Causal models of rate-independent damping in insect exoskeleta

In insect locomotion, the transmission of energy from muscles to motion is a process within which there are many sources of dissipation. One significant but understudied source is the structural damping within the insect exoskeleton itself: the thorax and limbs. Experimental evidence suggests that exoskeletal damping shows frequency (or rate) independence, but investigation into its nature and implications has been hampered by a lack methods for simulating the time-domain behaviour of this damping. Here, synergising and extending results across applied mathematics and seismic analysis, I provide these methods. Existing models of exoskeletal rate-independent damping are equivalent to an important singular integral in time: the Hilbert transform. However, these models are strongly noncausal, violating the directionality of time. I derive the unique causal analogue of these existing exoskeletal damping models, as well as an accessible approximation to them, as Hadamard finite-part integrals in time, and provide methods for simulating them. These methods are demonstrated on several current problems in insect biomechanics. Finally, I demonstrate, for the first time, that these rate-independent damping models show counterintuitive energetic properties - in certain cases, extending to violation of conservation of energy. This work resolves a key methodological impasse in the understanding of insect exoskeletal dynamics and offers new insights into the micro-structural origins of rate-independent damping as well as the directions required to resolve violations of causality and the conservation of energy in existing models.