The spider as an artist
Has never been employed
Though his surpassing merit
Is freely certified
Emily Dickinson
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The natural frequencies and mode shapes of multibody structures can be found by collecting the connection constraints into a nonlinear eigenvalue problem (NLEVP). This project provides tools for constructing such NLEVPs for networks of elastic strings, spiderwebs being typical examples. The NLEVPs in this project are similar to dynamic stiffness matrices, but they do not need to be symmetric and their derivation does not require algebra to eliminate displacement constraints.
An NLEVP maps complex scalars to square matrices. When the matrix is singular, the input is called an eigenvalue, and all corresponding null vectors of the matrix are its eigenvectors. The NLEVPs produced by this project have eigenvalues at natural frequencies of the related multibody system. With some work, the corresponding mode shapes can be derived from the eigenvectors.
The primary purpose of this project is to provide interesting problem examples, not to provide solution methods.
- Download the project and update MatLab's working directory (or search path)
- Run
demo_tritare- Figure 1 should display a wiggly Y-shape that is a modal vibration of a "tritare" string
- Examine the code of
demo_tritare.mto see how the outputs of NLEVP-producing functions liketritareare used - Modify
demo_tritare.mto choose a different mode (choose a value ofjfrom 1 to 24) - Rerun
demo_tritareto see a different mode shape
tritare.m is the only file in the project that provides pre-calculated eigenvaleus.
In order to draw the mode shapes of the other systems, you will need to calculate the eigenvalues yourself.
This project runs in base MatLab without any additional packages needed. It was developed with versions 2020a and 2021b but is expected to work in much older versions.