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

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2021-05-15 DOI:10.21203/RS.3.RS-563354/V1

Zofia Wr'oblewska, P. Kowalczyk, Lukasz Plociniczak

引用次数: 0

Abstract

We consider a classical spring-mass model of human running which is built upon an inverted elastic pendulum. Based on our previous results concerning asymptotic solutions for large spring constant (or small angle of attack), we construct analytical approximations of solutions in the considered model. The model itself consists of two sets of differential equations - one set describes the motion of the centre of mass of a runner in contact with the ground (support phase), and the second set describes the phase of no contact with the ground (flight phase). By appropriately concatenating asymptotic solutions for the two phases we are able to reduce the dynamics to a one-dimensional apex to apex return map. We find sufficient conditions for this map to have a unique stable fixed point. By numerical continuation of fixed points with respect to energy, we find a transcritical bifurcation in our model system.

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弹簧质量运行模型近似解中不动点的稳定性

我们考虑了一个建立在倒立弹性摆上的经典人体跑步弹簧质量模型。基于我们之前关于大弹簧常数（或小迎角）渐近解的结果，我们构造了所考虑模型中解的解析近似。模型本身由两组微分方程组成——一组描述了与地面接触的转轮质心的运动（支撑阶段），第二组描述了不与地面接触阶段（飞行阶段）。通过适当地串联两个阶段的渐近解，我们能够将动力学简化为一维顶点到顶点的返回图。我们找到了这个映射具有唯一稳定不动点的充分条件。通过对不动点相对于能量的数值延拓，我们发现了模型系统中的跨临界分岔。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.