-Elastic strain energy (duty factor decreases at higher speeds). Higher forces act on feet. More tendon stretch. More energy stored in tendon.

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1 As velocity increases ( ) (i.e. increasing Froude number v 2 / gl) the component of the energy cost of transport associated with: -Internal kinetic energy (limbs accelerated to higher angular velocity). -Gravitational potential energy (duration of floating phase decreases). -Elastic strain energy (duty factor decreases at higher speeds). Higher forces act on feet. More tendon stretch. More energy stored in tendon. Confirmed experimentally for mammals kg (smaller mammals have a higher cost of transport) Metabolic Power Consumption vs Running Speed Approximately linear relation between metabolic power consumption and running speed. P(v) P(0) + C.v where P(v) rate of metabolic energy consumption when running at speed v. P(0) rate of metabolic energy consumption when standing. C.v extra rate of energy use (power) for running at speed v. C extra rate of energy consumption per unit velocity (energy cost per unit distance). 40

2 Determined from many species of mammals running on treadmill (ignore effect of different gaits). Energy cost of transport (running mammals). C (mass) 0.68 C/m (mass) Large animals are more economical. Explanation is uncertain: - Possibly due to energy cost of exerting force being more important than cost of performing work. 41

3 2.5 HUMAN LOCOMOTION Walking - Is a unique two legged style. Straightest legs of any animal, with an erect spine. - At least one foot in contact with the ground at all times (usually duty factor β = ). Walking at a constant speed: It would seem that we would requireonly: - Vertical forces to support body weight. - Horizontal force to overcome air resistance (usually negligible). Actual walking technique is quite different. People do not walk with F vert = body weight (no vertical acceleration, i.e with the center of gravity level) Walking technique: - Knee is almost straight when in contact with the ground. - COG moves in arcs of a circle (rises and falls ~ 35 mm during stride). - F vert body weight (at all times). - Moments of force about knee are small Little muscle activation Low metabolic energy cost (High energy efficiency locomotion) 42

4 Additional factors: - Pelvic rotation, pelvic tilt, stance, leg flexion, ankle flexion. Smooth the arc of the COG. Does not require infinite force for change in direction of velocity of COG at midstance. Limbs as pendulums: - Motion of the legs as (passive) swinging pendulums. - Leg-swing half-period: T 0.35 sec (leg fixed at hip, knee allowed to bend freely, allow for pelvic motion) observed swing period for fast walk, v 2.0 m/s. - Economical transport (Little muscular activity in legs during walking from EMG studies). - Adjust stride length with walking speed to maintain a passive leg swing. 43

5 Maximum Speed of Walking Walking model: - Assume all body mass located at hip. - Body on straight leg behaves like an inverted pendulum (mass mounted on top of the pendulum). - Torque exerted at hip. - Need to maintain contact with the ground with at least one leg at all times (feet cannot pull down on the ground). Requires that (centripetal force) (body weight) during the motion of the COG along the arc of a circle to avoid flying off at tangent to arc. Leg length l, walking speed v; m v 2 l mg v 2 (Froude number) = 1 gl v max = gl Result: Longer legs faster maximum walking speed. Examples: - Adult human: l = 0.9 m v max = 3.0 m/s - Child: l = 0.5 m v max = 2.2 m/s Child has slower maximum speed due to shorter legs. Starts running at a lower velocity than adult. 44

6 Running Human running: - Duty factor usually β Must run to attain speeds above the maximum walking speed - Abrupt change of gait from walk to run at a critical speed. - Legs bent during support phase. - Ground reaction force in line with leg. - Large muscular forces high metabolic cost - KE and PE - Low at midstance - High at midstride Muscles behave like springs: - Motion of legs not like pendulums, COG motion more like bouncing ball or pogo stick. - Muscles of knee (quadruceps) and ankle (gastrocnemius and soleus) behave like springs. - Muscles activated when foot on ground. Energy cost of running is mainly due to the horizontal component of ground force. 45

7 Ground Reaction Forces Recorded with a force platform. Some important features are: - Ground reaction force is in line with the leg and accelerates / retards body during stride. Horizontal force: - Forward then backward force (retard, accelerate). - Represented by superposition of 2 sine terms. Vertical force: - Greatly exceeds body weight when running. - Represented by the superposition of 2 cosine terms. Impact peak: - Damped oscillation superimposed on ground reaction force due to body mass on a leg spring. - Provides impulse to halt motion of foot (mass 4 kg in 25 ms). - Compliant foot pad moderates the impact force, improves "road holding" by preventing "chatter" (vibrations in which the foot repeatedly leaves and returns to the ground before settling). 46

8 MetabolicEnergyCost Most economical speed for walking; v 1.3 m/s. Running more economical than walking at v 2.3 m/s. - Humans change walk run at Fr = i.e. when v 2.5 m/s for adult human. v 2 gl 0.7 (Froude number) Human walking: Human running: - More economical than walking for an animal of similar size. - Relatively uneconomical. Energy storage in humans: - Much of KE lost in running stride is stored as elastic strain energy in stretched tendons and ligaments. Achilles tendon: - Store 35 J of elastic strain energy (1/3 of KE and PE lost during running stride). - 93% elastic recoil (7% dissipated). - Thin tendon in proportion to strength of muscles (low stiffness, k) Large stretch. Large energy storage F F = k x, E = k x = 2 2k - Calf muscles do not have to lengthen and shorten as much, or as fast. Use more economical muscle type. Ligaments in arch of foot: - Store 17 J of elastic strain energy - 80% elastic recoil (20% dissipated). Running shoes: - Compress 10 mm - Store 7 J of elastic strain energy % elastic recoil (30-50% dissipated) 47

9 2.6 JUMPING Mode of locomotion used: - To capture prey. - Escape predators. - For locomotion in trees. Peak height of jump: (conservation of energy) PE = KE 1 2 m g h = m v 2 h = 2 v 2g where h = Increase in height of COG from take-off topeak of jump (m). v = Velocity of COG at instant of take-off (m/s) m = Mass of animal (kg). Work performed by animal during take-off: (assuming velocity at start of take-off = 0) ( F mg) dy = KE = PE 1 2 ( F mg) s = m v = mg h Fs h = s 2 mg where F = Average take-off force (N). s = Take-off distance (m). For geometrically similar animals: -Body mass, m (length) 3 -Take-off distance, s length -Max take-off force, F Cross-sectional area (length) 2 Height of jump h -(length) -(mass) 1/3 - Jump height decreases with increasing size - Maximum size of jumping animal. 48

10 Ground reaction force generated in take-off depends on: - Moment of muscle force about joint - Shortening speed of muscle (less force exerted as shortening speed increases). Stretch-shorten cycle: - Higher jumps when preceded by motion in opposite direction. - Advantage due to: - Muscle pre-tensing - Enhanced force-velocity (stretching of muscle fibres). - Return of elastic energy stored in stretched tendon and muscle. Measurements of dog jumping from force platform: 49

11 Jumping by Small Animals For small animals: -The effect of aerodynamic drag is not negligible (work done against drag during flight phase). -For a vertical jump: dv m dt = mg or F drag dv dt = F g m drag where F drag Frontal area S f For same take-off velocity - Smaller animals cannot jump as high. (F drag / m S f / m is larger) More energy lost to aerodynamic drag. 50