River Rapids Conference Proceedings of the Society for Experimental Mechanics Series Nonlinear Dynamics, Volume 1 Gaetan Kerschen Proceedings of the 34th IMAC, A Conference and Exposition on Structural Dynamics 2016 River Publishers
Conference Proceedings of the Society for Experimental Mechanics Series Series Editor Kristin B. Zimmerman, Ph.D. Society for Experimental Mechanics, Inc., Bethel, CT, USA
River Publishers Gaetan Kerschen Editor Nonlinear Dynamics, Volume 1 Proceedings of the 34th IMAC, A Conference and Exposition on Structural Dynamics 2016
Published, sold and distributed by: River Publishers Broagervej 10 9260 Gistrup Denmark www.riverpublishers.com ISBN 978-87-7004-925-2 (eBook) Conference Proceedings of the Society for Experimental Mechanics An imprint of River Publishers © The Society for Experimental Mechanics, Inc. 2016 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, or reproduction in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Preface Nonlinear Dynamics represents one of ten volumes of technical papers presented at the 34th IMAC, A Conference and Exposition on Structural Dynamics, organized by the Society for Experimental Mechanics and held in Orlando, Florida, on January 25–28, 2016. The full proceedings also include volumes on Dynamics of Civil Structures; Model Validation and Uncertainty Quantification; Dynamics of Coupled Structures; Sensors and Instrumentation; Special Topics in Structural Dynamics; Structural Health Monitoring, Damage Detection & Mechatronics; Rotating Machinery, Hybrid Test Methods, Vibro-Acoustics and Laser Vibrometry; Shock & Vibration, Aircraft/Aerospace, Energy Harvesting, Acoustics & Optics; and Topics in Modal Analysis &Testing. Each collection presents early findings from experimental and computational investigations on an important area within structural dynamics. Nonlinearity is one of these areas. The vast majority of real engineering structures behave nonlinearly. Therefore, it is necessary to include nonlinear effects in all the steps of the engineering design: in the experimental analysis tools (so that the nonlinear parameters can be correctly identified) and in the mathematical and numerical models of the structure (in order to run accurate simulations). In so doing, it will be possible to create a model representative of the reality which, once validated, can be used for better predictions. Several nonlinear papers address theoretical and numerical aspects of nonlinear dynamics (covering rigorous theoretical formulations and robust computational algorithms) as well as experimental techniques and analysis methods. There are also papers dedicated to nonlinearity in practice where real-life examples of nonlinear structures will be discussed. The organizers would like to thank the authors, presenters, session organizers, and session chairs for their participation in this track. Liege, Belgium Gaetan Kerschen v
Contents 1 Nonlinear Vibrations of a Beam with a Breathing Edge Crack Using Multiple Trial Functions ............... 1 Ali C. Batihan and Ender Cigeroglu 2 Enforcing Linear Dynamics Through the Addition of Nonlinearity............................................... 11 G. Habib, C. Grappasonni, and G. Kerschen 3 Experimental Analysis of a Softening-Hardening Nonlinear Oscillator Using Control-Based Continuation ........................................................................................... 19 L. Renson, D.A.W. Barton, and S.A. Neild 4 Experimental Nonlinear Dynamics of Laminated Quasi-Isotropic Thin Composite Plates..................... 29 H.G. Kim and R. Wiebe 5 Experimental Identification of a Structure with Internal Resonance.............................................. 37 Alexander D. Shaw, Tom L. Hill, Simon A. Neild, and Michael I. Friswell 6 Shock Response of an Antenna Structure Considering Geometric Nonlinearity................................. 47 Yunus Emre Ozcelik, Ender Cigeroglu, and Mehmet Caliskan 7 Investigation on Friction-Excited Vibration of Flexibly Supported Shafting System............................ 61 Wenyuan Qin, Zhenguo Zhang, Suining Hu, and Zhiyi Zhang 8 Resonant Analysis of Systems Equipped with Nonlinear Displacement-Dependent (NDD) Dampers ......... 67 Javad Jahanpour, Shahab Ilbeigi, and Mojtaba Porghoveh 9 Performance Comparison Between a Nonlinear Energy Sink and a Linear Tuned Vibration Absorber for Broadband Control...................................................................................... 83 Etienne Gourc, Lamberto Dell Elce, Gaetan Kerschen, Guilhem Michon, Gwenaelle Aridon, and Aurelien Hot 10 Experimental and Numerical Investigation of the Nonlinear Bending-Torsion Coupling of a Clamped-Clamped Beam with Centre Masses .................................................................. 97 David A. Ehrhardt, Simon A. Neild, and Jonathan E. Cooper 11 Tracking of Backbone Curves of Nonlinear Systems Using Phase-Locked-Loops ............................... 107 Simon Peter, Robin Riethmüller, and Remco I. Leine 12 The Importance of Phase-Locking in Nonlinear Modal Interactions.............................................. 121 T.L. Hill, A. Cammarano, S.A. Neild, and D.J. Wagg 13 A Study of the Modal Interaction Amongst Three Nonlinear Normal Modes Using a Backbone Curve Approach ......................................................................................................... 131 X. Liu, A. Cammarano, D.J. Wagg, and S.A. Neild 14 Investigating Nonlinear Modal Energy Transfer in a Random Load Environment.............................. 141 Joseph D. Schoneman and Matthew S. Allen vii
viii Contents 15 Nonlinear Modal Testing Performed by Pulsed-Air Jet Excitation System....................................... 155 M. Piraccini, D. Di Maio, and R. Di Sante 16 EMA-FEA Correlation and Updating for Nonlinear Behaviour of an Automotive Heat-Shield ............... 171 Elvio Bonisoli, Marco Brino, and Giuseppe Credo 17 Tutorial on Nonlinear System Identification.......................................................................... 185 G. Kerschen 18 Higher-Order Frequency Response Functions for Hysteretic Systems............................................ 191 G. Manson and K. Worden 19 Model Upgrading T0 Augment Linear Model Capabilities into Nonlinear Regions............................. 203 S.B. Cooper, A. delli Carri, and D. Di Maio 20 Obtaining Nonlinear Frequency Responses from Broadband Testing............................................. 219 Etienne Gourc, Chiara Grappasonni, Jean-Philippe Noël, Thibaut Detroux, and Gaëtan Kerschen 21 Experimental Study of Isolated Response Curves in a Two-Degree-of-Freedom Nonlinear System........... 229 T. Detroux, J.P. Noël, G. Kerschen, and L.N. Virgin 22 Nonlinear Response of a Thin Panel in a Multi-Discipline Environment: Part I—Experimental Results..... 237 T.J. Beberniss, S.M. Spottswood, R.A. Perez, and T.G. Eason 23 Nonlinear Dynamic Response Prediction of a Thin Panel in a Multi-Discipline Environment: Part II—Numerical Predictions........................................................................................ 249 R.A. Perez, S.M. Spottswood, T.J. Beberniss, G.W. Bartram, and T.G. Eason 24 Stability Analysis of Curved Panels ................................................................................... 259 Ilinca Stanciulescu, Yang Zhou, and Mihaela Nistor 25 Optimal Representation of a Varying Temperature Field for Coupling with a Structural Reduced Order Model ............................................................................................................. 267 Raghavendra Murthy, Andrew K. Matney, X.Q. Wang, and Marc P. Mignolet 26 Basis Identification for Nonlinear Dynamical Systems Using Sparse Coding .................................... 279 Rohit Deshmukh, Zongxian Liang, and Jack J. McNamara 27 Interaction Between Aerothermally Compliant Structures and Boundary Layer Transition .................. 295 Zachary B. Riley and Jack J. McNamara 28 Simultaneous Vibration Isolation and Energy Harvesting: Simulation and Experiment ....................... 305 R. Benjamin Davis and Matthew D. McDowell 29 Nonlinear Dynamic Interaction in a Coupled Electro-Magneto-Mechanical System: Experimental Study ..................................................................................................... 317 I.T. Georgiou and F. Romeo 30 Hysteresis Identification Using Nonlinear State-Space Models .................................................... 323 J.P. Noël, A.F. Esfahani, G. Kerschen, and J. Schoukens 31 Nonholonomically Constrained Dynamics of Rolling Isolation Systems .......................................... 339 Karah C. Kelly and Henri P. Gavin 32 Parameter Estimation on Nonlinear Systems Using Orthogonal and Algebraic Techniques ................... 347 L.G. Trujillo-Franco, G. Silva-Navarro, and F. Beltrán-Carbajal 33 Online State and Parameter Estimation of a Nonlinear Gear Transmission System............................ 355 Dimitrios Giagopoulos, Vasilis Dertimanis, Eleni Chatzi, and Minas Spiridonakos 34 Model Updating of a Nonlinear System: Gun Barrel of a Battle Tank............................................ 365 Güvenç Canbalog˘lu and H. Nevzat Özgüven 35 Experimental Passive Flutter Mitigation Using a Linear Tuned Vibrations Absorber.......................... 389 E. Verstraelen, G. Habib, G. Kerschen, and G. Dimitriadis
Contents ix 36 Adaptive Harmonic Balance Analysis of Dry Friction Damped Systems ......................................... 405 Dominik Süß, Martin Jerschl, and Kai Willner 37 Dynamics of an MDOF Rotor Stator Contact System............................................................... 415 Alexander D. Shaw, David A.W. Barton, Alan R. Champneys, and Michael I. Friswell
Chapter1 Nonlinear Vibrations of a Beam with a Breathing Edge Crack Using Multiple Trial Functions Ali C. Batihan and Ender Cigeroglu Abstract In this paper, a beam like structure with a single edge crack is modeled and analyzed in order to study the nonlinear effects of breathing crack on transverse vibrations of a beam. In literature, edge cracks are generally modeled as open cracks, in which the beam is separated into two pieces at the crack location and these pieces are connected to each other with a rotational spring to represent the effect of crack. The open edge crack model is a widely used assumption; however, it does not consider the nonlinear behavior due to opening and closing of the crack region. In this paper, partial differential equation of motion obtained by Euler-Bernoulli beam theory is converted into nonlinear ordinary differential equations by using Galerkin’s method with multiple trial functions. The nonlinear behavior of the crack region is represented as a bilinear stiffness matrix. The nonlinear ordinary differential equations are converted into a set of nonlinear algebraic equations by using harmonic balance method (HBM) with multi harmonics. Under the action of a harmonic forcing, the effect of crack parameters on the vibrational behavior of the cracked beam is studied. Keywords Breathing crack • Euler-Bernoulli beam • Galerkin’s method • Harmonic balance method • Nonlinear vibrations 1.1 Introduction Identification of cracks and determination of their location is an important consideration, since crack propagation may cause unexpected failure. Therefore, beams with edge cracks has been an interesting area of research. Dimarogonas [1] provides a review paper in which studies on open breathing crack, continuous crack beam theories and vibration of cracked plates are covered. Aydin [2], carried out a study considering arbitrary number of cracks and axial loads applied on a beam. Khiem and Lien [3], used a transfer matrix method in order to calculate natural frequencies of a beam with an arbitrary number of cracks. In order to see the effect of crack clearly, beam models with mass attachments are used by Mermertas¸ and Erol [4] and Zhong and Oyadiji [5]. Mazanog˘lu et al. [6] applied Rayleigh-Ritz and finite element methods in order to study vibrations of non-uniform cracked beams. In a study of Chondros et al. [7], flexibility due to crack region is distributed along the whole beam by developing a continuous theory of cracked beams. In another study of Chondros et al. [8], breathing edge crack was studied by combining vibration characteristics of open and closed period as a bi-linear model. Cheng et al. [9] represented the breathing crack with time dependent stiffness. Finite element method was used by Chati et al. [10] in order to study modal analysis of a beam with a breathing edge crack. Giannini et al. [11] also used finite element method to identify sub and super harmonics of a beam with a breathing edge crack. Baeza and Ouyang [12], developed an analytical approach by using beam modes to calculate a scale factor matrix which indicates crack location. In most of the studies available in literature, the beam is modeled by Euler-Bernoulli beam theory and the crack region is represented by a rotational spring whose stiffness is obtained by fracture mechanics methods. A significant number of studies are investigated the effect of crack parameters on natural frequencies. In addition, application of finite element methods for beams with breathing edge cracks is a common modeling approach. In this paper, based on the previous study of authors [13], beam is modeled using Euler-Bernoulli beam theory and breathing edge crack is modeled as a piecewise linear stiffness. In the analytical model, the state of the crack is determined by checking the slope difference at the crack location. The governing equations are obtained by Galerkin’s method utilizing multiple trial functions and the resulting set of nonlinear equations are solved by application of harmonic balance method with multi harmonics. A.C. Batihan • E. Cigeroglu ( ) Department of Mechanical Engineering, Middle East Technical University, Ankara 06800, Turkey e-mail: ender@metu.edu.tr © The Society for Experimental Mechanics, Inc. 2016 G. Kerschen (ed.), Nonlinear Dynamics, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, DOI 10.1007/978-3-319-29739-2_1 1
2 A.C. Batihan and E. Cigeroglu 1.2 Formulation of the Breathing Edge Crack Problem Equation of motion of a uniform beam vibrating in transverse direction under the action of an external point forcef (t) located at Lf can expressed by utilizing Euler-Bernoulli beam theory as follows EI @4w.x; t/ @x4 C c @w.x; t/ @t C m @2w.x; t/ @t2 D f .t/ı x Lf ; (1.1) wherew(x, t) is transverse displacement, EI is flexural rigidity, c is viscous damping coefficient andmis mass per unit length of the beam. Using expansion theorem, transverse displacement can be expressed as follows w.x; t/ DX j aj.t/ j.x/; (1.2) where j(x) is the j th mass normalized eigenfunction of a beam with an open edge crack andaj(t) is the corresponding modal coefficient. Substituting Eq. (1.2) into Eq. (1.1), the following expression is obtained X j EIaj.t/ d4 j.x/ dx4 CX j c :aj.t/ j.x/ CX j mRaj.t/ j.x/ Df .t/ı x Lf : (1.3) Multiplying Eq. (1.3) by i(x) and integrating over the spatial domain of the beam result in the following equation X j kij aj.t/ CX j cij : aj.t/ CX j mij Raj.t/ DFi.t/; (1.4) where kij D L Z 0 EI i.x/ d4 j.x/ dx4 dx; (1.5) cij D L Z 0 c i.x/ j.x/dx; (1.6) mij D L Z 0 m i.x/ j.x/dx; (1.7) Fi.t/ D i Lf f .t/: (1.8) Equation (1.4) can be rearranged as a matrix equation as follows ŒM fRagCŒC ˚ :a CŒK fagDfFg; (1.9) where [M], [C] and [K] are the corresponding mass, damping and stiffness matrices of a beam with an open edge crack. Figure 1.1 shows the deformed shape of a cantilever crack beam at two different time instants. During vibration, due to breathing effect of the edge crack, the beam behaves as if it is an undamaged beam for some period of cycle; whereas, in the rest of the cycle it behaves as a beam with an open edge crack. Therefore, beam with a breathing edge crack can be represented as a combination of two linear systems as shown in Fig. 1.1. In order to consider the breathing effect, two different sets of mass normalized eigenfunctions are utilized as trial functions. For the time instant when the crack is closed, mass, damping and stiffness matrices are calculated by utilizing first few mass normalized eigenfunctions of the undamaged beam; whereas mass normalized eigenfunctions of a beam with an open crack are used in case the crack is in open state. The details and derivation of these eigenfunctions can be found in [14]. As the
1 Nonlinear Vibrations of a Beam with a Breathing Edge Crack Using Multiple Trial Functions 3 Fig. 1.1 Cantilever beam with a breathing edge crack crack switches from open state to closed state, mass and damping matrices remain unchanged; however, the stiffness matrix increases when the crack closes. The increase in the stiffness matrix is given by the following relation ŒB DŒKc ŒK ; (1.10) where [Kc] is the stiffness matrix of the beam when the crack is at the closed state. In this study, in order to identify whether the crack is open or not, the slope difference at the crack location is checked. Negative slope difference states that the crack is open and the nonlinear forcing term is zero; whereas positive slope difference indicates that the crack is closed and the nonlinear forcing term is nonzero. The periodic change in the stiffness of the beam which results from the breathing effect of the crack, leads to the following nonlinear forcing term fR.fag/gD 8 < : ŒB fag if @w.x;t/ @x ˇ ˇ ˇxDLc 0 f0g if @w.x;t/ @x ˇ ˇ ˇxDLc <0 : (1.11) Adding the nonlinear forcing termfR(fag)g, into Eq. (1.9) leads to the following equation in which the breathing effect of the crack is taken into consideration ŒM fRagCŒC ˚ : a CŒK fagCfR.fag/gDfFg: (1.12) 1.3 Harmonic Balance Method In order to utilize harmonic balance method with multi harmonics, modal coefficient of each trial function is expressed as follows aj.t/ Daj0 CX p ajcp cos.p!t/ CX p ajsp sin.p!t/; (1.13)
4 A.C. Batihan and E. Cigeroglu where aj0 is the bias term of the j th modal coefficient, ajcp and ajsp are the coefficients of cosine and sine components of the pth harmonic of the jth modal coefficient. Letting D!t and rearranging Eq. (1.13), the modal coefficients can be written in vector form as fagDfa0gCŒac fhc . /gCŒas fhc . /g; (1.14) where Œac D fac1g ˚acp ; (1.15) Œas D fas1g ˚asp ; (1.16) andfhc( )g &fhs( )g are the following vectors whose lengths are equal to the number harmonics (p) used fhc . /gD 8 ˆ< ˆ: cos. / : : : cos.p / 9 >= >; ; (1.17) fhs . /gD 8 ˆ< ˆ: sin. / : : : sin.p / 9 >= >; : (1.18) The periodic nonlinear forcing termfR(fag)g can be expressed by Fourier series as fRgD 1 2 f R0gCŒRc fhc . /gCŒRs fhs . /g: (1.19) where ŒRc D fRc1g ˚Rcp ; (1.20) ŒRs D fRs1g ˚Rsp ; (1.21) and Rj0 D 1 2 Z 0 Rj . /d ; (1.22) Rjcp D 1 2 Z 0 Rj . /cos.p /d ; (1.23) Rjsp D 1 2 2 Z 0 Rj . /sin.p /d : (1.24) Similarly external forcing, f (t) can be represented by Fourier series as follows f .t/ D˚fcp 0 fhc . /gC˚fsp 0 fhs . /g: (1.25) Substituting Eqs. (1.14), (1.19) and (1.25) into Eq. (1.12) and collecting sine and cosine terms leads to the following set of nonlinear equations
1 Nonlinear Vibrations of a Beam with a Breathing Edge Crack Using Multiple Trial Functions 5 0 B @ 2 6 4 ŒK 0 : : : 0 ŒK 3 7 5 !2 2 6 4 ŒM 0 : : : 0 ŒM 3 7 5 2 6 4 ŒI : : : p2 ŒI 3 7 5 1 C A 8 ˆ< ˆ: fac1g : : : ˚acp 9 >= >; C 8 ˆ< ˆ: fRc1g : : : ˚Rcp 9 >= >; C! 2 6 4 ŒC 0 : : : 0 ŒC 3 7 5 2 6 4 ŒI : : : pŒI 3 7 5 8 ˆ< ˆ: fas1g : : : ˚asp 9 >= >; 2 6 4 Lf 0 : : : 0 Lf 3 7 5 8 ˆ< ˆ: fc1 fIg : : : fcp fIg 9 >= >; D 8 ˆ< ˆ: f0g : : : f0g 9 >= >; ; (1.26) 0 B @ 2 6 4 ŒK 0 : : : 0 ŒK 3 7 5 !2 2 6 4 ŒM 0 : : : 0 ŒM 3 7 5 2 6 4 ŒI : : : p2 ŒI 3 7 5 1 C A 8 ˆ< ˆ: fas1g : : : ˚asp 9 >= >; C 8 ˆ< ˆ: fRs1g : : : ˚Rsp 9 >= >; C! 2 6 4 ŒC 0 : : : 0 ŒC 3 7 5 2 6 4 ŒI : : : pŒI 3 7 5 8 ˆ< ˆ: fac1g : : : ˚acp 9 >= >; 2 6 4 Lf 0 : : : 0 Lf 3 7 5 8 ˆ< ˆ: fs1 fIg : : : fsp fIg 9 >= >; D 8 ˆ< ˆ: f0g : : : f0g 9 >= >; ; (1.27) ŒK fa0gC 1 2 f R0gDf0g; (1.28) where [I] is the identity matrix, fIg is a vector elements of which are all 1, and Lf D 2 6 4 1 Lf 0 : : : 0 j Lf 3 7 5 : (1.29) 1.4 Case Study and Discussion Effect of breathing edge crack for different crack parameters is studied by using a cantilever beam model with the following properties: L D 1 m, I D 2:667 10 8 m4, D 7850 kg/m3, A D 8 10 4 m2, E D 206 GPa, D 0:07, Lf D 0:1 m and f .t/ D100cos.!t/ N. Galerkin’s method is applied by utilizing first three modes of the beam and the resulting set of nonlinear ordinary differential equations are converted into a set of nonlinear algebraic equations by using harmonic balance method with multi harmonics. According to [13], as crack ratio increases fundamental frequency decreases. Similarly, crack location also affects the fundamental frequency depending on the boundary conditions of the beam. The effect of different crack ratio and crack location on natural frequencies is given in Fig. 1.2. It is observed from the results that the effect of crack parameters on natural frequencies are not significant. Therefore, alternative features should be considered in crack detection problems. In Fig. 1.3, absolute value of bias term of each modal coefficient as function of frequency is given. It is observed that bias term of each modal coefficient is influenced by both crack location and crack ratio. In Fig. 1.3a, bias term of each modal coefficient is plotted for different crack locations and it is observed that depending on the crack location bias term of different modal coefficient becomes dominant. Bias terms are also affected by the crack ratio, however the effect of crack ratio is not similar to the effect of crack location. For instance, from Fig. 1.3b it is observed that, an increase in crack depth shifts all the plots upward and the order of dominancy is conserved. The amplitude of pth harmonic of jth modal coefficient is expressed by the following equation ajp D qajcp 2 Casjp 2: (1.30) In Fig. 1.4, amplitudes of the first harmonic of modal coefficients with respect to frequency are given for different crack location and ratio. From the plots, it is observed that first harmonics are slightly influenced by crack parameters. However, the first harmonic is the most dominant harmonic on the total response among the higher harmonics. This fact also explains why crack parameters have negligible effect on the total response.
6 A.C. Batihan and E. Cigeroglu 0 200 400 600 800 1000 1200 1400 1600 1800 2000 0 0.2 0.4 0.6 0.8 1 1.2 1.4 a b x 10-5 Frequency, w, [rad/s] Frequency, w, [rad/s] Amplitude, [m] Linear a=0.1 a=0.4 a=0.5 0 200 400 600 800 1000 1200 1400 1600 1800 2000 0 0.2 0.4 0.6 0.8 1 1.2 1.4 x 10-5 Amplitude, [m] Linear L c =0.2 L c =0.4 L c =0.6 Fig. 1.2 Transverse displacement of a point at location Lp D0:1 m vs. frequency. (a) For Lc D0:2 m and different crack ratio. (b) For ˛ D0:5 and different crack location 0 200 400 600 800 1000 1200 1400 1600 1800 2000 10-14 10-12 10-10 10-8 10-6 10-4 Bias Term, aj0 [m] a 10 , L c =0.2 a 10 , L c =0.4 a 20 , L c =0.2 a 20 , L c =0.4 a 30 , L c =0.2 a 30 , L c =0.4 0 200 400 600 800 1000 1200 1400 1600 1800 2000 10-14 10-12 10-10 10-8 10-6 10-4 Frequency, w, [rad/s] Frequency, w, [rad/s] Bias Term, aj0 [m] a 10 , a=0.2 a 10 , a=0.5 a 20 , a=0.2 a 20 , a=0.5 a 30 , a=0.2 a 30 , a=0.5 a b Fig. 1.3 Bias term of each modal coefficient vs. frequency. (a) For ˛ D0:5 and different crack location. (b) For Lc D0:2 m and different crack ratio Higher harmonics of different trial functions for different crack parameters are shown in Figs. 1.5, 1.6, 1.7, and 1.8. Studying these figures, it is observed that higher harmonics can be grouped as even and odd harmonics depending on the similarities of the plots. Observing Figs. 1.5 and 1.6, it is seen that effect of crack parameters on even harmonics is similar to the effect of crack parameters on bias terms. As crack location changes, even harmonic of a different trial function becomes more dominant; whereas increase in crack depth causes an increase in the overall amplitude of the even harmonics. Effect of crack parameters can also be observed in the odd harmonics. As depicted in Figs. 1.7 and 1.8, effect of crack location on odd harmonics is similar to the effect of crack location on bias terms and even harmonics. Studying the same figures, it is also observed that, odd harmonics of the first modal coefficient are significantly affected by crack depth. However, getting information from the odd harmonics of the second and the third modal coefficient is a more challenging due to scattered pattern. Therefore, analyzing the even harmonics can be used as means for crack detection problems. In Table 1.1,
1 Nonlinear Vibrations of a Beam with a Breathing Edge Crack Using Multiple Trial Functions 7 0 200 400 600 800 1000 1200 1400 1600 1800 2000 10-7 10-6 10-5 10-4 10-3 Frequency, w, [rad/s] Frequency, w, [rad/s] First Harmonic, aj1 [m] a 11 , L c =0.2 a 11 , L c =0.4 a 21 , L c =0.2 a 21 , L c =0.4 a 31 , L c =0.2 a 31 , L c =0.4 0 200 400 600 800 1000 1200 1400 1600 1800 2000 10-7 10-6 10-5 10-4 10-3 First Harmonic, aj1 [m] a 11 , a=0.2 a 11 , a=0.5 a 21 , a=0.2 a 21 , a=0.5 a 31 , a=0.2 a 31 , a=0.5 a b Fig. 1.4 First harmonic of each modal coefficient vs. frequency. (a) For ˛ D0:5 and different crack location. (b) For Lc D0:2 m and different crack ratio 0 200 400 600 800 1000 1200 1400 1600 1800 2000 10-12 10-10 10-8 10-6 10-4 Frequency, w, [rad/s] Frequency, w, [rad/s] Second Harmonic, aj2 [m] a 12 , L c =0.2 a 12 , L c =0.4 a 22 , L c =0.2 a 22 , L c =0.4 a 32 , L c =0.2 a 32 , L c =0.4 0 200 400 600 800 1000 1200 1400 1600 1800 2000 10-14 10-12 10-10 10-8 10-6 10-4 Second Harmonic, aj2 [m] a 12 , a=0.2 a 12 , a=0.5 a 22 , a=0.2 a 22 , a=0.5 a 32 , a=0.2 a 32 , a=0.5 a b Fig. 1.5 Second harmonic of each modal coefficient vs. frequency. (a) For ˛ D0:5and different crack location. (b) For Lc D0:2m and different crack ratio ratio of maximum amplitudes of the even harmonics are provided. The ratio is obtained by dividing maximum amplitude of pth harmonic of each modal coefficient to the maximum amplitude of pth harmonic of the first modal coefficient. Studying Table 1.1, significant effect of crack location on other trial functions can be observed. In order to study the effect of crack depth on harmonics of different modal coefficients Table 1.2 is prepared in a similar way. In Figs. 1.5b and 1.6b it is observed that the increase in crack depth also increases all the plots with a similar magnitude. As a result of this fact, order of magnitude of the harmonics do not change. Therefore, in Table 1.2, it is observed that the maximum amplitude ratio is not influenced by the crack ratio.
8 A.C. Batihan and E. Cigeroglu 0 200 400 600 800 1000 1200 1400 1600 1800 2000 10-12 10-10 10-8 10-6 10-4 Frequency, w, [rad/s] Frequency, w, [rad/s] Fourth Harmonic, aj4 [m] a 14 , L c =0.2 a 14 , L c =0.4 a 24 , L c =0.2 a 24 , L c =0.4 a 34 , L c =0.2 a 34 , L c =0.4 0 200 400 600 800 1000 1200 1400 1600 1800 2000 10-14 10-12 10-10 10-8 10-6 10-4 Fourth Harmonic, aj4 [m] a 14 , a=0.2 a 14 , a=0.5 a 24 , a=0.2 a 24 , a=0.5 a 34 , a=0.2 a 34 , a=0.5 a b Fig. 1.6 Fourth harmonic of each modal coefficient vs. frequency. (a) For ˛ D0:5 and different crack location. (b) For Lc D0:2 m and different crack ratio 0 200 400 600 800 1000 1200 1400 1600 1800 2000 10-14 10-12 10-10 10-8 10-6 10-4 Frequency, w, [rad/s] Frequency, w, [rad/s] Third Harmonic, aj3 [m] a 13 , L c =0.2 a 13 , L c =0.4 a 23 , L c =0.2 a 23 , L c =0.4 a 33 , L c =0.2 a 33 , L c =0.4 0 200 400 600 800 1000 1200 1400 1600 1800 2000 10-16 10-14 10-12 10-10 10-8 10-6 10-4 Third Harmonic, aj3 [m] a 13 , a=0.2 a 13 , a=0.5 a 23 , a=0.2 a 23 , a=0.5 a 33 , a=0.2 a 33 , a=0.5 a b Fig. 1.7 Third harmonic of each modal coefficient vs. frequency. (a) For ˛ D0:5 and different crack location. (b) For Lc D0:2 m and different crack ratio 1.5 Conclusion In this study, beam with breathing edge crack is modelled by using Euler-Bernoulli beam theory and nonlinear piecewise linear stiffness. Multiple trial functions are used to represent the response in Galerkin’s method where a piecewise linear stiffness matrix based on the slope difference at the crack location is introduced. Harmonic Balance Method with multiple harmonics is used to convert nonlinear ordinary differential equation into a set of nonlinear algebraic equations. It is observed from the results that effect of crack parameters on the natural (resonance) frequency of the cracked beam is insignificant. However, it is observed that harmonics of the response are affected from the crack parameters significantly; hence, this information can be used for the crack detection. Both crack depth and crack location affects the amplitudes of the harmonics of each modal coefficient. As crack ratio increases the amplitudes of the harmonics also increase, however the order of magnitudes of the harmonics are not affected by the crack ratio. The crack location affects amplitudes of the harmonics as well as the order of magnitudes of the harmonics. Depending on the crack location, a harmonic of a different modal coefficient becomes dominant. This fact explains the necessity of using multiple trial functions.
1 Nonlinear Vibrations of a Beam with a Breathing Edge Crack Using Multiple Trial Functions 9 0 200 400 600 800 1000 1200 1400 1600 1800 2000 10-15 10-10 10-5 Frequency, w, [rad/s] Frequency, w, [rad/s] Fifth Harmonic, aj5 [m] a 15 , L c =0.2 a 15 , L c =0.4 a 25 , L c =0.2 a 25 , L c =0.4 a 35 , L c =0.2 a 35 , L c =0.4 0 200 400 600 800 1000 1200 1400 1600 1800 2000 10-16 10-14 10-12 10-10 10-8 10-6 Fifth Harmonic, aj5 [m] a 15 , a=0.2 a 15 , a=0.5 a 25 , a=0.2 a 25 , a=0.5 a 35 , a=0.2 a 35 , a=0.5 a b Fig. 1.8 Fifth harmonic of each modal coefficient vs. frequency. (a) For ˛ D0:5 and different crack location. (b) For Lc D0:2 m and different crack ratio Table 1.1 Ratio of maximum amplitudes of even harmonics of each modal coefficient for different crack location ˛ D0:5 Lc a12/a12 a22/a12 a32/a12 a14/a14 a24/a14 a34/a14 0.2 1 0.00128 0.01438 1 0.00141 0.0151 0.4 1 0.18786 0.05545 1 0.22663 0.04163 Table 1.2 Ratio of maximum amplitudes of even harmonics of each modal coefficient for different crack depth Lc D0:2 ˛ a12/a12 a22/a12 a32/a12 a14/a14 a24/a14 a34/a14 0.2 1 0.00132 0.01327 1 0.00147 0.01385 0.5 1 0.00128 0.01437 1 0.00141 0.01507 References 1. Dimarogonas, A.D.: Vibration of cracked structures: a state of the art review. Eng. Fract. Mech. 55(5), 831–857 (1996) 2. Aydın, K.: Vibratory characteristics of Euler-Bernoulli beams with arbitrary number of cracks subjected to axial load. J. Vib. Control. 14(4), 485–510 (2008) 3. Khiem, N.T., Lien, T.V.: A simplified method for natural frequency analysis of multiple cracked beam. J. Sound Vib. 254(4), 737–751 (2001) 4. Mermertas¸, V., Erol, H.: Effect of mass attachment on the free vibration of cracked beam. In: The 8th International Congress on Sound and Vibration (2001) 5. Zhong, S., Oyadiji, S.O.: Analytical predictions of natural frequencies of cracked simply supported beam with stationary roving mass. J. Sound Vib. 311, 328–352 (2008) 6. Mazanog˘lu, K., Yes¸ilyurt, I., Sabuncu, M.: Vibration analysis of multiple cracked non-uniform beams. J. Sound Vib. 320, 977–989 (2009) 7. Chondros, T.G., Dimarogonas, A.D., Yao, J.: A continuous cracked beam vibration theory. J. Sound Vib. 215(1), 17–34 (1998) 8. Chondros, T.G., Dimarogonas, A.D., Yao, J.: Vibration of a beam with a breathing crack. J. Sound Vib. 239(1), 57–67 (2001) 9. Cheng, S.M., Wu, X.J., Wallace, W.: Vibrational response of a beam with a breathing crack. J. Sound Vib. 225(1), 201–208 (1999) 10. Chati, M., Rand, R., Mukherjee, S.: Modal analysis of a cracked beam. J. Sound Vib. 207(2), 249–270 (1997) 11. Giannini, O., Casini, P., Vestroni, F.: Nonlinear harmonic identification of breathing cracks in beams. Comput. Struct. 129, 166–177 (2013) 12. Baeza, L., Ouyang, H.: Modal approach for forced vibration of beams with a breathing crack. Key Eng. Mater. 413–414, 39–46 (2009) 13. Batihan, A.C., Cigeroglu, E.: Nonlinear vibrations of a beam with a breathing edge crack. In: IMAC XXXIII (2015) 14. Batihan, A.C.: Vibration analysis of cracked beams on elastic foundation using Timoshenko beam theory. Master Thesis, Middle East Technical University (2011)
Chapter2 Enforcing Linear Dynamics Through the Addition of Nonlinearity G. Habib, C. Grappasonni, and G. Kerschen Abstract The current trend of developing more slender structures is increasing the importance of nonlinearities in engineering design, which, in turn, gives rise to complicated dynamical phenomena. In this study, we evidence the somewhat paradoxical result that adding purposefully nonlinearity to an already nonlinear structure renders the behavior more linear. Isochronicity, i.e., the invariance of natural frequencies with respect to oscillation amplitude, and the force-displacement proportionality are two key properties of linear systems that are lost for nonlinear systems. The objective of this research is to investigate how these properties can be enforced in a nonlinear system through the addition of nonlinearity. To this end, we exploit the nonlinear normal modes theory to derive simple rules, yet applicable to real structures, for the compensation of nonlinear effects. The developments are illustrated using numerical experiments on a cantilever beam possessing a geometrically nonlinear boundary condition. Keywords Nonlinear normal modes • Linearization • Compensation of nonlinearity • Isochronicity • Perturbation 2.1 Introduction Many engineering applications, as for instance tuned vibration absorbers [1], ultrasensitive mass and force sensing devices [2], time keeping devices [3], nanoscale imaging systems [4] and many others, rely on linear properties of mechanical systems, such as force-displacement proportionality and invariance of the resonant frequency. However, if high excitation amplitudes are considered, nonlinearities are activated, invalidating linear properties. This situation is particularly relevant for nano- and micro-electromechanical systems, where nonlinearities are activated already at moderate forcing amplitudes [5]. Furthermore, the current trend of developing more slender structures increases the importance of nonlinearities also in macro systems. The loss of force-displacement proportionality, the dependence of the resonant frequency on the amplitude, the appearance of quasiperiodic or chaotic solutions, variations in stability properties, coexistence of different solutions, boundedness of basins of attraction, appearance of bifurcations are some of the effects typically generated by nonlinearities [6] that have no linear counterpart. Most of these phenomena have been studied in depth during the last decades, and it is now possible to predict the consequences of many different types of nonlinearities, although unexpected behaviors are always possible. Nevertheless, there are few studies that attempt to eliminate these usually unwanted phenomena. Most existing studies deal with the implementation of active controllers [7, 8], which is referred to as feedback linearization. The objective of this paper is to enforce linear properties in a nonlinear system through the addition of passive nonlinear elements. The two target properties are the force-displacement proportionality and the invariance of the resonant frequency with respect to the amplitude, which are generally lost even at low level of excitation. The developed procedure exploits the nonlinear normal modes (NNMs) [9] of undamped, unforced systems, because they give a good approximation of the system’s backbone curves. Thanks to the energy balance criterion, the undamped, unforced dynamics can be related to the forced damped dynamics, thus giving complete (but approximate) information about the location of the resonant peaks in the force-frequency space. The resulting equations are solved through reduction to a single harmonics and a standard perturbation technique, which allows to derive equations that can be solved explicitly. G. Habib ( ) • C. Grappasonni • G. Kerschen Department of Aerospace and Mechanical Engineering, Space Structures and Systems Laboratory (S3L), Structural Dynamics Research Group, University of Liège, Liège, Belgium e-mail: giuseppe.habib@ulg.ac.be © The Society for Experimental Mechanics, Inc. 2016 G. Kerschen (ed.), Nonlinear Dynamics, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, DOI 10.1007/978-3-319-29739-2_2 11
12 G. Habib et al. This procedure prepares the ground for the definition of linear equations, whose solution is directly related to the target linear property. The analytical developments are validated using a two-degree-of-freedom (DoF) reduced-order model of a cantilever beam possessing a geometrically nonlinear boundary condition and a nonlinear attachment. 2.2 Model We consider a general n-DoF mechanical system with concentrated polynomial nonlinearities of odd orders, subject to harmonic excitation. The system has the form MRxCDPxCKxC Qb.x/ D p" Qvf cos.!t/ (2.1) where M, Dand Kare the mass, damping and stiffness matrices, x is the position vector, Qb contains the nonlinear terms, Qv indicates which DoFs are excited, f is the forcing amplitude, !is the excitation frequency and " is a small parameter, while t is time. M, DandKare assumed to be symmetric, real and positive-definite, while Qbhas the generic form QbD 2 6 6 4 : : : PmD3;5;::: Ph1C:::ChnDm Q bjh1:::hn Qn iD1 xhi i : : : 3 7 7 5 ; (2.2) where j D1; n, such that, for example, for a 2-DoF system the cubic terms of the first raw of Qb are Qb130x3 1 C Qb121x2 1x2 C Qb112x1x2 2 C Qb103x3 2. In order to decouple the linear part of the system, we apply classical modal analysis, i.e., denotingUthe matrix containing the eigenvectors of M 1K, we apply the transformation x DUy to Eq. (2.1) and we pre-multiply it by UT. The resulting system has the form UTMURyCUTDUPyCUTKUyCUTQbD p"UTQvf cos.!t/ ; (2.3) where UTMU and UTKU are diagonal. Pre-multiplying then the system by the inverse of UTMU and applying the transformationy D p"f q, we have RqCCPqC qCbDvcos.!t/ ; (2.4) where Cis not symmetric (differently fromD), and is diagonal and contains the squares of the natural frequencies of the different modes of vibration. , Candbhave the general form D 2 6 6 4 : : : 0 2 j 0 : : : 3 7 7 5 ; CD 2 6 4 c11 : : : c1n : : : : : : cn1 : : : cnn 3 7 5 ; bD 2 6 6 4 : : : PmD3;5;::: " m 1 2 f m 1 Ph1C:::ChnDm bjh1:::hn Qn iD1 qhi i : : : 3 7 7 5 : (2.5) The forcing amplitude f is contained uniquely in b, i.e. the system depends on the forcing amplitude only through the coefficients of the nonlinear terms. This clearly illustrates that considering small nonlinearities or small forcing amplitudes is equivalent, in many cases. Furthermore, terms of order mare proportional to " m 1 2 (see b), which prepares the ground for the perturbation procedure implemented in the following sections.
2 Enforcing Linear Dynamics Through the Addition of Nonlinearity 13 For the sake of simplicity, we consider only cubic nonlinearities in this analytical development. Neglecting terms higher than the third order allows us to develop the equations up to order "1. Nevertheless, the procedure can be extended to higherorder nonlinear terms. 2.3 Calculation of Unforced, Undamped Response Using Nonlinear Normal Modes NNMs give a good approximation of the system backbone curves. In order to calculate them, we consider the undamped unforced equations of motion RqC qCbD0: (2.6) Adopting a standard perturbation procedure, the solution of Eq. (2.6) can be approximated to a single harmonic by qD q0 C"q1 CO " 2 sin.!t/ ; (2.7) where q0 D 2 6 6 4 : : : qj0 : : : 3 7 7 5 ; q1 D 2 6 6 4 : : : qj1 : : : 3 7 7 5 and ! D!0 C"!1 CO " 2 ; (2.8) which is valid for small values of ". Substituting Eq. (2.7) into Eq. (2.6) and adopting the standard single harmonic approximation sin3 .!t/ 3=4sin.!t/, we obtainnequations of the form !2 0 C2"!1!0 qj0 C"qj1 C 2 j qj0 C"qj1 C 3 4 " 0 @ X h1C:::ChnD3 bjh1:::hn n Y iD1 qhi i 1 A CO "2 D0: (2.9) Considering the terms of order "0, related to the underlying linear system, we have !2 0qj0 C 2 j qj0 D0: (2.10) In order to obtain the NNM associated with the lth mode of vibration, we impose that the linear part of all other modes have zero amplitude, i.e. forj ¤l )qj0 D0; (2.11) while ql0 ¤0, such that the we refer to the lth mode of vibration and!0 D l. Considering now the terms of order "1 of Eq. (2.9), we obtain j ¤l 2 l qj1 C 2 j qj1 C 3 4 bj0:::3:::0q 3 l0 D0 (2.12) j Dl 2 l ql1 2!1 lql0 C 2 l ql1 C 3 4 bl0:::3:::0q 3 l0 D0: (2.13) (bj0:::3:::0, for example in a 4-DoF system where l D2, wouldbe bj0300). Thus, from Eq. (2.12) we have qj1 D 3 4 bj0:::3:::0q3 l0 2 l 2 j ; j ¤l; (2.14)
14 G. Habib et al. that indicates how the modes not directly excited by the force (j ¤ l) are excited by the nonlinear coupling. While from Eq. (2.13) we have !1 D 3 4 bl0:::3:::0q2 l0 2 l ; (2.15) where !1 represents the variation of the lth natural frequency with respect to the amplitude of oscillations, in first approximation. 2.4 Calculation of Forced, Damped Response Using Energy Balance The energy balance criterion can be used to relate the undamped, unforced dynamics of the NNMs to the forced damped dynamics [10, 11], thus obtaining complete information of the resonant peaks. Given a general linearly-damped mechanical system, the balance between the dissipated and input energies is expressed by the equation Z T 0 P x.t/ TDPx.t/dt D Z T 0 P x.t/ Tf.t/dt; (2.16) whereTis the period of vibration andf is the external force. In the case of harmonic excitationf D Qvf cos.!t/, approximating the solution to a single harmonics x.t/ x0 sin.!t/ in resonant conditions, Eq. (2.16) has the form !2x T 0Dx0 Z T 0 cos.!t/2 dt !x T 0 Qvf Z T 0 cos.!t/2 dt ) !x T 0Dx0 x T 0 Qvf : (2.17) Inserting in Eq. (2.17) the solution of the undamped unforced system, it is possible to estimate the ratio between the forcing amplitude and the amplitude of oscillation. Applying the aforementioned procedure, considering the system in Eq. (2.4) and the tentative solution (2.7), we obtain the energy balance equation . l C"!1/ 0 B B B B B B B @ 2 6 6 6 6 6 6 4 0 : : : ql0 : : : 0 3 7 7 7 7 7 7 5 T C" 2 6 4 q11 : : : qn1 3 7 5 T 1 C C C C C C C A C 0 B B B B B B @ 2 6 6 6 6 6 6 4 0 : : : ql0 : : : 0 3 7 7 7 7 7 7 5 C" 2 6 4 q11 : : : qn1 3 7 5 1 C C C C C C A D 0 B B B B B B B @ 2 6 6 6 6 6 6 4 0 : : : ql0 : : : 0 3 7 7 7 7 7 7 5 T C" 2 6 4 q11 : : : qn1 3 7 5 T 1 C C C C C C C A 2 6 4 v1 : : : vn 3 7 5C O "2 : (2.18) Collecting terms of order "0 we obtain lq 2 l0cll Dql0vl ) ql0 D vl lcll ; (2.19) which yields the relation in the linear range between the forcing amplitude (here normalized) and the oscillation amplitude at resonance as a function of the modal damping.
2 Enforcing Linear Dynamics Through the Addition of Nonlinearity 15 Collecting terms of order "1 of Eq. (2.18) we have lql0 0 B @ n X jD1 j¤l cljqj1 C n X jD1 j¤l cjlqj1 1 C AC2 lql0cllql1 C!1q 2 l0cll D n X jD1 j¤l qj1vj Cq1lvl; (2.20) thus ql1 D lql0 Pn jD1 j¤l cljqj1 CPn jD1 j¤l cjlqj1 C!1q2 l0cll Pn jD1 j¤l qj1vj vl 2 lql0cll ; (2.21) which indicates the variation of the modal amplitude of oscillation due to nonlinearity. 2.5 Enforcement of Linear Properties The set of Eqs. (2.11), (2.14), (2.15), (2.19) and (2.21) characterize the NNMs and forced resonant response of the lth structural mode. Equations (2.11) and (2.19) refer to the underlying linear system; Eq. (2.11) is due to the imposed resonant condition, while Eq. (2.19) gives the relationship between the forcing amplitude and the amplitude of oscillation in the linear case. Equations (2.14), (2.15) and (2.21) refer to the nonlinear properties of the system. qj1 .j D 1n; j ¤ l/ are directly proportional to bj0:::3:::0, !1 is directly proportional to bl0:::3:::0, while ql1 depends linearly on all the coefficients bj0:::3:::0 (j D1; n), which are the coefficients of the solely nonlinear terms relevant at order "1, when the lth resonance is excited. If the system includes higher-order nonlinear terms, other coefficients come into play. The objective of this section is to show that, through an appropriate tuning of the coefficients bj0:::3:::0, the dynamics of the nonlinear system can resemble that of a linear system. 2.5.1 Enforcing Force-Displacement Proportionality We consider the general case for which the objective is to keep the force-displacement proportionality typical of linear systems for the kth DoF (xk in the physical coordinate system), while the system vibrates at the lth resonant frequency. Recalling that x D p"f Uqand considering the lth resonance, it follows that xk D p"f 0 @ uklql0 C" 0 @ n X jD1 ukjqj1 1 AC O "2 1 A : (2.22) where uij is an element of matrixU. If we impose that n X jD1 ukjqj1 D0 (2.23) xk obeys, in first approximation, a force-displacement proportionality. Equation (2.23) depends on the n coefficients bj0:::3:::0 (j D1n), which, in turn, depend on the parameters characterizing the physical nonlinearities of the system. 2.5.2 Enforcing Straight Line Frequency Backbone Another property, typical of linear systems and generally not satisfied in nonlinear systems, is the invariance of the resonance frequencies with respect to forcing amplitude, giving rise to straight backbone curves. Hardening (softening) nonlinearities
16 G. Habib et al. shift resonance frequencies toward greater (lower) values for increasing amplitudes of oscillation. To enforce a straight backbone curve for the lth resonance, !1 should be set to 0, and, hence, bl0:::3:::0 D0. An important feature of the proposed approach is that the final equations are fully explicit. Thus, in spite of their complexity, they can be rapidly implemented. 2.6 Beam Example To validate the previous theoretical developments, a nonlinear cantilever beam with a nonlinear attachment, similar to that studied in [12, 13], is considered. The two nonlinearities of the attachment, knl2 andknl3, are designed so as to enforce linear properties in the coupled system. A 2-DoF reduced-order model of this system is 0:46 0 0 0:069 Rx1 Rx2 C 0:52 0:15 0:15 0:25 Px1 Px2 C 14065 1709 1709 1709 x1 x2 C knl1x3 1 Cknl2 .x1 x2/ 3 knl2 .x2 x1/ 3 Ck nl3x3 2 D f 1 0 : (2.24) The cantilever beam has a single concentrated nonlinearity, knl1 D3:3 10 9 N/m3. Figure 2.1a illustrates the normalized frequency response of the first degree of freedom of the system without the additional nonlinearities, i.e., knl2 D knl3 D 0, for three forcing amplitudes. A substantial hardening effect is observed, while the amplitude of the first (second) resonance decreases (increases) when the forcing amplitude increases. The displacement x1 around the first resonance obeys force-displacement proportionality if 0:0128C1:39 10 10k nl2 C2:17 10 10k nl3 D0: (2.25) which is verified if knl2 D9:2 10 7 N/m3 and k nl3 D0. Figure 2.1b that depicts the corresponding normalized frequency response confirms that the amplitude of the first resonance is almost identical for the three forcing amplitudes. The linearization of the force-displacement relation can be further improved if a fifth-order spring is added between the two lumped masses. This is evidenced in Fig. 2.1c, which compares the envelopes of the resonant peaks in the different considered cases. Aiming now to enforce isochronicity of the first resonance, i.e., !1 D0, we obtain 62:6C7:39 10 7k nl2 C2:84 10 6k nl3 D0: (2.26) 150 200 250 0 2 4 x 10−3 ω [rad/s] x1/f [m/N] x1/f [m/N] x1/f [m/N] a 150 200 250 0 2 4 x 10−3 ω [rad/s] b 0 0.1 0.2 0.3 0.4 0.5 0 2 4 x 10−3 f [N] c Fig. 2.1 (a), (b) frequency response of the system in Eq. (2.24) for forcing amplitudes f !0, f D0:3 and f D0:5N, for knl2 Dknl3 D0 (a) and for knl2 D9:2 10 7 N/m3 and k nl3 D0 (b); (c) envelope of the first resonant peak with respect to the forcing amplitude. Dashed line: knl2 Dknl3 D0; solid line: knl2 D9:2 10 7 N/m3 and k nl3 D0; dash-dotted line: knl2 D9:2 10 7 N/m3, k nl3 D0 and additional quintic spring knl4 D3:9 10 12 N/m5
2 Enforcing Linear Dynamics Through the Addition of Nonlinearity 17 150 200 250 0 2 4 x 10−3 ω [rad/s] x1/f [m/N] 150 200 250 0 2 4 x 10−3 ω [rad/s] x1/f [m/N] 0 0.1 0.2 0.3 0.4 0.5 0 2 4 x 10−3 x1/f [m/N] f [N] (2) (1) a b c Fig. 2.2 Frequency response of the system in Eq. (2.24) for forcing amplitudes f !0, f D0:3 and f D0:5N, for knl2 D2:11 10 7 and knl3 D 7:7 10 7 (a) and for k nl2 D2:75 10 7 and k nl3 D4:14 10 7 (b); envelope of the two resonant peaks with respect to the forcing amplitude. Dashed lines: knl2 Dknl3 D0; solid lines: knl2 D2:11 10 8 and k nl3 D 7:7 10 7; the numbers in brackets indicate the first (1) and the second (2) resonant peak Equations (2.25) and (2.26) can be simultaneously satisfied if and only if knl2 D2:11 10 8 and k nl3 D 7:7 10 7. Doing so, the first resonance peak can be made practically unchanged with respect to the linear resonance, as plotted in Fig. 2.2a. The developed framework can go beyond operating on a single resonance. For instance, force-displacement proportionality of the second peak (with respect to x1) can be imposed through 0:00654 1:53 10 10k nl2 5:63 10 11k nl3 D0: (2.27) For knl2 D 2:75 10 7 and k nl3 D 4:14 10 7, Eqs. (2.25) and (2.27) are simultaneously verified, which means that both resonances obey approximately force-displacement proportionality. The corresponding normalized frequency response is illustrated in Fig. 2.2b. Figure 2.2c depicts the envelopes of the two peaks with respect to the forcing amplitude. The dashed lines, referring to the original system (knl2 Dknl3 D0), either decrease (first peak) or increase (second peak) whereas the solid lines are almost horizontal for a large range of forcing amplitudes. 2.7 Conclusions This paper has demonstrated how it is possible to design nonlinearities of a mechanical system so that its dynamics resemble that of a linear system, which include force-displacement proportionality and isochronocity. The developments exploited a standard perturbation technique, combined with NNM theory and energy balance of periodic solutions at resonance, and were validated using a 2-DoF reduced-order model of a nonlinear cantilever beam. Acknowledgements The authors G. Habib, C. Grappasonni and G. Kerschen would like to acknowledge the financial support of the European Union (ERC Starting Grant NoVib 307265). References 1. Hartog, J.D.: Mechanical Vibrations. Courier Corporation, Chelmsford (1985) 2. Ono, T., Li, X., Miyashita, H., Esashi, M.: Mass sensing of adsorbed molecules in sub-picogram sample with ultrathin silicon resonator. Rev. Sci. Instrum. 74(3), 1240–1243 (2003) 3. Piazza, G., Stephanou, P.J., Pisano, A.P.: Piezoelectric aluminum nitride vibrating contour-mode MEMS resonators. J. Microelectromech. Syst. 15(6), 1406–1418 (2006) 4. Rega, G., Settimi, V.: Bifurcation, response scenarios and dynamic integrity in a single-mode model of noncontact atomic force microscopy. Nonlinear Dyn. 73(1–2), 101–123 (2013)
RkJQdWJsaXNoZXIy MTMzNzEzMQ==