In order to gain a thorough understanding and insight into established knowledge regarding the topics of interest, previous studies related to the scope of this research are reviewed. Numerous publications regarding the research methodology have been found useful to the present study. Among the selected literature, the studies including experimental data and entirely theoretical efforts are subdivided and listed below.

2.1 Experimental Studies

The experimental studies of interest can be divided into two categories: site testing on existing large-scale masonry structures and testing on small-scale laboratory specimens under controlled conditions.

2.1.1 Experimental Studies on Existing Masonry Structures

Armstrong et al. (1995) apply the techniques of modal analysis to masonry arch bridges in order to identify the defects in the structure. The structure is excited by an impact hammer and the response is captured by accelerometers located throughout the bridge. Frequency response function (FRF) graphs are obtained for the original structure and then the procedure is repeated after the system is artificially mass loaded. Due to stress dependent characteristics of the infill soil, the variations of natural frequency are not well explained. However, significant attention is given to distortions in mode shapes. This paper indicates that we cannot expect to identify damage in masonry structures by using experimental modal analysis, especially without a baseline measurement.

Sortis, Antonacci and Vestroni (2003) emphasize that the dynamic identification procedure widely used for steel and reinforced concrete structures can also be effectively applied to masonry buildings. They conduct a survey on a two-story masonry structure by subjecting it to low amplitude vibratory forces induced by multiple shakers. They mention that dynamic behavior of the system is dependent on the exciter location and excitation type due to small nonlinearities in the structure. Both sweep and sinusoidal tests are employed, and the latter is concluded to be more reliable. They also adopt a modal identification algorithm and then use the identified modal parameters to estimate the physical parameters of a corresponding finite element model. After various error minimization steps, the FRF obtained by the experiments and the updated FE model gives satisfactory agreement. In spite of the conclusion that the sinusoidal shaker excitation is most desirable, practical considerations in the following research dictate other means of excitation.

Bonato et al. (1998) published a paper on cross-time frequency techniques for inspection of masonry buildings. They examine the dynamic behavior of the Roccaverano bell tower by placing accelerometers on landings in the horizontal direction, under different unknown excitation conditions. The authors measure accelerations caused by bell tolling in different directions, pulses applied to the bell, and finally the ambient vibration caused by the environmental noise. In the paper, it is concluded that the excited mode shapes are disturbed and concealed by the strong harmonic components of bell tolling, which dominates the frequency spectrum. The environmental noise is therefore selected as the excitation source in their experiments. By employing timefrequency estimators, the amplitude and phase data are obtained. They note a better quality of phase estimator, as it is originated from instantaneous cross-correlations during which noise is filtered more effectively. The updating procedure they use is rudimentary, but possibly adaptable to this research. The structures under investigation in this research, however, have a much broader excitation spectrum and FRFs including input and output will prove more useful.

Moreover, the three-dimensional finite element model of the bell tower is built by simplifying the structure as a space truss whose horizontal members correspond to flexural stiffness while the diagonal members represent the shear stiffness. According to experimentally obtained natural frequencies and mode shapes, the FE model is updated. They present time-frequency estimators as a valuable technique to identify the material properties and structural schemes of masonry buildings, especially for cases when the source of excitation is not known.

Chaigne et al. (2002) experimentally and numerically investigate the nonlinear vibrations of free-edged shallow spherical shells in order to identify the dynamics of percussion instruments. Assuming the specimen to be perfectly isotropic and homogenous, shallow brass shells are subjected to harmonic excitation with a constant frequency but varying amplitude. The first phase is defined as “periodic” dominated by the harmonic spectrum of the forcing function. The bifurcation initiates the second phase, called “quasi-periodic,” after which the “chaotic” regime would start.

In their study, various shallow spherical freely suspended caps are tested under the excitation by a filtered white noise, and the data is analyzed to understand the dynamics of gongs. They discuss the existence of numerous asymmetric modes lower than the first axisymmetric mode. This fact is explained by the high inherent stiffness of the double curved domain, especially on the central part, and owing to the homogeneity of the material, they describe the asymmetric modes as a doublet of close frequencies. They also evaluate the already developed standard formula of hemispherical thin shell theory and compare the obtained frequencies. They obtain close agreement between numeric solutions and experimental results and explain the discrepancies caused by the replication of the free edge condition. The authors conclude that, due to the geometric nonlinearity of the hemispherical shells, the large amplitudes of vibrations would mostly be responsible for the nonlinearities of the response. In the following, asymmetric modes with lower frequencies than the first axisymmetric mode are identified with the validated FE model.

Erdogmus (2004), in her dissertation, studies the axisymmetric mode shapes of the three masonry vaults especially focusing on the movement of the crown. Three different excitation sources, heel-drop, shaker, and impact hammer, are employed, but an impact hammer is considered the most suitable while testing on large-scale masonry buildings. The experimentally acquired data are then used to calibrate an FE model of the vaults, particularly the boundary conditions, and the material properties. In her studies on the National Cathedral in Washington, D.C., she obtains a natural frequency correlation for one mode–including crown movement–between experimental data and analytical estimates. After validation of the FEM model, she discusses the static behavior of different types of vault constructions. Although Erdogmus presents a means of model validation, due to the low number of measurement points and the limited analysis of mode shapes, her results are limited to one mode, not necessarily incorporating the fundamental mode.

2.1.2 Experimental Studies on Spherical Laboratory Specimens

Perhaps the most extensive study on the vibratory behavior of hemispherical shells is done by Hwang (1966). In his study, experiments are performed on an aluminum hemispherical shell with a radius of 15 cm (six inches). Excitation forces are generated by an electromagnet without physically interfering with the test structure. The shell response is acquired by accelerometers attached on the shell surface and by a microphone pickup. The analog signals are manually digitized and then analyzed by Fourier transform. Obtained test results–natural frequencies and mode shapes–are then used to thoroughly compare different analytical approaches like inextensional theory and elastic theory of thin shells. He achieves an overall agreement between analytical and experimental results. With this study, Hwang experimentally confirms the applicability of thin elastic shell theory to the spherical segmental structures.

Mal’tsev, Konon and Adishchev (1984) address the discrepancies between analytically and experimentally obtained shell stresses occurring on the spherical thin shell structure due to a blast loading. A spherical steel shell structure with 2 m diameter and 8 mm thickness is excited by blasting of concentrated charges produced by highly explosive ammonite. The strain of the shell is monitored by wire resistance strain gauges. The time history data obtained from gauges are used to identify the fundamental frequency. A close agreement with the analytical radial vibration solution of spherical shells is obtained. They mention the presence of at least three major modes in the strain curves. By evaluating the beating phenomena, they intend to identify the higher order modes. The authors explain the existence of parametric instability in spherical shell structures causing energy transfers between initially excited modes and the others involved in the vibration of the shell. This study is unique in terms of the utilized excitation technique: the action of the blast wave.

Ogihara (1985) carries out another vibration test on silicone rubber spherical shell structure in order to evaluate the stiffness representation of the structure for a lumped mass model. Stepped sinusoidal excitation is created by a uniaxial shaking table and the response of the sphere is measured by an optical displacement meter. Computation of moment of inertia and shear area from the experimental data by the help of strain energy principles is studied for a simplified lumped mass FEM model. Then, the fundamental frequencies and mode shapes are computed by eigenvalue analysis of FEM model and are found to show a reasonable accuracy. In this study, Ogihara highlights the applicability of dynamic response analysis for stiffness evaluation of the simplified lumped mass
models.

Souza and Croll (1983) underline the fact that geometric imperfections or thickness variations of test specimens may cause substantial uncertainties on the acquired data. For the testing, they follow a well-defined production procedure to control the geometric quality of the truly hemispherical five test specimens varying from shallow to deep and thin to very thin. In their comprehensive paper, the importance of the test set up, which does not interfere with the dynamic behavior of the system, is also mentioned. A loudspeaker is used as source of vibration on the free edge hemispherical specimen and capacitive type displacement transducers observed the motion. Fundamental frequencies and mode shapes obtained from experimental data are compared with the estimates of thin shell theory. Although the lack of a filter during the experiments degrades the quality of the data, they obtain a very close agreement between experimental and analytical results. The small variations in the mode shapes are explained as the contributions of non-resonant modes. They also indicate the linear response of the system with the unique displacement response vs. input energy ratio.

Ross (1996), in his paper on vibrations of thin-walled domes, focuses on vibration modes of hemi-ellipsoid shells, in particular the axisymmetric modes. The displacement functions along the meridian of the shell are established as linear variations of in-plane displacements and parabolic variations of out-of-plane displacements. He continues by placing the circumferential displacements in a sinusoidal form, which reduces the computational work. The computation along the meridional line is done by Gauss numeric integration. He also completes a series of experiments on solid urethane plastic hemi-ellipsoidal domes with a base diameter of 0.2 m and a thickness of 0.002 m. The experimental data is compared to the analytical estimates and a good correlation is obtained for the meridional and circumferential displacement between the two. His study provides a remarkable basis for vibration solutions of thin hemi-ellipsoid shells.

2.2 Analytical Studies

Within the scope of this study, established formulations for segmental spherical shells are utilized and the obtained solutions are compared to the experimental data and the FEA estimates. A thorough survey on previous studies of analytical solutions for spherical shell vibrations is presented.

Liew and Lim (1995) complete the most extensive studies on curved shell vibrations, discussing the vibratory behavior of double curved shallow shells of curvilinear planform. Focusing on isotropic, homogenous thin shells, they employ the Ritz principle of minimum potential energy to obtain the general eigenvalue equation. Two-dimensional, mathematically complete, orthogonal polynomials represent the inplane and transverse deflections while a developed basic function stands for the boundary conditions. They investigate the effects of Gaussian curvature–negative for hyperbolic paraboloids, zero for cylinders and positive for spherical shells. Furthermore, the effects of different curvilinear plan shapes are examined–elliptical or circular. In this extensive paper, valuable information regarding solutions for hemispherical thin shells free and clamped edge conditions are presented. The highest frequency, signifying highest stiffness, is observed when the Gaussian curvature of the shell is set to one for all plan types. They also present a complete set of results for the effects of the curvature ratios for elliptical forms. This study has significant importance as it presents comprehensive results on the vibration mode shapes of hemispherical shells with simply supported and fixed boundary conditions. The following is another important paper on the topic of vibration solutions for thin segmental spherical shells.

Kunieda (1982) discusses the complicated nature of the exact expressions developed for axisymmetric free vibration of spherical shells and provides an approximate method to simplify this tedious computational work. He mentions, that despite common belief, the “so-called” fundamental mode is not the first mode. He develops torsionless mode shapes of a spherical shell with one free edge and closed apex by a combination of three Legendre functions with real fractional order and with complex conjugate orders. He notes that the deviation in the natural frequencies of hinge and clamped edge are relatively insignificant where spherical shells are concerned. In his numeric calculations, he is not able to obtain the fundamental mode–which is estimated to be purely radial. He believes in the absence of this mode for case of domes in 30-90 degrees conical range. His study has significance as he presents the issue of the clustered, closely spaced nature of the natural frequencies for spherical shells. He also notes that the assumption of solely flexural vibration is appropriate in analysis since no major difference are obtained between the natural frequencies of normal vibration or flexural vibration in his study. This study has been an important source for the work presented here.

Pesciullesi, Rapallini and Tralli (1997) investigate spherical masonry domes of uniform strength subjected to their own weight under both infinite and limited compressive strength conditions. As the geometry and all the applied loading are both independent of circumferential angle, throughout the analysis all the quantities are considered axisymmetric. The neutral surface of the spherical shell is assumed to remain spherical and masonry is assumed a no-tension material. In context of the membrane theory, a function denoting the variation of thickness with respect to meridional angle is obtained for uniform strength spherical shells. The study is extrapolated for closed spherical domes under uniformly distributed loading. The decrease in admissible span and formation of tensile circumferential stresses due to additional loading is observed. This paper also presents the solutions of the spherical domes with an oculus. They state that the presence of a hole at the apex is seen to increase the admissible span for cases where the size of the hole is significant.

Kang and Leissa (2004) recently published a paper on three-dimensional vibration analyses of spherical shells. They mention the well-known two-dimensional shell analysis to be accurate for thin structures, in particular around low frequency ranges. According to this paper, the assumptions established regarding the variations throughout the thickness are not applicable for higher frequencies. They focus on both solid and hollow shells, with or without conical openings. The thickness variation throughout the shell is also investigated for hollow spheres. The displacement components are established to be sinusoidal in time, periodic in circumferential angle φ, algebraic polynomials in meridional angle θ. Based on three-dimensional equations of elasticity, the formulated total potential energy equation is minimized by the Ritz principle and then an eigenvalue solution is adopted to obtain the natural frequency estimates. They mention the convergence to the exact results with the increase of the degree of polynomials used. This paper points to the necessity of three-dimensional analysis for higher frequencies and confirms the accuracy of well-known two-dimensional shell analyses around low frequencies, which is the frequency range of interest in the present research.