Walking and Running BIO-39 October 30, 2018 BACKGROUND REVIEW From Oct. 25, 2018 Planar Pendulum Equation of motion (for small θ) 2 d g 0 2 dt Solution is Where: g is acceleration due to gravity l is pendulum length Ω is the natural frequency ( t) sin t 1
Pendulum as time keeper T 2 g T is period g is acceleration due to gravity l is pendulum length Stride Energetically: walking is like a planar pendulum 2
Kinetic and potential energy Total energy in a moving mechanical system is equal to sum of kinetic and potential energy Potential energy is proportional to height of center of mass above the walking surface Kinetic energy is proportional to the speed at which the center of mass is moving parallel to the walking surface Better model for walking is an inverted pendulum IP PP Walking is a plane pendulum (PP) from point of view of hip, but inverted pendulum (IP) from point of foot IP model is better since in order to walk center of mass must be moved outside base of support: so is an unstable motion Inverted Pendulum model of gait Maximum speed at which you can walk and still maintain the circular path for the center of mass is limited by gravity 2 d g 0 2 dt 3
Can predict walking speeds as a function of leg length People with shorter legs walk slower than people with longer legs Olympic walk racers tend to have long legs Centripetal force = Froude Number 2 mv Maximum speed at which we an walk (v) and still maintain the circular path for the center of mass is limited by gravity Froude number = 2 mv v mg g 2 / Froude number is dimensionless number that is ratio of these two forces Froude number: Facts Cannot walk if Froude number >1 (strictly speaking for normal gravity) Many bipeds, including humans and birds, prefer to switch from a walk to a run at a Froude number of ~0.5 4
Evidence for pendulum model: Kram, et al (1997) Speed of walk-run transition expected to decrease if acceleration due to gravity declines so that Froude number remains at ~0.5 Used a suspension harness to mimic decreased gravity At very low g s, swinging of arms and legs contributes and hence gravity is not the only force: Hence get a departure from expected (change in Froude number at transition). TODAY s Lecture Locomotion: Gait patterns 2-legged animals: Walking Running 4-legged animals Walking Running Galloping 5
Stride Classical distinction between walking and running Walking One foot always in contact with ground throughout the whole gait cycle Running All feet can be off the ground at certain phases of gait cycle Locomotion: PE vs KE Walking Changes in PE are out of phase with changes in KE Running Changes in PE are in phase with changes in KE 6
Energetically: Walking is like a pendulum Running is different Walking: COM is highest at mid-stride Running: COM is highest end of stride A runner or a walker? 7
Dinosaurs can t run! Walk-run transitions: energetic considerations Gait transitions: energetic considerations Humans 4-legged animals 8
Elephants: run or walk? 11 km/hr 40 km/hr Motion studies indicate that at least one foot is always in contact with ground Have 4 knee joints groucho running Who was Groucho Marx? Quadrupeds: walking vs running Trotting and galloping are running gaits with footfall patterns that are distinct from walking An aerial phase (no foot touches ground) often marks transition from walking to running Run is defined as any gait with a duty factor (fraction of time of sequence of footfalls for which a given foot is in contact with ground) of less than 0.5 9
Energetic picture of elephant movement Bottom part of figure Shoulder motion Stance phase (dark triangles) resembles walking: moves upward and then shoulder downwards when forefoot is on ground Hip motion hip Stance phase (dark circles) resembles running: moving Big symbols: stance; small symbols swing downwards then upwards Note: elephant motion pattern does not change as Froude number changes. Do elephants run? Time (% stride period) Do elephant s have four knees? Standing elephants, angle between femur and tibia is close to 180 degrees Differs from half-bent posture in most mammals Similar extended knee posture occurs only in bipedal humans 10
Elephant legs & 4 knees Obviously an elephant cannot have four knees, but rather has two elbows (forelimbs) and two knees (hind limbs) Another common error to assume that the forelimb knee joint is actually more like a wrist joint However, by comparing horse and elephant skeletons we see that the elephant forelimb knee joint corresponds to an elbow joint Question to ask yourselves Do elephants walk like Groucho Marx? OR Is there something wrong with our definition of walking and running? Running is different Walking: COM is highest at mid-stride Running: COM is highest end of stride 11
Running mysteries Muscle contraction at times related to foot contact with surface 12
Turkeys and Runners Tendons: Biological springs Work equal to area under curve Human tendons Work to cause tendon to stretch is almost completely recovered on recoil Summary Running Walking 13
Better to walk or run? Should we wear this space suit or one that Dr. Spock wore? Answer: Run 14