Vibration Modal Density and Number of Modes in a Cylinder

Scripts to calculate the vibration modal density and number of modes in a cylinder, for use in Statistical Energy Analysis.

When conducting Statistical Energy Analysis simulations involving a cylinder, it is often useful to calculate the vibration modal density. The modal density can be calculated using a formula for the number of vibration modes below a certain frequency, and using this formula to calculate the difference in number of modes between an upper and lower frequency limit, then dividing the difference by the frequency bandwidth between the upper and lower frequencies.

The attached Matlab scripts compare the vibration modal density of an example cylinder calculated using three theories:

1. R.S. Langley (1994), "The Modal Density and Mode Count of Thin Cylinders and Curved Panels", Journal of Sound and Vibration, Volume 169, Issue 1, Pages 43-53,https://doi.org/10.1006/jsvi.1994.1005, https://www.sciencedirect.com/science/article/pii/S0022460X84710054
2. E. Szechenyi (1971), "Modal densities and radiation efficiencies of unstiffened cylinders using statistical methods", Journal of Sound and Vibration, Volume 19, Issue 1, Pages 65-81, https://doi.org/10.1016/0022-460X(71)90423-8, https://www.sciencedirect.com/science/article/pii/0022460X71904238
3. R.H. Lyon and R.G. DeJong (1995), "Theory and Application of Statistical Energy Analysis", Elsevier Inc., London, UK, Second Edition, ISBN 0-7506-9111-5, page 195, https://www.elsevier.com/books/theory-and-application-of-statistical-energy-analysis/lyon/978-0-7506-9111-6

The three Matlab files in the attached .zip file are:

1. compare_langley_vs_szechenyi_vs_lyon.m - a script that is used to calculate and plot the vibration modal density of an example cylinder from the paper by Langley (1994), using the three theories listed above.
2. langley.m - a Matlab function that calculates the vibration modal density using the theory presented in Langley (1994).
3. modal_density_cylinder_langley.m - a well documented script that can be 'published' in Matlab, which contains extensive Latex equations, to calculate the vibration modal density of the example cylinder in Langley (1994), and plots Figure 5 from Langley (1994), and Figure 6a from Szechenyi (1971).

Note there are several typos in the published paper by Langley (1994), which would cause difficulties for researchers, and these typos have been corrected in the attached Matlab scripts.

The work by Langley (1994) calculates the vibration modal density using the integration of elliptic functions, which is complicated. Whereas the work by Szechenyi (1971) uses a semi-empirical approach, where he calculated the modal densities of a number of cylinders, plotted them on a graph, and then determined a semi-empirical formula to fit the data points, resulting in formulas that are simpler to implement compared with Langley (1994). The expression from Lyon and DeJong (1995) is also simple to implement.

These three theories are used to calculate the vibration modal density of an example cylinder described in Langley (1994), and the modal densities are plotted to show that the results are similar, except near the 'ring frequency'.

It is important to note that in the published literature, there are various interpretations and definitions for the 'ring frequency' of a cylinder, so one needs to ensure that the formula used is appropriate for the corresponding theory for the vibration modal density of a cylinder.

These formulas are described in the upcoming textbook:

Bies, Hansen, Howard, Hansen (2023), "Engineering Noise Control", CRC Press, Sixth Edition.