Walking and Running
BIO-39 March 28, 2019
BACKGROUND REVIEW
From March 26, 2019
Planar Pendulum
Equation of motion (for small θ)
d 2 g 0 dt 2
Solution is (t) sint
Where: g is acceleration due to gravity l is pendulum length Ω is the natural frequency
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
d 2 g 0 dt 2
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
Froude Number 2 • Centripetal force = mv • Maximum speed at which we an walk (v) and still maintain the circular path for the center of mass is limited by gravity
mv 2 / v 2 Froude number = mg g
• 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 • Running: COM is highest at mid-stride 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 • Running: COM is highest at mid-stride 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
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