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Highlights • The proposed quasi-static method computes effective natural frequency and damping. • The method enables updating of Iwan element parameters in a bolted joint model. • Modal coupling is addressed by updating to measurements where one mode is dominant. • The updated model is verified to accurately predict modal coupling in measurements. Abstract Structures with bolted joints are known to exhibit amplitude-dependent shifts in the natural frequency and damping of its modes that are challenging to model accurately. These two properties could be derived from dynamic simulations (for example, by computing the response of the structure to an impulsive force), but such simulations are expensive. Any updating strategy based on dynamic simulation therefore becomes cumbersome. A solution was provided by Festjens et al. (2013) whereby the joint is treated as a static subcomponent in an otherwise dynamic global model. This work proposes a few modifications to their theory, resulting in a highly-efficient quasi-static algorithm that can be used to compute the amplitude-dependent frequency and damping by loading the finite element model in the shape of one of its modes. This new quasi-static method was then utilized to update a set of Iwan joint parameters so that the shift in the frequencies and damping seen in the finite element model matched those from measurements on an experimental beam containing three bolted joints. The updated model was then verified to be capable of capturing with good accuracy the effects of modal coupling seen in the impact response of the beam. [ABSTRACT FROM AUTHOR]

Finite element models can be used to model and predict the hysteresis and energy dissipation exhibited by nonlinear joints in structures. As a result of the nonlinearity, the frequency and damping of a mode is dependent on excitation amplitude, and when the modes remain uncoupled, quasi-static modal analysis has been shown to efficiently predict this behavior. However, in some cases the modes have been observed to couple such that the frequency and damping of one mode is dependent on the amplitude of other modes. To model the interactions between modes, one must integrate the dynamic equations in time, which is several orders of magnitude more expensive than quasi-static analysis. This work explores an alternative where quasi-static forces are applied in the shapes of two or more modes of vibration simultaneously, and the resulting load–displacement curves are used to deduce the effect of other modes on the effective frequency and damping of the mode in question. This methodology is demonstrated on a simple 2D cantilever beam structure with a single bolted joint which exhibits micro-slip nonlinearity over a range of vibration amplitudes. The predicted frequency and damping are compared with those extracted from a few expensive dynamic simulations of the structure, showing that the quasi-static approach produces reasonable albeit highly conservative bounds on the observed dynamics. This framework is also demonstrated on a 3D structure where dynamic simulations are infeasible. [ABSTRACT FROM AUTHOR]

Abstract: The nonlinear normal modes of a dynamical system provide a modal framework in which the dynamics of a structure can be readily understood. Current numerical approaches use continuation to find a nonlinear normal mode branch that initiates at a low energy, linearized mode. The predictor-corrector based approach follows the periodic solutions as the response amplitude increases, forming the nonlinear normal mode. This method uses the Jacobian of the shooting function in a Newton–Raphson algorithm to find the initial conditions and integration period that result in a periodic response of the conservative equations of motion. Large scale finite element models require that the Jacobian be computed using finite differences since the closed form equations are not explicitly available. The Jacobian must be computed with respect to all of the states, making the algorithm prohibitively expensive for models with many degrees-of-freedom. In this paper, the initial conditions of each periodic solution are determined based on a subset of the linear modes of a geometrically nonlinear finite element model. The first approach, termed enforced modal displacement, sets the initial conditions as a linear combination of linear mode shapes. The second approach, here called the applied modal force method, applies a static load to the structure in a combination of applied forces that would excite a single linear mode, computes the static response to that load, and uses that to set the initial conditions. Both of these algorithms greatly reduce the number of variables that are iterated on during continuation. As a result, the cost of computing each solution along the nonlinear normal mode is only on the order of ten times the cost required to integrate the finite element model over one period of the response. The algorithm is initiated with only one linear mode and additional modes are added in a systematic way as they become important to the periodic solutions along the nonlinear mode branch. The approach is demonstrated on two geometrically nonlinear finite element models, showing a dramatic reduction in the computational cost required to obtain the nonlinear normal mode. [Copyright &y& Elsevier]

Abstract: Continuous-Scan Laser Doppler Vibrometry (CSLDV) is a technique where the measurement point continuously sweeps over a structure while measuring, capturing both spatial and temporal information. The continuous-scan approach can greatly accelerate measurements, allowing one to capture spatially detailed mode shapes in the same amount of time that conventional methods require to measure the response at a single point. The method is especially beneficial when testing large structures, such as wind turbines, that have low natural frequencies and hence may require very long time records at each measurement point. Several CSLDV methods have been presented that use sinusoidal excitation or impulse excitation, but CSLDV has not previously been employed with an unmeasured, broadband random input. This work extends CSLDV to that class of input, developing an Output-only Modal Analysis method (OMA-CSLDV). A recently developed algorithm for linear time-periodic system identification, which makes use of harmonic power spectra and the harmonic transfer function concept developed by Wereley [17], is used in conjunction with CSLDV measurements. One key consideration, the choice of the scan frequency, is explored. The proposed method is validated on a randomly excited free-free beam, where one-dimensional mode shapes are captured by scanning the laser along the length of the beam. The first seven natural frequencies and mode shapes are extracted from the harmonic power spectrum of the vibrometer signal and show good agreement with the analytically-derived modes of the beam. The method is then applied to identify the mode shapes of a parked 20kW wind turbine using a ground based laser and with only a light breeze providing excitation. [Copyright &y& Elsevier]

Abstract: Many important systems, such as wind turbines, helicopters and turbomachinery, must be modeled with linear time-periodic equations of motion to correctly predict resonance phenomena. Time periodic effects in wind turbines might arise due to blade-to-blade manufacturing variations, stratification in the velocity of the wind with height and changes in the aerodynamics of the blades as they pass the tower. These effects may cause parametric resonance or other unexpected phenomena, so it is important to properly characterize them so that these machines can be designed to achieve high reliability, safety, and to produce economical power. This work presents a system identification methodology that can be used to identify models for linear, periodically time-varying systems when the input forces are unmeasured, broadband and random. The methodology is demonstrated for the well-known Mathieu oscillator and then used to interrogate simulated measurements from a rotating wind turbine. The measurements were simulated for a 5MW turbine modeled in the HAWC2 simulation code, which includes both structural dynamic and aerodynamic effects. This simulated system identification provides insights into the test and measurement requirements and the potential pitfalls, and simulated experiments such as this may be useful to obtain a set of time-periodic equations of motion from a numerical model, since a closed form model is not readily available by other means due to the way in which the aeroelastic effects are treated in the simulation code. [Copyright &y& Elsevier]

Abstract: A scanning laser Doppler vibrometer (LDV) can acquire non-contact vibration measurements from a structure with high spatial detail in an automated manner; one only need redirect the laser via computer-controlled mirrors to acquire measurements at additional points. However, since most LDV systems are only capable of measuring one point at a time, conventional scanning vibrometry cannot be effectively employed in some situations, for example when the time record is long at each measurement point or when the structure changes with time. Conventional scanning LDV systems are also difficult to employ with impact excitation because there is considerable variation in the impact location, angle and the character of the impacts, which leads to errors in the mode shapes that are extracted from the measurements. This paper presents a method by which one can determine the mode shapes, natural frequencies and damping ratios of a structure from as little as one response record by sweeping the laser continuously over the vibrating structure as the measurement is acquired. A novel resampling approach is presented that transforms the continuous-scan measurements into pseudo-frequency response functions, so they can be processed using standard identification routines to find the modal parameters of the structure. Specifically, this work employs a standard multi-input–multi-output identification routine and the complex mode indicator function to the continuous-scan laser Doppler vibrometry (CSLDV) measurements. The method makes no assumptions regarding the shape or properties of the surface and only requires that the laser scan periodically and that the structure vibrate freely. The method is demonstrated experimentally on a free–free beam, identifying the first nine mode shapes of the beam at hundreds of points from a few time histories. For this system, this represents a two-order of magnitude reduction in the time needed to acquire measurements with the LDV. [Copyright &y& Elsevier]

Abstract: An extension of the inverse structural filter (ISF) force reconstruction algorithm is presented that utilizes data from multiple time steps simultaneously to improve the accuracy and robustness of the ISF. The ISF algorithm uses a discrete-time system model of a structure and the measured response to estimate the forces causing the response. The proposed algorithm, dubbed the delayed, multi-step ISF (DMISF), is compared with the original ISF and with the sum of weighted accelerations technique (SWAT) and the classical frequency domain (FD) inverse method in terms of both accuracy and sensitivity to errors in the forward system model. The SWAT and ISF algorithms are capable of estimating the forces acting on a structure in real time, or when time data is available over such a short duration that FD methods cannot be applied effectively. The new DMISF can be created from a forward system model identified by any standard modal analysis algorithm, so one can leverage expertise with a particular system identification methodology. In contrast, the previously presented ISF was derived directly from experimental data using a proscribed technique. The theory behind the algorithms is presented, after which their performance is demonstrated using laboratory test data. The results of a Monte Carlo simulation are also presented, illustrating the nature of the sensitivity of the methods to errors in the modal parameters of the forward system. The DMISF algorithm is shown to yield a stable inverse system for the structure of interest whereas the traditional ISF is unstable, and hence gives erroneous estimates of the input forces. [Copyright &y& Elsevier]

Although flip chips have received wide acceptance as an integrated circuit package, significant manufacturing problems exist with the integrity of the connection between the package and the printed circuit board (PCB). Conventional X-ray, ultrasonic and electronic testing systems have been used to assess the integrity of this connection, however, none of these have proven suitable for detecting open solder bumps between the chip and the board. The inability to detect open solder bumps with traditional methods merits the investigation of new, nondestructive methods for detecting defects in a manufacturing environment. This work assesses the feasibility of monitoring the vibration characteristics of flip chips to detect open solder joints. Test vehicles with open solder joints were created, and a nondestructive laser ultrasonic system was used to measure the free vibration response of the chips attached to the printed circuit board. The algorithm of mode isolation (AMI) was applied to the vibration response data in order to extract the modal parameters of the chip. The statistical differences between the modal parameters of sets of damaged and undamaged chips were assessed, revealing the ability of the method to determine the location and severity of these defects in the presence of experimental scatter and manufacturing variation. The parameters of the first mode of vibration, especially its mode shape, were found to be much more sensitive to damage than those of a higher frequency mode. [ABSTRACT FROM AUTHOR]

When modeling dynamic structures, the quasi-static modal analysis (QSMA) approach seeks to very accurately resolve the force-displacement behavior of each individual mode but neglects dynamic modal coupling, while many other reduced order modeling (ROM) methods take an opposite view, capturing static and dynamic modal coupling but using polynomials that limit the fidelity with which the force-displacement behavior of each individual mode can be captured. This work contrasts these two approaches both theoretically and by applying them to geometrically nonlinear structures of varying complexity. The comparison reveals a potential deficiency in the QSMA approach in which the secant approximation that was typically used to estimate the effective natural frequency from the load-displacement curves is found to be inaccurate for some of the strongly nonlinear structures considered in this work. Conversely, the examples demonstrate that the low-order polynomial typically employed by ROM methods such as implicit condensation and expansion (ICE) is often insufficient to describe the force-displacement behavior with adequate fidelity when a single mode is used. These examples prompt the creation of a new method that is a hybrid between QSMA and ICE, which fits a high order polynomial to the modal response curve of a single mode. Several case studies show that the resulting single-mode ICE ROM, here dubbed a SICE-ROM, can capture the resonant behavior in the vicinity of the mode very accurately, but to do so the force-displacement relationship must be captured with extremely high accuracy. These comparisons highlight the importance of static coupling in these structures, suggesting that it is often far more important than dynamic coupling even in cases where the latter had been previously thought to be very important. The geometrically nonlinear structures studied in this work include FE models of a flat beam and an exhaust cover plate and a highly nonlinear curved beam where the Riks method is used to obtain the force-displacement curves through the snap-through regime. [ABSTRACT FROM AUTHOR]

• Static reduction used to accelerate contact analysis. • Algorithm iterates on groups of nodes, improving efficiency for 3D problems • Used to predict natural frequency and damping of modes of 2D and 3D FE models. • The algorithm resulted in dramatic computational savings while providing good accuracy. While a bolted joint can be modeled in commercial FE codes, even quasi-static analysis is extremely expensive due to the level of detail required to capture slip at the bolt interface. This is even worse for 3D cases where the solver must deal with the two-dimensional friction problem, making convergence more expensive. In order to address this issue, we present a quasi-static algorithm that greatly accelerates the analysis of 2D and 3D problems. This algorithm uses static reduction along with a new numerical method which is a hybrid between Block-Gauss-Seidel and Newton-Rhapson. A relaxed form of 2D frictional problem is presented in this algorithm which increases the efficiency of the method. In order to evaluate the performance of the algorithm, we apply the method to perform QSMA on 2D and 3D cases. QSMA uses a series of static loadings to predict the effective natural frequency and damping of each mode of a bolted structure as a function of vibration amplitude. In the cases studied here the algorithm resulted in dramatic computational savings while providing good accuracy as compared to simulations in commercial finite element software, both for the effective natural frequencies and damping and for the contact pressure distributions. To explore the robustness of the algorithm, the mesh density, preloads, and quasi-static force amplitudes are explored. The effect of the size of the reduced model on the accuracy of the result and the efficiency of the method is investigated in this paper. [ABSTRACT FROM AUTHOR]

Bolted joints not only provide critical connections in assembled structures but are also a significant source of damping. The damping induced by joints is often nonlinear, which is currently not captured by typical modeling practices. A method that was recently proposed by Festjens, Chevallier, and Dion addresses this by using a quasi-static loading to infer the effective stiffness and damping of each mode of a structure. Allen and Lacayo further developed this method, demonstrated its efficacy for structures where the joints were represented with discrete Iwan elements, and dubbed the method quasi-static modal analysis. This work expands on those studies by applying quasi-static modal analysis to 2-D and 3-D FE models in which the geometry, contact pressure, and friction in the joint are modeled in detail. Coulomb friction is assumed between the nominally flat contacting surfaces. In order to obtain confidence in the predictions, the mesh density, contact laws, and other solver settings are explored to understand what is needed to obtain accurate results for this type of problem. The studies are culminated with a first ever comparison between measurements and quasi-static predictions of the amplitude dependent natural frequency and damping ratio of a mode of a real structure. The results to date are promising but highlight areas in which further research is needed. [ABSTRACT FROM AUTHOR]

Bolted joints are a significant source of damping and nonlinearity in built up structures. In the micro-slip regime (i.e. when the bolts do not slip completely) the log of the damping tends to increase linearly with the log of the vibration amplitude, the so-called power-law behavior. The natural frequency also tends to decrease slightly with vibration amplitude. While many works have successfully tuned phenomenological models to capture these behaviors, far fewer have sought to predict the nonlinearity in a bolted joint and validated the predictions with measurements over a range of bolt preloads and vibration amplitudes. This work presents a step towards such a prediction, using a detailed finite element model of a structure, including the preload forces in the bolts and Coulomb friction in the contact, to seek to predict the nonlinear damping and stiffness of the structure and how they vary with preload. The structure studied consists of two beams bolted together at their free ends, the so-called S4 Beam. While the contact interfaces were machined to be nominally flat, the micron-scale curvature in the contact interface is included in the model, approximated in two different ways, to seek to understand what fidelity is needed at the contact interface to successfully capture its dynamic behavior. The recently presented Quasi-Static Modal Analysis (QSMA) method is employed, where the amplitude dependent damping and natural frequency can be predicted from a single quasi-static simulation, avoiding the expense of numerical integration. The predicted damping and natural frequency are compared with measurements at various preloads, showing reasonable agreement if a Coulomb friction coefficient of 0.2 is used for all simulations. The simulations also revealed that, while the micron-scale curvature of the interfaces was important, the results were not very sensitive to how it was approximated. • Joint modeled with a detailed Finite Element model and Coulomb friction. • Effective natural frequency and damping ratio predicted for two modes. • Predictions compared to measurements at various joint preloads. • Linear natural frequencies predicted as a function of joint preload. • Two modes are studied, involving shear and torsion of the joint. [ABSTRACT FROM AUTHOR]