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Kinematics / Kinetics
Linear Kinetics
Hamill & Knutzen (Ch 10)
Hay (Ch. 5), Hay & Ried (Ch. 11),
Kreighbaum and Barthels (Module F & G)
or Hall (Ch. 12)
Force
  is a push, pull, rub
(friction), or blow Usually drawn as an
arrow indicating direction
(impact)
and magnitude.
  causes or tends to
cause motion or
change in shape
of a body
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.
An understanding of
why humans move
(kinematics) cannot be
obtained if you do not
understand kinetics
(forces, torques and
inertial properties)
Properties of Forces
  Magnitude
  Direction
  Point of application
  Line of application
  Angle of application
θ
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.
1
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”.
Mechanical Analysis
  Instantaneous Force
 F = ma
Ft∞Δmv
F∞Δ mv
F∞m Δv
t
Δt
F∞ma
F = ma
Powerlifting (F = ma)
  Maximum Force Production.
  All joints simultaneously (not quite ……but
the idea is mechanically correct
  Impulse – Momentum
 Ft = Δmv
  Work – Energy
 Fd = Δenergy (linear kinetic, rotational
kinetic, potential)
Throwing & Striking
(Ft = Δmv)
Many (most) movements
are a combination
Use muscle joint
systems in sequence
Mechanics versus Biomechanics
2
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.”
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
“Law of action and reaction”
Contact Forces
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
  Ground Reaction Force (GRF)
 Pressure
  Friction
  Fluid Resistance
  Elastic Force
  Muscle Force
  Joint Reaction Force
Momentum
Momentum =
mass x velocity
Mechanical Impulse
F=
Δmv
t
Ft = Δmv
The quantity of motion.
Ft = m(vf-vi)
NFL football running
backs. Rugby forwards.
Anthropometry.
Examples:
Generating velocity
Trapping a soccer ball
Protective equipment
3
d
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
Vertical Jump
d
Think of Net Force
Net negative
impulse
a
600
c
e
b
b
Point
Time (s)
Force (N)
Sample Problem
  Given the following approximate
force profile (next slide) of a vertical
jump from rest, calculate the
subject’s take-off velocity.
a
0.0
600
f
b
0.2
150
c
0.3
600
d
e
f
0.5 0.55 0.6
2500 600 0
d
Think of Net Force
  F = ma
Mass of subject = 600 N
Area of triangle = 0.5 x base x height
Net positive
impulse
a
600
c
e
b
ANSWER
f
Impulse = area under curve
  Net force profile (Force - body weight)
a
0.0
0
b
0.2
-450
c
0.3
0
d
0.5
1900
e
0.55
0
  Then integrate the curve.
f
0.6
-600
Force (N)
1500
Point
Time
Net Force
Integration!
1000
BW
500
Integration was discussed in more
detail in the linear kinematics chapter
Running speed = 5 m/s
0
0
0.05
0.10
0.15
Time (s)
0.20
0.25
4
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
  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
1500
Force (N)
  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
1000
BW
500
Running speed = 5 m/s
0
0
0.05
0.10
0.15
Time (s)
0.20
0.25
5
Pressure (P = F/area)
Pressure Plots
  Force distribution is an important concept,
especially in impact and other tissue
loading situations.
  Pressure plots are essentially collected from a
large number of small force transducers.
  Orthotic design is moving in this direction.
Foot Pressure Plots
Seat Pan Pressure Distribution
2-dimensional
3-dimensional
2-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.
3-dimensional
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.
6
Vertical Ground Reaction Force
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 Force (BW)
Time course of the GRF Impulse
3
2
1
0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Time (seconds)
GRF vs. Running Styles
Force (N)
1500
1000
BW
500
Does Nike® Air
really work?
0
0
20
40
60
80
100
Percent of Support
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
7
FIGURE 10-37 B
Friction Force
Ground reaction force for running.
Note the difference in magnitude between the
vertical component and the shear components
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
wt
Frictional Coefficients
R
Coefficient of Friction =
µ  = Fy
Fz
Friction Force
Normal Force
wt
Fa
R Fs
wt
Fa
R Fm
Static (max) Friction
Sliding (kinetic) Friction
wt
Fa
R Fk
Friction Force
Static
Fs
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
Push or Pull?
Fm
Friction
Force
No applied horizontal force
No friction
No motion
Pull (400 N)
Dynamic
µ = Fy
Fk
Push (400 N)
Fz
Applied Force
8
Rolling Friction
Fluid Resistance
  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)
  We
will look at these
forces later.
Joint Force
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.
Femur
Fj
Vj
In this case the shank
is pushing the femur
forwards and upwards.
Vj
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.
Coefficient of Restitution
F = kΔs
  "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
9
If one of the bodies is stationary (i.e
impact with the floor).
Then
Depends on:
-e = v1/u1
2
As
vf
sub into
Coefficient of Restitution
v
2
= vi + 2ad
f
  the nature of both contacting surfaces.
= 2ad
  the temperature of the surfaces.
-e = v1/u1
−e =
2ah
2ah
r
=
d
Also in non-uniform materials (e.g.
baseball, golf ball) e may change with
the speed of contact.
h
h
r
d
Hysteresis Loops
Elastic Recoil
  Hysteresis loops are basically forcedisplacement curves
  The area between the two parts of the
loop represent the energy lost.
  Springboard diving, pole
vault.
  Stretch-shortening cycle.
Force
  Elastic recoil is important
in locomotion (especially
for kangaroos!)
Displacement
Baseball Hysteresis Loops
Centripetal Force
Force (N)
Wooden
9000
Bat
Forces Occurring Along a Curved Path
Aluminum
Bat
Golf
Ball
7250
4500
Baseball
2750
0
0
0.25
0.5
0.75
1.0
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
10
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.
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
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?
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?
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
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)
11
Work
Mechanical Work,
Energy and Power
(segment models)
Hamill & Knutzen Chapters 10 & 11
Winter 1979 Chapter 5
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
12
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.
F = m⋅a
F = m ⋅ dv
Work-Energy Relationship
Kinetic Energy (horizontal)
dt
F = m ⋅ dv
ds
F = m ⋅ v ⋅ dv
⋅ ds
dt
ds
F ⋅ ds = m ⋅ v ⋅ dv
∫ Fds = m∫ vdv
Work = m( v )
1
2
2
ma = mvf2/2d
F = mvf2/2d
Fd = ½mvf2
13
Back to the Vertical Jump
Work-Energy Problem
Additional Question
Vertical Jump Power (Kin 142 & 343)
Power = 2.21× Wt × d
Power = 2.21× 600 × 0.327
Power = 758 ⋅ W (J /s)
€
Power = force x velocity
From vf2 = 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
  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?
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
€
14
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 - dualmotion 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)
Human Power Output Intensity
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.)
Graph from “Champion Athletes”
Wilkie 1960
Sustaining 375 Watts
for 30 minutes?
Impressive!
15
http://www.crossfit.com/
Estimate of
Thruster
Average Work
and Power
Calculations
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 "windup" or "back-swing".
However, this term is
misleading.
Pre-stretch (plyometrics)
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
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?
16
Power to Weight Ratio
  In many sports it is
not just about how
much power you
output ….it is also
about how much you
weigh.
  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.
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
Muscle Moment Power
Inverse Dynamic Analysis
ΣFx = max
ΣFy = may
ΣM = Igα
ay
Flex.
Muscle
Moment
Ex.
Flex.
Ang.
Vel.
+
ax
α
Ex.
Muscle
Power
-
Mechanical Work of Muscles
t2
Wm =
∫ Pm. dt
t1
t2
Wm =
∫ M ω . dt
j
t1
j
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 forcedisplacement through the joint centre).
  Transfer of energy is very important in
improving the overall efficiency of human
movement patterns.
17
Joint Force Power
Seg1
Vastus Lateralis
Level
Uphill
Fj1
Fj1Vjcosθ is
positive
θ1
Gastrocnemius
Level
Uphill
Vj
Vj
Level
Uphill
θ2
Seg2
Fj2Vjcosθ is
negative
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
Total Instantaneous
Energy of a Body
ET = ½mv2 + mgh + ½Iω2
Soleus
Glycogen Usage
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
P
s
=
M ω +M ω +F v +F v
p
p
p
d
right heel contact
d
right toe off
d
right heel contact
Swing Phase
15
Energy of the Foot
Energy (J)
Fj2
10
5
0
0
20
40
60
Percent of Stride
80
100
18
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.
Flow of Energy
Causes of Inefficient Movement
  Co-contraction
  Isometric Contractions Against Gravity
  Example of hands out straight. No
mechanical work being done!
  Jerky Movements
maintenance
heat
Metabolic
Energy
 high accelerations & decelerations waste
energy compared to gradual acceleration
  Generation of energy at one joint and
absorption at another (walking example)
  Joint friction (small)
isometric work
against gravity
O2 uptake
CO2 expired
mechanical energy
(muscle tension)
joint friction
Body
segment
energy
heats of
contraction
loss due to co-contraction
or absorption by negative
work at another joint
External
work
19