Reconstruction of governing equations from vibration measurements for geometrically nonlinear systems

Data-driven system identification procedures have recently enabled the reconstruction of governing differential equations from vibration signal recordings. In this contribution, the sparse identification of nonlinear dynamics is applied to structural dynamics of a geometrically nonlinear system. First, the methodology is validated against the forced Duffing oscillator to evaluate its robustness against noise and limited data. Then, differential equations governing the dynamics of two weakly coupled cantilever beams with base excitation are reconstructed from experimental data. Results indicate the appealing abilities of data-driven system identification: underlying equations are successfully reconstructed and (non-)linear dynamic terms are identified for two experimental setups which are comprised of a quasi-linear system and a system with impacts to replicate a piecewise hardening behavior, as commonly observed in contacts.

[1]  M. S. Agbabian,et al.  System identification approach to detection of structural changes , 1991 .

[2]  F. Fontanela,et al.  Multistability and localization in forced cyclic symmetric structures modelled by weakly-coupled Duffing oscillators , 2019, Journal of Sound and Vibration.

[3]  D. J. Ewins,et al.  A method for modal identification of lightly damped structures , 1982 .

[4]  Laurent Baillet,et al.  Contact surface topography and system dynamics of brake squeal , 2008 .

[5]  Christoph W. Schwingshackl,et al.  DIC Measurement of the Kinematics of a Friction Damper for Turbine Applications , 2017 .

[6]  Christoph W. Schwingshackl,et al.  A method for non-parametric identification of non-linear vibration systems with asymmetric restoring forces from a resonant decay response , 2019, Mechanical Systems and Signal Processing.

[7]  K K Padmanabhan,et al.  Damping in Structural Joints Subjected to Tangential Loads , 1991 .

[8]  Annalisa Fregolent,et al.  Instability scenarios between elastic media under frictional contact , 2013 .

[9]  Laurent Baillet,et al.  Interaction between contact behaviour and vibrational response for dry contact system , 2018, Mechanical Systems and Signal Processing.

[10]  Annalisa Fregolent,et al.  Dynamic Substructuring with a Sliding Contact Interface , 2018 .

[11]  M. Ciavarella,et al.  Effect of normal load variation on the frictional behavior of a simple Coulomb frictional oscillator , 2015 .

[12]  Sebastian Oberst,et al.  Recovery of Differential Equations from Impulse Response Time Series Data for Model Identification and Feature Extraction , 2018, Vibration.

[13]  Michele Ciavarella,et al.  Snaking bifurcations in a self-excited oscillator chain with cyclic symmetry , 2017, Commun. Nonlinear Sci. Numer. Simul..

[14]  Michele Ciavarella,et al.  Multiple spatially localized dynamical states in friction-excited oscillator chains , 2018 .

[15]  F. Takens.  Detecting strange attractors in turbulence , 1981 .

[16]  Christoph W. Schwingshackl,et al.  Structural Design with Joints for Maximum Dissipation , 2016 .

[17]  M. Brake.  The Mechanics of Jointed Structures , 2018 .

[18]  Aldo Sestieri,et al.  Brake squeal: Linear and nonlinear numerical approaches , 2007 .

[19]  D. P. Hess,et al.  Evaluating Surface Roughness From Contact Vibrations , 1995 .

[20]  A. Polycarpou,et al.  Measurement and Modeling of Normal Contact Stiffness and Contact Damping at the Meso Scale , 2005 .

[21]  Michele Ciavarella,et al.  Load-separation curves for the contact of self-affine rough surfaces , 2017, Scientific Reports.

[22]  Jean-Philippe Noël,et al.  Nonlinear system identification in structural dynamics: 10 more years of progress , 2017 .

[23]  Merten Tiedemann,et al.  Determining growth rates of instabilities from time-series vibration data: Methods and applications for brake squeal , 2019, Mechanical Systems and Signal Processing.

[24]  Rick Chartrand,et al.  Numerical Differentiation of Noisy, Nonsmooth Data , 2011 .

[25]  Sebastian Oberst,et al.  A statistical approach to estimate the Lyapunov spectrum in disc brake squeal , 2015 .

[26]  Mehmet Kurt,et al.  Nonlinear system identification of the dynamics of a vibro-impact beam: numerical results , 2012 .

[27]  C. Farrar,et al.  SYSTEM IDENTIFICATION FROM AMBIENT VIBRATION MEASUREMENTS ON A BRIDGE , 1997 .

[28]  F. Fontanela,et al.  Dark solitons, modulation instability and breathers in a chain of weakly nonlinear oscillators with cyclic symmetry , 2017, 1704.00695.

[29]  Christoph W. Schwingshackl,et al.  A method for detection and characterisation of structural non-linearities using the Hilbert transform and neural networks , 2017 .

[30]  L. Tsimring,et al.  The analysis of observed chaotic data in physical systems , 1993 .

[31]  Norbert Hoffmann,et al.  Dominant damping effects in friction brake noise, vibration and harshness: the relevance of joints , 2015 .

[32]  Sebastian Oberst,et al.  Complex machine dynamics: systematic recurrence quantification analysis of disk brake vibration data , 2019, Nonlinear Dynamics.

[33]  H. Abarbanel,et al.  Determining embedding dimension for phase-space reconstruction using a geometrical construction. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[34]  S. Brunton,et al.  Discovering governing equations from data by sparse identification of nonlinear dynamical systems , 2015, Proceedings of the National Academy of Sciences.