Alyssa T. Liem Department of Mechanical Engineering, Boston University, Boston, MA 02215 e-mail: atliem@bu.edu J. Gregory McDaniel Department of Mechanical Engineering, Boston University, Boston, MA 02215 e-mail: jgm@bu.edu Andrew S. Wixom Applied Research Laboratory, Structural Acoustics Department, Pennsylvania State University, State College, PA 16804 e-mail: axw274@psu.edu Improving Model Parameters in Vibrating Systems Using Neumann Series A method is presented to improve the estimates of material properties, dimensions, and other model parameters for linear vibrating systems. The method improves the estimates of a single model parameter of interest by finding parameter values that bring model pre- dictions into agreement with experimental measurements. A truncated Neumann series is used to approximate the inverse of the dynamic stiffness matrix. This approximation avoids the need to directly solve the equations of motion for each parameter variation. The Neumman series is shown to be equivalent to a Taylor series expansion about nomi- nal parameter values. A recursive scheme is presented for computing the associated derivatives, which are interpreted as sensitivities of displacements to parameter varia- tions. The convergence of the Neumman series is studied in the context of vibrating sys- tems, and it is found that the spectral radius is strongly dependent on system resonances. A homogeneous viscoelastic bar in longitudinal vibration is chosen as a test specimen, and the complex-valued Young’s modulus is chosen as an uncertain parameter. The method is demonstrated on simulated experimental measurements computed from the model. These demonstrations show that parameter values estimated by the method agree with those used to simulate the experiment when enough terms are included in the Neu- mann series. Similar results are obtained for the case of an elastic plate with clamped boundary conditions. The method is also demonstrated on experimental data, where it produces improved parameter estimates that bring the model predictions into agreement with the measured response to within 1% at a point on the bar across a frequency range that includes three resonance frequencies. [DOI: 10.1115/1.4041217] 1 Introduction Within the past few decades, finite element models have become heavily relied upon to predict the dynamic responses of built structures. They provide valuable information about the structure’s characteristics that are used for a variety of applica- tions, ranging from design optimization to safety analysis. With the current predominance of predicting and analyzing structural responses, a precedence has been set on constructing accurate models. However, due to variations in manufacturing processes or unforeseeable alterations, it is often the case that as-built struc- tures differ from initial designs and models. Such differences may also occur when attempting to model damping in the structure, which often requires additional analyses to accurately model the energy dissipated in the built structure [1]. As a result, any dis- crepancy between the as-built structure and model results in model responses that do not agree with measurements and conse- quently an inaccurate finite element model. In an attempt to bring model responses into agreement with measurements, efforts have been made to develop a methodology for correcting finite element models, whose inaccuracies are mainly attributed to inaccurate input parameters. These parame- ters include, but are not limited to, dimensions, connection charac- terizations, and material properties. Classically, models have been corrected through exhaustive searches over parameter spaces; however, such processes are very computationally expensive and at times unfeasible [2]. These iterative processes also become very inefficient when the structure has hysteretic material proper- ties, due to the required new iteration at each frequency. With the increasing complexity and size of structures, a more efficient and robust method for correcting finite element models of damped structures is needed. To solve this problem, this paper proposes a methodology that utilizes the sensitivities of a structure. By definition, the sensitiv- ities of a system determine how the variation in the output of a model can be apportioned to different sources of variation in the model input [3]. In other words, the sensitivities characterize the change in model output due to variation in the model input param- eters. When the sensitivities of a finite element model are known, necessary corrections to inaccurate input parameters may be easily determined in order to bring model responses into agreement with measurements. Currently, an analytical expression has been found to determine the sensitivities of a system by differentiating the eigenvalue problem [4]. This approach, however, is limited to structures with either proportional damping or no damping at all. In a more gen- eral case, structures with hysteretic damping may be numerically analyzed using the finite difference method, which estimates the sensitivity as a first-order derivate found from either a forward, backward, or central difference evaluation [5]. Due to the addi- tional evaluations of the system to estimate the first-order or second-order approximation, this method suffers from high com- putational costs [6]. Additionally, the numerical precision and accuracy of the finite difference method are highly sensitive to the step size used in the approximation. To overcome the disadvantages of the finite difference method and iterative processes, the method presented in the paper offers a robust and efficient analysis for accurately determining the sensi- tivities of a structure with hysteresis in order to bring model response into agreement with measurements. The method utilizes the Neumann series approximation to allow for a convenient expression for a model response that can be exactly differentiated. The Neumann series approximation has been widely adopted in structural mechanics for various applications such as model reduc- tion [7], damage detection [8], analysis of mistuned blades [9–11], and interval analysis for bounded responses [12–20]. The preva- lent use of the approximation is attributed to its computational efficiency when approximating the response of a perturbed Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received April 12, 2018; final manuscript received August 6, 2018; published online September 17, 2018. Assoc. Editor: Jeffrey F. Rhoads. Journal of Vibration and Acoustics FEBRUARY 2019, Vol. 141 / 011017-1 Copyright V C 2019 by ASME Downloaded from https://asmedigitalcollection.asme.org/vibrationacoustics/article-pdf/141/1/011017/6380459/vib_141_01_011017.pdf by Boston University user on 26 February 2020