3.1 Introduction

Experimental modal analysis (EMA), a powerful interdisciplinary technique utilized to understand the dynamic behavior of a system, is used in this study as a validation method for analytical models of masonry domes. By the use of response transducers and signal processing techniques, a structure can be described in terms of its dynamic characteristics such as damping ratios, natural frequencies and mode shapes. The focus of this chapter is to review the main topics associated with modal testing as well as the user preferences adopted during the experiments conducted on domical masonry structures.

3.2 Experimental Modal Analysis

EMA is concerned with identifying the natural frequencies and mode shapes of an actual structure. In general, this is accomplished by exciting vibrations in the structure and capturing time dependent inputs from the system–displacements, accelerations or velocities. These inputs are transformed into the frequency domain and analyzed by various techniques of signal processing to obtain usable estimates of the vibration parameters of the structure.

3.2.1 Underlying Theory

When a system is subjected to a time dependent force, its response is dynamic. The dynamic behavior of the system shows differences from the static behavior since it involves the effects of inertia and damping forces, which participate in the equilibrium of the system as expressed in the equation 3-1.

(Equation 3-1)

The system dynamics are based on the spatial distribution of mass, stiffness, and damping. These three components are first converted into a modal model consisting of the defined modal mass, modal stiffness and modal damping values [Equation 3-2]. This conversion is accomplished through operations with the eigenvector matrix on the original equation. The coupled equations are transformed into uncoupled equations by mass normalization in modal coordinates. The assumption of proportional damping yields a simple solution consisting of diagonal matrices. After the solution of the uncoupled equations in modal coordinates, the system response is transformed back into physical coordinates again by using the same eigenvector matrices, or the results are used to obtain the response of the system in frequency domain. This method of transforming back and forth into the modal model is the origin of the term “modal analysis” (Hatch 2000).

 (Equation 3-2)

All real structures exhibit some degree of energy dissipation, which causes velocity dependent constraint forces and attenuates the system response over time. Structures that oscillate before they reach a steady state condition are called underdamped systems. For this specific study, as in practically all building structural systems, the structures of interest are underdamped. Additionally, for simplicity, the damping characteristics of the system is represented as a linear combination of mass and stiffness matrices, known as proportional damping. Throughout the paper, the discussion is restricted to underdamped systems with proportional damping.

There are some basic assumptions established while employing modal analysis throughout this study:

• The deflections are small so that the structure presents a linear dynamic response.

• The structural behavior is time invariant.

• The structural behavior is temperature invariant.

• Inertia, elastic and dissipative forces are independent from each other.

The frequency response function (FRF), the most important concept of modal analysis, is basically the ratio of the system response to the excitation force, examined in the frequency domain. The three physical parameters of equations of motion, mass, stiffness and damping ratio, are again necessary to analytically construct the FRF graph. Alternatively the FRF graph can be constructed from the division of the Fast Fourier Transform (FFT) of the experimentally obtained displacement response and forcingfunctions. Experimental modal analysis uses the theoretical equilibrium between two sides of the Equation 3-3.

 (Equation 3-3)

The FRF can also be written for velocity and acceleration. The preference for the independent variable of the FRF must be consistent with the transducers used during the testing. Equation 3-4 presents the development of FRFs for velocity and acceleration.

(Equation 3-4)

The physical characteristics, mass and stiffness, are the main parameters to determine the natural frequencies and mode shapes of a structure. Consider a simple system of mass and spring loaded by a time dependent sinusoidal force with a constant magnitude. If the rate of oscillation of the force is swept through a frequency range, the FRF graph displays an asymptotic behavior at both low and high frequencies, as seen in Figure 3-1. When the forcing frequency is much lower than the natural frequency, the system stiffness dominates the response. When excitation frequency draws near the natural frequency, the response of the structure displays a sharp maximum whose height and width in frequency space is determined by the damping in the system. At much higher excitation frequencies, the FRF is determined by the mass of the system.

In multiple degree of freedom (MDOF) systems, the FRF again displays sharp peaks at resonant frequencies. Due to Maxwell’s reciprocity theorem, MDOF systems have symmetric stiffness and mass matrices, which yields a symmetric FRF matrix. In practice, a symmetric FRF means that an excitation at point A causes the same response at point B as the response at A produced by an excitation at B. Reciprocity requires the system to be linearly elastic.

A full knowledge of theoretical foundations of modal analysis and postprocessing techniques is essential to obtain good, quality data. Although the theory behind the experiments is important, it is available in various references and will be omitted here (Ewins 2000, Hatch 2000, Maia and Silva 1997). The emphasis of the following section is the practical aspects of modal testing particular to this study.

Modal testing consists of three main phases: test set-up, data acquisition phase, and system identification.

3.2.2 Experimental Setup and Equipment

In the following subsections, three main components of modal analysis, the transducer, the excitation source, and the data acquisition system, are described and an introductory review of the variables of each mechanism, as well as their influence on the final data, is provided [Figure 3-2].

3.2.2.1 Transducer

Accelerometers convert the response of the system into a voltage signal which is converted back into physical units by calibration factors. The sensitivity of the accelerometer is defined by the calibration factor, and is an important concern in selection of accelerometer type.

Perhaps the most important part of the setup is the placement of accelerometers. The accelerometer locations must be arranged such that the significant geometric features of the structure are defined. The array of data points must have sufficient density to prevent misrepresentation of higher order mode shapes as lower order ones, known as spatial aliasing. At the same time, for practical reasons, it is crucial to keep the number of test points as low as possible. Depending on the goals of the test, a study on a preliminary FE model to determine mode shapes is a useful tool to assist in the development of testing plan. Yet, the analyst should not be too confident about the preliminary FE model results, and take precautions to be able to capture the mode shapes if different from the FE estimates.

Since it is initially assumed that the accelerometer motion is identical to that of system, particular care must be given while mounting the transducers to assure a good bond (Maia and Silva 1997). Although there are numerous methods of mounting accelerometers (stud mounting, magnetic mounting or adhesive mounting), the contact surface material usually dictates the method to be used. In order to achieve a proper mounting, securing the cable on the vibration surface and releasing the strain on the cable is necessary [Figure 3-3].

3.2.2.2 Exciter

Although it is possible to obtain the dynamic parameters of the structures by output-only (operational modal analysis) techniques utilizing ambient vibration, forced excitation of the structure with some type of excitation device typically yields better quality results if the exciter is capable of fully and evenly exciting the structure under consideration.

The most common excitation devices are shakers, and impulse (impact) hammers. Roving the accelerometers while having a stationary shaker excitation gives the results in one column in FRF matrix, while roving impact hammer with stationary accelerometers gives one row. Since the FRF matrix is symmetric, theory states that shaker and hammer testing give identical outcomes, however there are some basic, practical aspects that might produce minor differences.

An impact hammer is instrumented with a force transducer in its head, which measures the applied force while exciting the structure in a broadband frequency range. The impact excitation technique makes use of the fact that, when a system is excited by a Dirac delta function, all the natural frequencies contribute to the response of the system. In practice, a Dirac impulse is not possible since the theoretical contact duration of the impact is zero. The hammer excitation yields a triangular-shaped impact whose width, height and shape dictate the usable frequency range and level of the input spectrum. Increasing the impact duration decreases the range of frequencies excited. Compared to a shaker, an instrumented hammer has less control over the excited frequency range. However, using hammer tips with varying hardness and mass, partially controlling the impact duration and excitation of a variety of frequency ranges is possible.

Unlike shaker exciters, impact hammers avoid mass loading on the structure while testing small specimens. However, where large-scale structures are being tested, the additional mass of the person swinging the hammer, as well as the unknown reaction forces due to the swinging action, are unavoidable in most cases. These factors affect the dynamic behavior of the system, but are not included in the measurement captured by the hammer. The existence of the hammer operator also has the potential to introduce nonproportional damping to the structure and increase complexity of the mode shapes (Reynolds and Pavic 2000, Raebel 2000).

In some cases, the mode shapes include stationary points, also called nodal lines. Placing the excitation point at such stationary locations, does not reveal the corresponding mode shape. Hence, a preliminary FEM study to predict the possible nodes of the structure is an efficient and time saving method prior to determining the excitation locations.

As in many experimental situations, the quality of the modal test data is dependent on the proficiency of the test operator. If the hammer swinger accidentally hits the structure twice, also known as a double-hit, the energy input reduces in much lower frequencies and degrades the FRF. Additionally the response due to an unnecessarily hard impact may exceed the voltage limits of the data acquisition frame, which is referred to as overloading the measurement system. The clipped data, which is not included in the post processing, may be another source of inaccurate results.

The accuracy and quality of the results are affected by the ratio of response signal to the existing ambient vibration of the system. Obtaining a system response sufficiently higher than both the noise and the ambient vibration is necessary to acquire reliable test data for any experimental modal analysis test. However, keeping the system behavior in the linear range and providing an even distribution of energy in the structure are other concerns for linear analysis that affect the choice of excitation level.

Since the energy introduced by the impact is localized, vibrations are absorbed and damped by the structure before they propagate from the excitation point to the distant accelerometer locations. Due to the inherent non-proportional damping, a time lag occurs between the captured responses at different locations. Therefore, all data points are not necessarily 180° in-phase or out-of-phase (Avitabile 2001).

3.2.2.3 Data Acquisition and Signal Processing

The data acquisition and signal processing systems obtain analog time domain signals from transducers and digitize them in the frequency domain. The procedure starts with filtering the analog signals by anti-aliasing filters to ensure that no spurious higher frequencies are introduced into the data. After higher frequency signals are removed, a digital multi-channel Fourier analyzer or spectrum analyzer processes the data. The Fourier analyzer performs FFT algorithms and transforms the measured time domain data into the frequency domain. This process yields the complex valued frequency response functions (Avitabile 2001).

The FFT algorithm computes the frequency content of a waveform, which is obtained by the periodic repetition of the captured data. If the signal is not fully captured, this nonperiodic signal sample causes discontinuities and therefore results in the improper representation of the data. In this case, the energy associated with a particular frequency leaks over adjacent frequencies, and the output is degraded. This phenomenon is called leakage and must be avoided during the testing. One method of controlling leakage is the use of weighting functions, also known as windows, multiplied by the signal before FFT processing, in order to force the end conditions of the signal to zero. In this way, a periodic signal is repeated by the FFT and, distortion caused by leakage is reduced, although not completely eliminated.

Most data acquisition systems have a trigger option, which requires a specific slope or a percentage peak value to be reached to initiate recording of the data. If a pretrigger delay is not provided, all the data before the predetermined slope value is lost. The lack of a pre-trigger causes distortion in the spectrum as well as leakage problems (Schwarz at al. 1999).

Since there are always issues to challenge the accuracy of the experimental output, averaging of data is done to reduce the effects of ambient noise as well as to obtain statistical consistency (Ewins 2000). Raebel found that averaging over three and twenty steps for chirp excitation yields the same outcome, and he also mentions that as few as three averages may produce accurate results under controlled conditions (Raebel 2000).

After obtaining statistical data, the coherence function is developed to quantify the quality of the test data. The coherence function quantifies the relationship of the output response to input force over the whole frequency range of interest and rates the linearity of the correlation on a scale of 0-1 (Hanagan et al. 2003). Since the measure of quality is derived from the repeatability of averages, a single FRF measure gives a meaningless coherence function.

3.2.2.4 System Identification

System identification is the last step in modal analysis, where the acquired FRF data is investigated to obtain the frequencies and mode shapes of the system. The complex-valued FRF can be displayed as either real and imaginary parts vs. frequency, or magnitude and phase parts vs. frequency.

In the magnitude portion of an FRF graph, the system response is observed to amplify and attenuate at certain frequencies. The peaks in the plots depict the phenomenon of resonance with the natural frequencies of the structure, while the attenuations occur at anti-resonance frequencies. An alternative way of identifying the natural frequencies is the Nyquist plot, which is an imaginary vs. real component of FRF. It theoretically gives a perfect circle for displacement measurements and a distorted circle for velocity and acceleration measurements whose distortion depends on the damping.

The natural frequencies are then matched with particular deformation patterns called mode shapes. The experimental parameters obtained in all tests deliver the actual response of the structure due to the input excitation. However, they give reasonable approximation to the predicted mode shapes (Avitabile 2001). Therefore, once the acceleration FRF graphs are obtained, by examining the peaks in magnitude plot, or the zeros in the real part, the damped natural frequencies can be determined, which are a close approximation of the natural frequencies. The imaginary part of the FRF reveals the relative motion of one particular data point to the other points for each natural frequency. The global mode shapes are found by representing the relative locations of data points on a geometrical model. It should be noted that if a velocity transducer is used for data acquisiton, the real part of the FRF gives the same data as the imaginary part of the acceleration and displacement FRF. This modal parameter extraction method is called quadrature response analysis. It is a single degree of freedom system identification method, which estimates modal parameters one mode at a time (Shwarz et al. 1999). Although it requires a clean, high quality transfer function and assumes the MDOF FRF to be a superposition of multiple SDOF FRFs, it still is a fast and convenient way to find mode shapes and natural frequencies when modes are well separated and the system is lightly damped. In quadrature response analysis, the damping parameter is not provided. However if an estimate of damping is required the analyst can apply the halfpower method. Since the main purpose of this study, is the validation of the boundary condition and material property entry to FE model, the identification of modal damping is not necessary.

It is also possible to derive the system characteristics from the experimental data by application of mathematical algorithms. This type of parameter identification, typically referred to as curvefitting, can be done by employing commercially available computer softwares. Many software packages are available to accurately perform this curvefitting.

3.3 Impact-Echo Method

In principle, impact-echo (IE) testing is no different from experimental modal analysis with an impact excitation. The main distinction is that IE seeks local mode shapes while EMA seeks global ones. Since these two have very different frequency content, the in plane (bending waves) and out of plane (axial waves) oscillation waves can be identified by focusing separately on higher and lower frequencies (Sansalone 1997).

There are three types of elastic waves that propagate within a solid: primary (P-waves), shear (S-waves), and Rayleigh waves (R-waves). The P-waves generate tensile and compressive stresses in the direction of the propagation, while the S-waves cause particle motion perpendicular to the direction of propagation and result in shear stresses. The R-waves progress along the surface of the material or structure (Sansalone 1997).

In the impact-echo method, a very short duration impact produces broadband excitation up to approximate maximum of 80 kHz. The introduced stress pulse is dominated by a P-wave, which propagates, in a spherical pattern within the object, and maintains the same velocity in all directions.

Due to the larger wavelengths, typically greater than 10 cm, diffusion of the wave through aggregates, cracks, and pores is less significant than in ultrasonic pulse-echo testing where frequencies between 60 and 200 kHz are used (Schubert, Wiggenhauser and Lausch 2004, Colla and Lausch 2002).

The waves propagating through the thickness of the material are reflected when they come across a change in medium, in this case, a change from masonry to air. The multiple reflections of the stress waves between the internal and external boundaries of the test specimen are captured by the accelerometer, and this data can be analyzed to determine the thickness of the material based on its material properties.

Beneath the impact source, the amplitude of P-wave displacement is at its maximum while the amplitude of S-wave displacement is zero. Theoretically, it is possible to obtain a pure P-wave, however from a practical perspective, the excitation source and the response transducer cannot be located at the exact same point. Moreover, impacting too close to the transducer can cause R-waves of unnecessarily large amplitude. It has been found that keeping the distance between the response transducer and exciter less than 0.4 times the thickness of the medium, the contribution of S-waves is eliminated (Sansalone and Streett 1997).

The acquired time history response exhibits periodic characteristics. As in modal analysis data, analysis is processed by transforming the time domain response to frequency domain with an FFT algorithm. The frequency response function is then used to evaluate the frequency peaks existing in the spectrum. In the impact-echo technique, the satisfactory identification of the frequencies is more important than the amplitudes.

(Equation 3-5)

The frequency range excited is based on the contact duration between the exciter and the specimen. This duration is influenced by the use of the impacting technique and impact device. The most common devices are hardened steel balls and pencil size hammers with metal tips. In almost all cases, the commercial impact hammers have the specifications for maximum excitation frequency; however, for steel hardened spheres it is necessary to employ Equation 3-5 to find the maximum frequency excited (Sansalone 1997).

 (Equation 3-6)

Once the main resonant frequency of the local axial mode is acquired from the experimental data, the overall time required for one cycle can be obtained. As the waves are reflected backwards from the exterior edges of the material, the distance traveled by the wave is two times the thickness of the test specimen. By knowing or estimating the P-wave velocity [Equation 3-6], the thickness can be revealed as half of the product of the total time and the speed of sound in the material [Equation 3-7] (Sadri 2003).

 (Equation 3-7)

3.4 Experimental Procedures as Applied to Existing Masonry Domes 

Vibration characteristics of the test specimen may be sensitive to geometric imperfections, the precise nature of the boundary conditions, stiffness changes due to temperature or cracks in the structure. However, when testing on existing masonry structures, the level of desired accuracy is not as high as the tests conducted under fully controlled conditions in laboratories. Although the aspects listed above have an effect on the measured data, it has been shown by Erdogmus (2004) that the system behavior is not substantially altered, and the approach described here is capable of giving sufficiently accurate results. In the following sections, appropriate parameters for the application of experimental modal analysis to masonry domes is discussed and followed by the test results obtained from structures in this study.

3.4.1 Particulars of Experimental Setup

Each individual testing case poses its own potential problems in obtaining reliable output while conducting modal testing. Although a large body of literature describes testing on various forms of laboratory specimens, modal testing on an existing large-scale masonry structure has completely different practical issues and challenges. A degree of compromise is involved and it is necessary to seek the best testing plan to obtain the sufficiently accurate data in a reasonable time without disturbing the structure in use.

Establishing a clear definition of specific goals of the study is the first step and this affects every subsequent phase in the modal testing process. Based on these test objectives, the upper limit of the frequency range of interest and the required frequency resolution are determined. The equipment selection and the data acquisition settings are also dictated by the test objectives. In this study, a validation experiment that is planned, executed, and evaluated to determine and calibrate the quality of the computational model, is implemented. A validation experiment is designed to capture all the data of interest necessary while building the FE model (Garcia 2001, Oberkampf 2001).

3.4.1.1 Data Acquisition Settings

In this study, the FFT-based data acquisition and signal-processing system The DSP Technology, Inc. Siglab model 20-42 dynamic signal analyzers are utilized to obtain and post-process the vibratory data. Siglab presents bandwidth and record length as variables, that when combined, define the frequency resolution and the data capture time or frame size. The decision of which value to use depends on the behavior of the system. In order to prevent leakage problems and avoid the use of a window, the data capture time must be adjusted so that the system response attenuates over the data capture time window (Avitabile 2001). In this study, the response attenuated in approximately one second, and accordingly, the frame size was adjusted to two seconds.

Allemeng states that when comparing two alternate methods, FEA and EMA, assuming minimal errors and sufficient test experience, the first 10 modes show a reasonable agreement, and higher modes present more difficulty (Allemang et al. 1993). Since the specific purpose of the experiments is verifying the FEM model, the first 10 modes, particularly those involving crown movement, are the most important. In the preliminary FEM model developed in this study, the natural frequencies are observed to start as low as 50 Hz and the frequencies associated with the first 10 mode shapes were predicted to be less than 200 Hz. Based on this knowledge, an upper limit of frequency range, also known as the bandwidth, of 200 Hz is used for the experiments conducted on masonry domes.

In previous sections, the window functions are mentioned as a means of addressing the leakage phenomena. The use of a window introduces artificial damping to the structure, which widens the resonant peaks and potentially causes some lower magnitude peaks to be dominated by higher magnitude ones (Reynolds and Pavic 2000). In many cases, the resonant peaks obtained from testing on masonry structures already have wide skirts. A further widening would make identification of resonant frequencies and mode shapes more difficult. Therefore, the above-mentioned process of establishing an appropriate time interval for testing was utilized to avoid the use of windows.

3.4.1.2 Transducer

Accelerometers with a sensitivity between 1-10 V/g deliver sufficiently accurate results for frequencies between 0.5 Hz and 2000 Hz. In this study, model 393A03 accelerometers, manufactured by PCB Piezotronics, Inc., with a calibration value of approximately 1 V/g, are used.

A preliminary FEM model is developed to observe the dominant mode shapes. Accelerometer locations are determined such that as many low modes as possible are captured. The comparison of experimental results to FEM results is facilitated by capturing the vertical accelerations. Therefore, the accelerometers are mounted so that their axes are vertical. The double curvature of the domed structure presents difficulties in mounting the accelerometers. Hanagan states that mounting the accelerometer directly on the structure with wax, or clay yields acceptable results in low frequency testing (Hanagan et al. 2003). In this study, the modeling clay is utilized as a means of leveling the base of the accelerometer and coupling it to the masonry structure.

3.4.1.3 Excitation Source

An impact hammer, being very portable and easily utilized, is the most suitable and practical excitation method for the scope of this study. Mode shape estimates of the preliminary FEM model are used in determination of the optimum excitation locations.

A Piezotronics Inc. PCB model 086D20 instrumented impulse hammer, capable of applying a peak force of 22 kN, is used. Among the four available options for tip hardness, the softest, a nylon tip, is preferred since it offers the lowest frequency range of 0-400 Hz. The mass and surface stiffness of the impacted structure also contribute in defining the frequency range excited (Avitabile 2001). It is observed that while testing masonry structures, the nylon tip is capable of exciting frequencies in the range of 0-200Hz.

In transient analysis, the hammer operator has a major responsibility for the accuracy of the results. Since domed structures give very complicated mode shapes in higher orders, it is important to hit the same point with approximately the same driving force over all of the excitation locations and structures. Even a few centimeters difference in excitation point may cause a significant difference in the response. The hammer operator should avoid applying excessively high excitation amplitudes to the test structure while simultaneously keeping the signal to noise amplitude ratio above 100 (40db) (Hanagan et al. 2003). During the tests, it is preferable to have the hammer operator standing off the dome webbing since the additional mass and unknown reaction forces increase the complexity of the mode shapes.

3.4.1.4 System Identification

The preliminary identification of natural frequencies and corresponding mode shapes is done by quadrature response analysis. The initial results obtained with this manual method are then compared to those obtained with system identification software. In quadrature response analysis, the damping parameter is not obtained and the analyst is expected to apply the half-power method. However, for the specific purpose of this study, damping ratio extraction is not a concern.

In this study the Spectral Dynamics, Inc. STARModal system identification software, which uses a global curve fitting technique, is employed. By defining the bandwidth and the number of the modes to be extracted, the program fits a polynomial of the user defined order to that section of the FRF. In some instances, increasing the order of the polynomial may be useful to obtain better quality data when modes are closely coupled and when the system is highly damped. In this case, some computational modes are created, which must be elimininated by the analyst.

In order to assure the quality of the obtained mode shapes STARModal provides a Modal Assurance Criterion (MAC) matrix to examine the cross-correlation between the modes. The magnitude of off-diagonal terms is an indication of the uniqueness of the mode shapes, and of the absence of spatial aliasing.

3.5 Applications to SEB and CCB

The application of the techniques of modal testing to the structures in this study produced FRF data with corresponding coherence functions that were sufficiently high. The data is examined to obtain the natural frequencies and global mode shapes. During the evaluation process, the reciprocity and repeatability checks are completed. Moreover, the symmetric behavior of the system is inspected. The following sections provide brief descriptions of the testing and the results obtained for two example structures. Although the experiment on CCB was done earlier than those on SEB, the less extensive data on CCB are presented later in this chapter.

3.5.1 State Education Building (SEB)

The experiments on State Education Building [Figure 3-4] are conducted in four main steps:

• Simulated Modal Testing
• Preliminary Modal Testing (9 data points)
• Final Modal Testing (25 data points)
• Impact-Echo Testing

3.5.1.1 Simulated Modal Testing

A preliminary mathematical simulation based on the available knowledge on the definition of the boundary conditions, material properties, constitutive equations and the physical geometry of the system, is performed. In order to gain an understanding of the dynamics of the structure before the knowledge of the experimental results is available, a preliminary FE model is developed in ANSYS 8.0. The initial estimates of the lowest five mode shapes and natural frequencies are obtained [Figure 3-5]. The frequencies start at approximately 20 Hz. He mode shapes involve multiple domes, rather than isolated individual domes.

3.5.1.2 Preliminary Modal Testing

The preliminary test was conducted on September 6, 2005. Based on the knowledge gained from the preliminary FE analysis, a test plan for simultaneously capturing nine measurement data points was developed. The accelerometer placement, excitation locations, and bandwidth were selected according to the initial FE estimates. Figure 3-6 presents the scheme of accelerometer placement and excitation locations in the preliminary test.

Although it is important to follow a predetermined test plan, it is also prudent to conduct a few initial sample tests to check the quality of the data based on the test plan. For example, the structure might have a higher natural frequency range than the preliminary FE model estimated. In these instances, an immediate modification must be done to the test plan.

In SEB, domes have an oculus of 30 cm radius. Although the top of the dome is completely load-free, it still is not accessible. HVAC and electrical equipment, which are supported from the ceiling, rest on the apex. Therefore, measuring the response at the apex is not possible; instead, accelerometer #2 is located in the closest possible position in the direction of accelerometer #7 [Figure 3-6].

The accelerometers are mounted with modeling clay as seen in Figure 3-7. Tape is used to secure the cables. As the upper limit of frequency interest is 100 Hz, the nylon tip is preferred for the impulse hammer. The hammer blows are located approximately
10 cm below the accelerometer locations.

The appreciable system response is primarily due to the thinness of the shells. Since the domes are concealed within the building and are almost detached from the upper floor, the noise level recorded is determined to be insignificant. The response to noise ratio is calculated as approximately 60 db. Based on this ratio, the impulse data are averaged five times. At all excitation locations, the tests are repeated two times to check the repeatability of the data.

In the preliminary test, the FRF data obtained in SEB has very high coherence. The driving point magnitude FRF plot for point #7 is presented in Figure 3-8. The high coherence can be attributed to the following:

• The hammer operator is able to avoid standing on the dome shell. Thus, the additional non-proportional damping and unknown reaction forces do not exist in the experimental data for this specific test.

• The domes are small in dimension. Even and adequate excitation is possible with the available impact hammer. Therefore, a high signal to noise ratio is obtained.
• The consistency of the hammer operator also improved the quality of the data, especially for reciprocity and symmetry checks.
  

During the physical model creation in the FE software, the construction imperfections and small dimension variations are assumed insignificant and ignored. The validation of this assumption requires the investigation of the symmetric behavior of the dome. During the symmetry check, care must be exercised in defining the basis of comparison. The response of the accelerometers at axisymmetric modes is a good tool to compare the symmetric behavior. Although the dome is geometrically symmetric, it exhibits many unsymmetrical mode shapes. For those, the symmetry comparisons may be deceiving. The preliminary symmetry check between point #3 and #7 showed a good agreement at 41 Hz [Figure 3-9]. It will later be shown in this chapter that the mode shape at 41 Hz exhibits a symmetric behavior for point #3 and #7. The discrepancies at higher frequencies still exist due to the complicated nature of the higher order modes along with the non-symmetric location of the accelerometers and non-proportional damping.

Additionally, reciprocity checks are necessary in order to verify the assumption of linearly elastic material properties for the tile and mortar composite used in the construction of this building. The FRF data obtained at point #7, while exciting point #9, displayed very good agreement with the FRF data acquired at point #9 while exciting point #7. The ΔFRF graph, the difference between the two FRF plots, displays an average deviation of 3% [Figure 3-10].

The linearity of the system can also be checked by exciting the structure with different impact levels. Since the response of a linear system is proportional to its excitation, different excitation levels are expected to result in the same FRF. The FRF magnitude data showed almost no variation for excitation levels of 100 lb, 160 lb and 230 lb [Figure 3-11]. However, an improvement in the coherence function for the frequencies lower than 25 Hz is observed as the excitation force is increased. Recalling the uniqueness of the FRF for linear systems, this study once again confirms the linearity of the system.

The stability of the system behavior is checked by a repeatability analysis. Figure 3-12 presents almost identical FRF plots of test #1 and test #2 at point 7 and confirms the stability of the obtained data in the frequency range of interest.

3.5.1.3 Final Modal Testing

The final test is conducted on January 23, 2006. To confirm the validity of the preliminary experimental data and to obtain a more complete set of results, a final experiment plan is developed. The primary purpose of the final testing plan is improving the resolution by increasing the number of data points along the meridians as well as investigating the dynamic interaction between different parts of the structure [Figure 3-13]. Additionally, particular efforts are made to obtain the dome shell thickness experimentally.

The testing procedure is initiated with a test plan including 26 data points and utilizing the same equipment as in the previous test [Figure 3-13]. Capturing the higher order, axisymmetric modes enables a comparison between experimental and analytical result. This is the intention in improving the meridional resolution. Points #3 through #10 are kept in the exact location as the previous experiment to obtain a repeatability check. Two additional rings of test points are introduced along the same meridians. Three Siglab units are combined and 11 data points are measured simultaneously. In each step, the accelerometers at each ring level are relocated while the accelerometer at point #2 and #27 are held stationary. This multi-run test plan is executed in three steps.

One disadvantage of this test setup is that the hammer operator is required to stand on the dome in order to excite the data points located closer to the apex. [Figure 3-14]. Since the entire set of nodal points is not measured simultaneously, the combination of the three separate sets of data requires scaling based on the reference points. Despite these challenging issues, high coherence FRF data is captured. The driving point FRF and coherence plot at point #15 are illustrated in Figure 3-15 in order to provide a general sense of the data quality.

STARModal software is employed in order to determine the natural frequencies and corresponding mode shapes. Polynomial and coincidence curve fitting techniques are utilized [Figure 3-16]. The modal parameter estimates of the STARModal software are then further investigated by utilizing the manual techniques of quadrature response analysis. The obtained resonant frequencies are examined by visual inspection of the FRF magnitude graphs and the resulting mode shapes and are confirmed by investigating the FRF imaginary plots [Figure 3-17]. The list of the natural frequencies used to validate the FE model is presented in Table 3-1, while the animated mode shapes are provided in the Appendix A.

Table 3-1: The experimentally identified natural frequencies, SEB, NY.

In FE modeling, it is necessary to model the entire structure participating in the dynamic response. When the system is complex, as in this particular structure [Figure 3-18], it is not straightforward to decide what structural elements-for instance, surcharge volume, adjacent domes, buttresses, steel frames-must be included in the FE model. In this study, the techniques of dynamic testing are utilized to determine the existing constraints at boundary conditions and the interaction between adjacent members.

The estimation of the dynamic interaction between the steel truss and the tile domes is difficult, as the boundary conditions are dependent on physical properties and configuration of the material. A set of experiments is executed in order to monitor the reaction of the steel members to an impact excitation on the masonry dome. As can be seen in Figure 3-19, the results reveal that steel members exhibit almost no response. Based on this observation, the steel frame is excluded from the FE modeling and is discussed in Chapter 5.

In order to recognize the restraint induced by the buttresses, the accelerometers are also located on the buttresses [Figure 3-20]. The vertical response due to an excitation on the crown is captured. It is noted that the even numbered buttresses display a relatively greater response in vertical direction compared to the odd numbered (diagonal) buttresses. This observation, which affects the definition of the boundary conditions, is discussed in Chapter 5.

The influence of the adjacent domes on each other is also studied by mounting the accelerometers on the apices of the eight domes adjacent to the center dome [Figure 3-20]. The contribution of the adjacent domes is noticeable in frequencies higher than 90 Hz. However, in the frequency range of interest, the captured results do not present a significant dynamic interaction between the domes. Therefore, a single dome FE model is considered adequate for the analysis.

3.5.1.4 Impact-Echo Testing

As discussed in section 3.3, the identification of the local modes can reveal the thickness of the dome shell. When observing the local modes, the frequency range of interest is much higher than the upper limit for global mode shapes.

A Tektronix oscilloscope with a 500 kHz upper frequency limit is utilized. A pencil type impact hammer, which can excite up to 15 kHz, along with 15 mm diameter hardened steel balls that can excite approximately up to 20 kHz are employed as exciters. The balls are thrown on the masonry surface and captured immediately after the first bounce. A model 8630C50 accelerometer with a natural frequency of 22 kHz, manufactured by Kistler Instrument Corp., with 100 mV/g sensitivity is mounted with a thin layer of modeling clay directly on the dome shell.

A data capture time of 500μs is assigned, enabling a bandwidth of 50 kHz. Although the system response did not attenuate in 500μs, the time window is intentionally not increased because, increasing the time window decreases the upper frequency range. The captured time domain results are multiplied by an exponential window in order to prevent leakage issues before transforming into frequency domain by the use of a MATLAB code. The results with and without the application of the exponential window are illustrated in Figure 3-21. Both of the FRF graphs exhibit an evident frequency peak at 8 KHz.

Based on the tuning of the material sample test results through the calibration process described in Chapter 4, a velocity of wave propagation of approximately 2100 m/s is calculated. Inserting the Cp and f values in equation 7, reveals the thickness of dome shell to be 13.5 cm, which corresponds to four layers of 19 mm thick tiles.

3.5.2 City County Building (CCB)

The experimental data obtained from CCB is inferior to SEB, since only preliminary testing is accomplished. The experimental testing is conducted in two main steps:

• Simulated Modal Testing
• Preliminary Modal Testing (5 data points)

3.5.2.1 Simulated Modal Testing

Similar to SEB, in CCB a preliminary FE model is built to execute a virtual test prior to the actual experiment [Figure 3-22]. The most efficient locations for locating the response and force transducers are determined based on the preliminary FE mode shape estimates.

3.5.2.2 Preliminary Modal Testing

The preliminary test is conducted on July 22, 2005. Based on the virtual test results, the preliminary test plan is developed [Figure 3-23].

The test is conducted with Dactron data acquisition system, which is manufactured, by LDS Test and Measurement Ltd. Due to the limitation of the input channels in the Dactron unit, only three data locations are recorded simultaneously. The first two channels are kept stationary at point #1 and point #2, and the third channel is used to record point #3, #4, and #5 in three separate tests. The graphs of magnitude FRF and coherence along with imaginary FRF and real FRF for point #1 are given in Figure 3-24.

The upper floor is built in such a way that it contacts the dome apex, where point #1 is located. Therefore, locating the accelerometer at the exact apex is not possible, so point #1 is placed approximately half meter away from the apex. The hammer excitation point is located approximately 1 meter away. Due to the geometry of the dome, keeping the additional mass of the hammer swinger off the dome is not possible for any of the trials.

In cases where multiple channels are utilized, defining the natural frequencies can be tedious. The fact that the impact operator had to stand on the dome caused small shifts in the natural frequencies [Figure 3-25]. The summation of the FRF data of all channels and the summation of all measured driving point in this study are used as an auxiliary tool to determine the natural frequencies.

Reciprocity is checked between the diagonal points #2 and #5. The two driving point FRF graphs reveal fair agreement. The average discrepancy is around to 2×10-4 gs/lb, which is an average of 12% of the maximum response [Figure 3-26]. The possible sources for the deviation between the two FRF are the additional mass of the hammer swinger, the reaction forces induced by the swinging action as well as the nonlinearities induced by the existing cracks discussed in section 5.4.2.6. For the purpose of this study, this level of correlation for the reciprocity check is accurate enough in the frequency range of interest, and the further study on this building proceeds with linearly elastic material assumption.

Repeatability checks are necessary to be able to assess the stability of the structural behavior of the system. The peaks of ΔFRF must be compared with the resonant frequencies of the original transfer function to control the frequency shifts. The repeatability checks are done for point #2. The driving point data at point #2 presents a close correspondence between the two tests [Figure 3-27].

Three-dimensional Nyquist plots, describing the imaginary vs. real parts of the FRF, are prepared and studied throughout the frequency range of interest. The plot shows evident circles at expected frequency ranges [Figure 3-28]. Since for multiple degree of freedom systems, Nyquist plots are hard to read in two-dimensions, the frequency content is segmented and partially plotted for each resonant region through the system identification process. The list of identified natural frequencies through the preliminary testing is presented in Table 3-2.

Table 3-2: The experimentally identified natural frequencies, CCB, PA.

In the last phase of the field-testing, a test setup to identify the interaction between the two adjacent domes is prepared as seen in Figure 3-29. The accelerometer mounted on point #2 showed almost zero response, due to the force applied on point #1. This indicates the isolated behavior of the domes and lead towards the modeling of a single dome.

3.6 Concluding Remarks and Future Recommendations

With this study, the applicability of the techniques of EMA to large-scale masonry structures is confirmed. Provided that the variables of modal testing are adequately selected, acquiring appreciably high-quality data is possible even when dealing with a challenging material like masonry. Moreover, promising results are also gained from the application of the IE procedure to determine the thickness of a multi-layer masonry shell construction. In the following paragraphs, recommendations for the future applications of these two modern engineering techniques to historic masonry building are provided based on the findings of this study.

Experimental Modal Analysis: The PCB model 086D20 instrumented impulse hammer provide sufficient excitation for masonry structures with a mass around 10 tons. For larger and more massive structures, utilizing the PCB model 086D50 sledgehammer is recommended to achieve higher energy input. The PCB model 393A03 accelerometers or its equivalent can be adopted as response transducers to similar future studies. These accelerometers have a measurement range of 0.5–2000 Hz, which fully encompasses the possible natural frequencies of historic masonry domes. The applicable accelerometer mounting techniques are limited when experimenting on masonry buildings. In this research, modeling clay is observed to provide satisfactory results. The use of this material is recommended for future studies as long as the temperature limitations of the material are not violated.

While planning the test setup, the accelerometer placement and excitation locations are best determined with a preliminary FE analysis. If such an analysis is not possible, the solutions obtained from the preliminary and validated FE models of SEB an