Vibrations and Nonlinear Dynamics Laboratory, IIT-Madras
Lab Research Website

• (<2025-02-01 Sat>) Proposal for Nonlinear Normal Modes in FEAST, NAFED 08, IITH.
• (<2025-02-11 Tue>) Nonlinear Modal Synthesis under Narrow-band Random Excitation, IMAC 2025, Orlando, FL.
• (<2025-02-12 Wed>) Epistemic Insights on Dry-Friction From Micro-Slip Measurements, IMAC 2025, Orlando, FL.
• (<2025-02-13 Thu>) Averaging-based Stability Certification for Harmonic Balance applied to Fold and Neimark-Sacker bifurcations, IMAC 2025, Orlando, FL.

In a lot of practical problems involving systems undergoing periodic or nearly periodic excitation/response, the response can be described as a periodic solution with a slowly evolving envelope. Some examples included gradual degradation of rotating systems, heat-induced property change in vibration machinery, mechanical joint degradation due to wear, laminar fluid-field excitation on bluff body (von Karman vortex streets), narrow-banded random excitation, etc. In all such situations, time-scale separability between fastly evolving vibrations and slowly evolving envelopes is readily possible.

While vibratory systems need to be sampled higher than the Nyquist frequency (usually several times more), if all that one is interested in is health monitoring, it is sufficient to identify and track the envelope of the process. Being slowly evolving processes, it must be possible to sample these at much smaller intervals. The research question here is: Knowing the periodicity properties of a system, can we reconstruct the vibration envelope using undersampled data?

Links

• (Image) A schematic view of time-scale decomposition.

A fundamental observation that can be made in thermo-mechanical vibration problems is that the elastic/mechanical portion of the problem is governed by a hyperbolic PDE that leads to oscillatory/harmonic solutions, while the heat equation is parabolic with considerably larger time constants.
Application of time-scale decomposition is a promising framework for dealing with such coupled problems. For example in frictional systems, the vibration response will be accounted for by the fast dynamics and the energy dissipated as heat will be accounted for by slowly evolving envelopes. Thermal fluxes from the environment can also be considered under such a framework as having a direct influence on the vibrations.
Mathematically this will be achieved by using a singular perturbation framework leading to harmonic balance for the vibrations and a first order system for the slow dynamics.

Links

• (Image) A schematic view of time-scale decomposition.

Modeling rough surfaces as stochastic processes has been shown to lead to scale-independent descriptions of rough surfaces. We argue here that if such parameters are scale independent and therefore "features" of the surface, it MUST be possible to relate characteristic constitutive quantities to these features.

Furthermore, stochastic process models of surfaces seem to imply that one can write down a Stochastic Difference Equation (SDE) describing the structure of the surface features with length-scale being the independent variable. Preliminary studies in the literature indicate that this implies a "hierarchical" structure, with the largest length-scale features (read: macro-scale) influencing the intermediate scales (meso-scale) influencing the smallest scales (micro-scale roughness).

With the pursuit of this project, I hope to develop a clearer description of the different scales in the description of a rough surface and their relative contributions to the overall ("macro") dynamics of a jointed system.

Links

• (Image) The 2D Structure (SF) and Auto-Correlation Functions (ACF) of a machined (milled) surface.

In most applications involving frictional interfaces coupled to geometrically nonlinear sub-domains (panels riveted/bolted at its ends, etc.), it can be observed that the former are not geometrically nonlinear and latter (large displacement/strain) do not undergo contact under nominal operation. Therefore it makes sense to employ smooth reduced order modeling approaches (orthogonal polynomial projections, etc.) for the latter sub-domains and non-smooth approaches for the former sub-domain.

It has been a standard observation that contact interfaces show non-trivial evolution wherefore employing smooth polynomial fields is not the best way to go. It is my impression that strategies based on adaptive mesh-refinement are perhaps the way to go to improve computational performance in contact problems. Unlike earlier studies, I hope to directly use the governing partial differential equations for conducting the mesh-refinements, rather than using a finely resolved finite element mesh as reference.

Links

• (Images) Whole joint and mesh-coarsening reduced order representations for a jointed interface.

Nonlinear Modal Analysis has gradually started becoming reasonably mature in academic circles over the last two decades. Unlike linear modes, nonlinear modes allow characterization of modal characteristics (natural frequency, damping factor, mode-shape) as response-amplitude dependent features of the system. Substituting the employment of classical linear modal analysis in the industrial setting has all indications of being the next paradigm in the community.

Computational and experimental advances have ensured that analysis as well as experimental characterization of nonlinear modes do not pose prohibitive expenses (as compared to conducting linear analyses/experiments) any longer. Since the scientific advances have matured, technology adoption is the next step. This research will involve the following aspects (which will ideally be achieved by at least 3 candidates):

• Implementation and validation of nonlinear modal solvers in industrial finite-element software;
• Implementation and validation of nonlinear modal testing methods (Phase Locked-Loops, Control-Based Continuation) for industrial test cases;
• Addressing lingering theoretical questions pertaining to mode-mixing, modal interference, etc.

Links

• (Image) Experimental setup for nonlinear modal testing of a jointed fixed-free beam.

We recently developed a suite of wave-based analysis tools (see here for a MATLAB/OCTAVE toolbox) that are generally applicable for periodic and quasi-periodic responses of 1D continuum structures. The main advantage here is that the approximations obtained are spatially exact, with errors coming only from the temporal (Fourier) approximations. This results in a computational approach that is several orders of magnitude faster than conventional techniques such as finite elements, etc., while assuring superior accuracy.

We are looking to demonstrate the applicability of this method to

• Rotordynamic problems with nonlinear bearings, contact, etc.,
• Spatial frame structures with industrial applications.

Theoretically, we are also trying to see if an averaging concept is applicable to analyze the stability of the resulting solutions.

Links

• Research to be conducted in collaboration with Prof. Mike Leamy at Georgia Tech.
• (Image) The wave-based description for a jointed bar and (Image) modified AFT procedure for this.
• Entry on past work on this area.

Self excitation is often observed in aeroelastic systems where the aerodynamic influences lead to energy being added into the sytem in a state-dependent fashion. The resulting instabilities are saturated, in practical applications, by mechanical contacts undergoing frictional dissipation. We develop a novel fully frequency domain approach for the bifurcation analysis of such systems. An imporant aspect of the approach here is that no theoretical conditions are violated for the stability analysis eventhough the systems are fundamentally non-smooth (dry friction modeled using non-smooth relationships).

Links

• Research conducted in collaboration with Prof. Malte Krack at the University of Stuttgart.
• (Images) Forced response and eigenvalues of stability for a self-excited oscillator with frictional supports showing the periodic (main) and quasi-periodic branches.

Thus far, nonlinear modal methods have found applications only in periodic response regimes. We are trying to understand the efficacy in random response regimes in this projects.

The evolution of the slow envelope is written in terms of a Stochastic Differential Equation (SDE), and solving the corresponding Fokker Planck Equations (FPE) allows us to identify the diffusion of the probability density of the envelopes directly.

Links

• Research conducted in collaboration with Prof. Malte Krack at the University of Stuttgart.
• (Image) Comparison of synthesized responses and experimental measurements for the system shown above.

We deal with the fundamental problem of identifying hysteretic models that best fit measurements made from a tribometer developed at the University of Stuttgart. Although several models exist in the literature, robust identification poses several challenges due to epistemic issues characteristic of friciton modeling.

Links

• Research conducted in collaboration with Prof. Malte Krack at the University of Stuttgart.
• (Image) Example of measurements (blue) and identification (red).

Pursuing the spirit of the method of multiple scales and time-scale decomposition, we have shown that it is indeed possible to rigorously write down slow-time evolution equations for the Fourier coefficients of a nonlinear oscillator. Unlike classical Floquet theory which requires smoothness and is thereby inapplicable to frictional contact problems, the averaging approach has lesser theoretical constraints, making it more generally applicable.

Links

• Research conducted in collaboration with Prof. Malte Krack at the University of Stuttgart.
• (Image) Stability characteristics of a unilateral spring system computed using the averaging approach.

Accounting for the coexistence of multiple nonlinear modes in a single response has been an enduring problem in the nonlinear modal analysis literature. It is becoming clear that ignoring multi-modal effects in system identification leads to several inconsistencies. We are conducting a theoretical analysis to understand this phenomenon better.

Links

• Research conducted in collaboration with Prof. D. Dane Quinn at the University of Akron.

Hysteretic forces (specifically rate-independent laws) are often represented as differential relationships. Evaluation of these requires an incremental implementation based on discretizing the differential operators. For periodic response calculations (using Harmonic Balance, for instance), imposing periodicity as a constraint leads to an iterative implementation that is repeated until convergence. In this work we generalized this idea to the multi-frequency "quasi-periodic" case.

Links

• Research conducted as part of PostDoc with Prof. Malte Krack at the University of Stuttgart.
• Link to paper.
• (Image) Flow of time in 2 dimensions (torus coordinates) and QP forced response of a frictional oscillator.

Decoupling the dynamics of a weakly nonlinear oscillator into fast-varying dynamics and slowly-varying amplitudes leads to a theoretical framework referred to as quasi-linearization. When the amplitude is kept constant, the system dynamics ends up behaving in a linear manner. Conducting testing (forced response, for instance) while controlling the response amplitude (adjusting force appropriately) leads to fully linear responses. Repeating such tests for multiple amplitude levels has been shown to allow one to paint an accurate picture of the dynamics of the system.

In this work, we used this framework to develop a novel transfer function identification methodology. The transfer function is dependent, apart from the \(s\) parameter, also on the response amplitude. Solving the resulting problem using sparse regression leads to a computationally attractive system identification method for weakly non-linear systems.

Links

• Research conducted in collaboration with Prof. Matthew Brake at Rice University.
• (Image) A geometrically nonlinear beam used for the study and its measured forced response.

In this work we developed a suite of techniques, based on wave propagation representations on 1D bars and beams, to analyse the periodic responses and stability of jointed 1D structures. The main advantage with the wave-based approach is that it is spatially exact and the approximations are introduced from the time-representation only. A semi-analytical approach based on singular perturbation and a numerical approach based on plane-wave expansions were formulated and applied.

For stability analysis, a generalized strained-parameter approach using singular perturbations and a numerical approach based on sign changes in the determinant are proposed. It must be noted that these stability analysis approaches are capable of detecting simple bifurcations (fold-type, period-doubling) but will not work for Hopf-type bifurcations (Neimark-Sacker). A MATLAB/OCTAVE toolbox named WaveVib has been created based on this work, which is aimed to allow fast prototyping with a versatile "front-end".

Links

• Research conducted in collaboration with Prof. Matthew Brake at Rice University and Prof. Mike Leamy at Georgia Tech.
• (Image) Wave based description and the PWE-AFT procedure.
• Links to paper: Part-1 and Part-2.

Digital Image Correlation (DIC) has emerged as the new standard in video-based vibration testing. In this work, we established, firstly, a framework for conducting Experimental Modal Analysis (EMA) using DIC measurements for a system under electrodynamic excitation. Although DIC provides full-field data, the main drawback holding it from wide-spread applicability is that post-processing is a rather expensive process. We therefore adapted (fine-tuned) an existing deep Convolutional Neural Network (CNN) to substitute the DIC computation in the EMA application and demonstrated that such Neural Networks can indeed substitute DIC, with some examples showing even a 15x increase in computational performance.

Links

• Research conducted as part of PostDoc with Prof. Matthew Brake at Rice University.
• (Image) The CNN architecture used (from Yang et al. 2021).

Frictional interfaces are classically known to be statically indeterminate under nominal loads (until the "slip limit"). For structures with multiple joints wherein the tangential restoring forces from one joint influences the normal loading conditions of another joint, this indeterminacy manifests in a very pronounced manner, making the system exhibit non-unique solutions. Employing a mathematical parameterization of this effect for the elastic dry-friction element, we analyzed this effect from a stochastic perspective (using PCE) in the context of other possible uncertainties in frictional joints. It was observed that the influence of this non-uniqueness is most pronounced in the micro-slip regime.

Links

• Research conducted in collaboration with Dr. Erhan Ferhatoglu.
• (Image) Modal backbones along with the computed first order Sobol' indices.
• Paper under preparation.

Wear and friction related damage is a rather obvious expectation from mechanical joints. Capturing such phenomena quantitatively is, however, an enduring challenge in the joints community. We conducted a 12-hour experimental campaign studying the evolution of the dynamics of a jointed system. Frequency response and hammer impact-based nonlinear modal tests were conducted to characterize the dynamics. Interferometry was used to obtain height maps of the interfaces to characterize roughness. Continuous periodic excitation was provided at a near-resonant condition to induce wear in the interface.

Links

• The collected dataset is hosted here.
• Research conducted in collaboration with Prof. Matthew Brake at Rice University and Dr. Scott A. Smith at Norwich University.
• Conference paper presented at IMAC 2023.
• (Image) The Brake-Reuss Beam used for the studies.
• (Image) Evolution of selected regions of the interface over the campaign.
• Paper under review in MSSP.

We applied a statistical rough contact approach to conduct predictive nonlinear modal analysis of a bolted structure and compared this against experimental measurements. Frequency accuracy within 5% and damping accuracy within 25% were achieved. We also employed Polynomial Chaos Expansion for uncertainty quantification.

Links

• Research conducted as part of PhD under Prof. Matthew Brake at Rice University.
• (Image) The bolt-jointed benchmark used for the study and predicted dissipation fluxes over the interfaces at a nonlinear resonance point.
• Link to PhD thesis: "Dissipative Dynamics of Bolted Joints".
• Link to MS thesis: "Multi-Scale Modeling in Bolted Interfaces".
• Paper under preparation.

Hyper-reduction in the context of Reduced Order Modeling (ROM) refers to the development of ROMs wherein the model evaluation can be conducted completely without transforming the system back to the Full Order Model (FOM). Most projection based ROMs require one to first transform the reduced Degrees-of-Freedoms (DoFs) to the FOM domain for the evaluation of nonlinear forces (interfacial friction, for instance). We developed two novel hyper-reduction approaches for interfacial reduction of jointed systems.

The first technique is based on the so-called Spider elements/distributing coupling elements. These formulations are modified to be tailor-fit to flat interfaces and relevant constitutive models are also developed. The resulting "whole-joint model" provides a highly compressed representation of the original system, wherein the forces and moments on the "patches" represent specific traction distributions on the original interface.

The second technique is based on remeshing the interface and constructing the projection matrices in a manner that allows one to evaluate the nonlinear forces fully in the reduced mesh. The way we do this here is to evaluate the projection operators completely based on quadrature points of the reduced mesh. While this leads to a special left projector matrix for the vector of nonlinear forces, it completely avoids having to transform the system back to the original domain.

Links

• Research conducted as part of PhD under Prof. Matthew Brake at Rice University in collaboration with Prof. Malte Krack at the University of Stuttgart.
• (Images) Whole joint and mesh-coarsening reduced order representations for a jointed interface.
• Link to paper.

The most popular techniques for nonlinear modal analysis involve either transient shooting or Harmonic balance (which also involves transient evaluation). More recently quasi-static simulations, under certain assumptions, have been shown to lead to an approximate but efficient approach for nonlinear modal analysis. These were termed Quasi-Static Modal Analysis (QSMA) and have found applications in several practical examples.

In this work we went back to linear vibration theory where linear modes are defined as the minimizers of Rayleigh quotients, which are quadratic functions of the states. For weakly nonlinear systems, making a near-quadratic assumption for a generalized Rayleigh quotient leads to a completely new concept of nonlinear modes. Here, we apply the principle of virtual work (since energy can't be defined consistently for dissipative systems) and obtain a set of quasi-static equations with a scalar constraint. The constraint introduces a Lagrange multiplier as an additional unknown, which is interpreted in congruence with linear vibrations (where \(\lambda=\omega_n^2\)).

Links

• Research conducted as part of PhD under Prof. Matthew Brake at Rice University.
• (Image) Low and high amplitude mode-shapes of a phenomenological model of a bolt-jointed system.
• Link to paper.