The spring-mass model for running and hopping

J Biomech. 1989;22(11-12):1217-27. doi: 10.1016/0021-9290(89)90224-8.


A simple spring-mass model consisting of a massless spring attached to a point mass describes the interdependency of mechanical parameters characterizing running and hopping of humans as a function of speed. The bouncing mechanism itself results in a confinement of the free parameter space where solutions can be found. In particular, bouncing frequency and vertical displacement are closely related. Only a few parameters, such as the vector of the specific landing velocity and the specific leg length, are sufficient to determine the point of operation of the system. There are more physiological constraints than independent parameters. As constraints limit the parameter space where hopping is possible, they must be tuned to each other in order to allow for hopping at all. Within the range of physiologically possible hopping frequencies, a human hopper selects a frequency where the largest amount of energy can be delivered and still be stored elastically. During running and hopping animals use flat angles of the landing velocity resulting in maximum contact length. In this situation ground reaction force is proportional to specific contact time and total displacement is proportional to the square of the step duration. Contact time and hopping frequency are not simply determined by the natural frequency of the spring-mass system, but are influenced largely by the vector of the landing velocity. Differences in the aerial phase or in the angle of the landing velocity result in the different kinematic and dynamic patterns observed during running and hopping. Despite these differences, the model predicts the mass specific energy fluctuations of the center of mass per distance to be similar for runners and hoppers and similar to empirical data obtained for animals of various size.

Publication types

  • Research Support, Non-U.S. Gov't
  • Research Support, U.S. Gov't, P.H.S.

MeSH terms

  • Animals
  • Biomechanical Phenomena
  • Body Weight
  • Humans
  • Locomotion*
  • Models, Biological
  • Musculoskeletal Physiological Phenomena*
  • Running*