Outline
definition and units of angular
displacement, angular velocity, and
angular acceleration
! constant angular acceleration equations
! relations between linear and angular
kinematic quantities
! relative motion
! techniques for measuring and describing
joint angles
!
ANGULAR
KINEMATICS
Ozkaya and Nordin
p. 275 - 291
Stephen Robinovitch, Ph.D.
KIN 201
2007-1
Lecture 10: Linear Kinematics II
2
Most motion involves translation and
rotation
=
Angular displacement
• counter-clockwise
rotation is positive,
clockwise rotation is
negative
+
(a)
(b)
=
+
• 1 revolution = 360
degrees = 2! radians
! (radians) =
(c)
Y
+250o
3
"
! (degrees)
180
X
-110o
The importance of !t in numerical
differentiation
Angular velocity and acceleration
!
!
Angular velocity is denoted with the symbol #,
measured in units of rad/s, and defined as
d"
! = "˙ =
dt
The graph at left shows the
actual variation in x(t) and the
instantaneous (tangent)
velocity dx dt at time t1.
Angular acceleration is denoted with the
symbol ", measured in units of rad/s2, and
defined as
d# d 2" d $ d" '
˙
˙
! = " = #˙ =
=
= & )
dt
dt
dt % dt (
!
In the laboratory, x(t) can be
sampled every !t=(t1-t0), or
every !t/2, !t/4, or !t/8.
What sampling interval is
required to obtain a reasonable
estimate of dx dt ?
If we are using numerical differentiation:
" # " I #1
! # ! I #1
! i = I +1
% i = I +1
2$t
2$t
6
Angular Kinematics Example
Kinematic Quantities: Units
quantity
linear
time
second
displacement meter
!t
angular
(s)
second
(s)
(m)
radians
(rad)
velocity
meters per second (m!s-1) radians per
second
acceleration
meters per second (m!s-2) radians per
(rad!s-2)
second
per
second
per second
For above picture, if pointing to 12 represents an
angular displacement of zero and clockwise rotation is
positive,
(rad!s-1)
•angular displacement of minute hand =
•angular displacement of hour hand =
•angular velocity of second hand =
•angular accelerations of all hands =
7
8
Constant Angular Acceleration
Equations
Relationship between linear and
angular quantities
s = r!
1
(1) ! = ! 0 + " 0 t + #t 2
2
(2) " = " 0 + #t
2
$
r
s
(3) " $ " 0 = 2# (! $ ! 0 )
9
Any point P on a rigid body rotating
about a fixed axis O has:
• a velocity that is tangential to the
circular path of motion
• an acceleration with tangential and
normal (inwardly-directed) components
s
r
(with ! measured
in radians)
2
Relationship between linear and
angular quantities
ds
d!
" !=
All points on the hammer
travel through the same
angular displacement.
However, linear displacement
(and velocity) increases
linearly with distance from
the axis of rotation.
d2
d1
$
Angular velocities of limb segments
govern linear velocity of ball
vt =
=r
= r"
dt
dt
dv
d"
at = t = r
= r#
dt
dt
vt 2
an =
= r" 2
r
12
Relative motion: COG velocity
also affects linear velocity of ball
How is the release
velocity of the ball
affected by the
horizontal velocity
of the whole-body
COG?
Relative versus absolute angles
#arm
!
vCOG
!
r
r
r
rHAND = rCOG + rHAND/ COG
r
r
r
vHAND = vCOG + vHAND/ COG
r
r
r
a HAND = aCOG + a HAND/ COG
An absolute angle is
measured from an
external frame of
reference.
$abs
A relative angle is the
angle formed between
two limb segments.
$rel
13
Relative Angles
Measuring lower extremity joint angles
The preferred method for
describing a relative joint position
is degrees of flexion or extension.
Joint position may
alternatively be described as
the angle formed at the
articulation.
(180 " $rel)
$rel
Diagram at left shows
X,Y coordinates (acquired
with a motion
measurement system) of
the greater trochanter,
lateral femoral
epicondyle, and lateral
malleolus.
Knowing these values,
how would we determine
$knee?
Y
$knee
(XH,YH)
(XK,YK)
(XA,YA)
X
Instantaneous joint centres of
rotation
Measuring lower extremity joint angles
(contd)
#y
tan" =
#x
Yk % YA
Xk % XA
Y % YK
= tan -1 H
X H % XK
$knee
$thigh
$ " shin = tan -1
" thigh
In most biomechanical
analyses, we assume that joints
have fixed axes of rotation
! This assumption usually
introduces minimal error
! However, for many joints,
changes in flexion angle cause
corresponding changes in the
location of the centre of
rotation (due to rolling and
sliding between articulating
surfaces)
!
Y
(XH,YH)
" knee = " thigh % " shin
(XK,YK)
$shin
(XA,YA)
X
17
!
Review Questions
How can the sampling interval (!t) affect the
validity of parameters obtained through numerical
differentiation?
! What is the relationship between linear and angular
displacement, velocity, and acceleration?
! If a rigid body rotates with constant angular
velocity, what is its acceleration?
! Given the (X,Y) positions of the hip, knee, and
ankle, how can we determine knee flexion angle?
!
19