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We saw in the previous lecture that the components of a fixed vector with respect to
a frame that has been rotated, are related to the components of the original system
as follows:
๐‘ฅ๐‘ฅ
๐‘ฆ๐‘ฆ
๐‘ง๐‘ง
= ๐ด๐ด
๐‘‹๐‘‹
๐‘Œ๐‘Œ
๐‘๐‘
where ๐ด๐ด is the โ€˜Matrix of Direction Cosinesโ€™:
๐ด๐ด =
Cos(๐‘ฅ๐‘ฅ๐‘ฅ๐‘ฅ๐‘ฅ๐‘ฅ) Cos(๐‘ฅ๐‘ฅ๐‘ฅ๐‘ฅ๐‘ฅ๐‘ฅ) Cos(๐‘ฅ๐‘ฅ๐‘ฅ๐‘ฅ๐‘ฅ๐‘ฅ)
Cos(๐‘ฆ๐‘ฆ๐‘ฆ๐‘ฆ๐‘ฆ๐‘ฆ) Cos(๐‘ฆ๐‘ฆ๐‘ฆ๐‘ฆ๐‘ฆ๐‘ฆ) Cos(๐‘ฆ๐‘ฆ๐‘ฆ๐‘ฆ๐‘ฆ๐‘ฆ)
Cos(๐‘ง๐‘ง๐‘ง๐‘ง๐‘ง๐‘ง) Cos(๐‘ง๐‘ง๐‘ง๐‘ง๐‘ง๐‘ง) Cos(๐‘ง๐‘ง๐‘ง๐‘ง๐‘ง๐‘ง)
where for ๐‘ฅ๐‘ฅ๐‘ฅ๐‘ฅ๐‘ฅ๐‘ฅ is the angle between the x and the X axes, ๐‘ฅ๐‘ฅ๐‘ฅ๐‘ฅ๐‘ฅ๐‘ฅ is the angle between
the x and the Y axes etc.
Rotation of Axes
Advanced Kinematic Analysis
We will now prove this from geometry.
X
Y
y x
A 2D Rotation
We also saw that the transformation for a 2D rotation about the z axis
simplifies to:
๐‘ฅ๐‘ฅ
๐‘ฆ๐‘ฆ
๐‘ง๐‘ง
=
Cos(๐‘ฅ๐‘ฅ๐‘ฅ๐‘ฅ๐‘ฅ๐‘ฅ) Cos(๐‘ฅ๐‘ฅ๐‘ฅ๐‘ฅ๐‘ฅ๐‘ฅ) 0
Cos(๐‘ฆ๐‘ฆ๐‘ฆ๐‘ฆ๐‘ฆ๐‘ฆ) Cos(๐‘ฆ๐‘ฆ๐‘ฆ๐‘ฆ๐‘ฆ๐‘ฆ) 0
0 0 1
๐‘‹๐‘‹
๐‘Œ๐‘Œ
๐‘๐‘
=
๐ถ๐ถ๐ถ๐ถ๐ถ๐ถ๐ถ๐ถ ๐‘†๐‘†๐‘†๐‘†๐‘†๐‘†๐‘†๐‘† 0
โˆ’๐‘†๐‘†๐‘†๐‘†๐‘†๐‘†๐‘†๐‘† ๐ถ๐ถ๐ถ๐ถ๐ถ๐ถ๐ถ๐ถ 0
0 0 1
๐‘‹๐‘‹
๐‘Œ๐‘Œ
๐‘๐‘
i.e.:
๐‘ฅ๐‘ฅ
๐‘ฆ๐‘ฆ =
๐ถ๐ถ๐ถ๐ถ๐ถ๐ถ๐ถ๐ถ ๐‘†๐‘†๐‘†๐‘†๐‘†๐‘†๐‘†๐‘†
โˆ’๐‘†๐‘†๐‘†๐‘†๐‘†๐‘†๐‘†๐‘† ๐ถ๐ถ๐ถ๐ถ๐ถ๐ถ๐ถ๐ถ
๐‘‹๐‘‹
๐‘Œ๐‘Œ
Advanced Kinematic Analysis
Advanced Kinematic Analysis
2D Transformation - proof from geometry.
A 2D Rotation
Y
X
y
Y
X
y
x
And from the figure (using similar triangles) it is therefore evident that:
๐‘ฅ๐‘ฅ
๐‘ฆ๐‘ฆ =
cos ๐œƒ๐œƒ sin ๐œƒ๐œƒ
โˆ’ sin ๐œƒ๐œƒ cos ๐œƒ๐œƒ
๐‘‹๐‘‹
๐‘Œ๐‘Œ
end of proof
x
Advanced Kinematic Analysis
3D rotation of axes achieved by 3 successive 2D rotations
In general, we can always achieve any 3D rotation by 3 successive 2D rotations
(about the appropriate axes using the appropriate (3 x 3) 2D rotation matrix of
Direction Cosines) i.e.
i.e. ๐‘ฅ๐‘ฅโ€ฒ = ๐ด๐ด๐‘‹๐‘‹ โŸน ๐‘ฅ๐‘ฅโ€ฒโ€ฒ = ๐ด๐ดโ€ฒ๐‘ฅ๐‘ฅโ€ฒ โŸน ๐‘ฅ๐‘ฅโ€ฒโ€ฒโ€ฒ = ๐ด๐ดโ€ฒโ€ฒ๐‘ฅ๐‘ฅโ€ฒโ€ฒ
where the direction cosine matrices in each case (๐ด๐ด, ๐ด๐ดโ€ฒ
, and ๐ด๐ดโ€ฒโ€ฒ
) are 2D
rotations about corresponding axes.
Orthogonality of matrix A
๐ด๐ด๐‘‡๐‘‡ = ๐ด๐ดโˆ’1 ๐‘–๐‘–. ๐‘’๐‘’. ๐ด๐ด๐‘‡๐‘‡๐ด๐ด = ๐ผ๐ผ (the unit matrix)
Advanced Kinematic Analysis
A Physical rotation
A physical rotation can be obtained by keeping the axes fixed but rotating a
vector. Consider a point P on a disc. If the disc is rotated through angle ฮธ,
the new position vector P* can be obtained by multiplying vector P by ๐ด๐ดโˆ’1
e.g.: Y
X
P
P*
i.e.
๐‘ƒ๐‘ƒโˆ— = ๐ด๐ดโˆ’1๐‘ƒ๐‘ƒ =
cos ๐œƒ๐œƒ โˆ’ sin ๐œƒ๐œƒ
sin ๐œƒ๐œƒ cos ๐œƒ๐œƒ
๐‘ƒ๐‘ƒ๐‘ฅ๐‘ฅ
๐‘ƒ๐‘ƒ๐‘ฆ๐‘ฆ
Advanced Kinematic Analysis
KINEMATICS OF A PARTICLE OBTAINED BY ROTATION OF AXES
Here we return to the original task, namely the development of tools that
enable us to obtain the derivatives of vectors (particularly velocity and
acceleration) when the position vector is described in terms of a frame of
reference that is moving (i.e. a rotating frame). To do this, we initially
approach the problem in a โ€˜sledge-hammerโ€™ way by rotation of axes (which,
from the previous section, we now know how to do).
Advanced Kinematic Analysis
KINEMATICS OF A PARTICLE OBTAINED BY ROTATION OF AXES
Consider a particle P, with position vector r, that is moving arbitrarily in the (fixed) XY
plane as described in the following figure where Pโ€™ is a new position. Here the particle is
โ€˜trackedโ€™ by a frame of reference xy such that the x axis always points straight at the
particle. The xy axes are therefore moving polar coordinates. The question is: what are
the absolute velocity and acceleration vector for particle P? We will answer this
question using a rotation of axes.
P
X
Y
y
Pโ€™
x
r
P moves anyway in
the plane
Pโ€™ is a new position
Particle P moving arbitrarily in the XY Plane (where the XY frame is fixed). In addition, a (polar)
coordinate system xy is chosen as a special case to track particle P - the xy frame is therefore
moving.
Advanced Kinematic Analysis
KINEMATICS OF A PARTICLE OBTAINED BY ROTATION OF AXES
The position vector
Note the position vector ๐‘Ÿ๐‘Ÿ of P is:
๐‘Ÿ๐‘Ÿ = ๐‘‹๐‘‹ ๐‘ก๐‘ก ๐ผ๐ผ + ๐‘Œ๐‘Œ ๐‘ก๐‘ก ๐ฝ๐ฝ = ๐‘Ÿ๐‘Ÿ ๐‘ก๐‘ก cos ๐œƒ๐œƒ ๐‘ก๐‘ก ๐ผ๐ผ + ๐‘Ÿ๐‘Ÿ ๐‘ก๐‘ก sin ๐œƒ๐œƒ(๐‘ก๐‘ก)๐ฝ๐ฝ ๐ผ๐ผ ๐‘Ž๐‘Ž๐‘Ž๐‘Ž๐‘Ž๐‘Ž ๐ฝ๐ฝ ๐‘“๐‘“๐‘“๐‘“๐‘“๐‘“๐‘“๐‘“๐‘“๐‘“
The velocity vector
The velocity vector can be obtained by differentiation of the position vector with respect
to the fixed frame of reference, i.e.:
๐‘‰๐‘‰ = ฬ‡
๐‘Ÿ๐‘Ÿ =
๐‘‘๐‘‘
๐‘‘๐‘‘๐‘‘๐‘‘
๐‘Ÿ๐‘Ÿ cos ๐œƒ๐œƒ ๐ผ๐ผ + ๐‘Ÿ๐‘Ÿ sin ๐œƒ๐œƒ ๐ฝ๐ฝ = ฬ‡
๐‘Ÿ๐‘Ÿ cos ๐œƒ๐œƒ โˆ’ ๐‘Ÿ๐‘Ÿ sin ๐œƒ๐œƒ ฬ‡
๐œƒ๐œƒ ๐ผ๐ผ + ฬ‡
๐‘Ÿ๐‘Ÿ sin ๐œƒ๐œƒ + ๐‘Ÿ๐‘Ÿ cos ๐œƒ๐œƒ ฬ‡
๐œƒ๐œƒ ๐ฝ๐ฝ
(i.e. in the fixed system)
Advanced Kinematic Analysis
KINEMATICS OF A PARTICLE OBTAINED BY ROTATION OF AXES
The acceleration vector
The acceleration vector can be obtained again by differentiation of the velocity
vector with respect to the fixed frame of reference, i.e.:
๐‘Ž๐‘Ž = ฬˆ
๐‘Ÿ๐‘Ÿ = ฬˆ
๐‘‹๐‘‹๐ผ๐ผ + ฬˆ
๐‘Œ๐‘Œ๐ฝ๐ฝ
= ฬˆ
๐‘Ÿ๐‘Ÿ cos ๐œƒ๐œƒ โˆ’ ฬ‡
๐‘Ÿ๐‘Ÿ sin ๐œƒ๐œƒ ฬ‡
๐œƒ๐œƒ โˆ’ ฬ‡
๐‘Ÿ๐‘Ÿ sin ๐œƒ๐œƒ ฬ‡
๐œƒ๐œƒ โˆ’ ๐‘Ÿ๐‘Ÿ cos ๐œƒ๐œƒ ฬ‡
๐œƒ๐œƒ2 โˆ’ ๐‘Ÿ๐‘Ÿ sin ๐œƒ๐œƒ ฬˆ
๐œƒ๐œƒ ๐ผ๐ผ
+ ฬˆ
๐‘Ÿ๐‘Ÿ sin ๐œƒ๐œƒ + ฬ‡
๐‘Ÿ๐‘Ÿ cos ๐œƒ๐œƒ ฬ‡
๐œƒ๐œƒ + ฬ‡
๐‘Ÿ๐‘Ÿ cos ๐œƒ๐œƒ ฬ‡
๐œƒ๐œƒ + ๐‘Ÿ๐‘Ÿ cos ๐œƒ๐œƒ ฬˆ
๐œƒ๐œƒ โˆ’ ๐‘Ÿ๐‘Ÿ sin ๐œƒ๐œƒ ฬ‡
๐œƒ๐œƒ2 ๐ฝ๐ฝ
(i.e. again in the fixed system)
Advanced Kinematic Analysis
KINEMATICS OF A PARTICLE OBTAINED BY ROTATION OF AXES
The position, velocity, and acceleration vectors in the moving system
The components of the position vector ๐‘Ÿ๐‘Ÿ in the moving (polar) system can be obtained
by a 2D rotation matrix i.e.:
๐‘ฅ๐‘ฅ
๐‘ฆ๐‘ฆ =
cos ๐œƒ๐œƒ sin ๐œƒ๐œƒ
โˆ’ sin ๐œƒ๐œƒ cos ๐œƒ๐œƒ
๐‘‹๐‘‹(๐‘ก๐‘ก)
๐‘Œ๐‘Œ(๐‘ก๐‘ก)
i.e. since ๐‘Ÿ๐‘Ÿ = ๐‘‹๐‘‹ ๐‘ก๐‘ก ๐ผ๐ผ + ๐‘Œ๐‘Œ ๐‘ก๐‘ก ๐ฝ๐ฝ = ๐‘Ÿ๐‘Ÿ ๐‘ก๐‘ก cos ๐œƒ๐œƒ ๐‘ก๐‘ก ๐ผ๐ผ + ๐‘Ÿ๐‘Ÿ ๐‘ก๐‘ก sin ๐œƒ๐œƒ(๐‘ก๐‘ก)๐ฝ๐ฝ :
๐‘ฅ๐‘ฅ
๐‘ฆ๐‘ฆ =
cos ๐œƒ๐œƒ sin ๐œƒ๐œƒ
โˆ’ sin ๐œƒ๐œƒ cos ๐œƒ๐œƒ
๐‘Ÿ๐‘Ÿ cos ๐œƒ๐œƒ
๐‘Ÿ๐‘Ÿ sin ๐œƒ๐œƒ
And by noting that ๐‘๐‘๐‘๐‘๐‘๐‘2๐œƒ๐œƒ + ๐‘ ๐‘ ๐‘ ๐‘ ๐‘ ๐‘ 2๐œƒ๐œƒ = 1, we get:
๐‘ฅ๐‘ฅ
๐‘ฆ๐‘ฆ =
๐‘Ÿ๐‘Ÿ
0
i.e. ๐‘Ÿ๐‘Ÿ = ๐‘Ÿ๐‘Ÿ๐‘–๐‘– (where ๐‘–๐‘– is moving with angular velocity ฬ‡
๐œƒ๐œƒ).
This result is obvious because the x axis always points straight at the particle so the
frame of reference xy (polar coordinates) is defined precisely to โ€˜trackโ€™ the particle.
Advanced Kinematic Analysis
The Velocity vector ๐‘‰๐‘‰ in the moving system
The components of the velocity vector obtained by a 2D rotation matrix i.e.:
๐‘‰๐‘‰ = ฬ‡
๐‘‹๐‘‹๐ผ๐ผ + ฬ‡
๐‘Œ๐‘Œ๐ฝ๐ฝ = ฬ‡
๐‘Ÿ๐‘Ÿ cos ๐œƒ๐œƒ โˆ’ ๐‘Ÿ๐‘Ÿ sin ๐œƒ๐œƒ ฬ‡
๐œƒ๐œƒ ๐ผ๐ผ + ฬ‡
๐‘Ÿ๐‘Ÿ sin ๐œƒ๐œƒ + ๐‘Ÿ๐‘Ÿ cos ๐œƒ๐œƒ ฬ‡
๐œƒ๐œƒ ๐ฝ๐ฝ
ฬ‡
๐‘ฅ๐‘ฅ
ฬ‡
๐‘ฆ๐‘ฆ
=
cos ๐œƒ๐œƒ sin ๐œƒ๐œƒ
โˆ’ sin ๐œƒ๐œƒ cos ๐œƒ๐œƒ
ฬ‡
๐‘‹๐‘‹
ฬ‡
๐‘Œ๐‘Œ
=
ฬ‡
๐‘Ÿ๐‘Ÿ
๐‘Ÿ๐‘Ÿ ฬ‡
๐œƒ๐œƒ
i.e. the velocity vector in the moving system is:
ฬ‡
๐‘Ÿ๐‘Ÿ = ฬ‡
๐‘Ÿ๐‘Ÿ๐‘–๐‘– + ๐‘Ÿ๐‘Ÿ ฬ‡
๐œƒ๐œƒ๐‘—๐‘—
KINEMATICS OF A PARTICLE OBTAINED BY ROTATION OF AXES
Advanced Kinematic Analysis
KINEMATICS OF A PARTICLE OBTAINED BY ROTATION OF AXES
Acceleration vector ๐’‚๐’‚ in the moving system
The components of the acceleration vector also obtained by a 2D rotation matrix are:
๐‘Ž๐‘Ž = ฬˆ
๐‘‹๐‘‹๐ผ๐ผ + ฬˆ
๐‘Œ๐‘Œ๐ฝ๐ฝ = ฬˆ
๐‘Ÿ๐‘Ÿ cos ๐œƒ๐œƒ โˆ’ ฬ‡
๐‘Ÿ๐‘Ÿ sin ๐œƒ๐œƒ ฬ‡
๐œƒ๐œƒ โˆ’ ฬ‡
๐‘Ÿ๐‘Ÿ sin ๐œƒ๐œƒ ฬ‡
๐œƒ๐œƒ โˆ’ ๐‘Ÿ๐‘Ÿ cos ๐œƒ๐œƒ ฬ‡
๐œƒ๐œƒ2
โˆ’ ๐‘Ÿ๐‘Ÿ sin ๐œƒ๐œƒ ฬˆ
๐œƒ๐œƒ ๐ผ๐ผ
+ ฬˆ
๐‘Ÿ๐‘Ÿ sin ๐œƒ๐œƒ + ฬ‡
๐‘Ÿ๐‘Ÿ cos ๐œƒ๐œƒ ฬ‡
๐œƒ๐œƒ + ฬ‡
๐‘Ÿ๐‘Ÿ cos ๐œƒ๐œƒ ฬ‡
๐œƒ๐œƒ + ๐‘Ÿ๐‘Ÿ cos ๐œƒ๐œƒ ฬˆ
๐œƒ๐œƒ โˆ’ ๐‘Ÿ๐‘Ÿ sin ๐œƒ๐œƒ ฬ‡
๐œƒ๐œƒ2 ๐ฝ๐ฝ
And in terms of the xy frame:
ฬˆ
๐‘ฅ๐‘ฅ
ฬˆ
๐‘ฆ๐‘ฆ
=
cos ๐œƒ๐œƒ sin ๐œƒ๐œƒ
โˆ’ sin ๐œƒ๐œƒ cos ๐œƒ๐œƒ
ฬˆ
๐‘‹๐‘‹
ฬˆ
๐‘Œ๐‘Œ
And after some manipulation we get:
= ฬˆ
๐‘Ÿ๐‘Ÿ โˆ’ ๐‘Ÿ๐‘Ÿ ฬ‡
๐œƒ๐œƒ2
2 ฬ‡
๐‘Ÿ๐‘Ÿ ฬ‡
๐œƒ๐œƒ + ๐‘Ÿ๐‘Ÿ ฬˆ
๐œƒ๐œƒ
i.e. the acceleration vector in the moving system is:
๐‘Ž๐‘Ž = ( ฬˆ
๐‘Ÿ๐‘Ÿ โˆ’ ๐‘Ÿ๐‘Ÿ ฬ‡
๐œƒ๐œƒ2)๐‘–๐‘– + (2 ฬ‡
๐‘Ÿ๐‘Ÿ ฬ‡
๐œƒ๐œƒ + ๐‘Ÿ๐‘Ÿ ฬˆ
๐œƒ๐œƒ)๐‘—๐‘—
Advanced Kinematic Analysis
Physical Interpretation of Acceleration Terms
The components of the acceleration vector are now shown in the figure below where the
unit vectors ๐‘–๐‘– , ๐‘—๐‘— are moving.
P
X
Y
y
x
r=ri
r
ฬˆ
๐‘Ÿ๐‘Ÿ: is the radial acceleration.
โˆ’๐‘Ÿ๐‘Ÿ ฬ‡
๐œƒ๐œƒ2: is the centripetal acceleration.
๐‘Ÿ๐‘Ÿ ฬˆ
๐œƒ๐œƒ: is the tangential acceleration.
2 ฬ‡
๐‘Ÿ๐‘Ÿ ฬ‡
๐œƒ๐œƒ: is the coriolis component.
Advanced Kinematic Analysis
The Coriolis acceleration stems from the combined radial and angular motion.
Imagine moving radially outwards on a spinning disc (e.g. a carousel or roundabout)
with constant angular velocity ฯ‰. At radius r1, the tangential velocity is v1= ฯ‰r1. At
radius r2, the tangential velocity is v2= ฯ‰r2. Since r2 > r1 the tangential velocity must
increase, representing an acceleration component in the tangential direction.
ฯ‰r1
๐‘Ÿ๐‘Ÿ2 > ๐‘Ÿ๐‘Ÿ1
ฬ‡
๐œƒ๐œƒ = ๐œ”๐œ” ฬ‡
๐œ”๐œ” = 0
ฯ‰r2
r1 r2

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DOMV No 12 CONTINUED ADVANCED KINEMATIC ANALYSIS v2.pdf

  • 1. We saw in the previous lecture that the components of a fixed vector with respect to a frame that has been rotated, are related to the components of the original system as follows: ๐‘ฅ๐‘ฅ ๐‘ฆ๐‘ฆ ๐‘ง๐‘ง = ๐ด๐ด ๐‘‹๐‘‹ ๐‘Œ๐‘Œ ๐‘๐‘ where ๐ด๐ด is the โ€˜Matrix of Direction Cosinesโ€™: ๐ด๐ด = Cos(๐‘ฅ๐‘ฅ๐‘ฅ๐‘ฅ๐‘ฅ๐‘ฅ) Cos(๐‘ฅ๐‘ฅ๐‘ฅ๐‘ฅ๐‘ฅ๐‘ฅ) Cos(๐‘ฅ๐‘ฅ๐‘ฅ๐‘ฅ๐‘ฅ๐‘ฅ) Cos(๐‘ฆ๐‘ฆ๐‘ฆ๐‘ฆ๐‘ฆ๐‘ฆ) Cos(๐‘ฆ๐‘ฆ๐‘ฆ๐‘ฆ๐‘ฆ๐‘ฆ) Cos(๐‘ฆ๐‘ฆ๐‘ฆ๐‘ฆ๐‘ฆ๐‘ฆ) Cos(๐‘ง๐‘ง๐‘ง๐‘ง๐‘ง๐‘ง) Cos(๐‘ง๐‘ง๐‘ง๐‘ง๐‘ง๐‘ง) Cos(๐‘ง๐‘ง๐‘ง๐‘ง๐‘ง๐‘ง) where for ๐‘ฅ๐‘ฅ๐‘ฅ๐‘ฅ๐‘ฅ๐‘ฅ is the angle between the x and the X axes, ๐‘ฅ๐‘ฅ๐‘ฅ๐‘ฅ๐‘ฅ๐‘ฅ is the angle between the x and the Y axes etc. Rotation of Axes Advanced Kinematic Analysis
  • 2. We will now prove this from geometry. X Y y x A 2D Rotation We also saw that the transformation for a 2D rotation about the z axis simplifies to: ๐‘ฅ๐‘ฅ ๐‘ฆ๐‘ฆ ๐‘ง๐‘ง = Cos(๐‘ฅ๐‘ฅ๐‘ฅ๐‘ฅ๐‘ฅ๐‘ฅ) Cos(๐‘ฅ๐‘ฅ๐‘ฅ๐‘ฅ๐‘ฅ๐‘ฅ) 0 Cos(๐‘ฆ๐‘ฆ๐‘ฆ๐‘ฆ๐‘ฆ๐‘ฆ) Cos(๐‘ฆ๐‘ฆ๐‘ฆ๐‘ฆ๐‘ฆ๐‘ฆ) 0 0 0 1 ๐‘‹๐‘‹ ๐‘Œ๐‘Œ ๐‘๐‘ = ๐ถ๐ถ๐ถ๐ถ๐ถ๐ถ๐ถ๐ถ ๐‘†๐‘†๐‘†๐‘†๐‘†๐‘†๐‘†๐‘† 0 โˆ’๐‘†๐‘†๐‘†๐‘†๐‘†๐‘†๐‘†๐‘† ๐ถ๐ถ๐ถ๐ถ๐ถ๐ถ๐ถ๐ถ 0 0 0 1 ๐‘‹๐‘‹ ๐‘Œ๐‘Œ ๐‘๐‘ i.e.: ๐‘ฅ๐‘ฅ ๐‘ฆ๐‘ฆ = ๐ถ๐ถ๐ถ๐ถ๐ถ๐ถ๐ถ๐ถ ๐‘†๐‘†๐‘†๐‘†๐‘†๐‘†๐‘†๐‘† โˆ’๐‘†๐‘†๐‘†๐‘†๐‘†๐‘†๐‘†๐‘† ๐ถ๐ถ๐ถ๐ถ๐ถ๐ถ๐ถ๐ถ ๐‘‹๐‘‹ ๐‘Œ๐‘Œ Advanced Kinematic Analysis
  • 3. Advanced Kinematic Analysis 2D Transformation - proof from geometry. A 2D Rotation Y X y Y X y x And from the figure (using similar triangles) it is therefore evident that: ๐‘ฅ๐‘ฅ ๐‘ฆ๐‘ฆ = cos ๐œƒ๐œƒ sin ๐œƒ๐œƒ โˆ’ sin ๐œƒ๐œƒ cos ๐œƒ๐œƒ ๐‘‹๐‘‹ ๐‘Œ๐‘Œ end of proof x
  • 4. Advanced Kinematic Analysis 3D rotation of axes achieved by 3 successive 2D rotations In general, we can always achieve any 3D rotation by 3 successive 2D rotations (about the appropriate axes using the appropriate (3 x 3) 2D rotation matrix of Direction Cosines) i.e. i.e. ๐‘ฅ๐‘ฅโ€ฒ = ๐ด๐ด๐‘‹๐‘‹ โŸน ๐‘ฅ๐‘ฅโ€ฒโ€ฒ = ๐ด๐ดโ€ฒ๐‘ฅ๐‘ฅโ€ฒ โŸน ๐‘ฅ๐‘ฅโ€ฒโ€ฒโ€ฒ = ๐ด๐ดโ€ฒโ€ฒ๐‘ฅ๐‘ฅโ€ฒโ€ฒ where the direction cosine matrices in each case (๐ด๐ด, ๐ด๐ดโ€ฒ , and ๐ด๐ดโ€ฒโ€ฒ ) are 2D rotations about corresponding axes. Orthogonality of matrix A ๐ด๐ด๐‘‡๐‘‡ = ๐ด๐ดโˆ’1 ๐‘–๐‘–. ๐‘’๐‘’. ๐ด๐ด๐‘‡๐‘‡๐ด๐ด = ๐ผ๐ผ (the unit matrix)
  • 5. Advanced Kinematic Analysis A Physical rotation A physical rotation can be obtained by keeping the axes fixed but rotating a vector. Consider a point P on a disc. If the disc is rotated through angle ฮธ, the new position vector P* can be obtained by multiplying vector P by ๐ด๐ดโˆ’1 e.g.: Y X P P* i.e. ๐‘ƒ๐‘ƒโˆ— = ๐ด๐ดโˆ’1๐‘ƒ๐‘ƒ = cos ๐œƒ๐œƒ โˆ’ sin ๐œƒ๐œƒ sin ๐œƒ๐œƒ cos ๐œƒ๐œƒ ๐‘ƒ๐‘ƒ๐‘ฅ๐‘ฅ ๐‘ƒ๐‘ƒ๐‘ฆ๐‘ฆ
  • 6. Advanced Kinematic Analysis KINEMATICS OF A PARTICLE OBTAINED BY ROTATION OF AXES Here we return to the original task, namely the development of tools that enable us to obtain the derivatives of vectors (particularly velocity and acceleration) when the position vector is described in terms of a frame of reference that is moving (i.e. a rotating frame). To do this, we initially approach the problem in a โ€˜sledge-hammerโ€™ way by rotation of axes (which, from the previous section, we now know how to do).
  • 7. Advanced Kinematic Analysis KINEMATICS OF A PARTICLE OBTAINED BY ROTATION OF AXES Consider a particle P, with position vector r, that is moving arbitrarily in the (fixed) XY plane as described in the following figure where Pโ€™ is a new position. Here the particle is โ€˜trackedโ€™ by a frame of reference xy such that the x axis always points straight at the particle. The xy axes are therefore moving polar coordinates. The question is: what are the absolute velocity and acceleration vector for particle P? We will answer this question using a rotation of axes. P X Y y Pโ€™ x r P moves anyway in the plane Pโ€™ is a new position Particle P moving arbitrarily in the XY Plane (where the XY frame is fixed). In addition, a (polar) coordinate system xy is chosen as a special case to track particle P - the xy frame is therefore moving.
  • 8. Advanced Kinematic Analysis KINEMATICS OF A PARTICLE OBTAINED BY ROTATION OF AXES The position vector Note the position vector ๐‘Ÿ๐‘Ÿ of P is: ๐‘Ÿ๐‘Ÿ = ๐‘‹๐‘‹ ๐‘ก๐‘ก ๐ผ๐ผ + ๐‘Œ๐‘Œ ๐‘ก๐‘ก ๐ฝ๐ฝ = ๐‘Ÿ๐‘Ÿ ๐‘ก๐‘ก cos ๐œƒ๐œƒ ๐‘ก๐‘ก ๐ผ๐ผ + ๐‘Ÿ๐‘Ÿ ๐‘ก๐‘ก sin ๐œƒ๐œƒ(๐‘ก๐‘ก)๐ฝ๐ฝ ๐ผ๐ผ ๐‘Ž๐‘Ž๐‘Ž๐‘Ž๐‘Ž๐‘Ž ๐ฝ๐ฝ ๐‘“๐‘“๐‘“๐‘“๐‘“๐‘“๐‘“๐‘“๐‘“๐‘“ The velocity vector The velocity vector can be obtained by differentiation of the position vector with respect to the fixed frame of reference, i.e.: ๐‘‰๐‘‰ = ฬ‡ ๐‘Ÿ๐‘Ÿ = ๐‘‘๐‘‘ ๐‘‘๐‘‘๐‘‘๐‘‘ ๐‘Ÿ๐‘Ÿ cos ๐œƒ๐œƒ ๐ผ๐ผ + ๐‘Ÿ๐‘Ÿ sin ๐œƒ๐œƒ ๐ฝ๐ฝ = ฬ‡ ๐‘Ÿ๐‘Ÿ cos ๐œƒ๐œƒ โˆ’ ๐‘Ÿ๐‘Ÿ sin ๐œƒ๐œƒ ฬ‡ ๐œƒ๐œƒ ๐ผ๐ผ + ฬ‡ ๐‘Ÿ๐‘Ÿ sin ๐œƒ๐œƒ + ๐‘Ÿ๐‘Ÿ cos ๐œƒ๐œƒ ฬ‡ ๐œƒ๐œƒ ๐ฝ๐ฝ (i.e. in the fixed system)
  • 9. Advanced Kinematic Analysis KINEMATICS OF A PARTICLE OBTAINED BY ROTATION OF AXES The acceleration vector The acceleration vector can be obtained again by differentiation of the velocity vector with respect to the fixed frame of reference, i.e.: ๐‘Ž๐‘Ž = ฬˆ ๐‘Ÿ๐‘Ÿ = ฬˆ ๐‘‹๐‘‹๐ผ๐ผ + ฬˆ ๐‘Œ๐‘Œ๐ฝ๐ฝ = ฬˆ ๐‘Ÿ๐‘Ÿ cos ๐œƒ๐œƒ โˆ’ ฬ‡ ๐‘Ÿ๐‘Ÿ sin ๐œƒ๐œƒ ฬ‡ ๐œƒ๐œƒ โˆ’ ฬ‡ ๐‘Ÿ๐‘Ÿ sin ๐œƒ๐œƒ ฬ‡ ๐œƒ๐œƒ โˆ’ ๐‘Ÿ๐‘Ÿ cos ๐œƒ๐œƒ ฬ‡ ๐œƒ๐œƒ2 โˆ’ ๐‘Ÿ๐‘Ÿ sin ๐œƒ๐œƒ ฬˆ ๐œƒ๐œƒ ๐ผ๐ผ + ฬˆ ๐‘Ÿ๐‘Ÿ sin ๐œƒ๐œƒ + ฬ‡ ๐‘Ÿ๐‘Ÿ cos ๐œƒ๐œƒ ฬ‡ ๐œƒ๐œƒ + ฬ‡ ๐‘Ÿ๐‘Ÿ cos ๐œƒ๐œƒ ฬ‡ ๐œƒ๐œƒ + ๐‘Ÿ๐‘Ÿ cos ๐œƒ๐œƒ ฬˆ ๐œƒ๐œƒ โˆ’ ๐‘Ÿ๐‘Ÿ sin ๐œƒ๐œƒ ฬ‡ ๐œƒ๐œƒ2 ๐ฝ๐ฝ (i.e. again in the fixed system)
  • 10. Advanced Kinematic Analysis KINEMATICS OF A PARTICLE OBTAINED BY ROTATION OF AXES The position, velocity, and acceleration vectors in the moving system The components of the position vector ๐‘Ÿ๐‘Ÿ in the moving (polar) system can be obtained by a 2D rotation matrix i.e.: ๐‘ฅ๐‘ฅ ๐‘ฆ๐‘ฆ = cos ๐œƒ๐œƒ sin ๐œƒ๐œƒ โˆ’ sin ๐œƒ๐œƒ cos ๐œƒ๐œƒ ๐‘‹๐‘‹(๐‘ก๐‘ก) ๐‘Œ๐‘Œ(๐‘ก๐‘ก) i.e. since ๐‘Ÿ๐‘Ÿ = ๐‘‹๐‘‹ ๐‘ก๐‘ก ๐ผ๐ผ + ๐‘Œ๐‘Œ ๐‘ก๐‘ก ๐ฝ๐ฝ = ๐‘Ÿ๐‘Ÿ ๐‘ก๐‘ก cos ๐œƒ๐œƒ ๐‘ก๐‘ก ๐ผ๐ผ + ๐‘Ÿ๐‘Ÿ ๐‘ก๐‘ก sin ๐œƒ๐œƒ(๐‘ก๐‘ก)๐ฝ๐ฝ : ๐‘ฅ๐‘ฅ ๐‘ฆ๐‘ฆ = cos ๐œƒ๐œƒ sin ๐œƒ๐œƒ โˆ’ sin ๐œƒ๐œƒ cos ๐œƒ๐œƒ ๐‘Ÿ๐‘Ÿ cos ๐œƒ๐œƒ ๐‘Ÿ๐‘Ÿ sin ๐œƒ๐œƒ And by noting that ๐‘๐‘๐‘๐‘๐‘๐‘2๐œƒ๐œƒ + ๐‘ ๐‘ ๐‘ ๐‘ ๐‘ ๐‘ 2๐œƒ๐œƒ = 1, we get: ๐‘ฅ๐‘ฅ ๐‘ฆ๐‘ฆ = ๐‘Ÿ๐‘Ÿ 0 i.e. ๐‘Ÿ๐‘Ÿ = ๐‘Ÿ๐‘Ÿ๐‘–๐‘– (where ๐‘–๐‘– is moving with angular velocity ฬ‡ ๐œƒ๐œƒ). This result is obvious because the x axis always points straight at the particle so the frame of reference xy (polar coordinates) is defined precisely to โ€˜trackโ€™ the particle.
  • 11. Advanced Kinematic Analysis The Velocity vector ๐‘‰๐‘‰ in the moving system The components of the velocity vector obtained by a 2D rotation matrix i.e.: ๐‘‰๐‘‰ = ฬ‡ ๐‘‹๐‘‹๐ผ๐ผ + ฬ‡ ๐‘Œ๐‘Œ๐ฝ๐ฝ = ฬ‡ ๐‘Ÿ๐‘Ÿ cos ๐œƒ๐œƒ โˆ’ ๐‘Ÿ๐‘Ÿ sin ๐œƒ๐œƒ ฬ‡ ๐œƒ๐œƒ ๐ผ๐ผ + ฬ‡ ๐‘Ÿ๐‘Ÿ sin ๐œƒ๐œƒ + ๐‘Ÿ๐‘Ÿ cos ๐œƒ๐œƒ ฬ‡ ๐œƒ๐œƒ ๐ฝ๐ฝ ฬ‡ ๐‘ฅ๐‘ฅ ฬ‡ ๐‘ฆ๐‘ฆ = cos ๐œƒ๐œƒ sin ๐œƒ๐œƒ โˆ’ sin ๐œƒ๐œƒ cos ๐œƒ๐œƒ ฬ‡ ๐‘‹๐‘‹ ฬ‡ ๐‘Œ๐‘Œ = ฬ‡ ๐‘Ÿ๐‘Ÿ ๐‘Ÿ๐‘Ÿ ฬ‡ ๐œƒ๐œƒ i.e. the velocity vector in the moving system is: ฬ‡ ๐‘Ÿ๐‘Ÿ = ฬ‡ ๐‘Ÿ๐‘Ÿ๐‘–๐‘– + ๐‘Ÿ๐‘Ÿ ฬ‡ ๐œƒ๐œƒ๐‘—๐‘— KINEMATICS OF A PARTICLE OBTAINED BY ROTATION OF AXES
  • 12. Advanced Kinematic Analysis KINEMATICS OF A PARTICLE OBTAINED BY ROTATION OF AXES Acceleration vector ๐’‚๐’‚ in the moving system The components of the acceleration vector also obtained by a 2D rotation matrix are: ๐‘Ž๐‘Ž = ฬˆ ๐‘‹๐‘‹๐ผ๐ผ + ฬˆ ๐‘Œ๐‘Œ๐ฝ๐ฝ = ฬˆ ๐‘Ÿ๐‘Ÿ cos ๐œƒ๐œƒ โˆ’ ฬ‡ ๐‘Ÿ๐‘Ÿ sin ๐œƒ๐œƒ ฬ‡ ๐œƒ๐œƒ โˆ’ ฬ‡ ๐‘Ÿ๐‘Ÿ sin ๐œƒ๐œƒ ฬ‡ ๐œƒ๐œƒ โˆ’ ๐‘Ÿ๐‘Ÿ cos ๐œƒ๐œƒ ฬ‡ ๐œƒ๐œƒ2 โˆ’ ๐‘Ÿ๐‘Ÿ sin ๐œƒ๐œƒ ฬˆ ๐œƒ๐œƒ ๐ผ๐ผ + ฬˆ ๐‘Ÿ๐‘Ÿ sin ๐œƒ๐œƒ + ฬ‡ ๐‘Ÿ๐‘Ÿ cos ๐œƒ๐œƒ ฬ‡ ๐œƒ๐œƒ + ฬ‡ ๐‘Ÿ๐‘Ÿ cos ๐œƒ๐œƒ ฬ‡ ๐œƒ๐œƒ + ๐‘Ÿ๐‘Ÿ cos ๐œƒ๐œƒ ฬˆ ๐œƒ๐œƒ โˆ’ ๐‘Ÿ๐‘Ÿ sin ๐œƒ๐œƒ ฬ‡ ๐œƒ๐œƒ2 ๐ฝ๐ฝ And in terms of the xy frame: ฬˆ ๐‘ฅ๐‘ฅ ฬˆ ๐‘ฆ๐‘ฆ = cos ๐œƒ๐œƒ sin ๐œƒ๐œƒ โˆ’ sin ๐œƒ๐œƒ cos ๐œƒ๐œƒ ฬˆ ๐‘‹๐‘‹ ฬˆ ๐‘Œ๐‘Œ And after some manipulation we get: = ฬˆ ๐‘Ÿ๐‘Ÿ โˆ’ ๐‘Ÿ๐‘Ÿ ฬ‡ ๐œƒ๐œƒ2 2 ฬ‡ ๐‘Ÿ๐‘Ÿ ฬ‡ ๐œƒ๐œƒ + ๐‘Ÿ๐‘Ÿ ฬˆ ๐œƒ๐œƒ i.e. the acceleration vector in the moving system is: ๐‘Ž๐‘Ž = ( ฬˆ ๐‘Ÿ๐‘Ÿ โˆ’ ๐‘Ÿ๐‘Ÿ ฬ‡ ๐œƒ๐œƒ2)๐‘–๐‘– + (2 ฬ‡ ๐‘Ÿ๐‘Ÿ ฬ‡ ๐œƒ๐œƒ + ๐‘Ÿ๐‘Ÿ ฬˆ ๐œƒ๐œƒ)๐‘—๐‘—
  • 13. Advanced Kinematic Analysis Physical Interpretation of Acceleration Terms The components of the acceleration vector are now shown in the figure below where the unit vectors ๐‘–๐‘– , ๐‘—๐‘— are moving. P X Y y x r=ri r ฬˆ ๐‘Ÿ๐‘Ÿ: is the radial acceleration. โˆ’๐‘Ÿ๐‘Ÿ ฬ‡ ๐œƒ๐œƒ2: is the centripetal acceleration. ๐‘Ÿ๐‘Ÿ ฬˆ ๐œƒ๐œƒ: is the tangential acceleration. 2 ฬ‡ ๐‘Ÿ๐‘Ÿ ฬ‡ ๐œƒ๐œƒ: is the coriolis component.
  • 14. Advanced Kinematic Analysis The Coriolis acceleration stems from the combined radial and angular motion. Imagine moving radially outwards on a spinning disc (e.g. a carousel or roundabout) with constant angular velocity ฯ‰. At radius r1, the tangential velocity is v1= ฯ‰r1. At radius r2, the tangential velocity is v2= ฯ‰r2. Since r2 > r1 the tangential velocity must increase, representing an acceleration component in the tangential direction. ฯ‰r1 ๐‘Ÿ๐‘Ÿ2 > ๐‘Ÿ๐‘Ÿ1 ฬ‡ ๐œƒ๐œƒ = ๐œ”๐œ” ฬ‡ ๐œ”๐œ” = 0 ฯ‰r2 r1 r2