# Solution and asymptotic analysis of a boundary value problem in the spring-mass model of running

## TL;DR

This paper rigorously analyzes a boundary value problem in the spring-mass model of running, deriving asymptotic solutions and proving the existence and uniqueness of the solution, with implications for understanding stiffness dependence.

## Contribution

It provides a rigorous asymptotic analysis of the boundary value problem in the spring-mass running model, including existence, uniqueness, and approximation of stiffness.

## Key findings

- Existence and uniqueness of the solution to the boundary value problem.
- Asymptotic estimates for the stiffness based on initial conditions.
- Numerical validation of theoretical results.

## Abstract

We consider the classic spring-mass model of running which is built upon an inverted elastic pendulum. In a natural way, there arises an interesting boundary value problem for the governing system of two nonlinear ordinary differential equations. It requires us to choose the stiffness to ascertain that after a complete step, the spring returns to its equilibrium position. Motivated by numerical calculations and real data we conduct a rigorous asymptotic analysis in terms of the Poicar\'e-Lindstedt series. The perturbation expansion is furnished by an interplay of two time scales what has an significant impact on the order of convergence. Further, we use these asymptotic estimates to prove that there exists a unique solution to the aforementioned boundary value problem and provide an approximation to the sought stiffness. Our results rigorously explain several observations made by other researchers concerning the dependence of stiffness on the initial angle of the stride and its velocity. The theory is illustrated with a number of numerical calculations.