How to modify beam frequencies to desired frequencies by adding masses to the beam

What is it about?

Given a beam, it has certain natural frequencies of vibrations. Adding a mass or several somewhere along the beam changes those frequencies. Suppose we want the beam to have certain specific, chosen frequencies of vibrations. Can we take a given beam and add masses to it somewhere so that we can get the frequencies we want? This paper shows how to solve this problem.

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Why is it important?

This paper gives a structured approach to solving this type of inverse problem exactly and analytically. Engineering structures such as a beam can have their frequencies changed ('tuned') to more desired frequencies. We also show that there are many solutions to a given type of inverse problem and it will be up to the engineer to decide which one is best - the math simply returns the list of possible solutions.