Abstract: In this research, the free vibration analysis of cylindrical shells with circumferential stiffeners, i.e. rings with non-uniform stiffeners eccentricity and unequal stiffeners spacing is investigated using analytical, experimental and finite elements (FE) methods. Ritz method is applied in analytical solution while stiffeners treated as discrete elements. The polynomial functions are used for Ritz functions and natural frequency results for simply supported stiffened cylindrical shell with equal rings spacing and constant eccentricity is compared with other's analytical and experimental results, which showed good agreement. Also, a stiffened shell with unequal rings spacing and non-uniform eccentricity with free–free boundary condition is considered using analytical, experimental and FE methods. In experimental method, modal testing is performed to obtain modal parameters, including natural frequencies, mode shapes and damping in each mode. In FE method, two types of modeling, including shell and beam elements and solid element are used, applying ANSYS software. The analytical and the FE results are compared with the experimental one, showing good agreements. Because of insufficient experimental modal data for non-uniformly stiffeners distribution, the results of modal testing obtained in this study could be as useful reference for validating the accuracy of other analytical and numerical methods for free vibration analysis.

Abstract: In ballasted railway tracks, one of the im- portant components that supports the rails and distrib- utes wheel/rail loading onto the ballast supporting formation is a railway sleeper (sometimes is also called a Brailway tie^). This paper presents results of an experimental modal analysis of prestressed concrete sleepers in both free-free and in-situ conditions, incor- porating the dynamic influence of sleeper/ballast inter- action. Dynamic interaction between concrete sleepers and ballast support is crucial for the development of a dynamic model of railway track capable of predicting its responses to impact loads due to wheel flats, wheel burns, irregularities of the rail, etc. In this study, four types of prestressed concrete sleepers were in-kind provided by the Australian manufacturers. The concrete sleepers were tested using an impact hammer excitation technique over the frequency range of interest, 0-1600 Hz. Frequency response functions (FRFs) were measured using PULSE modal testing system. The FRFs were processed using STAR modal analysis package to identify natural frequencies and the corresponding mode shapes for the sleepers. The conclusions are presented about the effect of the sleeper/ballast interaction on the dynamic properties of prestressed concrete sleepers and their use for predicting railway track dynamic responses.

Abstract: Vibration problems in stadia are becoming more common due to increased structural slenderness and more lively dynamic crowd excitation. Unfortunately, there is very little guidance available to design engineers dealing with the assessment and design of stadia structures. This paper presents unique data from a program of modal testing and in-service monitoring of a large contemporary cantilever grandstand in the United Kingdom. The in-service monitoring was carried out during an international football match, during which the stadium was full to capacity. Modal properties obtained from the testing on the empty structure are presented and the results from in-service monitoring are described. It is found that crowd occupation can significantly alter the modal properties of a stadium, and that the changes can vary according to the crowd configuration. Additionally, previously proposed methods for assessment of vibration serviceability have been applied and it has been shown that they can lead to inconsistent results, which is a result of their sensitivity to the data acquisition and analysis techniques used. It is concluded that it is very important that consistent methods of data acquisition, analysis, and vibration serviceability assessment are utilized by future researchers and practitioners. Also, further research is required to define vibration serviceability limits using the state-of-the-art vibration dose approach.

Abstract: Based on the theoretical derivations of the free vibration problem for a cable-TMD system in the companion paper, a parametric study is carried out in the present paper to investigate the influence on the solutions, and thus the system modal damping and frequency, of the cable geometry-elasticity parameter, cable inclination, damper position, mass ratio, frequency ratio, damper damping ratio, and tuning mode. Emphasis is put on the system modal damping, as this is an important index of the performance of the TMD. The features and mechanisms of energy transfer and damping redistribution from the TMD to each mode of the cable-TMD system are also investigated and discussed.

Abstract: The simple modal damping identification model of Hart and Vasudevan is generalized. The model works in the frequency domain and provides time-invariant modal damping ratios of building structures under seismic excitations in terms of modal participation factors and the roof-to-basement transfer function. Translational as well as torsional modes of vibration are considered. The performance of the model is assessed through a small, yet indicative, number of numerical examples involving steel plane and space frames under seismic excitations and on the basis of a number of criteria an ideal identification model should satisfy. It is concluded that the presented model, in spite of its simplicity, gives very good results for low-amplitude seismic excitations resulting in linear elastic structural behavior (with damping) even for cases of closely spaced modes, local modes, and very small or large amounts of damping. Some numerical pitfalls regarding the application of the model are mentioned and carefully treated. The limitations of the model when used in conjunction with inelastic structural behavior are determined and discussed. Experimental verification of the model is also provided.

Abstract: Low-frequency floor and footbridge vibration serviceability problems typically arise when the structure is excited in resonance due to a walking excitation and the resulting accelerations exceed human comfort levels. The measures required to resolve an annoying vibration problem after the structure is constructed can be very difficult and expensive to implement. In most cases, the costs of fixing the problem in-situ are much greater than tackling the problem in the design phase, prior to the structure's construction, considering the potential cost to building owners from possible legal expenses, loss of rental revenue, and consultation fees. Design guidance for composite steel framed floor systems and footbridges is available in the AISC/CISC Steel Design Guide Series 11: Floor Vibrations Due to Human Acitivity . Although the current design guidance is generally acceptable, there is a need to continue characterizing the often-complicated vibration behavior of these structures in an effort to refine current design and analysis techniques, particularly as researchers gain a better understanding of behavior by collecting high-quality experimental data on in-situ floor and footbridge structures. This paper presents observations from such efforts to further characterize behavior through experimental modal testing of a large in-situ composite steel office floor and a laboratory constructed multi-span footbridge. While not entirely inclusive, some general observations are noted on the dynamic behavior, problems encountered, and the consistency/reliability of the applied testing/analysis techniques employed by the researchers.

Abstract: When modal testing a structure for model validation, free boundary-conditions are frequently approximated in the lab to compare with free boundary-condition analyses. Free conditions are used because they are normally easy to simulate analytically and easier to approximate experimentally than boundary conditions with fixed conditions. However, the free conditions can only be approximated in the lab because the structure must be supported in some manner. This paper investigates and quantifies the effects of the support conditions on both the measured modal frequencies and damping factors. The investigation has determined that the measured modal damping is significantly more sensitive to the support system (stiffness and damping) than the measured modal frequency. Included in the paper are simple formulas which can be used to predict the effect on the measured modal parameters given the support stiffness and damping.

Abstract: An extensive program of full-scale ambient vibration testing has been conducted to measure the dynamic response of a 240 meter cable-stayed bridge - Gi-Lu Bridge in Nan-Tou County, Taiwan. A MEMS-based wireless sensor system and a traditional microcomputer-based system were used to collect and analyze ambient vibration data. A total of four bridge modal frequencies and associated mode shapes were identified for cables and the deck structure within the frequency range of 0~2Hz. The experimental data clearly indicated the occurrence of many closely spaced modal frequencies. Most of the deck modes were found to be associated with the cable modes, implying a considerable interaction between the deck and cables. The results of the ambient vibration survey were compared to modal frequencies and mode shapes computed using three-dimensional finite element modeling of the bridge. For most modes, the analytical and the experimental modal frequencies and mode shapes compare quite well. Based on the findings of this study, a linear elastic finite element model for deck structures and beam element with P-Delta effect for the cables appear to be capable of capturing much of the complex dynamic behavior of the bridge with good accuracy.

Abstract: There are several significant modal testing challenges to overcome when striving for the goal of an “adequate linear estimate” of the structural dynamics model. Shaker excitation often has an advantage over impact excitation in meeting this goal. In many cases, shaker testing can reduce the effects of nonlinear response, provide better signal to noise ratio, eliminate overload problems associated with out-of-band force input, provide more consistent data, and provide faster testing by utilizing MIMO acquisition. There are tradeoffs between FRF linearization, FRF distortion, signal to noise ratio, and testing time depending on the shaker forcing function. Some of the major tradeoffs have been addressed for random, pseudorandom, periodic random, burst random, chirp, and stepped sine testing. Shaker mounting and fixturing requires special considerations to mitigate undesirable frequency and damping effects of the shaker and its hardware upon the test article and to minimize side loads and moments on the force gage. Certain steps can be taken to ensure the shaker fixturing is adequate, and many diagnostics are available to ensure the goal of obtaining an “adequate linear estimate” of the structural dynamics model.

Abstract: Dynamic testing of an inflatable thin-film torus structure with regular formed convex dome surface features has been performed to evaluate structural natural frequencies, mode shapes, and modal damping ratios. The structure presented two unique challenges with respect to modal testing. First, it is extremely lightweight, flexible, and highly damped. Second, the thin film that provides the integrity to the structure is not smooth and flat, but has a pattern of hexagonal dome structures formed into it that increases the local stiffness of the thin film to the point that the structure is self-supporting in the gravity environment of Earth when the internal pressure is released. The dynamic testing was performed in this self-supporting state. In the modal test a loudspeaker provided acoustic excitation, and a laser displacement sensor was used to measure the vibration response at various points on the torus surface. The acoustic excitation and the laser displacement measuring technique were chosen because they are all noncontact methods that avoid mass loading of the structure as would happen if accelerometers and a shaker were used. Three in-plane and three out-of-plane modes were extracted using this approach. The experimental results indicate that the noncontact modal data-acquisition approach for extremely lightweight structures is suitable and effective. The modal identification procedure found modes that are analogous to elastic ring modes as well as widely spaced repeated modes.

Abstract: The study of modal sensors and actuators is motivated by their ability to control the resonance frequency and the vibration modes of mechanical structures. In this work, a non-homogeneous poling method to obtain a radial modal sensor from a piezoelectric disc is presented. The construction method for three modal elements is described and their electrical behavior is presented and analyzed in comparison to a uniform poled disc.

Abstract: : The presented research examined three areas: best practices in high quality dynamic testing of in-situ floor systems, extensive dynamic testing of three bare (non-fit out) in-situ multi-bay steel composite floors to estimate their dynamic parameters/response and to identify trends in dynamic behavior, and development of a set of fundamental finite element (FE) modeling techniques to adequately represent the dynamic response of steel composite floors for the purpose of evaluating vibration serviceability. The measurement, analysis, and computation of a floor's accelerance frequency response function (FRF) is the core premise linking all areas of the presented research. The burst chirp signal using an electrodynamic shaker is recommended as the most accurate and consistent source of excitation for acquiring high quality measurements suitable for use in parameter estimation, operating deflection shape animation, and calibration/validation of FE models. A reduced mid-bay testing scheme is recommended as a time-saving alternative to modal testing over a full coverage area, provided the only desired estimated parameters are frequencies, damping, and mid-bay acceleration response. Accelerance FRFs were measured with an electrodynamic shaker located within 23 unique bays on the three tested floors. Dominant frequencies ranged from 4.85 Hz to 9 Hz and measured estimates of damping varied considerably, ranging from 0.44% to 2.4% of critical (0.5%-1.15% was typical). Testing showed several mode shapes were localized to just a few bays and not all modes were adequately excited by forcing at a single location. The quality of the estimated mode shapes was significantly improved using multi-reference modal testing.

Abstract: The objective of this paper is to find the relatively low-frequency (200–2,000 Hz) mode shapes of a single-winding gradient coil cylinder with intermediate wall thickness. The dynamic behavior of a gradient coil cylinder plays a crucial role in determining and controlling the vibroacoustic performance of magnetic resonance imaging (MRI) scanners. Modal analyses of the gradient coil cylinder were carried out under different boundary conditions to obtain the various mode shapes. Theoretical modes, predicted by using modified Love’s governing equations, and numerical modes simulated using a finite element method show close agreement with experimental modal results and reveal the mode shapes for both free-end and fixed-end boundary conditions. These results were further compared to in situ measurements of the mode shapes of the gradient coil cylinder insert during scanning in a 4 Tesla MRI. The general agreement among the analytical, numerical, and experimental mode shapes indicates that a linear combination of basic beam vibration and ring vibration patterns occupy the dynamic vibration modes in the low frequency range. The in situ vibration measurements show that the forcing function developed by the distributed Lorentz forces on the surface of the single-winding gradient coil results in predominantly beam-type bending mode patterns in the low frequency range.

Abstract: Damping is a mechanism that dissipates vibration energy in dynamic systems and plays a key role in dynamic response prediction, vibration control as well as in structural health monitoring during service. In this paper a time domain and a time-scale domain approaches are used for damping estimation of engineering structures, using ambient response data only. The use of tests under ambient vibration is increasingly popular today because they allow to measure the structural response in service. In this paper we consider two engineering structures excited by ambient forces. The first structure is the 310 m tall TV tower recently constructed in the city of Nanjing in China. The second example concerns the Jinma cable-stayed bridge that connects Guangzhou and Zhaoqing in China. It is a single tower, double row cable-stayed bridge supported by 112 stay cables. Ambient vibration of each cable is carried out using accelerometers. From output data only, the modal parameter are extracted using a subspace method and the wavelet transform method.

Abstract: Multiple references are often used to excite a structure in modal testing programs. This is necessary to excite all the modes and to extract accurate mode shapes when closely spaced roots are present. An algorithm known as SMAC (Synthesize Modes And Correlate), based on principles of modal filtering, has been in development for several years. This extraction technique calculates reciprocal modal vectors based on frequency response function (FRF) measurements. SMAC was developed to accurately extract modes from structures with moderately damped modes and/or high modal density. In the past SMAC has only worked with single reference data. This paper presents an extension of SMAC to work with multiple reference data. If roots are truly perfectly repeated, the mode shapes extracted by any method will be a linear combination of the "true" shapes. However, most closely spaced roots are not perfectly repeated but have some small difference in frequency and/or damping. SMAC exploits these very small differences. The multi-reference capability of SMAC begins with an evaluation of the MMIF (Multivariate Mode Indicator Function) or CMIF (Complex Mode Indicator Function) from the starting frequency list to determine which roots are likely repeated. Several seed roots are scattered in the region of the suspected multiple roots and convergence is obtained. Mode shapes are then created from each of the references individually. The final set of mode shapes are selected based on one of three different selection techniques. Each of these is presented in this paper. SMAC has long included synthesis of FRFs and MIFs from the roots and residues to check extraction quality against the original data, but the capability to include residual effects has been minimal. Its capabilities for including residual vectors to account for out-of-band modes have now been greatly enhanced. The ability to resynthesize FRFs and mode indicator functions from the final mode shapes and residual information has also been developed. Examples are provided utilizing the SMAC package on multi-reference experimental data from two different systems. NOMENCLATURE FRF: Frequency Response Function SDOF: Single Degree-of-Freedom ωr: natural frequency of r mode ωj: j frequency line of FRF ζr: damping coefficient of r mode ψ: weighting vector HX(ω): experimental FRF HA(ω): analytical generalized coordinate FRF HP(ω): predicted generalized coordinate FRF MAC: Modal Assurance Criteria MIF: Mode Indicator Function CMIF: Complex Mode Indicator Function MMIF: Multivariate Mode Indicator Function NMIF: Normal Mode Indicator Function NS: Number of Spectral Lines {ω}: vector of all frequency components Ar: modal residue coefficient j: 1 − MOTIVATION For a variety of reasons, multiple references are often used to excite the structure in a modal testing program. One reason is to excite the structure in enough locations to excite all of the modes of interest. Another reason is to allow extraction of modes with closely spaced roots. SMAC has previously been a single-reference curve-fitting algorithm. Since most modal tests are performed with multiple references, the capabilities of SMAC have been extended to simultaneously handle multi-reference data. SMAC has had the capability to synthesize FRFs and MIFs to compare with the experimental data as a parameter quality checking capability, but the ability to synthesize the effects of out-of-band modes was minimal. For some applications, such as admittance modeling, the accurate reconstruction of all parts of the FRF are required, not just the frequency band around each resonance. Residual terms that represent the effect of out-ofband modes are important for these applications. SMAC now includes more options for characterizing the residual effects in the synthesis phase. In addition, the ability to resynthesize FRFs from mode shapes and residual vectors that have been saved has been added for "after the fact" modal parameter quality checking and special applications previously mentioned. The residual capabilities were enhanced by modifying the calculation method and by providing additional residual terms to the synthesis choices. ROOT EXTRACATION THEORY The SMAC algorithm is based upon the modal filtering approach rather than an assumed matrix polynomial form. In the strictest sense this means that there must be at least as many response measurements as there are active modes in the frequency band of interest. The sensors should be placed so that the associated experimental mode shape matrix is well conditioned for inversion. Since the algorithm is not based on a matrix polynomial, there are no computational roots, eliminating a major set of decisions the analyst must make in deciding on the true system roots. The theory for the SMAC approach has been presented previously. For review, a few equations and an intuitive explanation will be provided here regarding the SMAC theory. The modal substitution allows one to reproduce the responses of a system up to the frequency of the highest mode as given here. { } [ ]{ } q X Φ = (1) where {X} is a vector of responses, [Φ] is the mode shape matrix and {q} are uncoupled modal coordinates. If {X} and [Φ] are known, {q} can be determined from the pseudo-inverse of [Φ]. [ ] { } { } q X = Φ + (2) Now if we are only computing the ith modal coordinate in the vector {q}, we only need one row of [Φ] which we can call {Ψ} . Then we can write { } { } Ψ = T i q X (3) which holds true for time histories, Fourier transforms or FRFs. Assume that a set of FRFs for one reference input are measured, and the functions are arranged in a matrix so that each column represents a different sensor, and each row contains the complex values at each frequency line. This matrix is called [HX]. Then a value for a natural frequency, ωr, and a damping ratio, ζr, of the system is arbitrarily selected to create an analytical FRF representing an assumed modal coordinate. We call this FRF {HA}, and it is a vector calculated using the selected frequency and damping values, for a SDOF FRF with arbitrary amplitude and a length of NS spectral lines. Equation (4) gives the analytical vector based on a real mode assumption.

Abstract: The head‐gimbal assembly suspension is a cantilever‐like structure that holds the heads on a hard drive. We will discuss a noncontact method for modal testing, of suspensions in air, that utilizes the radiation force at the difference frequency generated by two intersecting ultrasound beams. The resulting low‐frequency excitations were measured using a scanning vibrometer. This excitation technique has been demonstrated for MEMS and other small devices. There are several unique advantages of the ultrasound radiation force relative to mechanical shakers. Since the ultrasound radiation force is noncontact, a specialized test fixture was not needed; the technique was relatively insensitive to distracting resonances of fixtures and support structures. Another advantage is broadband excitation; a 550‐kHz confocal ultrasound transducer excited suspension resonance frequencies from under 1 kHz to over 50 kHz. Other advantages include the ability to selectively excite different modes. For example, the amplitude of the suspension’s 2.28‐kHz transverse mode was suppressed by an order of magnitude by shifting the modulation phase between the two ultrasound beams by 90. In another test, the amplitude of the 6.01‐kHz torsional mode was doubled by moving the ultrasound focus point from near the center to near the edge of the suspension.

Abstract: Multiple-input, multiple-output (MIMO) experimental modal testing is often used for large structures. The data collected are used in multiple-reference reduction schemes to find the best set of modal parameters to describe the system. Often several or many of the reference shakers do not adequately excite all of the modes from each reference location. When this is the case, using all of the reference data may produce modes that are not optimum. A careful selection of references for generating modal parameters is critical for developing a good modal database for design, analysis, simulation and correlation efforts. While this is true of earlier modal parameter estimation algorithms, the latest PolyMAX estimation algorithm has significant advantages over historically used techniques. Experimental modal tests are often conducted using a multiple–input, multiple-output testing strategy. Depending on the complexity of the structure to be tested, two or more shakers may be used for the excitation of the system. Many times it is very difficult, if not impossible, to have all the shakers excite all the modes of the system equally. This is especially true when the structure exhibits directional global modes or when the structure has an abundance of local modes due to appendage or subcomponent modal energy. When this is the case,

Abstract: Provided is a method and apparatus for generating and sensing torsional vibrations using magnetostriction, in order to perform modal testing on a bar, a pipe, a shaft, a beam, or the like. The apparatus includes a torsional vibration generating part transmitting torsional vibrations to a test part having an arbitrary cross-section and a predetermined length, and a torsional vibration sensing part sensing the torsional vibrations generated from the torsional vibration generating part. The torsional vibration generating part and/or the torsional vibration sensing part includes a magnetostrictive body attached around the test part and made of a magnetostriction material, a first magnetic field forming part forming a magnetic field around the test part in a circumferential direction of the magnetostrictive body, and a second magnetic field forming part forming a magnetic field to the magnetostrictive body in a direction substantially perpendicular to the direction of the magnetic field formed by the first magnetic field forming part and in parallel with a longitudinal direction of the test part.

Abstract: This paper presents long-term dynamic characteristics of a cable-stayed bridge where installed SHM (Structural Health Monitoring) system. Modal parameters such as natural frequencies and mode shapes are identified by modal analysis using three dimensional finite element model. The developed baseline model has a good correlation with measured natural frequencies identified from field ambient vibrations. By statistical data processing between measured natural frequencies and temperatures, it is demonstrated that the natural frequency is in linearly inverse proportion to the temperature. The estimation of temperature effects against frequency variations is performed. Mode shapes are identified from the TDD (Time Domain Decomposition) technique for ambient vibration measurements. Finally, these results demonstrate that the TDD method can apply to identify modal parameters of a cable-stayed bridge.