![Page 1: ME 208 DYNAMICS · 2021. 7. 20. · ðÔŠ =ðÔŠ +ðŒÔŠÃ ÔŠ âð2 ÔŠ +2ðÃð£ÔŠ +ðÔŠ InterpretationofTerms Assume the moving coordinate system x-y be formed of a plate](https://reader036.vdocument.in/reader036/viewer/2022081623/6146f8c3f4263007b13584aa/html5/thumbnails/1.jpg)
ME 208 DYNAMICS
Dr. Ergin TÃNÃK
Department of Mechanical Engineering
Graduate Program of Biomedical Engineering
http://tonuk.me.metu.edu.tr
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Relative Velocity
ÔŠððŽ = ÔŠððµ + ð¥ Æžð + ðŠ Æžðá¶ÔŠððŽ = á¶ÔŠððµ + á¶ð¥ Æžð + á¶ðŠ Æžð + ð¥ á¶Æžð + ðŠ á¶Æžð
ÔŠð£ðŽ = ÔŠð£ðµ + ÔŠð£ððð + ð¥ á¶Æžð + ðŠ á¶Æžð
á¶Æžð = ð à ƞðá¶Æžð = ð à ƞð
ÔŠð£ðŽ = ÔŠð£ðµ + ÔŠð£ððð + ð¥ð à ƞð + ðŠð à ƞð
ÔŠð£ðŽ = ÔŠð£ðµ + ÔŠð£ððð + ð à ð¥ Æžð + ðŠ Æžð
ÔŠð£ðŽ = ÔŠð£ðµ + ÔŠð£ððð + ð à Ԋðððð
ÔŠð£ðŽ = ÔŠð£ðµ + ð à Ԋðððð + ÔŠð£ððð
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ÔŠð£ðŽ = ÔŠð£ðµ +ð à Ԋðððð + ÔŠð£ðððInterpretation of Terms
Assume the moving coordinate system x-y be formed
of a plate with a slot in it where particle A can move
in relative to the moving coordinate system.
Assume a point P, instantly coincident with particle A
fixed on the moving plane.
ÔŠð£ðŽ = ÔŠð£ðµ +ð à Ԋðððð + ÔŠð£ðððÔŠð£ðŽ = ÔŠð£ðµ + ÔŠð£ð/ðµ + ÔŠð£ðŽ/ð
ÔŠð£ðŽ = ÔŠð£ð + ÔŠð£ðŽ/ð
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Transformation of a Time Derivative
(Transport or Coriolis Theorem)
Let ð be any vector quantity
ð = ðð¥ Æžð + ððŠ Æžð
The time derivative of this quantity in X-Y coordinates is
ðð
ðð¡ ðâð= á¶ðð¥ Æžð + á¶ððŠ Æžð + ðð¥ á¶Æžð + ððŠ á¶Æžð
ðð
ðð¡ ðâð=
ðð
ðð¡ ð¥âðŠ+ ð à ð
The term ð Ã ð is the difference between the time
derivative of ð in the rotating (x-y) and non-rotating
(X-Y) coordinate frames.
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Relative Acceleration
Relative acceleration equation may be obtained either
by taking time derivative of relative velocity equation
ÔŠð£ðŽ = ÔŠð£ðµ + ð à Ԋðððð + ÔŠð£ðððor by applying Transport/Coriolis Theorem.ð
ðð¡ÔŠð£ðŽ =
ð
ðð¡ÔŠð£ðµ +
ð
ðð¡ð Ã ÔŠðððð +
ð
ðð¡ÔŠð£ððð
ÔŠððŽ = ÔŠððµ +á¶ð à Ԋðððð + ð Ã
ð
ðð¡ð¥ Æžð + ðŠ Æžð +
ð
ðð¡á¶ð¥ Æžð + á¶ðŠ Æžð
ÔŠððŽ = ÔŠððµ + ÔŠðŒ à Ԋðððð +ð à á¶ð¥ Æžð + á¶ðŠ Æžð + ð¥ á¶Æžð + ðŠ á¶Æžð + á·ð¥ Æžð + á·ðŠ Æžð + á¶ð¥ á¶Æžð + á¶ðŠ á¶Æžð
ÔŠððŽ = ÔŠððµ + ÔŠðŒ à Ԋðððð âð2 ÔŠðððð + 2ð à Ԋð£ððð + ÔŠðððð
á¶Æžð = ð à ƞð, á¶Æžð = ð à ƞð
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Relative AccelerationRelative acceleration equation may be obtained either
by taking time derivative of relative velocity equation
ÔŠð£ðŽ = ÔŠð£ðµ + ð à Ԋðððð + ÔŠð£ðððor by applying Transport/Coriolis Theorem.ð
ðð¡ÔŠð£ðŽ =
ð
ðð¡ÔŠð£ðµ +
ð
ðð¡ð Ã ÔŠðððð +
ð
ðð¡ÔŠð£ððð
ÔŠððŽ = ÔŠððµ +á¶ð à Ԋðððð + ð à á¶ÔŠðððð +
ð
ðð¡ÔŠð£ððð
ÔŠððŽ = ÔŠððµ + ÔŠðŒ à Ԋðððð +ð à ð à Ԋðððð + ÔŠð£ððð + ð à Ԋð£ððð +á¶ÔŠð£ððð
ÔŠððŽ = ÔŠððµ + ÔŠðŒ à Ԋðððð âð2 ÔŠðððð + 2ð à Ԋð£ððð + ÔŠðððð
ðð
ðð¡ ðâð= ð Ã ð +
ðð
ðð¡ ð¥âðŠ
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ÔŠððŽ = ÔŠððµ + ÔŠðŒ à Ԋðððð âð2 ÔŠðððð + 2ð à Ԋð£ððð + ÔŠððððInterpretation of Terms
Assume the moving coordinate system x-y be formed
of a plate with a slot in it where particle A can move
in relative to the moving coordinate system.
Assume a point P, instantly coincident with particle A
is fixed on the moving plane.ÔŠððŽ = ÔŠððµ + ÔŠðŒ à Ԋðððð âð2 ÔŠðððð + 2ð à Ԋð£ððð + ÔŠðððð
ÔŠððŽ = ÔŠððµ + ÔŠðð/ðµ + ÔŠððŽ/ð
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ÔŠððŽ = ÔŠððµ + ÔŠðŒ à Ԋðððð âð2 ÔŠðððð + 2ð à Ԋð£ððð + ÔŠððððInterpretation of Terms
Assume the moving coordinate system x-y be formed
of a plate with a slot in it where particle A can move
in relative to the moving coordinate system.
Assume a point P, instantly coincident with particle A
is fixed on the moving plane.ÔŠððŽ = ÔŠððµ + ÔŠðŒ à Ԋðððð âð2 ÔŠðððð + 2ð à Ԋð£ððð + ÔŠðððð
ÔŠððŽ = ÔŠððµ + ÔŠðð/ðµ + ÔŠððŽ/ð
ÔŠððŽ = ÔŠðð + ÔŠððŽ/ð
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ÔŠððŽ = ÔŠððµ + ÔŠðŒ à Ԋðððð âð2 ÔŠðððð + 2ð à Ԋð£ððð + ÔŠððððInterpretation of Terms
Assume the moving coordinate system x-y be formed
of a plate with a slot in it where particle A can move
in relative to the moving coordinate system.
Assume a point P, instantly coincident with particle A
is fixed on the moving plane.ÔŠððŽ = ÔŠððµ + ÔŠðŒ à Ԋðððð âð2 ÔŠðððð + 2ð à Ԋð£ððð + ÔŠðððð
ÔŠððŽ = ÔŠððµ + ÔŠðð/ðµ + ÔŠððŽ/ð
ÔŠððŽ = ÔŠðð + ÔŠððŽ/ð
ÔŠððŽ = ÔŠððµ + ÔŠððŽ/ðµ
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ÔŠððŽ = ÔŠððµ + ÔŠðŒ à Ԋðððð âð2 ÔŠðððð + ðð à ðððð + ÔŠððððCoriolis Acceleration
Non-rotating frame: ð = 0
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ÔŠððŽ = ÔŠððµ + ÔŠðŒ à Ԋðððð âð2 ÔŠðððð + ðð à ðððð + ÔŠððððCoriolis Acceleration
Rotating frame: ð â 0
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ÔŠððŽ = ÔŠððµ + ÔŠðŒ à Ԋðððð âð2 ÔŠðððð + ðð à ðððð + ÔŠððððCoriolis Acceleration
Rotating frame: ð â 0
https://www.youtube.com/watch?v=mPsLanVS1Q8
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ÔŠððŽ = ÔŠððµ + ÔŠðŒ à Ԋðððð âð2 ÔŠðððð + ðð à ðððð + ÔŠððððCoriolis Acceleration
Rotating frame: ð â 0
https://www.youtube.com/watch?v=Wda7azMvabE
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ÔŠððŽ = ÔŠððµ + ÔŠðŒ à Ԋðððð âð2 ÔŠðððð + ðð à ðððð + ÔŠððððCoriolis Acceleration
Rotating frame: ð â 0
https://www.youtube.com/watch?v=Wda7azMvabE
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ÔŠð£ðŽ = ÔŠð£ðµ +ð à Ԋðððð + ÔŠð£ðððInterpretation of Terms
Assume the moving coordinate system x-y be formed
of a plate with a slot in it where particle A can move
in relative to the moving coordinate system.
Assume a point P, instantly coincident with particle A
fixed on the moving plane.
ÔŠð£ðŽ = ÔŠð£ðµ +ð à Ԋðððð + ÔŠð£ðððÔŠð£ðŽ = ÔŠð£ðµ + ÔŠð£ð/ðµ + ÔŠð£ðŽ/ð
ÔŠð£ðŽ = ÔŠð£ð + ÔŠð£ðŽ/ð
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ÔŠððŽ = ÔŠððµ + ÔŠðŒ à Ԋðððð âð2 ÔŠðððð + 2ð à Ԋð£ððð + ÔŠððððInterpretation of Terms
Assume the moving coordinate system x-y be formed
of a plate with a slot in it where particle A can move
in relative to the moving coordinate system.
Assume a point P, instantly coincident with particle A
is fixed on the moving plane.ÔŠððŽ = ÔŠððµ + ÔŠðŒ à Ԋðððð âð2 ÔŠðððð + 2ð à Ԋð£ððð + ÔŠðððð
ÔŠððŽ = ÔŠððµ + ÔŠðð/ðµ + ÔŠððŽ/ð
ÔŠððŽ = ÔŠðð + ÔŠððŽ/ð
ÔŠððŽ = ÔŠððµ + ÔŠððŽ/ðµ
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Sample Problem 5/16
At the instant represented, the disk with radial slot is
rotating about O with a counterclockwise angular
velocity of 4 rad/s which is decreasing at the rate of 10
rad/s2. The motion of slider A is separately controlled,
and at this instant, r is 150 mm, increasing with á¶ð =
125 mm/s and á·ð = 2025 mm/s2. Determine the
absolute velocity and acceleration of A for this position.
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In the solution presented by the textbook the formula is
directly applied. However one may think point P fixed on the
disk and instantly coincident with the slider A and obtain
absolute velocity and acceleration of point A.
ÔŠð£ðŽ = ÔŠð£ð + ÔŠð£ðŽ/ðSince point P fixed on the rotating disk it makes fixed axis
rotation about O,
ÔŠð£ð = ð à Ԋðð = 4ð à 0.15 Æžð = 0.600 Æžð ð/ð
ÔŠð£ðŽ/ð = á¶ÔŠð = 0.125 Æžð ð/ð
ÔŠð£ðŽ = 0.6 Æžð + 0.125 Æžð = 0.1250 Æžð + 0.600 Æžð ð/ð ÔŠððŽ = ÔŠðð + 2ð à Ԋð£ððð + ÔŠððððÔŠðð = ÔŠðŒ à Ԋðððð â ð2 ÔŠðððð = â10ð à 0.15 Æžð â 420.15 ÆžðÔŠðð = 2.40 Æžð â 1.500 Æžð ð/ð 2
ÔŠððŽ = 2.40 Æžð â 1.500 Æžð + 2 â 4ð à 0.125 Æžð + 2.025 ÆžðÔŠððŽ = â0.375 Æžð â 0.500 Æžð ð/ð 2
x
y
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5/150 (4th), 5/151 (5th), None (6th), 5/159 (7th), None (8th)
The disk rotates about a fixed axis through O with angular
velocity = 5 rad/s and angular acceleration = 3 rad/s2
at the instant represented, in the directions shown. The
slider A moves in the straight slot. Determine the absolute
velocity and acceleration of A for the same instant, when y
= 250 mm, á¶ðŠ = â600 mm/s and á·ðŠ = 750 mm/s2.
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Again one may assume point P fixed on the disk, instantly
coincident with A (or may apply the formula directly).
Velocity analysis is required first since acceleration
equations contain velocities too.
ÔŠð£ðŽ = ÔŠð£ð + ÔŠð£ðŽ/ðSince point P on the disk makes fixed axis rotation about O,
ÔŠð£ð = ð à Ԋðð = 5ð à â0.15 Æžð + 0.25 Æžð = â1.25 Æžð â 0.75 Æžð ð/ð ÔŠð£ðŽ/ð = á¶ðŠ Æžð = â0.6 Æžð ð/ð
ÔŠð£ðŽ = â1.25 Æžð â 0.75 Æžð â 0.6 Æžð = â1.250 Æžð â 1.350 Æžð ð/ð ÔŠððŽ = ÔŠðð + 2ð à Ԋð£ððð + ÔŠððððÔŠðð = ÔŠðŒ à Ԋðððð â ð2 ÔŠððððÔŠðð = â3ð à â0.15 Æžð + 0.25 Æžð â 52 â0.15 Æžð + 0.25 ÆžðÔŠðð = 4.50 Æžð â 5.80 Æžð ð/ð 2
ÔŠððŽ = 4.5 Æžð â 5.8 Æžð + 2 â 5ð à â0.6 Æžð + 0.75 ÆžðÔŠððŽ = â10.50 Æžð â 5.05 Æžð ð/ð 2
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5/155 (4th), 5/156 (5th), 5/165 (6th), None (7th), 5/163 (8th)
Car B is rounding the curve with a constant speed of 54
km/h, and car A is approaching car B in the intersection
with a constant speed of 72 km/h. Determine the velocity
which car A appears to have to an observer riding and
turning with car B. The x-y axes are attached to car B. Is
this apparent velocity the negative of velocity that B
appears to have to a nonrotating observer in car A? The
distance separating two cars at the instant depicted is 40
m.
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Here use of a point P fixed in the moving coordinate system and instantly
coincident with A is hard to visualize and direct application of formula is
straight forward.
ð£ðŽ = 72 ðð/â â¡ 20 ð/ð ð£ðµ = 54 ðð/â â¡ 15 ð/ð When motion is observed through B using x-y coordinates it is rotating
about the center of curvature of the road therefore
ð =ð£ðµð=
15
100= 0.1500 ððð/ð (ð¶ð¶ð)
ÔŠð£ðŽ = ÔŠð£ðµ + ð à Ԋðððð + ÔŠð£ðððÔŠðððð = ÔŠððŽ/ðµ = â40 Æžð ð
20 Æžð = 15 Æžð + 0.15ð à â40 Æžð + ÔŠð£ðððÔŠð£ððð = 20.0 Æžð â 9.00 Æžð ð/ð Also consider car A at the center of curvature where ÔŠðððð = ÔŠððŽ/ðµ =
â 100 Æžð ð and about to hit car B where ÔŠðððð = ÔŠððŽ/ðµ = 0
For center of curvature:
20 Æžð = 15 Æžð + 0.15ð à â100 Æžð + ÔŠð£ðððÔŠð£ððð = 20.0 Æžð â 0.00 Æžð ð/ð At the instant of hit:
20 Æžð = 15 Æžð + 0.15ð à 0 + ÔŠð£ðððÔŠð£ððð = 20.0 Æžð â 15.00 Æžð ð/ð
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ÔŠððŽ = ÔŠððµ + ÔŠðŒ à Ԋðððð âð2 ÔŠðððð + 2ð à Ԋð£ððð + ÔŠððððÔŠððŽ = 0ÔŠððµ = ð2 ÔŠð = â0.152 â 100 Æžð = â2.25 Æžð ð/ð 2
ÔŠðŒ = 0, ð = ðððð ð¡0 = â2.25 Æžð + 0 â 0.152 â â40 Æžð + 2 â 0.15ð à 20.0 Æžð â 9.00 Æžð + ÔŠððððÔŠðððð = 1.350 Æžð â 6.00 Æžð ð/ð 2
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5/162 (4th), None (5th), None (6th), None (7th), None (8th)
The slotted disk sector rotates with a constant
counterclockwise angular velocity = 3 rad/s.
Simultaneously the slotted arm OC oscillates about line OB
(fixed to the disk) so that changes at a constant rate of 2
rad/s except at the extremities of the oscillation during
reversal of direction. Determine the total acceleration of the
pin when = 30° and its first rate is positive (clockwise).
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One may assume a point P fixed on the disk and instantly coincident with A
ÔŠððŽ = ÔŠðð + 2ð à Ԋð£ððð + ÔŠððððÔŠðððð = 0.15ð¡ððð Æžð + 0.15 Æžð = 0.0866 Æžð + 0.15 Æžð ðÔŠðð = ÔŠðŒððð ð à Ԋðððð â ðððð ð
2 ÔŠððððÔŠðð = 0 Ã 0.0866 Æžð + 0.15 Æžð â 32 0.0866 Æžð + 0.15 ÆžðÔŠðð = â0.779 Æžð â 1.350 Æžð ð/ð 2
ÔŠð£ððð =á¶ÔŠðððð = á¶ð¥ Æžð =
ð
ðð¡0.15ð¡ððð Æžð + 0.15 Æžð
ÔŠð£ððð = 0.15 á¶ðð ðð2ð Æžð = 0.15 â 2 â ð ðð230° Æžð = 0.400 Æžð ð/ð
ÔŠðððð =á¶ÔŠð£ððð = á·ð¥ Æžð =
ð
ðð¡0.15 á¶ðð ðð2ð Æžð = 2 â 0.15 á¶ðð ðð2ðð¡ððð = 0.923 Æžð ð/ð for á·ð = 0
ÔŠððŽ = â0.779 Æžð â 1.350 Æžð + 2 â2ð à 0.4 Æžð + 0.923 Æžð = 0.1230 Æžð â 1.050 Æžð ð/ð 2
Alternatively you could directly apply the formula with the same terms.
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5/163 (4th), 5/168 (5th), 5/178 (6th), 5/176 (7th), 5/179 (8th)
For the instant represented, link CB is rotating
counterclockwise at a constant rate N = 4 rad/s, and its pin
A causes a clockwise rotation of the slotted member ODE.
Determine the angular velocity and angular acceleration
of ODE for this instant.
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Assume a point P fixed on body ODE instantly
coincident with A.
ÔŠð£ð = ÔŠð£ðŽ + ÔŠð£ð/ðŽÔŠð£ð = ððð·ðž à Ԋðð/ð = ððð·ðž
ð Ã 0.12 Æžð = 0.12ððð·ðž Æžð
ÔŠð£ðŽ = ðð à ԊððŽ/ð¶ = 4ð à â0.12 Æžð = 0.48 Æžð ð/ð
ÔŠð£ð/ðŽ = ð£ð/ðŽ ððð 45° Æžð + ð ðð45° Æžð
0.12ððð·ðž Æžð = 0.48 Æžð + ð£ð/ðŽ ððð 45° Æžð + ð ðð45° Æžð
Æžð: 0 = 0.48 + ð£ð/ðŽððð 45°, ð£ð/ðŽ = â0.679 ð/ð
Æžð: 0.12ððð·ðž = ð£ð/ðŽð ðð45°, ððð·ðž = â4 ððð/ð
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Assume a point P fixed on body ODE instantly
coincident with A.ÔŠðð = ÔŠððŽ + ÔŠðð/ðŽÔŠðð = ÔŠððð¡ + ÔŠðððÔŠðð = ÔŠðŒðð·ðž à Ԋðð/ð â ððð·ðž
2 ÔŠðð/ðÔŠðð = ðŒðð·ðž ð à 0.12 Æžð â 42 â 0.12 ÆžðÔŠðð = â1.920 Æžð + 0.12ðŒðð·ðž Æžð ð/ð 2
ÔŠððŽ = ÔŠððŽð¡ + ÔŠððŽð = ÔŠðŒð¶ðŽ à ԊððŽ/ð¶ âð2 ÔŠððŽ/ð¶
ÔŠððŽ = 0 à â0.12 Æžð â 42 â â0.12 Æžð = 1.920 Æžð ð/ð 2
ÔŠðð/ðŽ = 2ððð·ðž à Ԋð£ð/ðŽ + ÔŠððððÔŠðð/ðŽ = 2 â â4ð à â0.679 ððð 45° Æžð + ð ðð45° Æžð + ðððð ððð 45° Æžð + ð ðð45° Æžð
ÔŠðð/ðŽ = 3.84 â Æžð + Æžð + ðððð ððð 45° Æžð + ð ðð45° Æžðâ1.92 Æžð + 0.12ðŒðð·ðž Æžð = 1.92 Æžð + 3.84 â Æžð + Æžð + ðððð ððð 45° Æžð + ð ðð45° Æžð
Æžð: â1.92 = â3.84 + ððððððð 45°ðððð = 2.72 ð/ð 2
Æžð: 0.12ðŒðð·ðž = 1.92 + 3.84 + 2.72ð ðð45°ðŒðð·ðž = 64.0 ððð/ð 2