© B. Guilbert et al., Published by EDP Sciences 2025

1 Introduction

Planthoppers are champion jumpers. Adult specimens are propelled by rapid movements of the posterior extension in a matter of microseconds, which can launch them a horizontal distance of 1.1 m with a take-off speed of 5.5 m/s [1,2]. The mechanism required to achieve such a performance by insects in general and planthoppers in particular has long been studied. Bennet-Clark and Lucey [3] observed the jump of a Spiloplyllus cuniculus (a rabbit flea) with a high-speed camera and studied its trajectory. They were able to see all the phases of the jump and study the kinematics of the tendons and muscles. They concluded that an energy storage mechanism was absolutely necessary, as the jump is performed over a shorter time than that required for muscle contraction [4]. Moreover, energy storage is a mechanism that had been observed in other insects (Uropetala carovei Wh., Libellula spp., Orthetrum sp., Sympetrum spp., Aeshna spp., Agrion spp [5]). Later, Bennet-Clark [6] was able to quantify the energy required for such a jump in another insect, the grasshopper. He concluded that the loss of energy in the rotation of the femur was quite small and that most of the energy in the jump was well stored beforehand. Likewise, Burrows [7] focused on the anatomy and jumping phases of froghopper insects (Hemiptera, Coercopoidea, closely related to planthoppers) and found a similar energy storage and release mechanism. Likewise, Burrows et al. [8] then studied the storage mechanism of the planthopper, demonstrating that it, like froghoppers, fleas, and grasshoppers, also uses an energy storage structure for jumping. In all of the above insects, this composite structure combines the rigidity of the chitinous cuticles with the elasticity of the resilient materials, enabling the elastic structure to perform repeated jumps.

Further studies were carried out by Burrows [9], Burrows and Bräunig [10] and Sutton and Burrows [11], quantifying the jumping performance of planthoppers found out these jumps required precise synchronization between the legs, with differences in timing of no more than 30 μs between its hind legs. They also found that without an efficient synchronization mechanism, the result of the planthopper and froghopper jump is a pirouette [12], with a single leg jumps in froghoppers spinning six times faster than a froghoppers with both legs [12]. Burrows, in [13], found that adult planthoppers (Issus Coleoptratus) synchronize their legs by having a friction-pad like structure that ensures each leg extends at the same time.

The nymphs of Issus Coleoptratus, however, do not use a friction pad. For Issus nymphs, the synchronization required for jumping is achieved by a pair of cuticular gears located on the insect’s trochanters [14] (Fig. 1). These cuticular gears show spectacular geometric differences from the classic profile of a man-made tooth, the involute (Fig. 2).

The mechanics of man-made involute gears are well studied and understood. This specific man-made tooth profile has the particularity of creating a homokinetic power transmission and thus avoiding any problems that might result from assembly deviation, for example. It has been the subject of extensive research over the years, as its involute shape allows the contact between the teeth to be represented as a contact line located on the base plane of the gear. Numerous light and fast models have been produced to study its dynamic behavior [15–21]. It has also been the subject of numerous experimental studies, for example on the FZG test rig. This test rig features a closed mechanical power loop which allows high power to flow through the gearbox, while the motor generates only the energy required to compensate for the system’s mechanical losses. Two gearboxes are mounted back-to-back. To be able to withstand the closed loop high load, resisting ISO [22] type gears are used. This test rig is a reference for gear research, and the mounteed gear geometries have been the object of a high number of papers. Martins et al. [23] used this test rig to characterize two industrial gear oils. They studied their physical properties, wear properties, chemical content and were able to compare them in terms of power dissipation when used for gear applications. Fernandes et al. [24] characterized oils for wind turbine gears. Durand de Grevigney et al. [25] used the same test bench to validate a power loss model. Navet et al. [26] studied its behavior under starvation conditions and Touret et al. [27] studied the influence of micropitting on power loss and temperature. Hong et al. [28] studied tooth bending failure and compared tests carried out on a single tooth bending test rig and on a complete gear rotation test rig. From a number of tests, they extracted a report linking these two experiments.

However, this widely used classical ISO FZG geometry encounters limitations, as optimisations by modifying the profile of the teeth were constantly necessary to ensure correct dynamic behaviour of the transmission at higher speeds [29–31]. Another known limitation of classic ISO geometry is the point of weakness in terms of resistance presented by the root of the tooth where the lever arm is exerted by the meshing load applied to the tooth flank. In an attempt to improve its resistance, new designs have emerged which present involute profiles with asymmetric characteristics (Fig. 3). Cavdar et al. [32] have shown both by FEM analyses and a developped code that asymmetric design improves the mechanical strengh performances with respect to both classical symmetric designs and symmetric design aimed for stress reduction at tooth root. The improvements in bending stress for gears are confirmed by the work conducted by Pedersen [33]. More recently, the study of such gear design has been extended to meshing parameters such as contact ratio that is known to have a huge influence on load capacity, wear, effciency, noise and vibration [34]. And lately, Mo et al. [35] have worked on the meshing characteritics of such gears, leading the steps of new model developments able to include such gears.

The objective of the paper is to study the characteristics of the Issus tooth profile and compare it to the known properties of the involute gear shape, with the eventual benefit that the Issus geometry could bring to classical man-made gear. To this end, an observation study will be conducted on several Issus gears from various geographical origin to highlight the gear geometry characteristics of the specie and find a representative specimen for a more in-depth analysis. Thanks to the previous part conclusions, a comparative study of this specimen with the geometry of a man-made FZG gear will be conducted with identical materials in order to enlight the potential interest that such specific gear geometry could bring. This study will be conducted numerically by a static finite element analysis. Both tooth profiles will be submitted to the meshing static load at the higher meshing point and the teeth displacement and stresses will be analysed and compared. Finally, an attempt to understand the insect gear role in nature will be made by submitting the chosen specimen profile to the load that could be transmitted during an Issus badly engaged jump and determine if the gear would be able to save such a bad attempt and therefore present an evolutive advantage for the insect.

Fig. 1 Gears on the hind trochantera of Issus nymphs [14].

Fig. 2 Profile of gear tooth in Issus (left) compared to man-made involute tooth (right) [14].

Fig. 3 Symmetric and asymmetric tooth profiles and corresponding gears.

2 Preliminary observation of the insect

2.1 Issus specimen observation − comparison to the literature and geometrical characterization

Issus Coleoptratus (Hemiptera:Fugoroidea:Issidae) is an insect commonly found in Europe, the Middle East, Asia and North Africa [36]. In order to study the characteristics of the insect relative to the specie, the following observation were conducted on French specimen and compared to their English counterparts to account for geographical variety. For this study, insects were gathered in the south-east of the country during May/June in a woody area. Several nymphs and adults of Issus Coleoptratus were collected. They were observed using a 3D digital microscope. An example of an adult specimen (Fig. 4a) under the microscope shows the distinctive dorsal pattern of the species [37], which enabled it to be identified. Observation of the nymph (Fig. 4b) when in the jumping position shows the distinctive shape (trochanter, coxa and gear) of the hind legs.

For the jump mechanism study, a close-up of the attachment of the hind legs (Fig. 5) is taken at several time-steps during the jump, with video capture under the microscope. To do this, the nymph is attached to the microscope table by its back and air is blown onto the hooked hind legs to provoke the jump. During this jump, the rapid (a few ms) and synchronized depression of the hind legs is observed. During the jump, the coxa (upper part, dark green) lowers with respect to the nymph’s body, and the tendons connecting this coxa to the trochantera contract, inducing the downward movement of the legs and thus the jump. The movement observed is similar to those previously described by Burrows and Sutton [14]. The gear studied meshes (dark brown in the centre of the image) throughout the jump phase, synchronizing the legs. The first image (Fig. 5a) shows that meshing starts at the first tooth on both sides. However, the final image (Fig. 5c) clearly shows the remaining teeth on the right side.

Superimposing the beginning and end of the jump (Fig. 6) main outlines drawn from the pictures Figures 5a and 5c shows that the whole body is elastically deformed. As the tendons contract, the trochantera tilt and the coxa and hind legs descend. The most static point during this jump is the contact between the two trochantera, i.e. the point of engagement (noted “gear” Fig. 6). There is no other immobile point indicating rotation in the plane of observation. It is therefore impossible, on the basis of 2D image capture and observation, to recover the 3D kinematics of the system.

A close-up, with 3D reconstruction using the 3D digital microscope to avoid depth-of-field problems, is then performed on a trochanter of French specimens (Fig. 7a). The image obtained is compared with a Micro-CT scan of another trochanter performed on an Issus Coleoptratus found in England (Fig. 7b). Pixel size is 0.1 μm.

The shapes of the teeth in the two photos are very similar: they have the same number of teeth, and the characteristic shape with large teeth on one side decreasing in size towards the other is clearly visible. On very close inspection, some differences are observed. On Figure 7a, the 5th tooth from the left is altered, as if it had broken off in the middle. On Figure 7b, this tooth is also altered, but in the form of light wear visible in the upper central part of the tooth. These alterations could either prove that the shape can vary from one individual to another, or that tooth shape can be changed by wear caused by the insect’s jumping. From [14] onwards, adults no longer moult and therefore need a durable means of synchronizing their hind legs. These adults do not possess the gears that have been observed on nymphs. This could lead to the conclusion that the teeth are not strong enough to last the lifetime of the adult insect. They can break, which in the case of the nymph is not synonymous with a death sentence, since they will be replaced during the next moult, a phenomenon impossible for the adult.

Fig. 4 3D digital microscope close up on (a) adult dorsal view and (b) nymph ventral view when crocked up for jump.

Fig. 5 3D digital microscope observation of the Issus gear during jump.

Fig. 6 Diagram of the initial (dotted blue, Fig. 5a) and final (plain red, Fig. 5c) positions of the hind legs, coxa and gear.

Fig. 7 Teeth from Issus (a) French origin with 3D image reconstructed from the 3D digital microscope observation and (b) English origin (4.5 mg animal) from Micro-CT scan. Both (a) and (b) are from a 5th instar Issus nymph.

2.2 Gear observation − gear meshing and contact ratio

Two Micro-CT scans were obtained (one side example Fig. 7b), one for each side of the Issus studied. These were converted into CAD and cut surfaces were created on each trochanter with a plane passing through the midline of the teeth. The resulting surfaces were extruded for 3D printing and hand manipulation. Attempts were made to position them in different mesh stages (Fig. 8).

The first observation to be made is the difference of tooth size on each trochanter. This difference is present and symmetrical on both trochantera. However, the number of teeth is different on both sides, with two teeth remaining on the right side after meshing. This finding is identical to that of the video extraction in Section 2.1, as well as to previous studies. In view of the few observations made, this difference in the number of teeth seems to be fairly common and apparently does not induce any jumping problems for the Issus nymph. On the other hand, all the images (Figs. 8a–8c) show that the two sides mesh smoothly at different positions with initial gearing on the respective first teeth. They also show that at these different time steps, there are always around 3 teeth in contact − a significant difference from artificial gears. Indeed, on conventional man-made spur gears, except for very specific geometries, the number of teeth in contact is one or two. This geometry, with higher meshing ratio, would therefore enable a better distribution of loads between the teeth in contact.

Fig. 8 Issus Gear cut meshing at three meshing-steps (scale 260/1).

2.3 Observation limits

The above study was carried out on just under 10 nymphs of French origin. In particular, the observation of jumping mechanics was made on a nymph very close in size (around 3.5 mm long) and number of teeth to studies already carried out [14]. However, other Issus were observed under the 3D digital microscope. Two of them (example Fig. 9) are around 2.3 mm long and have 9 teeth on each trochanter. This indicates that the number of teeth on the gear varies with size and that, potentially, not only does the Issus grow with molting, but it is also able to replace and add more teeth to its jumping gear if these nymphs are smaller in size due to their less advanced stage of development than those studied previously.

Fig. 9 Smaller nymph (a) ventral view and (b) close up on tooth number.

3 Tooth strength analysis − comparison to FZG design

3.1 Man-made gear comparison

Gears (Tab. 1, FZG C30 from [26]) are classic man-made involute gear design that has been widely used to study gear behavior. They feature characteristics that meet ISO [15] criteria and can be used as a control sample to assess the strength limit of Issus gears.

One side of the flat section of the Issus gear (see Sect. 2.2) was 3D printed at 530/1 scale, and an attempt was made to position it in mesh with the type C FZG gear (Fig. 10) printed at 1/1 scale. Although the Issus teeth are longer and thinner than conventional involute teeth, their positioning at different mesh levels shows that they present geometrical proportions (tooth pitch and height) that make the meshing possible. They therefore have roughly the same size/spacing ratio as the classic involute teeth of the FZG. In these images, too, the meshing ratio appears to be between two and three teeth, rather than the usual one to two. This confirms that Issus teeth have a geometry that tends to increase the meshing ratio.

A static analysis is carried out to compare the strength of the FZG tooth (wheel Z = 24 side) and the Issus tooth. To this end, an FE model is built with Table 1 gearing. The Issus tooth cut previously used for meshing tests is extruded to obtain a gear with the same tooth width as the FZG (b = 30 mm). It is mounted on a circle passing through the tooth root. As the teeth are not all of the same size, a tooth in the middle of the largest zone is chosen for this study. A standard steel (E = 2.1e11 Pa, nu = 0.3) is chosen for all the following analyses.

A 5-tooth sector is retained and a bore is made in the centre of the gear (Fig. 11). The radius of the bore and the angular cut are chosen far enough away so that the results of the static analysis are not affected by these cuts. The mesh is hexahedral quadratic to take into account tooth curvature as accurately as possible, and refined in the area of interest:

• For the FZG gear, the upper single contact point creates the most critical load case and the position of the 30° tangent used in the ISO [15] standard for calculating tooth root stress (Fig. 12a). The mesh is refined in these two zones.

• For the Issus gear, two loading cases are chosen (L and R) to take account of tooth asymmetry (Figs. 12b and 12c). Here again, the mesh is refined over the loading zone (chosen to have the same proportionality as the FZG gear) and the tangent at 30° to the root. To take account of the particular, non-symmetrical profile of the flank, the mesh is also refined between these two points and on the opposite side of the L case.

The load is applied to a very fine surface (∼0.6mm large) over the entire width of the tooth, to obtain a loading profile close to the contact line. The model is clamped on both sides of the sector and on the central bore (Fig. 11).

Fig. 10 Meshing of the cut (scale 530/1) with module 4.5 mm gear from FZG test rig.

Fig. 11 FE model with in blue clamped area (A) and red loading area and direction (along normal x).

Fig. 12 Mesh side view with applied load and thinner mesh for stress concentration of (a) FZG tooth, (b) Left (L) loading case of Issus and (c) Right (R) loading case of Issus.

3.2 Tooth strength

The maximum displacement (Tab. 2) obtained after resolution shows that the FZG gear is more resistant under an identical load than the Issus gear. The maximum Von-Mises stress of the FZG tooth is indeed as expected at the tooth root (Fig. 13). The stress at loading is excluded from this analysis following S^t Verant’s principel. The corresponding material has been removed from the visualisation Figure 13 to avoid scale issue as the local stress at this point is much higher and would be delt with local material specificities (surface hardening etc…) and not general tooth geometry. What is more, the value calculated by FE is very close to the ISO [15] calculation performed by KISSsoft® (approx. 6.5% difference). The maximum principal stress is even closer (less than 2% difference). This is due to the nature of the ISO calculation, which is carried out analytically on the assumption that the tooth is a beam embedded at its base. The stress is then calculated only in the direction of the tooth height, leaving out the other components. The principal stress in this direction gives a good estimate of the overall stress because it dominates the others, but explains the differences compared with the Von-Mises which includes all the components. For subsequent calculations, the Von-Mises criterion will nevertheless be retained, as it gives a better account of the overall stresses used for a strength calculation, especially for non-ISO teeth that present geometry differences with the standard such as the Issus tooth.

The Von-Mises stress of the FZG tooth is also lower (just over half) than that of the Issus gear. It should be noted that the Issus gear shows a reasonable difference in stress depending on the loading case (just under 8%). Loading from the right (Issus R) is therefore the dimensioning side.

The stress distribution on the Issus tooth for cases L. and R. (Figs. 14a and 14b respectively) shows a different stress zone. In both analyses, the stress location is higher on the tooth and more widely distributed than on the conventional FZG gear. This is a potentially interesting result, as one of the disadvantages of the ISO gear is that the fracture that occurs at the root of the tooth due to the stress concentration can cause a crack that runs down into the gear body and can lead to gear failure. A fracture arriving at a higher location on the tooth would allow the fracture to occur only on the tooth itself, thus preserving the gear body and thus the integrity of the system.

When only one tooth is in contact, the FZG gear is still more interesting, as it can withstand a higher load better than the Issus gear, despite the latter’s interesting stress distribution. However, this conclusion is reversed when the gear’s contact ratio is taken into account in the analysis. The classic man-made spur gear (FZG) has between one and two teeth in contact at any given time, whereas, as mentioned in the previous section, the Issus gear always seems to have around three.

The same analysis is carried out including the effect of the contact ratio: the load applied to the tooth is divided by the minimum number of teeth in contact (assuming that all teeth share equal load). This analysis is carried out on the assumption that the stresses of a loaded tooth remain localized on that same tooth, and that the teeth therefore behave mechanically independently of each other. The results (Tab. 3) show that this new analysis changes the conclusion. The high contact ratio on the Issus gear logically divides the displacement by 3 (elastic static model). The resulting maximum Von-Mises stress on the Issus gear is now lower than on the FZG tooth.

Fig. 13 Stress for the FZG gear, material around the loading area is not displayed, creating a visual indentation, due to the non-physical high stress concentration.

Fig. 14 Stress for the (a) L and (b) R Issus tooth loading case, material at the loading point is not displayed due to the non-physical high stress concentration.

4 Quantifying insect transmitted load

4.1 Quantifying tooth load during jump

An energy study of the insect during the jump is carried out to quantify the load actually borne by the Issus tooth and the resulting stress. To this end, the jump data listed in the literature are used (Fig. 15, [12]). For this study, the hind leg mechanism is assimilated to two springs attached to each end of the Coxa (Fig. 15). The center of rotation of the jumping movement, although of unknown location, is considered to lie within the Trochanter. In addition, the springs are classically elastic, as the pleural arch deforms elastically during the jump (Sect. 2.1). No energy dissipation is assumed. All deformation energy is transferred to the leg, which participates in the jump by pushing on the ground. In Figure 16, taken from [9], five jumps from the same Issus are represented by the different curves. Each jump is divided into 3 phases, the first of which is on the ground, when the Issus prepares for the jump and its gear mechanism moves into the hooking position. In the second phase, the Issus executes the jump while still on the ground. Its hind legs move from the hooked position (phase 1) to full extension. Finally, in phase 3, the insect has lifted off the ground and is in flight. The gear mechanism (Fig. 15) is active during phase 2, when the hind legs make the jump and push off the ground. It is at this moment that the following energy study is carried out.

During this phase, the legs go from the fully elevated to depressed state (Fig. 15). The Coxa travel the angle θ_c and its extremities the distances x₁ and x₂. From the literature, an estimation of the covered distance of the Coxa end is x₂ = 100 µ m [12] and the Coxa maximum angle is θ_c = 55° [12]. Using Figure 15b, the trigonometry gives access to the rest of the data: $$\{\begin{array}{l}{x}_2=2d\sin \left(\frac{{\theta }_c}{2}\right)=100\mu \text{m}\iff d=108\mu \text{m}\\ x_1=2\left(L_c-d\right)\sin \left(\frac{{\theta }_c}{2}\right)=574\mu \text{m}\end{array}.$$(1)

In order to calculate the load going through the system during phase 2, the average speed over the phase is retained. It is estimated at V = 2.5m/s (Fig. 16). From this, using an average value [12] of the insect’s weight (M = 20 mg), the kinetic energy is calculated: $$E_k=\frac{1}{2}M{V}^2=62.5\mu J.$$(2)

The energy necessary to this jump is stored in the Issus jumping system (Fig. 15) as elastic energy, ready to be released to perform the jump. It has the following shape for one leg: $$E_v=\frac{1}{2}\left(k_1{x}_1^2+k_2{x}_2^2\right)$$(3)

with x₁, x₂ displacements of the Coxa extremities and k₁, k₂ spring stiffness for respectively pleural arch and tendon.

And therefore, the equality between the jumping energy (kinetic energy, E_k) and stored energy of both legs (elastic energy, E_v) can be written: $$E_k=2.E_v\Rightarrow E_k=k_1{x}_1^2+k_2{x}_2^2.$$(4)

From there, it is assumed that the pleural arch distributes the load evenly on both sides of the Coxa (F₁ = F₂), the following assumption can be made: $$k_1{x}_1=F_1=k_2{x}_2=F_2$$(5)

with F₁ and F₂ loads applied on the Coxa ends.

Using equations (4) and (5), the kinetic energy equation can be simplified to calculate the stiffness k₁: $$E_k=k_1{x}_1^2+k_2{x}_2^2\Rightarrow k_1=\frac{E_k}{x_1\left(x_1+x_2\right)}=161.55\text{N/m}$$(6)

with x₂ = 100µm as stated before and x₁ = 574µm from (1).

This stiffness is then used to calculate the loads F₁ and F₂ necessary to the x₁ and x₂ displacement for the jump: $$F_1=F_2=92.73\text{mN}.$$(7)

The F₁ load is on the inner side of the trochanter, pushing in a downward direction on the scheme Figure. 15. This load is therefore located at the meshing, tangential to the gear meshing. If it is now assumed that the insect has had a problem and that only one side (one thigh) is able to push. In order to be able to jump in a straight line, the load F₁ direction must be divided equally between the two hind legs thanks to the meshing gear. In this most unfavourable scenario for the insect, the gear teeth will be subjected to the load $F_c=\frac{F_1}{2}=46.37\text{mN}$. This value represents the maximum load the gear can be expected to transmit. The calculations in the following section will therefore be based on this value, in order to be conservative and check the strength of the teeth in all situations.

Fig. 15 Simplified kinematics model of (a) the Issus nymph hind leg during the jump and (b) close up on Coxa, length value from [12], see Figure 6 for movement representation.

Fig. 16 Picture of jump phases, Issus speed evolution during jump, modified from [12].

4.2 Static FE model

To verify the strength of the Issus tooth under the maximum applied load, the static FE model of the Issus tooth load case (case R.) from part 3 (scale 530/1) is retained. An adaptation of the insect material and the actual load F_c is made.

Vincent et al. [38] have carried out an in-depth study of insect cuticles. They present (Fig. 17) the range covered by cuticles and compare it with several other organic materials in terms of Young’s modulus and tensile strength scaled by their density. For the present study, two extreme cases of these materials are extracted from the literature (Tab. 4). They are used for finite element analysis under real load. The properties of conventional steel are shown in the table for comparison purposes.

Next, the load F_c and material constant defined for the insect are adapted from the scale of the tooth to match the scale of the FE model (530/1). The adaptation is made from [39] with the aim of preserving the geometric, kinematic and dynamic isometry of the insect during jumping. Four constants are defined to enable scaling (Eq. (8)). $$\begin{array}{c}{L}_r={\kappa }_{G}L\\ F_r={\kappa }_{F}F\\ m_r={\kappa }_{m}m\\ t_r={\kappa }_{t}t\end{array}$$(8)

with κ_L geometry, κ_F load, κ_m mass and κ_t time proportional constants between replica (r) and reality.

The geometrical constant is known from the previous section: κ_G = 530. The time constant is taken to κ_t = 1 and the density is set to unchanged between reality and replica (κ_m = r³, with r scaling ratio) as the studied material constants (Tab. 5) are all defined versus their density. These assumptions lead to Table 5 definition.

Fig. 17 A material property chart for natural materials, plotting specific Young’s modulus against specific strength. Guide-lines identify materials which make good elastic hinges, and which store the most elastic energy per unit weight. (Figure created using the Natural Materials Selector, Wegst, 2004.), with in red two extreme materials of the cuticle range chosen for the study, from [38].

4.3 Issus tooth strength results

The elastic model logically gives the same results in terms of deflection and stress distribution. Only the amplitudes are modified. The displacements, recalculated on the scale of the insect (Tab. 6), are very high in relation to the size of the tooth (which is around 20 μm high according to [14]. The maximum stress on the tooth flank is well above the material’s tensile strength limit. Material A cannot transmit the required power. Material B, on the other hand, performs better. The deflection of the tooth under loading, both unitary and distributed over the three teeth of the contact ratio, remains high but consistent with the size of the tooth. What’s more, material B is capable of withstanding the load without breaking when the load is distributed over 3 teeth.

Quantified finite-element analysis shows that, with a well-chosen material between cases A and B, the insect would potentially be able to catch up with any poorly initiated jump. The gear would be able to withstand the resulting load. However, this study focuses solely on the strength limit of the material and does not take fatigue into account. This limitation does not seem particularly relevant, however, as the gears are only present in the pupal state and are therefore replaced at each moult [14]. Interestingly, adults have no such gears. One of the reasons given by Burrows and Sutton is the inability of these mechanisms to withstand tooth fatigue, as adults do not molt and therefore cannot replace a broken tooth. The results of FE analysis are therefore in line with the biological findings from the literature.

5 Conclusions

Jumping nymphal planthoppers (Issus Coleoptratus) achieve precise (within 30 μs) synchrony of their legs through the use of a pair of trochanteral gears, which interlock the left and right metathoracic legs, ensuring simultaneous leg extension [12,14]. The geometry and action of these biological gears and combined kinematic and FEA analysis were analysed to compare them with involute ISO standard man-made gears. In comparison with man-made gears, we found a few striking differences between the Issus gears and ISO standard gears.

Firstly, when engaged, the profile shape of the of the Issus gears results in a contact-ratio between 2 and 3, allowing transmission of torque over a larger number of teeth than a man-made involute gear (which have transmission ratios of 1–2). The distribution of torque along a larger number of gear teeth would reduce the loading on each individual gear tooth, and thus an individual Issus gear tooth would experience less torque than an equivalently sized involute gear. Surprisingly, however, the shape, spacing, and size of the Issus gears allows meshing with an equivalently sized ISO standard involute gear (Fig. 10). While this compatibility between Issus gears and involute shaped ISO standard gears is not biologically important for the animal, it is a neat detail that Issus gears and involute shaped ISO standard gears share enough fundamental geometric principles of gearing that they can interface together.

Secondly, finite element analysis of the Issus gears shows that, for an equivalent torque, the maximum stress on an Issus gear is higher than the maximum stress on an involute shaped ISO standard gear (Figs. 13 and 14). The point of maximum stress, however, for the Issus gear is on the flank of the tooth, as opposed to an ISO standard gear where the point of maximum stress is at the base of the tooth. This may make an Issus gear more resilient to torque because the base of a gear tooth has a stress concentration, and thus moving the point of maximum stress to the flank focuses stress to a point on the gear tooth that is more resistant to failure. Combined with the larger contact ratio, it is possible that the shape of the Issus gears may make them more resistant to wear than gears shaped like an ISO standard gear.

Lastly, the shape of the Issus gear teeth allows smooth transmission even if the gears are slightly mis-aligned, allowing an animal to cleanly jump even if the gears are engaged such that the legs aren't perfectly parallel. This resistance to errors in engagement combined with the high gear ratio and flank-focused location of maximum stress all suggest that the Issus gear shape is a more stable, error resistant, and fatigue resistant gear shape than an equivalently sized ISO standard involute gear shape would be. The Issus gear shape thus provides a bio-inspired design for a high-speed fatigue-resistant gear.

Acknowledgments

Parts of this work were conducted with the help of several student projects done at INSA Lyon, in the Mechanical Engineering department. The authors would like to thanks Lucas CERONI, Noémie PEFFERKORN, Mounia BOUGHALEB and Mathieu SAUVEUR whose work contributed to this research and the results presented in this paper. The authors would also like to thanks Abderrahman KHILA from laboratory IGFL at ENS Lyon for his advices for the insect capture.

Funding

This research was funded by INSA Lyon and its Mechanical Engineering department through the student projects that were conducted.

Conflicts of interest

The authors certify that they have no financial conflicts of interest in connection with this article.

Data availability statement

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Author contribution statement

Conceptualization, Bérengère GUILBERT, Marie WINGER, Gregory SUTTON, Richard JOHNSTON and Fabrice VILLE; Methodology, Bérengère GUILBERT, Marie WINGER, Gregory SUTTON, Richard JOHNSTON and Fabrice VILLE; Software, Bérengère GUILBERT; Validation, Bérengère GUILBERT and Fabrice VILLE; Formal Analysis, Bérengère GUILBERT, Gregory SUTTON and Fabrice VILLE; Investigation, Bérengère GUILBERT, Marie WINGER and Jérôme CAVORET; Resources, Fabrice VILLE; Data Curation, Bérengère GUILBERT, Richard JOHNSTON; Writing − Original Draft Preparation, Bérengère GUILBERT.; Writing − Review & Editing, Bérengère GUILBERT, Marie WINGER, Gregory SUTTON and Fabrice VILLE; Visualization, Bérengère GUILBERT, Marie WINGER and Jérôme CAVORET; Supervision, Fabrice VILLE and Gregory SUTTON; Project Administration, Fabrice VILLE; Funding Acquisition, Bérengère GUILBERT and Fabrice VILLE.

References

All Tables

All Figures

	Fig. 1 Gears on the hind trochantera of Issus nymphs [14].
In the text

	Fig. 2 Profile of gear tooth in Issus (left) compared to man-made involute tooth (right) [14].
In the text

	Fig. 3 Symmetric and asymmetric tooth profiles and corresponding gears.
In the text

	Fig. 4 3D digital microscope close up on (a) adult dorsal view and (b) nymph ventral view when crocked up for jump.
In the text

	Fig. 5 3D digital microscope observation of the Issus gear during jump.
In the text

	Fig. 6 Diagram of the initial (dotted blue, Fig. 5a) and final (plain red, Fig. 5c) positions of the hind legs, coxa and gear.
In the text

	Fig. 7 Teeth from Issus (a) French origin with 3D image reconstructed from the 3D digital microscope observation and (b) English origin (4.5 mg animal) from Micro-CT scan. Both (a) and (b) are from a 5th instar Issus nymph.
In the text

	Fig. 8 Issus Gear cut meshing at three meshing-steps (scale 260/1).
In the text

	Fig. 9 Smaller nymph (a) ventral view and (b) close up on tooth number.
In the text

	Fig. 10 Meshing of the cut (scale 530/1) with module 4.5 mm gear from FZG test rig.
In the text

	Fig. 11 FE model with in blue clamped area (A) and red loading area and direction (along normal x).
In the text

	Fig. 12 Mesh side view with applied load and thinner mesh for stress concentration of (a) FZG tooth, (b) Left (L) loading case of Issus and (c) Right (R) loading case of Issus.
In the text

	Fig. 13 Stress for the FZG gear, material around the loading area is not displayed, creating a visual indentation, due to the non-physical high stress concentration.
In the text

	Fig. 14 Stress for the (a) L and (b) R Issus tooth loading case, material at the loading point is not displayed due to the non-physical high stress concentration.
In the text

	Fig. 15 Simplified kinematics model of (a) the Issus nymph hind leg during the jump and (b) close up on Coxa, length value from [12], see Figure 6 for movement representation.
In the text

	Fig. 16 Picture of jump phases, Issus speed evolution during jump, modified from [12].
In the text

	Fig. 17 A material property chart for natural materials, plotting specific Young’s modulus against specific strength. Guide-lines identify materials which make good elastic hinges, and which store the most elastic energy per unit weight. (Figure created using the Natural Materials Selector, Wegst, 2004.), with in red two extreme materials of the cuticle range chosen for the study, from [38].
In the text