REVIEW AND EVALUATION OF SHAPE EXPANSION METHODS Etienne Balmès École Centrale Paris, MSSMat 92295 Châtenay-Malabry, France balmes@mss.ecp.fr ABSTRACT Correlation criteria and modeshape expansion techniques deal with the spatial incompatibility linked to the measurement of modeshapes through a limited set of physical sensors and their analytical prediction at a (larger) number of finite element (FE) degrees of freedom (DOFs). Expansion techniques have two conflicting objectives : estimate the motion at all FE DOFs and smooth test errors. The paper first proposes a unified per- spective that covers most existing expansion methods. Exam- ples, based on a model of the GARTEUR SM-AG-19 testbed, are used then to analyze the performance of major expan- sion methods (modal, static, dynamic, minimum residual, and minimum residual with test error). The impact of three usual sources of errors is then considered : inaccuracies in the way true sensor locations are taken into account; error in the test data; errors in the FE model used for the expansion. 1 INTRODUCTION Expansion methods seek to estimate the motion at all DOFs of a finite element model based on measured information (mode- shapes or frequency response functions) and prior, but not necessarily accurate, information about the structure under test in the form of a reference finite element model. The paper first seeks to give a unified perspective on expan- sion methods allowing their classification. The proposed clas- sification is based on how various methods combine informa- tion about and errors. Section 2.1 addresses subspace methods, which use the model to define a subspace of possible FE deformation, and define a projector giving a di- rect mapping between test and expanded shapes based on a minimization of test errors. These expansion methods include modal/SEREP [1, 2] , static (based on Guyan reduction [3] ), dynamic [4] and hybrid [5, 6] . Section 2.2 addresses model based methods that combine test and modeling error mea- sures [7, 8] . Finally, section 2.3 addresses the distinction be- tween sensors and DOFs, and model reduction techniques which are needed for a general and numerically acceptable implementation of expansion methods. The second part of the paper uses a model of the GAR- TEUR SM-AG-19 testbed [9] to analyze standard sources of error and evaluate the qualities of major expansion meth- ods: modal, static, dynamic, minimum residual, and minimum residual with test error. Three major sources of errors are suc- cessively considered : the effect of inaccuracies in the way true sensor locations are taken into account; the impact of er- ror in the test data; the impact of errors in the FE model used for the expansion. 2 UNIFIED PERSPECTIVE ON EXISTING METHODS 2.1 Subspace expansion methods : interpolation, modal/SEREP, static A large class of methods, called here, only use modeling information to select a subspace of possible dis- placements with dimensions inferior or equal to the number of sensors. If [T ] N×NR is a basis of this subspace, one assumes that the full displacement is of the form {q Ex} =[T ]{qR} (modeling error is minimized for responses within the sub- space). An estimate of the full response is then simply ob- tained by minimizing test error (distance between the test data and the associated response for the expanded shape). The minimum is generally obtained by solving the least-squares problem {qR} = arg min ||{yT est }- [c][T ]{qR}|| 2 2 where the observation matrix c is defined in section 2.3 for readers not familiar with it. Wire-frame representations are the most trivial form of sub- space expansion method : they assume that on the line be- tween two connected test nodes the motion varies linearly be- tween the values of motion taken at each end. The considered subspace corresponds to linear responses along each line to a unit displacements of each sensor. In this case the sub- space dimension is equal to the number of sensors so that {yT est } =[c]{qEx}. Spline interpolations are a way to extend a geometrical con- struction of the subspace but they are ill suited for complex geometries. If one has a FEM model of the structure under test (even a poor one), the easiest approach to select a subspace is to use this model. The two natural subspaces in modal analysis are the low frequency modeshapes and the static responses to loads or displacements applied at sensors. Expansion based on the subspace of low frequency modes