13.3. Free Vibration Analysis of MDOF Systems

Assumption of no damping and no external force.

In free vibration analysis, one of the key assumptions is that there is no damping in the system and that no external forces are acting on it. This simplifies the analysis because damping typically dissipates energy, making it more complex to analyze the system's response. By assuming no external forces, we isolate the natural oscillatory behavior of the system, focusing solely on its inherent characteristics.

Imagine a swing in a park. If you push the swing (an external force) or if the swing has some friction (damping), it doesn't oscillate in a simple manner. However, if you let the swing go without pushing it or if it were perfectly smooth (no friction), it would swing back and forth in a predictable, regular pattern. This is similar to the concept of free vibration in engineering.

Solution using harmonic motion: {X(t)} = {ϕ}sin(ωt)

To solve the free vibration problem of MDOF systems, we assume that the motion of the system can be described using harmonic functions, specifically sinusoidal functions. Here, {X(t)} represents the displacement of the system as a function of time, {ϕ} stands for the mode shape of the system, and ω denotes the angular frequency of the oscillation. This approach allows us to express the complex motion of the system in terms of simpler oscillatory modes, which are easier to analyze.

Think of tuning forks. Each fork vibrates at a specific frequency, producing a clear tone when struck. If you represent how the fork vibrates over time, that motion can be described using sinusoidal functions. Similarly, by expressing a complex vibrating system into its fundamental modes of motion using harmonic functions, engineers can better understand and predict the system's behavior.

Substituting into equations of motion leads to: ([K]−ω²[M]){ϕ} = 0

When we substitute the harmonic motion solution into the equations of motion for the MDOF system, we arrive at the expression ([K]−ω²[M]){ϕ} = 0. This forms an eigenvalue problem, where the goal is to find the values of ω² (which represent squared natural frequencies) and the corresponding mode shapes {ϕ}. This is a critical step because solving this eigenvalue problem yields the natural frequencies at which the system tends to oscillate freely.

Imagine a set of swings in a playground. Each swing can be set into motion at specific frequencies based on its length and weight. The frequencies at which these swings resonate are akin to the natural frequencies we find in the mathematical model. By determining those frequencies (eigenvalues), we can understand how the swings (mode shapes) will behave when set in motion.

Procedure: a. Formulate [M] and [K] b. Solve characteristic equation det([K]−ω²[M])=0 c. Obtain natural frequencies ω and corresponding mode shapes {ϕ}**

The analysis of free vibrations of MDOF systems can be systematically approached through the following steps:

1. First, we need to define the mass matrix [M] and stiffness matrix [K] that characterize the system's properties.
2. Next, we solve the characteristic equation, which involves finding the determinant of the matrix ([K]−ω²[M]) and setting it equal to zero. This step indicates the values of ω for which the system can oscillate.
3. Finally, using these natural frequencies, we can determine the associated mode shapes. This provides us with a complete picture of how the system will behave in free vibration.

Think of building a musical instrument, like a guitar. First, you determine the materials and dimensions (analogous to formulating [M] and [K]). Then, you figure out the notes it plays as you pluck it (analogous to solving the characteristic equation for natural frequencies). Lastly, you visualize or hear the specific sounds each note makes (determining mode shapes). Following these steps leads you to understand the complete sound profile of the guitar.

--