16.4.1. Eigenvalue Problem

Solving the free vibration problem (no damping, no external force) gives: [M]{u¨(t)}+[K]{u(t)}=0 Assuming a harmonic solution {u(t)}={ϕ}sin(ωt), we get:

In the analysis of multi-degree-of-freedom systems subjected to free vibrations, we start with the fundamental equation that describes the motion of the system, which is given by [M]{u¨(t)}+[K]{u(t)}=0. This equation represents a scenario where there are no external forces acting on the system, and there is no damping involved. To find the solutions for this equation, we assume a harmonic solution where the displacement {u(t)} varies sinusoidally with time, expressed as {u(t)}={ϕ}sin(ωt). Here, ϕ represents the mode shape, and ω is the angular frequency of the oscillation.

Think of a swing at a playground. When you push it, it starts oscillating back and forth in a smooth, repetitive motion. The fancy terms used in physics, like 'harmonic solutions', are essentially describing similar repetitive motions we see in everyday life, like the swing's back-and-forth movement.

([K]−ω2 [M]){ϕ}=0 This leads to the eigenvalue problem, where:

• ω2 are the eigenvalues (squared natural frequencies)
• {ϕ} are the mode shapes (eigenvectors)

After substituting our assumed harmonic solution into the original motion equation, we rearrange it to form the equation ([K]−ω²[M]){ϕ}=0. This equation represents the eigenvalue problem. In this context, ω² corresponds to the eigenvalues, which represent the squared natural frequencies of the system indicating how fast the system can oscillate in its natural state. The mode shapes {ϕ} are the eigenvectors, showing the specific patterns or shapes that the structure takes when it vibrates at those natural frequencies.

When you pluck a guitar string, it vibrates at specific frequencies, creating different musical notes. The frequencies that the string can vibrate at are akin to the eigenvalues, while the various shapes that different sections of the string take during its vibrations represent the mode shapes. Just like the guitar string, buildings and bridges have their own frequencies and modes that determine how they respond to forces.

The system has n natural frequencies and mode shapes for n DOFs.

In a multi-degree-of-freedom system, there are typically 'n' degrees of freedom (DOFs). For each degree of freedom, there is a corresponding natural frequency (how fast it can vibrate naturally) and a mode shape (how the structure deforms). Thus, a system with 'n' DOFs will yield 'n' eigenvalues (natural frequencies) and 'n' eigenvectors (mode shapes). Understanding these natural frequencies is crucial because they highlight how the system will behave when subjected to dynamic loads, like during an earthquake.

Consider a crowded stadium. When everyone stands and jumps at the same beat—this synchronized action creates a wave-like motion through the crowd. Each wave pattern you see represents a mode shape, while the rhythm of jumping represents the natural frequency. Just like the stadium crowd responds to the music, buildings respond to seismic waves, following their natural frequencies and mode shapes to dictate their motion.

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