Verge and Foliot Clock Escapement: Mathematical Appendix
Pallet speed
Analysis of the simple verge and foliot clock of figure 1 (main text) is straightforward at each stage, requiring no advanced concepts or calculation techniques. However, the overall motion is made up of several such stages, occurring sequentially, and so care is needed to present the analysis in a clear fashion. To further ease presentation we will assume, initially, that friction is absent from the mechanism. We will also assume initially that momentum is conserved during each collision, when a gear wheel pin strikes a pallet. The collision is like that of a ballistic pendulum; the rotary motion is considered at the instant of impact to be linear, with linear momentum conserved. We assign an effective mass to pin and pallet as follows:
, (1)
Thus the effective mass is that which, concentrated at the radius of contact, yields the moment of inertia of the actual mass distribution. In Eqs. (1) IG is the moment of inertia of the gear wheels. Also, foliot moment of inertia is taken to be that of the two end masses—the bar itself is assumed to make a negligible contribution.1
The pallet and peg are moving in opposite directions prior to the collision and so linear momentum conservation takes the form
(2)
We have assumed here that the collision is inelastic (energy is not conserved, as we will see) and that the pallet and peg move together with speed u(vpeg, vpallet) immediately after the collision.
Now that we have determined the effects of each collision, we are in a position to study the escapement action in detail. Consider Fig. 2, which illustrates the pallet motion relative to the pegs on the gear wheels. In Fig. 2a a peg on the right gear wheel with effective momentum strikes a pallet with momentum . Immediately after this first collision, which we consider to have been instantaneous, the pallet (indeed, the entire escapement assembly) reverses direction and moves with the peg and gear wheel assembly with a momentum of . This momentum increases due to the action of the suspended mass, which exerts a torque that is equivalent to a force F acting at the peg of
(3)
as is easily seen. Thus the momentum of the system following the first collision is
. (4)
Equation (4) applies for an interval , at which point the pallets have rotated through an angle and the left pallet is just about to collide with a peg on the left wheel, as shown in figure 2b. Substituting Eq. (2) into Eq. (4) we see that the pallet and peg speed just before the second collision are related to the speed v0 just before the first collision via , and in general for the nth collision
(5)
where
, . (6)
Equations (5) are the result presented in the main text, as Eq. (1). We can determine the intervals n (the intervals between “ticks” of successive collisions) by noting that during this time the peg advances a distance (see Fig. 2b). This distance is just the velocity integrated over the interval n, i.e. . Inverting yields
. (7)
This is Eq. (4) of the main text.
Energy and friction
Energy is continually input to the clock mechanism by the falling mass M, yet the mechanism does not gain energy (averaged over one cycle of motion); we have demonstrated that it reaches a limit cycle which is stable against small perturbations. Consequently, energy must be dissipated for this system, as for others that exhibit stable motion. In most dynamical systems of physical interest it is necessary to include the detailed effects of friction in order to model this energy dissipation—indeed, friction is sometimes considered to be an essential component for the stability of dynamical systems2—but including friction in the equation of motion introduces mathematical complexity that can be off-putting to the physics student. The verge and foliot clock escapement mechanism analyzed here dissipates energy via inelastic collisions. This fact means that we can approximate the motion as being frictionless (thus simplifying the mathematics) without losing the essential dynamics. Consequently an analysis of this dynamical system can be performed with less advanced mathematics than is required for other dynamical systems.
For real clocks, friction is an important effect; we have neglected it so far for ease of presentation. In fact the approach adopted here permits us to include friction effects in a general way. Let us rewrite Eq. (5) as
, (8)
where e1,2 represent dissipative effects. If momentum is conserved then , else e1 assumes a value between zero and one. If there is no friction in the gears then else . Note that we do not need to specify the form taken by the friction force, or of the dissipative force that is responsible for loss of momentum; Eq. (8) simply notes the effects of these forces upon the clock mechanism speed from one cycle to the next. If we repeat our limit cycle analysis based upon Eq. (8), instead of Eq. (5), then we find that a limit cycle still exists (though the equilibrium speed changes). Stability is seen to require (assuming for simplicity that ) which is always true and so this limit cycle is stable.
1 If the student is not familiar with moments of inertia, then a simpler derivation of the effective masses Meff and meff is possible. For example, by assuming that the gear wheel masses are negligible compared with the suspended weight, we can obtain Meff via a more elementary consideration of moments.
2 The classic example of limit cycle stability, which demonstrated the essential role played by friction, is that of the Watt flyball governor. The governor regulates steam power via a centrifugal flyball mechanism. It worked well in the early years of steam power (the late eighteenth and early nineteenth centuries) but by the middle of the nineteenth century the flyball governor system exhibited instability in thousands of the newer, better-engineered steam engines. Detailed analysis by James Clerk Maxwell and (independently) by the Russian engineer I. A. Vyshnegradskii showed that the improved engineering of the new machines had reduced friction to an extent that, combined with other changes in engine design, led to instability. For a detailed mathematical analysis of this system see L. S. Pontryagin, Ordinary Differential Equations (Addison-Wesley, Reading, MA, 1962) or Mark Denny, “Watt steam governor stability”, Eur. J. Phys. 23, 339-351 (2002). For a simplified presentation and a historical summary of the centrifugal governor problem, see Mark Denny, Ingenium: Five machines that changed the world (Johns Hopkins University Press, Baltimore, 2007, chapter 5).
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