Biomechanics: Efficient Runner Model Rev 1.0

File has since been updated, please see:  Biomechanics:  Efficient Runner Model Rev 1.1

Mathematical Analysis utilizing biomechanics and basic physics can be used to determine the energy cost of various running factors.  When the energy cost is known and compared to empirical data the time penalty associated with these factors can be determined.  Running variables, combinations thereof, and their effect on performance that can be readily analyzed include:

Shoe weight

Shoe cushioning

Over-stride/cadence

Excess body weight

Foot strike

This post establishes the basic model that will be used for the separate analysis of each of these factors.  Assumptions and initial conditions are also defined.  This is a crude model intended to capture the gross movements, which account for the majority of energy cost.  This model can be refined in the future to better capture all the energy costs associated with running.

Efficient Runner Model

A visual examination of efficient runners and their form was necessary to create a proper model.  This link contains the video of the barefoot heel and forefoot strikes that aided in creating this model.  Cadence is 180 fpm (foot strikes per minute) or t_STRIDE = 0.333s.  A forefoot strike is shown though the gross movements for a heel strike and forefoot strike were found to be the same.   Notable differences exist in the impact and power times.

Footstrike	tIMPACT	tPOWER	tAIR
Heel	0.030s	0.136s	0.160s
Forefoot	0.064s	0.106s	0.160s

The two gross movements which constitute the majority of energy expenditure are striding and propelling the center of mass up and forward.

Striding

The energy cost of striding can be calculated using Newton’s laws of motion and by treating the legs and shoes as point masses.  A single stride is shown in the figure starting at time t=0.  The lead right foot in black has just impacted the ground.  The trailing left foot in grey is at the height of the knee ∆d_V.  The left foot is accelerated forward to become the lead foot as the right foot pushes off of the ground to become trailing foot.  This cycle repeats itself for each stride alternating left/right.  When analyzing striding, the power generated to push off the ground is ignored, as it is captured by the center of mass and conversion of energy calculations.

Center of Mass and Conservation of Energy

The energy cost related to center of mass displacement can be calculated using a spring mass model and Hooke’s law.  The center of mass m_C follows the dashed sinusoidal path of amplitude ∆h.  When cadence and pace are known ∆h can be readily calculated using equations of motion.  Additionally the shoe, foot, leg, and torso can all be modeled as one equivalent spring or separate springs in series. 

Central to calculating center of mass energy costs is the conservation of energy.  Each stride transfers some energy to the next, loses energy to the environment, and adds additional energy from muscle contractions.  To maintain a steady pace the energy lost to the environment is equal to the propulsive energy of muscle contractions.

Please feel free to refute any part of this post or add detail that has been missed.  While my intent is to fully explain things I do not always succeed and some nuances have been omitted in the interest of brevity.  I am happy to discuss in further detail as required.