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Linear Kinetics

Hamill & Knutzen (Ch 10) Hay (Ch. 5), Hay & Ried (Ch. 11),

Kreighbaum and Barthels (Module F & G) or Hall (Ch. 12)

Kinematics / Kinetics An understanding of why humans move (kinematics) cannot be obtained if you do not understand kinetics (forces, torques and inertial properties)

Force   is a push, pull, rub

(friction), or blow (impact)

  causes or tends to cause motion or change in shape of a body

Usually drawn as an arrow indicating direction and magnitude.

Properties of Forces  Magnitude  Direction

 Point of application  Line of application  Angle of application θ

Mass

The quantity of matter contained in an object. Units: kilograms

Inertia

The tendency of a body to maintain a motionless state or a state of constant velocity. Proportional to mass.

Newton’s First Law “Every body continues in its state of

rest or motion (constant velocity) in a straight line unless compelled to change that state by external forces exerted upon it.”

This relates to the concept of inertia.

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Newton’s Second Law The rate of change of momentum of a

body (or the acceleration for a body of constant mass) is proportional to the force causing it and the change takes place in the direction in which the force acts.

Incorrectly described as “Newton’s law of acceleration” by Hamill & Knutzen. It is in fact the “law of momentum”.

maFmaF

tvmFt

mvF

mvFt

=

ΔΔ

Δ

Δ

∞

∞

∞

∞

Mechanical Analysis   Instantaneous Force

 F = ma   Impulse – Momentum

 Ft = Δmv  Work – Energy

 Fd = Δenergy (linear kinetic, rotational kinetic, potential)

Powerlifting (F = ma)  Maximum Force Production.   All joints simultaneously (not quite ……but

the idea is mechanically correct

Throwing & Striking (Ft = Δmv)

Use muscle joint systems in sequence

Many (most) movements are a combination

Mechanics versus Biomechanics

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Newton’s Third Law

“ For every force applied by one body on a second, the second body applies an equal and oppositely directed force on the first.”

“Law of action and reaction”

Non-Contact Forces   The force of gravity is inversely

proportional to the square of the distance between the centre of gravities of attracting objects and proportional to the product of their masses.

F = Gm1m2 r2

Weight The attractive force that the earth exerts on a

body (the earth's gravitational pull). W = mg Units: Newtons!!??

Acceleration due to Gravity

The acceleration of a body due to the gravitational force of the earth is considered to be constant at -9.81 m/s2

Contact Forces   Ground Reaction Force (GRF)

 Pressure   Friction   Fluid Resistance   Elastic Force  Muscle Force   Joint Reaction Force

Momentum

Momentum = mass x velocity

The quantity of motion.

NFL football running backs. Rugby forwards. Anthropometry.

Mechanical Impulse

F = Δmv t

Ft = Δmv

Ft = m(vf-vi)

Examples: Generating velocity Trapping a soccer ball Protective equipment

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Mechanical Impulse   Effect of a force applied over a period of time   Analysing human effort aimed at producing

maximal velocity (maximal impulse) has been a focus of numerous studies

  However, the effect of different material in running shoes and other injury prevention issues can also be investigated by studying force-time profiles

  In the vertical jump example (numerical integration) we started with a force-time graph

Think of Net Force

a

b

c

d

f

e 600

Net negative impulse

Net positive impulse

Vertical Jump

b

d

Sample Problem  Given the following approximate

force profile (next slide) of a vertical jump from rest, calculate the subject’s take-off velocity.

 F = ma Mass of subject = 600 N Area of triangle = 0.5 x base x height

Think of Net Force

a

b

c

d

f

e 600

Point a b c d e f Time (s) 0.0 0.2 0.3 0.5 0.55 0.6

Force (N) 600 150 600 2500 600 0

ANSWER   Net force profile (Force - body weight)

  Then integrate the curve.

Point a b c d e fTime 0.0 0.2 0.3 0.5 0.55 0.6

Net Force 0 -450 0 1900 0 -600

Integration was discussed in more detail in the linear kinematics chapter

Impulse = area under curve

Time (s) 0 0.05 0.10 0.15 0.20 0.25

Forc

e (N

)

1500

1000

500

0

BW

Integration!

Running speed = 5 m/s

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Newton’s Third Law

“ For every force applied by one body on a second, the second body applies an equal and oppositely directed force on the first.”

“Law of action and reaction”

Conservation of Momentum   Following on from Newton’s law is the law

of Conservation of Momentum.

  “In a system of bodies that exert forces on each other, the total momentum in any direction remains constant unless some external force acts on the system in that direction”.

Contact Forces   The non-contact force of gravity already

covered   Ground Reaction Force (Pressure)   Joint Reaction Force (already covered)   Friction   Fluid Resistance  Muscle Force (already covered)   Elastic Force

Force Platforms

  Force platforms are a sophisticated and expensive type of force transducer.

  Forces are calculated in x, y and z planes as are moments.

  Centre of pressure can also be calculated.

Ground Reaction Force

Ground Reaction Force

Time (s) 0 0.05 0.10 0.15 0.20 0.25

Forc

e (N

)

1500

1000

500

0

BW

Running speed = 5 m/s

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Pressure (P = F/area)

  Force distribution is an important concept, especially in impact and other tissue loading situations.

Pressure Plots

  Pressure plots are essentially collected from a large number of small force transducers.

  Orthotic design is moving in this direction.

Foot Pressure Plots

2-dimensional

3-dimensional

Seat Pan Pressure Distribution

2-dimensional

3-dimensional

Backrest Comfort

  In addition to reducing pressure in the disk a good backrest should provide firm support across a wide area of the back (no pressure points).

  Opposite is a back rest pressure distribution.

Force Transducers

  This is a pinch grip force transducer.   A wide variety of force transducers are available.   Simple strain gauge systems can also be very

effective.

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Magnitude of GRF

 Walking = 1 to 1.2 x Body Weight  Running = 3 to 5 x Body Weight (Hamill

& Knutzen 1995)  Squats = up to 7.6 x Body Weight at

patello-femoral joint (Reilly & Matens 1972)  Hamill & Knutzen text has 7 graphs of

GRF’s during different types of human movement (pages 400-401).

Vertical Ground Reaction Force Time course of the GRF Impulse

Time (seconds) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Vert

ical

For

ce (B

W) 3

2

1

0

GRF vs. Running Styles

Percent of Support 0 20 40 60 80 100

Forc

e (N

)

1500

1000

500

0

BW

Does Nike® Air really work?

FIGURE 10-35 Center of pressure patterns for the left foot. A. Heel-toe footfall pattern runner.

B. Mid-foot foot strike pattern runner.

FIGURE 10-37 A Ground reaction force for walking.

Note the difference in magnitude between the vertical component and the shear components

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FIGURE 10-37 B Ground reaction force for running.

Note the difference in magnitude between the vertical component and the shear components

Friction Force

Friction is the force created between two contacting surfaces that tend to rub or slide past each other.

Note: There can be friction without movement

Frictional Coefficients

Coefficient of Friction = Friction Force Normal Force

Static (max) Friction Sliding (kinetic) Friction

µ  = Fy Fz

Fa

Fa

Fa

No applied horizontal force No friction No motion

Small applied force Small friction force Fa = Fs No motion

Larger applied force Maximum static friction Fa = Fm Pending motion

Larger applied force Fa > Fk Motion Occurring

R

wt

wt

wt

wt

R

R

R

Fs

Fm

Fk

Friction Force

Static Dynamic

Applied Force

Friction Force

Fs

Fm

Fk

Push or Pull?

Pull (400 N)

Push (400 N)

µ = Fy Fz

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Rolling Friction

  Coefficients of sliding and limiting friction are normally within a range of 0.1 to 1.0

  Rolling friction is generally of a magnitude around 0.001 (100 to 1000 times less than sliding and limiting friction

  Synovial fluid and articular cartilage? (0.01 to 0.003)

Fluid Resistance

 We will look at these forces later.

Inertial Force

  The force exerted due to the movement (inertia) of a body.

  Note a true force? Do not include on free body diagrams

The muscles crossing a joint are not the only way forces are exerted on adjacent segments.

In this case the shank is pushing the femur forwards and upwards.

Vj

Vj

Fj

Joint Force

Femur

Shank

Elastic Force

When a falling ball hits the ground the reaction force compresses it until its C of g stops its downward motion. The elastic recoil of the ball back to its round shape causes it to push against the ground, generating a ground reaction force that moves ball upwards.

F = kΔs Coefficient of Restitution

  "When two bodies undergo a direct collision, the difference between their velocities after impact is proportional to the difference between their velocities before impact."

v1 - v2 = -e(u1 - u2)

or -e = v1 - v2 u1 - u2

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If one of the bodies is stationary (i.e impact with the floor).

Then -e = v1/u1

As vf2 = vi

2 + 2ad

sub into -e = v1/u1

fv ad= 2

− = =e r

d

r

d

ahah

hh

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Coefficient of Restitution

Depends on:

  the nature of both contacting surfaces.

  the temperature of the surfaces.

Also in non-uniform materials (e.g. baseball, golf ball) e may change with the speed of contact.

Elastic Recoil   Springboard diving, pole

vault.

  Stretch-shortening cycle.

  Elastic recoil is important in locomotion (especially for kangaroos!)

Hysteresis Loops   Hysteresis loops are basically force-

displacement curves   The area between the two parts of the

loop represent the energy lost.

Displacement

Forc

e

Baseball Hysteresis Loops

0 0.25 0.5 0.75 1.0

9000

7250

4500

2750

0

Forc

e (N

)

Baseball Golf Ball

Aluminum Bat

Wooden Bat

Centripetal Force Forces Occurring Along a Curved Path

If the car goes around the corner an external force must be exerted against it. You are forced in the same direction if wearing a seat belt.

Fcp

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Centripetal & Centrifugal Forces Whenever a body moves in a circular force it

must be experiencing a force pushing or pulling it towards the centre of its path (axis). This Centripetal (centre seeking) force has an equal and opposite reaction (often called Centrifugal force although it is often inertial resistance).

These forces are just special cases of an external force and the reaction force to that original force.

Centripetal & Centrifugal Forces

Which comes first the Centrifugal or the Centripetal force?

Sprinter running around curve? Hammer rotating around the thrower in the

hammer throw? Cyclist negotiating a bend?

Magnitude of Centripetal Force

Fc = mv2/r Therefore, the

centripetal force is higher if the mass

and/or speed of the cyclist is increased and/or the radius of the curve is decreased

Why do Cyclists Lean into the Curve?

  This is not a situation of static equilibrium, why?

  However, if no rotation in the frontal plane is occurring, the net torque must equal zero.

ΣΤ = 0

Sample Final Question? Leaning in towards the

centre of rotation is common in many sports.

Could you explain how these skaters do not fall inwards?

What affects how much they have to lean?

Why do we bank the track?

  If the track is not banked all of the centripetal force (reaction) must be obtained from friction.

  If the track is banked some of the centripetal force can be obtained via a normal ground reaction force (90o to frictional force)

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Mechanical Work, Energy and Power (segment models)

Hamill & Knutzen Chapters 10 & 11 Winter 1979 Chapter 5

Work

Work Power

Power = Δwork Δtime

= (force) x Δdistance Δtime

= force x velocity

Units => Watts (J/s)

Energy Definition: “The ability to do work”

Kinetic Energy = ½mv2

Gravitational Potential Energy = mgh (h is measured from the objects position to ground

and therefore is negative, hence PE is positive)

Elastic Strain Energy = ½kx2

Units => Joules

Units   F x d => MLT-2 x L => ML2T-2

 ½mv2 => M(LT-1)2 => ML2T-2

 mgh => MLT-2 x L => ML2T-2

 What are the units of the spring constant in the equation for strain energy (½kx2)?

 MT-2

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Error in Hamill & Knutzen text?   Force = kΔs   Elastic Strain Energy = ½kΔx This is

wrong (see Andrew’s slides also).   k is the same constant? The authors refer

to it as the stiffness constant in both the section on elastic force and energy.

  F => MLT-2 Therefore units of k => MT-2   Energy => MLT-2 ??????   Elastic Strain Energy = ½kx2

Conservation of Energy   The total energy of a closed system is

constant since energy does not enter or leave a closed system.

  This only occurs in human movement when the object is a projectile and we neglect air resistance. Then the total energy of the system (TE) = PE + KE.

  Note that gravity does not change the total energy of the system.

Work-Energy Relationship (staying with Linear Kinetics)

Work-Energy Relationship

This is not a new mechanical concept. It can be derived from Newton’s second law.

( )vmWork

vdvmFds

dvvmdsFds

dvvmFdt

dsds

dvmFdt

dvmF

amF

221=

=

⋅⋅=⋅

⋅⋅=

⋅⋅=

⋅=

⋅=

∫ ∫ ma = mvf2/2d

F = mvf2/2d

Fd = ½mvf2

Work-Energy Relationship Kinetic Energy (horizontal)

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Back to the Vertical Jump Work-Energy Problem

Additional Question Vertical Jump Power (Kin 142 & 343)

  If you used body mass (61.2 kg) instead of body weight (600 N) you should have calculated and answer of 77.3 kgm.s-1

 Where does the above equation come from? €

Power = 2.21×Wt × d

Power = 2.21× 600 × 0.327Power = 758 ⋅W (J /s)

Power = force x velocity From vf

2 = vi2 + 2ad we can calculate the velocity

of take-off and, as we started from zero velocity, the average velocity during take-off.

€

Vto = 2ad = 2a × d

Vto = 19.62 × d = 4.42 d

Average velocity ≈ 2.21× d

Power = Force ×Velocity

Power = 2.21×mass× g × d

Physiologists & Mechanical Units!   You will come across a lot of physiology

texts that report the power output from such tests in kg.m.s-1.

  This is not a unit of power.  Without being too pedantic, I wonder why

they cannot multiply the result by g (9.81 m.s-2) to get the correct units of; kgm2.s-3, or Joules/sec (J/s) or Watts.

  Fundamental units: ML2T-3

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Sayers Equation   Average power is not ideally the attribute we

wish to measure in a vertical jump.   The Sayers equation is an estimate of peak

leg power.   Peak Leg Power (Watts) = [60.7 x jump

height (cm)] + [45.3 x body mass (kg)] – 2055   Do it for the subject we just used (jump height

= 0.327 meters, body mass = 61.2 kg   Compare to average power calc. (758 Watts)

Bowflex Treadclimber   “Reduce your workout time - dual-

motion treadles let you step forward like a treadmill and up like a stair climber so you get more exercise in less time”

  “TreadClimber® machine burns up to 2 TIMES more calories than a treadmill - at the same speed!”

  “Studies were conducted at the prestigious Human Performance Laboratory at New York's Adelphi University. The results were dramatic! In 22 separate trials, the TreadClimber® machine burned up to 2 times more calories in 30 minutes than a treadmill at the same speed!”

  Company Website Sep-2006

http://www.treadclimber.com/trc_microsite/fitnessbenefits.jsp

Work is Work (Power Output is…)   Sure it is possible to burn twice the calories but

……………it would be twice as difficult   TV commercial “burn twice the calories in one easy

motion”   “What do you get when you combine the best aerobic

features of the stairclimber, treadmill, and elliptical trainer? Quite simply, you get a triple-charged cardio workout “ Bowflex Website Sep-2006

  Top CrossFit athletes ≅ 400 watts sustained for 2¾ min   Approx equivalent to 80 RPM at 7.5 kp (kg-Force) on a

Monark Bike. (although using less muscle mass so it would be very difficult to generate that much power for that long on a bike.

  Wingate test (30 seconds maximal output) top performers ≅ 700 Watts.

  Lance Armstrong can generate about 500 watts for 20 minutes (a typical 25-yr-old could last for 30 seconds)

Next Slide   The relationship of metabolic power

produced in skeletal muscle to the mechanical power of activity. (Adapted from H.G. Knuttgen, Strength Training and Aerobic Exercise: Comparison and Contrast, Journal of Strength and Conditioning Research 21, no. 3 (2007): 973-978.)

Human Power Output Intensity Graph from “Champion Athletes” Wilkie 1960

Sustaining 375 Watts for 30 minutes? Impressive!

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Estimate of Thruster Average Work and Power Calculations

http://www.crossfit.com/

Seems simple enough – but what is the problem with relating the external work done in such movements to the metabolic cost to the athlete?

The “Back-Swing” or “Wind-Up” Movements that cause a muscle to shorten immediately after a period of stretching are often referred to as a "wind-up" or "back-swing". However, this term is misleading.

Stretch-Shortening Cycle Enhancement of Positive Work

  Return of stored energy from passive elastic structures within the muscle (cross-bridges and connective tissue (70-75% of increase?)

  Prior activation (time to develop force reduced)   Initial increased force potentiation (eccentric

contraction)   Reflex augmentation (stretch reflex)

Small amplitude – high velocity – no delay

Pre-stretch (plyometrics) Olympic Lifting and Powerlifting Power Outputs

Jerk ≈ 2,140 W (56 kg) ≈ 4,786 W (110 kg) Second pull Average power output from transition to

maximum vertical velocity ≈ 5,600 Watts (100 kg male); 2,900 Watts (75 kg female).

Average Power (Powerlifting) •  bench ≈ 300 W •  squat ≈ 1,000 W •  deadlift ≈ 1,100 W

•  Why are “Powerlifting” events less powerful?

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Power to Weight Ratio

  For events like the Tour de France it is a matter of watts per kilogram of body weight, that is, the specific power output at lactate threshold - the amount of power/weight that the body can sustainably generate. It turns out that 6.7 is more or less a magic number - the power/weight ratio required to win the TDF.

  In many sports it is not just about how much power you output ….it is also about how much you weigh.

Energy/Power Analysis   The previous is OK for a

fitness test or an estimate of workrate (power) during exercise.

  However, to calculate energy change (power) segment by segment we need to do a dynamic analysis.

  We need to take accelerations into account if the movement is too dynamic for a static analysis

Inverse Dynamic Analysis

α ax

ΣFx = max ΣFy = may ΣM = Igα

ay

Muscle Moment Power

Muscle Power

Ang.

Vel.

Muscle Moment

Flex.

Ex.

Flex.

Ex.

+

-

Mechanical Work of Muscles

Wm Pmt

tdt= ∫

1

2

.

Wm Mt

tdtj j= ∫ ω

1

2

.

Mechanical Energy Transfer Between Segments

 Muscles can obviously do work on a segment (muscle moment power).

  However, if there is translational movement of the joints there is mechanical energy transfer between segments. (i.e. one segment does work on an adjacent segment by force-displacement through the joint centre).

  Transfer of energy is very important in improving the overall efficiency of human movement patterns.

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Vj

Vj

θ1

Fj1

Fj2

Joint Force Power

Seg1

Seg2

Fj1Vjcosθ is positive

θ2

Fj2Vjcosθ is negative

Level

Level

Level

Uphill

Uphill

Uphill

Vastus Lateralis

Soleus

Gastrocnemius

Glycogen Usage

Human Energy Harvesting   Biomechanical

Energy Harvesting: Generating Electricity During Walking with Minimal User Effort

  J. M. Donelan,1* Q. Li,1 V. Naing,1 J. A. Hoffer,1 D. J. Weber,2 A. D. Kuo3

  Science 8 February 2008: Vol. 319. no. 5864, pp. 807 - 810

Rate of change of the energy of a segment (power) [Ps]

 Muscle moment power for the proximal joint  Muscle moment power for the distal joint   Joint force power for the proximal joint   Joint force power for the distal joint

s p d p p d dP M M F v F v = ω ω+ + +

Total Instantaneous Energy of a Body

ET = ½mv2 + mgh + ½Iω2

Ener

gy (J

)

15

10

5

0

Swing Phase right heel contact right toe off right heel contact

Percent of Stride 0 20 40 60 80 100

Energy of the Foot

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Efficiency  Metabolic efficiency is a measure of the

muscles ability to convert metabolic energy to tension.

  A high metabolic efficiency does not necessarily mean that an efficient movement is taking place (e.g. cerebral palsy).

  The ability of the central nervous system to control the tension patterns is what influences the mechanical efficiency.

Overall Muscular Efficiency Muscular Eff. = Net mechanical work Net metabolic energy

Net mechanical work = Internal work + External work

  Internal work: Work done by muscles in moving body segments.

  External work: Work done by muscles to move external masses or work against external resistance.

  Aprrox. 20-25% efficiency.

Contraction time related to force -velocity curve

Efficiency All efficiency calculations involve some measure of mechanical output divided by a measure of metabolic input. Metabolic work is not too difficult to estimate if we do gas analysis. External work also easy to calculate. But we need to calculate internal mechanical work. Clearly we must at least calculate absolute energy changes (negative work is still an energy cost to the body). However, isometric contractions against gravity still a problem.

Causes of Inefficient Movement   Co-contraction   Isometric Contractions Against Gravity

  Example of hands out straight. No mechanical work being done!

  Jerky Movements  high accelerations & decelerations waste

energy compared to gradual acceleration   Generation of energy at one joint and

absorption at another (walking example)   Joint friction (small)

Flow of Energy

MetabolicEnergy

Body segment energy

O2 uptake

CO2 expired External work

mechanical energy (muscle tension)

maintenance heat

heats of contraction

isometric work against gravity

loss due to co-contraction or absorption by negative work at another joint

joint friction