Vibration, Volume 3, Issue 3 (September 2020) – 10 articles

In the present article, we follow up our recent work on the experimental assessment of two data-driven nonlinear system identification methodologies. The first methodology constructs a single nonlinear-mode model from periodic vibration data obtained under phase-controlled harmonic excitation. The second methodology constructs a state-space model with polynomial nonlinear terms from vibration data obtained under uncontrolled broadband random excitation. The conclusions drawn from our previous work (experimental) were limited by uncertainties inherent to the specimen, instrumentation, and signal processing. To avoid these uncertainties in the present work, we pursued a completely numerical approach based on synthetic measurement data obtained from simulated experiments. Three benchmarks are considered, which feature geometric, unilateral contact, and dry friction nonlinearity, respectively. As in our previous work, we assessed the prediction accuracy of the identified models with a focus on the regime near a particular resonance. This way, we confirmed our findings on the strengths and weaknesses of the two methodologies and derive several new findings: First, the state-space method struggles even for polynomial nonlinearities if the training data is chaotic. Second, the polynomial state-space models can reach high accuracy only in a rather limited range of vibration levels for systems with non-polynomial nonlinearities. Such cases demonstrate the sensitivity to training data inherent in the method, as model errors are inevitable here. Third, although the excitation does not perfectly isolate the nonlinear mode (exciter-structure interaction, uncontrolled higher harmonics, local instead of distributed excitation), the modal properties are identified with high accuracy. Full article

The present paper proposes a linearised hybrid finite element-statistical energy analysis (FE-SEA) formulation for built-up systems with nonlinear joints and excited by random, as well as harmonic, loadings. The new formulation was validated via an ad-hoc developed stochastic benchmark model. The latter was derived through the combination of the Lagrange-Rayleigh-Ritz method (LRRM) and the Monte Carlo simulation (MCS). Within the build-up plate systems, each plate component was modelled by using the classical Kirchhoff’s thin-plate theory. The linearisation processes were carried out according to the loading-type. In the case of random loading, the statistical linearisation (SL) was employed, while, in the case of harmonic loading, the method of harmonic balance (MHB) was used. To demonstrate the effectiveness of the proposed hybrid FE-SEA formulation, three different case studies, made-up of built-up systems with localized cubic nonlinearities, were considered. Both translational and torsional springs, as joint components, were employed. Four different types of loadings were taken into account: harmonic/random point and distributed loadings. The response of the dynamic systems was investigated in terms of ensemble average of the time-averaged energy. Full article

This work suggests a solution for joint input-state estimation for nonlinear systems. The task is to recover the internal states of a nonlinear oscillator, the displacement and velocity of the system, and the unmeasured external forces applied. To do this, a Gaussian process latent force model is developed for nonlinear systems. The model places a Gaussian process prior over the unknown input forces for the system, converts this into a state-space form and then augments the nonlinear system with these additional hidden states. To perform inference over this nonlinear state-space model a particle Gibbs approach is used combining a “Particle Gibbs with Ancestor Sampling” Markov kernel for the states and a Metropolis-Hastings update for the hyperparameters of the Gaussian process. This approach is shown to be effective in a numerical case study on a Duffing oscillator where the internal states and the unknown forcing are recovered, each with a normalised mean-squared error less than 0.5%. It is also shown how this Bayesian approach allows uncertainty quantification of the estimates of the states and inputs which can be invaluable in further engineering analyses. Full article

In stepped-sine testing of strongly nonlinear structures with the classical force-control strategy, corrective force perturbations of a standard controller used to capture the reference signal in the proximity of turning points of frequency response curves may often lead to a premature jump before reaching the actual resonance peak. Accordingly, a classical force-control approach is not suitable to identify backbone curves of strongly nonlinear structures. This paper shows that currently available commercial modal test equipment can accurately identify backbone curves of strongly nonlinear structures by using Response-Controlled stepped-sine Testing (RCT) and the Harmonic Force Surface (HFS) concept, both recently proposed by the authors. These methods can be applied to systems where there are many nonlinearities at several different (and even unknown) locations. However, these techniques are not applicable to systems where internal resonances occur. In RCT, the displacement amplitude of the driving point, rather than the amplitude of the applied force, is kept constant during the stepped-sine testing. Spectra of the harmonic excitation force measured at several different displacement amplitude levels are used to build up a smooth HFS. Isocurves of constant amplitude forcing on the HFS lead to constant-force frequency response curves with accurately measured turning points and unstable branches (if there are any), which makes it possible to identify backbone curves of strongly nonlinear structures experimentally. The validation of the proposed approach is demonstrated with numerical and experimental case studies. A five degree-of-freedom (DOF) lumped system with five cubic stiffness elements, which create strong conservative nonlinearity, is used in the numerical example. Experimental case studies consist of a cantilever beam and a control fin actuation mechanism of a real missile structure. The cantilever beam is supported at its free-end by two metal strips constrained at both ends to create strong stiffening nonlinearity. The control fin actuation mechanism exhibits very complex and strong nonlinearity due to backlash and friction. Full article

A digital twin is a powerful new concept in computational modelling that aims to produce a one-to-one mapping of a physical structure, operating in a specific context, into the digital domain. The development of a digital twin provides clear benefits in improved predictive performance and in aiding robust decision making for operators and asset managers. One key feature of a digital twin is the ability to improve the predictive performance over time, via improvements of the digital twin. An important secondary function is the ability to inform the user when predictive performance will be poor. If regions of poor performance are identified, the digital twin must offer a course of action for improving its predictive capabilities. In this paper three sources of improvement are investigated; (i) better estimates of the model parameters, (ii) adding/updating a data-based component to model unknown physics, and (iii) the addition of more physics-based modelling into the digital twin. These three courses of actions (along with taking no further action) are investigated through a probabilistic modelling approach, where the confidence of the current digital twin is used to inform when an action is required. In addition to addressing how a digital twin targets improvement in predictive performance, this paper also considers the implications of utilising a digital twin in a control context, particularly when the digital twin identifies poor performance of the underlying modelling assumptions. The framework is applied to a three-storey shear structure, where the objective is to construct a digital twin that predicts the acceleration response at each of the three floors given an unknown (and hence, unmodelled) structural state, caused by a contact nonlinearity between the upper two floors. This is intended to represent a realistic challenge for a digital twin, the case where the physical twin will degrade with age and the digital twin will have to make predictions in the presence of unforeseen physics at the time of the original model development phase. Full article

The experimental characterisation of a nonlinear structure is a challenging process, particularly for multiple degree of freedom and continuous structures. Despite attracting much attention from academia, there is much work needed to create processes that can achieve characterisation in timescales suitable for industry, and a key to this is the design of the testing procedure itself. This work proposes a passive testing method that seeks a desired degree of resonance between forcing and response. In this manner, the process automatically seeks data that reveals greater detail of the underlying nonlinear normal modes than a traditional stepped sine method. Furthermore, the method can target multiple harmonics of the fundamental forcing frequency, and is therefore suitable for structures with complex modal interactions. The method is presented with some experimental examples, using a structure with a 3:1 internal resonance. Full article

High-rate dynamic systems are defined as engineering systems experiencing dynamic events of typical amplitudes higher than 100 g${}_n$ for a duration of less than 100 ms. The implementation of feedback decision mechanisms in high-rate systems could improve their operations and safety, and even be critical to their deployment. However, these systems are characterized by large uncertainties, high non-stationarities, and unmodeled dynamics, and it follows that the design of real-time state-estimators for such purpose is difficult. In this paper, we compare the promise of five time-frequency representation (TFR) methods at conducting real-time state estimation for high-rate systems, with the objective of providing a path to designing implementable algorithms. In particular, we examine the performance of the short-time Fourier transform (STFT), wavelet transformation (WT), Wigner–Ville distribution (WVD), synchrosqueezed transform (SST), and multi-synchrosqueezed transform (MSST) methods. This study is conducted using experimental data from the DROPBEAR (Dynamic Reproduction of Projectiles in Ballistic Environments for Advanced Research) testbed, consisting of a rapidly moving cart on a cantilever beam that acts as a moving boundary condition. The capability of each method at extracting the beam’s fundamental frequency is evaluated in terms of precision, spectral energy concentration, computation speed, and convergence speed. It is found that both the STFT and WT methods are promising methods due to their fast computation speed, with the WT showing particular promise due to its faster convergence, but at the cost of lower precision on the estimation depending on circumstances. Full article

This work aims at investigating the free torsional vibration of one-directional nanostructures with an elliptical shape, under different boundary conditions. The equation of motion is derived from Hamilton’s principle, where Eringen’s nonlocal theory is applied to analyze the small-scale effects. The analytical Galerkin method is employed to rewrite the equation of motion as an ordinary differential equation (ODE). After a preliminary validation check of the proposed formulation, a systematic study investigates the influence of the nonlocal parameters, boundary conditions, geometrical and mechanical parameters on the natural frequency of nanorods; the objective is to provide useful findings for design and optimization purposes of many nanotechnology applications, such as, nanodevices, actuators, sensors, rods, nanocables, and nanostructured aerospace systems. Full article

Dynamic behaviors of composite railway sleepers and bearers in railway switches and crossings are not well-known and have never been thoroughly investigated. In fact, the dynamic properties of the full-scale composite sleepers and bearers are not available in practice. Importantly, the deteriorated condition or even the failure of composite materials and components in the railway system can affect the functional limitations or serviceability of the switches and crossings. Especially, it is important to identify the dynamic modal parameters of Fiber-reinforced Foamed Urethane (FFU) composite railway sleepers and bearers so that track engineers can adequately design and optimize the structural components with their superior properties, for benchmarking with the conventional sleepers and bearers. This paper is the world’s first to investigate the vibration characteristics of full-scaled FFU composite beams in healthy and damaged conditions, using the impact hammer excitation technique. This study also determines the dynamic elastic modulus of FFU composite beams from experimental dynamic measurements. It is found that the first bending mode in a vertical plane obviously is the first dominant mode of resonance under a free-free condition. The dynamic modal parameters reduce when damages occur. In this study, finite-element modeling has been used to establish a realistic dynamic model of the railway track incorporating FFU composite sleepers and bearers. Then, numerical simulations and experimental campaigns have been performed to enable new insights into the dynamic behaviors of composite sleepers and bearers. These insights are fundamental to the performance benchmarking as well as the development of vibration-based condition monitoring and inspection for predictive track maintenance. Full article

Suspension dampers are extremely critical for modern automobiles for absorbing vibrational energy while in operation. For years now, the viscous passive damper has been dominant. However, there is a constant need to improve and revolutionize the damping technology to adapt to modern road conditions and for better performance. Controlled shock absorbers capable of adapting to uneven road profiles are required to meet this challenge and enhance the passenger comfort level. Among the many types of modern damping solutions, magnetorheological (MR) dampers have gained prominence, considering their damping force control capability, fast adjustable response, and low energy consumption. Advancements in energy-harvesting technologies allow for the regeneration of a portion of energy dissipated in automotive dampers. While the amount of regenerated energy is often insufficient for regular automobiles, it could prove to be vital to support lightweight battery-operated vehicles. In battery-operated vehicles, this regenerated energy can be used for powering several secondary systems, including lighting, heating, air conditioning, and so on. This research focuses on developing a hybrid smart suspension system that combines the MR damping technology along with an electromagnetic induction (EMI)-based energy-harvesting system for applications in lightweight battery-operated vehicles. The research involves the extensive designing, numerical simulation, fabrication, and testing of the proposed smart suspension system. The development of the proposed damping system would help advance the harvesting of clean energy and enhance the performance and affordability of future battery-operated vehicles. Full article