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An Introduction to Angular Kinematics. Angular Kinematics. Aim. The aim of these slides is to introduce the variables used in angular kinematics These slides include: An introduction to conventions used to measure angles and vectors used to represent angular quantities - PowerPoint PPT Presentation
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B i o L a b - B i o m e c h a n i c s T e a c h i n g & L e a r n i n g T o o l B o xB i o L a b - B i o m e c h a n i c s T e a c h i n g & L e a r n i n g T o o l B o x
Angular KinematicsAngular Kinematics
An Introduction to Angular Kinematics
An Introduction to Angular Kinematics
AimAim
• The aim of these slides is to introduce the variables used in angular kinematics
• These slides include:
– An introduction to conventions used to measure angles and vectors used to represent angular quantities
– Definitions and calculations of angular displacement, velocity and acceleration
– Application and interpretation of angular displacement, velocity and acceleration in simple motion
• Angular Kinematics– Description of the circular motion or rotation of a body
• Motion described in terms of (variables):– Angular position and displacement– Angular velocity– Angular acceleration
• Rotation of body segments – e.g. Flexion of forearm about transverse axis through elbow joint
centre
• Rotation of whole body– e.g. Rotation of body around centre of mass (CM) during
somersaulting
Angular Kinematic AnalysisAngular Kinematic Analysis
Absolute and Relative AnglesAbsolute and Relative Angles
• Absolute angles– Angle of a single body
segment, relative to (normally) a right horizontal line (e.g. trunk, head, thigh)
• Relative Angles– Angle of one segment
relative to another (e.g. knee, elbow, ankle)
Units of MeasurementUnits of Measurement• Angles are expressed in one of the
following units:
• Revolutions (Rev)– Normally used to quantify body
rotations in diving, gymnastics etc.– 1 rev = 360º or 2 π radians
• Degrees (º)– Normally used to quantify angular
position, distance and displacement
• Radians (rad)– Normally used to quantify angular
velocity and acceleration– Convert degrees to radians by
dividing by 57.3
θ
radius (r)
arc (d)
d = =
rθ 1 radian
Angular Motion VectorsAngular Motion Vectors
• Right-hand thumb rule
– Fingers of right hand curled in direction of rotation
– Direction of extended thumb coincides with the direction of the angular motion vector
– Counter clockwise rotations are positive
– Clockwise rotations are negative
Angular Distance and DisplacementAngular Distance and Displacement
• Angular distance:– The sum of all the angular
changes of a rotating body between its initial and final positions
• Denoted by ∆θ
• Angular displacement– The difference between the
final and initial positions of a rotating body
• Calculated by θ2 - θ1
• Denoted by ∆θ
11
θ1 = 90º
22
θ2 = 110º
Angular Displacement:
∆θ = 110º - 90º = 20º
Angular Displacement:
∆θ = 110º - 90º = 20º
Angular VelocityAngular Velocity
• Angular velocity (ω) is equal to the angular displacement (∆θ) divided by change in time (∆t)
t =
t=2 1ω
- θ θ
θ
• Units:
º/s or º·s-1
rad/s or rad·s-1
Example:
∆θ = 45º ∆t = 0.6 s
-1 -1ω45
= = 75 º·s 1.31 rad= 6
·s0.
Angular AccelerationAngular Acceleration
• Angular acceleration (α) is equal to change in angular velocity (∆ω) divided by change in time (∆t)
t =
t=2 1α
- ω ω
ω
• Units:
º/s/s or º·s-2
rad/s/s or rad·s-2
Example:
∆ω = 1.31 rad·s-2 ∆t = 0.5 s
-21.31= = α 2.18 rad
6·s
0.
11
θ = 170º
t1 = 0 s
22
θ = 91º
t2 = 0.5 s
θ = 185º
t3 = 0.8 s
33
Standing Vertical Jump – DisplacementStanding Vertical Jump – Displacement
• Angular displacement (∆θ)
– During countermovement= θ2 - θ1 or ∆θ
= 91 - 170
= -79º
– During upward movement= θ2 - θ1 or θ
= 185 - 91
= 94º
N.B.
Flexion = -ve displacement
Extension = +ve displacement
11
θ = 170º
t1 = 0 s
22
θ = 91º
t2 = 0.5 s
• Angular velocity (ω)
– During countermovement
ω = -158º·s-1 or -2.76 rad·s-1
– During upward movement
ω = 313º·s-1 or 5.47 rad·s-1
N.B.
Flexion = -ve velocity
Extension = +ve velocity
= t
θ -79ω =
0.5
= t
θ 94ω =
0.3
Standing Vertical Jump - VelocityStanding Vertical Jump - Velocity
θ = 185º
t3 = 0.8 s
33
Standing Vertical Jump - AccelerationStanding Vertical Jump - Acceleration
• Angular acceleration (α)
– Between countermovement and upward movement
α = 27.4 rad·s-2
N.B.
Positive angular acceleration decreases negative angular velocity to zero at bottom of countermovement and increases positive angular velocity from bottom of countermovement
= t
ωα
= (5.47 - (-2.76)) 8.22
α = 0.3 0.3
SummarySummary
• Angular distance is the angle between two bodies
• Angular displacement is the angle through which a body has been rotated
• Average angular velocity is the angular displacement divided by the change in time
• Average angular acceleration is the change in angular velocity divided by the change in time
• Enoka, R.M. (2002). Neuromechanics of Human Movement (3rd edition). Champaign, IL.: Human Kinetics. Pages 3-10 & 27-33.
• Grimshaw, P., Lees, A., Fowler, N. & Burden, A. (2006). Sport and Exercise Biomechanics. New York: Taylor & Francis. Pages 22-29.
• Hamill, J. & Knutzen, K.M. (2003). Biomechanical Basis of Human Movement (2nd edition). Philadelphia: Lippincott Williams & Wilkins. Pages 309-336.
• McGinnis, P.M. (2005). Biomechanics of Sport and Exercise (2nd edition). Champaign, IL.: Human Kinetics. Pages 147-158.
Recommended ReadingRecommended Reading