Linear Kinetics - SFU.caleyland/Kin201 Files/Linear Kinetics.pdf · 3 Newton’s Third Law “ For...
Transcript of Linear Kinetics - SFU.caleyland/Kin201 Files/Linear Kinetics.pdf · 3 Newton’s Third Law “ For...
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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