1

Linear Kinematics

Hamill & Knutzen (Ch 8) Hay (Ch. 2 & 3)

Hay & Ried (Ch. 8 & 9) Kreighbaum & Barthels (Module F)

or Hall (Ch. 2 & 10)

Translational Motion Figure 8.1

Quadrants in a Two-Dimensional Reference System Figure 8.3

Quadrant I (+,+) Quadrant II (-,+)

Quadrant III (-,-) Quadrant IV (+,-)

+y

-y

+x -x

Position & Displacement

 Position defines an object’s location in space.

 Displacement defines the change in position that occurs over a given period of time.

 Displacement is a vector  Distance is a scalar

Movements Occur Over Time

Knowledge of the temporal patterns of a movement is critical in a kinematic analysis since changes in position occur over time.

Speed

Speed is a scalar (m/s)

Speed = distance Δtime

Δ = change in

2

Velocity

Velocity is a vector (m/s)

Velocity = Δposition(displacement) Δtime

Velocity is designated by lowercase v Time is designated by

lowercase t

Velocity

Fundamental units = LT-1

If we plot our displacement data on a graph we are calculating the slope of the line when we calculate velocity.

Acceleration

Acceleration = Δvelocity Δtime

Acceleration is designated by lowercase a

It is used for both scalar and vector quantities.

Units of Acceleration

 Units are m/s 2 or m.s-2

 Fundamental units => LT -2

Acceleration

  If my velocity in the x-direction goes from 3 m/s to 2 m/s in 0.05 seconds what would my acceleration be?

 Answer: -20 m/s2

 Be careful of the term “deceleration”

Data Acquisition

Not covered in any detail in most texts on

reserve.

3

Where am I?

 The accurate and complete answer to this question is not as simple as it may seem.

Photographs

  If taken in the correct plane photographs can allow for later evaluation of angles and hence for a static kinetic analysis.

  In dynamic situations how do you know you have the extreme posture?

Video Systems   There is a limited amount of quantitative

data that can be gleaned from a full-motion video system.

  Stop frame capability does however allow for a reasonably accurate assessment of posture.

  60 frames/sec is more than adequate for most movements but the real problem is identifying joint centres of rotation and calibration.

Opto-electronic systems The location of the

joint centres of rotation is entered directly into the computer.

Systems usually come with software that will calculate velocity and acceleration.

Q-Track and Force Plate Data

Previous Data Acquisition System in Dr. Robinovitch’s Lab

Q-trac markers

4

Vectors and Scalars

  Scalars can be described by magnitude  E.g., mass, distance, speed, volume

  Vectors have both magnitude and direction  E.g., velocity, force, acceleration

  Vectors are represented by arrows   See pages 305-308 for vector addition,

subtraction, and multiplication

Vector Components Figure 8.9   a = original vector

acomponentxa

componenty

−=

−=

θ

θ

cos

sin

θx-component

y-co

mpo

nent

a

Displacement => Velocity

Time

Displ.

A

B

Δt = run

Δx = rise

Displacement => Velocity

v = positionfinal - position initial time at final displ. - time at initial displ.

v = xf - xi = xf - xi tf - ti Δt

As (tf - ti ) is usually constant we just use Δt.

Finite Differentiation

x2 x3 x4 x1 x5

t1 t2 t3 t4 t5

Δx

Δt

Finite Differentiation

Time

Displ.

A (x1, y1)

B (x2 , y2)

5

Finite Differentiation

tyyVy

Δ

−=

125.1

txxVx

Δ

−=

125.1

Finite Differentiation

tyyy

Δ

−=

125.1

txxx

Δ

−=

125.1

Sample Data

(0.15 - 0.00) / 0.0167 = 8.98

Frame Time (s) Vert.Pos (y) Vel. (vy)

1 0.0000 0.00 8.98

2 0.0167 0.15

3 0.0334 0.22

4 0.0501 0.27

Finite Differentiation

(0.22 - 0.15) / 0.0167 = 4.19

Frame Time (s) Vert.Pos (y) Vel. (vy)

1 0.0000 0.00 8.98

2 0.0167 0.15 4.19

3 0.0334 0.22 2.99

4 0.0501 0.27

Finite Differentiation Finite Differentiation

x2 x3 x4 x1 x5

x2 x3 x4 x5

Δt

2Δt

x1

V2-3 V3

6

Finite Differentiation

txxx Δ

−= 2

132

Finite Differentiation First central difference method

(0.22 - 0.00) / 0.0334 = 6.59

Frame Time (s) Vert.Pos (y) Vel. (vy)

1 0.0000 0.00 0.00

2 0.0167 0.15 6.59

3 0.0334 0.22 3.59

4 0.0501 0.27

Acceleration

  Again if we are using coordinate systems we use the following convention.

yonacceleratiVerticalxonacceleratiHorizontal

⇒⋅

⇒⋅

Figure 8.18

Figure 8.19

Sample Problem Time (s) Displ. (m)

0.0 0.000 0.5 0.857 1.0 3.160 1.5 6.484

2.0 10.564 2.5 15.210

10.5 105.514

7

Time Displ.

0.0 0.000 0.5 0.857 1.0 3.160 1.5 6.484 2.0 10.56 2.5 15.21 3.0 20.26 3.5 25.60 4.0 31.130 4.5 36.77 5.0 42.480

Time Displ. 5.5 48.21 6.0 53.95 6.5 59.68 7.0 65.42 7.5 71.16 8.0 76.89 8.5 82.63 9.0 88.37 9.5 94.11 10.0 99.84 10.5 105.51

Draw the following graphs (do not use 1st central difference)

d vs t v vs t a vs t F vs v

Assume mass of runner = 70 kg

Sampling Theory

Winter, 1979 (page 22-39)

How small should Δt be? Instantaneous Velocity

Time

Displ.

Tangent

This line would be a poor estimate of the tangent for this section of the curve.

Δt   Obviously the smaller Δt is the more

accurate you estimate of instantaneous velocity.

  However, the smaller you try to get Δt the more expensive it is going to be!

  Regular video at 60 frames/sec is good for most applications.

Sampling Theory

8

Analogue to Digital Synchronization (A to D)

  If you have force platform, video and EMG data there can be a problem in synchronizing the data.

  How do you know the time frames on each data acquisition system match?

  No problem if all collected by computer, but if some is collected on video and some on the computer!

Aliasing Error

Signal 1

Signal 2

Fourier Transformation

f2

Sampling Theory Filtering Raw Data

Differentiation and Noise

Red stars = true location of markers Yellow stars = location of markers due to “noise”

The differential of the line between these markers is much larger than the difference in their location.

9

Signal vs Noise

Integration

  Differentiating positional data to get velocity and acceleration has been covered.

  However, acceleration may be collected in a biomechanical analysis.

  In this case, you may want to calculate velocity and displacement data.

  This is the opposite of differentiation and is known as integration.

Accelerometers

The Basic Accelerometer: A classical second order mass-spring mechanical system with damping and applied force

Tri-Axial Accelerometers   Accelerometers vary

considerably in resolution and max. acceleration.

  Must be sure of planar acceleration if using uni-axial accelerometers.

  Tri-axial accelerometers are bulkier and much more expensive.

  These can be rented rather than purchased.

Vibration   Vibration is measured using accelerometers

and then various mathematical and statistical techniques are used to quantify and interpret the signal.

Force Platform Data

If you have force – and obviously F = ma, then you can easily calculate the acceleration of the body’s centre of gravity.

10

Finite Integration   Finite differentiation methods are used with

digital data.   Similarly, finite integration methods are used

with digital data.   Finite differentiation calculates the slope of

the curve.   Finite integration calculates the area under

the curve.   Most often used with force-time curves – area

under the curve is mechanical impulse (more in Linear Kinetics lecture)

Example

  Area A equals 3m/s2 x 6s = 18 m/s

  Units! LT-2 x T = LT-1   This is change in

velocity from 0-6 s.   Area B is 14 m/s.   Total change in

velocity from 0-8 s is 32 m/s.

7

3

0 6 8

acc.

Time

Finite Integration

€

If :→ a = ΔvΔt

€

Then :→ Δv = aΔt Acc

eler

atio

n

Time

€

Hence area under curve = aΔt

Riemann Sum   Finite differentiation

approximates the area under curves as a series of rectangles

  This is called the Riemann sum (see equation opposite)

  If Δt is small enough this is an accurate approximation

v dt dst

t

ds (v *dt)

xi

1

30

xi

i 1

30

=

=

∫

∑=

Example above: Horizontal velocity time curve with 30 time intervals. Integral equals change in displacement.

Integration is less sensitive to errors due to “noise”

The slope of curve A varies greatly but the area under the curve is not that different from curve B.

A B

High frequency “noise” present

Kinematics of Running Hamill & Knutzen, Chapter 8

(pages 319-323)

How fast can we go?

11

Stride Rate vs Stride Length Running Kinematics   Stride length (SL) and stride frequency (SF) are very

commonly studied kinematic parameters.   Both SR & SL increase linearly (approx.) from a slow

jog up until 7 m/s.   After this SR increases much more than SL.   Support and non-support phases are also of interest.   Support Phase: Jogging 68%, moderate sprint 54%,

full-sprint 47%

Mechanical Efficiency (Figure 8-27)

02 consumption PSF = preferred stride frequency

100 m vs 200 m

  The world record for the 100m is?

 Women 10.49s (1988)  Men 9.58s

  The world record for the 200m is?

 Women 21.34s (1988)  Men 19.19s

4 x 100 m Relay   Best male 4 x 100m relay time = 37.10s   This an average of 9.275s per 100m!   Best female 4 x 100m relay time = 41.37s   This an average of 10.34s per 100m!

  This is possible due to the fact an acceleration phase is allowed within the 2nd, 3rd and 4th 100m segments.

Projectile Motion

12

Equations of Constant Acceleration

1).......vf = vi + at This is a re-arrangment of: a = Δv a = vf - vi Δt t

2).......d = vit + 0.5at2 3).......vf

2 = vi2 + 2ad

Forces Influencing Projectiles

  Gravity   Air resistance   No other forces can

influence the flight (trajectory) of the projectile

  Air resistance is often negligible

  Air resistance is considered negligible in this section of the text

Without air resistance

With air resistance

Air Resistance

  Can often be ignored but is often a considerable factor.

  Name a few examples where air (or fluid) resistance in considerable.

  Baseball, cycling, swimming, skydiving!   Name a few where it is negligible   Shot-put, long jump (possibly)?

Maximum Vertical Displacement

  Relatively simplistic

  Height centre of gravity (CG) reaches will be determined by height of CG at take-off and vertical velocity of CG at take-off.

Maximum Vertical Reach Actual reach will be affected

by body anthropometry and position.

In what ways?

Projecting for Horizontal

Distance   It is a very common

performance objective to project an object, or the body, for maximum horizontal distance.

  Long jump, triple jump, golf drive, football punts.

13

Factors Affecting Trajectory

Projection Height

Projection Angle

Projection Speed

Optimum Angle of Projection You need a large horizontal velocity, but if you sacrifice vertical velocity you have little time in the air.

Optimum = 45o (without air resistance)

Range Equation

  This can be arrived at via the use of the equation, d = vit + 0.5at2 and knowing the solution to a quadratic.

  There are many problems to work through in the texts on reserve and the course workbook

€

Range =

2v × sinθ × cosθ + xv y2v + 2gh

g

Vertical & Horizontal Components are Independent

Easier way to calculate range

Vertical

Horizontal €

Time =− yv ± y

2v + 2gh

g

€

Range = xv × time

Projecting for Accuracy

  Optimize velocity of release, rather than maximize, in sports like darts, slow-pitch softball

  In other sports like baseball, tennis, squash and golf drives it is desirable to have high release velocities.

14

Optimum Angle of Projection If the target is above the release point then the optimum angle is steeper (too low and you don’t get there!

If the target is below the release point then the optimum angle is shallower.

Vertical Plane Targets

  Ideally would like projectile projected to target at 90o.

  However, the further the projection distance, for any given velocity, the more arc (parabola) needed on the projectile.

Horizontal Plane Targets

  Ideally a vertical descent into the target area is desirable.

  Again the horizontal distance from the target will determine how closely one can achieve this ideal.

Projecting the Body for Accuracy

  Point targets in space are often used rather than a physical target.

  Best examples are in gymnastics and other tumbling activities.

Speed and Accuracy

 Many sports require both accuracy & speed (spike volleyball serve, tennis serve, lacrosse shot, etc.).

  One cannot maximize projection speed in some cases, but this requirement cannot be ignored.

Long Jumper’s Angle of Take-Off