Research on Fuzzy Structures

What is a Fuzzy Structure?

A complex mechanical system, like a submarine (you know, one of those cylindrical shells with hemispherical endcaps, submerged in water, strengthened with ribs and bulkheads, and loaded with lots of classified stuff), is impossible to completely model for two reasons. Firstly, the total number of degrees-of-freedom is way to large for any computer in existence to handle efficiently or economically. Secondly, even if it was computationally feasible to model such a structure, it is impossible to obtain complete, detailed information about all internal substructure details. The theory of fuzzy structures is a method of predicting the vibration of, and radiation and scattering from such a complex structure without having to know all the internal details. The part of the structure which can be modelled conventionally (shell, ribs, bulkheads, etc.) is called the master structure. The remaining part of the structure, which cannot be modelled conventionally, is called the fuzzy substructure.

The Pierce, Sparrow, Russell theory of fuzzy structures

Currently, at Penn State, the fuzzy substructure is modelled as a system of 1-DOF attachments whose properties are governed by a principle of maximum ignorance. That is, the number of attachments is unknown, though it is assumed that there are a large number of them; their exact locations are unknown, though it is assumed that they are "smeared" over a given area; their individual masses, stiffnesses, and natural frequencies are unknown. What is assumed to be known is how the total mass of all attachments is distributed according to natural frequency. This information, termed the mass-frequency distribution, is the key to the whole theory.

It has been shown theoretically, that the primary effects of the fuzzy attachments are to add a frequency dependent damping and a frequency dependent mass loading to the master structure.

Example: Backscatter from a plate strip with attached fuzzy substructure

Consider a finite width, inifinite length plate strip, simply supported in a rigid baffle, with fluid above. Attached to this plate strip are a large number of fuzzy 1-DOF mass-spring systems whose masses are distributed with natural frequency according to a Rayleigh distribution. An acoustic plane wave in the fluid is incident upon the plate strip at 45 degrees form the normal.

The backscattered Target Strength, a measure of the signal that is reflected back in the direction of the source is plotted below. The dashed lines correspond to a plate strip without any attachments. The solid lines correspond to a plate strip with fuzzy attachments such that the most probable resonance frequency of an attachment is matched to the frequency of the second flexural resonance of the plate. It is obious (even to the most casual observer) that the effect of the fuzzy attachments are rather substantial, completely damping out the second and third resonances of the plate. In addition, the first resonance is shifted downward in frequency, and the higher modes are shifted upwards.

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