Improvement in Ballistae Design From Eutitonon to Palintonon: A Study on the Mechanical Advantages

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)

   $D=1.1·100·m3$

   (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

   $D=S/9$

   (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.

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.

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

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

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.

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

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

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

   $VC,eut=a·θ··[sinθ+sin2θ+2·ɛ·cosθ2μ2-(sinθ+ɛ)2]=a·θ··f3,eut(θ)$

   (13a)
2. (b)
   For the palintonon

   $VC,pal=a·θ··[sinθ+sin2θ-2·ɛ·cosθ2μ2-(ɛ-sinθ)2]=a·θ··f3,pal(θ)$

   (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.

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.

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:

θ_in = 20 deg,

θ_fin = 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 σ_e, as shown in Fig. 3.

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

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:

θ_in = 55 deg,

θ_fin = 140 deg, and

θ₀ = 165 deg.

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_m ≈ 35,000 J for the palintonon.

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.

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 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.

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.

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.