Predicting Dancers’ Jumping Patterns through the Health of Bone Cartilage Using Mathematical Modelling and Simulation

Abstract

Many dancers overwork their jumps without proper rest – causing the body’s cartilage (or shock absorbers, which reduce the impact of the jump on the joints) to become damaged or wear away. The affected cartilage can lead to various negative effects including limited movements such as not being able to jump as well. This study incorporated mass-spring-damper (MSD) models and for describing the equations of motion (EOM) using a system of coupled second order equations. These were then solved using the fourth order Runge-Kutta method to predict the dancer’s jumping patterns and to understand how losses of cartilage in specific places of the human body can affect a dancer’s jumping patterns for a specific initial velocity. The MSD model varied in complexity (ranging from one mass, one spring, and one damper to three masses, three springs, and three dampers) to modeling the entire human body. The masses represented specific parts of the body (lower leg, upper leg, and portion of body above hips), the springs representing joints (ankle, knee, and hip), and the dampers representing shock absorbers (hyaline/ankle cartilage, meniscus/knee cartilage, and articular/hip cartilage). This interdisciplinary research uses mathematical modeling to compare the graphs of different MSD models to understand how cartilage (healthy and deteriorated) affects a dancer’s jumping patterns.