DUDEK, DM; GOLDMAN, DI; FULL, RJ; Univ. of California, Berkeley: A Hysteretic Model Predicts a Biological Structure�s Response to Perturbations: A Test Case Using Insect Legs

Biological structures are often modeled using a linear spring (k) and viscous damper (c) in parallel (Voigt model). This model is commonly used because it is linear and analytical solutions for the equations of motion exist, but it has a major deficiency. While the energy absorbed by a viscous damper is directly proportional to oscillation frequency, energy lost per cycle in many biological materials is independent of frequency. Values for k and c calculated at one frequency cannot predict the behavior of the structure at another frequency or for other non-sinusoidal or transient perturbations. A hysteretic damping model maintains the frequency independence of energy absorption by using a complex stiffness term, k(1+i&gamma), where &gamma is the structural damping factor. Previously, we presented data on the sinusoidally oscillated hind limb of the Deathhead cockroach, Blaberus discoidalis, showing that energy lost per cycle is independent of frequency and proportional to amplitude squared. Both of these properties are characteristics of systems with hysteretic damping. We fit our leg oscillation data to the hysteretic model and found that over a 2.5 decade range of frequencies, stiffness rose only 1 N/m per decade increase of frequency (k=10 N/m at 1 Hz) while the structural damping factor was independent of frequency over this same range (&gamma=0.28). Moreover, when used to predict the response of the leg to a force impulse perturbation, these parameters gave a good approximation to actual results. With a hysteretic model, we describe a biological structure using only two numbers and predict its response to multiple types of perturbations. This predictive ability is a significant improvement over the viscous damping models traditionally used by biologists.