Keywords: frequency response function, spectral transfer matrix, viscous and hysteretic damping, proportional damping.

The frequency response function (FRF) is commonly used to analyze dynamic systems from the frequency-domain point of view. It is well known that the FRF gives a relevant characterization of the system behavior subjected to harmonic forced vibrations. Traditionally, in problems where the FRF is required, the finite element method (FEM) is used to assemble the mass, stiffness and damping matrices. From these matrices, different methods can be applied to the computation of the FRF, the most common being the modal space construction (up to a specific eigenvalue) [1]. This form will be named DIFRE (from "Discrete") because it is obtained from a finite element discretization. In light damped structures, it is common to use a Rayleigh model to model the damping matrix in DIFRE analysis [2]. Thus, two coefficients will be necessary, one for the mass and another for the stiffness matrix.

The work presented starts with the analysis of the motion partial differential equations (function of position and time) deduced from the Euler-Bernoulli beam theory. A set of transformations based on spectral methods [3], allows the FRF to be obtained as an ordinary differential equation solution derived from the motion, function of position and frequency. A method based on the Laplace transform is used to find the solution, involving the computation of a spectral transfer matrix [4]. The entries of this matrix contain the complete model information, allowing the development of a FRF analytical version called ANFRE, from "Analytical". ANFRE includes a mixed Kelvin-Voigt (KV) damping model based on a combined viscous and hysteretic [2].

The main objective of this paper is the comparison between ANFRE and the aforementioned DIFRE. For that, the question: what Rayleigh coefficients yield the best fitting between DIFRE and ANFRE curves? is answered in this paper, through the extraction of modal damping factors (eventually the Rayleigh coefficients) from both models. These factors will be calibrated with an optimization process, that in turn update a DIFRE model. The adjustment between both curves will be discussed as a function of the KV parameters.

1

K. Worden, G.R. Tomlinson, "Nonlinearity in Structural Dynamics", Institute of Physics Publishing, Bristol and Philadelphia, 2001.

2

C.W. de Silva, "Vibration Damping, Control, and Design", CRC Press, 2007.

3

S. Gopalakrishnan , A. Chakraborty, D.R. Mahapatra, "Spectral Finite Element Method", Springer, 2008.

4

U. Lee, "Vibration Analysis of one-dimensional structures using the spectral transfer matrix method", Engineering Structures, 22(6), 681-690, 2000. doi:10.1016/S0141-0296(99)00002-4

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