The Dynamic Behaviour of Some Bell Towers During Ringing J.M. Wilson

Home , Bell tower, Church bell, Full circle ringing, Ring of bells

The dynamic behaviour of some bell towers during ringing

J.M. Wilson, A.R. Selby, S.E. Ross

University of Durham, Durham, UK

Abstract.

When bells are rung to the English system they are swung full-circle through 360° from the mouth-up position. A body which is constrained to move around the arc of a circle experiences acceleration, and so a swinging bell imparts considerable force on to the bell frame and thence to the tower. This causes the tower to respond dynamically, primarily in swaying motion. The bell forces can be estimated from relatively simple measurements, and the motion of the tower can then be estimated computationally, using finite element methods. Three church towers in Northern England have been studied, and additionally, the dynamic movements of the towers have been measured for comparison with computed values.

Introduction

The traditional English church is of masonry construction and has a cruciform plan. A tower, sometimes surmounted by a spire, may be situated above the intersection of the nave and transepts or at the west end of the nave, or more rarely may be free-standing.

Masonry towers are generally of thick-wall construction incorporating a sandwich of faces of ashlar or coursed stone and a filling of loose rubble.

Large towers may be buttressed. The tower will normally house a ring of between four and 12 bells, depending on its size. The bells are carried in a bell frame, see Figure 1, and the ringers swing the bells by ropes from within a ringing chamber situated below. Traditionally bell frames were of timber, but modern frames are of steel construction or occasionally of reinforced concrete. In order to minimise the potentially damaging effects of the dynamic forces from the swinging bells, modern frames are relatively stiff, and are securely built into the tower structure.

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In the English system of bell ringing, Wilson 1, the bells are rung full circle in various patterns of sequences, known as methods. Bells also may be rung in simple fixed sequences known as rounds, two or more bells may be 'fired' simultaneously, and single bells may be tolled individually.

When a bell is swung full circle from the mouth-up position about an axis which is not coincident with its centre of mass it behaves as a compound pendulum undergoing non-linear oscillations. The forces transmitted to the bell frame and the tower depend on the mass of the bell and on the eccentricity of its centroid from the axis of suspension. A number of investigators have published work on the behaviour of bell towers in response to bell ringing. An important contribution to the practice of bell hanging and its consequences for bell towers was made by Heywood^, in which reference was made to the significant theoretical study of bell forces by Lewis^. More recently, Heyman and

ThrelfalH found good agreement between theoretical predictions of inertial forces from bell ringing and those measured in laboratory tests. They also developed a useful method for deducing the inertial properties of a bell from simple in-situ measurements, namely timing the frequency of small amplitude oscillations of the bell and measuring the static rotation of the bell due to weights attached to the bell rope. Wilson and Selby^ extended the analysis to include the initial impulse provided by the ringer.

While the horizontal and vertical force components are readily expressed mathematically as functions of angular position with respect to mouth-up, it becomes necessary to define the forces produced by the swinging bell as functions of time. An appropriate technique is the use of elliptic integrals evaluated numerically through standard computer algorithms, Wilson^. Typical time dependent forces are shown in Figure 2, for the complete oscillation of a bell involving two full circle swings during which the bell sounds twice. The duration of such an oscillation is typically of the order of five seconds with a ringing interval of half this time. For certain simple methods of ringing such as tolling or ringing rounds the forces acting on the tower are periodic and can be represented by Fourier series, Wilson^.

In this study, three towers in the north of England were investigated: St. Brandon's in Brancepeth, St. Oswald's in Durham and St. Cuthbert's in Benfieldside. In each case the towers were surveyed and the bell parameters were obtained using the method of Heyman and Threlfall. The dynamic forces computed from these measurements were imposed on to a finiteelemen t model of the masonry tower, to produce an estimate of the sway of the tower as a function of time. In addition, measurements were taken of the sway of each tower using sensitive velocity transducers, which gave specific information about the fundamental natural frequency, mode shape and damping. Comparison between measurements and estimated values gave an appreciation of the accuracy of the finite element representation of the tower structure.

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The Churches In The Study.

1. St Brandon's Church, Brancepeth. This church has been studied over several years, and measurements and analysis were reported by Selby and Wilson?. Additional measurements of tower sway response have since been made. The church dates from the 12th century and was built using local sandstone. The tower is located at the west end of the nave and is some 20m high. It houses a ring of eight bells, and the heaviest, tenor, bell has a mass of 693 kg.

2. St Oswald's Church, Durham. This church dates in part from the 12th century, although the tower and other parts of the structure were added in the 14th and 15th centuries. It was constructed from local yellow sandstone. The tower is of sandwich construction and is located at the west end of the nave. It is some 25 m high and square in cross section. Access to the ringing chamber is by stone spiral staircase, which allows measurements of sway at regular intervals as well as in the ringing chamber and above the bell frame. The ring of bells comprises eight bells, with the tenor having a mass of 651kg.

3. St. Cuthbert's Church, Benfieldside. St Cuthbert's was constructed during the 1840's of dressed masonry, and was consecrated in 1850. It stands on the side of a steep hillside, and because of this feature it is unusual in that the nave runs north-east to south-west. In this text, for convenience, the altar will be described as the 'east' end. The bell tower is at the true west corner of the masonry church. The square tower is small both in plan and elevation, and is surmounted by a masonry spire, being some 35m in height overall. The ring of six bells are hung on a steel frame on two levels, the heaviest bells being on the lower level.

Characterisation Of The Bells.

The first stage of each study was to estimate the properties of all the bells in each ring of bells. The masses, m, of the bells were available from foundry records for all three towers, and could be accepted as reliable; however a 20% addition was made in every case to make allowance for the headstock and ropewheel, Heyman and Threlfallt Weights of up to 600N were applied incrementally to each bell rope, and rotation of the bell was measured; from these measurements the distance from the support axis to the centroid of mass, h, of the bell was estimated. Next, small oscillations of the bell were then timed, over 50 cycles, to give the period, T, and thence an estimate of the moment of inertia, mk^. These parameters are recorded in Table 1, for the heaviest bells swinging E/W and N/S in each church, and can be used as input to a computer program to derive dynamic forces as a function of time, as in Figure 2.

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Table 1. Bell data.

Church Bell Mass (kg) k(m) h(m) T(s)

St. Brandon's 7(N/S) 498 0.390 0.338 1.780 Brancepeth 8(E/W) 693 0.426 0.387 1.855 St. Oswald's 7(N/S) 447 0368 0.282 1.750

Durham 8(E/W) 651 0.406 0.302 1.848 St. Cuthbert's 3(E/W) 338 0.329 0.322 1.627 Benfieldside 6(N/S) 643 0.406 0.313 1.837

Finite Element Modelling Of The Towers

The modelling of structural behaviour by the finite element method is now commonplace, but considerable care is necessary in constructing the model if realistic results are to be obtained, especially in transient dynamic analyses. A church tower can be modelled by either a vertical cantilever using Timoshenko beam elements or as a fully 3-dimensional structure using 'brick' elements. However it is necessary to make several assumptions for either model. First, the detailed behaviour of a sandwich wall of outer masonry skins with rubble infill is approximated as a solid wall which has an effective elastic modulus, Egg; much reduced from the elastic modulus, E, of intact sandstone; typically E is about 17 GPa for Durham sandstones, while an appropriate value of E^ff varies between 3 and 6GPa. A second major unknown factor is the rigidity of the foundation; the towers were probably built directly on to firm clay perhaps one or two metres below ground surface; the towers were assumed to be 'fixed' at ground level. The third area of uncertainty is the degree of restraint offered to the tower by the surrounding church walls and roof; the towers were assumed to be independent of the adjacent church structure.

Acceptable resonance analyses of the towers were achieved using the cantilever beam approach, with a tuned value of Egff so as to match the towers' fundamental frequencies in sway mode. The computed values of the horizontal force functions from tolling of the heaviest bells were then applied in a time- stepping analysis, to give estimated tower sway as a function of time.

Measurements Of Tower Response.

The dynamic response of each tower, during tolling of the heaviest and next bell was undertaken at each church. Several pieces of information could be derived.

Firstly, by tolling one bell then ceasing, a decaying trace was obtained which gave the fundamental natural frequency of the tower in that direction (either east/west or north/south), and also the damping factor. Table 2 summarises the

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natural frequencies, damping factors and full-circle ringing periods for the three towers in both directions.

Table 2. Frequencies and damping

Church Direction Natural Damping Full circle freq. (Hz) factor period (s)

St. Brandon's N/S 2.59 0.032 5.13 Brancepeth E/W 2.84 0.020 532 St. Oswald's N/S 2.08 0.025 5.20

Durham E/W 2.01 0.025 5.30 St. Cuthbert's N/S 2.25 0.026 4.86 Benfieldside E/W 2.27 0.027 4.83

It is also possible, by moving velocity transducers around the towers to deduce the mode shape of each tower. Logistically, this can present difficulties. St

Brandon's tower was reached by a long wooden stair, and had no accessible points on which to mount the transducers except at the ringing chamber, middle chamber and at the bell frame; consequently the mode shapes deduced for this tower are of limited accuracy. The structure of St Oswald's tower, however, contained a stone staircase, and so much more detailed mode shapes were obtained, with, interestingly, some movement at ground level. The mode shapes for each tower are displayed in Figure 3. Finally, the shapes of the time-based tower sway responses may be compared with the computed traces derived as described above.

Comparison Of Computed And Measured Tower Sway.

The primary objective of the work was to obtain correlation between computed and measured traces of tower sway during tolling of a heavy bell. Comparison between the computed and measured tower velocities (the parameter measured by the transducers) is demonstrated in Figures 4, 5, and 6, which include one example for each church in the study. Further examples are included in several internal reports of Durham University. The style of the response is clearly demonstrated in each case, with a major pulse of tower sway in response to each swing of the bell, with smaller amplitude decaying free oscillations between, when the bell is at rest. This characteristic form of response is clearly shown in both the computed and recorded traces for all three churches.

The absolute maximum values of tower velocity during ringing, in all three churches, for both recordings and computer analyses are summarised in table 3. In all cases, the measured values exceeded the computed values, by up to a factor of two. It can be deduced only that the uncertainties of the assumptions on which the computational models were based caused a consistent

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underestimate of the tower responses, despite the correlation between natural frequencies and mode shapes during free oscillation.

Table 4. Maximum tower responses.

Church Bell Max. recorded Max. computed

velocity (mm/s) velocity (mm/s) St. Brandon's 7(N/S) 4.4 2.1 8(E/W) 4.4 2.5

St. Oswald's 7(N/S) 3.3 2.2 8(E/W) 6.0 2.9 St. Cuthbert's 3(E/W) 1.6 0.6 6(N/S) 2.2 0.9

Conclusions.

The full-circle ringing of church bells imparts substantial dynamic forces onto the tower. These forces have been evaluated for three churches in north-east England. The forces can be applied to finite element models of the towers to

estimate dynamic sway response.

Measurements of tower sway during tolling gave information on the natural frequencies, damping factors, mode shapes and transient response of the towers.

Some correlation between computations and measurements was achieved within the constraints of several unknown factors of the tower structure.

Acknowledgements.

Permission to ring bells in the towers was kindly given by Canon Nugent, and

Rev. Reed. Russell Collingham patiently rang bells 'to order'.

References.

1. Wilson, W.G. The Art and Science of Change Ringing, Faber & Faber, London, 1965. 2. Heywood, A.P. Bell Towers and Bell Hanging - an Appeal to Architects, ms. in Central Council of Church Bell Ringers, 1914.

3. Lewis, EH Experiments at All Saints Church, Loughborough, ms. in Central Council of Church Bell Ringers, 1913. 4. Heyman, J and Threlfall, B.D. 'Inertia Forces Due to Bell Ringing', Int. J. of Mech. Sci. Vol. 18, pp 161-164, 1976.

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5. Wilson, J.M. and Selby, A.R., 'Durham Cathedral Tower Vibrations During Bell-Ringing', in press.

6. Wilson, J.M. Periodic Forces on Bell Towers Arising from Bell Ringing, internal report, Durham University, 1988. 7. Selby, A.R. and Wilson, J.M. The Dynamic Response of a Church Tower to Bell-Ringing', in STREMA2 (ed. Brebbia, C A) Vol. 2, pp 3-16, Proc. of

2nd Int. Conf. on Structural Repair and Maintenance of Historic Buildings, Seville, 1991. Comp.Mech.Publ. Southampton, 1991.

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Headstock Rope Wheel Stay

1 In—- Bearings

[p— Frame

\ -Sound Bow

-Clapper

Slider

Figure 1 Detail of bell, supports, and ringing gear (down position)

Force 3.0 Force s 2.op

2.0 1.0 Time(S) 0 -r 2.0 \ 3.0 40 1.0 -1.0 Time(S) 0 1.0 20 30 40

Figure 2. Normalised vertical and horizontal bell forces, as a function of time.

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Mode shapes, E/W and N/S, of St. Brandon's Church.

Ofl

Mode shapes, E/W and N/S, of St. Oswald's Church.

Mode shapes, E/W and N/S, of St. Cuthbert's Church.

Figure 3. Measured mode shapes of the three Churches.

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J...J-

r Li thl:

5 (sec) 0 5(sec)

Computed Recorded

Figure 4 Unsealed traces of transient velocity against time, belfry level, St Brandon's Church.

0 5(sec) 0 5(sec) Computed Recorded Figure 5 Unsealed traces of transient velocity against time, belfry level, St Oswald's Church.

0 5(sec) 0 5(sec) Computed Recorded. Figure 6 Unsealed traces of transient velocity against time, belfry level, St

Cuthbert's Church.