Each bell is supported on a pivot so that, for each note struck, it can rotate through a full circle. At some time during the swing, the clapper strikes the bell. However, there is not a single, clean impact as one might expect on a xylophone or kettledrum, or indeed on a carillon bell.
As will be shown by experimental results in the next section, the clapper bounces and undergoes a decreasing sequence of multiple impacts. Usually it has come to rest against the side of the bell by the time the swing is completed.
Under normal circumstances, the ringer pulls the rope again more or less immediately to start the reverse swing and the next strike. By making subtle changes in the amplitude of the swing near the top of the circle, just before or just after the moment when the swing stops and reverses, the ringer can make the necessary timing adjustments for controlling the position of the bell in the ringing sequence. This is the subject of this paper.
Experimental measurements on a relatively small bell in the laboratory, augmented by more limited tests on larger bells in a church tower, will be compared with the predictions of a numerical simulation.
The simulation program can be used for virtual design investigations into questions of importance to bell-hangers and ringers. Two quantities of interest are the bell angle at which the first clapper strike occurs and the velocity with which the clapper then strikes the bell. These affect the loudness of the bell and the ease with which the ringer can achieve accurate timing during change-ringing.
Both these quantities are determined by the first flight of the clapper after it leaves the surface of the bell; they are not influenced by the details of clapper bouncing.
It has been shown in earlier work King et al. This will be discussed in some detail in Section 3. Plotting the behaviour in the plane of these two parameters leads to useful design charts. Other relevant issues are influenced by the bouncing of the clapper. These include the frequency bandwidth of the impact, determining how many bell modes are excited, and the envelope of the vibration following the first strike.
These bring in two additional dimensionless parameters: the clapper-to-bell mass ratio and the coefficient of restitution on impact. It will be argued that this low value is caused primarily by the energy used to excite vibration of the bell, and an estimate of the excitation bandwidth will be derived by a simple argument based on energy conservation.
Two less obvious questions about bell dynamics will also be investigated. This then becomes the trailing edge for the next stroke. Some bells allow both types to occur, depending on the initial setup, while others allow only one or the other. The masses, rotational inertias, and geometric configuration of the bell and clapper make the difference, and it is useful to be able to predict the possible behaviour of a bell from knowledge of these details.
A design chart for this purpose, in the parameter plane described earlier, was produced in [ 3 ] based on calculations using a PACE analogue computer of the time. This chart is revisited using digital simulation. This confirms the accuracy of the earlier results and adds significant additional detail.
Given that there are always in fact multiple strikes because of clapper bouncing, the key question is why the effect is usually not heard. Some details of this bell are given in Table 1. It is a bronze bell of conventional shape, having a traditional wrought iron clapper with a spherical striking ball.
Sensors were used to give information about dynamic interactions; accelerometers were attached to both bell and clapper, and also a simple electrical circuit was designed in which the bell-clapper contact acted as a switch, allowing times of contact and noncontact to be detected directly. Typical results are shown in Figure 2 a.
At the left-hand side of Figure 2 a , the clapper is out of contact with the bell. At about , the clapper first makes contact, and both accelerometer traces show vibration starting at that moment. The contact trace and the clapper accelerometer then show very clearly a sequence of later impacts, with decreasing magnitude in the acceleration. By about , the clapper has ceased to bounce and is resting in contact with the bell. This continues until about , when the clapper lifts out of contact and is in flight ready for the next impact on the next swing of the bell.
Figure 2 b shows corresponding measurements on the next strike of the bell, in the reverse direction. The pattern is similar but the details are somewhat different; in particular, the time intervals between the first three bounces of the clapper are longer.
This will turn out to have a significant implication for the perception of the sound of this bell, as will be discussed in Section 4. The bell acceleration in Figures 2 a and 2 b does not show the complicated bouncing activity so clearly. The bell vibrates after the first impact, modified in detail by the later impacts. The bell vibration dies down somewhat during the multiple bouncing, and then just after the time the clapper has come to rest, the remaining vibration in the bell can be seen to die away more rapidly.
If the same bell is struck with a single clean impact, as shown in Figure 2 c , the acceleration trace shows a much longer and cleaner decay. This plot was obtained by ringing the bell at low amplitude rather than full-circle. More detail of the bell vibration is revealed by time-frequency analysis as shown in Figure 3. These plots were generated by short FFTs of overlapping segments of the time data, using a Hanning window on each see Hodges et al. The two plots correspond directly to Figures 2 a and 2 c.
Both show the strongest response at the same frequencies, representing resonant modes of the bell. The single-struck bell shows clear and accurate exponential decay for each of the strongly excited modes, modulated by some evidence of beating. In practice, the two modes are always slightly separated in frequency because of inevitable departures from perfect symmetry, leading to beating. Figure 3 a contains the same strong frequencies, but with profiles through time that are more complicated.
This clutter is presumably caused by the response to multiple impacts from the bouncing clapper, together with some additional input from forces generated at the bearings by the bell and clapper rotation.
This bell has an old-style clapper pivot, consisting of a leather strap around a somewhat worn and rusty iron rod. This generates higher and less steady friction forces than a modern bearing. The single-struck bell shows an exponential decay. Some comments can be made about possible mechanisms underlying this profile. First, there will be some energy loss associated with each impact; possible mechanisms could include local plastic deformation and air being squeezed out of the contact zone.
Second, each impact event will serve to redistribute energy across the frequency spectrum. If energy is transferred from a slowly decaying low frequency mode to higher modes with faster decay, the overall decay rate will be increased. These effects will influence the somewhat irregular initial period of decay. However, neither of these mechanisms can operate once the clapper has come to rest.
The strongest candidate for the observed dissipation after that time is friction. Bending vibration in the rather thick-walled bell will produce some tangential surface motion and thus cause sliding against the resting clapper. It is hard to see at the scale of the plot in Figure 2 , but the clapper accelerometer shows significantly raised levels during the period of the rapid decay of the bell, continuing for some 0.
It seems likely that the relatively long time during which the clapper bounces plays an important, and somewhat counterintuitive, role in the sound of the bell. This sound difference is influenced by factors not relevant to this study, especially the Doppler effect of the moving bell, but the behaviour just revealed surely plays a part. However, while the clapper is bouncing, the bell sound is able to ring on roughly as it would if chimed; it only switches to the faster decay when the clapper comes to rest.
The frictional sliding effect then damps the sound out rather abruptly, in time for the next strike to be heard clearly without much residual vibration from the previous strike. Bells are made of alloys with very low damping, which generally goes with high hardness because both are determined by the mobility of dislocations.
Although at first sight it seems reasonable that the constant hammering of the clapper against the bell would be responsible for the visible wear patch, in fact, frictional wear seems a stronger candidate. Note that matters are quite different for a carillon bell; the frictional effect is then negligible and impact is presumably the main mechanism for generating a wear patch, as has been investigated by Fletcher et al.
When the results illustrated in Figure 2 were first obtained, some scepticism was expressed about whether the clapper bouncing was normal behaviour or whether it only happened on atypically small bells like the one tested.
From a ring of 12 bells, the tenor the largest bell and the fifth bell were selected to give a large and a medium-sized bell to complement the results on the small laboratory bell. It was not possible to install the full range of instrumentation; fitting accelerometers safely on these large bells in a way that allowed full-circle ringing proved impractical.
However, the electrical contact measurements could be made simply and safely. This electrical measurement is sufficient to demonstrate that multiple clapper bouncing occurs during normal ringing with both bells. Example results are shown in Figure 5. The small laboratory bell had about 0. It seems that larger bells exhibit the effect more, not less, than the small bell.
Bouncing obviously needs to be taken into account in any realistic model of the dynamical behaviour of tower bells. The system to be analysed is shown in Figure 6. The bell plus its supporting hardware has mass and moment of inertia about its bearing, and the centre of mass lies a distance from its swing axis. The clapper has mass and moment of inertia about its pivot, and the centre of mass lies a distance from its swing axis.
The bell and clapper swing axes are separated by a distance , which is shown positive in Figure 6 but is sometimes made negative by hanging a bell in a different configuration.
Damping in the pivots is also ignored. Finally, the small-amplitude vibration of the bell and clapper is ignored in this modelling of the dynamics of the ringing process, although it will be seen later that the energy involved in vibration enters the problem indirectly through the value of the coefficient of restitution.
The potential energy of the system is where is the acceleration due to gravity. During free flight of bell and clapper, these generalised forces are zero, but during contact between clapper and bell, there is a contact force that must be expressed through the generalised forces. This will be discussed in the next subsection. These equations of free motion can be used in their full complexity for numerical calculations, but in order to understand the dominant aspects of the underlying physics, it is useful to note that they can be simplified if some reasonable approximations are made.
Equation 5 consists of an additive combination of terms proportional to these various masses and moments of inertia. If the terms involving and are neglected, this equation takes the far simpler form It is now useful to cast 6 and 7 into a nondimensional form. Define where are the lengths of the equivalent simple pendulums with periods of small oscillation matching those of the bell and clapper separately. This form makes it clear that the free motion depends only on two dimensionless parameters: and.
Both are easy to estimate for bells in situ in a tower; can be measured directly, and periods of small oscillation can be determined by timing a few cycles, then 9 can be used to obtain. The behaviour in the plane determined by these two parameters will be discussed in Section 4. The modelling just described is uncontroversial and represents the underlying physics of the swinging bell and clapper quite accurately.
However, the modelling of the detailed forces and deformations occurring near the contact during an impact is far less simple. Fortunately, it is not necessary to represent these details accurately in order to obtain a satisfactory simulation model for the purposes of this paper. As is clear from the results of Figure 2 , the contact time during a single bounce of the clapper is always very short compared to the time scales characterising bell and clapper motion.
However, for the purposes of numerical simulation, such an instantaneous event is not convenient. It is far more reliable to use an approach that gives a finite contact force and a continuous variation of the dynamical quantities , through the impact. To this end, the contact interaction between clapper and bell can be modelled as a simple linear contact spring with a stiffness defined so that the stored potential energy in the spring is where is the maximum value of before contact occurs.
There is a corresponding form for negative. During the first part of each contact, the clapper compresses this spring, then it bounces back as the spring extends again. As first explained by Rayleigh when talking about the impact of a piano hammer with a string [ 8 ], the time of contact is then simply a half-period of this contact resonance oscillation. The reason is that after a half period, the spring force would become tensile , so in practice this is the moment when loss of contact occurs.
To model the observed short contact time, a very large value of is required. However, when becomes sufficiently big, the actual value ceases to matter from the perspective of the predicted motion of bell and clapper; the overall result converges to what would have been obtained with an instantaneous contact using energy and momentum considerations. The choice of for simulation purposes can thus be based on computational considerations rather on a need to represent the detailed physics accurately.
If the value is too big, it can cause problems of excessive computation time because the time step has to be short enough to resolve the contact resonance oscillation. A convenient value was determined using convergence experiments; the lowest stiffness compatible with reproducing the instantaneous bouncing behaviour was chosen.
The usual virtual work argument for calculating generalised forces can be applied to expression 12 , with the result that An interesting consequence of 11 and 13 can be deduced immediately.
At some stage, it lifts off the bell surface into free flight. The angle at which this occurs can be called the lift-off angle, and it shows an interesting pattern of behaviour. This is revealed most clearly for the case. In order to satisfy both of 11 including the nonzero generalised force suggested by 13 , the value of this generalised force needs to be The moment of lift-off is defined by this generalised force changing sign from compressive to tensile, and so with a little rearrangement, the lift-off angle must satisfy The corresponding angle for ringing wrong follows from assuming that , and so when the angles are expressed in degrees.
The two lift-off angles are plotted as a function of in Figure 7. There is another important issue concerning contact modeling. As will be confirmed by results to be shown later, it is important to take some account of the energy lost during impact. This can be done using a simple approach due to Stronge [ 9 ]; the values 13 are applied during the compression phase of contact, but during the rebound phase the stiffness is reduced to , where is a factor less than unity.
This reduction factor directly represents the fraction of stored energy in the contact spring which is lost. During the rebound phase, the generalised forces are given by 13 but with the stiffness replaced by the reduced value. The factor is the energy-based coefficient of restitution introduced by Stronge to resolve paradoxical predictions of earlier theories [ 10 ]. To use this contact model for numerical simulations, a suitable value of is needed, so a simple experiment was carried out.
As in the ringing tests described earlier, the measurement method was first developed on the laboratory bell and then applied to additional tower bells. This set covered the common range of sizes of tower bells and also included clappers of two types: traditional ones made of wrought iron and more modern ones made of SG spheroidal graphite cast iron.
The test procedure was simple. An accelerometer was fitted to the clapper using a clamping device, as in the ringing tests described earlier. The accelerometer signal was recorded by a PC-based data logger and numerically integrated to give a velocity signal.
Each impact between clapper and bell was visible as a jump in the velocity, and these jumps were measured for the first few impacts. The jumps could be determined reliably, but the absolute velocities before and after impact could not because of difficulties associated with the instrumental hardware and with numerical integration.
Absolute calibration of the sensor was not necessary, because the coefficient of restitution can be estimated using only ratios of the velocity jumps, as will now be shown.
Suppose that in the initial impact the clapper falls with velocity defined positive downwards and rebounds from the stationary bell with velocity.
In the absence of friction at the clapper pivot, it will arrive back for the second impact with velocity and rebound with velocity.
The first two measured velocity jumps are then Later velocity jumps can be similarly labelled ,. The factor defined previously is the ratio of kinetic energies after and before the impact, so that Then,. Some assumption must now be made about the proportion of energy lost in the second impact.
At first sight, it seems natural to assume that this is the same fraction as in the first impact, which would give so that could be estimated from However, the effective coefficient of restitution that is being used for the bell simulation model is a rather more slippery quantity than this argument acknowledges; the observed pattern of velocity jumps on the real bells is quite different.
The results are summarised in Figure 8 for all five bells tested and for a range of drop heights of the clapper on each bell. Every individual velocity jump ratio is shown as a star, and the mean of each set is shown as a circle.
The ratio was always quite small and quite consistent in value, whereas and subsequent ratios were much more variable and generally significantly bigger, in many cases bigger than unity. It is remarkable how similar the five bells were in this respect. Bell size and clapper material seem to produce no clear effect. There is a simple interpretation of this pattern. It suggests that a lot of energy is lost on the first impact because it is converted into vibrational energy of the bell—this is, after all, the purpose of the clapper striking the bell.
At later impacts, the clapper encounters a bell that is already vibrating with significant velocity normal to the surface at the striking point, in a phase that is effectively random because it is determined by the vibration frequencies, independent of the time between impacts.
The bell surface may be approaching or receding from the clapper at the moment of impact, and clapper bounce as observed by the attached accelerometer will be correspondingly enhanced or reduced. There is obviously no such thing as a single coefficient of restitution that applies to all impacts; the first impact must be treated differently. Pulling on the Sally is called the "hand stroke" and pulling on the tail end is called the "back stroke".
The reason why so few countries ring "full circle" is because, once the bell get up to the top of the swing then, unless the ringer is careful, the bell could go right over the top and, with bells weighing hundredweights or even tons, the bell would then be out of control and dangerous.
A safety device is therefore fitted called a stay. This is a piece of wood, called a "stay" fitted to the headstock which, when the bell goes just over the top, rests on a slider and prevents the bell going further. The figure on the right shows this in operation.
There is now, however, a danger that, if the ringer is over-enthusiastic, the force of the stay hitting the slider could shake the whole frame and even damage the whole tower!
This can be frightening but rarely does any harm. Practically all learners break a stay at some point - it is part of learning the right strength to use pulling the rope and the tower should carry a stock of spare stays which are not too difficult to fit again.
The bell ringers usually stand on the ground floor, or first floor if there is a lobby below. At the top of the tower, the bells are hung in a wooden or metal frame with each bell fixed to the axle of a large wooden wheel that pivots in ball bearings on the frame. A rope is tied to the wheel spokes, runs partly round the rim and falls through holes and pulleys to the ringing chamber below. Clock hammers sit adjacent to the bells and are used to strike the clock chimes.
Most bell towers contain six or eight bells, with many also having five, ten or twelve bells. There are also towers which have four or less, although they are not generally used for change ringing. These baffle-boarded sound windows are seen on the outside of the tower and help to spread the sound. Behind the wide, decorative faces of church clocks are large mechanisms.
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