Engineered structures are present in almost all aspects of daily life. The houses and buildings in which we live and work, the cars we drive, aircraft, power plants, bridges, and dams are among the structures which are integral to society.
In order to design and build structures which will serve efficiently and safely, engineers make use of computer models called finite element models, or simply FEM. These detailed models can represent, to a high degree of fidelity, the complex geometric features of structures and the various materials which comprise these structures.
Engineers rely on these models to predict important structural responses arising from the dynamic environment in which structures serve. When an airplane touches down on the runway, engineers need to know that the stress levels in the wings and fuselage do not exceed allowable levels inherent to the materials making up the structure of the airplane. It is also important to predict the deflection of the wings under the aerodynamic forces which act on them. The swaying of a high-rise building in the wind, or due to an earthquake, must be predicted and accounted for in the design phase of the building.
These computer models, which are essentially sets of equations, can grow to be very large. FEM comprising millions of equations are routinely solved using computational algorithms developed specifically for such large sets of equations. The model building and solution process is referred to generally as finite element analysis or FEA.
Given that FEM are used for making critical design decisions, it is important to answer the question of the accuracy of the FEM. Does the FEM accurately represent the structure such that its predictions of motion and stress are acceptably close to reality? If not, what are the discrepancies between the predictions of the FEM and the responses of actual structure?
In order to answer these questions, a test is performed of either the newly-built structure (such as a building or a bridge) or a prototype, such as an aircraft, where several or many will ultimately be built. This test is designed not to reproduce the actual excitations found in the real world but applies a class of elemental dynamic excitations to the structure, which is designed to elicit a set of signature dynamic responses from the structure.
The FEA is used to calculate these signature responses, and the measured and calculated responses are compared. The degree of discrepancy is assessed, and the next, more challenging step, is to determine how to correct or improve the FEM such that its updated responses more closely represent the measured signature dynamic responses, to an acceptable degree.
There are many technical challenges associated with these tasks. Representing the structure in a FEM requires adhering to good modeling practices while accounting for the fact that the behavior of the actual structure is ultimately more complex than can be represented by the FEM, adding to the need to correct the FEM once it has been built.
Experimentally, the structure may need to be supported in a way that facilitates an accurate test. For example, an aircraft is supported in a way that mimics free flight, by providing a very soft suspension of the aircraft. The structure is outfitted with a number of transducers, which are devices which convert dynamic motions (e.g. vibration) to electrical signals. Other devices are used to apply the elemental dynamic excitations to the structure. A host of instrumentation and computers are used to collect the dynamic data applied to the structure. This includes measuring the forces applied to the structure (the excitations) and the responses of the structure which arise due to the excitations.
The creation of the FEM and testing of the structure are costly endeavors and involve efforts lasting weeks or months or more. Bringing a new product to market or into service is ideally done expeditiously. The cost of developing the FEM and performing the dynamic test is often high. It is, therefore, of great advantage to obtain as much usable data from the test as possible.
As mentioned earlier, one of the big challenges is determining how to improve the FEM once the disparity of its dynamic response predictions with the measured responses has been determined. To understand this challenge, consider the simple scenario in which you have created a FEM of a guitar string. The sound a string produces when plucked is a function of the string tension, the free length of the string, and the axial stiffness of the string, which is dependent on the material of the string and its diameter.
The FEA is used to calculate the natural frequencies of this string, which correspond to the fundamental tone and overtones heard when the string is plucked. If the actual string is plucked and the resulting tones differ from the FEA predictions, the question now becomes, “In what ways do the FEM inaccurately represent the actual string? Is the tension in the string represented incorrectly? Is the material model inaccurate? Is the diameter inaccurate?” The measured sonic frequencies, in conjunction with the FEA prediction of these frequencies, must be used to determine which of these parameters are in need of adjustment.
Now consider a FEM of a complex structure such as an aircraft. There are many more parameters in the FEM of the aircraft that could be in need of adjustment. The dynamic test of the aircraft produces an analogous set of characteristic vibrational frequencies, similar to that of the string. However, this set of frequencies is relatively small in number compared to the large set of parameters defining the aircraft FEM.
This disparity in the amount of information produced by the dynamic test versus the FEA can significantly impede the effort to localize in the FEM the source of the discrepancy in response. Our research has focused on a unique way to extract a large number of additional frequencies from the same test data, without having to perform any additional testing, thereby significantly reducing or eliminating the disparity. Returning to our string example, the dynamic test involves plucking the string and measuring the fundamental frequency of the string and its overtone frequencies. Due to the practical limited bandwidth of testing, only a relatively small number of these frequencies can be measured.
In our research, our technique actually produces the natural frequencies of the string corresponding to different free lengths, without having to physically change the length of the string. With respect to the example of an aircraft, the dynamic test is performed in a simulated free-flight condition, as mentioned earlier. Using this technique, the dynamic data so collected can yield the natural frequencies of the aircraft under a large number of alternative “boundary conditions,” each set of boundary conditions producing a unique set of frequencies. These boundary condition sets could include one wing tip held fixed, both wing tips held fixed, the nose of the aircraft held fixed, etc.
It is important to emphasize again that these boundary conditions need not be physically applied, which would be expensive, difficult, and, in many cases, impossible. However, the natural frequencies of a structure, such as this aircraft, corresponding to these alternative boundary conditions can be extracted from the dynamic data taken from the aircraft in its as-tested simulated free-flight condition.
Finally, it should be noted that the research described is also applicable to the problem of structural damage detection. In this context, the FEM is built for an undamaged structure and is corrected using test data of the undamaged structure. The structure sees service and potentially experiences damage, which might not be visible. The structure is tested, and the method is applied with the intent of discovering the location and severity of damage in the structure. This is in contrast to the FEM updating problem, where the intention is to uncover the severity and location of the error in the FEM.
These findings are described in the article entitled Minimum condition number by orthogonal projection row selection of artificial boundary conditions for finite element model update and damage detection, recently published in the Journal of Sound and Vibration. This work was conducted by Joshua H. Gordis from the Naval Postgraduate School, Jae-Cheol Shin from the Agency for Defense Development, and Matthew D. Bouwense.
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