmosphere.

Assuming a harmonic solution of the form:

{u(t)}={ϕ}sin(ωt)

Substituting and simplifying leads to the eigenvalue problem:

([K]−ω²[M]){ϕ}={0}

• ω = natural frequency • {ϕ} = mode shape (eigenvector)

In this chunk, we assume that the displacement of the system can be described using harmonic motion, which is a sinusoidal function represented by {u(t)}={ϕ}sin(ωt). When we substitute this assumption into the governing equation, we arrive at an eigenvalue problem involving the stiffness matrix ([K]), the mass matrix ([M]), and the natural frequency (ω). The eigenvalues denote the square of the natural frequencies at which the system can oscillate, and the eigenvectors represent the mode shapes.

Think of a swing at a park; when you pull it back and let go, it swings back and forth in a predictable, repetitive motion (harmonic). The swing's motion can be described by its position at various points in time, similar to how the model captures the shape of the swing's path during its oscillation.

This formulation yields eigenvalues (natural frequencies squared) and eigenvectors (mode shapes).

After setting up the eigenvalue problem, solving it provides two important outputs: eigenvalues and eigenvectors. Eigenvalues are essentially the squares of the natural frequencies of the system, reflecting the types of oscillation the system can have. Eigenvectors, or mode shapes, describe how different parts of the structure will move relative to one another at those frequencies. Understanding the relationship between natural frequencies and their corresponding mode shapes is essential for predicting structural behavior under dynamic loads.

Consider a music note played by a guitar string; the pitch (frequency) can change depending on how tightly the string is pulled (akin to the stiffness of a structure). Each unique way the string vibrates when plucked corresponds to different eigenvectors (the mode shapes), where each shape might produce a different sound (or response in a structure).

--