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The objective of this publication is to propose a novel technique for black-box identification of structures with geometric nonlinearity. The technique is based on a recently introduced frequency-domain system identification method called Nonlinear Identification through eXtended Outputs (NIXO). The proposed algorithm first expresses the nonlinear part of modal equation of motion (EOM) as a general polynomial of high order, and then removes the terms which are determined to be irrelevant to the mechanical system's response. This division into dominant and irrelevant nonlinear terms relies on the values of two novel indicators that are particular to NIXO. The heuristic presented here was observed to work well only when the tested structure is excited with various swept-sine input signals of different magnitudes (so that the system oscillates at sufficiently distinctive amplitudes in different experimental tests). The technique is first demonstrated on a noise-free numerical case study employing a reduced model of a curved beam. The results obtained are verified via comparing the true Nonlinear Normal Mode (NNM) to the one computed using the modal EOM pointed out by NIXO. Then the method is demonstrated on experimental measurements collected on a 3D-printed flat beam exhibiting significant natural frequency shifts, and the outcomes are verified by overlaying the identified NNMs on the swept-sine responses. [ABSTRACT FROM AUTHOR]

Nonlinear reduced-order models are being investigated for use as digital twins for advanced aircraft and spacecraft. In concept, the digital twin would be tuned to accurately describe the behavior of the vehicle, and then it would be updated as the vehicle is modified or ages. Unfortunately, the linear model correlation and validation techniques that are commonly used in the aerospace industry are not valid for nonlinear models, so a new set of tools is needed. This work presents an algorithm to update geometrically nonlinear reduced-order models using nonlinear normal modes as a correlation metric. The nonlinear normal modes serve as a strong metric to correlate the numerical models because they can be extracted from experiments, and they describe the dynamics of the nonlinear system over a range of amplitudes and are independent of the loading applied to the system. This paper presents a novel method of computing analytical gradients of nonlinear normal mode solutions with respect to system parameters using the multiharmonic balance method. The procedure is first tested using numerical simulations and then applied to tune the reduced-order model of a curved beam based on nominal blueprints and idealized boundary conditions to match a nonlinear normal mode that has been measured experimentally from a three-dimensional printed test specimen. [ABSTRACT FROM AUTHOR]

This work presents a novel technique for nonlinear system identification that operates in the frequency domain and fits a model to measured spectra to estimate the parameters in a modal domain nonlinear equation of motion (EOM). Nonlinear terms are added to the linear EOM in the form of polynomials, and the proposed algorithm estimates the polynomial coefficients as well as the underlying linear Frequency Response Function (FRF). This method is an extension to a popular nonlinear system identification algorithm called NIFO, from Nonlinear Identification through Feedback of the Outputs. However, NIFO identifies the nonlinear parameters as complex numbers that may be different at each frequency line, even though the mechanical system is expected to be governed by an EOM in which the nonlinear parameters are real and constant with frequency. This might be problematic, because any variation in the identified nonlinear parameters will distort the linear FRFs estimated by NIFO, and those linear FRFs are important to tell the user whether all of the significant nonlinearity has been extracted from the system. The proposed algorithm, here dubbed Nonlinear Identification through eXtended Outputs (NIXO), estimates the nonlinear parameters as frequency-independent and real. Additionally, it is demonstrated that for the systems studied here that the algorithm works when random and swept-sine inputs are used to excite the tested structure, while NIFO only worked well when random inputs were used. The method is first evaluated numerically using benchmark case studies, starting with the SDOF equation and then reduced models of a clamped-clamped flat beam, and the results are compared to those obtained with NIFO. Then the algorithm is applied to swept-sine measurements from a 3D-printed flat beam and the results are validated by computing the primary nonlinear normal mode of the identified model and comparing it with measurements. [ABSTRACT FROM AUTHOR]

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]

Reduced order models, such as Hollkamp and Gordon's Implicit Condensation and Expansion (ICE) model, are a highly efficient alternative to full-order finite element models (FEM) of geometrically nonlinear structures. However, a reduced order model (ROM) is typically only valid for one FEM. It does not capture how each ROM coefficient changes due to variations in the FEM (e.g., design parameters or uncertainties), so if the FEM is updated then the ROM needs to be re-computed with a new set of static load–displacement data. This study presents a data-driven reduced order modeling approach that creates a single ROM that incorporates design variations in FEM. The proposed method applies Gaussian Process Regression (GPR) to the ICE approach, making each coefficient in an ICE ROM a regression model with respect to a collection of FEMs with varying material properties or geometric parameters. Once the GPR ROM has been identified, one can immediately produce an ICE ROM for a set of FEM parameters without a need to solve any static load–displacement cases on the full FEM. This dramatically enhances the computational efficiency and could be helpful when model uncertainty needs to be considered or when seeking to update a model to correlate with measurements. Additionally, the coefficients of a ROM can often change considerably if the scale on the load–displacement data changes, so it can be difficult to know whether the scaling that was chosen has really identified an accurate ROM. The proposed GPR ROM estimates the mean ROM coefficients for a range of load scaling as well as the uncertainty on each ROM coefficient with respect to the load level. This can be used to gauge the success of the ROM identification and to eliminate ROM coefficients that are unimportant and hence highly variable. The proposed GPR ROM approach is evaluated by applying it to flat and curved beam structures, revealing that the advantages outlined above can be realized with a relatively modest increase in cost relative to a traditional ICE ROM. [ABSTRACT FROM AUTHOR]

Substructuring methods have been widely used in structural dynamics to divide large, complicated finite element models into smaller substructures. For linear systems, many methods have been developed to reduce the subcomponents down to a low-order set of equations using a special set of component modes, and these are then assembled to approximate the dynamics of a large-scale model. In this paper, a substructuring approach is developed for coupling geometrically nonlinear structures, where each subcomponent is drastically reduced to a low-order set of nonlinear equations using a truncated set of fixed-interface and characteristic constraint modes. The method used to extract the coefficients of the nonlinear reduced-order model is nonintrusive, in that it does not require any modification to the commercial finite element code but computes the reduced-order model from the results of several nonlinear static analyses. The nonlinear reduced-order models are then assembled to approximate the nonlinear differential equations of the global assembly. The method is demonstrated on the coupling of two geometrically nonlinear plates with simple supports at all edges. The plates are joined at a continuous interface through the rotational degrees of freedom, and the nonlinear normal modes of the assembled equations are computed to validate the models. The proposed substructuring approach reduces a 12,861-degree-of-freedom model down to only 23 degrees of freedom while still accurately reproducing the nonlinear normal modes. [ABSTRACT FROM AUTHOR]

This paper presents a case study in which the finite element model for a curved circular plate is calibrated to reproduce both the linear and nonlinear dynamic response measured from two nominally identical samples. The linear dynamic response is described with the linear natural frequencies and mode shapes identified with a roving hammer test. Due to the uncertainty in the stiffness characteristics from the manufactured perforations, the linear natural frequencies are used to update the effective modulus of elasticity of the full order finite element model (FEM). The nonlinear dynamic response is described with nonlinear normal modes (NNMs) measured using force appropriation and high speed 3D digital image correlation (3D-DIC). The measured NNMs are used to update the boundary conditions of the full order FEM through comparison with NNMs calculated from a nonlinear reduced order model (NLROM). This comparison revealed that the nonlinear behavior could not be captured without accounting for the small curvature of the plate from manufacturing as confirmed in literature. So, 3D-DIC was also used to identify the initial static curvature of each plate and the resulting curvature was included in the full order FEM. The updated models are then used to understand how the stress distribution changes at large response amplitudes providing a possible explanation of failures observed during testing. [ABSTRACT FROM AUTHOR]

Spatially detailed dynamic measurements of thin, lightweight structures can be difficult to obtain due to the structure׳s low mass and complicated deformations under certain loading conditions. If traditional contacting sensors, such as accelerometers, strain gauges, displacement transducers, etc., are used, the total number of measurement locations available is limited by the weight added and the effect each sensor has on the local stiffness of the contact area. Other non-contacting sensors, such as Laser Doppler Vibrometers (LDV), laser triangulation sensors, proximity sensors, etc., do not affect the dynamics of a structure, but are limited to single point measurements. In contrast, a few recently developed non-contacting measurement techniques have been shown to be capable of simultaneously measuring the response over a wide measurement field. Two techniques are considered here: Continuous-Scan Laser Doppler Vibrometry (CSLDV) and high speed Three-Dimensional Digital Image Correlation (3D DIC). With the use of these techniques, unprecedented measurement resolution can be achieved. In this work, the linear and nonlinear deformations of a clamped, nominally flat beam and plate under steady state sinusoidal loading will be measured using both techniques. In order to assess their relative merits, the linear natural frequencies, mode shapes, and nonlinear deformation shapes measured with each method are compared. Both measurement systems give comparable results in many cases, although 3D DIC is more accurate for spatially complex deformations at large amplitudes and CSLDV is more accurate at low amplitudes and when the spatial deformation pattern is simpler. [ABSTRACT FROM AUTHOR]

This article presents measurements from a sub assembly of an off-the-shelf automotive exhaust system containing a bolted-flange connection and uses a recently proposed modal framework to develop a nonlinear dynamic model for the structure. The nonlinear identification and characterization methods used are reviewed to highlight the strengths of the current approach and the areas where further development is needed. This marks the first use of these new testing and nonlinear identification tools, and the associated modal framework, on production hardware with a realistic joint and realistic torque levels. To screen the measurements for nonlinearities, we make use of a time frequency analysis routine designed for transient responses called the zeroed early-time fast Fourier transform (ZEFFT). This tool typically reveals the small frequency shifts and distortions that tend to occur near each mode that is affected by the nonlinearity. The damping in this structure is found to be significantly nonlinear and a Hilbert transform is used to characterize the damping versus amplitude behavior. A model is presented that captures these effects for each mode individually (e.g. assuming negligible nonlinear coupling between modes), treating each mode as a single degree-of-freedom oscillator with a spring and viscous damping element in parallel with a four parameter Iwan model. The parameters of this model are identified for each of the structure׳s modes that exhibited nonlinearity and the resulting nonlinear model is shown to capture the stiffness and damping accurately over a large range of response amplitudes. [ABSTRACT FROM AUTHOR]

The ability to model nonlinear structures subject to random excitation is of key importance in designing hypersonic aircraft and other advanced aerospace vehicles. When a structure is linear, superposition can be used to construct its response to a known spectrum in terms of its linear modes. Superposition does not hold for a nonlinear system, but several works have shown that a system's dynamics can still be understood qualitatively in terms of its nonlinear normal modes (NNMs). This work investigates the connection between a structure's undamped nonlinear normal modes and the spectrum of its response to high amplitude random forcing. Two examples are investigated: a spring-mass system and a clamped-clamped beam modeled within a geometrically nonlinear finite element package. In both cases, an intimate connection is observed between the smeared peaks in the response spectrum and the frequency-energy dependence of the nonlinear normal modes. In order to understand the role of coupling between the underlying linear modes, reduced order models with and without modal coupling terms are used to separate the effect of each NNM's backbone from the nonlinear couplings that give rise to internal resonances. In the cases shown here, uncoupled, single-degree-of-freedom nonlinear models are found to predict major features in the response with reasonable accuracy; a highly inexpensive approximation such as this could be useful in design and optimization studies. More importantly, the results show that a reduced order model can be expected to give accurate results only if it is also capable of accurately predicting the frequency-energy dependence of the nonlinear modes that are excited. [ABSTRACT FROM AUTHOR]

Nonlinear Normal Modes (NNMs) offer tremendous insight into the dynamic behavior of a nonlinear system, extending many concepts that are familiar in linear modal analysis. Hence there is interest in developing methods to experimentally and numerically determine a system׳s NNMs for model updating or simply to characterize its dynamic response. Previous experimental work has shown that a mono-harmonic excitation can be used to isolate a system׳s dynamic response in the neighborhood of a NNM along the main backbones of a system. This work shows that a multi-harmonic excitation is needed to isolate a NNM when well separated linear modes of a structure couple to produce an internal resonance. It is shown that one can tune the multiple harmonics of the input excitation using a plot of the input force versus the response velocity until the area enclosed by the force–velocity curve is minimized. Once an appropriated NNM is measured, one can increase the force level and retune the frequency to obtain a NNM at a higher amplitude or remove the excitation and measure the structure׳s decay down a NNM backbone. This work explores both methods using simulations and measurements of a nominally-flat clamped–clamped beam excited at a single point with a magnetic force. Numerical simulations are used to validate the method in a well defined environment and to provide comparison with the experimentally measured NNMs. The experimental results seem to produce a good estimate of two NNMs along their backbone and part of an internal resonance branch. Full-field measurements are then used to further explore the couplings between the underlying linear modes along the identified NNMs. [ABSTRACT FROM AUTHOR]

The efficiency of a modal substructuring method depends on the component modes used to reduce each subcomponent model. Methods such as Craig-Bampton have been used extensively to reduce linear finite-element models with thousands or even millions of degrees of freedom down orders of magnitude while maintaining acceptable accuracy. A novel reduction method is proposed here for geometrically nonlinear finite-element models using the fixed-interface and constraint modes of the linearized system to reduce each subcomponent model. The geometric nonlinearity requires an additional cubic and quadratic polynomial function in the modal equations, and the nonlinear stiffness coefficients are determined by applying a series of static loads and using the finite-element code to compute the response. The geometrically nonlinear, reduced modal equations for each subcomponent are then coupled by satisfying compatibility and force equilibrium. This modal substructuring approach is an extension of the Craig-Bampton method and is readily applied to geometrically nonlinear models built directly within commercial finite-element packages. The efficiency of this new approach is demonstrated on two example problems: one that couples two geometrically nonlinear beams at a shared rotational degree of freedom, and another that couples an axial spring element to the axial degree of freedom of a geometrically nonlinear beam. The nonlinear normal modes of the assembled models are compared with those of a truth model to assess the accuracy of the novel modal substructuring approach. [ABSTRACT FROM AUTHOR]

In many applications it would be advantageous to be able to compute the nonlinear Frequency Response Functions (FRF) of structures containing bolted interfaces, but to do this with time integration is expensive so more efficient methods are desired. Harmonic Balance methods are typically more efficient, yet they are difficult to apply to systems with joints because one must track the state of many points on a surface and their highly nonlinear evolution from stick to slip in each cycle of vibration. This paper proposes an adaptation of the alternating frequency time harmonic balance method that is able to compute the steady-state response and consequently the nonlinear FRFs of systems with Iwan elements in an accurate and efficient manner. Rather than including the state of all of the sliders in the state vector, the points at which the displacement across the joint reverses are identified and used as a surrogate, greatly reducing the number of state variables and making the state variables more continuous. Furthermore, the nonlinear force and the Jacobian only need to be computed for the degrees of the freedom related to the joints, improving the efficiency of the approach. While the damping in the hysteretic system is not velocity dependent, an effective damping is introduced and added to the Jacobian to improve convergence of the Newton–Raphson algorithm for frequencies near the resonance. The proposed method, dubbed as "Hysteresis Identification via Reversal Points or (HIRP)", is tested on various models with multiple DOF and Iwan elements, including a relatively high order model of two beams with two bolted joints. The results obtained show the capability of the proposed method and its efficiency and accuracy for realistic, industrially relevant models. [Display omitted] • "Hysteresis Identification via Reversal Points or (HIRP)" is proposed to compute the nonlinear frequency response of complex, hysteretic systems. • The stick/slip state of many sliders is captured using a few reversal points in the response of each joint. • The nonlinear force and Jacobian are computed only for the nodes corresponding to joints, providing huge computational saving. • The effect of damping is considered in the computation of Jacobian. • The method is found to be capable of computing the steady-state response of complicated 3D systems with multiple Iwan joints. [ABSTRACT FROM AUTHOR]

Abstract: The authors recently presented a new nonlinear system identification method, here dubbed the NL-LTP method, in which the system of interest is forced harmonically so that it responds in a stable periodic orbit, and then it is perturbed slightly and its response is recorded as it returns to the orbit. Under mild assumptions the response about the periodic orbit can be approximated using a linear time periodic system model, which can be identified from the measurements using techniques that are akin to linear modal analysis. While the prior work focused on simulated measurements from single degree-of-freedom systems, this work presents several tools that are needed in order to use this approach on multi-degree-of-freedom systems and focuses on applying the method to experimental hardware. The proposed system identification methodology is unique in that it identifies both the order of the nonlinear system and a mathematical model for the nonlinear restoring forces without assuming the mathematical form for the nonlinearities a priori. Towards these ends, this work explains how to extract the underlying nonlinear system model, or nonlinear restoring force versus displacement relationships, from the time periodic model that governs deviations of the system from its periodic orbit, and presents various metrics that can be used to determine which terms in the model are meaningful. These new tools are used to apply the identification method to a continuous, multi-degree-of-freedom structure with a discrete geometric nonlinearity, using both simulated and experimental measurements. The experimental hardware consists of a cantilever beam with a nonlinear spring attached to its tip, which is driven in a periodic limit cycle by an electromagnetic shaker. [Copyright &y& Elsevier]

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) can greatly accelerate modal testing by continuously sweeping the measuring laser over a structure, effectively capturing its response at tens or even hundreds of points along the laser path. Several techniques have been devised to process CSLDV measurements from structures under controlled input. The authors recently extended CSLDV to the case where the input forces are unmeasured random white noise, using the harmonic power spectrum of a time periodic system. The harmonic power spectrum is analogous to the power spectrum used in the identification of time invariant systems, but with many additional harmonics for each mode, requiring an additional effort in modal parameter estimation. This paper presents a variant on the harmonic power spectrum, proposing a simplified algorithm based on the lifting approach. Lifting causes all sideband peaks in the harmonic power spectrum to collapse into a single peak in the range from zero to half of the scan frequency, so the spectra are far easier to interpret. The proposed algorithm is first evaluated on a simulated beam, and found to give results that are comparable with those obtained by the harmonic power spectrum method, yet the data reduction with the lifting approach is much simpler. This algorithm is then employed to identify the first several modes of a parked wind turbine under wind excitation, using a new long range remote sensing vibrometer. The speckle noise is found to be remarkably small even at a standoff distance of 77m and a surface scan velocity of 500m/s without any surface treatment. [Copyright &y& Elsevier]

Abstract: Experimental-to-analytical substructuring has been investigated as a way to avoid costly tests and analyses on large systems by experimentally characterizing certain subcomponents and then coupling them to an analytical model for the rest of the structure. Unfortunately, substructuring has sometimes proven difficult to implement because the substructuring calculations can be sensitive to modal truncation and experimental noise and often require the measurement of rotational motions at the interface. These issues have led to decreased confidence in substructuring calculations, causing it to remain underutilized in industry. This work proposes performing a small, inexpensive validation test to give confidence or reveal flaws in an experimentally derived model. This is only meaningful if the error in the validation test is characteristic of the error in an application of interest; this work explores this assumption. The validation test is evaluated in two scenarios which represent substructuring realities. The first scenario investigates measurement errors by applying normally distributed noise to a simple beam structure and evaluating how well the validation test compares to the application of interest. The second scenario simulates substructuring of a complicated engine–generator system with a statically indeterminate connection, where the effect of modal truncation is not easy to anticipate. [Copyright &y& Elsevier]

This work presents a new technique for nonlinear system identification that utilizes near-resonant steady-state harmonically excited vibration measurements to estimate the Nonlinear Normal Mode backbones. The algorithm is based on the previously proposed Single Nonlinear Resonant Mode formula and uses it in a new and more effective way to estimate one point on the nonlinear mode from only one steady-state measurement collected near the resonance. A by-product of this work is a derivation of a novel formula expressing how the damping ratio changes with the motion amplitude. Several measurements at various forcing amplitudes can be combined to estimate the nonlinear mode and damping as a function of amplitude, which can be further used to predict the forced steady-state response of the structure in the vicinity of the mode of interest. Compared to existing phase resonance methods, the proposed technique can reduce the time required to obtain measurements and avoids difficulties due to e.g. the premature jump phenomenon. The algorithm assumes that the modes are well-separated and no internal resonances are present in the system. Additionally, it requires the accurate identification of linear modes in the low-amplitude vibration tests and it assumes that the nonlinear normal mode shape does not change significantly with response amplitude The method is first evaluated numerically using reduced models of clamped–clamped flat and curved beams that exhibit both stiffening and softening–stiffening responses, respectively. Then the method is employed experimentally to measure the NNM backbones of beams that were manufactured from polylactide using a 3D printer and experience significant eigen-frequency shifts when the motion amplitude increases. The results are validated against measurements collected using the traditional phase resonance testing approach. • A new technique to utilize the Single Nonlinear Resonant Mode formula is presented. • The method uses near-resonant steady-state measurements to estimate the Nonlinear Normal Modes of a structure. • A novel formula expressing how the damping ratio changes with the motion amplitude is derived. • The experimental data is collected with a phase resonance method. • The method is applied to identify the NNMs of 3D printed flat and curved beams. [ABSTRACT FROM AUTHOR]

• Experimental measurements show coupling between modes due to friction in bolted joints. • One-way coupling is observed where Mode 2 affects Mode 1 but not the reverse. • Micro-scale geometry is carefully characterized and included in finite element models. • Nonlinear contact simulations predict natural frequency versus preload. The goal of this paper is to present a set of measurements from a benchmark structure containing two bolted joints to support future efforts to predict the damping due to the joints and to model nonlinear coupling between the first two elastic modes. Bolted joints introduce nonlinearities in structures, typically causing a softening in the natural frequency and an increase in damping because of frictional slip between the contact interfaces within the joint. These nonlinearities pose significant challenges when characterizing the response of the structure under a large range of load amplitudes, especially when the modal responses become coupled, causing the effective damping and natural frequency to not only depend on the excitation amplitude of the targeted mode, but also the relative amplitudes of other modes. In this work, two nominally identical benchmark structures, known in some prior works as the S4 beam, are tested to characterize their nonlinear properties for the first two elastic modes. Detailed surface measurements are presented and validated through finite element analysis and reveal distinct contact interactions between the two sets of beams. The free-free test structures are excited with an impact hammer and the transient response is analyzed to extract the damping and frequency backbone curves. A range of impact amplitudes and drive points are used to isolate a single mode or to excite both modes simultaneously. Differences in the nonlinear response correlate with the relative strength of the modes that are excited, allowing one to characterize mode coupling. Each of the beams shows different nonlinear properties for each mode, which is attributed to the different contact pressure distributions between the parts, although the mode coupling relationship is found to be consistent between the two. The test data key finding are presented in this paper and the supporting data is available on a public repository for interested researchers. [ABSTRACT FROM AUTHOR]

Abstract: When testing to identify an experimental model of a substructure, it is often beneficial to exercise the interface by attaching a fixture such as a rigid mass. An analytical representation of the fixture must then be used to subtract its effects from the assembly. However, this can cause the mass and stiffness matrices to become indefinite in some situations. This paper presents two new metrics that can be used by the analyst to determine the cause of indefinite mass or stiffness matrices after substructure uncoupling. The metrics rank the experimental and fixture modes based upon their contribution to offending negative eigenvalues. Once the troublesome modes have been identified, they can be inspected and often reveal why the mass has become negative. Two examples are presented to demonstrate the metrics and to illustrate the physical phenomena that they reveal. [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: This work presents a new method for identifying models of nonlinear systems from experimental measurements. The system is first forced to oscillate in stable periodic orbit, and then a small impulsive disturbance force is used to perturb the system slightly from that orbit. One then measures the response until the system returns to the periodic orbit. If the nonlinearities in the system are sufficiently smooth and the perturbation from the periodic orbit is sufficiently small, then one can linearize the perturbed response about the periodic orbit and approximate the system as linear time periodic. One of a variety of methods can then be used to extract the time varying modal model of the system from the response. The extracted modes can be used to construct a time periodic state transition matrix and state coefficient matrix, which describe the system''s nonlinear dynamics over a range of the states. The resulting model for the nonlinear system encompasses that portion of the state space is traversed by the system during its periodic orbit. [Copyright &y& Elsevier]

Abstract: Fixed boundary conditions are often difficult if not impossible to simulate experimentally, but they are important to consider in many applications. In principle, modal substructuring or impedance coupling approaches can be used to predict the fixed base modes of a system from tests where the system has some other boundary condition if the motion at the connection point can be measured, but this approach can be highly sensitive to imperfections in the experimental measurements. This work presents two alternatives that reduce the sensitivity to experimental errors, capitalizing on recent works where additional degrees of freedom are used to improve the robustness of substructure uncoupling. The system of interest is tested while mounted on a stiff fixture, where some modes of the fixture inevitably interact with those of the system of interest. The modes of the system–fixture assembly are extracted using a modal test and then a modal substructuring approach is used to apply constraints to eliminate the motion of the fixture. Two types of constraints are proposed, one based on the modes of the fixture and the other on a singular value decomposition of the fixture motion that was observed during the test. Neither approach requires an estimate of the displacements or rotations at the points where the system of interest is connected to the fixture. The methods are validated by applying them to experimental measurements from a simple test system meant to mimic a flexible satellite on a stiff shaker table. A finite element model of the subcomponents was also created and the method is applied to its modes in order to separate the effects of measurement errors and modal truncation. The proposed method produces excellent predictions of the first several modes of the fixed-base structure, so long as modal truncation is minimized. The proposed approach is also applied to experimental measurements from a wind turbine blade mounted in a stiff frame and found to produce reasonable results. [Copyright &y& Elsevier]

Questions: Most modern fire-prone landscapes have experienced disruptions of their historic fire regimes. Are the primary tallgrass prairies of the Flint Hills reflective of a history of continuous fire occurrence? Did fire frequency, severity, size and seasonality change in connection with changes in land use? Has fire occurrence been related to drought conditions? Location: Edges of Cross Timbers forest stands at the Tallgrass Prairie Preserve (TGPP) in the Flint Hills of Osage County, Oklahoma, USA. Methods: Cross-sections of 76 Quercus stellata were collected from Cross Timbers stands at or near the grassland edge in the TGPP. Dendrochronological methods were used to identify years of formation for tree rings and fire scars. Superposed epoch analysis was used to evaluate the effect of drought conditions on fire occurrence. Results: Fires were recorded in 46.6% of the years between 1729 and 2005. In 41 cross-sections at one site, the mean fire interval between 1759 and 2003 was 2.59 years, with fire interval decreasing from a mean fire interval of 3.76 years in the early part of the record to 2.13 years in modern times. No extended periods without fire were recorded in the study area. Drought conditions had no significant effect on fire occurrence. Conclusions: In contrast with many fire-prone landscapes worldwide, the prairies of the Flint Hills have experienced no recent fire suppression or exclusion. Changes in fire frequency mark transitions in land use, primarily from being traditionally used by Native Americans to being managed for cattle production. [ABSTRACT FROM AUTHOR]

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: Modal substructuring or component mode synthesis (CMS) has been standard practice for many decades in the analytical realm, yet a number of significant difficulties have been encountered when attempting to combine experimentally derived modal models with analytical ones or when predicting the effect of structural modifications using experimental measurements. This work presents a new method that removes the effects of a flexible fixture from an experimentally obtained modal model. It can be viewed as an extension to the approach where rigid masses are removed from a structure. The approach presented here improves the modal basis of the substructure, so that it can be used to more accurately estimate the modal parameters of the built-up system. New types of constraints are also presented, which constrain the modal degrees of freedom of the substructures, avoiding the need to estimate the connection point displacements and rotations. These constraints together with the use of a flexible fixture enable a new approach for joining structures, especially those with statically indeterminate multi-point connections, such as two circular flanges that are joined by many more bolts than required to enforce compatibility if the substructures were rigid. Fixture design is discussed, one objective of which is to achieve a mass-loaded boundary condition that exercises the substructure at the connection point as it is in the built up system. The proposed approach is demonstrated with two examples using experimental measurements from laboratory systems. The first is a simple problem of joining two beams of differing lengths, while the second consists of a three-dimensional structure comprising a circular plate that is bolted at eight locations to a flange on a cylindrical structure. In both cases frequency response functions predicted by the substructuring methods agree well with those of the actual coupled structures over a significant range of frequencies. [Copyright &y& Elsevier]

Abstract: This work presents time-frequency signal processing methods for detecting and characterizing nonlinearity in transient response measurements. The methods are intended for systems whose response becomes increasingly linear as the response amplitude decays. The discrete Fourier transform of the response data is found with various sections of the initial response set to zero. These frequency responses, dubbed zeroed early-time fast Fourier transforms (ZEFFTs), acquire the usual shape of linear frequency response functions (FRFs) as more of the initial nonlinear response is nullified. Hence, nonlinearity is evidenced by a qualitative change in the shape of the ZEFFT as the length of the initial nullified section is varied. These spectra are shown to be sensitive to nonlinearity, revealing its presence even if it is active in only the first few cycles of a response, as may be the case with macro-slip in mechanical joints. They also give insight into the character of the nonlinearity, potentially revealing nonlinear energy transfer between modes or the modal amplitudes below which a system behaves linearly. In some cases one can identify a linear model from the late time, linear response, and use it to reconstruct the response that the system would have executed at previous times if it had been linear. This gives an indication of the severity of the nonlinearity and its effect on the measured response. The methods are demonstrated on both analytical and experimental data from systems with slip and impact nonlinearities. [Copyright &y& Elsevier]

Abstract: The industry standard method used to validate finite element models involves correlation of test and analysis mode shapes using reduced Test-Analysis Models (TAMs). Some organizations even require this model validation approach. Considerable effort is required to choose sensor locations and to create a suitable TAM so that the test and analysis mode shapes will be orthogonal to within the required tolerance. This work uses a probabilistic framework to understand and quantify the effect of small errors in the test mode shapes on test-analysis orthogonality. Using the proposed framework, test-orthogonality is a probabilistic metric and the problem becomes one of choosing sensor placement and TAM generation techniques that assure that the orthogonality has a high probability of being within an acceptable range if the model is correct, even though the test measurements are contaminated with random errors. A simple analytical metric is derived that is shown to give a good estimate of the sensitivity of a TAM to errors in the test mode shapes for a certain noise model. These ideas are then applied to a generic satellite system, using TAMs generated by the Static, Modal and Improved Reduced System (IRS) reduction methods. Experimental errors are simulated for a set of mode shapes and Monte Carlo simulation is used to estimate the probability that the orthogonality metric exceeds a threshold due to experimental error alone. For the satellite system considered here, the orthogonality calculation is highly sensitive to experimental errors, so a set of noisy mode shapes has a small probability of passing the orthogonality criteria for some of the TAMs. A number of sensor placement techniques are used in this study, and the comparison reveals that, for this system, the Modal TAM is twice as sensitive to errors on the test mode shapes when it is created on a sensor set optimized for the Static TAM rather than one that was optimized specifically for the Modal TAM. These findings are evaluated in light of previously published studies of TAM sensitivity, and special attention is given to Gordis''s theory, which suggest that TAM sensitivity is related to the natural frequencies of the structure when all measurement points are fixed. Some aspects of TAM sensitivity are problem dependent, so this one work cannot achieve a conclusive ranking of all of the available methodologies. Instead, this work focuses on presenting a set of tools and a probabilistic framework that can be used to correctly quantify TAM sensitivity and demonstrating the approach for one dynamic system and for a particular probabilistic model for the errors contaminating the test mode shapes. [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]

Coulomb sliders are used widely in order to model frictional interfaces. In many applications, the nonlinear Frequency Response Functions (FRFs) of these systems is desired, but it is extremely computationally expensive to use time-domain integration methods to compute the steady-state response. The nonlinear force and stiffness of these devices depends on the position of sliders, and the implicit nature of the slider positions makes these systems hysteretic and therefore non-trivial to address using conventional continuation methods. This paper proposes a novel numerical method, dubbed "Hysteresis Identification via Reversal Points or (HIRP)" that can compute the steady-state harmonically forced response of frictional nonlinear systems. For this purpose, a quasi-static algorithm is introduced which can evaluate the state of sliders at any time based on the reversal points over a period. This reduces the number of state variables needed in the continuation routine by at least an order of magnitude. The method has been tested by computing the harmonic response of a hysteretic system with multiple Coulomb sliders. The method is evaluated on multi-degree of freedom systems with Iwan joints, demonstrating that the method can be a valuable tool for predicting the nonlinear, steady-state response of hysteretic systems. [ABSTRACT FROM AUTHOR]

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]

Abstract: This work presents a global, single-input–multi-output (SIMO) extension of the algorithm of mode isolation (AMI) in which the frequency response functions (FRFs) from all measurement points are considered simultaneously, resulting in global estimates for natural frequencies and damping ratios, and consistent mode vectors. This allows a large number of response points to be processed quickly and with minimal user interaction and makes determining the number of modes active in a frequency band easier than with a SISO algorithm. To achieve a global, SIMO AMI algorithm, a fast new linear-least-squares, frequency domain, global, SIMO (or common denominator MIMO), multi-degree of freedom (MDOF) fitting algorithm, similar to the LSCF algorithm is derived and implemented in AMI. This implementation of AMI is then compared with the stochastic subspace identification algorithm (SSI) on noise contaminated analytical data. Some practical issues in applying the SSI algorithm are discussed. Monte Carlo simulations are used to compare the natural frequencies and damping ratios found by AMI and SSI for noise contaminated, synthetic data for a frame structure. The results for the analytical structure also illustrate the difficulty in processing data from a structure having modes with a large range of decay rates. Finally, application to data from the Z24 highway bridge in Switzerland demonstrates the speed and flexibility of the algorithm, and validates its performance on experimental data. [Copyright &y& Elsevier]

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]

Structures with mechanical joints are difficult to model accurately. Even if the natural frequencies of the system remain essentially constant, the damping introduced by the joints is often observed to change dramatically with amplitude. Although models for individual joints have been employed with some success, accurately modeling a structure with many joints remains a significant obstacle. To this end, Segalman proposed a modal Iwan model, which simplifies the analysis by modeling a system with a linear superposition of weakly-nonlinear, uncoupled single degree-of-freedom systems or modes. Given a simulation model with discrete joints, one can identify the model for each mode by selectively exciting each mode one at a time and observing how the transient response decays. However, in the environment of interest several modes may be excited simultaneously, such as in an experiment when an impulse is applied at a discrete point. In this work, the modal Iwan model framework is assessed numerically to understand how well it captures the dynamic response of typical structures with joints when they are excited with impulsive forces applied at point locations. This is done by comparing the effective natural frequency and modal damping of the uncoupled modal models with those of truth models that include nonlinear modal coupling. These concepts are explored for two structures, a simple spring-mass system and a finite element model of a beam, both of which contain physical Iwan elements to model joint nonlinearity. The results show that modal Iwan models can effectively capture the variations in frequency and damping with amplitude, which, for damping, can increase by as much as two orders of magnitude in the microslip regime. However, even in the microslip regime the accuracy of a modal Iwan model is found to depend on whether the mode in question is dominant in the response; in some cases the effective damping that the uncoupled model predicts is found to be in error by tens of percent. Nonetheless, the modal model captures the response qualitatively and is still far superior to a linear model. [ABSTRACT FROM AUTHOR]

Quantitative studies of material properties and interfaces using the atomic force microscope (AFM) have important applications in engineering, biotechnology, and chemistry. Contrary to what the name suggests, the AFM actually measures the displacement of a microscale probe, so one must determine the stiffness of the probe to find the force exerted on a sample. Numerous methods have been proposed for determining the spring constant of AFM cantilever probes, yet most neglect the mass of the probe tip. This work explores the effect of the tip mass on AFM calibration using the method of Sader (1995, "Method for the Calibration of Atomic Force Microscope Cantilevers," Rev. Sci. Instrum., 66, pp. 3789) and extends that method to account for a massive, rigid tip. One can use this modified method to estimate the spring constant of a cantilever from the measured natural frequency and Q-factor for any mode of the probe. This may be helpful when the fundamental mode is difficult to measure or to check for inaccuracies in the calibration obtained with the fundamental mode. The error analysis presented here shows that if the tip is not considered, then the error in the static stiffness is roughly of the same order as the ratio of the tip's mass to the cantilever beam's. The area density of the AFM probe is also misestimated if the tip mass is not accounted for, although the trends are different. The model presented here can be used to identify the mass of a probe tip from measurements of the natural frequencies of the probe. These concepts are applied to six low spring-constant, contact-mode AFM cantilevers, and the results suggest that some of the probes are well modeled by an Euler-Bernoulli beam with a constant cross section and a rigid tip, while others are not. One probe is examined in detail, using scanning electron microscopy to quantify the size of the tip and the thickness uniformity of the probe, and laser Doppler vibrometry is used to measure the first four mode shapes. The results suggest that this probe's thickness is significantly nonuniform, so the models upon which dynamic calibration is based may not be appropriate for this probe. [ABSTRACT FROM AUTHOR]

It has recently been shown that shear wave speed in tendons is directly dependent on axial stress. Hence, wave speed could be used to infer tendon load provided that the wave speed-stress relationship can be calibrated and remains robust across loading conditions. The purpose of this study was to investigate the effects of loading rate and fluid immersion on the wave speed-stress relationship in ex vivo tendons, and to assess potential calibration techniques. Tendon wave speed and axial stress were measured in 20 porcine digital flexor tendons during cyclic (0.5, 1.0 and 2.0 Hz) or static axial loading. Squared wave speed was highly correlated to stress (r2 avg = 0.98) and was insensitive to loading rate (p = 0.57). The constant of proportionality is the effective density, which reflects the density of the tendon tissue and additional effective mass added by the adjacent fluid. Effective densities of tendons vibrating in a saline bath averaged 1680 kg/m3 and added mass effects caused wave speeds to be 22% lower on average in a saline bath than in air. The root-mean-square error between predicted and measured stress was 0.67 MPa (6.7% of maximum stress) when using tendon-specific calibration parameters. These errors increased to 1.31 MPa (13.1% of maximum stress) when calibrating based on group-compiled data from ten tendons. These results support the feasibility of calculating absolute tendon stresses from wave speed squared based on linear calibration relationships. [ABSTRACT FROM AUTHOR]

The Hurty/Craig-Bampton method in structural dynamics represents the interior dynamics of each subcomponent in a substructured system with a truncated set of normal modes and retains all of the physical degrees of freedom at the substructure interfaces. This makes the assembly of substructures into a reduced-order system model relatively simple, but means that the reduced-order assembly will have as many interface degrees of freedom as the full model. When the full-model mesh is highly refined, and/or when the system is divided into many subcomponents, this can lead to an unacceptably large system of equations of motion. To overcome this, interface reduction methods aim to reduce the size of the Hurty/Craig-Bampton model by reducing the number of interface degrees of freedom. This research presents a survey of interface reduction methods for Hurty/Craig-Bampton models, and proposes improvements and generalizations to some of the methods. Some of these interface reductions operate on the assembled system-level matrices while others perform reduction locally by considering the uncoupled substructures. The advantages and disadvantages of these methods are highlighted and assessed through comparisons of results obtained from a variety of representative linear FE models. [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]

Several reduced-order modeling strategies have been developed to create low-order models of geometrically nonlinear structures from detailed finite element models, allowing one to compute the dynamic response of the structure at a dramatically reduced cost. However, the parameters of these reduced-order models are estimated by applying a series of static loads to the finite element model, and the quality of the reduced-order model can be highly sensitive to the amplitudes of the static load cases used and to the type/number of modes used in the basis. This paper proposes to combine reduced-order modeling and numerical continuation to estimate the nonlinear normal modes of geometrically nonlinear finite element models. Not only does this make it possible to compute the nonlinear normal modes far more quickly than existing approaches, but the nonlinear normal modes are also shown to be an excellent metric by which the quality of the reduced-order model can be assessed. Hence, the second contribution of this work is to demonstrate how nonlinear normal modes can be used as a metric by which nonlinear reduced-order models can be compared. Various reduced-order models with hardening nonlinearities are compared for two different structures to demonstrate these concepts: a clamped-clamped beam model, and a more complicated finite element model of an exhaust panel cover. [ABSTRACT FROM AUTHOR]

• Two different nonlinear models are identified for two experimental test rigs. • The models are used to predict dynamic behavior under sine (-sweep) excitations. • The modal model is accurate for sine excitation close to resonance. • The modal model is sensitive for uncontrolled sweeps if the excitation force drops. • Polynomial nonlinear state-space models highly depend on the used training data. In the present paper, two existing nonlinear system identification methodologies are used to identify data-driven models. The first methodology focuses on identifying the system using steady-state excitations. To accomplish this, a phase-locked loop controller is implemented to acquire periodic oscillations near resonance and construct a nonlinear-mode model. This model is based on amplitude-dependent modal properties, i.e. does not require nonlinear basis functions. The second methodology exploits uncontrolled experiments with broadband random inputs to build polynomial nonlinear state-space models using advanced system identification tools. The methods are applied to two experimental test rigs, a magnetic cantilever beam and a free-free beam with a lap joint. The respective models obtained by either method for both specimens are then challenged to predict dynamic, near-resonant behavior observed under different sine and sine-sweep excitations. The vibration prediction of the nonlinear-mode and state-space models clearly highlight capabilities and limitations. The nonlinear-mode model, by design, yields a perfect match at resonance peaks and high accuracy in close vicinity. However, it is limited to well-spaced modes and sinusoidal excitation. The state-space model covers a wider dynamic range, including transient excitations. However, the real-life nonlinearities considered in this study can only be approximated by polynomial basis functions. Consequently, the identified state-space models are found to be highly input-dependent, in particular for sinusoidal excitations where they are found to lead to a low predictive capability. [ABSTRACT FROM AUTHOR]