INSTITUTE OF PHYSICS PUBLISHING Eur. J. Phys. 26 (2005) 561–577

EUROPEAN JOURNAL OF PHYSICS

doi:10.1088/0143-0807/26/4/002

Siege engine dynamics Mark Denny 5114 Sandgate Road, RR 1, Victoria, British Columbia, V9C 3Z2, Canada E-mail: [email protected]

Received 21 January 2005, in final form 23 February 2005 Published 14 April 2005 Online at stacks.iop.org/EJP/26/561 Abstract

The medieval siege engine is a historically important machine that has latterly been adopted for the purpose of physics instruction. Here we analyse the historical developments and show why these engines ultimately evolved into the highly efficient trebuchet.

1. Introduction

A siege engine1 is an ancient mechanical device designed to hurl rocks or other materials at or over the walls of a besieged fortress. It was usually constructed of wood such as oak, and often strapped with iron or rope for reinforcement. The engines of classical antiquity, such as the onager, derived their power from twisted cord or metal springs [1, 2]. These machines were still in use in medieval times, but by 1200 C.E. had been superseded by the trebuchet [3]. This is a counterpoise machine, powered by gravity. A projectile, connected to one end of a lever by a sling, was propelled by the action of a large counterweight (CW), attached to the other end of the lever. A medieval drawing of a trebuchet is shown in figure 1. Traction trebuchets were invented in China in the 5th–3rd centuries B.C.E. [1, 4, 5]. This early form lacked a CW; the motive power was provided by (up to 250 [1]) soldiers hauling on ropes. Despite their antiquity, the counterpoise engines are generally considered to be medieval devices, because they evolved considerably and grew very large as their use spread westward. Islamic, Byzantine and later western European engineers developed the trebuchet into a leviathan: some trebuchets threw a projectile (usually a cut stone sphere, but sometimes flaming tar or a dead horse, or human) weighing 1 ton or more [1] and were powered by the gravitational force of a 20 ton CW [1, 6, 7]. A more typical projectile mass was m = 100 kg, which could be thrown a distance of 150–350 m, depending upon the trebuchet design and upon which historical source we consult [1, 5, 7]. These engines dominated siege warfare until the 16th century C.E. when efficient cannon began to supplant them. The trebuchet grew large so that it could destroy the strong curtain walls and towers of medieval Islamic and European castles, which in turn grew larger partly to counter the trebuchet—an early ‘arms race’. 1 From medieval European vernacular ingenium (an ingenious contrivance), hence operated by ingeniators (engineers) [1].

c 2005 IOP Publishing Ltd Printed in the UK 0143-0807/05/040561+17$30.00


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Figure 1. Medieval illustration (about 1405 C.E.) of a trebuchet. Note the long sling and hinged counterweight. The wheels on the left are probably part of a treadmill winch, for raising the counterweight. The projectile range in this case must have been less than that of an arrow, judging by the shield attached to the front.

The development of siege engines over a wide area and long period has led to a profusion and confusion of terminology2 . Here we shall follow common but not universal practice and refer to counterpoise engines with hinged CW as trebuchets, and those with CW attached directly to one end of the lever arm (the beam) as mangonels. Counterpoise siege engines in their various forms are of great interest to physicists. Many university physics departments have built trebuchets or mangonels as part of their undergraduate programmes [8]. The pedagogical value for physics and engineering students is considerable. Here we shall emphasize the physical principles involved: potential and kinetic energy and the transfer from one form to the other, Lagrangian mechanics, velocity-dependent forces, constraints and Lagrange multipliers, pumped oscillator, double pendulum and normal modes. These concepts are made particularly clear when applied to counterpoise siege engines because we can realistically assume that the frame and beams of these machines are ideal rigid structures. Thus the physics student need not be concerned with deformation characteristics of the structure material; (to a good approximation) the only force involved is gravity. Also siege engines are readily appreciated and historically important machines; analysis of them illustrates to the student how we can gain insight by the application of physics to the real (in this case medieval) world. Physicists and engineers apart, the wider public is interested in these machines, judging by the number of hobbyists in the US and of museums and historical societies in Europe that build small and large scale machines. The hobbyists have given rise to commercial trebuchet builders, and to trebuchet simulators, with detailed technical analyses [9]. The historical societies (in Denmark, Sweden, England and, particularly, France) have reconstructed siege engines [10], some very large, and these provide a wealth of data, against which we can compare our predictions. 2 Different European names for this machine include trebuchet, mangonel, bricolle, couillard, pierriere, petrarie, matafunda, robinet, biffa and tormentum. These and other cognates, including Arabic nicknames (such as ‘Daughter of the earthquake’), are discussed in great detail in [4].

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Our analysis will employ the concepts listed above, while providing an explanation for the historical developments of counterpoise engines. We will see why adding a sling increases the projectile range (longer lever arm—fairly obvious), and why the hinged-CW trebuchet performs better than the fixed-CW mangonel (pumped oscillator—less obvious). In addition, we shall calculate the efficiency of these siege engines. This is an instructive exercise, of practical importance for two reasons. Firstly a more efficient machine will have a greater range. This mattered because the crew were vulnerable to arrows fired from besieged castle ramparts, so it was clearly desirable to have trebuchet range exceed bow range. Secondly an inefficient engine will retain significant kinetic energy after the projectile has been released. This results in a very slow rate of fire if the crew wait until the energy is dissipated by friction. On the other hand, stopping the beam suddenly may cause damage or may cause the frame to shift. (To destroy a tower or curtain wall, a siege engine usually needed to aim many projectiles at the same location. If the engine frame shifted, it would need to be reset after every shot.) One of the significant advantages claimed for the trebuchet, compared with classical engines such as the onager3 , is that the motion was smooth and gentle. A large and efficient trebuchet could fire twice every hour, and the projectile trajectories were reproducible. One modern reconstruction was able to group shots within a 6 m × 6 m square at a range of 180 m [1, 5]. Our siege engine analysis begins with a simple lever and adds components—hinged CW and sling—so that we can determine the physical significance of each component for the overall performance of the fully-developed large trebuchet that dominated European and Middle Eastern siege warfare for over three centuries.

2. Projectile launch speed and range

It is natural to measure siege engine performance by the attainable range. This depends upon aerodynamical drag, and the projectile launch angle α. In figure 2, we show how projectile range depends upon launch speed u and upon α. Here we have assumed a drag force of the standard form [11] FD = 12 cD ρa Av 2 where v is the projectile speed (hence v = u when the projectile is launched), A is the projectile area, ρ a is the air density and cD is the dimensionless drag coefficient. For a trebuchet projectile, assumed here to be a stone sphere, we have cD ≈ 0.5. The projectile trajectory for such a quadratic (in speed) drag force can be determined only numerically. The calculations are well known (e.g. [11]) and so shall not be repeated here. Because drag is much less significant than gravity, FD  mg, we see that the iso-range contours of figure 2 are nearly symmetrical about α = 45◦ (the symmetry would be exact in the absence of drag). Thus the optimum launch angle, which leads to maximum projectile range for a given projectile launch speed u, is close to α = 45◦ . For the counterpoise siege engines investigated here, the optimum launch angle cannot be simply read off from the contours of figure 2, however, because for these engines u is dependent upon α. For the simplest engine of section 4 and for a typical fully developed trebuchet of section 7 the functions u(α) are plotted in figure 2. From these curves it is clear that the optimum launch angle is in the region 40◦ –45◦ . Figure 2 is a simplification of range calculations, for two reasons. Firstly the iso-range contours depend slightly upon projectile mass m (in figure 2 they are shown for m = 100 kg) because of the drag force, and secondly the dependence of u upon α varies with siege engine parameters. However, for realistic parameter values and projectile masses we find that the optimum launch angle is always close to 45◦ ; we shall assume henceforth in all cases 3

Onager = wild ass due to the kicking motion (recoil) [2].

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investigated here that the best launch angle is α = 45◦ ; this is close to optimum for all the siege engine designs considered. Thus when investigating different siege engine designs we need only calculate u (α = 45◦ ); to obtain a range estimate we then simply consult figure 2 in all cases. 3. Outline of the calculations

In order to understand why the historical development of counterpoise siege engines constituted improved performance we analyse different versions of the machine, starting from the simplest. In figures 3(a), (b) we illustrate a hypothetical siege engine with fixed CW (therefore it is a mangonel) and with no sling. The beam of length L +  is initially set at an angle θ = −θ0 . (Note that, for the convenience of our calculations, θ is oriented clockwise with a negative initial angle, i.e. θ 0 is positive.) With M  mL the beam swings as indicated; a trigger mechanism releases the projectile at launch angle α. The motion of this simplest engine (herein denoted by case I ) is described by a single variable θ . This is the only case that can be investigated analytically. In figure 3(c) we allow the CW to swing—it is hinged. Thus the machine (again hypothetical—we know of no such historical engine) is a trebuchet with no sling. Compared with case I this engine (case II ) is more complicated to analyse (the motion is described by two variables: beam angle θ and CW hinge angle φ) and more efficient. We shall see in section 5 that, with suitable choice of parameters, the hinged CW can provide an extra impetus to the projectile just prior to release.

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Figure 3. Siege engine designs. (a), (b) Case I—the basic lever arm engine, (c) case II—add a

hinged counterweight, (d), (e) case III—the mangonel: fixed counterweight, plus sling (f) case IV— the trebuchet: hinged counterweight plus sling. The trebuchet projectile is here constrained (it may not fall below its initial height) whereas the mangonel projectile is here unconstrained. In all cases m = projectile, M = counterweight, L +  is the beam length, pivots are represented by open circles, beam angle θ has initial value −θ 0, h is the counterweight hinge length, r is the sling length, φ is the hinge angle, ψ is the sling angle, and α is the launch angle.

Case III is illustrated in figures 3(d), (e). This is the mangonel: fixed CW with sling. The motion is described by two variables: beam angle θ and sling angle ψ. To simplify calculations we do not constrain the projectile. For real mangonels the projectile was usually constrained to travel along a horizontal trough until the vertical acceleration exceeded zero. Here we permit the projectile in the sling to swing freely. This implies in practice that a trench be dug beneath the mangonel to permit a free swing (there are historical references showing that this was done occasionally [4]). We shall see that the sling improves performance significantly compared with case II. Launch speed increases because the effective lever arm length is greater than for cases I and II, as we shall see. Efficiency is also increased. Case IV (figure 3(f)) is the most complicated considered here. This is the fully developed trebuchet, with hinged CW and projectile sling. The motion is described by three variables (beam angle θ , hinge angle φ and sling angle ψ) and we shall constrain the projectile so that initially it slides along a trough, as indicated in figure 3(f). We shall see in section 7 that this machine is the most efficient design, with the highest launch speed and hence longest range. Energy is efficiently transferred from CW to projectile via the beam.

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In addition to the three variables there are six parameters for case IV: the masses m, M, Mb of the projectile, CW and beam, respectively, and the lengths , L and r shown in figure 3. We shall assume a ‘standard’ set of parameters for all cases, as follows: L +  = 12 m

m = 100 kg

M = 10 000 kg

Mb = 2000 kg.

(1)

This corresponds to a large siege engine, but entirely realistic. The remaining parameters are L, θ 0, hinge length h (for cases II and IV) and the sling length r (for cases III and IV). These will be chosen for optimum performance, subject to practical limitations discussed later. We note here that our claims for ‘optimum’ performance are not proven, since we estimate performance numerically. We have not calculated the launch speed for all possible parameter values—the parameter space is too large for this to be done. So, where we refer to optimum launch speed herein we mean the largest value obtained by integrating a subset of parameter values. The local peak thus obtained is, in general, rather broad. Thus, varying θ 0 by 5◦ , or varying L, r by 0.1 m causes u to fall by less than 1 m s−1, and usually much less than this amount. Where the peak is not broad we shall say so, since our claim to optimum performance is less sure for such instances. To simplify the analysis we make a number of reasonable idealizations. Thus, we shall assume no friction prior to projectile release—the calculations will prove to be quite complicated enough without friction (especially for the constrained trebuchet, case IV). We do not expect that this idealization will significantly influence the motion. For case IV we shall investigate the engine motion after the projectile has been released, and this requires that friction forces be included. Another idealization adopted in our calculations is beam rigidity. We shall not consider the engineering properties of the siege engine frame, beam4 or sling. These are important in practice, but here we shall regard the frame and beam as perfectly rigid, and the sling as perfectly inelastic (no stretching). The sling is considered to be effectively massless, as is the hinge (the rigid structure of length h in figure 3(f), linking the CW to the main beam). The two trigger mechanisms (one initiates beam motion, and the other releases the projectile from the sling) are assumed to work perfectly, and will not be considered further. 4. Case I: mangonel without sling

The motion of this simple machine is readily determined by considering the energy EI, which is conserved if we neglect friction. Thus, from figures 3(a), (b): EI = V sin(θ0 ) = −V sin(θ ) + 12 I θ˙ 2 where we have used Newton’s dot notation for time derivatives, and where 1 V ≡ Mg − mgL − Mb g(L − ) 2 
3 L + 3 1 . Ib = Mb I ≡ M2 + mL2 + Ib 3 L+

(2)

(3)

The beam angle θ is initially (t = 0) set at –θ 0, and increases with time to a value π /2 – α, at which point the projectile is released. In (3), we have calculated the beam moment of inertia Ib about the pivot point (figures 3(a), (b)) assuming that the beam is a rod of uniform cross-section We have implicitly taken some account of beam material properties with our choice of beam mass Mb = 2000 kg. For such a mass, a 12 m long oak beam would have a cross-sectional area of 0.2–0.25 m2, which is enough for the beam to remain rigid when acted upon by the bending moments that arise.

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length L, assuming a total beam length of 12 m. For all plots M = 10 000 kg, θ 0 = 60◦ , α = 45◦ . The bold lines apply for the parameters (m, Mb) = (100 kg, 2000 kg). The other lines correspond to: a = (20, 2000), b = (100, 1000), c = (20, 1000).

along its length. In practice this may be only approximately true. Modern reconstructions usually have a tapered beam [8], to reduce beam mass and moment of inertia. Some medieval depictions indicate tapered beams (e.g. figure 1); others suggest uniform beams. The projectile speed prior to release is u(θ ) = θ˙ L and so the launch speed is  V u(α) = L 2 (sin(θ0 ) + cos(α)). (4) I This is plotted in figure 4, from which we note that the optimum choice for lever arm ratio is 3 < L/ < 5. For the parameters of equation (1) the maximum launch speed is u = 22.2 m s−1, for θ 0 = 90◦ . For a more realistic θ 0 = 60◦ we have u = 20.3 m s−1. This corresponds to a range of about 45 m (from figure 2), much shorter than the ranges obtained by ancient and modern siege engines. We here define siege engine efficiency in two ways. Firstly the kinetic efficiency is the ratio of projectile kinetic energy at launch to engine kinetic energy. For the case I engine this is mL2 (5) εK = I which is plotted in figure 4 as a function of L. We see that the kinetic efficiency is very low; εK = 5% for the optimum choice of L = 9 m (with the parameters of equation (1)), so that much kinetic energy remains in the beam and CW after the projectile is released. This energy must be dissipated before the engine can be loaded again. The quickest way to achieve this would be to place a barrier, padded to avoid breakage, so that the beam is stopped at θ = π/2 − α. Such ‘stops’ were necessary for classical torsion engines [2] but are undesirable. The large forces that arise cause the engine frame to shift, and shorten machine life.

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The second measure of efficiency that we define is the ratio of projectile kinetic energy to input CW potential energy. This engine efficiency is a measure of how successfully the available CW potential energy is converted to projectile motion. We define the input CW potential energy Ein as the work performed in raising the CW from the natural rest position at θ = π/2 to the firing position at θ = −θ0 . Thus Ein = Mg(1 + sin(θ0 )), and so the engine efficiency is ε=

mu2 . 2Mg(1 + sin(θ0 ))

(6)

This is also shown in figure 4: it is 4% for the optimum L and the parameters of equation (1). In figure 4 we see the effect of different beam and projectile masses. Unsurprisingly the launch speed increases (though not much) as m decreases, but efficiency is reduced. Reducing Mb increases both launch speed and efficiency. 5. Case II: trebuchet without sling

Here the fixed CW of case I is replaced by a hinged CW, as shown in figure 3(c). We will see how this development leads to increased launch speed (and hence range) and increased efficiency. Also we will gain insight into the transfer of energy from CW to projectile. The most convenient method of analysis henceforth is Lagrangian mechanics. From figure 3(c) we obtain the Lagrangian (density) LII = 12 I θ˙ 2 + 12 Mh2 φ˙ 2 − Mhθ˙ φ˙ sin(θ + φ) + V sin(θ ) + Mgh cos(φ)

(7)

where V and I are defined in (3). The equations of motion are found from the Euler–Lagrange equation ∂L d ∂L = ∂q dt ∂ q˙

(8)

for q = θ, φ and L = LII. This yields I θ¨ = Mhφ¨ sin(θ + φ) + Mhφ˙ 2 cos(θ + φ) + V cos(θ ) hφ¨ = θ¨ sin(θ + φ) + θ˙ 2 cos(θ + φ) − g sin(φ).

(9)

We have integrated these equations numerically5 , assuming the standard masses of equation (1), and for different values of the remaining parameters θ 0, L and h. Results are shown in figures 5 and 6, for the optimum (in the sense discussed in section 3) choice of θ 0, L, h; these yield a launch speed of u = 41.6 m s−1, corresponding to a maximum range of about 160 m, from figure 2. The launch speed falls off rapidly if either L or h differs much from the values of figure 5. For the optimum choice, the kinetic efficiency is εK = 14% whereas the engine efficiency is ε = 10%. Thus range and efficiency are improved compared with case I, for the same beam length and for the same masses. From figures 5 and 6, we can see why this happens. The CW drops more vertically than for case I—less energy is wasted in horizontal motion. At launch the projectile is rotating clockwise about the end of the beam (see figure 3(c)), and the CW is rotating counterclockwise. Thus the CW angular momentum provides an extra boost to the projectile just at the right time (from figure 5 note the rapid increase in projectile speed, and corresponding increase in hinge angular velocity, just prior to launch). The CW moment about 5 Since we are concerned here mainly with physical principles, and not mathematics, we omit the rather lengthy manipulations necessary to render equation (9), and equations (13) and (15), into a form suitable for numerical integration. In the same spirit we omit discussion of numerical integration technique.

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time, obtained by numerical integration. The parameter values are θ 0 = 55◦ , L = 7.3 m, h = 1.4 m, and the masses of equation (1). These parameters yield the maximum launch speed.

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the main pivot (figure 6) is a maximum when hinge and beam are co-linear, so we expect the optimum parameters to be those that align hinge and beam at the time of launch. The case II engine motion is like the first cycle of a pumped or self-excited oscillator [12], say a child ‘pumping’ a playground swing. Our understanding of the reason why case II is more efficient than case I can be applied to give us an approximate analytical expression for the launch speed. The siege engine energy is EII = V sin(θ0 ) − Mgh = 12 I θ˙ 2 + 12 Mh2 φ˙ 2 − Mhθ˙ φ˙ sin(θ + φ) − V sin(θ ) − Mgh cos(φ). (10) At launch we have θ = π/2 − α. From our discussion of the last paragraph we expect, for optimum performance, φ ≈ α (so hinge and beam align) and hφ˙ ≈ θ˙ at launch. Substituting into (10) yields  V (sin(θ0 ) + cos(α)) − Mgh(1 − cos(α)) u≈L . (11) 1 (mL2 + Ib ) 2 For the optimum parameter values this yields u ≈ 44.7 m s−1, which is close to the numerical value. A similar calculation leads to an estimate for kinetic efficiency: εK ≈ mL2 / (mL2 + Ib ) = 16%, again close to the value found by numerical integration. From this study of the hypothetical case I and case II counterpoise engines, we see why real siege engines with hinged CW performed better than those with fixed CW: trebuchets outperformed mangonels because less CW potential energy was wasted in developing CW kinetic energy. 6. Case III: mangonel

The mangonel is shown schematically in figures 3(d), (e). The beam has a fixed CW at one end, and a sling of length r at the other. The Lagrangian for this system is LIII = 12 I θ˙ 2 + 12 mr 2 ψ˙ 2 − mLr θ˙ ψ˙ cos(ψ − θ ) + V sin(θ ) + mgr sin(ψ).

(12)

Here ψ is the sling angle. The equations of motion are found from (8), with variables q = θ, ψ: I θ¨ = mLr ψ¨ cos(ψ − θ ) − mLr ψ˙ 2 sin(ψ − θ ) + V cos(θ ) r ψ¨ = Lθ¨ cos(ψ − θ ) + Lθ˙ 2 sin(ψ − θ ) + g cos(ψ).

(13)

If we fix the sling length at r = 5 m then numerical integration leads to optimum performance for θ 0 = 20◦ and L = 8.5 m (with the remaining parameters given in equation (1)). Launch speed is u = 38 m s−1, significantly greater than the optimum value of 22 m s−1 found for case I without the sling. Efficiency is found to be εK = 21% and ε = 16%, again much better than for case I. Projectile acceleration is found to be much less variable than for case II (figure 5), and the engine motion altogether smoother. The time evolution of this system is shown in figure 7. Note that the projectile in the sling initially loses height; it is important that it does not strike the ground, since this would impede movement unless a trench is dug beneath the mangonel frame. We may assume that the CW at its lowest point is very close to the ground; note that for the sling length of figure 7, the projectile falls only 0.6 m below the lowest point of the CW centre of mass. The mangonel parameters are optimized for r = 5 m, but performance is improved significantly if we increase the sling length. For instance, if r = 10 m then a maximum launch speed of u = 56 m s−1 is obtained for θ 0 = 50◦ and L = 8.5 m. Efficiency is increased to

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Case III. Mangonel with sling length r = 5 m and the standard parameters of equation (1). The maximum launch speed arises for θ 0 = 20◦ and L = 8.5 m (shown). A longer sling would yield a higher launch speed, but would require a trench beneath the engine. Engine geometry is shown at t = 0, t = T/3, t = 2T/3, and t = T, with T = 1.3 s. Figure 7.

Table 1. Comparison of launch speeds for mangonel and lever.

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εK = 32% and ε = 26%. Unfortunately, such a long sling means that the projectile initially falls well below ground level (over 6 m) and so is impractical. The benefit of long slings was well known to medieval engineers (see, e.g., figure 1), and so a trough or slide was added beneath the siege engine frame to accommodate the projectile at the initial stages of its trajectory, before it became airborne. Most of the improvement in launch speed compared with case I can be attributed to increased mechanical advantage of the mangonel lever. If, as indicated in figure 7, optimum performance occurs for beam and sling more or less co-linear at launch, then the mangonel at launch may be viewed simply as a case I engine with lever arm length increased from L to L + r. We can show quantitatively that this simple lever arm argument accounts for most of the increased launch speed, as follows. Consider two mangonels with sling lengths 2 ≡ R12 . From r1,2 and optimum launch speeds6 u1,2. Our argument suggests that uu21 ≈ L+r L+r1 the numerical results presented above we can construct table 1. (Here we have assumed L = 8.5 m, as in figure 7.) The similarity of R12u1 and u2 shows that our simple physical explanation accounts for most of the increased range of the mangonel, compared with case I. The improvement is significant; this is why real mangonels were constructed with slings, rather than as the hypothetical case I engines. 6

We note here that, for counterpoise engines with a sling, the angle θ at launch is not necessarily exactly equal to π /2 − α (compare case I). For our simple lever arm approximation, however, the equality holds.

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7. Case IV: trebuchet

We shall now put together all the pieces of the fully developed large medieval siege engine— the trebuchet. This siege engine is shown schematically in figure 3(f). The CW is hinged, the projectile is supported in a sling, and initially is constrained to slide in a trough. The benefits of the individual components, discussed for cases I–III, combine here to produce an efficient and effective machine. While the projectile is constrained to slide along the trough, the trebuchet Lagrangian is LIV = 12 I θ˙ 2 + 12 Mh2 φ˙ 2 + 12 mr 2 ψ˙ 2 − Mhθ˙ φ˙ sin(θ + φ) − mLr θ˙ ψ˙ cos(ψ − θ ) + V sin(θ ) + Mgh cos(φ) + mgr sin(ψ) − λ[L sin(θ ) − r sin(ψ) + L sin(θ0 )]. (14) The term in square brackets is the constraint z = 0, where z is the projectile height above ground. The Lagrange multiplier λ is the normal force that must be applied to the projectile (by the trough) to impose the constraint. We might instead have applied the constraint z˙ = 0, in which case λ would physically be a momentum (see, for example, [13] for a discussion of the interpretation of Lagrange multipliers). While (14) applies, the equations of motion are found from (8) assuming four variables: q = θ, φ, ψ, λ. The constrained equations of motion are found to be I θ¨ = Mhφ¨ sin(θ + φ) + mLr ψ¨ cos(ψ − θ ) + Mhφ˙ 2 cos(θ + φ) − mLr ψ˙ 2 sin(ψ − θ ) + (V − λL) cos(θ ) hφ¨ = θ¨ sin(θ + φ) + θ˙ 2 cos(θ + φ) − g sin(φ) 
(15) λ cos(ψ) r ψ¨ = Lθ¨ cos(ψ − θ) + Lθ˙ 2 sin(ψ − θ) + g + m λ sin(ψ) = L(sin(θ ) + sin(θ0 )). We integrate these equations numerically, from t = 0 to t = tλ, during which interval λ increases from –mg to 0. At this point the constraint no longer applies—the projectile is airborne. The equations of motion are then given by the first three equations of (15) with λ = 0; these are integrated numerically from t = tλ to t = T, at which point the projectile is released from the sling. Again we adopt the parameters of equation (1), and adjust the remaining parameters (θ 0, L, r, h) to find an optimum design. We do not have to be concerned about the length of the sling here, because we have allowed for a trough. However the length h of the CW hinge is limited—we do not want the CW to strike the ground as it descends. This limits hinge length to h < L sin(θ0 ) − . Numerical calculation shows that the maximum value yields the best results. For the standard parameters of equation (1), and with θ 0 = 45◦ , we find a local maximum launch speed for (L, r, h) = (8.7 m, 7.9 m, 2.4 m) of u = 63 m s−1, corresponding to a range of about 360 m. The efficiency of this machine is relatively high: εK = 58% and ε = 36%. The time evolution of the trebuchet motion is shown in figure 8. We note the features observed in cases II and III apply here: the CW swing causes the beam to accelerate just before releasing the projectile, while the sling extends the effective beam length. The motion is reminiscent of a whiplash in that energy is transferred along a flexible body from the heavy end to the light end, increasing the body velocity as it proceeds. Here, though, the continuously flexible whip is replaced by rigid rods (due to tension we may regard the sling as rigid) connected by pivots7 . This transfer of energy is made clear in figure 9. As well as plotting the hinge, beam and sling angular velocities we show how the CW, beam and 7

In [4] the word trebuchet is traced to Medieval Latin tribracho, a device with three arms.

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with the standard parameters of equation (1), we find maximum launch speed for L = 8.7 m, r = 7.9 m and h = 2.4 m. Engine geometry is shown at t = 0, t = T/3, t = 2T/3, and t = T, with T = 1.63 s.

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˙ ˙ and hence beam and CW are θ˙ (solid line), φ˙ (dash) and ψ˙ (dash dot). Note that both θand φ, motion, slow down markedly just prior to projectile release, whereas the sling velocity increases. Bottom. Normalized kinetic energy versus time: CW (dash dot), beam (dash) and projectile (solid line). Note how, as the motion unfolds, kinetic energy moves from CW to beam to projectile. Also shown (bold line) is the ratio of projectile kinetic energy to potential energy given up by the CW; at launch this is the engine efficiency ε.

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projectile share the total kinetic energy. Note how kinetic energy is transferred from the heavy CW through the beam to the projectile. Note also how the projectile kinetic energy is drawn ultimately from CW potential energy. If we increase the initial beam angle, then trebuchet range increases. Thus, for the standard parameters and θ 0 = 55◦ we find a local maximum projectile speed for (L, r, h) = (8.5 m, 8 m, 3 m) of u = 67 m s−1, say a range of 380 m. These large beam angles appear not to have been adopted historically. The illustration of figure 1, and many other medieval illustrations of large trebuchets (and modern reconstructions) favour much lower beam angles. This may be due to stability of the frame, or construction cost (for larger beam angles the frame is necessarily larger, and therefore heavier and more difficult to make and transport). With θ 0 = 30◦ we find u = 48 m s−1 (220 m range) for (L, r, h) = (8.9 m, 6.8 m, 1 m). All of these ranges fall within the claims made for historical trebuchet ranges, and for ranges found by modern machines. 8. Trebuchet recoil

After the projectile has been released, the beam and CW move erratically until dissipative forces bring them to rest. We model this here. A convenient method of including velocitydependent forces in the Lagrangian formalism is via the Rayleigh dissipation function [14]. If we model trebuchet friction by velocity-dependent forces at the two pivots (we may neglect the sling here) then the dissipation function is R = 12 (bθ θ˙ 2 + bφ φ˙ 2 ).

(16)

The Euler–Lagrange equations are modified from form (8) to d ∂L ∂L ∂R − = ∂q ∂ q˙ dt ∂ q˙

(17)

and the equations of motion are (M2 cos2 (θ + φ) + Ib )θ¨ = M2 θ˙ 2 sin(θ + φ) cos(θ + φ) + Mhφ˙ 2 cos(θ + φ)   − Mg sin(φ) sin(θ + φ) + Mg − 12 Mb g(L − ) cos(θ )  (18) − bφ φ˙ sin(θ + φ) − bθ θ˙ h bφ ˙ hφ¨ = θ¨ sin(θ + φ) + θ˙ 2 cos(θ + φ) − g sin(φ) − φ. Mh To estimate the friction coefficients bθ,φ we have examined video footage of the motion of a large trebuchet reconstructed in Sweden [10]. The captured motion includes more than one oscillation cycle after the projectile has been released, and shows that the oscillation amplitude is attenuated by 10–15% per cycle. From this we have chosen friction coefficients that yield the same attenuation. Integrating numerically we obtain the recoil motion plotted in figure 10, for our optimum trebuchet of figures 8 and 9. The beam angle eventually settles down to an equilibrium value of θ = 90◦ while the hinge angle settles to φ = 0◦ , as we would expect. The motion is that of a double pendulum; for small φ, β (where θ = π/2 + β) we can linearize (18) and solve by the method of normal modes. We leave it as an exercise for the interested student to show that, for small angles, β ≈ β0 cos(ωt), φ ≈ φ0 cos(ωt) where the frequency is given by the solutions to Ib hω4 − (Ib g + V0 h + Mg2 )ω2 + V0 g = 0,

V0 ≡ Mg − 12 Mb g(L − )

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time (s) Figure 10. Trebuchet recoil. Top: for the trebuchet of figures 8 and 9, the beam angle θ (solid

line) and hinge angle φ (dash) are plotted versus time. Friction coefficients are bφ = bθ = 5000 js. Bottom: the same plot for a trebuchet with L = 10.15 m, r = 10.1 m, h = 5 m and with all other parameters unchanged. Note the reduced amplitudes. For these parameters the efficiency is increased to εK = 76% and ε = 68%.

and where the amplitudes are related by ω2 β0 = (ω2 h − g)φ0 .

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(The normal mode approach is applied to the double pendulum in [13], pp 207–9.) There are two normal modes here; the first oscillates with beam and hinge co-linear, the second oscillates at higher frequency with beam and hinge ‘out of phase’. This second mode permits energy to be dissipated without large beam amplitude swings—mangonels cannot dissipate energy this way. This analysis may provide an explanation for the statement made by medieval writers that the best choice for lever arm ratio L/ is 5.5 or 6 [4]. In this paper we have found for all types of counterpoise siege engine that a lower ratio, between 2 and 4, yields maximum launch speed. (Modern trebuchets are constructed with these lower L/ ratios [10].) However, our calculations reveal that machines with larger lever arm ratios are more efficient, and thus the recoil is less. We illustrate this point in figure 10, for a ratio of 5.5: note how, compared with our ‘optimum’ trebuchet, it oscillates with smaller amplitudes. The price paid is reduced range (180 m versus 360 m). 9. Summary and discussion

We have examined the physics of medieval counterpoise siege engines, and learned why they evolved in the way they did. Progressing from the simplest hypothetical lever arm engine via the mangonel to the trebuchet we found that launch speed and engine efficiency increased, and that post-launch motion (as measured by increasing kinetic efficiency) became

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smoother. The engine efficiency (projectile kinetic energy divided by input CW potential energy) depends upon projectile mass and other engine design parameters. We have found values of εK ≈ 60–80% to be typical for large trebuchets with heavy projectiles. Between 13 and 12 of CW potential energy is converted into projectile kinetic energy. The trebuchet is of interest to physics teachers because of the number of techniques that we can apply to the analysis (including Newtonian and Lagrangian mechanics, constraints, normal modes), and for the varied phenomena exhibited by trebuchet motion (including energy transfer, pumped oscillation, double pendulum). There are a number of other interesting features of these machines that we leave for the student to investigate. 1. Dimensional analysis. This is interesting to those, for example, who wish to know how to design a large engine, having already constructed a smaller one. If lengths scale by a factor k, so that (L, , r, h) → (kL, k, kr, kh), then it is straightforward to show from the equations of motion that time and launch speed scale as (t, u) → (k 1/2 t, k 1/2 u), if masses are unchanged. However, beam mass will increase as Mb → k 3 Mb . The other masses may scale differently, since the strength of the beam depends upon cross-sectional area, so for example (m, M) → (k 2 m, k 2 M). So how does siege engine performance change with k? 2. Wheels. Sometimes smaller siege engines were placed on wheels, and it was found that this improved performance. We might understand this by considering the CW trajectory— the horizontal movement will diminish if the siege engine frame is mounted on wheels. Analyse the simple case I engine assuming that is on wheels. Such an analysis will also apply to the floating-arm trebuchet—a modern development (e.g. [9]. Performance is enhanced if the beam pivot is permitted to move horizontally during engine motion. Thus the beam, rather than the whole siege engine, is mounted on wheels). 3. Propped counterweight. Some historical [1] and modern [10] trebuchets were designed so that the CW hinge angle was not initially zero φ(t = 0) > 0. Show how this enhances performance. 4. Constraints. There are a couple of exercises that arise here. Because we have neglected friction prior to releasing the projectile, energy is conserved. Confirm from numerical integration that, for the case IV constrained trebuchet, energy is independent of time. This is because the constraint performs no work. There is a simple analytical exercise for students who wish to understand the physical significance of Lagrange multipliers. Consider a rigid rod initially lying horizontally upon a table. One end is raised vertically at constant speed. Examine the motion using Lagrangian mechanics and show that the Lagrange multiplier is just the normal force acting upon the rod. Repeat this exercise for the case IV trebuchet. Acknowledgments

The author is grateful to Amaury Mouchet for proposing an analysis of these impressive machines, and to Patric Djurfeldt for confirming our estimation of the recoil of the Swedish trebuchet [10]. References [1] Chevedden P E, Eigenbrod L, Foley V and Soedel W 1995 Sci. Am. 273 66–71 [2] Landels J G 1978 Engineering in the Ancient World (London: Constable) chapter 5 [3] Keen M H (ed) 1999 Medieval Warfare: A History (Oxford: Oxford University Press) p 115

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[4] Chevedden P E 2000 The Invention of the Counterweight Trebuchet: A Study in Cultural Diffusion (Dumbarton Oaks Papers vol 54) ed A-M Talbot pp 71–114 [5] Hansen P V 1992 Acta Archaeol. 63 189–268 [6] Gravett C 1990 Medieval Siege Warfare (Oxford: Osprey) p 61 [7] Payne-Gallwey R 1907 Projectile-Throwing Engines of the Ancients (London: Longman) pp 29–30 [8] See, for example, the following websites, where construction details and calculations are presented: http:// www2.carthage.edu/departments/physics/trebpaper.doc, http://www.physics.siu.edu/˜baer/trebuchet.htm and http://www.geocities.com/SiliconValley/Park/6461/trebuch.html [9] See http://www.algobeautytreb.com/ or http://www.tasigh.org/ingenium/physics.html for simulators and analyses [10] The following excellent websites include photographs and video footage of modern reconstructed trebuchets: http://xenophongroup.com/montjoie/treb etc.htm, http://home.swipnet.se/˜w-64205/artillery.html and http:// members.iinet.net.au/∼rmine/middel.html [11] For example, Massey B S 1997 Mechanics of Fluids (London: Chapman and Hall) chapter 8 [12] For example, Jordan D W and Smith P 1999 Nonlinear Ordinary Differential Equations (Oxford: Oxford University Press) p 39 [13] Kibble T W B and Berkshire F H 1996 Classical Mechanics (London: Longman) p 194 [14] Leech J W 1965 Classical Mechanics (London: Methuen) pp 24–25