In order to analyse and simulate the behaviour of a swinging church bell, we first need the governing equations. This account draws heavily on the published description [1]. The bell and clapper are sketched in Fig. 1. The bell (including its supporting hardware and the wheel that the rope goes round) has mass $M$ and moment of inertia $I_b$ about its bearing, and its centre of mass $G_b$ lies a distance $a$ from its swing axis. The clapper has mass $m$ and moment of inertia $I_c$ about its pivot, and its centre of mass $G_c$ lies a distance $b$ from its swing axis. The bell and clapper swing axes are separated by a distance $r$, which is shown positive in Fig. 1 but is sometimes made negative by hanging a bell in a different configuration. Motion is described by two generalised coordinates: the angle $\theta$ between the bell’s axis and the downward vertical, and the angle $\phi$ between the clapper and the bell’s axis.
Slightly surprisingly, to analyse the key behaviour of the bell and clapper we do not need to take detailed account of the small-amplitude vibration of the bell. The aim here is not to perform sound synthesis of a ringing bell, but to understand how the clapper moves in relation to the bell during full-circle ringing. For that purpose, the only important role of the bell’s vibration is to extract energy from the swinging clapper and thus determine how strongly the clapper rebounds after the first impact. We will represent this via a coefficient of restitution, drawing on experimental measurements and also on the analysis from section 12.1.2 of vibration excitation by a bouncing mass.
We will obtain governing equations by an energy-based approach, making use of Lagrange’s equations. It is easy to write down an expression for the potential energy of the bell-clapper system. The only source of stored energy is from lifting the two centres of mass against gravity, so the energy is
$$V=Mga[1 – \cos \theta] +mgr [ 1 – \cos \theta] + mgb [1-\cos (\theta + \phi)] . \tag{1}$$
For the kinetic energy, we need to add contributions from the linear motion of the two centres of mass and rotational motion about those centres of mass. To obtain the linear kinetic energy of the clapper, the simplest approach is to write down Cartesian components of the velocity of the centre of mass $G_c$ with respect to horizontal and vertical axes: the results are
$$v_x = r \dot{\theta} \cos \theta +b (\dot{\theta} + \dot{\phi}) \cos (\theta+ \phi) \tag{2}$$
and
$$v_y = r \dot{\theta} \sin \theta +b (\dot{\theta} + \dot{\phi}) \sin (\theta+ \phi) \tag{3}$$
respectively. The total kinetic energy is then
$$T =\dfrac{1}{2} (I_b \\-\\ M a^2) \dot{\theta}^2 + \dfrac{1}{2} M a^2 \dot{\theta}^2$$
$$+\dfrac{1}{2} (I_c \\-\\ m b^2) (\dot{\theta} + \dot{\phi})^2 + \dfrac{1}{2} m (v_x^2 + v_y^2) \tag{4}$$
where the parallel axis theorem has been used to obtain the two moments of inertia about the centres of mass.
This expression can be simplified down to
$$T=\dfrac{1}{2} I_b \dot{\theta}^2 + \dfrac{1}{2} I_c (\dot{\theta}^2 + \dot{\phi}^2 + 2 \dot{\theta}\dot{\phi})$$
$$+\dfrac{1}{2} m r^2 \dot{\theta}^2 + mrb (\dot{\theta}^2 + \dot{\theta}\dot{\phi}) \cos \phi . \tag{5}$$
Combining this with equation (1) and cranking the handle on Lagrange’s equations gives the two governing equations
$$[I_b + I_c + mr^2 + 2mrb \cos \phi] \ddot{\theta} +[I_c + mrb \cos \phi] \ddot{\phi}$$
$$-mrb \dot{\phi} (2 \dot{\theta}+\dot{\phi}) \sin \phi + Mga \sin \theta + mgr \sin \phi$$
$$+ mgb \sin (\theta + \phi) = Q_\theta \tag{6}$$
and
$$I_c [\ddot{\theta} + \ddot{\phi}] + mrb \ddot{\theta} \cos \phi + mrb \dot{\theta}^2 \sin \phi$$
$$+ mgb \sin (\theta + \phi) = Q_\phi \tag{7}$$
where $Q_\theta$ and $Q_\phi$ are the generalised forces associated with $\theta$ and $\phi$ motion. These can be used to express all the effects that have not been taken into account by the energy expressions: contact force during impact, frictional forces in the bell and clapper bearings, and any force that may be applied to the wheel by the rope (either from its own weight or from the ringer pulling it).
To allow for energy dissipation in the bearings, assuming linear viscous damping, we can use
$$Q_\theta=-\alpha \dot{\theta} \tag{8}$$
for the bell bearing and
$$Q_\phi=-\beta \dot{\phi} \tag{9}$$
for the clapper bearing, with suitable values of the constants $\alpha$ and $\beta$. By using different expressions, other sources of dissipation (such as Coulomb friction) could be modelled.
Rope force $F$ could be included with another component of $Q_\theta$ of the form
$$Q_\theta = H(\theta) F R \tag{10}$$
where $R$ is the radius of the wheel, and $H(\theta)$ is either $+1$ or $-1$ depending on which direction the rope is pulling the wheel: the rope force must always be positive, of course, but the wheel can be pulled in different directions at different stages in the ringing cycle depending on the position of the “garter hole” in the wheel relative to the pulley (refer back to Fig. 1 of section 12.4 if this is not clear).
Contact between bell and clapper occurs when $\phi$ reaches a limiting value $\phi_{max}$ determined by the geometry — or on the other side of the bell, when $\phi$ reaches $-\phi_{max}$ assuming that the bell and clapper mounting are both symmetrical. As we have done in other problems, we can model the contact force with a contact spring. We know from observations that the time of contact is always very short, so it is sufficiently accurate to use a linear contact spring with a spring constant chosen to be high enough to guarantee a fast bounce. With a suitable definition of contact stiffness $k$, the effect can be captured by a component of generalised force
$$Q_\phi = -k (\phi \\-\\ \phi_{max}) \mathrm{~~~~for~~~~} \phi > \phi_{max} \tag{11}$$
and a corresponding result for $\phi \le -\phi_{max}$.
We can modify this generalised force to incorporate a coefficient of restitution during bouncing, using Stronge’s approach [2] as was described in section 12.1.1. During the rebound phase of contact, when $\phi$ is decreasing after the point of maximum compression of the contact spring, the constant $k$ is replaced by the reduced value $Rk$ where $R$ is the energy-based coefficient of restitution.
To find a suitable value of $R$, we need to look at some experimental results. The test procedure was simple. An accelerometer was fitted to the clapper, as in the ringing tests described in section 12.4. With the bell stationary in the “up” position the clapper was lifted a small distance off the surface of the bell and dropped. The accelerometer signal was 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 absolute velocities before and after impact could not be determined reliably, but this does not matter: the coefficient of restitution can be estimated using only ratios of the velocity jumps. Combined results for several bells are summarised in Fig. 2. Each individual velocity jump ratio is shown as a star, and the mean of each set is shown as a red circle. The first ratio, on the left-hand side of the plot, was always quite small and quite consistent in value. However, subsequent ratios were much more variable and generally significantly bigger: in many cases, bigger than unity. Five different bells were tested, and it is remarkable how similar the results are: bell size and clapper material seem to produce no clear effect.
There is a simple interpretation of the pattern shown in Fig. 2. A lot of energy is lost on the first impact because it is converted into vibrational energy of the bell, but for later impacts the clapper meets a bell that is already vibrating 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.
The observed velocity ratios can be used to estimate the value of $R$. Two different approaches were described in reference [1], which gave upper and lower limits of 0.12 and 0.045. The correct value of $R$ will lie somewhere in between, probably closer to the lower limit, but in any case it is clear that almost all the energy of the initial clapper motion is absorbed at the first impact, and a very low value of $R$ should be used in simulations. Simulations have been run using $R=0.1$ and $R=0.05$.
There is a final twist in this story. The value of $R$ just estimated is only relevant to the first impact. Later impacts will have less energy loss into vibration, but they could only be treated accurately if the bell vibration were explicitly taken into account so that the true relative velocity of bell and clapper could be determined at each impact. In the simplified analysis based on the equations developed here, vibration is not modelled except through the value of $R$, and in practice the same value will be used for all impacts. This obviously conflicts with the results of Fig. 2, but fortunately the value of $R$ is so low that the use of the same $R$ for later impacts does not make much difference to the dynamical interactions of bell and clapper, since so little energy is left after the first impact.
The equations given above can be used in their full complexity for numerical calculations, but in order to understand the dominant aspects of the underlying physics it is very useful to note that they can be simplified if some reasonable approximations are made. The mass and moment of inertia of the bell are invariably much greater than those of the clapper: for the small laboratory bell the mass ratio was 28:1, and even larger values are found in many tower bells. Equation (6) contains an additive combination of terms proportional to these various masses and moments of inertia. If the terms involving $m$ and $I_c$ are neglected, this equation takes the far simpler form
$$I_b \ddot{\theta} +Mga \sin \theta \approx Q_\theta , \tag{12}$$
and if we ignore the rope force and the damping force in the bell bearing, $Q_\theta = 0$.
It is now useful to cast equations (7) and (12) into a non-dimensional form. Define
$$L_b=\dfrac{I_b}{Ma} \mathrm{~~~and~~~} L_c=\dfrac{I_c}{mb} \tag{13}$$
where $L_b$ and $L_c$ are the lengths of the equivalent simple pendulums with periods of small oscillation matching those of the bell and clapper separately. These periods $T_b, T_c$ are given by
$$T_b=2 \pi \sqrt{\dfrac{L_b}{g}} \mathrm{~~~and~~~} T_c=2 \pi \sqrt{\dfrac{L_c}{g}} . \tag{14}$$
Now introduce a non-dimensional time-like variable based on the bell’s period:
$$\tau=t \sqrt{\dfrac{g}{L_b}} . \tag{15}$$
Combining these substitutions, the equations during free flight of the bell and clapper become
$$\theta^{\prime \prime} + \sin \theta \approx 0 \tag{16}$$
and
$$\theta^{\prime \prime} + \phi^{\prime \prime} + \dfrac{L_b}{L_c} \sin (\theta+\phi)$$
$$ + \dfrac{r}{L_c} \theta^{\prime 2} \sin \phi + \dfrac{r}{L_c}\theta^{\prime \prime} \cos \phi = 0 \tag{17}$$
where prime denotes differentiation with respect to $\tau$. This form makes it clear that the free motion depends only on two dimensionless parameters: $L_b/L_c$ and $r/L_c$. Both are easy to estimate for bells in situ in a tower: $r$ can be measured directly, and periods of small oscillation can be determined by timing a few cycles then equation (14) can be used to obtain $L_b$ and $L_c$. The behaviour in the plane determined by these two parameters is discussed in some detail in Section 12.4.
For that study, simulations were run using the equations given above, for a grid of values of the two parameters $L_b/L_c$ and $r/L_c$. The equations are complicated and nonlinear, so there are no tricks or short cuts for simulation: the equations must be tackled directly using a general-purpose solver for ordinary differential equations. I used the routine ODE45 in Matlab for this purpose.
[1] J. Woodhouse, J. C. Rene, C. S. Hall, L. T. W. Smith, F. H. King and J. W. McClenahan, “The dynamics of a ringing church bell”, Advances in Acoustics and Vibration 681787 (2012). The article is available here: http://www.hindawi.com/journals/aav/2012/681787/
[2] W. J. Stronge: Impact mechanics, Cambridge University Press (2000)