Abstract
This paper presents the experimental investigations conducted on the carillon tower of the Santissimo Crocifisso Sanctuary in Castel San Pietro (Bologna, Italy) and the analysis of data collected by using velocimeters and accelerometers installed on the structure. The main goal is to assess the effects of the swinging bells on the dynamic behaviour of the tower. The paper’s novelty relies on the kind of structure monitored and the originality of the experiments. The structure is a rare example of a carillon tower, with fifty-five bells of different sizes, subjected to a careful measurement campaign never carried out before. Six experiments were conducted selectively by activating the bells to measure the tower’s response induced by different vibration sources and determine the peak velocities recorded by using instruments at different heights. Two ambient vibration tests complemented the six experiments. The carillon’s action induces low velocities on the tower, while experiments involving the bells swinging in the upper chamber produce the highest velocity values in the swinging direction; these values are more significant than those induced by the carillon alone. The most robust action is induced on the tower when all the bells (carillon plus swinging bells) ring. The experimental results are complemented by numerical simulations of the dynamic behaviour of the tower subjected to the action of a swinging bell.
1. Introduction
Many problems can undermine the safety of historical masonry towers and slender structures, including soil settlements, materials’ deterioration and ageing, damages due to high compressive stresses, seismic actions, and excessive bell vibrations. Since the action of swinging bells repeated over time may damage the masonry of ancient towers, accurate experimental campaigns and numerical modelling are crucial to evaluate the performance of such structures under dynamic loads and assess their safety and plan maintenance.
This paper presents the results of the experimental investigations and numerical analyses conducted on the tower of the Santissimo Crocifisso Sanctuary in Castel San Pietro, focused on assessing the effects of the swinging bells on the dynamic behaviour of the structure.
Studies on the dynamic interaction between bells and masonry towers are quite scarce. A seminal study dates back to the seventies [1] when the first experiments were set up to determine the inertia forces induced by the bell ringing. Further thorough investigations were conducted for the millennium celebrations in England [2]. Relevant outcomes are reported in [3–14]. The papers by Bennati et al. [3, 4] present the experimental tests carried out on the bell tower of the San Miniato Cathedral, provide an analytical expression of dynamic actions transmitted to the tower by the swinging bells, and describe the motion of the tower as well. The dynamic behaviour of a bell tower in Valencia is the subject of the paper by Ivorra and Pallares [5], where possible dynamic amplification phenomena connected to the closeness of excitation and bell tower frequencies are investigated. A resonance phenomenon between the first natural frequency of a modern bell tower and some harmonic components of dynamical actions transmitted by the swinging bells is highlighted in [6], where the authors show that this phenomenon is significantly reduced by stiffening the structure. A similar analysis on a modern concrete bell tower is also shown in [7]. The outcomes of the studies conducted on the Soncino Civic Tower subjected to the bells’ action are reported in [8], where the model of the tower is calibrated via experimental data and the effect of retrofitting on the stress field is assessed. Another investigation on the effects produced by the bells’ swinging has been carried out on the SS. Medici Bell Tower (Apulia, Italy) [9], a framed reinforced concrete structure with masonry walls. High accelerations are recorded on the bell tower. When bells swing in the north-south direction, a coupled behaviour between north-south and east-west deformations has been detected, explaining the presence of cracks in the structure. The interaction between the structure’s natural frequencies and forces generated by bells is analysed in [10] for masonry bell towers, considering different bell arrangement and turning speeds along with the tower’s dynamic characteristics. The dynamic behaviour of a Tuscan masonry bell tower subjected to forces generated by the swinging bell is addressed in [11]. A parametric analysis of the dynamic interaction between harmonic bell forces and fundamental tower modes is conducted, assuming swing angles, velocities, position, and direction of bells as parameters and using the dynamic amplification factor [15], under the hypothesis of linear elasticity. Vincenzi et al. [12] assessed the dynamic response of a bell tower in North Italy subjected to the oscillations of swinging bells under the hypothesis of linear elasticity. The influence of swinging bells on the crack distribution in a bell tower in Lithuania is analysed in the paper [13], where the modal curvature approach is used to detect damage in the tower. Finally, Nochebuena-Mora et al. [14] investigated possible resonance effects by comparing the natural frequencies of a bell tower in Portugal with those of the bells’ actions; they also presented the results of nonlinear dynamic analysis in terms of displacements and cracking.
The present paper, which adopts approaches followed in previous papers, aims to improve the knowledge of the dynamic behaviour of masonry bell towers. The structure under examination is a challenging, complex, and never investigated case study, a carillon masonry bell tower hosting fifty-five bells, which in 2021 was instrumented by the authors to measure the velocities induced by the bells and determine the structure’s dynamic properties. The carillon mechanism, which is driven by using a keyboard at the tower’s base, and the number and size of the bells involved represent an example of a bell tower rare in Europe and unique in Italy.
The study’s novelty also relies on (i) the use of high-sensitivity instrumentation (triaxial velocimeters) generally employed in geophysics and seismology and still uncommon in civil engineering, (ii) the experimental campaign conducted, consisting of the measurement of the velocities induced in the bell tower both by environmental vibrations and by the oscillation of single bells having different masses (to which different oscillation frequencies correspond) or by the action of all the swinging bells plus the carillon, and (iii) the possibility of comparing experimental and numerical results coming from finite element dynamic analyses conducted with NOSA-ITACA [16], freely downloadable software entirely developed by the authors, both in the linear and nonlinear field.
The paper is organized as follows. In Section 2, we describe the tower of the Santissimo Crocifisso Sanctuary and the carillon system. Section 3 provides a detailed description of the experimental tests conducted on the tower in 2021 to assess the system’s dynamic response to the action of bells. Six experiments were carried out selectively by activating the bells to measure the tower’s response induced by different vibration sources and determine the peak velocities recorded by using instruments at different heights. Two ambient vibration tests complemented the six experiments. These two tests were conducted with the bells at rest, the former before and the latter at the end of the bells’ activation, and allowed to determine the structure’s experimental frequencies, mode shapes and damping ratios via operational modal analysis (OMA) techniques. Section 4 is devoted to Finite Element (FE) simulations. An FE model of the tower is calibrated via the NOSA-ITACA code by minimizing the discrepancy between experimental and numerical frequencies. Linear and nonlinear dynamic analyses of the calibrated model subjected to the action transmitted by the biggest of the swinging bells are then conducted with NOSA-ITACA. The numerical velocities are compared with their experimental counterparts recorded in some selected points during the tests. The conclusions are drawn in Section 5.
2. The Carillon Tower and Its Fifty-Five Bells
The main building of the Santissimo Crocifisso Sanctuary in San Pietro Terme (Central Italy, Bologna) was built in the first half of the eighteenth century and enlarged at the beginning of the past century with the construction of the pronaos (Figure 1). In the same years (1926–1930), Giulio Gollini built the present masonry bell tower (Figure 2) and designed one of the most famous and complex bell carillons in Italy and Europe, consisting of fifty-five bells connected to a keyboard placed at the tower’s base via an electropneumatic system. The bell tower overlooks a large public square. The music of the carillon is often played during religious and civil ceremonies, and local authorities organize concerts in which the sound of the carillon accompanies the civic band on the square.
The renowned Brighenti foundry in Bologna produced bells hosted at different levels inside the tower. The lower bell chamber (Figure 3(a)), at about 15 m in height, hosts forty-eight bells hung by hammers, activated in turn by the carillonneur at the keyboard. These bells do not move and are suspended from a steel structure, which rests on the chamber’s floor.
(a)
(b)
The upper bell chamber (Figure 3(b)), at about 18 m in height, hosts seven swinging bells: three small bells (no. 8, 9, and 10) are activated via long ropes going downward from the bell’s floor to the tower’s base; bell ringers swing the remaining four bells (52–55) in particular circumstances (Figure 4(b)). These four bells, suspended from a steel frame supported by the bell chamber’s floor and masonry, can swing and also be stroked using hammers. The bronze bells are balanced by wooden counterweights that allow the system to fulfil large oscillations, up to 160° from the rest position, following the Central Europe system known as “Alla Romana” [17]. The bells can also overturn and remain upright with their mouths up until bell ringers push them down again (English system [1]). This configuration is occasionally reached at the end of the bells’ concerts for very few cycles. The heaviest bell is no. 51, with a diameter of about 1200 mm; it is located in the lower bell chamber (Figure 3(a)) and is struck by using a hammer (Figure 4(a)). Among the swinging bells located in the upper chamber, the heaviest is no. 54 (Figure 3(b)), with a diameter of about 880 mm. Figure 4(b) shows the bells swung by bell ringers during the experiments.
As far as geometry is concerned, the tower is about 31 m high, with a square transverse section of about 5.1 m × 5.1 m. The masonry walls have a thickness of 0.8 m, constant for first 15 m. The bell chambers’ floor is supported by metallic beams and small masonry vaults. The tower ends with a dome, which takes up the last 7 m of the structure. No experimental and historical information is available on the construction techniques and the mechanical characteristics of the constituent materials. Upon visual inspection, the masonry appears to be made up of regular stone blocks and bricks at the base, while relatively homogeneous brick masonry is visible in the upper part; all the floors consist of masonry vaults supported by steel beams. The bell tower is adjacent to the sanctuary, whose walls presumably constitute a horizontal constraint for the tower for first 4.5 m.
3. Dynamic Monitoring and Identification
This section describes the experimental tests and the analysis of data recorded by using instruments installed on the tower. The bells were activated selectively during the experiments to measure the tower’s response induced by their dynamic actions. The ambient vibration tests conducted with the bells at rest allowed the dynamic identification of the tower.
3.1. Experimental Tests
Following an inspection of the tower motivated by a perceived increase in the structure’s vibration amplitudes, the authorities in charge of the Sanctuary’s maintenance requested experimental investigations. On 17 May 2021, the authors carried out the measurements. The instrumentation consisted of seven seismic stations produced by SARA Electronic Instruments and two triaxial accelerometers. The authors installed four SS45 (eigenfrequency 4.5 Hz) and three SS20 (eigenfrequency 2 Hz) triaxial velocimeters, each coupled with a 24-bit digitizer (SL06) (https://www.sara.pg.it). The velocimeters’ transfer functions allowed enlarging the frequency bandwidth, thus giving a correct estimation of the measured velocities. The instrumental set was completed by using two broadband triaxial accelerometers Guralp CMG5T (https://www.guralp.com/) coupled with two 24-bit digitizers RefTek 72A07. The sensors’ arrangement is shown in Figures 5 and 6(a). The y axis of all instruments coincided with the swinging direction of the bells in the upper chamber (north direction). The sampling frequency was set to 100 Hz. All stations were synchronized via GPS signal receivers.
The experiments were designed to consider different kinds of actions: vibrations induced by the only sound of the carillon, by the swinging of the major bells in case of a free or fixed clapper and by the bells’ swinging and carillon acting together. In addition, some ambient vibration measurements were carried out before and after the experiments to identify the modal properties of the tower and recognize potential damage. Table 1 describes the six experiments; Experiment 0 (OMA) and Experiment 7 (OMA) denote the initial and final ambient vibration test, respectively.
Figure 6(b) shows the velocities recorded by using the SS20 2045 triaxial velocimeter along the x, y, and z directions (top to bottom) during Experiment 0; the measured values, rather low, are about 0.05 mm/s along the x and y directions and 0.02 mm/s along z.
Figures 7 and 8 show the velocity measured by using the seismic stations during Experiments 2 and 3. The swinging of the four bells in succession is evident in the signals. The velocity recorded in Experiment 3 is about three times lower than that recorded during Experiment 2 when clappers are free to move and strike the bells.
The signals also show that when the bells’ clappers are free to move (Experiment 2), the heaviest bell (bell no. 54) induces the maximum velocities, as expected. When the clappers are not allowed to play, the maximum velocity’s magnitude is induced instead by bell no. 55. A visual inspection of the videos recorded during the experiment revealed that although bell ringers did not allow the clappers to ring, they sometimes rang anyway during the bells’ oscillations. During the swaying of bell no. 55, the clapper rang four times, two times during the swaying of bells no. 52 and no. 53, while only bell no. 54 oscillated without strokes. The clappers’ strokes are, in fact, evident in the experiment’s signals.
Figure 9 provides further information on the tower’s behaviour over the six experiments in the band [0, 15] Hz; it shows the spectrogram of the signals recorded in the x and y directions by using the SS45 2897 sensor, placed in the lower bell chamber at a height of 15.10 m. The tower’s natural frequencies, identified in Subsection 3.2, are visible in the band [0, 9] Hz (horizontal lines). As a result of bell swinging, the power spectral density increases in the entire band during all experiments, particularly during Experiment 5. It is worth noting that the maximum increase in the power spectral density during the experiments is visible in the band [1, 3] Hz. Figure 10 shows, for the six experiments, the velocities in the x and y directions recorded over the time by using a couple of sensors 2045–2897 and 0944–2896 placed at different quotes and aligned along the z axis (Figure 5). The dashed line in each chart represents linear regression, reported along with the squared correlation index R2.
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(b)
Except for Experiments 1 and 6 (carillon only), the R2 values are high, indicating a good correlation between the velocities and a linear trend alongside the tower during bell swinging. Interestingly, when the bells’ clappers are not allowed to play (during Experiment 3), the R2 index, calculated between the velocities recorded in the y direction, increases compared to that of Experiment 2 for both pairs of instruments.
Table 2 reports the peak component particle velocities (PCPVs) for Experiments 1–6 (PCPV denotes the maximum absolute value of the velocities recorded in each direction by an instrument during the experiment). The components that exceed a limit value of 8 mm/s suggested for historical buildings by many technical rules [18, 19] are highlighted in bold. The PCPV values versus the tower’s height are plotted in Figure 11 for the x and y components; the black dash-dot lines in the charts represent a limit value of 8 mm/s. The analysis of the previous figures and tables allows one to make the following remarks:(i)The carillon’s sound (Experiments 1 and 6) can be modelled as an impulsive action on the tower and induces velocities in the order of 0.5 mm/s, both in the x and y directions.(ii)Experiments in which the bells in the upper chamber swing (Experiments from 2–5) produce the highest velocity values in the swinging direction (y). These values are more significant than those induced by the carillon alone.(iii)The most robust action is induced on the tower in Experiment 5 when all the bells (carillon plus swinging bells) ring. The highest measured values are located over the vault overlooking the upper bell chamber and reach 30 mm/s.(iv)Swinging of only bell no. 54 (Experiment (2) gives velocity values (up to 20 mm/s) comparable with those induced by all the bells playing.(v)Experiment 3, in which the bells swing without clappers, induces velocity values three times lower than those measured in Experiment 2 when the clappers are free to move and strike the bell.(vi)The carillon’s sound (Experiments 1 and 6) induces velocities that vary linearly with the tower’s height.(vii)The experiments with the bells swinging (Experiments from 2–5) result in a linear velocity trend alongside the tower up to the bell chamber’s floor (18 m), with values in a limit of 8 mm/s. Robust amplification of the velocity is shown instead in the upper portions of the tower. This behaviour reflects the different stiffness of the bell chamber with respect to the lower tower’s structure and the fact that the reaction forces induced by swinging are applied to the tower in the high section of the bell chamber. It might also indicate that nonlinear behaviour occurs in the upper part of the structure, as confirmed by the presence of cracks in the dome and the numerical outcomes.
It is worth noting that the three instruments located in the upper bell chamber (at 18 m) and the vault (at 24 m) showed some signal saturation phenomena in the y direction during most energetic Experiment 5. Based on the recorded waveforms, we can estimate in the y direction a decrease of about 15% in the PCPV magnitude. Moreover, the SS45 2542 station at the tower’s base stopped recording during Experiment 6 and was reset during Experiment 7 (OMA).
3.2. Dynamic Identification of the Tower
The dynamic identification of the tower is performed by employing the ambient vibration measurements (Experiments 0 and 7, Table 1) carried out before and after the experiments involving the bells and carillon. All data recorded by using the seven triaxial velocimeters SS45 and SS20 (Figure 5) are processed by the covariance-driven stochastic subspace identification method (cov-SSI) [20] implemented in the MACEC 3.4 toolbox [21]. Both OMA experiments are characterized by a time window longer than 2000 times T1 [22, 23], where T1 = 0.513 s is the structure fundamental period assessed onsite by the fast Fourier transform of the signals from an instrument placed at the tower top.
Table 3 shows the value of the first six frequencies f identified for Experiments 0 and 7, together with the corresponding damping ratios ξ and the modal phase collinearity (MPC) values. MPC is a parameter variable between 0 and 1, characterizing the “complexity” of an eigenvector; MPC is equal to 1 for real eigenvectors [24].
The last two columns of the table show the relative differences Δ in the frequencies gathered by Experiments 0 and 7 and the modal assurance criterion value (MAC) [20] calculated between the mode shapes ϕ0 extracted from Experiment 0 and corresponding ϕ7 of Experiment 7.
Figure 12 shows the experimental mode shapes. The first two shapes represent bending modes along the y and x directions. The third and sixth ones are torsional mode shapes, while the fourth and fifth are bending mode shapes along the diagonal direction of the cross-section of the bell tower.
Some remarks follow from the results reported in Table 3:(i)The maximum value of Δ is about 1.0% and has the same order of the parameter assumed by the MACEC code for clustering the frequencies in the stabilization diagram.(ii)OMA Experiment 7 was carried out immediately after the experiments involving the bells and carillon; thus, the input acting on the bell tower could not fully satisfy the assumption of the white noise signal, which is the fundamental hypothesis underlying the OMA techniques.(iii)The third mode shape is not identified in Experiment 7, probably due to the influence of previous Experiment 6, which mainly brings out the tower’s bending mode shapes instead of the torsional ones.(iv)There are no significant changes in mode shapes, with the MAC values being greater than or equal to 0.97.(v)There are no significant variations in the MPC values [25].
Therefore, it follows that the slight variation in the bell tower’s frequencies may likely be attributed to the choice of the clustering parameters and modelling uncertainties and cannot be associated with any structural damage caused by the bell drive.
4. Numerical Simulations
This section is devoted to calibrating the finite element model of the tower and simulating its dynamic behaviour when subjected to the action of the biggest of the swinging bells.
To this aim, by following the approach described in numerous papers [8–11, 14], the FE model of the tower is subjected to the time-varying force transmitted by the bell and modelled via the approximated formulation proposed in [15]. A nonlinear dynamic analysis is conducted with the NOSA-ITACA code [16] by assuming that masonry materials constituting the tower have zero tensile strength and infinite compressive strength [26]. The velocities recorded in some selected points by the seismic stations installed on the tower are compared with the numerical velocities calculated by the code. For the sake of comparison, the results for the linear elastic case are also provided.
4.1. FE Model Updating
A refined FE model of the tower was created with NOSA-ITACA [16], software developed by ISTI-CNR for the analysis and calibration of masonry structures. The mesh of the tower, shown in Figure 13, consists of 67,747 isoparametric eight-node bricks and beam elements (element no. 8 and no. 9 of the NOSA-ITACA library) with 81,777 nodes, for a total of 245,331 degrees of freedom. Beams are used to model steel elements supporting the bells. The tower is assumed to be clamped at the base, and the presence of the adjacent sanctuary is considered via fixed constraints, up to a height of about 4.5 m on the southwest side.
The FE model has to be calibrated in order to determine the unknown tower’s characteristics (Young’s moduli of constituent materials) by matching the numerical results with those obtained via experimental vibration tests.
The calibration of historical constructions and towers has been addressed in many papers by using stochastic and deterministic approaches. Stochastic approaches take into account and assess uncertainties in experimental data and numerical models [27] and are mainly based on Bayesian parameter estimation methods [28]. Deterministic approaches to model updating range from the Douglas–Reid method [29] to the multistart optimization methods [30] aimed at avoiding possible local minima. The model updating procedure implemented in NOSA-ITACA described in [31] belongs to the realm of deterministic methods and calculates the global minimum point of the objective function in the feasible set.
The calibration of the tower is conducted by minimizing the distance between the first five numerical frequencies and their experimental counterparts inferred by Experiment 0 only. In fact, in Experiment 7, the third frequency is not identified, and calculation of the remaining dynamic features could be influenced by oscillations induced on the tower by previous Experiment 6. The choice to exclude the sixth frequency identified in Experiment 0 from the model updating process is justified by a high degree of uncertainty affecting this quantity and the corresponding mode shape.
For the purpose of calibration, the FE model is subdivided into four sets of materials (based on scarce information recovered from a visual inspection) with different Young’s moduli E1, E2, E3, and E4, as shown in Figure 13. The sets coincide with the higher portion of the tower including the dome, drum, and bell chamber (E4), the tower’s internal slabs (E3), the lower portion of the structure (E1), and, finally, the portion included between the lower bell chamber and the tower bottom (E2). The elastic moduli E1, E2, and E4 have been allowed to vary within the interval [1.0, 8.0] GPa, while E3 ranges within [1.0, 15.0] GPa, in consideration of different construction techniques and materials used for the slabs. The mass density and Poisson’s ratio of all materials are fixed at 1800 kg/m3 and 0.2, respectively; the bell’s mass is uniformly distributed at the lower (522.20 kg/m2) and upper (71.17 kg/m2) bell chamber levels.
Table 4 reports the optimal Young’s modulus values recovered . Furthermore, the table shows, for each optimal value, the parameters ζ and η calculated by NOSA-ITACA at the end of the optimization process using Jacobian of the numerical frequencies (fnum1, fnum2,fnum3,fnum4, and fnum5) with respect to (E1, E2, E3, and E4), calculated at the minimum point. The parameters ζ and η allow us to assess if the optimal parameters obtained are well-defined and robust against perturbations in the experimental data due to noise [31]. If the indices satisfy the inequalities 0 << η << ζ, then the optimal values recovered are reliably determined from the experimental measurements, even if subjected to noise (as it happens for E1, E2, and E4). The condition η << ζ << 1 (occurring for the parameter E3 of the slabs) means that the optimal value obtained cannot be reliably determined.
The table also reports the ζ−1 and η−1 quantities, which estimate the minimum and maximum percentage error in the assessment of the parameters’ optimal values, under the hypothesis of a 1% error in the identification of the experimental frequencies [31]. The table shows that, in the worst-case scenario, the estimated parameters E1, E2,and E4 will be affected, at most, by a 9.2% error (E1). This high percentage error can be justified by the uncertainties on the boundary conditions adopted to simulate the connection between the tower and the sanctuary. In the absence of accurate knowledge, this connection was modelled with discontinuous constraints in some nodes of the FE model. The same justification applies to the relatively high value of Young’s modulus E1 recovered. Concerning the slabs’ elastic modulus (E3), a very high percentage error means that this parameter cannot be reliably estimated and that it does not influence the frequencies, as confirmed by the results of the subsequent sensitivity analysis.
Table 5 summarizes the numerical frequencies of the tower corresponding to optimal Young’s modulus values and their relative error |Δf| with respect to the experimental counterparts and the MAC value calculated between the numerical and experimental mode shapes. The maximum value of the relative error |Δf| is 0.10% (for the first two frequencies). Figure 14 shows the numerical mode shapes calculated by NOSA-ITACA at the optimal point (E1, E2, E3, and E4).
To investigate how variation in Young’s modulus could influence the numerical frequencies, justify the choices made in the calibration described above, and thereby highlight the effectiveness and reliability of the sensitivity analysis implemented in NOSA- ITACA, a Global Sensitivity Analysis (GSA) has been carried out. Sensitivity analysis is generally performed to choose the number of updating parameters and exclude some uncertain parameters from the model updating process. In this case, we use the sensitivity analysis to highlight the potential of the NOSA-ITACA code, which allows estimating, with reduced computational costs, both the optimal values of the unknown parameters and their reliability, thus effectively guiding the user through the model updating process. To this aim, the FE model is subdivided into six sets of materials with different Young’s moduli (from E1 to E6, as shown in Figure 15), assuming that no information about the materials’ mechanical properties is available.
The GSA has been executed through the SAFE toolbox [32], linked to simulation models running in the NOSA-ITACA code. The elementary effects test (EET method, [33]) is used to evaluate sensitivity indices, assuming that Young’s moduli of the six materials shown in Figure 15 have a uniform probability distribution function, and adopting the Latin hypercube method [34] as a sampling strategy. From Figure 15, where the sensitivity indices calculated via the EET method are plotted, it is possible to deduce that Young’s moduli of materials 1, 2, and 4 affect the numerical frequencies much more than the remaining parameters. These results confirm substantially the information recovered by the quantities ζ and η as calculated by NOSA-ITACA and reported in Table 4, and they justify the choice of considering only four parameters instead of six. It is also worth noting that the computational cost of such a global sensitivity analysis is very high (Figure 15 shows the results of 560 FE modal analysis runs) with respect to the cost of the optimization procedure implemented in NOSA-ITACA, which provides both the global minimum point and an assessment of its reliability after 7 runs.
4.2. Dynamic Analysis
Since the experimental tests described in Section 2 did not directly measure the bells’ oscillations, the action of the bells has been assessed by adopting an empirical approach. Once the geometry of the bells is known, the approximated formulation proposed in [15] can be applied and allows us to estimate reaction forces transmitted by the bells to supports during swinging. In particular, the quantities of interest are the maximum amplitude of the reaction forces and their frequency content. A comparison of the frequencies of the bells’ oscillations and the natural frequencies of the tower may highlight possible resonance phenomena.
The analysis is applied to bell no. 54, the heaviest swinging bell. The geometry of the bell can be deduced, once the mouth’s diameter D is known, as a function of the “module” m = D/15. The bell note is also a function of D. The diameter of the bell’s transverse sections varies from D (mouth) to D/2 (top), and it is worth noting that this interval corresponds to the extension of a complete octave and produces the typical timbre of the bell sound. The diameter D of bell no. 54 is 882 mm and corresponds to A4. In the absence of a detailed geometrical survey, the geometry of the wooden counterweight has only been estimated herein. Table 6 reports the main features of the oscillating system sketched in Figure 16, with reference to [15]. In the table, m represents the system’s mass, the unbalance s measures the distance of the centre of gravity to the rotation axis, and Οs is the mass moment of inertia, evaluated with respect to an axis parallel to the rotation axis and passing across the centre of gravity. The shape coefficient c is thus defined as
The quantities in Table 6 are evaluated for the bronze bell, the wooden counterweight, and the global system. The effect of the counterweight on the global system’s unbalance is highlighted in the table.
The features of the bell’s oscillation were deduced by an inspection of some videos filmed during the experiments and some other liturgical ceremonies [15]. The swinging effects over the tower’s structure strongly depend on the maximum angle reached by the system with respect to the vertical line during oscillation. We investigated the effects of different values of , from 66° suggested by [15] for wholly unbalanced systems (bronze bell only) to 160°. As reported in similar case studies [3, 11], the axial stiffness of the tower is much greater than bending stiffness; the problem is therefore governed by the horizontal reaction as follows:
In equation (2), G is the weight (N) of the system and Ω is the circular frequency (rad/s). The equation (2) proposed in [14] approximates using the odd terms of a Fourier series, with the fundamental frequency related to the number of the bell revolutions per minute (Table 7). The larger the value of , the greater the superharmonic term contribution. Figure 17 shows the fast Fourier transform of the function for different maximum angle values. The dashed red line indicates the fundamental frequency of the tower (1.951 Hz), which corresponds to the bending mode in the swinging direction of the bell. When the swinging angle increases, the contribution of the superharmonic terms goes closer to the fundamental frequency, which is about 23% higher than the third-order harmonic term and 20% lower than the fifth term. It is worth noting from Figure 17 that the frequencies that give the main contribution to the harmonic spectrum of H (t) are in the range [1, 3] Hz, in agreement with the experimental behaviour detected in Figure 9. The maximum amplitude of also depends on . The maximum of the function , plotted in Figure 18 for (which has been detected many times by the video inspection), could represent a dynamic amplification factor of the bell’s weight, and it is substantially in agreement with those suggested in [35], where a simplified method is proposed based on equivalent static analysis and an amplification factor of 1.55 is reported. It is worth noting, however, that the static analysis does not allow for considering the interaction between the frequency content of the force induced by the swinging bells and the dynamic properties of the tower.
The numerical model calibrated above has been subjected to the dynamic action H (t) plotted in Figure 18, and the numerical results are compared with the experimental ones. The nonlinear behaviour of the tower is modelled by adopting the constitutive equation of masonry-like materials, implemented in NOSA-ITACA, which models masonry as an isotropic nonlinear elastic material with zero tensile strength and infinite compressive strength. Assumptions underlying the model are that the infinitesimal strain tensor E is the sum of an elastic part Ee and a fracture part Ef and that the stress tensor T depends linearly and isotropically on the elastic part. The fracture strain is positive semidefinite and satisfies suitable orthogonality conditions involving stress, which turns out to be a nonlinear function of the infinitesimal strain [26]. Results of the nonlinear analysis are finally compared with those obtained via classical linear analysis.
Before applying H (t) to the model, a nonlinear static analysis considering only the self-weight has been performed, obtaining the results summarized and shown in Figure 19:(i)The maximum value of the stress vertical component σzz is relatively low if compared to the values of compressive strength reported in the literature for masonry, and this justifies the use of a masonry-like material with infinite compression strength.(ii)The model is able to catch the crack pattern in the tower’s dome.(iii)The fracture strains in the lower part of the tower’s mesh can be attributed to the constraints applied to the model (no apparent cracks are visible in the actual masonry structure).
Afterward, some dynamic analyses with time integration are performed, using the H (t) force applied along the y direction, both in linear and nonlinear cases. Function H (t), evaluated for α = 90° and c = 0.79, is plotted in Figure 18. The damping matrix has been calculated according to the Rayleigh hypothesis, using the experimental damping ratios estimated for the first two frequencies in Experiment 0 and reported in Table 3.
A comparison between experimental and numerical results is provided in Figure 20. Figure 20(a) shows the maximum velocities in the y direction plotted versus the tower’s height, achieved through the dynamic analyses (dashed black line for the linear case and continuous green line for the nonlinear case); the experimental results are indicated by orange squares. Figure 20(b) reports the fast Fourier transform of the velocities recorded by using the SS20 0943 sensor and their numerical counterparts.
The analysis of the figure allows making the following observations:(i)The numerical models overestimate the maximum velocity values compared to the experimental ones, with regard to the upper portions of the structure; it is worth noting, however, that the upper bell chamber’s model, whose bending stiffness is substantially different from that of the tower’s frame, is affected by numerous uncertainties regarding the actual geometry of the iron structure supporting the bells’ system and the constituent materials of masonry pillars.(ii)The linear and nonlinear numerical results are practically coincident up to a height of about 18 meters, thus suggesting that the tower does not enter the nonlinear field as an effect of the bell’s oscillations; the results of the two analyses differ in the tower portion between 18 and 24 m, corresponding to the upper bell chamber.(iii)Both experimental and numerical results are characterized by a frequency response in the interval [1.9, 2.0] Hz, which also contains the first two natural frequencies of the tower. In addition, the numerical results show a predominant frequency of about 2.7 Hz, which also appears in the plot of the FFT of H (t) in Figure 17.
It should be noted that the expression of H (t) suggested by the technical rules [15] depends on many unknown parameters, all characterized by a high degree of uncertainty. At the same time, the choice of H (t) strongly influences the dynamic response of the numerical model. The results obtained in this paper show that by adopting the approach proposed in [15] and appropriately choosing the unknown parameters, the numerical model can capture some relevant characteristics of the dynamic behaviour of the tower. In particular, the numerical maximum velocities at selected points along the tower are substantially in good agreement with their experimental counterparts up to 18 m. These encouraging preliminary results underscore the importance of combining experimental tests and numerical simulations while investigating the influence of parameter uncertainties on the dynamic response of the FE model.
5. Conclusions
This paper describes the investigations conducted on the tower of the Santissimo Crocifisso Sanctuary in Castel San Pietro, aimed at assessing the effects of the swinging bells on the structure’s dynamic behaviour. The structure, featuring a carillon with fifty-five bells of different sizes, has been subjected to a careful, complex monitoring campaign consisting of six experiments in which the bells have been selectively activated, complemented by two ambient vibration tests.
The primary outcomes of the experimental campaign can be summarized as follows. The carillon’s sound can be modelled as an impulsive action on the tower and induces velocities in the order of 0.5 mm/s, both in the x and y direction, while experiments involving the bells in the upper chamber produce the highest velocity values in the swinging y direction; these values are more significant than those induced by the carillon alone. The most robust action is induced on the tower when all the bells (carillon plus swinging bells) ring; the highest measured values are located over the vault overlooking the upper bell chamber and reach 30 mm/s. Swinging of only bell no. 54 gives velocity values (up to 20 mm/s) comparable with those induced by all the bells playing.
The velocities induced by the swinging bells have a linear trend alongside the tower up to the bell chamber’s floor (18 m), with values in a limit of 8 mm/s. Robust amplification of the velocity is shown in the tower’s upper portions.
The FE numerical simulations conducted on the tower via the NOSA-ITACA code encompass a preliminary calibration of the tower model and dynamic analyses of the structure subjected to the action of the heaviest swinging bell. The results of the numerical simulation are affected by assumptions made for the structure supporting the bells, whose geometry and characteristics are not adequately known, and by several uncertainties regarding the action of the swinging bell and then the expression of the time-dependent load H (t). Despite these uncertainties, the numerical results, in terms of maximum velocities calculated at some selected points, are in reasonable agreement with the experimental ones. In addition, a comparison between the results of a linear and a nonlinear analysis, the latter conducted by modelling the masonry constituting the tower as a masonry-like material, shows that the tower behaves linearly under the action of the swinging bell up to a height of about 18 meters. The preliminary dynamic analysis conducted in this paper is the first step towards accurate numerical modelling, which considers the numerous unknown parameters influencing the tower’s response to time-dependent loads.
Data Availability
The experimental data used to support the findings of this study have not been made available because two research projects are still in progress related to this article.
Disclosure
The manuscript has already been published as a preprint based on the link https://hal.archives-ouvertes.fr/hal-03789905/document.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This research was partially supported by the Fondazione Cassa di Risparmio di Lucca (SOUL project, 2019–2022) and the Italian National Research Council (Revolution Project, Progetti di Ricerca @CNR, 2022–2024). The authors wish to thank Michele Naldi, Stelio Montebugnoli, Gilberto Dallavalle, and Giada Gasparini for the technical support in the experimental campaign. Open access funding was provided by CNR within the Revolution Project, Progetti di Ricerca @CNR, 2022–2024.