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    Topic review

    Nonlinear Damping Identification

    Subjects: Others
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    Submitted by: Tareq Al-hababi


    In recent decades, nonlinear damping identification (NDI) in structural dynamics has attracted wide research interests and intensive studies. Different NDI strategies, from conventional to more advanced, have been developed for a variety of structural types. With apparent advantages over classical linear methods, these strategies are able to quantify the nonlinear damping characteristics, providing powerful tools for the analysis and design of complex engineering structures.

    1. Introduction

    In recent decades, a huge number of engineering structures such as various civil structures (e.g., bridges, dams, buildings, etc.), rotating machines, aircraft, etc., have been designed and widely used in real life-services [1]. Such structures, with either simple or complex geometric or material properties, are subjected to different levels of vibrations from numerous sources, including earthquakes, wind loads, vehicle motions, imbalance of rotating machines, etc. [2]. Excessive levels of vibrations largely affect the performance, health condition, and serviceability of structures and even lead to structural instability and failures [3].

    In general, structural damping is a preferable dynamic characteristic able to reduce the degree of system vibrations to an acceptable level [4]. Damping and its variation occur due to the continuous dissipation of energy [5], both internally and externally, linked with factors such as material degradation, geometrical changes, boundary conditions, etc. [6]. With complex mechanisms, structural damping is commonly classified into three types. First, the fluid damping, which originates from hydrodynamic or aerodynamic forces surrounding the structures [7]. Second, the material damping, which appears due to complex atomic-molecular interactions inside materials [8]. Third, the structural damping, resulting from Coulomb friction between parts within a structural system [9]. For damping characterization, several simplified models have been suggested; for instance, viscous damping, hysteretic damping, and Coulomb frictional damping models [10]. In real applications, equivalent viscous damping is commonly used to model the overall behavior of damped systems [11].

    Most structural systems show a certain extent of nonlinearity associated with different sources [12][13]. However, neglecting the nonlinearity is acceptable in many cases for the sake of simplification of analysis [14][15]. In other cases, nonlinear behavior plays a dominant role. Nonlinearity neglection should be prevented in such cases, as it may lead to erroneous predictions of system behaviors [16]. Among the causes of system nonlinearity, nonlinear damping [17][18] is often regarded as the most influential; this complexity of which makes it challenging to perform system identification [19]. Moreover, to better accommodate the development of advanced material with strong nonlinear behavior, research on the impact of nonlinear damping is of increasing importance [20][21][22].

    Some damping identification approaches, theoretical and experimental, have been developed [23]. Experimental techniques for damping estimation show advantages in accuracy and reliability. Damping identification approaches provide straightforward explanations for damping properties as compared with theoretical approaches [24]. The enhancement technique of engineering structures is carried out by measuring inputs and outputs during the experiment of real structures. This enhancement technique is made to avoid unwanted behavior of the system subject to damping effects [25].

    Several hypotheses can be used for linear and nonlinear damping of low-damping systems [26][27][28]. High-damping systems usually involve structural damage [29]; however, nonlinear effects on vibrating systems cannot be neglected because of their significant impact on dynamic behaviors [30][31]. Linear damping approaches can provide precise structural numerical predictions [32][33]. Nevertheless, in numerous modern applications, the effectiveness of these methods cannot be guaranteed, especially when the structures are complex, e.g., composed of composite materials or operating in a hostile environment [34][35]. Therefore, linear methods should be limited to identifying damping in specific simple structures that operate under normal conditions [36]. These drawbacks have, in the last decade, contributed to the rapid development of nonlinear damping methods [37][38].

    Damping identification is a challenging task [39] performed by developing a variety of analytical [40][41][42] and experimental methods of linear and nonlinear systems [43][44][45]. Moreover, NDI is a practical aspect being conducted to prevent structural failure and tragic events caused by structural damages [46][47]. In addition to enhancing the safety and maintenance of key structures, it also contributes to the control of systems and predicting structural systems responses under nonlinear damping properly [48][49].

    NDI for dynamic systems can be classified into seven categories: Linearization methods, time-domain methods, frequency-domain methods, time-frequency methods, modal methods, black-box modeling, and model updating methods [1]. Classification can also be made from other perspectives, for example, parametric and non-parametric [50][51]. An example of linearization methods is the equivalent linearization approximation (ELA), which is a common method used in applications such as a spring-suspended sectional model system. ELA is utilized for bridges and aeroelastic systems and dampers and shock absorbers used in control systems [52].

    The time-domain methods are well-known methods such as the log decrement method used in lightly damped systems and aeroelastic systems [53][54]. In addition, the Hilbert transform (HT) is used to analyze vibration systems [55], heat exchangers, and damage detection of reinforced concrete (RC) structures and composite materials [56][57]. The frequency-domain methods are known by their mathematical simplicity and ability to provide insightful interpretation [58]. For example, harmonic balance nonlinearity identification (HBNID) was employed in systems such as civil engineering, actuators, bioengineering devices, sensors, and robotics [59]. Furthermore, the frequency response functions (FRFs) have been widely utilized to study nonlinear damping in adhesive joints [60][61].

    Time-frequency methods, e.g., those based on continuous wavelet transform (CWT), are powerful tools used in applications such as vibration absorbers that are broadly used in naval architecture, rotor-bearing systems, and constructions [62]. In addition, HT was employed in applications such as unbalance of rotating machines, ship movement control, and damage detection of RC beams based on free vibration measurements for nonlinear damping determination [63]. Modal methods are considered particularly useful in the field of structural dynamics and damage identification. The resonant decay method (RDM) was applied to investigate the nonlinear damping in civil, aircraft, and various types of dampers. The wavelet transform (WT) was used in instantaneous damping coefficient identification for damage detection in concrete, automotive, aerospace, and simple built-up structures comprising two bolted beams [64]. Black-box modeling (BBM) is an accurate and efficient method in describing the dynamic behaviors of structures. For example, a fuzzy wavelet neural network (FWNN) was used for nonlinear identification in systems such as vehicle magnetorheological (MR) fluid dampers, aeroelastic systems, modern industries, control systems, and military and defense equipment [65].

    Model updating methods include, for instance, the identification of structural damping using the FRF-based model updating method and damping identification for accurate prediction of the measured FRFs using finite element updated models of the mechanical system [66][67]. Recently, several studies were conducted on dynamic systems. These studies showed the high feasibility of the NDI methods compared to the linear methods because they give more reliable results despite their difficulty. The nonlinear research focuses primarily on the development of efficient and functional methods for reacting to nonlinear structural damping as a fundamental scientific and technological problem. A survey of available research in the engineering community has provided many studies of nonlinear damping, as shown in Figure 1. It is expected that interest from researchers in this field will continue.

    Figure 1. The number of publications for nonlinear damping studies (According to the Scopus engine system in the duration from January 1999 to November 2020).

    2. Nonlinear Damping Identification Methods

    The early study on the identification of nonlinearity of structural models can be traced back to 1970s. The objective of this section is to survey the NDI methods that evolved over the last few years as a result of the development of novel industrial materials and structures [68]. Various nonlinear factors have led to different systems’ behaviors, which means that each system has unique behavior and therefore requires a different approach [69]. Relevant studies were first focused on single-degree of freedom (SDOF) systems [70]. Then, with the advancement of computational techniques, studies were extended to more complex, multi-degree of freedom (MDOF) systems [71][72][73]. Specific structural types under investigation include bridges, tall buildings, aircraft, etc. [74]. Apart from previous review articles that are mainly about nonlinear system identification , the focus of this study resides on the development of various NDI methods. The methods can be classified into seven categories, as shown in Figure 2.

    Figure 2. Nonlinear damping identification methods.

    The entry is from 10.3390/s20247303


    1. Kerschen, G.; Worden, K.; Vakakis, A.F.; Golinval, J.C. Past, present and future of nonlinear system identification in structural dynamics. Mech. Syst. Signal Process. 2006, 20, 505–592, doi:10.1016/j.ymssp.2005.04.008.
    2. Cao, M.S.; Sha, G.G.; Gao, Y.F.; Ostachowicz, W. Structural damage identification using damping: A compendium of uses and features. Smart Mater. Struct. 2017, 26, 043001, doi:10.1088/1361-665X/aa550a.
    3. Blackwell, C.; Palazotto, A.; George, T.J.; Cross, C.J. The evaluation of the damping characteristics of a hard coating on titanium. Shock Vib. 2007, 14, 37–51, doi:10.1155/2007/260183.
    4. Eichler, A.; Moser, J.; Chaste, J.; Zdrojek, M.; Wilson-Rae, I.; Bachtold, A. Nonlinear damping in mechanical resonators made from carbon nanotubes and graphene. Nat. Nanotechnol. 2011, 6, 339–342, doi:10.1038/nnano.2011.71.
    5. Liang, J.W.; Feeny, B.F. Identifying Coulomb and viscous friction in forced dual-damped oscillators. J. Vib. Acoust. Trans. ASME 2004, 126, 118–125, doi:10.1115/1.1640356.
    6. Mevada, H.; Patel, D. Experimental Determination of Structural Damping of Different Materials. Proc. Eng. 2016, 144, 110–115, doi:10.1016/j.proeng.2016.05.013.
    7. Zhang, X.; Du, X.; Brownjohn, J. Frequency modulated empirical mode decomposition method for the identification of instantaneous modal parameters of aeroelastic systems. J. Wind Eng. Ind. Aerodyn. 2012, 101, 43–52, doi:10.1016/j.jweia.2011.12.005.
    8. Kulkarni, P.; Bhattacharjee, A.; Nanda, B.K. Study of damping in composite beams. Mater. Today Proc. 2018, 5, 7061–7067, doi:10.1016/j.matpr.2017.11.370.
    9. Chatterjee, A.; Chintha, H.P. Identification and parameter estimation of cubic nonlinear damping using harmonic probing and volterra series. Int. J. Non. Linear. Mech. 2020, 125, 103518, doi:10.1016/j.ijnonlinmec.2020.103518.
    10. Dou, C.; Fan, J.; Li, C.; Cao, J.; Gao, M. On discontinuous dynamics of a class of friction-influenced oscillators with nonlinear damping under bilateral rigid constraints. Mech. Mach. Theory 2020, 147, 103750, doi:10.1016/j.mechmachtheory.2019.103750.
    11. Heitz, T.; Giry, C.; Richard, B.; Ragueneau, F. Identification of an equivalent viscous damping function depending on engineering demand parameters. Eng. Struct. 2019, 188, 637–649, doi:10.1016/j.engstruct.2019.03.058.
    12. Ghiringhelli, G.L.; Terraneo, M. Analytically driven experimental characterisation of damping in viscoelastic materials. Aerosp. Sci. Technol. 2015, 40, 75–85, doi:10.1016/j.ast.2014.10.011.
    13. Amabili, M. Nonlinear damping in nonlinear vibrations of rectangular plates: Derivation from viscoelasticity and experimental validation. J. Mech. Phys. Solids 2018, 118, 275–292, doi:10.1016/j.jmps.2018.06.004.
    14. Anastasio, D.; Marchesiello, S.; Kerschen, G.; Noël, J.P. Experimental identification of distributed nonlinearities in the modal domain. J. Sound Vib. 2019, 458, 426–444, doi:10.1016/j.jsv.2019.07.005.
    15. Moradi, H.; Vossoughi, G.; Movahhedy, M.R. Experimental dynamic modelling of peripheral milling with process damping, structural and cutting force nonlinearities. J. Sound Vib. 2013, 332, 4709–4731, doi:10.1016/j.jsv.2013.03.029.
    16. Olejnik, P.; Awrejcewicz, J. Coupled oscillators in identification of nonlinear damping of a real parametric pendulum. Mech. Syst. Signal Process. 2018, 98, 91–107, doi:10.1016/j.ymssp.2017.04.037.
    17. Ciang, C.C.; Lee, J.R.; Bang, H.J. Structural health monitoring for a wind turbine system: A review of damage detection methods. Meas. Sci. Technol. 2008, 19, 12, doi:10.1088/0957-0233/19/12/122001.
    18. Kouris, L.A.S.; Penna, A.; Magenes, G. Seismic damage diagnosis of a masonry building using short-term damping measurements. J. Sound Vib. 2017, 394, 366–391, doi:10.1016/j.jsv.2017.02.001.
    19. Ji, H.; Qiu, J.; Zhu, K.; Badel, A. Two-mode vibration control of a beam using nonlinear synchronized switching damping based on the maximization of converted energy. J. Sound Vib. 2010, 329, 2751–2767, doi:10.1016/j.jsv.2010.01.012.
    20. Shariyat, M.; Jahangiri, M. Nonlinear impact and damping investigations of viscoporoelastic functionally graded plates with in-plane diffusion and partial supports. Compos. Struct. 2020, 245, 112345, doi:10.1016/j.compstruct.2020.112345.
    21. Peters, R.D. Nonlinear Damping of the ‘Linear’ Pendulum. arXiv 2003, arXiv:physics/0306081.
    22. Zaitsev, S.; Shtempluck, O.; Buks, E. Nonlinear damping in a micromechanical oscillator. 2012, 67, 859–883, doi:10.1007/s11071-011-0031-5.
    23. Noël, J.P.; Kerschen, G. 10 years of advances in nonlinear system identification in structural dynamics: A review. In Proceedings of ISMA 2016-International Conference on Noise and Vibration Engineering, Leuven, Belgium, 19–21 September 2016.
    24. Karaağaçlı, T.; Özgüven, H.N. A frequency domain nonparametric identification method for nonlinear structures: Describing surface method. Mech. Syst. Signal Process. 2020, 144, 106872, doi:10.1016/j.ymssp.2020.106872.
    25. Boltežar, M.; Slavič, J. Enhancements to the continuous wavelet transform for damping identifications on short signals. Mech. Syst. Signal Process. 2004, 18, 1065–1076, doi:10.1016/j.ymssp.2004.01.004.
    26. Heaney, P.S.; Bilgen, O. System identification of lumped parameter models for weakly nonlinear systems. J. Sound Vib. 2019, 450, 78–95, 2019, doi:10.1016/j.jsv.2019.03.010.
    27. Li, A.; Ma, L.; Keene, D.; Klingel, J.; Payne, M.; Wang, X. Forced oscillations with linear and nonlinear damping. Am. J. Phys. 2016, 84, 32–37, doi:10.1119/1.4935358.
    28. Lamarque, C.H.; Savadkoohi, A.T.; Charlemagne, S. Experimental results on the vibratory energy exchanges between a linear system and a chain of nonlinear oscillators. J. Sound Vib. 2018, 437, 97–109, doi:10.1016/j.jsv.2018.09.004.
    29. Nguyen, Q.T.; Tinard, V.; Fond, C. The modelling of nonlinear rheological behaviour and Mullins effect in High Damping Rubber. Int. J. Solids Struct. 2015, 75–76, 235–246, doi:10.1016/j.ijsolstr.2015.08.017.
    30. Haghdoust, P.; Conte, A.L.; Cinquemani, S.; Lecis, N. A numerical method to model non-linear damping behaviour of martensitic shape memory alloys. Materials (Basel) 2018, 11, 2178, doi:10.3390/ma11112178.
    31. Yuan, D.N. Dynamic modeling and analysis of an elastic mechanism with a nonlinear damping model. JVC/Journal Vib. Control 2013, 19, 508–516, doi:10.1177/1077546309356463.
    32. Phani, A.S.; Woodhouse, J. Viscous damping identification in linear vibration. J. Sound Vib. 2007, 303, 475–500, doi:10.1016/j.jsv.2006.12.031.
    33. Prandina, M.; Mottershead, J.E.; Bonisoli, E. An assessment of damping identification methods. J. Sound Vib. 2009, 323, 662–676, doi:10.1016/j.jsv.2009.01.022.
    34. Botelho, E.C.; Campos, A.N.; de Barros, E.; Pardini, L.C.; Rezende, M.C. Damping behavior of continuous fiber/metal composite materials by the free vibration method. Compos. Part B Eng. 2005, 37, 255–263, doi:10.1016/j.compositesb.2005.04.003.
    35. Berthelot, J.M.; Assarar, M.; Sefrani, Y.; el Mahi, A. Damping analysis of composite materials and structures. Compos. Struct. 2008, 85, 189–204, doi:10.1016/j.compstruct.2007.10.024.
    36. Guo, Z.; Sheng, M.; Ma, J.; Zhang, W. Damping identification in frequency domain using integral method. J. Sound Vib. 2015, 338, 237–249, doi:10.1016/j.jsv.2014.10.040.
    37. Jeong, B.; Cho, H.; Yu, M.F.; Vakakis, A.F.; McFarland, D.M.; Bergman, L.A. Modeling and measurement of geometrically nonlinear damping in a microcantilever-nanotube system. ACS Nano 2013, 7, 8547–8553, doi:10.1021/nn402479d.
    38. Ruderman, M.S. Nonlinear damped standing slow waves in hot coronal magnetic loops. Astron. Astrophys. 2013, 553, A23, doi:10.1051/0004-6361/201321175.
    39. Noël, J.P.; Renson, L.; Kerschen, G. Complex dynamics of a nonlinear aerospace structure: Experimental identification and modal interactions. J. Sound Vib. 2014, 333, 2588–2607, doi:10.1016/j.jsv.2014.01.024.
    40. Ahmadi, K. Analytical investigation of machining chatter by considering the nonlinearity of process damping. J. Sound Vib. 2017, 393, 252–264, doi:10.1016/j.jsv.2017.01.006.
    41. Saarenheimo, A.; Borgerhoff, M.; Calonius, K.; Darraba, A.; Hamelin, A.; Ghadimi Khasraghy, S.; Karbassi, A.; Schneeberger, C.; Stadler, M.; Tuomala, M.; et al. Numerical studies on vibration propagation and damping test V1. Raken. Mek. 2018, 51, 55–80, doi:10.23998/rm.68954.
    42. Cai, J.; Zhang, H. Efficient schemes for the damped nonlinear Schrödinger equation in high dimensions. Appl. Math. Lett. 2020, 102, 106158, doi:10.1016/j.aml.2019.106158.
    43. Lisitano, D.; Bonisoli, E. Direct identification of nonlinear damping: Application to a magnetic damped system. Mech. Syst. Signal Process. 2020, 146, 107038, doi:10.1016/j.ymssp.2020.107038.
    44. Srikantha Phani, A.; Woodhouse, J. Experimental identification of viscous damping in linear vibration. J. Sound Vib. 2009, 319, 832–849, doi:10.1016/j.jsv.2008.06.022.
    45. Singh, V.; Shevchuk, O.; Blanter, Y.M.; Steele, G.A. Negative nonlinear damping of a multilayer graphene mechanical resonator. Phys. Rev. B 2016, 93, 245407, doi:10.1103/PhysRevB.93.245407.
    46. Xu, B.; He, J.; Dyke, S.J. Model-free nonlinear restoring force identification for SMA dampers with double Chebyshev polynomials: Approach and validation. Nonlinear Dyn. 2015, 82, 1507–1522, doi:10.1007/s11071-015-2257-0.
    47. Bian, J.; Jing, X. Superior nonlinear passive damping characteristics of the bio-inspired limb-like or X-shaped structure. Mech. Syst. Signal Process. 2019, 125, 21–51, doi:10.1016/j.ymssp.2018.02.014.
    48. Yan, B.; Ma, H.; Yu, N.; Zhang, L.; Wu, C. Theoretical modeling and experimental analysis of nonlinear electromagnetic shunt damping. J. Sound Vib. 2020, 471, 115184, doi:10.1016/j.jsv.2020.115184.
    49. Yan, B.; Ma, H.; Zhang, L.; Zheng, W.; Wang, K.; Wu, C. A bistable vibration isolator with nonlinear electromagnetic shunt damping. Mech. Syst. Signal Process. 2020, 136, 106504, doi:10.1016/j.ymssp.2019.106504.
    50. Masri, S.F.; Caughey, T.K. A nonparametric identification technique for nonlinear dynamic problems. J. Appl. Mech. Trans. ASME 1979, 46, 433–447, doi:10.1115/1.3424568.
    51. Lisitano, D.; Bonisoli, E.; Mottershead, J.E. Experimental direct spatial damping identification by the Stabilised Layers Method. J. Sound Vib. 2018, 437, 325–339, doi:10.1016/j.jsv.2018.08.055.
    52. Lumori, M.L.D.; Schoukens, J.; Lataire, J. Identification and quantification of nonlinear stiffness and nonlinear damping in resonant circuits. Proc. ISMA 2010 Int. Conf. Noise Vib. Eng. Incl. USD 2010 2010, 24, 3205–3219, doi:10.1016/j.ymssp.2010.05.014.
    53. McNamara, J.J.; Friedmann, P.P. Flutter-boundary identification for time-domain computational aeroelasticity. AIAA J. 2007, 45, 1546–1555, doi:10.2514/1.26706.
    54. Sainsbury, M.G.; Ho, Y.K. Application of the time domain Fourier filter output (TDFFO) method to the identification of a lightly damped non-linear system with an odd-spring characteristic. Mech. Syst. Signal Process. 2001, 15, 357–366, doi:10.1006/mssp.2000.1308.
    55. Boz, U.; Eriten, M. Nonlinear system identification of soft materials based on Hilbert transform. J. Sound Vib. 2019, 447, 205–220, doi:10.1016/j.jsv.2019.01.025.
    56. Abramovich, H.; Govich, D.; Grunwald, A. Damping measurements of laminated composite materials and aluminum using the hysteresis loop method. Prog. Aerosp. Sci. 2015, 78, 8–18, doi:10.1016/j.paerosci.2015.05.006.
    57. Qu, H.; Li, T.; Chen, G. Multiple analytical mode decompositions (M-AMD) for high accuracy parameter identification of nonlinear oscillators from free vibration. Mech. Syst. Signal Process. 2019, 117, 483–497, doi:10.1016/j.ymssp.2018.08.012.
    58. Ruzzene, M.; Fasana, A.; Garibaldi, L.; Piombo, B. Natural frequencies and dampings identification using wavelet transform: Application to real data. Mech. Syst. Signal Process. 1997, 11, 207–218, doi:10.1006/mssp.1996.0078.
    59. Thothadri, M.; Moon, F.C. Nonlinear system identification of systems with periodic limit-cycle response. Nonlinear Dyn. 2005, 39, 63–77, doi:10.1007/s11071-005-1914-0.
    60. Ben, B.S.; Ben, B.A.; Vikram, K.A. Damping measurement in composite materials using combined finite element and frequency response method. Int. J. Eng. Sci. Invent. 2013, 2, 89–97.
    61. Naraghi, T.; Nobari, A.S. Identification of the dynamic characteristics of a viscoelastic, nonlinear adhesive joint. J. Sound Vib. 2015, 352, 92–102., doi:10.1016/j.jsv.2015.05.010.
    62. Ta, M.N.; Lardis, J. Identification of weak nonlinearities on damping and stiffness by the continuous wavelet transform. J. Sound Vib. 2006, 293, 16–37, doi:10.1016/j.jsv.2005.09.021.
    63. Feldman, M.; Bucher, I.; Rotberg, J. Experimental identification of nonlinearities under free and forced vibration using the Hilbert transform. JVC/Journal Vib. Control 2009, 15, 1563–1579., doi:10.1177/1077546308097270.
    64. Wang, S.; Li, J.; Luo, H.; Zhu, H. Damage identification in underground tunnel structures with wavelet based residual force vector. Eng. Struct. 2019, 178, 506–520, doi:10.1016/j.engstruct.2018.10.021.
    65. Ebrahimi, Z.; Asemani, M.H.; Safavi, A.A. Observer-based controller design for uncertain disturbed Takagi-Sugeno fuzzy systems: A fuzzy wavelet neural network approach. Int. J. Adapt. Control Signal Process. 2020, doi:10.1002/acs.3195.
    66. Tang, X.; Peng, F.; Yan, R.; Zhu, Z.; Li, Z.; Xin, S. Nonlinear process damping identification using finite amplitude stability and the influence analysis on five-axis milling stability. Int. J. Mech. Sci. 2020, 190, 106008, doi:10.1016/j.ijmecsci.2020.106008.
    67. Imregun, M.; Sanliturk, K.Y.; Ewins, D.J. Finite element model updating using frequency response function data—II. case study on a medium-size finite element model. Mech. Syst. Signal Process. 1995, 9, 203–213, doi:10.1006/mssp.1995.0016.
    68. Chung, D.D.L. Review: Materials for vibration damping. J. Mater. Sci. 2001, 36, 5733–5737, doi:10.1023/A:1012999616049.
    69. Kurt, M.; Eriten, M.; McFarland, D.M.; Bergman, L.A.; Vakakis, A.F. Strongly nonlinear beats in the dynamics of an elastic system with a strong local stiffness nonlinearity: Analysis and identification. J. Sound Vib. 2014, 333, 2054–2072, doi:10.1016/j.jsv.2013.11.021.
    70. Fujimura, M.; Maeda, J.; Morimoto, Y. Aerodynamic damping properties of transmission tower estimated by combining several identification methods. J. Wind Eng. Ind. Aerodyn. 2010, 74, 1949–1955, 2010, doi:10.3130/aijs.74.1949.
    71. Zhang, W. The Spectral Density of the Nonlinear Damping Model: Single DOF Case. IEEE Trans. Automat. Contr. 1990, 35, 1320–1329, doi:10.1109/9.61008.
    72. StaszewskI, W.J. Identification of Damping in Mdof Systems Using Time-Scale Decomposition. J. Sound Vib. 1997, 203, 283–305, doi:10.1006/jsvi.1996.0864.
    73. Wierschem, N.E.; Quinn, D.D.; Hubbard, S.A.; Al-Shudeifat, M.A.; McFarland, D.M.; Luo, J.; Fahnestock, L.A.; Spencer, B.F.; Vakakis, A.F.; Bergman, L.A. Passive damping enhancement of a two-degree-of-freedom system through a strongly nonlinear two-degree-of-freedom attachment. J. Sound Vib. 2012, 331, 5393–5407, doi:10.1016/j.jsv.2012.06.023.
    74. Wu, Y.; Chen, X. Identification of nonlinear aerodynamic damping from stochastic crosswind response of tall buildings using unscented Kalman filter technique. Eng. Struct. 2020, 220, 110791, doi:10.1016/j.engstruct.2020.110791.
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