## 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]:

- (a)For ballistae (Fig. 1)$D=1.1\xb7100\xb7m3$(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). - (b)For catapults$D=S/9$(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).

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 *σ _{e}*.
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.

## 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.

*E*stored in either of the two torsional motors is

_{m}where *E* is Young's modulus of the hair yarns, *l*_{0} is half of the bundle length, *R* is the radius of the hair yarns, and *θ* is the torsion of the hair yarns.

*f*

_{1}(

*θ*); hence, Eq. (9) can be simply written as follows

### 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 *S _{c}* 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:

*θ*_{in}= initial angular position when the machine is loaded,*θ*= generic arm position,*θ*_{fin}= final position when the projectile leaves the ballista, and*θ*_{0}= ideal position when the bundle is not preloaded.

*θ*. For the eutitonon

By differentiating the equations above, the projectile velocity is obtained as a
function of the arm position *θ* and arm velocity $\theta \xb7$.

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

- (a)For the eutitonon$VC,eut=a\xb7\theta \xb7\xb7[sin\theta +sin2\theta +2\xb7\u025b\xb7cos\theta 2\mu 2-(sin\theta +\u025b)2]=a\xb7\theta \xb7\xb7f3,eut(\theta )$(13a)
- (b)For the palintonon$VC,pal=a\xb7\theta \xb7\xb7[sin\theta +sin2\theta -2\xb7\u025b\xb7cos\theta 2\mu 2-(\u025b-sin\theta )2]=a\xb7\theta \xb7\xb7f3,pal(\theta )$(13b)

where *E*_{cin} is the kinetic energy of the moving
components of the machine, *E _{m}* is the elastic energy
of the bundle as calculated from Eq. (9), and

*E*

_{attr}is the energy lost due to friction between the projectile and its guide.

*m*is the projectile mass,

*I*is the mass moment of inertia for each arm,

_{b}*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:

Equation (20) allows the
projectile velocity to be calculated for a given arm angle *θ*.
Naturally, the quantities *f*_{1}, *f*_{2}, and *f*_{3} 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:

- (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:

*θ*_{in} = 20 deg,

*θ*_{fin} = 65 deg, and

*θ*_{0} = 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 *σ _{e}*, as shown in
Fig. 3.

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

*d*= 1/2

*D*where the cylinder was assumed to be made of beech wood, which has a density of 730 kg/m

^{3}

*θ*and is reported in Fig. 5

- (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:

*θ*_{in} = 55 deg,

*θ*_{fin} = 140 deg, and

*θ*_{0} = 165 deg.

*L*/

*D*must be 10.25. The distance

*b*was assumed according to reconstructions and archaeological finds [7–9,23,24] to be

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

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 *E*_{cin} ≈ 21,550 J for the eutitonon and *E*_{cin} ≈ 33,135 J for the palintonon.

Based on Eq. (9), the maximum
elastic energies stored in the hair bundles were *E _{m}* ≈ 22,500 J for the eutitonon and

*E*≈ 35,000 J for the palintonon.

_{m}If the efficiency is defined as *η* = *E*_{cin}/*E _{m}*,
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.

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 *E _{m}* ≈ 23,000 J. Because the kinetic energy
of the projectile is now

*E*

_{cin}≈ 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.

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/rad^{2}for the eutitonon*k*= 8000 Nm/rad^{2}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.