Dynamics of Coupled Structures, Volume 4

Experimental component mode synthesis (CMS) seeks to measure the fundamental modes of vibration of a substructure and develop a structural dynamics model of an as-built structural component through modal testing. Experimental CMS has the potential to circumvent laborious and costly substructure model development and calibration in lieu of a structural dynamics model obtained directly from experimental measurements. Previous efforts of interfacing an experimental CMS model with a production finite element code proved cumbersome. Recently an improved “Craig-Mayes” approach casts an experimental CMS model in the familiar Craig-Bampton form. This form is easily understood by analysts and more readily interfaced with non-trivial, discrete finite element models. The approach/work-flow for interfacing an experimental Craig-Mayes CMS model with the Sierra analysis framework is discussed and the procedure is demonstrated on a verification problem.

Cyber Physical Testing or Real Time Hybrid Testing is a Hardware-In-The-Loop approach allowing for tests of structural components of complex machines with realistic boundary conditions by coupling virtual components. The need to actuate the physical interface makes the tests on structural systems challenging. In order to deal with stability and accuracy issues, we propose the use of an Adaptive Feed-Forward Cancellation approach with a Recursive Least Squares (RLS) adaption law for interface synchronization of harmonically excited systems. The interface forces are generated from multiple harmonic components of the excitation force. A RLS adaption law sets the amplitudes and phases of the harmonic interface force components and minimizes the interface gap. One major practical advantage of using a RLS adaption law is that only one forgetting factor has to be chosen compared to other adaption algorithms with various tuning parameters. As a consequence, it is possible to test systems with multiple interface DoF. In order to illustrate the performance and robustness of the proposed testing algorithm, the contribution includes a numerical investigation on a lumped mass system.

In the design of mechanical systems, there are constraints imposed on the vibration of mechanical equipment to limit the vibration transmission into its support structure. To accurately predict the coupled system response, it is important to capture the coupled interaction of the two portions, i.e., the mechanical equipment and the support structure, of the mechanical system. Typically during a design, the analysis of the full mechanical system is not possible because a large part of the system may be non-existent. Existing methods known as Transfer Path Analysis and Frequency Based Substructuring are techniques for predicting the coupled response of vibrating mechanical systems. In this paper, a control based hybrid substructuring approach to Transfer Path Analysis is proposed. By recognizing the similarities between feedback control and dynamic substructuring, this paper demonstrates that this approach can accurately predict the coupled dynamic system response of multiple substructured systems including operating mechanical equipment with a complex vibration source. The main advantage of this method is that it uses blocked force measurements in the form of a power spectral density matrix measured uncoupled from the rest of the system. This substructuring method is demonstrated using a simplified case study comprised of a two-stage vibration isolation system and excited by operating mechanical equipment.

This paper addresses an inverse problem to determine dynamic forces acting on a structure from response data. Data-driven subspace algorithms and the linear regression are used to facilitate the estimation of the state sequences and system parameters. The force identification model is then reasonably established on the basis of the estimated system model. A weighted algorithm based on the homotopy analysis method is employed to discretize the well-known ill-posed problems. Moreover, a criterion based on L-curves is adopted for choosing the level of regularization. Finally, laboratory experiments are presented to demonstrate robustness and effectiveness of the proposed solution technique.

A modified Guyan reduction strategy for response degree-of-freedom (DOF) selection to map experimental normal modes is described and demonstrated. The method employs static load patches, rather than point loads, in regions defined by 3-D elastic elements and other problematic zones on a highly detailed finite element model (FEM). Three key benefits are realized by the methodology, namely (1) definition of a well-posed test-analysis mass (TAM) matrix, (2) application of a previously published residual kinetic energy matrix for definition of appropriate measurement DOFs, and (3) elimination of irrelevant modes from the measured mode set. Improved qualities of the modified Guyan reduction strategy are demonstrated with a problematic spacecraft-type FEM, which cannot be readily treated using classical Guyan reduction methodology.

This work proposes a means whereby weak nonlinearity in a substructure, as typically arises due to friction in bolted interfaces, can be captured experimentally on a mode-by-mode basis and then used to predict the nonlinear response of an assembly. The method relies on the fact that the modes of a weakly nonlinear structure tend to remain uncoupled so long as their natural frequencies are distinct and higher harmonics generated by the nonlinearity do not produce significant response in other modes. Recent experiments on industrial hardware with bolted joints has shown that this type of model can be quite effective, and that a single degree-of-freedom (DOF) system with an Iwan joint, which is known as a modal Iwan model, effectively captures the way in which the stiffness and damping depend on amplitude. Once the modal Iwan models have been identified for each mode of the subcomponent(s) of interest, they can be assembled using standard techniques and used with a numerical integration routine to compute the nonlinear transient response of the assembled structure. The proposed methods are demonstrated by coupling a modal model of a 3DOF system with three discrete Iwan joints to a linear model for a 2DOF system.

Some initial investigations have been published which simulate nonlinear response with almost traditional modal models: instead of connecting the modal mass to ground through the traditional spring and damper, a nonlinear Iwan element was added. This assumes that the mode shapes do not change with amplitude and there are no interactions between modal degrees of freedom. This work expands on these previous studies. An impact experiment is performed on a structure which exhibits typical structural dynamic nonlinear response, i.e. weak frequency dependence and strong damping dependence on the amplitude of vibration. Use of low level modal test results in combination with high level impacts are processed using various combinations of modal filtering, the Hilbert Transform and band-pass filtering to develop response data that are then fit with various nonlinear elements to create a nonlinear pseudo-modal model. Simulations of forced response are compared with high level experimental data for various nonlinear element assumptions.

Substructure decoupling consists in the identification of a dynamic model of a structural subsystem, starting from an experimental dynamic model (e.g. FRFs) of the assembled system and from a dynamic model of a known portion of it (the so-called residual subsystem). The degrees of freedom (DoFs) of the assembled system are partitioned into internal DoFs (not belonging to the couplings) and coupling DoFs. To achieve decoupling, a negative structure opposite to the residual subsystem is added to the assembled system, and compatibility and equilibrium conditions are enforced at interface DoFs. Interface DoFs can include coupling DoFs only (standard interface), additional internal DoFs of the residual subsystem (extended interface), subsets of coupling DoFs and internal DoFs (mixed interface), or a subset of internal DoFs only (pseudo interface). As shown in previous papers, the use of a mixed interface allows to replace some coupling DoFs (e.g. rotational DoFs) with a subset of internal DoFs. Furthermore, qualitative criteria for an appropriate selection of the internal DoFs used to replace unwanted coupling DoFs are stated. In this paper, a procedure to optimally replace coupling DoFs with internal DoFs is developed, using either the Frequency Response Function (FRF) or the transmissibility between internal and coupling DoFs. The procedure is tested on an assembled structure made by a cantilever column with two staggered short arms (residual substructure) coupled to a horizontal beam (unknown substructure).

The dynamic substructuring focus group of SEM organizes sessions on experimental substructuring each IMAC conference and has been doing so for a number of years. Over the last decade, the use of so-called transmission simulators has trended within the community. Transmission simulators are well-modeled parts that fit to the interface of the substructures to be coupled to allow distributed interfaces and relaxation of the coupling conditions by the transmission simulator’s analytical modes at the cost of adding a decoupling step to the substructuring problem. In this paper, the transmission simulator concept is adapted to state-space substructuring. Experimental-analytical substructuring of the focus group benchmark structure, the Ampair A600 wind turbine, is used to verify the methodology.

Predicting dynamic properties of modern industrial systems in their fully assembled state can be a challenging task. Numerical analysis or physical testing of the entire system is often not possible due to its sheer size or computational effort. Various substructuring and model reduction techniques make it possible to analyze the subcomponents separately and reunite their reduced representations to build the entire system later. However, models of all subcomponents are not always available or certain dynamic properties cannot be captured by FEM. Experimentally derived substructures become more and more important to overcome this problem. In this work the Transmission Simulator techniques are applied to a realistic FE model of the Ampair600 wind turbine to explore the chances and limitations of the method. The procedure to obtain the dynamic representation of the entire rotor derived from a hub-one-blade assembly is presented. Starting from an experimental free-free representation of the hub-one-blade assembly the hub is considered as flexible fixture and its FE representation is removed from the assembly. The obtained experimental Craig-Bampton-representation of one blade is then duplicated and circumferentially rotated to build the complete blading of the wind turbine. These three blades are finally coupled to the hub to complete the rotor.

This study is an experimental validation of dual assembly frequency based substructuring method for coupling the rotor components of the Ampair 600 wind turbine (the Substructuring Focus Group’s test bed). The method is applied for the assembly of the hub and three blades using a substitute (also called transmission simulator in modal based substructuring methods) in order to induce the effect of constrained interface of the blades. A method of virtual point transformation is used to tackle the problem of rotational degrees of freedom for interfaces. The hub is chosen to be the substitute and the frequency based substructuring approach is used for development of the full model. Finally, the results of substructuring with different substitutes are compared with the truth model of the rotor assembly.

Lightweight, long-span steel office floors of both new and old construction are commonly susceptible to objectionable footfall-induced vibrations. Frequently, older buildings are renovated for newer purposes that require the removal of partitions that had previously provided some vibration control through added structural damping and stiffness. It can be prohibitive, both economically and operationally, to either retrofit the original structure with more beams, or to add external damping devices such as Tuned Mass Dampers (TMDs). This paper provides three case studies of lightweight, long-span steel buildings, and demonstrates the effectiveness of using interstitial columns to vertically link two or more floors together. In doing so, the mass and/or stiffness of the floor will be increased, which serves to reduce footfall vibration levels.

The effects of occupants over the dynamic properties of a structure have been traditionally modeled showing the human body as lumped masses connected with spring and dampers. A new approach for modeling the effects of a standing person was developed 2 years ago based on control theory. The new approach starts from the assumption that the human body can input energy into the system, changing the response of the structural system. The control theory based models have shown to work for a single occupant. This work presents the modeling of groups of standing people using a controller model. The parameters of the controller are obtained using a probabilistic approach based on Bayesian model updating.

As an alternative to traditional wood framing in residential building construction, cold-formed steel (CFS) framing inherits many advantages of steel construction. However, if not appropriately designed, CFS floors with longer spans and lighter weight are likely to be susceptible to annoying vibrations induced by human activity such as walking. The fundamental frequency is a critical parameter for floor vibration serviceability. In current practice, the floor frequency is evaluated based on the simplification of a floor system to a simply-supported beam, which results in a considerable disparity in frequencies obtained from field tests and evaluation. In this research, based on current construction practice, the CFS floor system is modelled as an orthotropic plate with edges of joist-ends being partially restrained and edges that are parallel to floor joists being either freely or simply supported. The deflection of the partially restrained CFS floor joist is adopted as the admissible function of the plate in the derivation of equations to evaluate the fundamental frequency of the equivalent orthotropic plate. Simplified equations and equivalent rigidity properties are proposed for CFS floor systems.

Active vibration control (AVC) of human-induced vibrations in structures with proof-mass actuators has been subject to much research in recent years. This has predominantly focussed on footbridges and floors and there is some evidence that this research is paving the way for commercial installations of AVC where traditional vibration control measures are not appropriate. However, the design of an AVC system is a complex task because of the influence of actuator dynamics, the contributions from higher frequency modes of vibration and the effect of low and high pass filters that are required to make the control algorithm implementable. This puts the AVC design process beyond the abilities of the vast majority of civil design engineers, even at a scheming stage to approximate what sort of reductions could be achieved by such a system. This paper considers a generalised system and investigates what sort of performance can be achieved in theory by a perfect AVC system, then considers the added complexity of actuator dynamics to demonstrate how this degrades the performance from optimal.

For over 40 years, students, faculty, and staff have been living with an often disturbing level of floor vibration caused by pedestrian traffic in an open plan library area of a suburban middle school. The library extends just beyond a 47 × 52 ft bay framed with a 3 in. slab on metal deck supported by 28 in deep steel joists spanning the short direction and W36 rolled girders in the long direction. As part of an upcoming major renovation, there is interest in exploring cost effective alternatives to reducing the impact of the current conditions. To quantify and understand the existing behavior and make suggestions to alter the behavior, a research study was undertaken. This paper describes the vibration measurements taken, assesses the behavior found including the surprising impact of higher order modes, and explores possible options to mitigate the impact of vibration on the occupants.

Pedestrian bridges may experience significant vibrations under pedestrian traffic and wind loads. Design codes address the vibration limit state levels either by ensuring the frequency ranges associated with typical pedestrian passages are outside the lower fundamental frequencies of the structure or by restricting the maximum accelerations below the limits for pedestrian comfort. This paper discusses vibration serviceability assessment of a highly trafficked local pedestrian bridge based on the field dynamic tests. The selected bridge is a 60-m-long three-span steel structure with a continuous reinforced concrete slab supported on two longitudinal steel girders. First, a finite element model of the pedestrian bridge is developed to obtain the natural frequencies and mode shapes. Then, ambient vibration tests are conducted to validate the modal characteristics of the pedestrian bridge. Next, the dynamic response of the bridge in terms of peak accelerations is determined both experimentally and analytically under various pedestrian excitations. Finally, the implications of the results for the serviceability limit state assessment of the pedestrian bridge are discussed.

Lightweight buildings are sensitive to low-frequency vibrations, making it difficult to construct them in such a way that noise and disturbing vibrations are kept at an acceptable level. In the design of vibration reduction measures, it is desirable to have computational models for predicting the effects of structural modifications. Validations of the models to experimental data have to be performed to ensure reliable predictions. The experimental studies are simplified if full-scale models can be scaled down in size. In the paper, methods for designing scaled experimental models of building structures are discussed. An example, the scaling of a wooden building structure, is presented.

This paper suggests an approach to generate human jumping loads using wavelet decomposition and a database of individual jumping force records. A total of 1201 individual jumping force records of various frequencies were first collected. For each record, every single jumping impulse was extracted and decomposed by DB10 Wavelet into seven levels, and all the related decomposition information was stored into a database. The period of each jumping impulse in the same record was found to follow a normal distribution, and so does the contact ratio. In order to generate a jumping load time history having N impulses, Wavelet coefficients are first randomly selected from the database for different levels. They are then used to reconstruct N impulses by inverse wavelet transform. The periods and contract ratios are then randomly generated according to their probabilistic function. These parameters are assigned to each of the N impulses. The final jumping force time history is obtained by linking all the N impulses end to end. Examples are presented to show the simulation procedure. Due to the application of the Wavelet decomposition, the non-stationary features of the jumping load force in time-frequency domain can be preserved by the suggested simulation approach.

Motivated by the current demands in high-performance structural analysis, and by a desire to better model systems with localized nonlinearities, analysts have developed a number of different approaches for modelling and simulating the dynamics of a bolted-joint structure. However, the types of conditions that make one approach more effective than the others remains poorly understood due to the fact that these approaches are developed from fundamentally and phenomenologically different concepts. To better grasp their similarities and differences, this research presents a numerical round robin that assesses how well three different approaches predict and simulate a mechanical joint. These approaches are applied to analyze a system comprised of two linear beam structures with a bolted joint interface, and their strengths and shortcomings are assessed in order to determine the optimal conditions for their use.

Relative motion at bolted connections can occur for large shock loads as the internal shear force in the bolted connection overcomes the frictional resistive force. This macroslip in a structure dissipates energy and reduces the response of the components above the bolted connection. There is a need to be able to capture macroslip behavior in a structural dynamics model. A linear model and many nonlinear models are not able to predict marcoslip effectively. The proposed method to capture macroslip is to use the multi-body dynamics code ADAMS to model joints with 3-D contact at the bolted interfaces. This model includes both static and dynamic friction. The joints are preloaded and the pinning effect when a bolt shank impacts a through hole inside diameter is captured. Substructure representations of the components are included to account for component flexibility and dynamics. This method was applied to a simplified model of an aerospace structure and validation experiments were performed to test the adequacy of the method.

Bolted are joints are prevalent in most assembled structures; however, predictive models for the behavior of these joints do not yet exist. Many calibrated models have been proposed to represent the stiffness and energy dissipation characteristics of a bolted joint. In particular, the Iwan model put forth by Segalman and later extended by Mignolet has been shown to be able to predict the response of a jointed structure over a range of excitations once calibrated at a nominal load. The Iwan model, however, is not widely adopted due to the high computational expense of implementing it in a numerical simulation. To address this, an analytical, closed form representation of the Iwan model is derived under the hypothesis that upon a load reversal, the distribution of friction elements within the interface resembles a scaled version of the original distribution of friction elements. Additionally, the Iwan model is extended to include the pinning behavior inherent in a bolted joint.

Measured aero engine vibration responses vary in character and amplitude depending on the operating environment and prevailing loading conditions. When these responses are limited to linear dynamics range, existing analysis tools which are mostly targeted at such responses, can be used to analyse and characterise the underlying mechanisms. However when responses are nonlinear, use of the same tools may result in nontrivial errors or may not at all be applicable. Some methods have been put forward with the premise of dealing with nonlinearities in measured data. However, they are often very simplistic and cannot effectively deal with complexities an aero engine environment can produce. With designs getting more complex and non-traditional materials, such as composites, becoming more widely used; strong nonlinearities are becoming a more common occurrence. Therefore from industrial perspective, development of effective tools that can deal with identification, classification and eventually quantification of nonlinear features is a very real and present need.In this paper, a number of scenarios, in which sufficiently strong nonlinear responses have been measured, will be presented. Approaches available to an industrial practitioner in such cases will be discussed. Areas that need development of new methods, as well as the degree of robustness with which such methods need to be deployed, will be outlined from an industrial perspective.

Broadband impact excitation in structural dynamics is a common technique used to detect and characterize nonlinearities in mechanical systems since it excites many frequencies of a structure at once. Non-stationary time signals from transient ring-down measurements require time-frequency analysis tools to observe variations in frequency and energy dissipation as the response evolves. This work uses the short-time Fourier transform to estimate the instantaneous parameters from measured or simulated data. By combining the discrete Fourier transform with an expanding or contracting window function that moves along the time axis, the resulting spectra are used to estimate the instantaneous frequencies, damping ratios and complex Fourier coefficients. This method is demonstrated on a multi-degree-of-freedom beam with a cubic spring attachment. The amplitude-frequency dependence in the damped response is compared to the undamped nonlinear normal modes. A second example shows the results from experimental ring-down measurements taken on a beam with a lap joint, revealing how the mechanical interface introduces nonlinear frequency and damping parameters.

In many engineering applications the influence of a slipping contact interface has a major impact on the experienced damping in the structure. Predicting the generated damping is of uttermost importance to ensure an accurate analysis of the dynamic response of the system. Microslip, during which part of the contact is still stuck, and part is already slipping, plays a significant role in this damping, since many applications experience only this from of energy dissipation during operation. This paper investigates the possibility to capture microslip accurately with an explicit, quasi static modelling approach, where a large amount of traditional friction elements are distributed over a small contact area and a realistic pressure field is applied to reproduce the contact conditions. The resulting predicted hysteresis loops show microslip like behaviour, and the detailed contact mesh allows identifying the underlying nonlinear mechanism.

The work aims at performing the dynamic modal identification of the long-span laminated timber footbridge built on the Marecchia River near Rimini, Italy. A first sine vibration test has been performed adopting a mechanical shaker just after the footbridge construction in 2000 in order to check the structural behavior that has been assumed in the project. A second test has been replicated in 2005 using the same excitation and almost the same test set-up adopted in the first one. The dynamic modal extraction is performed on the FRFs through both the peak picking method, applied together with the half power bandwidth technique, and the circle-fit method. The orthogonal properties of the identified mode shapes are verified through the Modal Assurance Criterion (autoMAC). Finally, the results obtained for the two tests and through the two techniques are mutually compared. It is worth noting that the FRFs, evaluated for different intensity levels of the exciting force, reveal an inherent non-linear behavior of the footbridge. The analyses also show that the first natural frequencies of the structure are included in the frequency range of the human step and this could lead to unpleasant feelings for the pedestrians.

As a consequence of a paper presented by Michael Mistler at the VDI-Baudynamik-Tagung in Kassel, Germany, in April 2015, the authors checked the damping coefficients having been estimated for a footbridge in autumn 2014. Mistler stated that the critical damping ratio estimated from a halfpower bandwidth procedure to be dependent on frequency resolution for low frequency modes. Based on the data presented here this statement can be confirmed. The dependency on frequency resolution was found to be due to the leakage phenomenon on the spectral density. This fact may have been known in the academic world but not in the world of engineers applying OMA in practice. In this paper it is presented how the leakage on the spectral density estimate is affecting the damping estimation through OMA based frequency domain identification. Finally the paper compares the damping estimated in the time and frequency domain from ambient tests, with the damping estimated from the free decays. Unfortunately, bias error on damping values determined from analyses in the frequency domain is worst on low frequency modes usually being the most important ones when dealing with a resonance problem in practice.

This paper provides a critical analysis of the procedures available in the literature for the serviceability assessment of footbridges in unrestricted pedestrian traffic condition. Based on a full probabilistic model of walking-induced forces, Monte Carlo simulations are carried out and the numerically obtained dynamic response is compared with the one provided by simplified procedures. Furthermore, a new technical procedure is proposed which permits a direct and simple evaluation of footbridge maximum acceleration as a function of the expected average pedestrian flow and the modal properties of the footbridge.

Slender and lightweight structures are often unduly responsive to human excitation. The concerns of vibration comfort and safety are strengthened by unexplored human-structure interaction (HSI) phenomena. The present contribution proposes a numerical model for pedestrian excitation including HSI. In addition to the well-known forces induced by a pedestrian on a rigid floor, the pedestrian is represented by a linear mechanical model to simulate the interaction with the supporting structure. Inspired by the body postures assumed during the walking cycle, the mechanical properties are chosen to represent the dynamic behaviour of the human body with one or two legs slightly bent. The effect of HSI on the structural response is evaluated for various footbridge parameters and pedestrian densities. It is shown that by taking into account HSI, the structural response is reduced. Furthermore, it is demonstrated that the unrealistic high acceleration levels as predicted by simplified force models are not reached as the result of HSI. It is concluded that the mechanical interaction with the crowd is relevant for the vertical low-frequency dynamic behaviour of footbridges.

The nonlinear dynamics of a multi-mesh spur gear train is considered in this study. The gear train consists of three spur gears, with one of the gears in mesh with the other two. Dynamic model includes gear backlash in the form of clearance-type displacement functions and time variation of gear mesh stiffness. The system is reduced to a two-degree-of-freedom definite model by using the relative gear mesh displacements as the coordinates. The equations of motion are solved for periodic steady-state response by using Harmonic Balance Method (HBM). The accuracy of the HBM solutions is demonstrated by comparing them to direct numerical integration solutions. Floquet theory is applied to determine the stability of the steady-state solutions. Two different loading conditions, where the system is driven by the middle gear and driven by one of the end gears, are considered. Phase difference between the two gear meshes is determined under each loading condition and natural modes are predicted for each loading condition. The forced response due to the combination of parametric excitation and static transmission error excitation is obtained and effects of loading conditions and asymmetric positioning on the response are explored.

The physical mechanisms of energy dissipation in foam to metal interfaces must be understood in order to develop predictive models of systems with foam packaging common to many aerospace and aeronautical applications. Experimental data was obtained from hardware termed “Ministack”, which has large, unbonded interfaces held under compressive preload. This setup has a solid aluminum mass placed into two foam cups which are then inserted into an aluminum can and fastened with a known preload. Ministack was tested on a shaker using upward sine sweep base acceleration excitations to estimate the linearized natural frequency and energy dissipation of the first axial mode. The experimental system was disassembled and reassembled before each series of tests in order to observe the effects of the assembly to assembly variability on the dynamics. There are some important findings in the measured data: there is significant assembly to assembly variability, the order in which the sine sweeps are performed influence the dynamic response, and the system exhibits nontrivial damping and stiffness nonlinearities that must be accounted for in modeling efforts. A Craig-Bampton model connected with a four-parameter Iwan element and piecewise linear springs is developed and calibrated using test data with the intention of capturing the nonlinear energy dissipation and loss of stiffness observed in experiment.

We study stress wave propagation in two impulsively forced split Hopkinson bar systems: one with a prestressed interface and one with a frictional interface. We first consider only primary wave transmission and reflection, allowing for reduction of the problem to a first-order, strongly, nonlinear ordinary differential equation. A high-order finite element model is then developed and used to validate the results of the primary-pulse model. A spring that hardens with increasing preload is used to model the prestressed interface while an Iwan element is used to model the frictional interface. Using the primary-wave propagation model, we perform nonlinear system identification by matching simulation and experiment results and identify the nonlinear hardening characteristics for the prestressed interface and Iwan parameters for the frictional interface. These parameters are then used in the finite element model to compare the experimentally measured secondary effects with the simulated effects. Our results demonstrate that the primary-wave propagation model can be used as a reduced order model for nonlinear system identification at a fraction of the computational cost of higher-order models.

In the 1960s Rosenberg extended the definition of linear normal modes (LNM) for conservative systems to nonlinear systems: On a Nonlinear Normal Mode (NNM) every degree-of-freedom (DOF) vibrates in unison. Later Shaw and Pierre provided a definition for nonconservative systems and defined NNMs as invariant manifolds in phase space. If the system vibrates on such a manifold all other modes shall remain quiescent for all time. Nowadays there are many publications using the concept of NNMs to investigate systems with polynomial nonlinearities. But until now the upper mentioned definition is mostly used to investigate viscously damped systems. In this paper an oscillator containing a geometrically nonlinear (cubic) spring and a dry friction damper is considered. The system is driven into resonance and decay processes are evaluated. Wavelet analysis is used to identify which frequencies and harmonics remain during the decay process.

An investigation of the influence of constitutive friction laws on the damping behavior of dry friction joints is presented. The modeling starts at the micro- scale of the surface roughness leading to constitutive laws with regard to both the normal and tangential contact. These laws are used in a 3D finite element simulation of a simplified model joint. Numerical simulations show the influence of various levels of sophistication of the used contact laws. Here, the influence of high level contact laws proves to be limited in a spatially resolved joint model.

The objective of this paper is to present an occupant detection method through step-induced structural vibration. Occupant detection enables various smart building applications such as space/energy management. Ambient structural vibration monitoring provides a non-intrusive sensing approach to achieve that. The main challenges for structural vibration based occupant footstep detection include that (1) the ambient structural vibration noise may overwhelm the step-induced vibration and (2) there are various other impulse-like excitations that look similar to footstep excitations in the sensing environment (e.g., door closing, chair dragging, etc.), which increase the false alarm rate for occupant detection. To overcome these challenges, a two-stage step-induced signal detection algorithm is developed to (1) incorporate the structural characteristics by selecting the dominant frequencies of the structure to increase the signal-to-noise ratio in the vibration data and thus improve the detection performance and (2) perform footstep classification on detected events to distinguish step-induced floor vibrations from other impulse excitations. The method is validated experimentally in two different buildings with distinct structural properties and noise characteristics, Carnegie Mellon University (CMU) campus building and Vincentian Nursing Home deployments in Pittsburgh, PA. The occupant footstep detection F1 score shows up to 4X reduction in detection error compared to traditional thresholding method.

The advent of structural building instrumentation invites research into novel applications of such systems. Previous research has shown that the propagative nature of ground impacts on the floor of a building can be assimilated to a thin plate. This research presents results for time-difference of arrival (TDOA) and cross-correlation methods used for source localization in a sparsely instrumented plate Finite Element Model (FEM), where acceleration data is used. The overall accuracy is evaluated for various wave speeds with two different configurations of sensor positions, and the consistency of the perceived wave speeds are assessed by the coefficient of variance. The accuracy of localization is considered, with emphasis on the sign of arrival (SOA), as a method to provide directional inference. Peak-difference and cross-correlation techniques are found to be significantly erroneous, while the SOAs are consistently accurate, with less than 10 % of 126 SOAs being reported incorrectly for all cases. Repositioning the sensors closer to the boundary increased the errors for all methods.

The ability to automatically classify the gender of occupants in a building has far-reaching applications in multiple areas spanning security and threat detection, retail sales, and possibly biometric identification in smart buildings. While other classification techniques provide potential for gender classification, they face varied limitations such as invasion of privacy, occupant compliance, line of sight, and high sensor density. High-sensitivity accelerometers mounted under the floors provide a robust alternative for occupant classification. The authors take advantage of the highly-instrumented Goodwin Hall on the Virginia Tech campus to measure vibrations of the walking surface caused by individual walkers. A machine learning technique known as Support Vector Machines (SVMs) is used to classify gender. In this study, the gait (i.e. walking) of 15 individual walkers (eight male and seven female) was recorded as they, alone, walked down the instrumented hallway, in multiple trials. The trials were recorded via 14 accelerometers which were mounted underneath the walking surface with the placement of the sensors unknown to the walker. A tenfold-cross-validation method is used to comment on the validity of the algorithm’s ability to generalize to new walkers. This work demonstrates that a gender classification accuracy of 88 % is achievable using the underfloor vibration data from the Virginia Tech Goodwin Hall applying an SVM approach.

On civil engineering structures human occupancy is often modeled as a static load. Modeling humans as a static load is a simplification of matters, as will be demonstrated in the paper. The paper addresses the complexity of having both passive humans (sitting or standing) as well as active humans on a structure when evaluating the serviceability limit state of a structure carrying humans. The subject is addressed with offset in numerical modelling and experimental findings and is discussed in relation to existing design codes.

Structural engineers usually consider the building occupants as added weight (or mass). Even though this may be a reasonable assumption for the static analysis and design of structures, it can result in discrepancies between the predicted and measured dynamic responses of a structure. This paper presents the results of a research effort conducted to evaluate the effects of human bodies on the dynamic properties and responses of civil engineering structures. A laboratory test structure and a footbridge were used, and the Component Mode Synthesis technique was applied to compute the structural response. Using a parameter search technique, the dynamic properties of an equivalent single-degree-of-freedom lumped mass model to represent groups of people in different postures on a test structure, were found. The model was used to compute the dynamic properties and response of a footbridge when occupied by people. Comparison of the results with the measured properties and responses of the footbridge demonstrated the validity of the human dynamic model.

In a wake of Hillsborough disaster of 1989, all stadia hosting major sport championships in the UK were converted from terraced to all-seated. Driven by spectators’ demands for improving the quality of their experience and organisers’ interest in increasing the capacity of their venues, a debate has arisen recently about the possible introduction of safe standing areas. Some issues have been already highlighted, mainly related to security aspects and comfort. This study investigates how the introduction of safe standing areas and the expected increase of the density of a crowd could impact the dynamic loading induced by the spectators and the resulting structural response. To this end an experimental campaign has been conducted in a laboratory delving into the effects of common actions performed by seated and standing cheering spectators. The data on dynamic behaviour of a lively test structure in both conditions have been collected, simultaneously with data on behaviour of the spectators. The forces applied by the spectators have been inferred using inverse dynamics, by analysing the structural response. The results are presented in the context of human-to-structure interaction and human-to-human coordination.

Unlike most of civil engineering structures whose static and dynamic responses are estimated accurately through several codes and guidance, stadiums reserve a distinctive place especially when it comes to their dynamic behavior. This difference takes its source from several factors such as influence of crowd size, motion and slenderness of the structure. The most noticeable form of this difference shows itself as excessive vibration levels which is actually a threat to the serviceability of these structures. Eventually, it becomes essential to carefully evaluate several steps of this particular problem starting from correct representation of crowd activity through accurate loadings and human-structure interaction models to arranging acceptable vibration serviceability limits. This publication intends to point out the newly developed techniques and discovered issues on several stages of the problem during the last decade.

Developing reliable stochastic models of people walking and jumping is of crucial importance for accurate vibration serviceability assessment of structures such as footbridges, building floors and grandstands. To inform stochastic modelling, experiments which observe the kinetic and kinematic features of human actions should be conducted. The objective of this paper is to present a laboratory-based experimental programme designed to characterise walking and jumping actions performed on rigid surfaces by a population of 8–10 test subjects. A detailed characterisation of intra-subject variability was conducted in order to quantify randomness of parameters, such as pacing frequency, step length and step width in case of walking and the frequency, impulse area and contact ratio in case of jumping. The walking locomotion parameters on a lively surface were also monitored and compared against the benchmark data collected on a rigid surface. It was observed that an increase in the liveliness (in the vertical direction) of the supporting surface tends to lead to an increase in the intra-subject variability. In addition, it was shown that neglecting intra-subject randomness in the human-induced force could lead to significant error in calculation of the vibration response.

This study formulates and numerically solves the optimal restraint condition problem for the SID-IIs side impact crash dummy under given impact conditions. It extends a previous study (Shi, Y., Wu, J., Nusholtz, G.S.: ASME J. Dyn. Syst. Meas. Control 135, 031007-1–031007-8, 2013) on the optimal restraint for this dummy which has the peak of the thoracic compression as the objective function. This extension allows for the peak of the different responses of the dummy, or their weighted average to be used as the objective function to reflect different strategies for reducing the impact load on the dummy. The requirements of the FMVSS 214 regulation are considered with these formulations. At the center of these formulations is a spring-mass model of the SID-IIs dummy established previously. The loading on the dummy, i.e., the restraint condition, is optimized through a discretization scheme which reduces the problem to a linear programming problem. Numerical examples are presented which illustrate the numerical effectiveness of the formulation. The results from these identify the optimal restraint action under different objectives and different impact conditions. They also combine to provide insight on some fundamental response characteristics of this dummy, such as the relationship between thoracic loading and pelvic loading. Additional optimization results are presented which identify the minimum restraint space requirement under given impact conditions. Such information can be referenced in practical engineering of side impact protection.

In this paper, three common model updating approaches are considered to evaluate their efficiencies for model calibration on a component of an Ampair 600W wind turbine. This turbine is a test bed for the Substructuring Focus Group of the Society of Experimental Mechanics. The fin of the wind turbine is chosen to be investigated in updating procedures. Firstly, model updating method has been developed using an objective function based on extracted eigenfrequencies. The objective function is chosen as a sum of weighted eigenfrequencies with respect to their importance. In the second procedure, weighted Modal Assurance Criteria (MAC) are used to define the objective function. Finally, measured and simulated Frequency Response Functions (FRFs) are directly used to find the best correlated Finite Element (FE) model. The results of three updating procedures are then compared to measured eigenfrequencies, mode shapes and FRFs, and the evaluation of the performance of these three different model updating methods is discussed.

In many engineering application there are many instances in which it is convenient to be able to consider a complex engineering structure as an assembly of simpler components or substructures. Similarly, there exist applications in which, for model validation purposes, it might be important to identify the dynamic behavior of the structural subsystem starting from the known dynamic behavior of the coupled system and from information about the remaining part of the structural system. However, if the theoretical framework for Frequency Based Substructuring (FBS) has been widely studied and demonstrated, measurement errors, ill-conditioning and difficulties in measuring all required degrees of freedom—in particular at the connections—lead to poor results when trying to apply these techniques to real structures. This paper will focus on the analysis of the results obtained by applying Experimental Frequency-Based Substructuring on a test structure, both for coupling and decoupling applications and under different boundary conditions. The paper will particularly discuss the effects of typical measurement errors on the final results and potential techniques that could be used to improve the robustness and applicability of this methodology.

Substructuring methods are well known and are widely used in predicting dynamics of coupled structures. In theory, there is no reason why the same techniques could not be used in a reverse problem of predicting the dynamic behavior of a particular substructure from the knowledge of the dynamics of the coupled structure and of all the other substructures. However, the reverse problem, known as decoupling, usually requires matrix inversions, and therefore even small measurement errors may easily affect the accuracy of such methods. In this study two new FRF based approaches are presented for decoupling. The methods proposed require coupled system FRFs at coordinates that belong to the known subsystem as well as the measured or calculated FRFs of the known subsystem alone. Formulations are based on the reverse application of the structural coupling method proposed in a previous publication co-authored by one of the authors of this paper. The performances of the proposed methods are demonstrated and then compared with those of some well-known recent methods in the literature through a case study.

The focus of this paper is on continuing the experimental/modeling investigation of the Brake-Reuß beam which was initiated a year ago as part of the NOMAD program at Sandia National Labs. The ultimate goal of the overall effort is to (1) determine the parameters of joint models, in particular the Iwan model in its modal form, from well delineated tests and (2) extend this approach to identify statistical distributions of the model parameters to account for joint uncertainty. The present effort focused on free response of the beam resulting from an impact test. The use of this data in conjunction with the Hilbert transform is shown to provide a straightforward framework for the identification of the joint model parameters at the contrary of the forced response data used earlier. The resulting frequency and damping vs. amplitude curves are particularly conducive to a Iwan-type modeling which is demonstrated. The curves also show the effect of the bolt torque on the joint behavior, i.e., increase in natural frequency, linear limit, and macroslip threshold. Macroslip is shown to have occurred in some of the tests and it is concluded from ensuing testing that this event changed the nature of the jointed beams. Specifically, the linear natural frequency (observed under very low level impact test) shifted permanently by 20 Hz and, in one case, the linear natural frequency was observed to decrease with increasing bolt torque level in opposition to other beams and physical expectations. An analysis of the joint surface strongly suggest that a significant plastic zone developed during the macroslip phase which induced the above unusual behaviors.

This paper continues the investigation from a paper presented at IMAC XXXIII that looked into the influence of various experimental setups on the nonlinear measurements of structures with mechanical joints. The previous study reported how the system stiffness and damping was affected by the force input method, boundary conditions and measurement techniques. However, during the stepped sine excitation experiments the parameters for the control schemes were neglected. In this paper, different control strategies, namely force and acceleration control, are used to observe how the parameters affect the measurements at different levels of excitation. The experiments are conducted on bolted beams containing a lap joint with different boundary conditions. The beams are excited by a shaker using a stepped sine signal using narrow bandwidths around three of the natural frequencies. The results show that acceleration amplitude control can produce cleaner transfer functions compared to the force amplitude control method.

This paper examines the mechanical response of a simple bolted joint, the Brake–Reuß $$\ss$$ beam, under shock loading. This is done by creating a high-fidelity finite element model of the beam and subjecting it to a quasi-static bolt load followed by a dynamic shock load. The influence of several parameters on the beam’s response is studied, which include impact force, impact duration, impact location, and residual stress. The results indicate that when the energy input into the beam is held constant, the most influential parameter is the shock’s frequency and that increasing its frequency significantly increases dissipation. The next most influential parameter is the impact location, though its effect is frequency dependent and becomes stronger for higher frequencies. Finally, the results show that while residual stresses can significantly modify the contact-pressure distribution, they have minimal influence on the energy dissipated due to friction resulting from shock loading.

Experimental dynamic substructuring is a means whereby a mathematical model for a substructure can be obtained experimentally and then coupled to a model for the rest of the assembly to predict the response. Recently, various methods have been proposed that use a transmission simulator to overcome sensitivity to measurement errors and to exercise the interface between the substructures; including the Craig-Bampton, Dual Craig-Bampton, and Craig-Mayes methods. This work compares the advantages and disadvantages of these reduced order modeling strategies for two dynamic substructuring problems. The methods are first used on an analytical beam model to validate the methodologies. Then they are used to obtain an experimental model for structure consisting of a cylinder with several components inside connected to the outside case by foam with uncertain properties. This represents an exceedingly difficult structure to model and so experimental substructuring could be an attractive way to obtain a model of the system.