Abstract

This study investigated why the design of ancient throwing machines evolved from eutitonon (arms outside the mainframe) to palintonon (arms inside the mainframe) from the end of the first century B.C. to the first century A.D. and evaluated the mechanical advantages of the new design. Palintonon was first used for big machines; in the following centuries, it was also used for much smaller machines. Essentially, the palintonon design has several advantages: more elastic energy can be stored in the hair bundles representing the motors of these machines, heavier projectiles can be thrown with the same charging effort, projectiles are stressed by lower acceleration in the machine with the same muzzle velocity, and the throwing machines have higher efficiency. Results are also presented regarding the “internal ballistics” of these ancient throwing machines by using simulation software.

Introduction

Ancient engineers during the Greek and Roman period were capable of designing devices showing incredible modernity and ingenuity [1–7]. These devices are very interesting from a mechanical point of view and include throwing machines that clearly show very good knowledge of mechanics (from many points of view) and mechanical design. Some authors [8,9] think that the mesolabium (i.e., a cube root extractor [2]) was frequently used to calculate the main dimensions of a throwing machine: that is, the diameter of the flanges holding the bundle representing the torsional motors of the machine. (In this context, “torsional motor” indicates a device capable of storing elastic energy by means of torsion.) In the third century B.C., Greek and Roman engineers stated that the diameters of these flanges for throwing machines are as follows [2–4,9–18]:

  1. (a)
    For ballistae (Fig. 1)
    (1)

    where D is the diameter of the bundle in digits (1 digit ≈ 19.5 mm) and m is the mass of the projectile in minae (1 mina ≈ 431 g).

  2. (b)
    For catapults
    (2)

    where S is the length of the arrow.

Roman engineers named this flange modiolus, from which the words modulus and modular originate, because the design of those throwing machines was modular. In fact, in his treatise, Vitruvius [19] states that all of the most important dimensions are determined on the basis of the modiolus' diameter (Fig. 1).

According to Vitruvius, the ratios between the modiolus and all other main dimensions of the machine are as follows:
(3)
(4)
(5)
(6)
(7)
(8)

Figure 2 presents a reconstructed ballista and a drawing showing the torsional motor. The torsional spring of the latter was made of horsehair or (most frequently) women's hair [20]. The yarns were coiled from an upper iron bar (F in Fig. 2) to a lower one in order to create a bundle that was passed through an upper modiolus (M in Fig. 2) and a lower one to create a circular section, and the two modioli were locked on the machine head through two plates P.

Several investigations were carried out to characterize the performance of those ancient machines [21,22]; a recent study [4] described a model of a torsional motor to evaluate the elastic energy stored in the torsional spring(s). This study demonstrated that the hair bundle length according to Vitruvius allowed the elastic limit of the bundle fibers to be reached with a rotation of the arms typical of throwing machines having their arms mounted outside the mainframe.

The maximum elastic energy of the bundle clearly depends on the bundle torsion; the maximum value of the torsion exceeding the elastic limit of the bundle fibers depends on the L/D ratio between the bundle length and bundle diameter (i.e., modiolus diameter).

The model proposed in Ref. [4] can be used to calculate the twisting of the hair bundle that corresponds to stressing the hair on the bundle surface at its proportionality limit σe. Beyond this rotation, the hair will be quickly stressed beyond elastic behavior. Figure 3 shows the maximum twist as a function of the L/D ratio (Fig. 3(a)) and the stored elastic energy versus the bundle twist angle for several L/D ratios (Fig. 3(b)).

Figure 3 clearly shows that a higher L/D ratio means a wider arm rotation in order to store the maximum possible elastic energy in the bundle. Figure 3(b) shows that a lower L/D ratio means a higher slope of the stored energy curve. Higher slopes correspond to a faster release of energy when the projectile is thrown. This is similar to firearms: slow-burning powders are used for heavy projectiles, while quick-burning powders are used for light projectiles. This suggests that high L/D ratios for the bundle could have been used for machines that threw heavier projectiles with higher efficiency.

Around the second century B.C., Biton of Byzantium recorded an important improvement in throwing machine design. According to several authors [23–27], several machines begin to be built using a new design called palintonon; this is from the ancient Greek root πάλιν (palin) that means “newly.” In these “new” machines, the arms were mounted inside the mainframe, while in traditional machines (i.e., eutitonon) the arms were mounted outside the mainframe. The palintonon design obviously permits larger arm rotations with the probable advantages given above. Figure 4 presents schemes of the eutitonon and palintonon designs.

The palintonon design was probably first adopted for large stone-throwing machines and later used for handheld weapons [26,27]. Based on the reasons given above, an investigation to compare the two designs and evaluate the advantages of the palintonon design seemed to be of interest.

Throwing Machine Model

Both the theoretical (kinematic and dynamic) model of the throwing machines and a simulation model made with Working Model 2D™ are presented here.

Spring Model.

The spring model was previously presented in Ref. [4], so only the main results are presented here.

The elastic energy Em stored in either of the two torsional motors is
(9)

where E is Young's modulus of the hair yarns, l0 is half of the bundle length, R is the radius of the hair yarns, and θ is the torsion of the hair yarns.

For the rest of the paper, the quantity in curly brackets will be indicated as f1(θ); hence, Eq. (9) can be simply written as follows
(10)

Kinematic and Dynamic Model.

The elastic energy, stored in a torsional motor made from hair bundles having the given dimensions was evaluated in Ref. [4]; this can easily be used to calculate the torsional spring stiffness. In order to evaluate the projectile velocity, the schemes shown in Fig. 4 were used.

The arm angles were calculated from the (ideal) position, where the arm is parallel to the machine longitudinal axis and the quantity Sc represents the displacement of the back of the projectile as calculated from its position with the machine loaded. The arm rotation indicated in the figure is as follows:

  • θin = initial angular position when the machine is loaded,

  • θ = generic arm position,

  • θfin = final position when the projectile leaves the ballista, and

  • θ0 = ideal position when the bundle is not preloaded.

According to Fig. 4, the following expressions of
(11)
can be used to easily obtain the projectile displacement as a function of the arm position θ. For the eutitonon
(12a)
For the palintonon
(12b)

By differentiating the equations above, the projectile velocity is obtained as a function of the arm position θ and arm velocity θ·.

By considering θ as a function of time, the following hold:

  1. (a)
    For the eutitonon
    (13a)
  2. (b)
    For the palintonon
    (13b)
Hence
(14)
From the energy balance, it follows that
(15)

where Ecin is the kinetic energy of the moving components of the machine, Em is the elastic energy of the bundle as calculated from Eq. (9), and Eattr is the energy lost due to friction between the projectile and its guide.

Here, m is the projectile mass, Ib is the mass moment of inertia for each arm, f is the coefficient of friction between the projectile and guide. Thus, for a generic configuration defined by the arm rotation θ, the terms of Eq. (15) are defined as follows:
(16)
(17)
(18)
Thus
(19)
Hence
(20)

Equation (20) allows the projectile velocity to be calculated for a given arm angle θ. Naturally, the quantities f1, f2, and f3 are computed for the eutitonon or palintonon depending on which machine is considered.

Numerical Examples.

In order to compare the performances of a eutitonon and palintonon when throwing a 10-minae stone ( = 4.31 kg), the following machines were studied:

  1. (a)

    Eutitonon

According to Vitruvius, the main dimensions of this machine were as follows:

D = 11 digits ≅ 214.5 mm,

L = F + 2E + C ≅ 6.5 D ≅ 1394 mm,

a = 7 D ≅ 1500 mm.

Although there are two flanges, only the length of one was considered; this is because assuming that half of the length of the hair fibers are clamped to the frame in the modiolus.

The following arm rotations were considered:

θin = 20 deg,

θfin = 65 deg, and

θ0 = 90 deg.

With the values above, the maximum torsion of the bundle (arm rotation plus preload) was 70 deg; this corresponded to stressing the hair on the surface of the bundle at its proportional limit σe, as shown in Fig. 3.

In order to evaluate the rope length, the rope was assumed to be straight when the arms were in their final position; hence, Fig. 4 shows that
(21)
Hence
(22)
(23)

Parameter μ, which depends on the length of the rope, plays a rather important role.

The mass moment of inertia for each arm was determined as that of a cylinder having a mean diameter d = 1/2 D where the cylinder was assumed to be made of beech wood, which has a density of 730 kg/m3
The friction coefficient between the projectile and guide was assumed to be
Using the data above, the projectile velocity was calculated as a function of the arm position θ and is reported in Fig. 5 
Fig. 5

Projectile velocity versus arm position

Fig. 5

Projectile velocity versus arm position

Close modal
.
Figure 6 
Fig. 6

Force of inertia on the projectile versus arm rotation

Fig. 6

Force of inertia on the projectile versus arm rotation

Close modal
reports the force of inertia of the projectile as a function of the arm rotation.
  1. (b)

    Palintonon

In the design of the palintonon, it was also assumed to throw a projectile having the same mass (4.31 kg). The same modioli diameter and same arm length were assumed.

For the palintonon, the following arm rotations were considered:

θin = 55 deg,

θfin = 140 deg, and

θ0 = 165 deg.

Because the maximum torsional spring (hair bundle) rotation was 110 deg, Fig. 3 shows that L/D must be 10.25. The distance b was assumed according to reconstructions and archaeological finds [7–9,23,24] to be
(24)
The rope length was evaluated under the assumption that the rope was straight when the arms were in their final position, similar to the eutitonon. Figures 4 and 5 show that
(25)
Thus, the following were assumed
(26)
(27)

The parameter μ, which depends on the length of the rope, plays an important role in this machine.

Figure 7,
Fig. 7

Projectile velocity versus arm position

Fig. 7

Projectile velocity versus arm position

Close modal
presents the projectile velocity versus the arm position, and Fig. 8 
Fig. 8

Force of inertia on the projectile versus arm rotation

Fig. 8

Force of inertia on the projectile versus arm rotation

Close modal
presents the force of inertia on the projectile versus the arm rotation.

Figures 5 and 7 show that the maximum projectile velocities were ≈100 m/s for the eutitonon and ≈124 m/s for the palintonon. The greater maximum projectile velocity represents the main advantage of the palintonon.

The kinetic energies of the projectiles were Ecin ≈ 21,550 J for the eutitonon and Ecin ≈ 33,135 J for the palintonon.

Based on Eq. (9), the maximum elastic energies stored in the hair bundles were Em ≈ 22,500 J for the eutitonon and Em ≈ 35,000 J for the palintonon.

If the efficiency is defined as η = Ecin/Em, it comes: η ≈ 0.958 for the eutitonon and η ≈ 0.947 for the palintonon. Thus, the machines have practically the same efficiency.

A comparison of Figs. 6 and 8 shows that the force of inertia on the projectile thrown by the palintonon was a little higher than that thrown by the eutitonon.

To compare the two ballista designs, the palintonon torsional springs were lowered in order to obtain the same projectile initial velocity. This was achieved by considering bundles having the same diameter of modioli but an L/D = 11.8.

The results are shown in Figs. 9 and 10.

If the muzzle velocities of both designs were the same, the force of inertia acting on the palintonon's projectile was lower than the force of inertia on the eutitonon's projectile. Moreover, the maximum elastic energy stored in the hair bundle was Em ≈ 23,000 J. Because the kinetic energy of the projectile is now Ecin ≈ 21,550 J, the efficiency can be computed as η ≈ 0.937. This is just a little lower than the efficiency computed when the correct value of L/D = 10.25 was considered.

Model Made by Working Model 2D.

Multibody simulation software (Working Model 2D) was used to simulate the performance of these ballistae. The ballista dimensions in the simulations were the same as those given above. The hair bundle was modeled with a torsional spring giving a restoring moment that depended on the square of its rotation
(28)
This kind of restoring moment can be simply modeled in WM 2D. The constant k was calculated as follows. Equation (9) was used to calculate the elastic energy Em(θ) stored in either of the two torsional motors. The following equation was then substituted for Eq. (9)
(29)
where the trends are very similar. The elastic restoring torque was then determined to be
(30)
Hence
(31)

Finally, the correct value of the torsional spring preload was assigned.

The values of k used in the models were as follows:

  • k = 20,000 Nm/rad2 for the eutitonon

  • k = 8000 Nm/rad2 for the palintonon.

The ballistae were charged by a linear actuator acting on the rope up to the assigned arm rotation value; once this rotation was reached, the actuator was disabled.

Figures 11 and 12 present the eutitonon and palintonon models, respectively, with the projectile velocities as a function of time, as computed by Working Model. The velocities were 102 m/s for the palintonon and 122 m/s for the eutitonon. In other words, these results were very close to those obtained by the proposed mathematical model.

Conclusions

This study examined the behavior of two different designs of ancient ballistae. The study was based on a model of torsional springs made from hair bundles that was presented in a previous paper [4] and on a kinematic model of the machines. Numerical examples were presented; these were obtained by applying the proposed model to a palintonon and eutitonon, where the dimensions were taken from Vitruvius [19] for a machine throwing 4.31 kg projectiles. Both machines were simulated using Working Model 2D; the simulation produced substantially similar results to those obtained by the proposed mathematical model.

The study showed that the transition from the eutitonon to the palintonon design was due to several advantages. Essentially, the palintonon design allowed for higher projectile velocities and/or lower inertial forces on the projectile during the launch run. This explains why the palintonon, which appeared as a large machine for heavy projectiles, was later (in the early centuries A.D.) built in smaller sizes and even adopted for small arrow-throwing catapults.

A model was presented that can be used to evaluate the performance of ancient ballistae and catapults. The results of this investigation can also provide an evaluation tool for similar reconstructions while also offering an insight into design details and alternatives.

References

1.
Rossi
,
C.
, and
Pagano
,
S.
,
2011
, “
A Study on Possible Motors for Siege Towers
,”
ASME J. Mech. Des.
,
133
(
7
), pp.
1
8
.
2.
Rossi
,
C.
,
Russo
,
F.
, and
Russo
,
F.
,
2009
,
Ancient Engineers' Inventions, Precursors of the Present
,
Springer
,
Berlin
.
3.
Rossi
,
C.
, and
Russo
,
F.
,
2010
, “
A Reconstruction of the Greek-Roman Repeating Catapult
,”
Mech. Mach. Theory
,
45
(
1
), pp.
36
45
.
4.
Rossi
,
C.
,
2012
, “
Ancient Throwing Machines: A Method to Calculate Their Performance
,”
Mech. Mach. Theory
,
51
, pp.
1
13
.
5.
Russo
,
F.
,
Rossi
,
C.
,
Ceccarelli
,
M.
, and
Russo
,
F.
,
2008
, “
Devices for Distance and Time Measurement at the Time of Roman Empire
,”
Proceedings of HMM 2008 International Symposium on History of Machines and Mechanisms
,
Tainan, Taiwan
, pp.
101
114
.
6.
Chondros
,
T. G.
,
2010
, “
Archimedes Life Works and Machines
,”
Mech. Mach. Theory
,
45
(
11
), pp.
1766
1775
.
7.
Dimarogonas
,
A. D.
,
2001
, “
Introduction–The Machine: A Historical Design
,”
Machine Design: A CAD Approach
,
John Wiley and Sons
,
New York
, pp.
4
19
.
8.
Soedel
,
V.
, and
Foley
,
V.
,
1979
,
Scientific American: Ancient Catapults
,
240
(
3
), pp.
150
160
.
9.
Lahanas
,
M.
, “Ancient Greek Artillery Technology From Catapults to the Architronio Canon,” http://www.mlahanas.de/Greeks/war/CatapultTypes.htm
10.
Russo
,
F.
,
2004
, “
L'artiglieria delle legioni romane
,”
Ist. Poligrafico e Zecca dello Stato
, Rome, Italy.
11.
Russo
,
F.
,
2007
,
Tormenta Navalia, L'artiglieria navale romana
, USSM Italian Navy, Roma.
12.
Baldi
,
B.
,
1616
, Heronis Ctesibii Belopoeka, hoc est, Telifactiva, Augusta Vindelicorum, typu Davidu Frany.
13.
Shramm
,
E.
,
1918
,
Die antiken Geschütze der Saalburg
,
Reprint, Bad Homburg
,
Saalburg Museum
,
1980
.
14.
Marsden
,
E. W.
,
1969
,
Greek and Roman Artillery: Historical Development
,
Oxford University Press
,
New York
.
15.
Marsden
,
E. W.
,
1971
,
Greek and Roman Artillery: Technical Treatises
,
Oxford University Press
,
New York
, pp.
106
184
.
16.
Chondros
,
T. G.
,
2008
, “
The Development of Machine Design as a Science From Classical Times to Modern Era
,”
HMM 2008 International Symposium on History of Machines and Mechanisms
,
Tainan, Taiwan
, Nov. 11–14, Springer, Netherland.
17.
Baatz
,
D.
,
1978
, “
Recent Finds of Ancient Artillery
,”
Britannia
,
9
, pp.
1
17
.
18.
Bishop
,
M. C.
, and
Coulston
,
J. C. N.
,
1993
,
Roman Military Equipment From the Punic Wars to the Fall of Rome
,
London
, Reviewed by
Rankov
,
B.
,
1994
, “
Roman Military Equipment From the Punic Wars to the Fall of Rome
,”
The Classical Review (New Series)
, Vol.
44
, pp.
137
138
.
19.
Vitruvius
,
De Architectura
,
Liber X
. 1st Century A.D. Available at http://la.wikisource.org/wiki/De_architectura/Liber_X
20.
Appianus Alexandrinus
,
Wars Against Carthage
,
Liber VIII
. Available at http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.01.0230%3Atext%3DManuscripts
21.
Hart
,
V. G.
, and
Lewis
,
M. J. T.
,
1986
, “
Mechanics of the Onager
,”
J. Eng. Math.
,
20
(
4
), pp.
345
365
.
22.
Hutchison
,
W.
, and
Godfrey
,
S.
,
2004
, “
A Modern Reconstruction of Vitruvius' Scorpion
,” http://www.eg.bucknell.edu/∼whutchis/scorpion
23.
Russo
,
F.
,
2009
, La grande Balista di Hatra, Ed. ESA, Torre del Greco, Naples, Italy.
24.
Hart
, V
. G.
, and
Lewis
,
M. J. T.
,
2009
, “
The Hatra Ballista: A Secret Weapon of the Past?
,”
J. Eng. Math.
,
67
(
3
), pp.
261
273
.
25.
Iriarte
,
A.
,
2003
,
The Inswinging Theory, Gladius XXIII
, pp.
111
140
.
26.
Iriarte
,
A.
,
2000
, “
Pseudo-Heron's Cheiroballistra A(nother) Reconstruction: I. Theoretics
,”
J. Roman Mil. Equip. Stud.
,
11
, pp.
47
75
.
27.
Harpham
,
R.
, and
Stevenson
,
D. W. W.
,
1997
, “
Heron's Cheiroballistra (a Roman Torsion Crossbow)
,”
J. Soc. Archer-Antiquaries
,
40
, pp.
13
17
.