Causal models of rate-independent damping in insect exoskeleta Open Access

Exoskeletal structures play crucial roles in insect locomotion. In winged insects (Pterygota), the wings themselves are exoskeletal outgrowths originating from ancestral structures of uncertain form (Engel et al., 2013), and the thoracic exoskeleton contains not only the mechanisms that translate the action of the flight muscles into wingbeat motion (Melis et al., 2024; Walker et al., 2014), but also (in certain cases) structures that can store elastic energy and thereby reduce wingbeat power requirements (Gau et al., 2019; Warfvinge et al., 2017). In walking, jumping and hopping insects across orders, the exoskeleton of the legs and thorax provides contact and support forces, and can contribute to energy storage for use in jumps and bounds (Bolmin et al., 2021; Burrows, 2010). Studies of the energetics of insect locomotion focus often on the elastic properties of these exoskeletal structures – it is, after all, these elastic properties that enable improvements in flight efficiency (Gau et al., 2019; Pons et al., 2023) and jumping performance (Bolmin et al., 2021; Burrows, 2010). But damping is also at work: exoskeletal structures also dissipate energy via viscoelastic effects within the partially sclerotised chitin–protein matrix of which they are composed (Aberle et al., 2017). Empirical evidence from dynamic mechanical analysis (DMA) across a range of species and locomotor systems – including hawkmoth flight motors (Gau et al., 2019; Wold et al., 2023), beetle elytra (Lomakin et al., 2010) and cockroach legs (Dudek and Full, 2006, 2007) – indicates that this damping is often largely frequency- or rate-independent, a property consistent with the characteristics of chitin and other cuticle polymers (Aberle et al., 2017; Jensen and Weis-Fogh, 1962; Sun et al., 2016), and that can lead to favourable control properties (Dudek and Full, 2007; Wold et al., 2023).

Linear rate-independent damping – otherwise termed structural or frequency-independent damping – permits a straightforward and convenient model representation in the frequency domain: a constant imaginary value appended to the stiffness term within the force–displacement transfer function, often styled iγ (Dudek and Full, 2006; Gau et al., 2019; Wold et al., 2023). However, this straightforwardness conceals serious practical and analytical problems. Exact time-domain formulations of rate-independent damping are absent from biomechanical literature, and indeed, one finds this coefficient in no table of inverse Fourier transforms. Without such formulations, there is no accurate way to identify the classical damping parameter γ from time-domain exoskeletal mechanical data, particularly when nonlinear effects are present, and frequency-domain analysis breaks down (Lynch et al., 2021; Pons and Beatus, 2022a). Neither is there any way to simulate time-domain models of the exoskeleton, e.g. as part of a full bio-aero-structural model of an insect flight motor (Gau et al., 2023; Pons, 2023). However, parallel studies in the field of seismic analysis (Inaudi and Makris, 1996; Luo and Ikago, 2021) have made progress on time-domain modelling of same classes of damping model, and offer pathways for overcoming these biomechanical challenges. Despite the wide difference in field, vibrational deformation of built structures within earthquakes resembles that of exoskeletal structures during insect locomotion in more ways than one: in cantilever vibration in running (Dudek and Full, 2007), in complex modeshapes in the flight motor (Ando and Kanzaki, 2016) and in properties of terrestrial soil, which itself shows rate-independent damping (Kausel and Assimaki, 2002).

In this work, I address these issues by developing time-domain formulations of linear rate-independent damping and applying them to problems in the identification of insect exoskeletal damping, a process based on methods across applied mathematics, seismic analysis and biomechanics. First, leveraging techniques from seismic analysis (Inaudi and Makris, 1996; Makris, 2017), I present an exact time-domain formulation for rate-independent damping based on the singular integral known as the Hilbert transform. In doing so, I confirm this damping to be non-causal, that is, in violation of the directionality of time. Following and extending further existing results (Makris, 1997a; Pons, 2024), I resolve the causality violation and obtain a pair of approximate causal time-domain models, supported by new and accessible numerical methods for computing the singular integrals associated with these models. Next, I apply these causal and non-causal models to challenging problems in entomological literature, including the identification of rate-independent damping parameters from exoskeletal transient responses and DMA data, accounting for nonlinear stiffness and motion asymmetry (Dudek and Full, 2007; Wold et al., 2023), and time-domain simulation of structural models with rate-independent damping (Dudek and Full, 2007; Gau et al., 2023). Via these simulations, I reveal a crucial and previously unknown caveat of linear rate-independent damping: such damping is not always dissipative, that is, it is capable of releasing energy, as well as absorbing it. In doing so, I identify new restrictions on the use and validity of these damping models. Finally, I identify pathways toward more clearly understanding the mechanistic origins of and best-practice models for rate-independent damping. Synergising applied mathematics, seismic analysis and insect biomechanics, these results provide wider integrative studies of insect locomotion with tools to integrate rate-independent damping within time-domain models of insect locomotion processes, as well as a more detailed understanding of the limits of validity of such models and the ways in which they can break down.

In Eqn 1, the damping force is frequency dependent: scales by Ω (Fig. 1B). Empirical data for a range of insect exoskeletal structures contradict this viscous model (Dudek and Full, 2006, 2007; Gau et al., 2019; Lomakin et al., 2010; Wold et al., 2023), and indicate rather that this force amplitude (and associated metrics, such as the energy loss per oscillation) is largely independent of frequency, over several orders of magnitude. Several such data are illustrated in Fig. 1C.

As can be seen, the rate-independent damper itself has zero storage modulus, but this will not be the case for all rate-independent dampers, as shown in the next three subsections.

For illustration, Fig. 1 compares the response of a rate-independent damper (Eqn 7) with that of a viscous damper (Eqn 1), for a triangle wave input. Practically, evaluating the Hilbert transform is straightforward. A wide range of analytical results have been tabulated (King, 2009; Poularikas, 2018), and numerical implementations, in the form of the discrete Hilbert transform (DHT), are available in many scientific computing languages: in MATLAB, as hilbert(); in Python, as hilbert() within SciPy; and in R, via hilbert() within the gsignal package or HilbertTransform() within the hht package. An implementation in MATLAB is available within Dataset 1.

Evaluating Eqn 12 for general exoskeletal input data is challenging. A description and validation of novel numerical methods for doing so is presented in the Appendix. A derivation in Supplementary Materials and Methods 2 and an implementation in MATLAB within Dataset 1. These numerical methods enable the simulation illustrated in Fig. 2, which confirms that this rate-independent damping model is indeed causal. However, this causality has come at a cost. The damper now has a storage modulus (Eqn 9), which is not only contrary to the intent of a damping model, but is also negative-valued at frequencies below Ω_ref, and tends toward negative infinity in the quasistatic limit (Ω→0). As discussed in more detail later (see ‘Energetic properties of rate-independent damping’), Ω_ref can be set to ensure that any specified exoskeletal motion, x(t), will only elicit positive storage modulus. However, no Ω_ref can completely eliminate the trend to infinite negative storage modulus, and this raises a pair of new concerns. First, that infinite negative storage modulus is non-physical, and so this damper also can only be an idealisation (limit case) of a real physical structure. Second, that if the exoskeletal motion is not known in advance – e.g. in a simulation of an exoskeletal structure (see Gau et al., 2023) – then it is not possible to guarantee positive storage modulus. These concerns motivate a final alteration to this rate-independent damping model.

Finally, the Biot element is also physically realisable, as an infinite series of simple linear springs and viscous dampers, arranged in a generalized Maxwell configuration (Fig. 2C). More specifically, the Biot element arises in the case in which individual stiffnesses (k_i) are inversely proportional to relaxation times (r_i=k_i/d_i), k_i1/r_i, implying that individual damping coefficients show a slowly increasing trend with stiffness, ⁠. Details of this derivation are given in Caughey (1962) and Luo and Ikago (2021). In addition, based on the principle that the Biot element approaches the causal rate-independent damper as N_ref→∞, the latter should also be regarded as physically realisable (at least, in the limit). The generalized Maxwell construction of these models not only confirms the realisable nature of this damping, but also offers insight into one mechanism by which rate-independent damping can arise in an exoskeleton: as a bulk effect caused by a statistical distribution of elastic and viscous mesoscale or microscale structures. I discuss this, and other mechanisms, in the Discussion.

Above, I presented a series of three rate-independent damping models, defined in the time domain, and each addressing particular obstacles raised by the previous (causality, and infinite negative storage modulus). The end result is the Biot element, which suffers from none of the pathologies of the ideal rate-independent damper, but is considerably more complex, and comes with a storage modulus and a range of tuning parameters. However, for sinusoidal motion at any frequency Ω*, all three damping models can be made equivalent: the storage modulus of the causal damper and Biot element can be set to zero by setting Ω_ref=Ω*, and the Biot element will show only slightly smaller loss modulus (according to N_ref). The ideal damper is still non-causal, but in this circumstance, it behaves no different to its causal analogues. As such, for motion that is close to sinusoidal, the ideal rate-independent damper likely shows sufficient realism that it can be used as a convenient approximation. It is in the case of strongly non-sinusoidal motion that the other two damping models become relevant. Both in realism, and in a slightly more convenient evaluation, the Biot element is superior to the causal rate-independent damper: the former is merely the latter with an extra parameter (N_ref), and the latter is a limit case of the former with a specific connection to the original ideal rate-independent damper.

Using the time-domain models described above, progress can be made on several challenging problems in insect biomechanics that pertain to rate-independent damping. The first challenge was raised by Gau et al. (2019) and Wold et al. (2023) in the context of rate-independent damping within the exoskeleton of hawkmoths (Manduca sexta) within flight. The challenge is that existing frequency-domain methods for identifying rate-independent damping are restricted to a very specific form of data: loss moduli, measurable when a structure that behaves linearly is subjected to sinusoidal excitation during DMA. If loss moduli cannot be estimated owing to nonlinear effects – e.g. nonlinear elasticities, as per Gau et al. (2019) – then frequency-domain identification breaks down. And if we wish to study the case of non-sinusoidal excitation, as per Wold et al. (2023), to better understand thoracic damping effects during insect flight manoeuvres, then frequency-domain identification again breaks down, and worse, we have no method of confirming whether a rate-independent damping model remains valid.

The final challenge that is partly solved by these time-domain methods is that raised across Dudek and Full (2007), Lynch et al. (2022) and Gau et al. (2023): the challenge of simulating general insect exoskeletal models forward in time. In case of the insect flight motor, forward simulations of integrative motor under asynchronous muscle forcing have helped reveal deep properties of these motors, including their evolutionary history (Gau et al., 2023), but structural damping (e.g. as identified in Fig. 3) has been a missing component. A unique feature of the causal damping models is that they are compatible with structural models of the exoskeleton based on initial value problems in time. The causality of these models allows them to be simulated forward in time, with no knowledge of the future, as is typical for structural models, and as is representative of real exoskeletal behaviour.

Two features illustrate that these simulation results are a good representation of causal rate-independent behaviour. First, simulating the model at different natural frequencies (with equivalent tuned Ω_ref) yields free response profiles that are visually identical to those of Fig. 5B when scaled in time by T. This indicates that the damping within the model is frequency independent. Second, the damper forcing, –C_BIx_k, is a good approximation of the forcing of the ideal rate-independent damper (Fig. 5C). This indicates that the scale and waveform of forcing are representative of ideal rate independence. However, note that the numerical scheme is also subject to numerical dissipation; for greater precision, variational approaches could be devised (see Pons and Cirak, 2023).

These simulations reveal several interesting and concerning properties of the Biot element. Because the force response of the Biot element is an integral over past time, the entire system's response is strongly dependent on its long-term history, i.e. its behaviour before the initial condition. Fig. 5B compares this long-term memory effect for two prescribed long-term histories: the system held at x=1 mm over an infinite past, and the damper held at x=0 mm and then suddenly brought to x=1 mm right before the system is released. These two histories lead to free oscillations with large differences (∼100%) in oscillation amplitude. As discussed later, this presence or absence of memory effects in real exoskeleta is one characteristic that may help filter between differing models of rate-independent damping.

This memory effect can also lead to strange energetic properties. Fig. 5B,C compares the instantaneous power dissipation profiles of the Biot element (⁠⁠) under the two response motions. These power profiles contain regions of power output, or negative dissipation (P<0). Here, there are several challenging properties to disentangle. First, in a passive structure, instantaneous power output can still respect conservation of energy provided that this energy is stored and released via a conservative potential, i.e. structural elasticity. The Biot element can be realised as an array of linear dissipative and elastic elements (Fig. 2C), each of which respect conservation of energy, and so the Biot element itself must also respect conservation of energy: instantaneous power output must arise from the elastic elements of the element. However, from the work loops in Fig. 5C, we see that this power output is not consistent with any 1DOF linear or nonlinear parallel or series elasticity, as per Pons and Beatus (2022b, 2023): the Biot element, while conserving energy, behaves as a multiple-DOF rather than 1DOF structure, with internal elements of energy storage. However, by contrast, the ideal rate-independent damper appears to violate conservation of energy directly. This damper has no storage modulus or physical realisation that could provide energy storage and release, and more fundamentally, it is non-causal – without the directionality of time, conservation of energy appears meaningless. The topic of damper energetics leads to the next section.

First, the triangle pulse waveform in Figs 2 and 3 is purely dissipative. This applies to other isolated symmetric pulses, e.g. Gaussian, cosine and square pulses, but not to asymmetric versions of these pulses (half-Gaussian, half-cosine, etc.). Conveniently, these properties can be observed directly in tabulated Hilbert transform results (King, 2009); for pure dissipation, the Hilbert transform and pulse velocity always show opposite sign.

Second, dipteran wing stroke angle kinematics, being close to sinusoid, are dissipative for practical purposes. Fig. 6B illustrates this for several recent estimates of the wing stroke kinematics for hovering Drosophila melanogaster (Beatus and Cohen, 2015; Ben-Dov and Beatus, 2022; Maya et al., 2023). Strictly, however, these kinematics show a very small level of power output – approximately 0.1% of dissipative peak power.

Third, when a periodic wave has displacement extrema that are unequally spaced in time, non-dissipative behaviour can arise. Fig. 6C illustrates this effect for a triangle wave. In an insect flight motor, this unequal spacing represents, for example, the upstroke being slightly faster than the downstroke, as in D. melanogaster, partly explaining the power output described above and in Fig. 6B. However, the power output introduced by this effect is small, and the rate-independent damper may remain a reasonable approximation.

And fourth, noisy motion is not dissipative (Fig. 6D): the addition of high-frequency noise leads to power output in the response, as per the case with multiharmonic components (see Fig. 6A). The effect can be removed by filtering noise out of the signal, but notably this means that rate-independent dampers cannot effectively model a structure's response to noise excitation, e.g. the white noise motion that Wold et al. (2023) applied to hawkmoth thoraces.

The situation in the causal rate-independent damper and Biot element is more complex, as these structures have a nonzero storage modulus which itself will contribute to instantaneous power output. The construction of these dampers as the limit of generalised Maxwell configurations (see The Biot element and Fig. 2) gives the physical insight that this instantaneous power output arises from the combined action of different energy storage elements (stiffnesses) within the structure, which interact to output energy in a way that cannot be replicated by a single stiffness. To control this behaviour, the reference frequency, Ω_ref, is available as a free parameter. The role of Ω_ref can be described in two equivalent ways. First, Ω_ref tunes the damper to have zero storage modulus under sinusoidal motion at certain frequency, equating to pure power dissipation at this frequency. Second, Ω_ref defines a frequency threshold: above it, the damper storage modulus acts as positive stiffness; and below it, as negative stiffness (the latter may be unwanted behaviour). The question of whether Ω_ref is a real physical property, or just a threshold to exclude unwanted behaviour, is a challenging one (see the Discussion, Origins of rate-independent damping).

Broadly, however, the same classes of energetic behaviour identified in the previous subsection also apply to these dampers, because this behaviour arises from the loss modulus rather than the storage modulus. However, there is an additional energetic effect that is worth isolating. For sinusoidal motion at some frequency Ω, pure dissipation occurs at Ω_ref=Ω, a transparent energetic tuning relationship. However, for non-sinusoidal periodic waves at fundamental frequency Ω, the value of Ω_ref that exactly ensures pure dissipation (if any such value exists) may not be equal to Ω. Fig. 7 illustrates this phenomenon for the response of a causal rate-independent damper to a symmetric triangle wave. Denoting Ω₀ as the fundamental frequency of this triangle wave, a state of pure dissipation is reached at Ω_ref≈1.39Ω₀. If we were to select Ω_ref=Ω₀ regardless, power output would be present, to the level of ≈26% of the peak power dissipation and ≈0.35% of the overall absolute work. Several points on this effect should be noted.

Firstly, a general heuristic, following Fig. 7, is as follows: if the waveform is more like a triangle wave than a sinusoid, then the perfect tuning for Ω_ref will be higher than Ω₀, but not likely more than 40% higher (as this is the limit case for a pure triangle wave).

Secondly, for insect wingbeat kinematics under hovering flight, this effect is likely insignificant. For example, D. melanogaster wingbeat kinematics tend slightly toward a triangle wave (see Fig. 6B and Dickinson et al., 1999). However, this tendency is sufficiently small that the approximation Ω_ref=Ω₀ is suitable: for the kinematics of Maya et al. (2023), the peak power output under this approximation is only ≈0.1% of peak power dissipation.

Finally, two cases in which this effect may be significant are: (1) if the structure is undergoing a transient perturbation, e.g. the triangle perturbation of Fig. 2, in which case measurements of the perturbation timescale cannot be completely relied upon to determine Ω_ref; and (2) if the structure nominally undergoes sinusoid motion but then this waveform is altered, e.g. an insect flight motor during an extreme manoeuvre (see Gau et al., 2021). In the latter case, the only reliable option is to identify η and Ω_ref simultaneously for the data under consideration.

That rate-independent damping of some form is present in the insect exoskeleton is borne out by experimental data (Dudek and Full, 2006; Gau et al., 2019; Wold et al., 2023), but the physical mechanisms underlying it remain unclear. Several non-exclusive hypotheses can be made, and these hypotheses align in a revealing way with different models of rate-independent damping presented earlier.

The first hypothesis is that this damping represents structural hysteresis arising from elastoplastic behaviour: a transition between recoverable (elastic) and irrecoverable (plastic) deformation as strain increases (Giorgi and Morro, 2021; Zheng, 2019). In metals, the grain-related mesoscopic mechanisms behind this transition are well studied (Weng, 1983; Zheng, 2019). Harder materials within the exoskeleton may show elastoplastic behaviour: there is evidence for plasticity both of the chitin–polymer matrix of insect cuticle (Hillerton et al., 1982) and in chitin itself (Mir et al., 2008), but details are unclear. Elastoplasticity generates true quasistatic hysteresis: energy dissipation is maintained in the quasistatic limit (Ω→0). In a linear model, this corresponds to a constant loss modulus as Ω→0, which we now see is physically impossible: such a linear model is necessarily non-causal and/or shows infinite negative stiffness. This implication here is that true hysteresis is beyond the reach of linear analysis: it is a nonlinear phenomenon that is impossible to linearise without sacrificing basic physical principles. If the causal rate-independent damper or Biot element are used to approximate this hysteresis, then the parameter Ω_ref must be regarded as a threshold to exclude unwanted and inaccurate storage modulus trends in the quasistatic limit.

The second hypothesis is that this damping represents a viscoelastic effect that is largely rate independent but remains fundamentally dynamic rather than hysteretic. Here, the Biot element offers the insight that rate-independent damping can arise from an ensemble of mesoscale viscous dampers and linear elasticities (Fig. 2C), analogous to how weakly rate-dependent (fractional-order) damping can arise from a hierarchy of self-similar mesoscale elements within vertebrate cartilaginous tissues (Guo et al., 2021). In the exoskeleton, there are several possibilities for this mesoscale ensemble, extending from the mesoscopic composition of cuticle to the varying thickness and composition of the sclerites (Barbakadze et al., 2006; Casey et al., 2022). Yet, at its core, this viscoelastic damping remains an essentially dynamic process: it cannot show dissipation under true quasistatic motion, and the loss modulus must tend to zero as Ω→0. In this sense it is fundamentally different to true hysteresis, and can be modelled linearly, as per the construction and properties of the Biot element. Under this hypothesis, the Biot element's region of negative storage modulus (as per Ω_ref) is not necessarily non-physical. When the Biot element is combined with a sufficiently large linear (or nonlinear) stiffness, this negative storage modulus appears simply as decreased quasistatic stiffness, as appears, for example, in experimental DMA data for insect flight muscle (Viswanathan et al., 2020). In this context, Ω_ref might be regarded as a physical model parameter.

Distinguishing between these hypotheses does not appear possible with current data, and both effects may be present in the exoskeleton. Model-identification approaches may be an effective approach: each effect has naturally associated models (nonlinear hysteretic versus linear viscoelastic), and testable predictions are available for each. These are discussed below.

This study began with the classical frequency-domain exoskeletal damping model: the complex stiffness iγ. As time-domain formulations reveal, this model is only realistic under harmonic excitation; outside this narrow condition, it observably violates causality. Studying rate-independent damping under general exoskeletal motion requires alternative models, and in this work, two have already been suggested: the causal rate-independent damper and the Biot element. Future prospects for general models of rate-independent damping are inflected by the two hypotheses for its origin that have just been outlined.

In the direction of models for viscoelastic rate-independent damping, the Biot element is a promising candidate. Fractional-order models (Luo and Ikago, 2021) are an alternative with more pronounced rate scaling (i.e. weaker rate-independence), and are currently used in structural models of insect asynchronous muscle (Swank et al., 2006). Considered as a class, these viscoelastic models have a common characteristic: they show a storage modulus that varies with frequency. This varying storage modulus is a key testable prediction of viscoelastic modelling, and an understudied feature in existing structural damping studies. If linear viscoelastic damping is present, then a varying storage modulus will be observed, and if this damping is approximately rate independent, then the storage modulus scaling will be approximately logarithmic. Indeed, evidence for varying storage modulus may already be available: the raw time-series force data of Gau et al. (2019) show increasing peak force with frequency, which may indicate an increasing thoracic storage modulus.

In the direction models for general hysteretic rate-independent damping, the overwhelming consensus of our analysis is that linear hysteresis is non-physical: nonlinear modelling is required. Studies of structural hysteresis in other materials, such as rock (Guyer et al., 1995) and rubber (Penas et al., 2022), suggest several directions for nonlinear modelling, including thermodynamics-driven constitutive modelling with relaxation functions, and assemblies of hysteretic mesoscopic elements, or hysterons (Bertotti and Mayergoyz, 2006; Giorgi and Morro, 2021; Guyer et al., 1995). A key testable prediction of these models, and of quasistatic hysteresis generally, is the phenomenon of endpoint memory (Claytor et al., 2009; Vakhnenko et al., 2005), otherwise known as return-point memory (Zirka and Moroz, 1999), or Madelung's second rule (Brokate and Sprekels, 1996; Penas et al., 2022): the material remembers discrete information relating to recent displacement reversals, or endpoints.

The as-yet unresolved distinction between viscoelastic and hysteric forms of rate-independent damping in the exoskeleton has implications for understanding insect locomotion. Two points of relevance stand out. First, in analyses of exoskeletal mechanisms (e.g. the flight motor), energy dissipation and energy storage are phenomena that are typically considered, and modelled, independently (see Gau et al., 2022; Pons and Beatus, 2022a). Our analysis indicates that, in cases of structural damping, these two effects are closely intertwined. For instance, in the viscoelastic scenario (e.g. the Biot element), a rate-independent loss modulus directly implies a storage modulus that logarithmically increases with frequency. This in turn implies that tuning exoskeletal energy storage to particular locomotive behaviours might be more complex than it appears: the effective stiffness of the exoskeleton may itself vary across behaviours at different frequencies, and quasistatic stiffness measurements (see Jankauski, 2020), may not be completely representative of dynamic motion.

The second point of relevance concerns a commonality between viscoelastic and hysteretic rate-independent damping: the presence of structural memory. The Biot element shows continuous memory (Q_BI; Fig. 2A), whereas hysteretic models show discrete (endpoint) memory. Both memory effects would be testable with current exoskeletal DMA techniques (see Guyer et al., 1997 and Wold et al., 2023), and they bring to mind the topics of physical and/or mechanical intelligence (Blickhan et al., 2007; Sitti, 2021): forms of simple intelligence embodied in passive structure. Energetic memory effects could potentially play a significant role in biolocomotive efficiency. They are, for instance, a challenge to carefully constructed frameworks (Pons and Beatus, 2022b, 2023) that seek to identify the effects of insect wingbeat modulation (frequency and asymmetry) on the flight motor. Hypothetically, by using memory effects to output power at points within the stroke cycle that are not accessible to conventional elasticity, an insect flight motor could violate the elastic-bound conditions (Pons and Beatus, 2022b) and perform wingbeat modulations at lower energetic cost than would otherwise be possible. Structural data precise enough to test this hypothesis have not yet been reported in the literature, but these possibilities motivate further study of rate-independent damping.

In this work, I developed the first complete time-domain models of rate- (or frequency-) independent damping in insect exoskeleta. I began by translating an established frequency-domain exoskeletal damping model, the iγ model, into the time domain, identifying this time-domain form as the Hilbert transform, and revealing it to be strongly non-causal. Although this noncausality is not problematic under approximately sinusoidal motion, it is more serious under isolated perturbations; for these, significant anticipatory force generation occurs. In response, I extended results from seismic analysis to develop causal time-domain models of exoskeletal rate-independent damping, including an extended form of the Biot element. I developed and validated numerical quadrature techniques for the Hadamard finite-part integrals defining these models, allowing computation of responses to general motion, and simulated the damper within broader structural models. These techniques were applied to several problems in insect biomechanics: identifying thoracic damping from non-sinusoidal DMA data in the presence of nonlinear elasticity; identifying complete limb structural parameters from perturbation kinematics; and simulating exoskeletal structural models in time. In doing so, a key caveat of these models was revealed: they are not purely dissipative, but can output power in ways that are inconsistent with a single nonlinear elasticity. Together, these methods provide: (i) rigorous and complete methods to identify and simulate rate-independent damping in various exoskeletal contexts; (ii) testable predictions to distinguish different models of, and mechanisms behind, exoskeletal damping; and (iii) avenues for further improvement in modelling and understanding this understudied property of the insect exoskeleton.

Two further practical points regarding DHT implementation should be noted. First, existing implementations typically output not the Hilbert transform itself but the analytic signal, of which the Hilbert transform is the imaginary part: (King, 2009; Marple, 1999). Second, these implementations typically compute the Hilbert transform via the fast Fourier transform (FFT), and thereby assume the input signal to represent a single period of a longer periodic signal. This may be convenient in some contexts (e.g. periodic insect wingbeats), but inconvenient in others (e.g. perturbations of insect limbs). When non-periodic signals are under consideration, they must be zero padded (see Press, 2007).

Computing the finite-part integrals governing the causal rate-independent damper and Biot element is more complex, and requires customised numerical methods. Broadly, two types of method are available: those performed in the frequency domain, via FFT, and those performed directly in the time domain. Both have situational merits.

Frequency-domain computation of these causal damper responses is convenient; my implementations are illustrated, for validation, in Fig. A1. At some instant t, I: (i) FFT the input signal, ⁠; (ii) multiply by the relevant complex modulus to obtain output force, or ⁠; and (iii) inverse FFT, ⁠. In the case of the causal rate-independent damper, this process contains a weak (logarithmic) singularity in the storage modulus, at ⁠. Because this weak singularity is integrable, it suffices to approximate ⁠, where ΔΩ is the frequency step of the FFT data. Because of the integrability of the singularity, as ΔΩ→0, the damper's response will converge, in the manner of a principal value (p.v.). As a corollary, it is also suitable to perform the computation with an even-number FFT spectrum, not containing Ω=0. In the case of the Biot element, there is no singularity at all. Although this type of evaluation is convenient, its key drawback is that the FFT always takes the input motion, x(t), to be periodic. If this motion is non-periodic (e.g. Fig. 2), then extensive zero padding is required to prevent bleed over from the assumption of periodicity.

In Eqn A4, it is important that the same quadrature method is used as in Eqn A2, as these correctors can account for certain errors introduced via quadrature. In general, these numerical correctors are preferable: they are more accurate (Fig. A1C) and represent only a small increase in computational cost.