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

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 Poicaré-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.

Stability of Fixed Points in an Approximate Solution of the Spring-mass Running Model

“…The same approximation of K as ( 27) was obtained in [27] and [29]. Moreover, in [28] it was shown that the leading order of the expansion of…”

“…An asymptotic analysis of the main equations (2) with the use of the Poincaré -Lindstedt series was carried out in [27]. The solution of the problem is based on the perturbative expansion related to the significant spring stiffness (K → ∞).…”

“…The above problem can be easily solved numerically using the shooting method, as was done for example in [12]. Graphical illustrations confirming the validity (27) as an approximation of the solution to the boundary problem (28) are presented in [27] and [29]. In Fig.…”

“…Our approach is similar to the one conducted in [24] where the authors use a different approximation strategy stemming from the principles of mechanics, which, however, obscures the error one makes by necessarily approximating the solutions. In contrast to that, our approach is based on rigorous asymptotic analysis that clearly formulates the region of validity of the model [27,28]. Moreover, as our numerical studies indicate, the fixed point of the Poincaré map undergoes a transcritical bifurcation with increasing energy.…”

“…This has been done in the work of Geyer, Seifarth, and Blickhan [24] which is one of the basis of our subsequent results. Moreover, some approaches to analysis of the current model based on approximate solutions have been conducted in [25,26,22] and in our earlier works [27,28]. There, we have conducted a rigorous Poicaré-Lindstedt asymptotic analysis based on the perturbation theory in the regime of large spring stiffness and small angle of attack.…”