Structural Dynamics Applications Research Articles

Feasible Probabilistic Learning Method for Model-Form Uncertainty Quantification in Vibration Analysis

Recently, a novel probabilistic method for modeling and quantifying model-form uncertainties in a deterministic high-dimensional computational model was proposed and demonstrated for numerous applications in linear vibrations and nonlinear structural dynamics. The method relies on a stochastic projection-based reduced-order model (SPROM) grounded in a randomized reduced-order basis to accelerate the stochastic computations it incurs, and it accounts for all modeling uncertainties introduced by model reduction. It builds the probability model of the SPROM using a small number of hyperparameters, and it determines them so that the mean value and statistical fluctuations of quantities of interest predicted using the SPROM match target values obtained from available data. Hence, it can be interpreted as a probabilistic model learning or updating method. Unfortunately, even though it is performed using a reduced-order model, the hyperparameter identification process can be cost prohibitive. For this reason, the method is reformulated in this paper around the concept of hyper-reduction. Specifically, a reduced version of the mesh underlying the high-dimensional model is constructed and used to hyper-reduce the SPROM. The feasibility of the resulting approach is demonstrated for the model-form uncertainty quantification of generalized eigencomputations performed during what-if scenarios associated with shape changes of a jet engine nozzle.

Learning of model discrepancy for structural dynamics applications using Bayesian history matching

Calibration of computer models for structural dynamics is often an important task in creating valid predictions that match observational data. However, calibration alone will lead to biased estimates of system parameters when a mechanism for model discrepancy is not included. The definition of model discrepancy is the mismatch between observational data and the model when the ‘true’ parameters are known. This will occur due to the absence and/or simplification of certain physics in the computer model. Bayesian History Matching (BHM) is a ‘likelihood-free’ method for obtaining calibrated outputs whilst accounting for model discrepancies, typically via an additional variance term. The approach assesses the input space, using an emulator of the complex computer model, and identifies parameter sets that could have plausibly generated the target outputs. In this paper a more informative methodology is outlined where the functional form of the model discrepancy is inferred, improving predictive performance. The algorithm is applied to a case study for a representative five storey building structure with the objective of calibrating outputs of a finite element (FE) model. The results are discussed with appropriate validation metrics that consider the complete distribution.

Numerical Experiments on Time Step Selection in Linear Dynamic Analysis of Structures

ABSTRACT The complex nature of the exciting force in earthquake engineering applications mandates employing numerical methods to obtain the response of structures to ground motions. Several numerical methods are in use in the field of earthquake engineering and structural dynamics to solve the ordinary differential equation of motion. Previous studies assumed that the appropriate time step for numerical schemes can be selected as a constant fraction of natural period of the system disregarding the period range. This study is concerned with assessing, through a comprehensive numerical experiment; the numerical error of the most commonly used schemes in structural dynamics applications and assessing the appropriateness of each method for different excitations and damping conditions using different natural period ranges. The current investigation involved testing the central difference method, Newmark, HHT, and HHT1 methods. The four methods were tested for linear single degree of freedom (SDOF) oscillator. Three different exciting forces were used to test the schemes; half-sine pulse wave, harmonic force, and actual ground motion record. Three natural periods were used to conduct the experiment, representing short, medium, and long period systems including the resonant condition. Three structural damping ratios, representing lightly, moderately, and heavily damped systems, were used to assess the structural damping effect on the accuracy associated with selected time steps. A new displacement error norm was introduced to define the accuracy of numerical schemes. The results of this investigation indicated that the commonly used assumption of time step as a constant fraction of natural period of the system, even for SDOF linear oscillator, disregarding the period range could result in significant numerical errors. The study also confirmed the significant effect of structural damping ratio on the accuracy of the numerical schemes. The paper also presents a recommendation matrix for the most appropriate time step for each scheme for different applications and structural conditions. The results of the current investigation are strictly limited to linear systems.

Extraordinary wave manipulation characteristics of nonlinear inertant acoustic metamaterials

Since being postulated more than a decade ago, inerters have been successfully employed to enhance the dynamic performance of mechanical systems in several applications. Their ability to lend a high dynamic mass presence to systems that employ them with only a relatively small static device mass makes them unique among mechanical elements. This study explores the mechanical wave manipulation characteristics of nonlinear inertant acoustic metamaterial (NLIAM) configurations using analysis and simulations for their one-dimensional discrete element lattice representations. Firstly, based on notional concepts for nonlinear inertant devices, potential frequency-dependent and acceleration-dependent nonlinear inertant models are identified. Using an effective mass model for the NLIAM with frequency-dependent inertance in the local resonator attachment, the dispersion characteristics of inverse square law and power law inertance models are examined and contrasted with those for an acoustic metamaterial with frequency invariant inertance. While a tuned inverse square law inertance model ensures the existence of a band gap over almost the entire frequency bandwidth of interest even encompassing the extremely low frequency regime, the low and high frequency limits for this inertance law would not be realizable in practice. A potentially more practical power law approximation is proposed and shown to deliver a widening of the band gap by more than 100% towards frequencies below the lower bound of the band gap for the acoustic metamaterial with frequency invariant inertance. Further, drawing inspiration from the Duffing-type stiffness, an acceleration-dependent cubically nonlinear inertance model is proposed. First order corrections to the dispersion characteristics are obtained for an NLIAM with acceleration-dependent inertance using a perturbation approach. For weakly nonlinear cases, excitation amplitude-activated shifts in the dispersion curves are found to enable this NLIAM to act as a passive adaptive filter for mechanical waves based solely on their excitation amplitude. Practical manifestations of such NLIAM could therefore provide a means to realize extraordinary wave manipulation capabilities especially suitable for low frequency structural dynamic applications.

Structural vibration analysis with random fields using the hierarchical finite element method

Element-based techniques, like the finite element method, are the standard approach in industry for low-frequency applications in structural dynamics. However, mesh requirements can significantly increase the computational cost for increasing frequencies. In addition, randomness in system properties starts to play a significant role and its inclusion in the model further increases the computational cost. In this paper, a hierarchical finite element formulation is presented which incorporates spatially random properties. Polynomial and trigonometric hierarchical functions are used in the element formulation. Material and geometrical spatially correlated randomness are represented by the Karhunen–Loeve expansion, a series representation for random fields. It allows the element integration to be performed only once for each term of the series which has benefits for a sampling scheme and can be used for non-Gaussian distributions. Free vibration and forced response statistics are calculated using the proposed approach. Compared to the standard h-version, the hierarchical finite element approach produces smaller mass and stiffness matrices, without changing the number of nodes of the element, and tends to be computationally more efficient. These are key factors not only when considering solutions for higher frequencies but also in the calculation of response statistics using a sampling method such as Monte Carlo simulation.

Probabilistic learning for modeling and quantifying model‐form uncertainties in nonlinear computational mechanics

SummaryRecently, a novel nonparametric probabilistic method for modeling and quantifying model‐form uncertainties in nonlinear computational mechanics was proposed. Its potential was demonstrated through several uncertainty quantification (UQ) applications in vibration analysis and nonlinear computational structural dynamics. This method, which relies on projection‐based model order reduction to achieve computational feasibility, exhibits a vector‐valued hyperparameter in the probability model of the random reduced‐order basis and associated stochastic projection‐based reduced‐order model. It identifies this hyperparameter by formulating a statistical inverse problem, grounded in target quantities of interest, and solving the corresponding nonconvex optimization problem. For many practical applications, however, this identification approach is computationally intensive. For this reason, this paper presents a faster predictor‐corrector approach for determining the appropriate value of the vector‐valued hyperparameter that is based on a probabilistic learning on manifolds. It also demonstrates the computational advantages of this alternative identification approach through the UQ of two three‐dimensional nonlinear structural dynamics problems associated with two different configurations of a microelectromechanical systems device.

On the performance of direct piezoelectric rotational accelerometers in experimental structural dynamics

Rotational responses are comparing to the translational responses less frequently used in the experimental structural dynamics. Even though they represent half of all the existing degrees of freedom they are often omitted, since they are difficult to measure. Nevertheless, they have an important contribution in structural dynamic modifications, model validation, substructuring, etc. Therefore in this paper a performance evaluation of two direct piezoelectric rotational accelerometers is made in order to show the possibilities of their usage in the structural dynamic applications. Additionally, the results are compared to a commonly used method of approximated rotational responses obtained from two offset translational accelerometers. Sensors were tested with impact and sine-sweep excitations. Data is analyzed in the form of frequency response functions and compared to a numerical reference with a coherence criterion. The quality of directly obtained rotations are expected to have great potential in structural dynamics.

3D dynamic displacement-field measurement for structural health monitoring using inexpensive RGB-D based sensor

The advent of inexpensive digital cameras with depth sensing capabilities (RGB-D cameras) has opened the door to numerous useful applications that need quantitative measures of dynamic fields whose simultaneous time history quantification (at many points as dictated by the resolution of the camera) provides capabilities that were previously accessible only through expensive sensors (e.g., laser scanners). This paper presents a comprehensive experimental and computational study to evaluate the performance envelope of a representative RGB-D sensor (the first generation of Kinect sensor) with the aim of assessing its suitability for the class of problems encountered in the structural dynamics field, where reasonably accurate information of evolving displacement fields (as opposed to few discrete locations) that have simultaneous dynamic planar translational motion with significant rotational (torsional) components. This study investigated the influence of key system parameters of concern in selecting an appropriate sensor for such structural dynamic applications, such as amplitude range, spectral content of the dynamic displacements, location and orientation of sensors relative to target structure, fusing of measurements from multiple sensors, sensor noise effects, rolling-shutter effects, etc. The calibration results show that if the observed displacement field generates discrete (pixel) sensor measurements with sufficient resolution (observed displacements more than 10 mm) beyond the sensor noise floor, then the subject sensors can typically provide reasonable accuracy for transnational motion (about 5%) when the frequency range of the evolving field is within about 10 Hz. However, the expected error for torsional measurements is around 6% for static motion and 10% for dynamic rotation for measurements greater than 5°.

Excitation power quantities in phase resonance testing of nonlinear systems with phase-locked-loop excitation

Phase resonance testing is one method for the experimental extraction of nonlinear normal modes. This paper proposes a novel method for nonlinear phase resonance testing. Firstly, the issue of appropriate excitation is approached on the basis of excitation power considerations. Therefore, power quantities known from nonlinear systems theory in electrical engineering are transferred to nonlinear structural dynamics applications. A new power-based nonlinear mode indicator function is derived, which is generally applicable, reliable and easy to implement in experiments. Secondly, the tuning of the excitation phase is automated by the use of a Phase-Locked-Loop controller. This method provides a very user-friendly and fast way for obtaining the backbone curve. Furthermore, the method allows to exploit specific advantages of phase control such as the robustness for lightly damped systems and the stabilization of unstable branches of the frequency response. The reduced tuning time for the excitation makes the commonly used free-decay measurements for the extraction of backbone curves unnecessary. Instead, steady-state measurements for every point of the curve are obtained. In conjunction with the new mode indicator function, the correlation of every measured point with the associated nonlinear normal mode of the underlying conservative system can be evaluated. Moreover, it is shown that the analysis of the excitation power helps to locate sources of inaccuracies in the force appropriation process. The method is illustrated by a numerical example and its functionality in experiments is demonstrated on a benchmark beam structure.

Dynamic Modal Correlation of an Automotive Rear Subframe, with Particular Reference to the Modelling of Welded Joints

This paper presents a comparison between the experimental investigation and the Finite Element (FE) modal analysis of an automotive rear subframe. A modal correlation between the experimental data and the forecasts is performed. The present numerical model constitutes a predictive methodology able to forecast the experimental dynamic behaviour of the structure. The actual structure is excited with impact hammers and the modal response of the subframe is collected and evaluated by the PolyMAX algorithm. Both the FE model and the structural performance of the subframe are defined according to the Ferrari S.p.A. internal regulations. In addition, a novel modelling technique for welded joints is proposed that represents an extension of ACM2 approach, formulated for spot weld joints in dynamic analysis. Therefore, the Modal Assurance Criterion (MAC) is considered the optimal comparison index for the numerical-experimental correlation. In conclusion, a good numerical-experimental agreement from 50 Hz up to 500 Hz has been achieved by monitoring various dynamic parameters such as the natural frequencies, the mode shapes, and frequency response functions (FRFs) of the structure that represent a validation of this FE model for structural dynamic applications.

Bayesian optimal experimental design for parameter estimation and response predictions in complex dynamical systems

A Bayesian optimal experimental design (OED) framework is revisited and applied to a number of structural dynamics problems. The objective is to optimize the design of the experiment such that the most informative data are obtained for either for parameter estimation or response predictions. The Bayesian OED is based on maximizing the expected utility function taken as the Kullback-Leibler divergence between the prior and posterior distribution of the model parameters. Asymptotic approximations for the multi-dimensional integrals arising in the formulation of the expected utility function are proposed, valid for large number of data and small prediction errors. The OED based on these approximations are shown to be equivalent to the OED based on the robust information entropy introduced in the past for structural dynamics applications. Analytical expressions are developed to point out the effect of the variances of Bayesian Gaussian priors on the optimal design. The design variables may include the location of sensors, location of actuators or characteristics of the excitation such as amplitude variation and frequency content characteristics. A stochastic optimization algorithm is conveniently used to solve the optimization problem in the continuous physical domain of variation of the design variables. The proposed framework is applicable to complex linear and nonlinear dynamical systems. The asymptotic results are compared to the results obtained from accurate but computationally expensive sampling algorithms and are shown to be adequate for experimental design purposes. Two optimal experimental design problems illustrate the proposed methodology: 1) optimal sensor placement for load identification in nonlinear beam models, 2) optimal sensor placement for modal identification of bridges using complex FE models.

Improved formulation for a structure-dependent integration method

Structure-dependent integration methods seem promising for structural dynamics applications since they can integrate unconditional stability and explicit formulation together, which can enable the integration methods to save many computational efforts when compared to an implicit method. A newly developed structure-dependent integration method can inherit such numerical properties. However, an unusual overshooting behavior might be experienced as it is used to compute a forced vibration response. The root cause of this inaccuracy is thoroughly explored herein. In addition, a scheme is proposed to modify this family method to overcome this unusual overshooting behavior. In fact, two improved formulations are proposed by adjusting the difference equations. As a result, it is verified that the two improved formulations of the integration methods can effectively overcome the difficulty arising from the inaccurate integration of the steady-state response of a high frequency mode.

Virtual sensing of structural vibrations using dynamic substructuring

Virtual sensing techniques use information available from a limited set of physical sensors together with the finite element model to calculate an estimate of the quantity of interest. In structural dynamics applications, analytical mode shapes from the finite element model are typically used as a basis to estimate the response at unmeasured locations by an expansion algorithm. An alternative is to model only the interesting part of the structure using substructuring techniques, in which the natural modes are replaced by component modes consisting of a selected number of fixed interface modes plus the interface constraint modes. They are mutually independent and compose a valid subspace for estimating the unmeasured response. If the number of interface degrees of freedom is large, interface reduction is applied. The main advantage of the proposed approach is that the modelling effort can be substantially decreased, because only part of the structure is modelled and the modelling uncertainties, non-linearities, or changes in the omitted structure can be ignored. The method is validated by numerical simulations of three different structures under unknown excitation. Different types and locations of virtual sensors are studied. Also, the effects of noise and model errors are investigated. The most accurate estimation is obtained if the virtual sensor is located away from the interface and close to a physical sensor.

Computational studies of plasma lipoprotein lipids

Plasma lipoproteins are macromolecular assemblies of proteins and lipids found in the blood. The lipid components of lipoproteins are amphipathic lipids such as phospholipids (PLs), and unesterified cholesterols (UCs) and hydrophobic lipids such as cholesteryl esters (CEs) and triglycerides (TGs). Since lipoproteins are soft matter supramolecular assemblies easily deformable by thermal fluctuations and they also exist in varying densities and protein/lipid components, a detailed understanding of their structure/function is experimentally difficult. Molecular dynamics (MD) simulation has emerged as a particularly promising way to explore the structure and dynamics of lipoproteins. The purpose of this review is to survey the current status of computational studies of the lipid components of the lipoproteins. Computational studies aim to explore three levels of complexity for the 3-dimensional structural dynamics of lipoproteins at various metabolic stages: (i) lipoprotein particles consist of protein with minimal lipid; (ii) lipoprotein particles consist of PL-rich discoidal bilayer-like lipid particles; (iii) mature circulating lipoprotein particles consist of CE-rich or TG-rich spheroidal lipid-droplet-like particles. Due to energy barriers involved in conversion between these species, other biomolecules also participate in lipoprotein biological assembly. For example: (i) lipid-poor apolipoprotein A-I (apoA-I) interacts with ATP-binding cassette transporter A1 (ABCA1) to produce nascent discoidal high density lipoprotein (dHDL) particles; (ii) lecithin–cholesterol acyltransferase (LCAT) mediates the conversion of UC to CE in dHDL, driving spheroidal HDL (sHDL) formation; (iii) transfer proteins, cholesterol ester transfer protein (CETP) and phospholipid transfer protein (PLTP), transfer both CE and TG and PL, respectively, between lipoprotein particles. Computational studies have the potential to explore different lipoprotein particles at each metabolic stage in atomistic detail. This review discusses the current status of computational methods including all-atom MD (AAMD), coarse-grain MD (CGMD), and MD-simulated annealing (MDSA) and their applications in lipoprotein structural dynamics and biological assemblies. Results from MD simulations are discussed and compared across studies in order to identify key findings, controversies, issues and future directions. This article is part of a Special Issue entitled: Biosimulations edited by Ilpo Vattulainen and Tomasz Róg.

Photogrammetry and optical methods in structural dynamics – A review

In the last few decades, there has been a surge of research in the area of non-contact measurement techniques. Photogrammetry has received considerable attention due to its ability to achieve full-field measurement and its robustness to work in testing environments and on testing articles in which using other measurement techniques may not be practical. More recently, researchers have used this technique to study transient phenomena and to perform measurements on vibrating structures. The current paper reviews the most current trends in the photogrammetry technique (point tracking, digital image correlation, and target-less approaches) and compares the applications of photogrammetry to other measurement techniques used in structural dynamics (e.g. laser Doppler vibrometry and interferometry techniques). The paper does not present the theoretical background of the optical techniques, but instead presents the general principles of each approach and highlights the novel structural dynamic measurement concepts and applications that are enhanced by utilizing optical techniques.

Unprecedented Structural Skins. Experiments towards an Intelligent Tensegrity Skin

Pin-jointed prestressable structures -known as tensegrity systems- have been largely studied for over six decades, untouched in their complexity and fascination. Prior research has marginally offered interactive toolsets to design tensegrity: none without tedious procedures embedded in their resolution. Restrictions in terms of simultaneous design's topology and optimization - aggravated by lacking effective means to automate manufacturing and specially assembly of such system - have greatly disrupted and cornered their current applicability. We present results of unprecedented processes extending tensegrity systems applicability. Employing a novel approach to design's topology and optimization of arbitrary tensegrity systems, we developed means to automatically design and manufacture them. With advantages in tailoring design and production costs, we explore some speculative scenario.Therefore we present experiments towards novel processes and products aimed to revolutionized the market of lightweight structural systems, both in static and dynamic applications. Moving from tensegrity systems and their intrinsic transportability, material-economy, easy manufacture, customization and assembly -the latest expanded within our research- we propose a new standard for structural performing skin. We aim for an intelligent skin: free-form, lightweight, widely adaptable in use, and capable of embedding complex design features and functionalities. Central in addressing economical solutions to nowadays challenging tasks we present our findings, analyzing production processes and potential challenges further suggesting relevant scenarios of pertinence.Highlights: ▶Tensegrity automated production is possible linking design topology, optimization, manufacturing and assembling. ▶Two automated manufacturing approaches are outlined according to scale, functionality and use.▶Injection moulding is nowadays proofed for structural performances, cost competitiveness, easy multifunctional layer implementation.▶3D printing will favor in future robust automated customized production.

Assessment of explicit and semi‐explicit classes of model‐based algorithms for direct integration in structural dynamics

SummaryThe ‘model‐based’ algorithms available in the literature are primarily developed for the direct integration of the equations of motion for hybrid simulation in earthquake engineering, an experimental method where the system response is simulated by dividing it into a physical and an analytical domain. The term ‘model‐based’ indicates that the algorithmic parameters are functions of the complete model of the system to enable unconditional stability to be achieved within the framework of an explicit formulation. These two features make the model‐based algorithms also potential candidates for computations in structural dynamics. Based on the algorithmic difference equations, these algorithms can be classified as either explicit or semi‐explicit, where the former refers to the algorithms with explicit difference equations for both displacement and velocity, while the latter for displacement only. The algorithms pertaining to each class are reviewed, and a new family of second‐order unconditionally stable parametrically dissipative semi‐explicit algorithms is presented. Numerical characteristics of these two classes of algorithms are assessed under linear and nonlinear structural behavior. Representative numerical examples are presented to complement the analytical findings. The analysis and numerical examples demonstrate the advantages and limitations of these two classes of model‐based algorithms for applications in structural dynamics. Copyright © 2015 John Wiley & Sons, Ltd.

Inconsistent Stability of Newmark's Method in Structural Dynamics Applications

The stability of numerical time integrators, and of the physical systems to which they are applied, are normally studied independently. This conceals a very interesting phenomenon, here termed inconsistent stability, wherein a numerical time marching scheme predicts a stable response about an equilibrium configuration that is, in fact, unstable. In this paper, time integrator parameters leading to possible inconsistent stability are first found analytically for conservative systems (symmetric tangent stiffness matrices), then several structural arches with increasing complexity are used as numerical case studies. The intention of this work is to highlight the potential for this unexpected, and mostly unknown, behavior to researchers studying complex dynamical systems, especially through time marching of finite element models. To allow for direct interpretation of our results, the work is focused on the Newmark time integrator, which is commonly used in structural dynamics.

Correlation Measures and Their Applications in Structural Dynamics and Data Analyses

Correlation Measures and Their Applications in Structural Dynamics and Data Analyses

Stable force identification in structural dynamics using Kalman filtering and dummy-measurements

Many engineering applications require the knowledge of input forces to mechanical systems. However, in practice, it is quite difficult to measure these forces directly. In order to obtain an estimate of the input forces to structural systems, Kalman filtering based techniques have recently been introduced. These state-estimation techniques allow estimating the forces concurrent with the states of a system, based on a limited number of measurements. In practice, acceleration measurements are most convenient to use in structural dynamics applications. This paper proposes an analytical analysis of the stability of the Kalman based force estimation techniques and shows that only using acceleration measurements inherently leads to unreliable results. In order to circumvent this issue, the addition of dummy-measurements on a position level is proposed. These fictitious measurements dictate the estimator to return to an undeformed state and lead to a stable estimation approach. The proposed method is validated through both a numerical and a practical experiment. Both experiments show the inadequacy of the augmented Kalman filter based on only acceleration measurements to provide stable results. The estimator with dummy measurements on the other hand provides good results in the case of an unbiased external load.