11.6. Modal Analysis of Undamped MDOF Systems

Using the orthogonality of mode shapes, the equations of motion can be decoupled by expressing displacement as a sum of modal contributions:

n {u(t)}=∑ϕ q (t) i i i=1 Where:

• ϕ : Mode shape i
• q (t): Generalized (modal) coordinate i

In this part, we learn that the motion of an undamped Multiple Degree of Freedom (MDOF) system can be simplified. Instead of dealing with a complex system of equations, we express the overall displacement of the structure as a sum of its modes. Each mode has its own shape (ϕ) and corresponds to specific motion described by a generalized coordinate (q(t)). This method helps isolate the contributions of each mode to the overall motion, making the analysis simpler and more straightforward.

Think about a choir where each singer represents a mode. The overall harmony (the total displacement) is created by combining the individual voices (modal contributions) of each singer. Just like each singer has a unique voice (mode shape) and contributes a certain way (generalized coordinate) to the choir's performance, each mode adds its part to the system's overall response.

Substituting into the original equations yields n uncoupled SDOF equations:

q¨ (t)+ω2q (t)=0

i i i

After expressing the overall displacement in terms of the modal contributions, the original coupled equations transform into a series of uncoupled single degree of freedom (SDOF) equations. Each of these equations relates directly to the motion of a single mode, described by the term q¨(t) for acceleration and ω² for natural frequency. This simplification allows for independent analysis of the motions involved in different modes without interference from others.

Imagine that in a game of soccer, each player represents a mode. Instead of coordinating all players to work together on every play (which complicates things), you can assign each player (mode) to play their position independently. Each player runs and intercepts the ball based on their personal skills (natural frequencies) without overlapping with others, making the game simpler to analyze.

These can be solved independently, and the total response is obtained by summing all modal responses.

Once the uncoupled equations are derived, each equation can be solved separately, leading to results for each mode of the system. The total dynamic response of the MDOF system is then found by adding together the individual responses (modal responses) from all modes. This method of summation ensures that all contributions to the motion are accounted for, providing a comprehensive understanding of the structure's behavior.

Think of a group of musicians in a band. Each musician plays their instrument, contributing individually to the overall sound. After practicing, they come together for a performance. The entire performance (total response) is just the sum of each musician's individual contributions. If one musician were to play a solo, the others would still be playing their parts, and when combined, they create something larger than any single performance alone.

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