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A dynamical uncertain system is studied in this paper. Two kinds of uncertainties are addressed, where the uncertain parameters are described through random variables and/or fuzzy variables. A general framework is proposed to deal with both kinds of uncertainty using a polynomial chaos expansion (PCE). It is shown that fuzzy variables may be expanded in terms of polynomial chaos when Legendre polynomials are used. The components of the PCE are a solution of an equation that does not depend on the nature of uncertainty. Once this equation is solved, the post-processing of the data gives the moments of the random response when the uncertainties are random or gives the response interval when the variables are fuzzy. With the PCE approach, it is also possible to deal with mixed uncertainty, when some parameters are random and others are fuzzy. The results provide a fuzzy description of the response statistical moments. [ABSTRACT FROM AUTHOR]

Abstract: The use of modal response residuals and parameterized models forms the framework for parameter subset selection based damage detection. This research explores the novel approach in this class of methods which is characterized by the successive application of the homogeneous modal response residuals. The motivation behind this approach is to restrict the use of unknown weighting factors which are employed in cases with mixed response residuals. Particular attention is given to the parameter-effect symmetry issues and large nonlinear changes in response residuals due to increasing damage observed across multiple damage levels. A case study involving a real aluminum three-dimensional frame structure with a loose joint connection is used to demonstrate the approach and its ability to localize the damaged area. [Copyright &y& Elsevier]

Abstract: This paper is concerned with distributed sensors to measure the response of beam and plate structures. The design of modal sensors for beam structures is well established. The design for plate structures with constant thickness sensors may be achieved by optimizing the shape of the sensor boundary, or by optimizing the effectiveness of the electrode. Most applications consider vibration control, but this paper is concerned with structural health monitoring, where the sensor output is made sensitive to changes in key stiffness parameters, for example in joints. The procedure is based on finite element models of the structures, and thus distributed transducers may be designed for arbitrary beam and plate structures. Simulated examples of a beam and a plate excited by a rotating machine are used to demonstrate the approach. [Copyright &y& Elsevier]

Abstract: Discrepancies between experimentally measured data and computational predictions are unavoidable for complex engineering dynamical systems. To reduce this gap, model updating methods have been developed over the past three decades. Current methods for model updating often use discrete parameters, such as thickness or joint stiffness, for model updating. However, there are many parameters in a numerical model which are spatially distributed in nature. Such parameters include, but are not limited to, thickness, Poisson''s ratio, Young''s modulus, density and damping. In this paper a novel approach is proposed which takes account of the distributed nature of the parameters to be updated, by expressing the parameters as spatially correlated random fields. Based on this assumption, the random fields corresponding to the parameters to be updated have been expanded in a spectral decomposition known as the Karhunen–Loève (KL) expansion. Using the KL expansion, the mass and stiffness matrices are expanded in series in terms of discrete parameters. These parameters in turn are obtained using a sensitivity based optimization approach. A numerical example involving a beam with distributed updating parameters is used to illustrate this new idea. [Copyright &y& Elsevier]

Abstract: Vibrations in rotating machinery cause many problems such as fatigue of the rotating components, excessive noise, or transmission of vibration to the supporting structure. A major source of this vibration is out-of-balance forces and this paper proposes that the rotor response is reduced by suspending the machine on nonlinear springs. In the field of vibration isolation, nonlinear mounts have been proposed which have the same static stiffness as an equivalent linear support, i.e. load bearing capability, but at the same time offer a low dynamic stiffness, i.e. a lower natural frequency. Thus the isolator is effective over an increased frequency range. These mounts are known in the literature as high-static-low-dynamic-stiffness (HSLDS) mechanisms. In this paper, the rotor is suspended on a hardening HSLDS spring to considerably reduce the critical speeds to values far away from the operating speed. The advantages of the nonlinear supports are demonstrated using a simple two degree of freedom rotating machine model consisting of a rigid disk, and shafts, bearings and supports that are flexible but have negligible mass. Following a linear analysis to highlight the benefits of a low dynamic stiffness, an approximate analytical solution of the nonlinear equation of motion is presented. A comparison between the linear and nonlinear response shows the effectiveness of the nonlinear supports. Finally, the problems that occur if the nonlinearity is too strong are highlighted. [Copyright &y& Elsevier]

Abstract: Discrepancies between experimentally measured data and computational predictions are unavoidable for structural dynamic systems. Model updating methods have been developed over the past three decades to reduce this gap. Well established model updating methods exist when both the model and experimental measurements are deterministic in nature. However in reality, experimental results may contain uncertainty, for example arising due to unknown experimental errors, or variability in nominally identical structures. Over the past two decades probabilistic approaches have been developed to incorporate uncertainties in computational models. In this paper, the natural frequencies and the eigenvectors of the system are measured and assumed to be uncertain. A random matrix approach is proposed and closed-form expressions are derived for the mean matrix and the covariance matrix of the updated stiffness matrix. A perturbation technique is used to obtain a usable expression for the covariance matrix. The new method is illustrated by three numerical examples highlighting the influence of the eigenfrequency uncertainties on the mean matrix and the influence of the eigenvector uncertainties on the covariance matrix. [Copyright &y& Elsevier]

Over the decades, vibration-based condition monitoring has become well accepted and widely used to identify faults in rotating machines. However, the quantification of faults may require a significant number of tests to be carried out which may be time consuming and costly, if not impossible, by experiments alone. In the recent past, it has been observed that model-based identification has played a significant role in the rapid resolution and quantification of faults. The paper seeks to give an overview of the recent developments in this field which has considerable practical importance. [Copyright &y& Elsevier]

Abstract: The usual model updating method may be considered to be deterministic since it uses measurements from a single test system to correct a nominal finite element model. There may however be variability in seemingly identical test structures and uncertainties in the finite element model. Variability in test structures may arise from many sources including geometric tolerances and the manufacturing process, and modelling uncertainties may result from the use of nominal material properties, ill-defined joint stiffnesses and rigid boundary conditions. In this paper, the theory of stochastic model updating using a Monte-Carlo inverse procedure with multiple sets of experimental results is explained and then applied to the case of a simulated three degree-of-freedom system, which is used to fix ideas and also to illustrate some of the practical limitations of the method. In the companion paper, stochastic model updating is applied to a benchmark structure using a contact finite element model that includes common uncertainties in the modelling of the spot welds. [Copyright &y& Elsevier]

A series of recent papers has discussed the problem of parameter identification for rotating machinery. Two techniques have been developed to identify the foundation parameters, the unbalance distribution and the state of bend. In this paper, the method based on the non-linear optimisation of the fit between the measured and model behaviour is extended to evaluate the steady load in oil film bearings. This is an important practical problem since these steady loads have a strong influence on the bearing''s dynamic characteristics. After developing the theory, two distinct examples are simulated and corrupted with noise. In simulation it is shown that the technique is capable of yielding accurate results. It is demonstrated that a measurement of absolute shaft motion is required to give predictions that are sufficiently accurate to be useful. This is not an undue restriction since most modern machines are equipped with proximity probes. The experimental verification of the method is planned on a new rig, currently under construction. [Copyright &y& Elsevier]

• General nonlocal theory with two parameters is used for the porous FG nanobeam. • The nonlocal parameters cause softening and hardening behavior. • Reddy's beam theory is used to model higher-order transverse shear strains. • Material variations in the length and thickness directions are investigated. • The porosity volume fraction and the length to thickness ratio are varied. A comprehensive vibrational analysis of bi-directional functionally graded (2D-FG) rotating nanobeams with porosities is studied for the first time. The beam is modeled based on general nonlocal theory (GNT) where the beam governing equations are derived depending on two different nonlocal parameters. Unlike Eringen's conventional form of nonlocal theory, the general nonlocal theory can reveal both hardening and softening behaviors of the material. Here, the attenuation functions are altered in both transverse and longitudinal directions of 2D-FG nanobeam. This feature, which has a significant effect on the vibrational characteristics, has not been considered in previous studies. Moreover, to estimate the effects of the higher-order transverse shear strains on the vibration of the nanobeam, Reddy's beam theory (RBT), which includes higher-order shear deformation, is employed. The material properties of the 2D-FG rotating nanobeam vary both in the length and thickness directions according to a power law. The generalized differential quadrature method (GDQM) is used to predict the vibration response. Also, the effects of material variation along the length and thickness directions, the rotating velocity of the nanobeam, the porosity volume fraction and the length to thickness ratio of the rotating nanobeam are illustrated and discussed in detail. The investigations performed in this study expose new phenomena for the vibration of nanobeams. [ABSTRACT FROM AUTHOR]

• The effects of structural joint variability on modal parameters are experimentally studied. • A stochastic generic joint element is introduced to represent variability in joint. • Bayesian model updating is used to identify joint model parameters from measured data. • Bayesian model updating is used to identify joint parameters from measured data. Dynamic properties of the contact interfaces in joints and mechanical connections have a great influence on the overall dynamic properties of assembled structures. Uncertainty and nonlinearity are two major effects of contact interfaces which introduce challenges in accurate modeling. Randomness in surface roughness quality, surface finish and contact preload are the main sources of variability in the contact interfaces. On the other side, slip and slap are two mechanisms responsible for nonlinear behavior of joints. Stochastic linear/nonlinear models need to be developed for such uncertain structures to be used in dynamic response analysis or system parameter identification. In this paper, variability in linear behavior of an assembled structure containing a bolted lap-joint is investigated by using experimental results. A stochastic model is then constructed for the structure by employing a stochastic generic joint model and the uncertainty in the joint model parameters is identified by using a Bayesian identification approach. [ABSTRACT FROM AUTHOR]

• An arclength-based response matching method is proposed to match multivalued responses. • A model updating technique for structures with strong nonlinearities is presented. • The proposed method can handle the linear and strongly nonlinear responses at the same time. • Multi-valued frequency response data with both stable and unstable responses are used for model updating. Model updating of strongly nonlinear systems is a vital tool to establish the precise nonlinear model from the multivalued responses and one major step in the updating process is the matching of the predicted and measured multivalued responses. However, when the multivalued behavior of the predicted and measured responses varies widely, the traditional multivalued response matching methods perform poorly. In this paper, a novel arclength-based multivalued response matching technique is proposed, which can help to update the model from linear to strongly nonlinear and handle both stable and unstable responses. The arclength is obtained from the continuation and the multivalued response is parameterized and transferred from the frequency domain to the arclength domain. The Arclength-Response curve is single-valued and can be easily separated into several parts based on feature points. Transferring the separation results back to the frequency domain, the full multivalued response is divided to single-valued Frequency-Response curves, where single-valued predicted and tested responses may be directly matched. A 2 DOF strong nonlinear model is first updated for verification. Using the proposed method, the linear response is directly updated based on the multivalued response with both softening and hardening regions. The error in the final updated parameters is below 1% and the proposed updating method performs better than two traditional methods. A real 3 DOF nonlinear model with a non-smooth strong nonlinearity with the hardening region is then used for validation. The stable and unstable responses of the system are measured from a shaker voltage-based constant frequency test and model updating is then conducted from linear to strongly nonlinear. During the updating process, the complex nonlinear behavior has six bifurcation points on one predicted response curve, but the updated responses fit the real ones well after 18 iterations. These results show the validity and superiority of the proposed method. [ABSTRACT FROM AUTHOR]

• Updating strongly nonlinear models using fixed frequency tests is proposed. • Fixed frequency tests with continuation also measure unstable multivalued responses. • Multivalued responses of nonlinear models with fixed frequency excitation predicted. • Proposed updating method achieves higher precision than only using stable responses. Model updating using multivalued responses is an important approach to construct strongly nonlinear models. During updating, swept frequency tests are usually used to measure the multivalued responses, but the measured responses only contain the stable branches. Accordingly, updating mainly uses these stable responses for the nonlinear model construction. Unstable responses are rarely tested or participate in the updating process, which may lead to errors in the constructed model with strong nonlinearities. Recently, fixed frequency tests using a continuation scheme have been used to measure fully multivalued responses, including both the stable and unstable branches, for structures with the strong nonlinearity. However, the measured responses were then transformed to equivalent swept frequency tests for the existing updating schemes, where this transformation may induce errors in the updating process. In this paper, a novel method for strong nonlinear model updating using the results of the fixed frequency test is proposed. Such updating is based on the directly measured data of the fixed frequency tests and can handle both the stable and unstable responses. Fixed frequency continuation using the Multi-Harmonic Balance Method (MHBM) is developed to predict the multivalued responses and their sensitivities. The multivalued responses of the prediction and measurement are directly matched in sensitivity-based updating. The proposed method is numerically verified on a 2 DOF nonlinear model. Using the proposed method, the nonlinear model is updated from linear to strongly nonlinear and the updated parameters all converge to the simulated values after 7 iterations. The proposed method is then experimentally validated on a 3 DOF nonlinear system with a non-smooth nonlinearity. A nonlinear model with a 7th order polynomial stiffness represents the equivalent system and is directly updated based on the fixed frequency test results. The updated responses are generally a good fit to the tested responses. Compared with updating using only the stable responses, the proposed updating method achieves higher precision and its updated responses are closer to the tested responses. These results show the validity and superiority of the proposed method. [ABSTRACT FROM AUTHOR]

This work presents a conceptually simple experiment consisting of a cantilever beam with a nonlinear spring at the tip. The configuration allows manipulation of the relative spacing between the modal frequencies of the underlying linear structure, and this permits the deliberate introduction of internal resonance. A 3:1 resonance is studied in detail; the response around the first mode shows a classic stiffening response, with the addition of more complex dynamic behaviour and an isola region. Quasiperiodic responses are also observed but in this work the focus remains on periodic responses. Predictions using Normal Form analysis and continuation methods show good agreement with experimental observations. The experiment provides valuable insight into frequency responses of nonlinear modal structures, and the implications of nonlinearity for vibration tests. [ABSTRACT FROM AUTHOR]

In this paper the analytical analysis of an adaptively tuned piezoelectric vibration based energy harvester is presented. A bimorph piezoelectric energy harvester is suspended between two electrodes, subjected to a same DC voltage. The resonance frequency of the system is controllable by the applied DC voltage, and the harvested power is maximized by controlling the natural frequency of the system to cope with vibration sources which have varying excitation frequencies. The nonlinear governing differential equation of motion is derived based on Euler Bernoulli theory, and due to the softening nonlinearity of the electrostatic force, the harvester is capable of working over a broad frequency range. The steady state harmonic solution is obtained using the harmonic balance method and results are verified numerically. The results show that the harvester can be tuned to give a resonance response over a wide range of frequencies, and shows the great potential of this hybrid system. [ABSTRACT FROM AUTHOR]

Recent research in the field of finite element model updating (FEM) advocates the adoption of Bayesian analysis techniques to dealing with the uncertainties associated with these models. However, Bayesian formulations require the evaluation of the Posterior Distribution Function which may not be available in analytical form. This is the case in FEM updating. In such cases sampling methods can provide good approximations of the Posterior distribution when implemented in the Bayesian context. Markov Chain Monte Carlo (MCMC) algorithms are the most popular sampling tools used to sample probability distributions. However, the efficiency of these algorithms is affected by the complexity of the systems (the size of the parameter space). The Hybrid Monte Carlo (HMC) offers a very important MCMC approach to dealing with higher-dimensional complex problems. The HMC uses the molecular dynamics (MD) steps as the global Monte Carlo (MC) moves to reach areas of high probability where the gradient of the log-density of the Posterior acts as a guide during the search process. However, the acceptance rate of HMC is sensitive to the system size as well as the time step used to evaluate the MD trajectory. To overcome this limitation we propose the use of the Shadow Hybrid Monte Carlo (SHMC) algorithm. The SHMC algorithm is a modified version of the Hybrid Monte Carlo (HMC) and designed to improve sampling for large-system sizes and time steps. This is done by sampling from a modified Hamiltonian function instead of the normal Hamiltonian function. In this paper, the efficiency and accuracy of the SHMC method is tested on the updating of two real structures; an unsymmetrical H-shaped beam structure and a GARTEUR SM-AG19 structure and is compared to the application of the HMC algorithm on the same structures. [ABSTRACT FROM AUTHOR]

Abstract: The application of a stochastic model updating technique using Monte-Carlo inverse propagation and multivariate multiple regression to converge a set of analytical models with randomised updating parameters upon a set of nominally identical physical structures is considered. The structure in question is a short beam manufactured from two components, one of folded steel and the other flat. The two are connected by two rows of spot-welds. The main uncertainty in the model is concerned with the spot-weld but there is also considerable manufacturing variability, principally in the radii of the folds. [Copyright &y& Elsevier]

• Micro-slip behaviour of bolted joint is demonstrated using a detailed FE model. • An Equivalent model capable of predicting Micro-slip behaviour is developed. • An identification scheme based on the behaviour of the contact interface is proposed. • The parameters of equivalent model are identified using detailed model predictions. The dynamic response of individual components in an assembled structure shows high accuracy compared to experimental measurements of the system response. However, when it comes to assemblies, the conventional linear approaches fail to deliver good accuracy, due to the uncertain linear and nonlinear mechanisms in the contact interface of the joints. Therefore, the inherent dynamics of the contact interfaces needs to be considered in modeling assembled structures. In this paper the prediction of the nonlinear dynamic response in a bolted flange joint was obtained in two ways. First, a 3D detailed finite element model capable of representing the micro-slip mechanism was made using a quasi-static time stepping analysis. The linear characteristics and nonlinear mechanisms developing in the contact interface of a bolted joint are investigated by using the 3D detailed model. Moreover, the natural frequencies of the assembled structure (representing the linear response) and the micro-slip behavior in terms of hysteresis loops (representing the nonlinear response) are obtained using the detailed model. Second, an equivalent model composed of beam elements and an appropriate joint model is then constructed for the assembled structure. An identification approach is proposed, and the parameters of the joint model are identified using both linear and nonlinear characteristics, i.e. natural frequencies and hysteresis loops. Comparing the hysteresis loops obtained from the detailed and equivalent models verifies the accuracy of the joint model used to represent the contact interface and the identification approach proposed for parameter quantification. [ABSTRACT FROM AUTHOR]