Four degree-of-freedom lumped parameter model of the foot-ankle system exposed to vertical vibration from 10 to 60 Hz with varying centre of pressure conditions

Abstract

Modelling the foot-ankle system (FAS) while exposed to foot-transmitted vibration (FTV) is essential for designing inhibition methods to prevent the effects of vibration-induced whitefoot. K-means analysis was conducted on a data set containing vibration transmissibility from the floor to 24 anatomical locations on the right foot of 21 participants. The K-means analysis found three locations to be sufficient for summarising the FTV response. A three segment, four degrees-of-freedom lumped parameter model of the FAS was designed to model the transmissibility response at three locations when exposed to vertical vibration from 10 to 60 Hz. Reasonable results were found at the ankle, midfoot, and toes in the natural standing position (mean-squared error (e) ¼ 0.471, 0.089, 0.047) and forward centre of pressure (COP) (e ¼ 0.539, 0.058, 0.057). However, when the COP is backward, the model does not sufficiently capture the transmissibility response at the ankle (e ¼ 1.09, 0.219, 0.039). Practitioner summaryThe vibration transmissibility response of the foot-ankle system (FAS) was modelled with varying centre of pressure (COP) locations. Modelling the FAS using three transmissibility locations and two foot segments (rearfoot and forefoot) demonstrated reasonable results in a natural standing and forward COP position to test future intervention strategies.

The Effects of Altering the Center of Pressure in Standing Subjects Exposed to Foot-Transmitted Vibration on an Optimized Lumped-Parameter Model of the Foot

“…More specifically, there were three Kelvin-Voigt models associated with the soft tissue on the sole of the foot: at the rearfoot sole (i.e., heel or plantar) (k e and c e ), the metataral/phalange joint (i.e., distal head of the metatarsals) (k f and c f ) and the distal phalanges (i.e., tip of the toes) (k g and c g ). As previously suggested and anticipated, the stiffness of the heel fat pad increased when the COP was backward and decreased as the COP moved forward towards the toes [13]. Stiffness of the rearfoot fat pad (k B e = 3.5 × 10 5 N•m −1 , and k F e = 1.9 × 10 5 N•m −1 ) are in accordance with natural value k N e = 2.3 × 10 5 N•m −1 , comparable with the value reported by [30].…”

“…The opposite resulted when the COP was concentrated towards the rearfoot: the stiffness at the forefoot decreased, while the stiffness of rearfoot increased. This effect is reasonable, as the mass distribution on the foot regions changes accordingly to COP position [13]. For instance, in the forward COP position, the mass distribution is closer to the toes, causing a compression state in the musculature and ligaments of the superior portion of the forefoot (k b and k c ).…”

“…An improved and asymmetric version of this model investigated the role of the plantar fascia in the load-bearing aspect of the foot [9]. This hinge model was then used as the foundation for developing three lumped-parameter models that more specifically simulate the effects of FTV on the FAS [10,12,13].…”

“…The model structure was reduced to a four degrees-of-freedom (DOF) lumped-parameter model of the FAS and was tested using transmissibility measurements while participants were in three COP positions (natural, forward towards the toes and backward towards the rearfoot) [13]. Although the model performed well when the COP was natural or forward towards the toes, the optimization was inadequate when the COP was in the heel.…”

“…This model was selected as it performed well using natural standing position transmissibility and apparent mass data but has yet to be optimized when altering the COP. The study's central hypothesis is that variation in the COP will change the equivalent stiffness and damping values of the FAS, with this model more effectively simulating the transmissibility response when the COP is towards the heel [13]. It is anticipated that the stiffness parameters at the heel will increase when the COP is in the rearfoot and similarly for the toes when the COP is in the forefoot.…”