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Solution and asymptotic analysis of a boundary value problem in the spring-mass model of running. (English) Zbl 1434.34052
Summary: 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.
MSC:
34E10 Perturbations, asymptotics of solutions to ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
70K60 General perturbation schemes for nonlinear problems in mechanics
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