{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:39:22Z","timestamp":1760240362404,"version":"build-2065373602"},"reference-count":76,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2019,5,21]],"date-time":"2019-05-21T00:00:00Z","timestamp":1558396800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>Nonlinear modal analysis is a demanding yet imperative task to rigorously address real-life situations where the dynamics involved clearly exceed the limits of linear approximation. The specific case of geometric nonlinearities, where the effects induced by the second and higher-order terms in the strain\u2013displacement relationship cannot be neglected, is of great significance for structural engineering in most of its fields of application\u2014aerospace, civil construction, mechanical systems, and so on. However, this nonlinear behaviour is strongly affected by even small changes in stiffness or mass, e.g., by applying physically-attached sensors to the structure of interest. Indeed, the sensors placement introduces a certain amount of geometric hardening and mass variation, which becomes relevant for very flexible structures. The effects of mass loading, while highly recognised to be much larger in the nonlinear domain than in its linear counterpart, have seldom been explored experimentally. In this context, the aim of this paper is to perform a noncontact, full-field nonlinear investigation of the very light and very flexible XB-1 air wing prototype aluminum spar, applying the well-known resonance decay method. Video processing in general, and a high-speed, optical target tracking technique in particular, are proposed for this purpose; the methodology can be easily extended to any slender beam-like or plate-like element. Obtained results have been used to describe the first nonlinear normal mode of the spar in both unloaded and sensors-loaded conditions by means of their respective backbone curves. Noticeable changes were encountered between the two conditions when the structure undergoes large-amplitude flexural vibrations.<\/jats:p>","DOI":"10.3390\/s19102345","type":"journal-article","created":{"date-parts":[[2019,5,21]],"date-time":"2019-05-21T10:52:51Z","timestamp":1558435971000},"page":"2345","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":23,"title":["Using Video Processing for the Full-Field Identification of Backbone Curves in Case of Large Vibrations"],"prefix":"10.3390","volume":"19","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0414-7440","authenticated-orcid":false,"given":"Marco","family":"Civera","sequence":"first","affiliation":[{"name":"Department of Mechanical and Aerospace Engineering, Politecnico di Torino, 10129 Turin, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6269-5280","authenticated-orcid":false,"given":"Luca","family":"Zanotti Fragonara","sequence":"additional","affiliation":[{"name":"Centre for Autonomous and Cyber-Physical Systems, Cranfield University, Cranfield, Bedford MK43 0AL, UK"}]},{"given":"Cecilia","family":"Surace","sequence":"additional","affiliation":[{"name":"Department of Structural, Building and Geotechnical Engineering, Politecnico di Torino, 10129 Turin, Italy"}]}],"member":"1968","published-online":{"date-parts":[[2019,5,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Dussart, G., Portapas, V., Pontillo, A., and Lone, M. (2018). Flight Dynamic Modelling and Simulation of Large Flexible Aircraft. Flight Physics\u2014Models, Techniques and Technologies, IntechOpen.","DOI":"10.5772\/intechopen.71050"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Nguyen, N.T., Ting, E., and Lebofsky, S. (2016, January 4\u20138). Inertial Force Coupling to Nonlinear Aeroelasticity of Flexible Wing Aircraft. Proceedings of the 15th Dynamics Specialists Conference, San Diego, CA, USA.","DOI":"10.2514\/6.2016-1094"},{"key":"ref_3","unstructured":"Breitbach, E. (1978). Effects of Structural Non-Linearitles on Aircraft Vibration and Flutter, AGARD Advisory Group for Aerospace Research and Development."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"974","DOI":"10.2514\/1.5651","article-title":"Advanced Test Strategy for Identification and Characterization of Nonlinearities of Aerospace Structures","volume":"43","author":"Goege","year":"2005","journal-title":"AIAA J."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1409","DOI":"10.2514\/1.C031918","article-title":"Nonlinear Modal Analysis of a Full-Scale Aircraft","volume":"50","author":"Kerschen","year":"2013","journal-title":"J. Aircr."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"6785","DOI":"10.1016\/j.jsv.2013.08.006","article-title":"Nonlinear dynamic behaviors of a deploying-and-retreating wing with varying velocity","volume":"332","author":"Zhang","year":"2013","journal-title":"J. Sound"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"505","DOI":"10.1016\/j.ymssp.2005.04.008","article-title":"Past, present and future of nonlinear system identification in structural dynamics","volume":"20","author":"Kerschen","year":"2006","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"2","DOI":"10.1016\/j.ymssp.2016.07.020","article-title":"Nonlinear system identification in structural dynamics: 10 more years of progress","volume":"83","author":"Kerschen","year":"2017","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"187","DOI":"10.5545\/sv-jme.2015.2964","article-title":"Nonlinear Response of Cantilever Beams Due to Large Geometric Deformations: Experimental Validation","volume":"62","year":"2016","journal-title":"SV-JME"},{"key":"ref_10","unstructured":"Rossi, A., Orsini, F., Scorza, A., Botta, F., Leccese, F., Silva, E., Torokhtii, K., Bernabucci, I., and Sciuto, S.A. (2017, January 14\u201315). A preliminary performance validation of a MEMS accelerometer for blade vibration monitoring. Proceedings of the 22nd IMEKO TC4 International Symposium & 20th International Workshop on ADC Modelling and Testing Supporting World Development through Electrical & Electronic Measurements, Iasi, Romania."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Botta, F., Scorza, A., and Rossi, A. (2018). Optimal Piezoelectric Potential Distribution for Controlling Multimode Vibrations. Appl. Sci., 8.","DOI":"10.3390\/app8040551"},{"key":"ref_12","unstructured":"Botta, F., Rossi, A., Schinaia, L., Scorza, A., Orsini, F., Sciuto, S.A., and Belfiore, N.P. (, January September). Experimental validation on optimal placement of pzt plates for active beam multimode vibrations reduction. Proceedings of the 23rd Conference of the Italian Association of Theoretical and Applied Mechanics (AIMETA 2017), Rome, Italy."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"28","DOI":"10.1016\/j.ymssp.2015.09.020","article-title":"A framework for backbone experimental tracking : Piezoelectric actuators, stop-sine signal and Kalman filtering","volume":"78","author":"Peyret","year":"2016","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"2191","DOI":"10.1016\/j.ymssp.2011.01.018","article-title":"Comparison of FRF measurements and mode shapes determined using optically image based, laser, and accelerometer measurements","volume":"25","author":"Warren","year":"2011","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"58","DOI":"10.1016\/j.jsv.2015.01.024","article-title":"Modal identification of simple structures with high-speed video using motion magnification","volume":"345","author":"Chen","year":"2015","journal-title":"J. Sound"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Pontillo, A., Hayes, D., Dussart, G.X., Matos, G.E.L., Carrizales, M.A., Yusuf, S.Y., and Lone, M.M. (2018, January 25\u201329). Flexible High Aspect Ratio Wing: Low Cost Experimental Model and Computational Framework. Proceedings of the 2018 AIAA Atmospheric Flight Mechanics Conference, Atlanta, GA, USA.","DOI":"10.2514\/6.2018-1014"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Lacarbonara, W. (2013). Nonlinear Structural Mechanics, Springer.","DOI":"10.1007\/978-1-4419-1276-3"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"437","DOI":"10.1080\/03601217808907348","article-title":"Nonlinear Flexural-Flexural-Torsional Dynamics of Inextensional Beams. I. Equations of Motion","volume":"6","author":"Glynn","year":"1978","journal-title":"J. Struct. Mech."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"449","DOI":"10.1080\/03601217808907349","article-title":"Nonlinear Flexural-Flexural-Torsional Dynamics of Inextensional Beams. II. Forced Motions","volume":"6","author":"Glynn","year":"1978","journal-title":"J. Struct. Mech."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"70","DOI":"10.1002\/zamm.201600021","article-title":"A unified beam theory with strain gradient effect and the von K\u00e1rm\u00e1n nonlinearity","volume":"97","author":"Khodabakhshi","year":"2017","journal-title":"ZAMM J. Appl. Math. Mech."},{"key":"ref_21","first-page":"317","article-title":"A Parametric Identification Technique for Single-Degree-of-Freedom Weakly Nonlinear Systems with Cubic Nonlinearities","volume":"9","author":"Malatkar","year":"2003","journal-title":"Modal Anal."},{"key":"ref_22","first-page":"21","article-title":"Experimental Verification of the Importance of The Nonlinear Curvature in the Response of a Cantilever Beam","volume":"118","author":"Anderson","year":"1996","journal-title":"J. Acoust."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"4915","DOI":"10.1016\/S0020-7683(99)00197-3","article-title":"Influence of nonlinear boundary conditions on the single-mode response of a cantilever beam","volume":"37","author":"Tabaddor","year":"2000","journal-title":"Int. J. Solids Struct."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Nayfeh, A.H., Pai, P.F., and Perngjin, F. (2004). Linear and Nonlinear Structural Mechanics, Wiley.","DOI":"10.1002\/9783527617562"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"455","DOI":"10.1016\/0020-7462(90)90012-X","article-title":"Non-linear non-planar oscillations of a cantilever beam under lateral base excitations","volume":"25","author":"Pai","year":"1990","journal-title":"Int. J. Non-Linear Mech."},{"key":"ref_26","unstructured":"Hermann, G. (1956). Influence of Large Amplitudes on Flexural Motions of Elastic Plates, National Advisory Committee for Aeronautics."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"501","DOI":"10.1002\/zamm.19610411204","article-title":"Influence of Large Amplitudes on Flexural Vibrations of Elastic Plates","volume":"41","author":"Yamaki","year":"1961","journal-title":"Z. Angew. Math. Mech."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1016\/0045-7949(73)90081-3","article-title":"Finite element displacement method for large amplitude free flexural vibrations of beams and plates","volume":"3","author":"Mei","year":"1973","journal-title":"Comput. Struct."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"425","DOI":"10.1016\/0020-7403(63)90026-2","article-title":"Large amplitude flexural vibration of rectangular plates","volume":"5","author":"Wah","year":"1963","journal-title":"Int. J. Mech. Sci."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"139","DOI":"10.1115\/1.3629535","article-title":"The Normal Modes of Vibration of Certain Nonlinear Continuous Systems","volume":"31","author":"Wah","year":"1964","journal-title":"J. Appl. Mech."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"7","DOI":"10.1115\/1.3636501","article-title":"The Normal Modes of Nonlinear n-Degree-of-Freedom Systems","volume":"29","author":"Rosenberg","year":"1962","journal-title":"J. Appl. Mech."},{"key":"ref_32","unstructured":"Howell, L.L. (2001). Compliant Mechanisms, Wiley."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"272","DOI":"10.1090\/qam\/13360","article-title":"Large deflection of cantilever beams","volume":"3","author":"Bisshopp","year":"1945","journal-title":"Q. Appl. Math."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"021006","DOI":"10.1115\/1.4023558","article-title":"A Comprehensive Elliptic Integral Solution to the Large Deflection Problems of Thin Beams in Compliant Mechanisms","volume":"5","author":"Zhang","year":"2013","journal-title":"J. Mech. Robot."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1002\/sapm197352287","article-title":"On One-Dimensional Large-Displacement Finite-Strain Beam Theory","volume":"52","author":"Reissner","year":"1973","journal-title":"Stud. Appl. Math."},{"key":"ref_36","unstructured":"K\u00e1rm\u00e1n, T. (1910). Von Festigkeitsprobleme im Maschinenbau, Teubner."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"247","DOI":"10.1038\/079247c0","article-title":"Vorlesungen \u00fcber technische Mechanik","volume":"79","year":"1908","journal-title":"Nature"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"687","DOI":"10.1002\/cpa.3160200405","article-title":"On von k\u00e1rm\u00e1n\u2019s equations and the buckling of a thin elastic plate, I the clamped plate","volume":"20","author":"Berger","year":"1967","journal-title":"Commun. Pure Appl. Math."},{"key":"ref_39","unstructured":"Berger, H.M. (1954). A New Approach to the Analysis of Large Deflections of Plates. [Ph.D. Thesis, California Institute of Technology]."},{"key":"ref_40","unstructured":"Ewins, D.J. (2000). Modal Testing: Theory, Practice, and Application, Wiley."},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Worden, K., and Tomlinson, G.R. (2001). Nonlinearity in Structural Dynamics: Detection, Identification, and Modelling, Institute of Physics.","DOI":"10.1887\/0750303565"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"2335","DOI":"10.1016\/j.ymssp.2006.12.005","article-title":"Output-only modal analysis using blind source separation techniques","volume":"21","author":"Poncelet","year":"2007","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_43","unstructured":"Schwarz, B.J., and Richardson, M.H. (1999). Experimental Modal Analysis, CSI Reliability Week."},{"key":"ref_44","doi-asserted-by":"crossref","unstructured":"Kaplan, D., and Glass, L. (1995). Understanding Nonlinear Dynamics, Springer. Texts in Applied Mathematics.","DOI":"10.1007\/978-1-4612-0823-5"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"491","DOI":"10.1016\/j.automatica.2004.10.004","article-title":"Identification of linear systems with nonlinear distortions","volume":"41","author":"Schoukens","year":"2005","journal-title":"Automatica"},{"key":"ref_46","unstructured":"Storer, D.M. (1991). Dynamic Analysis of Non-Linear Structures Using Higher Order Frequency Response Functions. [Ph.D. Thesis, University of Manchester]."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"1614","DOI":"10.2514\/1.42024","article-title":"Identification of a Nonlinear Wing Structure Using an Extended Modal Model","volume":"46","author":"Platten","year":"2009","journal-title":"J. Aircr."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"2","DOI":"10.1109\/19.903871","article-title":"Practical choices in the FRF measurement in presence of nonlinear distortions","volume":"50","author":"Dobrowiecki","year":"2001","journal-title":"IEEE Trans. Instrum. Meas."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"184","DOI":"10.1016\/j.ymssp.2016.07.010","article-title":"Relationships between nonlinear normal modes and response to random inputs","volume":"84","author":"Schoneman","year":"2017","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"1011","DOI":"10.1016\/j.ymssp.2010.09.011","article-title":"Identifying and quantifying structural nonlinearities in engineering applications from measured frequency response functions","volume":"25","author":"Carrella","year":"2011","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1006\/jsvi.1993.1198","article-title":"Normal Modes for Non-Linear Vibratory Systems","volume":"164","author":"Shaw","year":"1993","journal-title":"J. Sound"},{"key":"ref_52","first-page":"293","article-title":"Elements of Nonlinear System Identification of Broad Applicability","volume":"555","author":"Vakakis","year":"2014","journal-title":"Mech. Behav. Soils Under Environ. Induc. Cycl. Loads"},{"key":"ref_53","doi-asserted-by":"crossref","unstructured":"Kerschen, G. (2014). Modal Analysis of Nonlinear Mechanical Systems, Springer.","DOI":"10.1007\/978-3-7091-1791-0"},{"key":"ref_54","first-page":"75","article-title":"Normal form theory and nonlinear normal modes: Theoretical settings and applications","volume":"555","year":"2014","journal-title":"Mech. Behav. Soils Under Environ. Induc. Cycl. Loads"},{"key":"ref_55","first-page":"47","article-title":"Invariant Manifold Representations of Nonlinear Modes of Vibration","volume":"555","author":"Shaw","year":"2014","journal-title":"Mech. Behav. Soils Under Environ. Induc. Cycl. Loads"},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1016\/j.jsv.2015.09.033","article-title":"Numerical computation of nonlinear normal modes in mechanical engineering","volume":"364","author":"Renson","year":"2016","journal-title":"J. Sound"},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1016\/j.compstruc.2015.03.008","article-title":"Nonlinear modal analysis of nonconservative systems: Extension of the periodic motion concept","volume":"154","author":"Krack","year":"2015","journal-title":"Comput. Struct."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"1493","DOI":"10.1007\/s11071-016-2974-z","article-title":"Nonlinear normal modes and spectral submanifolds: existence, uniqueness and use in model reduction","volume":"86","author":"Ponsioen","year":"2016","journal-title":"Nonlinear Dyn."},{"key":"ref_59","doi-asserted-by":"crossref","unstructured":"Cirillo, G.I., Mauroy, A., Renson, L., Kerschen, G., and Sepulchre, R. (2015, January 2\u20135). Global Parametrization of the Invariant Manifold Defining Nonlinear Normal Modes Using the Koopman Operator. Proceedings of the 39th Mechanisms and Robotics Conference, Boston, MA, USA.","DOI":"10.1115\/DETC2015-46366"},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"248","DOI":"10.1016\/j.jsv.2016.05.016","article-title":"A spectral characterization of nonlinear normal","volume":"377","author":"Cirillo","year":"2016","journal-title":"J. Sound Vib."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"486","DOI":"10.1016\/j.jsv.2010.08.028","article-title":"Dynamic testing of nonlinear vibrating structures using nonlinear normal modes","volume":"330","author":"Peeters","year":"2011","journal-title":"J. Sound"},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"224","DOI":"10.1016\/j.jsv.2015.03.015","article-title":"Identification of backbone curves of nonlinear systems from resonance decay responses","volume":"348","author":"Neild","year":"2015","journal-title":"J. Sound"},{"key":"ref_63","first-page":"298","article-title":"Identification of Multi-Degree of Freedom Systems With Nonproportional Damping Using the Resonant Decay Method","volume":"126","author":"Naylor","year":"2004","journal-title":"J. Acoust."},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"8","DOI":"10.1016\/j.ymssp.2007.11.016","article-title":"Identification of multi-degree of freedom non-linear systems using an extended modal space model","volume":"23","author":"Platten","year":"2009","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"116","DOI":"10.1016\/j.ymssp.2016.02.008","article-title":"Identification of systems containing nonlinear stiffnesses using backbone curves","volume":"84","author":"Cooper","year":"2017","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"1227","DOI":"10.1016\/j.ymssp.2010.11.006","article-title":"Modal testing of nonlinear vibrating structures based on nonlinear normal modes: Experimental demonstration","volume":"25","author":"Peeters","year":"2011","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_67","unstructured":"Nayfeh, A.H., and Mook, D.T. (1979). Nonlinear Oscillations, Wiley."},{"key":"ref_68","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1016\/j.jsv.2018.01.049","article-title":"Exact nonlinear model reduction for a von K\u00e1rm\u00e1n beam: Slow-fast decomposition and spectral submanifolds","volume":"423","author":"Jain","year":"2018","journal-title":"J. Sound"},{"key":"ref_69","unstructured":"Civera, M., Zanotti Fragonara, L., and Surace, C. (, January November). Video Processing Techniques for the Contactless Investigation of Large Oscillations. Proceedings of the proceedings of the AIVELA XXVI Meeting, Milan, Italy. in press."},{"key":"ref_70","doi-asserted-by":"crossref","first-page":"295","DOI":"10.1016\/0888-3270(90)90010-I","article-title":"Data processing and experiment design for the restoring force surface method, part I: integration and differentiation of measured time data","volume":"4","author":"Worden","year":"1990","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_71","unstructured":"Bay, H., Tuytelaars, T., and Gool, L. (, January May). Van SURF: Speeded Up Robust Features. Proceedings of the European Conference on Computer Vision, Graz, Austria."},{"key":"ref_72","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1016\/j.ymssp.2016.03.015","article-title":"Nonlinear vibrating system identification via Hilbert decomposition","volume":"84","author":"Feldman","year":"2017","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_73","doi-asserted-by":"crossref","first-page":"239","DOI":"10.1016\/j.ymssp.2018.05.010","article-title":"A method for non-parametric identification of non-linear vibration systems with asymmetric restoring forces from a resonant decay response","volume":"114","author":"Ondra","year":"2019","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_74","doi-asserted-by":"crossref","first-page":"231","DOI":"10.1016\/0020-7403(63)90051-1","article-title":"Applications of approximate expressions for complete elliptic integrals","volume":"5","year":"1963","journal-title":"Int. J. Mech. Sci."},{"key":"ref_75","doi-asserted-by":"crossref","first-page":"156","DOI":"10.1115\/1.2826101","article-title":"Parametric Deflection Approximations for End-Loaded, Large-Deflection Beams in Compliant Mechanisms","volume":"117","author":"Howell","year":"1995","journal-title":"J. Mech."},{"key":"ref_76","doi-asserted-by":"crossref","first-page":"2629","DOI":"10.1007\/s11012-016-0453-8","article-title":"Nonlinear normal modes for damage detection","volume":"51","author":"Lacarbonara","year":"2016","journal-title":"Meccanica"}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/19\/10\/2345\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T12:54:02Z","timestamp":1760187242000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/19\/10\/2345"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,5,21]]},"references-count":76,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2019,5]]}},"alternative-id":["s19102345"],"URL":"https:\/\/doi.org\/10.3390\/s19102345","relation":{},"ISSN":["1424-8220"],"issn-type":[{"type":"electronic","value":"1424-8220"}],"subject":[],"published":{"date-parts":[[2019,5,21]]}}}