McMahon T A, Cheng G C
Division of Applied Sciences, Harvard University, Cambridge, MA 02138.
J Biomech. 1990;23 Suppl 1:65-78. doi: 10.1016/0021-9290(90)90042-2.
A mathematical model for terrestrial running is presented, based on a leg with the properties of a simple spring. Experimental force-platform evidence is reviewed justifying the formulation of the model. The governing differential equations are given in dimensionless form to make the results representative of animals of all body sizes. The dimensionless input parameters are: U, a horizontal Froude number based on forward speed and leg length; V, a vertical Froude number based on vertical landing velocity and leg length, and KLEG, a dimensionless stiffness for the leg-spring. Results show that at high forward speed, KLEG is a nearly linear function of both U and V, while the effective vertical stiffness is a quadratic function of U. For each U, V pair, the simulation shows that the vertical force at mid-step may be minimized by the choice of a particular step length. A particularly useful specification of the theory occurs when both KLEG and V are assumed fixed. When KLEG = 15 and V = 0.18, the model makes predictions of relative stride length S and initial leg angle theta o that are in good agreement with experimental data obtained from the literature.
提出了一种基于具有简单弹簧特性的腿部的陆地跑步数学模型。回顾了实验测力平台的证据，证明了该模型的公式合理性。控制微分方程以无量纲形式给出，以使结果代表所有体型动物的情况。无量纲输入参数为：U，基于前进速度和腿长的水平弗劳德数；V，基于垂直着陆速度和腿长的垂直弗劳德数，以及KLEG，腿部弹簧的无量纲刚度。结果表明，在高前进速度下，KLEG几乎是U和V的线性函数，而有效垂直刚度是U的二次函数。对于每对U、V，模拟表明，通过选择特定的步长，可以使步中垂直力最小化。当假设KLEG和V都固定时，该理论会出现一个特别有用的规范。当KLEG = 15且V = 0.18时，该模型对相对步长S和初始腿角θo的预测与从文献中获得的实验数据高度吻合。
