Simple Quantitative Method to Compare Aircraft Wing Mode Shapes

Abstract

T HE importance of mode shapes is undeniable in contemporary flutter analyses. Not only are modes used to transform the governing equations from spatial to modal coordinates, but they are also used to transform flutter solution eigenvectors back to physical space to facilitate viewing of the critical flutter mechanism components. Mode shapes have even been used to gain insight into nonlinear aeroelastic behavior, such as limit cycle oscillations [1]. Mode shapes are so commonplace and easy to generate that their ability to overcome weaknesses inherent in flutter solution methods is not appreciated. In essence, modes are powerful tools that are not consistently exploited to their full potential. One cause for the underutilization of mode shapes could be the difficulty in quantitatively comparing them in an efficient manner with other modes. Quantitative comparison of modal frequencies is convenient, but a subjective assessment of the deflection mode shapes is the convention. Because of this subjectivity, the results of modal comparisons in this fashion could be regarded as less reliable or of limited utility. The situation is further impaired for complex or large models that are often cumbersome and sometimes obscure key shape features of interest. Anyone who has evaluated large quantities of flutter analyses is aware of the benefits of being able to compare mode shapes of different dynamical systems. Thesemode shapes can be indicative of energy-extraction and damping pathways within the structural system. They can provide insight into the similarity of predicted solution results, thereby reducing the number of critical analysis cases that may require flutter flight testing. They can also aid in the identification of critical flutter modes during flight testing, which is necessary for damping determination [2] and flutter margin [3,4] methods. Many methods are available to identify modes, but methods to explicitly compare one mode shape with another are not well documented in the literature. Themethods that are available formode shape comparisons are predominantly visual-based, such as 3-D still plots, animations, and node lines, although some work to extend these methods to quantitative approaches is available. Of particular interest to the flutter engineer is Maxwell’s work (as reported in [5]) to depict the wingtip deformation characteristics of the flutter mode shape and Northington’s [6] basis vector approach. There are also strain energy methods [7]. In short, it appears that many in-house methods for mode shape comparison likely exist, but they are only passed from generation to generation internally and are not disseminated widely to the engineering public. The development and dissemination of concise and distinctive mode shape description methodologies would allow the flutter engineering community to take better advantage of the inherent utility of mode shapes. Some applications that would become more practicable are sorting, tracking, and screening of modes for large quantities of analyses and using modes in alternative solution methodologies, such as artificial neural networks [8]. This Note describes a simple method in which curve-fit equations are used to depict the mode shapes of the wing. Some examples will be presented that compare the conventional 3-D still plots of the mode shape to the fitted curves as well as the coefficients of the curve-fit equations.