Hysteretic damping of structures vibrating at resonance: An iterative complex eigensolution method based on damping-stress relation George D. Gounaris, Eleftherios Antonakakis, Chris A. Papadopoulos * Machine Design Laboratory, Mechanical Engineering Department and Aeronautics, University of Patras, 26500-Patras, Greece Received 4 September 2006; accepted 19 February 2007 Available online 16 April 2007 Abstract In this paper the damping is examined as an engineering property used in analysis and design of structures and machines. The design engineer needs to know not only the stresses of his structure or machine, under steady state conditions but also the stresses under res- onance conditions. Then the material damping, as a function of the stress of the structure, has an important role to play and ignoring the damping the calculated stresses are far from reality. The nonlinearity here is due to the dependence of the hysteretic damping on the stress of the structure. Speciﬁcally here two problems are investigated in the following way: Firstly the direct problem is solved. The direct problem is to ﬁnd the maximum bending stress at the resonance when the relation of the dissipating energy (or of the hysteretic damping) vs. the bending stress is known in advance. To perform this calculation, a useful tool for the design engineer, the structure is modelled using the continuum mechanics analytical approach or the ﬁnite elements (FE) method. Then the eigenvalues are calculated and using an iterative procedure the real stress. The procedure presented here is called iterative com- plex eigensolution method (ICEM). Secondly the inverse problem is solved. The inverse problem is to ﬁnd the relation between the hys- teretic damping and the bending stress. For this purpose the logarithmic decrement is experimentally measured, the eigenvalues and the maximum bending stress of the structure, excited at the eigenvalue, when the damping is the same as the measured one, are computed using the ﬁnite elements method. Once the bending stresses are found in each discrete element of the structure, then the mathematical expression of the relation of the dissipating energy and the stresses can be speciﬁed by minimizing a suitably formed objective function. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Hysteretic damping; Damping-stress relation; Damping at resonance; Iterative complex eigensolution method (ICEM); Damping auto- determination; Hysteretic; Damping; Stress; Resonance; Eigensolution 1. Introduction 1.1. Hysteretic damping models Several models for the hysteretic damping are intro- duced last decade in order to ﬁnd the response in the time domain viscoelastic linear analysis, besides the well known Maxwell, Kelvin–Voigt and their diﬀerent combinations such as the standard linear solid model. These models are extremely useful studying i.e. the seismic response of struc- tures in the time domain. For completeness purposes these models are presented here. Non causal constant hysteretic model: The constant hysteretic model is a pathological model because it is non-causal. This means that the model responds prior to the application of an impulsive excitation. GðxÞ¼ G 1 þ iG 2 sgn x e  ¼ G 1 1 þ icsgn x e  h i ; ð1Þ where G 1 is the frequency-independent storage modulus, G 2 sgnðx=eÞ is the loss modulus, x is the frequency variable and e is an arbitrary positive real number with units of rad/s 0045-7949/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruc.2007.02.026 * Corresponding author. Tel.: +30 261 096 9426; fax: +30 261 099 6258. E-mail address: chris.papadopoulos@upatras.gr (C.A. Papadopoulos). www.elsevier.com/locate/compstruc Available online at www.sciencedirect.com Computers and Structures 85 (2007) 1858–1868