11.3. Mode Shapes and Natural Frequencies

To determine the natural behavior of the system, we assume a harmonic solution:

{u(t)}={ϕ}sin(ωt)

In dynamics, we often look for solutions that repeat over time, known as harmonic solutions. A common way to represent these types of solutions is to express the displacement of a system as a product of a mode shape, {ϕ}, and a sine function that describes a particular frequency, ω. The sine function oscillates, allowing us to explore how the system moves in response to vibrational stimuli.

Think of a swing at a playground. When you push a swing, it moves back and forth in a periodic manner. The way the swing moves can be described similarly to the harmonic solution, where the swing's position depends on the angle (the mode shape) and the frequency of your pushes.

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)

When we substitute our assumption into the equation of motion, we transform it into a specific type of mathematical problem known as an eigenvalue problem. In this context, ω² represents the natural frequencies of the system, while {ϕ} gives us the mode shapes, which are the specific patterns of motion the system can exhibit at these frequencies. Essentially, this means each natural frequency corresponds to a unique way the system can vibrate.

Consider a guitar string. When plucked, the string vibrates at different frequencies, producing different musical notes. Each note corresponds to a specific natural frequency of the string, and the way the string moves for each note corresponds to its mode shape.

Properties:

• There are n natural frequencies and n mode shapes.
• Mode shapes are orthogonal with respect to both [M] and [K].

For every system with n degrees of freedom, there will be n unique natural frequencies and mode shapes. One important property is orthogonality, which means that the different mode shapes do not interfere with each other when analyzed mathematically using mass and stiffness matrices. This property greatly simplifies the analysis because it allows us to consider each mode independently when assessing the system's dynamic behavior.

Imagine a symphony orchestra where each musician plays a unique instrument. Each instrument produces distinct sounds that do not overlap when played together, much like how each mode shape works independently during vibration analysis. This allows the 'orchestra' of vibrations in a structure to work together without disrupting each other.

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