Topics in Modal Analysis I, Volume 5

Modal analysis classicaly used signals that respect the

Shannon/Nyquist

theory. Compressive sampling (or Compressed Sampling, CS) is a recent development in digital signal processing that offers the potential of high resolution capture of physical signals from relatively few measurements, typically well below the number expected from the requirements of the

Shannon/Nyquist

sampling theorem. This technique combines two key ideas: sparse representation through an informed choice of linear basis for the class of signals under study; and incoherent (eg. pseudorandom) measurements of the signal to extract the maximum amount of information from the signal using a minimum amount of measurements. We propose one classical demonstration of CS in modal identification of a multi-harmonic impulse response function. Then one original application in modeshape reconstruction of a plate under vibration. Comparing classical

$$ {\ell_2} $$

inversion and

$$ {\ell_1} $$

optimization to recover sparse spatial data randomly localized sensors on the plate demonstrates the superiority of

$$ {\ell_1} $$

reconstruction (RMSE).

Given measured data as estimated frequency responses of a quasi-linear system, there is a variety of system identification methods that identify a state-space model that gives good correlation to the data. Such methods are the N4SID and the PolyMAX methods. Using these methods, a key problem is to select the proper model order. In this work we investigate a method for the automatic detection of proper model order. The method is based on the statistical evaluation of an ensemble of state-space models all identified from the same basic set of frequency response functions, but with different realizations based on a bootstrapping scheme. We apply the method to real test data.

Stochastic subspace identification methods are an efficient tool for system identification of mechanical systems in Operational Modal Analysis, where modal parameters (natural frequencies, damping ratios, mode shapes) are estimated from measured ambient vibration data of a structure. System identification is usually done for many successive model orders, as the true system order is unknown. Then, identification results at different model orders are compared to distinguish true structural modes from spurious modes in so-called stabilization diagrams. These diagrams are a popular GUI-assisted way to select the identified system model, as the true structural modes tend to be stable for successive model orders, fulfilling certain stabilization criteria that are evaluated in an automated procedure. In Operational Modal Analysis of large structures the number modes of interest as well as the number of used sensors can be very large, thus leading to high model orders that have to be considered for system identification. This also means a big computational burden. Recently, an efficient approach to estimate system matrices at multiple model orders in Stochastic Subspace Identification was proposed. In this paper it is shown how this new “Fast SSI” improves the computation of the stabilization diagrams, leading to much faster system identification results for large systems. The Fast SSI is applied to the system identification of some relevant large scale industrial examples.

Damping in mechanical systems is understood to be the irreversible transition of mechanical energy into other forms of energy as found in time-dependent processes. Damping is mostly associated with the change of mechanical energy into thermal energy. Damping can also be caused by releasing energy into a surrounding medium. Electromagnetic and piezoelectric energy conversion can also give rise to damping if the energy converted is not returned to the mechanical system.

The tutorial classifies damping phenomena, explains their physical cause and provides phenomenological descriptions starting from the level of constitutive equations with parameters, that need to be identified from experimental techniques. The step from material properties to members with damping properties is outlined. Damping in assemblies such as laminated members, damping in structural joints, damping due to fluids and damping by squeezing are prescribed. Models for damped structures are provided by the Finite Element Method and the Boundary Element Method.

Experimental techniques for the determination of damping characteristics including Experimental Modal Analysis and experimental techniques for the damping measurement of subsoil are described.

The tutorial incorporates the Guideline VDI 3830 “Damping of Materials and Members” and points out by citing 5 peer reviewed journal articles written by the tutor L. Gaul and his coworkers, where progress in damping description has been achieved after the Guideline was finished.

The practical application of damping description is provided by the design of turbogenerators in which the tutor was involved. It includes to distinguish between internal and external damping and the difference in stability of a rotor. Contact-surface damping is considered between shrink connections of shaft and disc, as well as between disc and blades. Damping in journal bearings is described. Interface damping in bolted joints split into microslip and macroslip behavior between the engine housing and the super plate of a turbogenerator frame foundation is modeled and compared to material damping of concrete and steel frame foundations. Finally the radiation (geometrical) damping caused by waves emanating from vibrating base plate into the subsoil by longitudinal and transverse body waves and Rayleigh surface waves are considered.

Finite element analyses are a powerful engineering tool that can yield insight into the behavior of engineering systems that experimentation may not accurately yield. Including damping in these analyses can yield more accurate models, but damping is often neglected due to difficulties in determining which damping model to use, and the appropriate parameters for the models. This research explored the relationship between physical damping and its implementation in Abaqus, with a focus on investigating the different effects of different damping models. The three damping models investigated were mass-proportional, stiffness-proportional, and viscoelastic. The parameters of the models were calibrated for a particular material with experimental data from a simple “calibration structure.” Preliminary testing on the validity of the models was completed by comparing experimental results of a more complex “validation structure,” made of the same material, with simulation results generated using the calibrated damping parameters.

Damping and loss factor assessment of high damped materials is a challenging task addressed in the past by several researchers, one of the most important is H. Oberst who defined a standard method to tackle the issue.

In this paper a sandwich beam composed of two aluminium layers separated by the damping material has been studied. First the beam made only of aluminium layer has been tested and then the composite beam has been assembled by using as damping layer a sheet of styrene-butadiene rubber (SBR). The highly damped behaviour of the test sample makes the test very difficult; great care must be taken in the setting up the test itself.

In particular the excitation system has been studied in depth by comparing the results obtained with a traditional non-contact impact test and an electro-dynamic system. Being the impact excitation invasive as well when considering light structures as the object under test, a non-contact electro-magnetic system has been developed in order to limit the influence of the excitation device on the structure behaviour.

Issues regarding occupancy comfort in vibration-sensitive structures are the motivation of this study concerning wind-induced vibrations in the European Court Towers in Luxembourg. In one of the two identical towers tuned liquid dampers (TLD) have been installed. Recent studies investigate the changes in the dynamic behavior of the artificially damped tower, before and after the activation of TLD's. This paper widens these investigations and exploits the benefit of having two similar full-scale structures with dissimilar damping characteristics. The main subject of interest in the investigation is the difference in vibration level as well as the modal damping ratios of the towers at various excitation levels. The experimental measurements that form the basis of this paper were conducted during a period of two weeks in the fall of 2010.

The modal parameters are extracted using operational modal analysis at different wind speeds and directions. This paper presents the experimental setup and the results of the analysis regarding the differences in response and modal damping ratios of the two towers under varying excitation levels.

Under natural stochastic excitations, responses of time-varying structures are always nonstationary stochastic signals, of which spectra change with time. This paper studies the time-dependent power spectrum density based on time-frequency analysis. Based on the time-dependent power spectrum density, a mathematical model of the time-frequency-domain two-step least square modal parameter estimator for time-varying structures is presented. In the first-step estimation, the modal parameters at each time instant are estimated using the least square complex frequency-domain method. Furthermore, the estimated modal parameters are sifted and sorted. Based on the sifted and sorted modal parameters, the time-varying resonance frequency, damping ratio and operational mode shapes are estimated in the second-step estimation. A numerical simulation example and a group of experiments validate this two-step least square modal parameter estimator for time-varying structures.

Experimental modal testing has been performed for several decades as an approach to identify system characteristics. This involves a controlled test where the applied force and response are measured to identify frequency response functions which are processed to obtain modal parameters. More recently there has been a trend to utilize output only measurement approaches to extract similar information. The main advantage is that the structure need not be brought to the test lab and operating responses can be used to extract similar modal parameters. However, the operating approaches may not always obtain the same modes as those obtained from the traditional modal test.

This paper presents a study of the modal parameters obtained from both traditional modal testing and operational modal analysis approaches to compare the two sets of estimated parameters. Data from several structures are compared to show the differences between the two approaches.

While the multiple-input, multiple-output, multiple degree of freedom, modal parameter estimation algorithm implemented on a stability (or consistency) diagram is the workhorse of today’s “curve-fitting” software, there are still occasions on which a simple single degree of freedom method can be useful—for example, when extracting a weakly excited mode hiding among the towering peaks, or while tracking a heavily damped control surface mode as the force level increases. The method described in this paper is a single-input, single-output, single degree of freedom, frequency domain, rational fraction polynomial algorithm that uses a generalized residual polynomial to account for the other modes near the mode of interest. A consistency diagram is generated not by varying the model order of the characteristic equation, which would defeat the purpose of an SDOF method, but by varying the order of the residual polynomial. This algorithm is implemented as a supplement to a traditional multiple-reference, MDOF method, and the SDOF results can be combined with its results for computing multiple reference residues. The algorithm for obtaining the modal participation factors from the SDOF method is also given.

When the modal parameters are changed quickly with time, they need to be identified at many moments to obtain the changing trend. To complete all the analysis manually is a very tremendous work, so the autonomous analysis is obligatory. Besides, for the test in which parameters are always changed with time, the data for computing FRF is short and excitation force is continuous. All these things make it difficult to obtain the precise FRFs which are necessary for autonomous modal analysis. To obtain the precise FRFs, in the paper, a direct time domain devolution method is presented to calculate the Impulse Response Functions (IRFs) for the first time. In the devolution algorithm, time-consuming inversion of matrix with large size is necessary. To avoid the matrix inversion, an effective iterative algorithm is put forward which can speed the calculation greatly at the cost of little accuracy reduction. In order to complete the modal analysis automatically, first, the modal analysis is completed manually at initial time and this analysis result will be as a preliminary reference model, the modal analysis of other times are completed total automatically. In the process, the previous analysis result is as the reference model, the stability diagram of current time is obtained by some methods such as ERA, PRCE, PolyMAX. Some poles are selected automatically by computing the pole weighted Modal Assurance Criterion (pwMAC) with the reference model. At the end of this paper, the real engineering example is introduced and the automatic analysis results are promising.

Impulse response functions (IRFs) and frequency response functions (FRFs) are bases for modal parameter identification of single-input, single-output (SISO) and multiple-input, multiple-out (MIMO) systems, and the two functions can be transformed from each other using the fast Fourier transform and the inverse fast Fourier transform. An efficient iterative algorithm is developed in this work to directly and accurately calculate the IRFs of SISO and MIMO systems in the time domain using relatively short input and output data series. The iterative algorithm can avoid the time-consuming inversion of a large matrix in the conventional least-square method for calculating an IRF, greatly reducing the computation time. In addition, a fitting index and an error energy decreasing coefficient are introduced to evaluate the accuracy in calculating an IRF and to provide the termination criterion for the iterative algorithm. A new coherence function is also introduced to evaluate the accuracy of calculated IRFs and FRFs at different spectral lines. Two examples are given to illustrate the effectiveness and efficiency of the methodology.

A local solve method will be presented for extracting modal parameters from inconsistent data. By definition global parameter estimation methods cannot handle inconsistent Frequency Response Function (FRF) data (frequency shifts, non-linearity’s, etc.) and in practice it is very difficult to select appropriate poles from the stability or consistency diagrams presented in commercially available modal parameter estimation methods. The typical way to resolve this issue is to employ measurement techniques that acquire all FRF’s simultaneously, requiring shakers, numerous accelerometers and a large channel count acquisition system; or performing a Roving hammer, Multiple Reference Impact Test (MRIT). The reality is that sometimes FRF data is not acquired in a consistent manner. This paper presents a “local solve” method that performs a global solve on individual or groups of consistent FRF’s and combines the end result into a set of global modal parameters.

The notion of transmissibility, in what mechanical vibrations are concerned, may lead to various potential applications. Depending on the related quantities, responses or forces, it is possible to use different formulations and to deduce important properties. All these aspects are overviewed in the present article, where transmissibility can be oriented to the evaluation of unmeasured frequency response functions, to the estimation of reaction forces and, with special focus here, to the detection of damage in a structure. This important application of the transmissibility concept leads to the damage detection phase. To accomplish such a goal, the authors present some indicators and numerical simulations are given to illustrate the theoretical developments.

Mechanical measurements are commonly performed by discretizing a continuum in a set of acquisition positions at which a transducer can measure a response to a stimulus. Mechanical behaviours of a test piece can be represented and understood by information obtained from these discrete positions. The discretization of a continuum is required because of the measurement methods, which are often developed using transducers at fixed locations both with contact and contact-less types. Measurement systems, such as Scanning Laser Doppler Vibrometer (SLDV) one, are becoming more used in dynamic testing thanks to the ability of measuring vibrations remotely by moving the laser spot anywhere on a test structure. Contact-less transducers of this type can measure responses either at fixed positions or by sweeping continuously a test area. The latter measurement technique is called Continuous Scanning. This work is focussed on the advanced modal testing using Continuous Scanning LDV methods, where the modal properties of a test structure can be measured very efficiently and rapidly by using these methods. The continuous scanning is a measurement method which can move away from the discretization of a measurement space thanks to the capability of measuring meaningful parameters.

Detailed linear finite element simulations and accurate modal testing techniques ensure a reliable validation of linear dynamic response predictions of engineering structures. The good agreement between simulation and measurement for single components is often diminished when an assembly is considered, since nonlinear effects of the joints influence the response behaviour. To re-establish the agreement between analysis and measurement the nonlinear behaviour must be included in the simulation. Analysis tools are available today to take these nonlinear effects into account which require accurate input parameters, to represent the nonlinear contact interface.

Research at Imperial College London has focused on the reliable measurement of the dynamic friction contact parameters for over a decade. The extraction of the dynamic friction coefficient, μ, and the tangential contact stiffness, k

t

, requires the measurement of the nonlinear hysteresis loop with a minimum interference from the test rig. A newly developed friction, with a high test accuracy and a large test range will be presented and its behaviour compared to previous data.

The poly-reference Least Squares Complex Frequency-domain (pLSCF) estimator –commercially known as the LMS PolyMAX method- has introduced an improvement in the field of modal analysis. The main advantages are its computational speed and the very clear stabilization diagrams it yields even in the case of highly damped systems and noisy FRF measurements. Moreover, the numerical stability of the algorithm allows for a large-bandwidth and high-model order analysis and makes it suitable both for lowly- and highly-damped structures.

Regardless of the very clear stabilization diagrams, it has been observed that the damping estimates are not always reliable for highly damped modes when the FRFs are very noisy. In this paper, a two-step procedure is introduced to improve the damping estimates while maintaining the very clear stabilization diagrams. In the first step, a kind of smoothing is used to remove the noise from the data and in the second step; the pLSCF estimator is applied to the smoothed data resulting in improved (damping) estimates. In addition, the proposed procedure properly deals with uncertainty: the data variance due to measurement noise is taken into account to compute confidence bounds on the modal parameter estimates. The procedure is validated by means of simulated as well as experimental data. The presented procedure to process highly damped noisy vibration data leads to very accurate estimates in comparison to the traditional pLSCF approach.

Cepstral methods of modal analysis offer two advantages with respect to conventional methods. The first is that they give both poles and zeros of the transfer function, and thus most of the information about the relative scaling of the residues of adjacent modes. Fully scaled modes can be obtained using a minimum of extraneous information, which can be provided for example by a finite element (FE) model of the structure, which does not have to be very accurate. The other advantage is that for single input, multiple output (SIMO) systems, the cepstrum of the responses is the sum of the cepstra of the forcing and transfer functions, and provided the spectrum of the force is reasonably smooth (on a log scale) the corresponding cepstrum is very short and the higher quefrency part of the cepstrum is completely dominated by the transfer function and can be curve-fitted for its poles and zeros. This is a much weaker restriction than the assumption of most techniques that the excitation is white. The above properties of the cepstrum apply only to SIMO systems and in the normal MIMO situation one possibility is to separate the responses to a single input at each measurement point. The methods available for this include blind source separation (BSS) techniques, for convolutively mixed systems. An exciting possibility is where there is just one second order cyclostationary source with a particular cyclic frequency such as with a diesel railcar. The responses to this single source can be separated in the cepstrum of the spectral correlation function. A very recent development is the possibility of performing editing of time signals using the real cepstrum instead of the complex cepstrum. The latter contains all information about the phase, but only if it can be unwrapped to a continuous function of frequency, and this is not possible for stationary forcing or response functions. For many applications, the original phase can be combined with the edited amplitude information from the real cepstrum, even for general stationary responses.

Second Order Blind Source Separation (SO-BSS) techniques possess several mathematical characteristics making them a viable option for Operational Modal Analysis (OMA). However, on closer scrutiny it is revealed that there are certain subtleties that limit their direct application to OMA applications. This paper continues from past work of the authors, which focussed on understanding SO-BSS techniques from a perspective of OMA applicability and developing SO-BSS based algorithm for OMA. In this paper, a new algorithm is proposed that overcomes the inherent limitations of SO-BSS algorithms with regards to their applicability to OMA. These limitations include applicability to heavily damped systems, identification of complex modes, and applicability to scenarios where number of available sensors is lesser than the number of modes to be estimated, etc. The algorithm’s advantage over original form of SO-BSS is demonstrated by means of an analytical example.

An effective identification method is developed for the determination of modal parameters of a structure from its measured ambient nonstationary vibration data. It has been shown in a previous paper of the authors that by assuming the ambient excitation to be nonstationary white noise in the form of a product model, the nonstationary response signals can be converted into free-vibration data via the correlation technique. In the present paper, if the ambient excitation can be modeled as a nonstationary white noise in the form of a product model, then the

nonstationary

cross random decrement signatures of structural response evaluated at any fixed time instant are shown theoretically to be proportional to the nonstationary cross-correlation functions. The practical problem of insufficient data samples available for evaluating nonstationary random decrement signatures can be approximately resolved by first extracting the amplitude-modulating function from the response and then transforming the nonstationary responses into stationary ones. Modal-parameter identification can then be performed using the Ibrahim time-domain technique, which is effective at identifying closely spaced modes. Numerical simulations confirm the validity of the proposed method for identification of modal parameters from nonstationary ambient response data.

Extraction of robust, damage-sensitive, low-dimensional features lies at the heart of any structural health monitoring (SHM) process. Symbolic dynamics provides a largely unexplored framework for extracting features applicable to multiple damage types and structural platforms. Dynamic systems theory dictates that dynamical systems (continuous-valued in continuous time) have a fully isomorphic (same structure) representation in symbolic dynamics (discrete-time, discrete-alphabet digital streams). Discretization occurs through a process called partitioning, and practical algorithms for partitioning noisy data have been recently developed. The symbolic transformation greatly reduces data-handling requirements while maintaining full data fidelity, and allows the use of sophisticated algorithms from information theory and communications engineering for statistically classifying the features. Thus, utilization of symbolic data and analysis techniques in SHM systems can greatly reduce memory requirements and necessary computational power which can be of great benefit for use in SHM network nodes employing wireless data transfer and energy harvesting technologies. In this preliminary study, a symbolic dynamics-based damage detection algorithm using a partitioning scheme rooted in extreme value statistics is proposed and experimentally validated.

The fundamental period of vibration plays a primary role for the assessment of the seismic demand on structures. It can be evaluated by numerical analyses, or even according to basic formulations provided by building codes for common structural typologies.

In the present paper, the results of an extensive experimental campaign based on output-only modal identification of masonry towers in the Molise region (Southern Italy) are presented. The collected data, including information about geometry and material properties of the towers, have been analyzed and combined with those derived from an extensive literature review, to identify the main parameters influencing the value of the fundamental period of Italian masonry towers. The resulting database has been useful first of all to check the effectiveness of available formulations for prediction of the fundamental period of masonry towers reported in building codes. Moreover, correlations between the fundamental period of masonry towers and relevant geometric parameters are illustrated and discussed in order to develop a novel formulation providing the fundamental period of Italian masonry towers as a function of the height.

The launch environment is a challenging regime to work due to changing system dynamics, changing environmental loading, joint compression loads that cannot be easily applied on the ground, and control effects. Operational testing is one of the few feasible approaches to capture system level dynamics since ground testing cannot reproduce all of these conditions easily. However, the most successful applications of Operational Modal Testing involve systems with good stationarity and long data acquisition times.

This paper covers an ongoing effort to understand the launch environment and the utility of current operational modal tools. This work is expected to produce a collection of operational tools that can be applied to non-stationary launch environment, experience dealing with launch data, and an expanding database of flight parameters such as damping. This paper reports on recent efforts to build a software framework for the data processing utilizing existing and specialty tools; understand the limits of current tools; assess a wider variety of current tools; and expand the experience with additional datasets as well as to begin to address issues raised in earlier launch analysis studies.

In recent years, a number of studies have addressed the possibility of replacing the conventional rigid mirrors that are used in space-based telescopes with optical membranes. Weight reduction, reduced cost of transportation and ability to provide a continuous surface for the attenuation of wave front aberrations are some of the benefits given by optical membranes. Given the harsh environmental loading conditions represented by thermal radiation, debris impact and slewing maneuvers, the ability to characterise fully the dynamics of such low-density thin-film membranes is essential. Nevertheless, the testing of membrane-like structures has proven to be a non-trivial process, and requires a considerable amount of work to achieve accurate results, as well as validating experiments against numerical models. Despite typical testing, the inherent low-density feature of membranes requires the use of non-invasive, non-contact sensors and excitation capability. The work presented here, investigates the possibility of using Operational Modal Analysis (OMA) techniques to extract modal parameters from an acoustically-exited membrane, where responses are collected by using a Scanning Laser Doppler Vibrometer (SLDV) as a non-contact velocity transducer. Results from this experiment are validated against an impedance-based numerical model.

The dynamic performance of helmets in terms of the transmitted force from the helmet to the head is investigated in this paper using techniques in nonlinear vibration analysis. Impedance modeling is used to describe the interaction between the helmet and head-neck subsystems, which are each assumed to behave linearly. The focus of this study is on the influence of padding inside the helmets on the attenuation or amplification of forces applied to the helmet as these forces are transmitted through the padding into the head. The experimental results for vertical only excitation and response measurements indicate that dynamic stiffnesses (including both static and dynamic transfer impedance) for three types of padding were different with certain pads exhibiting less dynamic transfer impedance in certain frequency ranges. The experiments also indicate that these dynamic stiffnesses vary as a function of the amplitude of the forcing function suggesting that the padding behaves nonlinearly with respect to the amplitude of excitation.

In this paper, several important design issues for viscoelastic material characterization test systems which utilize dynamic stiffness measurements are discussed. These discussions are focused on structural dynamics aspects of the design of these test systems. These test systems are used to experimentally obtain the complex modulus of viscoelastic solids such as rubber, plastics, etc. Various standards exist on dynamic stiffness-based viscoelastic material characterization test methods, which give general guidelines on possible test procedures, associated theory, and limitations of recommended test procedures. In these standards however, there is not enough guidance on how to design the details of such a test system, specifically the mechanical structure of the test system. In this paper, authors’ past experiences on the structural design of such test systems are presented which may be utilized as design guidelines for design of similar test systems. Main design issues discussed in this paper include the effect of fixture compliance on the accuracy of calculated material data, frequency limitations, selection of actuators and sensors, and edge (Poisson’s) effect considerations for material specimens.

Modal parameters extracted from test articles with known boundary conditions are useful for model correlation and updating but are often not obtained because of the time and cost associated with moving and installing the test article on a rigid or isolated seismic mass and performing a separate modal test. It would be advantageous if frequency response functions from known boundary conditions, such as fixed base frequency response functions, could be obtained while the structure undergoes testing on fixtures that contain compliance or dynamics in the frequency range of interest, such as base shake tables.

This paper proposes a method that uses response measurements at the structure’s boundary to mathematically constrain the test article to produce equivalent fixed base frequency response functions. Fixed base frequency response functions are obtained by using the boundary measurements as additional references to mathematically constrain these degrees of freedom in the frequency range of interest.

This paper provides analytical justification for using the proposed methodology, discusses assumptions associated with boundary deformation, and finally presents an analytical example using a lumped parameter system.

Modal parameters extracted from test articles with known boundary conditions are useful for model correlation and updating but are often not obtained because of the time and cost associated with moving and installing the test article on a rigid or isolated seismic mass and performing a separate modal test. It would be advantageous if frequency response functions from known boundary conditions, such as fixed base frequency response functions, could be obtained while the structure undergoes testing on fixtures that contain compliance or dynamics in the frequency range of interest, such as base shake tables.

A method that uses response measurements at the structure’s boundary to mathematically constrain the test article and produce equivalent fixed base frequency response functions is proposed. Fixed base frequency response functions can be obtained by using the boundary measurements as additional references to mathematically constrain these degrees of freedom in the frequency range of interest.

This paper presents the experimental results of applying the developed methodology to two test articles: a cantilever beam and a mass simulator mounted on a flexible bracket. Each test article was mounted to an aluminum block that was supported in two ways either resting on flexible foam supports or rigidly attached to ground. The mathematically constrained frequency response functions from the flexible boundary conditions compared favorably to those obtained when the structure was rigidly attached to ground, which demonstrates the potential of the developed method.

Impact testing has proven to be a valuable tool for performing modal tests, especially when the frequency response functions from many single-input impact tests are combined together to help extract all observable modes from a structure. This type of testing is historically known as multiple reference impact testing (MRIT) when multiple response measurements are acquired for each test. In each of these single-input tests, the impact hammer is used to excite the structure once per frame of data and all response signals are allowed to decay to zero before the frame ends.

Impact testing may also involve situations where there is more than one impact per frame, whether the multiple impacts are due to a single impact hammer impacting the test article multiple times, multiple impact hammers impacting the structure once each per frame, or multiple impact hammers impacting the structure multiple times per frame. The potential reason for performing multiple impacts per frame of data is to introduce more energy into the test article. The potential reasons for using multiple impact hammers are, as with multiple shaker testing, to decrease testing time and to distribute energy throughout the structure.

This paper investigates and compares testing results from each of the testing methods described above, and then discusses the signal processing issues involved with each method. Single impact per acquisition frame methods provide the highest coherence levels at the peak frequencies. However, implementing different signal processing methods can significantly improve coherence at resonance in the case of the multiple impact per frame tests. Some clear benefits in frequency response function quality can be observed when comparing the various impact techniques.

Vibration testing is used in many different industries to qualify components and evaluate performance. Components can range widely in size, shape, and function. Typical vibration tests in the aerospace industry implement single degree of freedom (DOF) input (either translation or rotation) due to complexity in control while meeting specific specification levels. Previous efforts were presented in 2007 on a 6-DOF test bed using multiple electro-dynamic modal shakers to apply simultaneous vibration in all six rigid body degrees of freedom. An algorithm was originally developed for single sine inputs at a total of 17 frequencies simultaneously. This paper describes two major developments in this test bed and control algorithm.

The first development was the addition of two additional shakers to control a more complex test structure. A flexible bracket had been added to the unit under test, which added additional dynamics and complicated the original control; adding additional control shakers solved this issue.

The second development involved the extension of the control algorithm to allow for broadband random excitation in 6-DOF. The updated algorithm attempts to match both the magnitude and relative phase of all 6 DOF across the entire frequency range of interest.

Validation of finite element models using experimental data with unknown boundary conditions proves to be a significant obstacle. For this reason, the boundary conditions of an experiment are often limited to simple approximations such as free or mass loaded. This restriction means that vibration testing and modal analysis testing have typically required separate tests since vibration testing is often conducted on a shaker table with unknown boundary conditions. If modal parameters can be estimated while the test object is attached to a shaker table, it could eliminate the need for a separate modal test and result in a significant time and cost savings. This research focuses on a method to extract fixed base modal parameters for model validation from driven base experimental data. The feasibility of this method was studied on an Unholtz-Dickie T4000 shaker and slip table using a mock payload and compared with results from traditional modal analysis testing methods.

In prior work by the author and others [1–3], a new method was demonstrated to extract fixed base modes from a modal test performed on a test article mounted on a vibration slip table. This paper addresses uncertainty that was apparent in frequency and damping estimates in previous work [3]. After reviewing the method based on substructure coupling, additional testing indicates that some of the frequency error was due to different size attachment bolts in the seismic mass truth test and the slip table test. In the previous work, the largest errors in prediction of the truth data were associated with damping. A procedure to subtract significant low frequency slip table damping is implemented and the resulting corrected damping estimate presented.

A precision active vibration mount with a frictionless magnetic hinge system will be utilized to compensate for small attitude fluctuations to enhance the effectiveness of imaging and pointing systems onboard small Unmanned Aircraft Systems at the University of North Dakota. The system consists of a custom lightweight composite mounting plate, piezoelectric actuators, and a digital controller to actively control flight vibrations. Embedded magnets hold the composite mounting plate in place creating a conservative system, and piezoelectric stack actuators are used to precisely displace the plate. This lossless mount is an essential component in a project trying to establish point to point laser communication. Using the magnetic composite mounting plate reduces the number of components, creates a stiff and lightweight mounting surface, and removes friction from the vibration mount which creates a system with no hysteresis. The focus of this active vibration control system is to compensate for small deflections due to engine vibrations or air turbulence. Alongside a gimbal that compensates for large attitude changes, the mount creates a stable platform for precise imaging and pointing. Static and dynamic testing of the active vibration mount will verify the effectiveness of the stabilization system.

High Purity Germanium (HPGe) Detectors are the gold-standard sensor for nuclear spectroscopy applications. In order to make spectroscopic measurements, the detectors must be maintained at cryogenic temperatures. Cryogenic temperatures can be achieved using either liquid nitrogen or piston driven cryocoolers. Because of the bulky and transient nature of liquid nitrogen, the piston driven cryocoolers are preferable for remote, long term, or portable detector applications. These cryocoolers are a promising alternative due to the fact they only require a power source and heat dissipation for continuous operation. A major drawback however is that microphonic noise induced by the vibration of the piston reduces the resolution of the spectroscopic measurements. Passive damping techniques have been applied to this problem, An active damping control system is under development to significantly reduce the vibrations of the cryocooler by adapting to changes in boundary conditions and mass loading through time. Mitigating these vibrations will increase the resolution of portable HPGe detectors and facilitate the identification of nuclear materials.

Response limiting capabilities of vibration controllers can be insufficient to compensate for lightly-damped structures with high Q values and narrow resonance skirts during high sweep rate sine sweep tests. Structures with these characteristics are common in the aerospace industry. This paper outlines and exhibits an automated method developed and validated cooperatively by M + P International and engineers at Harris Corporation to predict and compensate for response limit overshoots. The method employed utilizes data from lower-level sweeps to modify response limit tables. It has been demonstrated to be effective for linear structures with constant transmissibilities in terms of amplitude and frequency.

Active vibration control using non-model based algorithm was developed for beam vibration suppression by the electromagnetic force. The algorithm detects the vibration frequencies in real-time using the laser displacement sensor signal, and then activates the electromagnets for vibration control. The control algorithm does not use any mathematical modeling of the structure. The experimental results show that the proposed method can be applied for the vibration control of beams with variable dynamic characteristics and excitation frequencies.

The adaptive neuro-fuzzy inference system (ANFIS) is a process for mapping from a given input to a single output using the fuzzy logic and neuro-adaptive learning algorithms. Using a given input–output data set, ANFIS constructs a Fuzzy Inference System (FIS) whose fuzzy membership function parameters are adjusted using combination of back propagation algorithm with a least square type of method. The feasibility of ANFIS as strong tool for predicting the severity of damage in a model steel girder bridge is examined in this research. Reduction in the structural stiffness produces changes in the dynamics properties, such as the natural frequencies and mode shapes. In this study, natural frequencies of a structure are applied as effective input parameters to train the ANFIS and the required data are obtained from experimental modal analysis. The performance of ANFIS model was assessed using Mean Square Error (MSE) and coefficient of determination (R

2

). The ANFIS model could predict the severity of damage with MSE of 0.0049 and correlation coefficient (R

2

) of 0.9976 for traing data sets. The results show the ability of an adaptive neuro-fuzzy inference system to predict the damage severity of the structure with high accuracy.

Estimation and correction of telescope vibration have proven to be crucial for the performance of astronomical interferometers. For this purpose the large binocular telescope (LBT) has been equipped with an optical path difference and vibration monitoring system (OVMS), which will serve to ensure conditions suitable for adaptive optics and interferometry. The vibration data is acquired with accelerometers build into each of the six main mirrors and at significant locations of the measurement equipment. The sensor signals are converted into tip-tilt and optical path difference data, which can be fed into the control loop of the adaptive optics and interferometers in order to permit the correction of structural vibrations at frequencies up to 100 Hz, and possibly beyond. In this paper the real-time estimation of the mirrors movement based on acceleration data is investigated. This includes the identification of a structural model from an operational modal analysis and identification of environmental excitations in terms of process noise needed for the design of a Kalman filter.

A heliostat is a structure whose function is to reflect sunlight to a target collector. Heliostat vibrations can degrade optical pointing accuracy and fatigue the structural components. This paper reports on an experimental and analytical program with a goal to improve understanding of the response to wind loading on heliostats. A modal test was performed on a heliostat located at the National Solar Thermal Testing Facility (NSTTF) at Sandia Labs in Albuquerque, New Mexico. Modal tests were performed with artificial and natural wind excitation. Strain and displacements were also measured under wind loading. The information gained from these tests has been used to evaluate and improve structural models that predict the deformations of the heliostat due to gravitational and dynamic wind loadings. The paper will provide an up-to-date summary of model validation work, evaluation of suitable sensors, and development of data-processing methods for long-term deformation monitoring.

In this paper, we focus on extending the subspace identification to the class of linear periodically time-varying (LPTV) systems. The Lyapunov-Floquet transformation is first applied to the system’s state-space model in order to get the monodromy matrix (MM) and, thus, a necessary and sufficient condition for system stability. Then, given two successive covariance-driven Hankel matrices, the MM matrix is extracted by some calculus of a simultaneous singular value decomposition (SVD) and a least square optimization. The method is illustrated by a simulation application to the model of a hinged-blades helicopter.

Dynamic characteristics of structural systems, in many cases, highly depend on joint properties. In order to predict the dynamic characteristics of assembled structures accurately, joint models are required. Due to the complexity of joints, it is extremely difficult to describe dynamic behavior of joints with analytical models. Reliable models are generally obtained using experimental measurements. In this study, substructure decoupling based on measured and calculated FRFs is used to model a structural joint. In the method proposed the FRFs of two substructures connected with a joint are measured, while the FRFs of the substructures are obtained analytically or experimentally. The joint properties are then calculated in terms of stiffness and damping values by using FRF based substructure decoupling. The method is verified with case studies using simulated experimental data. The limitations of the method in its application to engineering systems are discussed.

In the classical Kalman filter theory, one of the key assumptions is that a priori knowledge of the system model, which represents the actual system, is known without uncertainty. Our focus in this research is to estimate the state of a system that is subjected to stochastic disturbances by using an erroneous model along with the available stored measurements. We examine two approaches that take the effects of uncertain parameters into the account since these uncertain parameters degrade the estimate of the state. In the first approach, the errors in the nominal model, which are approximated by fictitious noise and covariance of the fictitious noise, are computed by using stored data. It is premised that the norm of discrepancy between correlation functions of the measurements and their estimates from the nominal model is minimum. The second approach involves the identification of a Kalman filter model on the premise that the norm of discrepancy between the measurements and their estimates is minimum. This paper reviews the two approaches and illustrates their performances numerically.

Ground vibration testing of a Predator® B Unmanned Aircraft System (UAS) developed by General Atomics Aeronautical Systems, Inc. (GA-ASI) was performed to validate analytic models and assess local and global structural dynamics. The aircraft was fitted with a unique sensor package: an up-looking turret chin-mounted on the forward fuselage. This sensor and its support structure participated in the global aircraft modes and required application-specific approaches to excite and measure all target modes. Specifically, testing was required to assess the structural dynamics of the chin-mount integration with a focus on any coupling with the airframe or strong local modes, either of which could potentially affect sensor performance. A combination of electrodynamic shaker positions, excitation types, and impact hammer inputs were employed to identify the modes of interest. A free-free boundary condition was simulated by suspending the aircraft overhead with a flexible bungee suspension.

Acoustic fluid-structure interaction is a common issue in automotive applications. An example is the pressure-induced structure-borne sound of piping and exhaust systems. Efficient model order reduction and substructuring techniques accelerate the finite element analysis and enable the vibro-acoustic optimization of such complex systems with acoustic fluid-structure interaction. This tutorial reviews the application of the Craig-Bampton and Rubin method to fluid-structure coupled systems and presents two automotive applications. First, a fluid-filled brake-pipe system is assembled by substructures according to the Craig-Bampton method. Fluid and structural partitions are fully coupled in order to capture the interaction between the pipe shell and the heavy fluid inside the pipe. Second, a rear muffler with an air-borne excitation is analyzed. Here, the Rubin and the Craig-Bampton method are used to separately compute the uncoupled component modes of both the acoustic and structural domain. These modes are then used to compute a reduced model which incorporates full acoustic-structure coupling. For both applications, transfer functions are computed and compared to the results of dynamic measurements.

The vibro-acoustic behavior of ship-like structures is noticeably influenced by the surrounding water and thus represents a multi-field problem. In this tutorial, fast boundary element methods are applied for the semi-infinite fluid domain. As an advantage, the Sommerfeld radiation condition is satisfied in an exact way and only the boundary, i.e. the ship hull, has to be discretized. To overcome the draw-back of fully populated matrices, fast boundary element methods are applied. The focus is on the comparison of the multipole method with hierarchical matrices, which are set up by adaptive cross approximation. In both cases, a half-space fundamental solution is used to incorporate the water surface, which is treated as pressure-free. The structural domain is discretized with the finite element method. A binary interface to the commercial finite element package ANSYS is used to import the mass and stiffness matrices. The coupled problems are formulated as Schur complements, which are solved by a combination of iterative and direct solvers. Depending on the applied fast boundary element method, different strategies arise for the preconditioning and the overall solution. The applicability of these approaches is demonstrated using a realistic model problem.

Noise and vibration has long been sought to be reduced in major industries: automotive, aerospace and marine to name a few. Products must be tested and pass certain levels of federally regulated standards before entering the market. Vibration measurements are commonly acquired using accelerometers; however limitations of this method create a need for alternative solutions. Two methods for non-contact vibration measurements are compared: Laser Vibrometry, which directly measures the surface velocity of the aluminum plate, and Nearfield Acoustic Holography (NAH), which measures sound pressure in the nearfield, and using Green’s Functions, reconstructs the surface velocity at the plate. The surface velocity from each method is then used in modal analysis to determine the comparability of frequency, damping and mode shapes. Frequency and mode shapes are also compared to an FEA model. Laser Vibrometry is a proven, direct method for determining surface velocity and subsequently calculating modal analysis results. NAH is an effective method in locating noise sources, especially those that are not well separated spatially. Little work has been done in incorporating NAH into modal analysis.

Rotor dynamics is the study of vibration behavior in axially symmetric rotating structures which is analyzed to improve the design and decrease the possibility of failure. Excess vibration can cause noise and cyclic stress. There are several phenomena which need to be detected such as centrifugal and gyroscopic effect adding to the complexity in the mathematical procedures in modal analysis. The experimental technique used thus far is called Modal Testing, a well known and widely used technique in research and industry to obtain the Modal and Dynamic response properties of structures. The technique has recently been applied to rotating structures and some research papers been published, however the full implementation of Modal Testing in active structures and the implications are not fully understood and are therefore in need of much further and more in depth investigations. The raw data obtained from experiment was used in finite element (FE) model for comparison, validation purposes. 3-D models result in large number of nodes and elements. This paper demonstrates how to extract a plane 2-D model from the 3-D model that can be used with fewer nodes and elements. The aims is to establish a system identification methodology using the analytical/computational techniques and update the model using experimental techniques already established for passive structures but to active rotating structures, which subsequently help to carry out health monitoring and further design and development

Often times, tests are performed on structures to attempt to simulate a built in condition at the attachment point on the structure – or at least some type of fixity is desired. The actual implementation of this type of connection is not always easy to achieve. The use of a “large attachment stiffness” is one approach and is a possibility when only the lower order, first few modes are of interest and the fixture/attachment stiffness is much higher than the lower order modes of interest. But when this is not possible, then the use of a “large mass attachment” is another approach but when the structure does not have a low center of gravity (such as wind turbine blade applications), the mass inertia can be significant to simulate a built in fixed condition.

Several conditions are studied for the simulation of a built in condition using a large mass approach for a simple beam type structure. Models are presented to show the need for careful evaluation of the test set up so that the proper boundary conditions are achieved; in addition, tests are performed to confirm the model results obtained. Models are extended to consider wind turbine blades to understand the importance of identifying the proper test set up to achieve a particular boundary condition of interest; several test results are also included.

Traditionally, the method of low stiffness suspension is a standard method for the modal test of structures. A requirement on the suspension stiffness is that the suspension frequency should be much lower than the first flexible natural frequency of the tested structure. Nevertheless, for large scale structure with very low first natural frequency, which could be as low as 0.1Hz, it will need very long suspension spring to achieve the required frequency. This may be not acceptable for many labs. Even if there is sufficient high room space for the suspension, the test is, strictly speaking, two dimensional, i.e. in the plane defined by the suspension. In this paper, we discuss existing methods for coping with these problems by replacing the low stiffness suspension with a group of supports that can have any stiffness. By removing the influence of the support stiffness from a later computation with measured frequency responses, the three dimension modal test can be done conveniently. Numerical examples are described in the paper.

It is often remarked that pure random noise cannot, or should not, be used for estimating frequency response on lightly damped structures. The reason given is that the bias errors caused by leakage would “deform” the estimated FRF in an unsuitable way for modal parameter processing. In this paper, traditional coherence-based methods are compared with the recently proposed local polynomium method. It is demonstrated that the bias error can be made arbitrarily small but that it requires rather long FFT blocksize, relative to what is needed with other excitation signals, such as burst random or periodic excitation signals. A main conclusion is that, while pure random is an inferior excitation signal for synthetic excitation, in cases where there is no choice, such as wind or traffic loaded structures, low-bias FRFs can very well be estimated, provided the measurement time can be long enough.

This paper studies the effect of shaker impedance and the way this parameter can affect the frequency response measurement (FRF) on structures. Three different shaker designs were used for this study to better understand the effect of the shaker impedance on the force transmissibility characteristic function. Analytical MKC models are used to derive equations and then the experiments were conducted to validate the model. Another effect that is studied here briefly is the cross-sensitivity of the sensors when subjected to lateral forces (or displacements) and bending. It is shown that this effect will lead to non-repeatable measurements, especially in the case of impedance heads.

Effects of structural flexibility on the dynamic performance of structures such as staircases, footbridges, and long span floors is becoming an increasingly important aspect of modern design. Cost reduction, improving efficiency of design, enhancement of aesthetic perception and, innovation in architectural forms often result in slender and lightweight structures that are significantly more flexible and vibration-prone than ever before. Consequently, meeting relevant vibration serviceability criteria, as opposed to ultimate strength requirements, is becoming the governing factor in the design of many new structures. Despite significant advances in numerical prediction of modal properties of structures using Finite Element (FE) modelling technique, there still exist challenges in accurate representation of the actual dynamic behaviour. This is mainly due to some inherent modelling uncertainties related to a lack of information on the as-built structures, such as uncertainties in boundary conditions, material properties and the effects of non-structural elements. This paper presents the results of a modal testing exercise carried out to assess the dynamic behaviour of a lively staircase structure. The assessment procedure includes a full-scale ambient vibration testing, modal identification and FE modelling and updating. In particular, the influence of boundary conditions and presence of handrails on dynamic properties of the structure are commented.

During actual operating conditions, mechanical systems, such as aerospace structures, may exhibit variations in their dynamic properties. Those variations need to be carefully tracked and identified to avoid interaction with excitation sources, leading to unstable behavior or even structural failures. In this paper, signals acquired during a static firing test on a Solid Rocket Motor will be analyzed and processed using Operational modal Analysis. To analyze the evolution of natural frequencies and modal damping as the propellant is burnt, time-histories are cut into shorter segments, which are then analyzed separately. Operational PolyMAX method is applied to identify structural properties of the system and correlation techniques are implemented to track the evolution of the modes. Finally, comparison and correlation with a numerical Finite Element model is also performed to evaluate the analysis.