me 208 dynamics · 2021. 7. 20. · đÔŠ =đÔŠ +đŒÔŠĂ ÔŠ âđ2 ÔŠ +2đĂđŁÔŠ +đÔŠ...
TRANSCRIPT
![Page 1: ME 208 DYNAMICS · 2021. 7. 20. · đÔŠ =đÔŠ +đŒÔŠĂ ÔŠ âđ2 ÔŠ +2đĂđŁÔŠ +đÔŠ 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 TĂNĂK
Department of Mechanical Engineering
Graduate Program of Biomedical Engineering
http://tonuk.me.metu.edu.tr
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Relative Velocity
ÔŠđđŽ = ÔŠđđ” + đ„ Æžđ + đŠ Æžđá¶ÔŠđđŽ = á¶ÔŠđđ” + á¶đ„ Æžđ + á¶đŠ Æžđ + đ„ á¶Æžđ + đŠ á¶Æžđ
ÔŠđŁđŽ = ÔŠđŁđ” + ÔŠđŁđđđ + đ„ á¶Æžđ + đŠ á¶Æžđ
á¶Æžđ = đ Ă Æžđá¶Æžđ = đ Ă Æžđ
ÔŠđŁđŽ = ÔŠđŁđ” + ÔŠđŁđđđ + đ„đ Ă Æžđ + đŠđ Ă Æžđ
ÔŠđŁđŽ = ÔŠđŁđ” + ÔŠđŁđđđ + đ Ă đ„ Æžđ + đŠ Æžđ
ÔŠđŁđŽ = ÔŠđŁđ” + ÔŠđŁđđđ + đ Ă ÔŠđđđđ
ÔŠđŁđŽ = ÔŠđŁđ” + đ Ă ÔŠđđđđ + ÔŠđŁđđđ
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ÔŠđŁđŽ = ÔŠđŁđ” +đ Ă ÔŠđđđđ + ÔŠđŁđđđInterpretation 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.
ÔŠđŁđŽ = ÔŠđŁđ” +đ Ă ÔŠđđđđ + ÔŠđŁđđđÔŠđŁđŽ = ÔŠđŁđ” + ÔŠđŁđ/đ” + ÔŠđŁđŽ/đ
ÔŠđŁđŽ = ÔŠđŁđ + ÔŠđŁđŽ/đ
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Transformation of a Time Derivative
(Transport or Coriolis Theorem)
Let đ be any vector quantity
đ = đđ„ Æžđ + đđŠ Æžđ
The time derivative of this quantity in X-Y coordinates is
đđ
đđĄ đâđ= á¶đđ„ Æžđ + á¶đđŠ Æžđ + đđ„ á¶Æžđ + đđŠ á¶Æžđ
đđ
đđĄ đâđ=
đđ
đđĄ đ„âđŠ+ đ Ă đ
The term đ Ă đ is the difference between the time
derivative of đ 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
ÔŠđŁđŽ = ÔŠđŁđ” + đ Ă ÔŠđđđđ + ÔŠđŁđđđor by applying Transport/Coriolis Theorem.đ
đđĄÔŠđŁđŽ =
đ
đđĄÔŠđŁđ” +
đ
đđĄđ Ă ÔŠđđđđ +
đ
đđĄÔŠđŁđđđ
ÔŠđđŽ = ÔŠđđ” +á¶đ Ă ÔŠđđđđ + đ Ă
đ
đđĄđ„ Æžđ + đŠ Æžđ +
đ
đđĄá¶đ„ Æžđ + á¶đŠ Æžđ
ÔŠđđŽ = ÔŠđđ” + ÔŠđŒ Ă ÔŠđđđđ +đ Ă á¶đ„ Æžđ + á¶đŠ Æžđ + đ„ á¶Æžđ + đŠ á¶Æžđ + á·đ„ Æžđ + á·đŠ Æžđ + á¶đ„ á¶Æžđ + á¶đŠ á¶Æžđ
ÔŠđđŽ = ÔŠđđ” + ÔŠđŒ Ă ÔŠđđđđ âđ2 ÔŠđđđđ + 2đ Ă ÔŠđŁđđđ + ÔŠđđđđ
á¶Æžđ = đ Ă Æžđ, á¶Æžđ = đ Ă Æžđ
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Relative AccelerationRelative acceleration equation may be obtained either
by taking time derivative of relative velocity equation
ÔŠđŁđŽ = ÔŠđŁđ” + đ Ă ÔŠđđđđ + ÔŠđŁđđđor by applying Transport/Coriolis Theorem.đ
đđĄÔŠđŁđŽ =
đ
đđĄÔŠđŁđ” +
đ
đđĄđ Ă ÔŠđđđđ +
đ
đđĄÔŠđŁđđđ
ÔŠđđŽ = ÔŠđđ” +á¶đ Ă ÔŠđđđđ + đ Ă á¶ÔŠđđđđ +
đ
đđĄÔŠđŁđđđ
ÔŠđđŽ = ÔŠđđ” + ÔŠđŒ Ă ÔŠđđđđ +đ Ă đ Ă ÔŠđđđđ + ÔŠđŁđđđ + đ Ă ÔŠđŁđđđ +á¶ÔŠđŁđđđ
ÔŠđđŽ = ÔŠđđ” + ÔŠđŒ Ă ÔŠđđđđ âđ2 ÔŠđđđđ + 2đ Ă ÔŠđŁđđđ + ÔŠđđđđ
đđ
đđĄ đâđ= đ Ă đ +
đđ
đđĄ đ„âđŠ
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ÔŠđđŽ = ÔŠđđ” + ÔŠđŒ Ă ÔŠđđđđ âđ2 ÔŠđđđđ + 2đ Ă ÔŠđŁđđđ + ÔŠđđđđInterpretation 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.ÔŠđđŽ = ÔŠđđ” + ÔŠđŒ Ă ÔŠđđđđ âđ2 ÔŠđđđđ + 2đ Ă ÔŠđŁđđđ + ÔŠđđđđ
ÔŠđđŽ = ÔŠđđ” + ÔŠđđ/đ” + ÔŠđđŽ/đ
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ÔŠđđŽ = ÔŠđđ” + ÔŠđŒ Ă ÔŠđđđđ âđ2 ÔŠđđđđ + 2đ Ă ÔŠđŁđđđ + ÔŠđđđđInterpretation 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.ÔŠđđŽ = ÔŠđđ” + ÔŠđŒ Ă ÔŠđđđđ âđ2 ÔŠđđđđ + 2đ Ă ÔŠđŁđđđ + ÔŠđđđđ
ÔŠđđŽ = ÔŠđđ” + ÔŠđđ/đ” + ÔŠđđŽ/đ
ÔŠđđŽ = ÔŠđđ + ÔŠđđŽ/đ
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ÔŠđđŽ = ÔŠđđ” + ÔŠđŒ Ă ÔŠđđđđ âđ2 ÔŠđđđđ + 2đ Ă ÔŠđŁđđđ + ÔŠđđđđInterpretation 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.ÔŠđđŽ = ÔŠđđ” + ÔŠđŒ Ă ÔŠđđđđ âđ2 ÔŠđđđđ + 2đ Ă ÔŠđŁđđđ + ÔŠđđđđ
ÔŠđđŽ = ÔŠđđ” + ÔŠđđ/đ” + ÔŠđđŽ/đ
ÔŠđđŽ = ÔŠđđ + ÔŠđđŽ/đ
ÔŠđđŽ = ÔŠđđ” + ÔŠđđŽ/đ”
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ÔŠđđŽ = ÔŠđđ” + ÔŠđŒ Ă ÔŠđđđđ âđ2 ÔŠđđđđ + đđ Ă đđđđ + ÔŠđđđđCoriolis Acceleration
Non-rotating frame: đ = 0
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ÔŠđđŽ = ÔŠđđ” + ÔŠđŒ Ă ÔŠđđđđ âđ2 ÔŠđđđđ + đđ Ă đđđđ + ÔŠđđđđCoriolis Acceleration
Rotating frame: đ â 0
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ÔŠđđŽ = ÔŠđđ” + ÔŠđŒ Ă ÔŠđđđđ âđ2 ÔŠđđđđ + đđ Ă đđđđ + ÔŠđđđđCoriolis Acceleration
Rotating frame: đ â 0
https://www.youtube.com/watch?v=mPsLanVS1Q8
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ÔŠđđŽ = ÔŠđđ” + ÔŠđŒ Ă ÔŠđđđđ âđ2 ÔŠđđđđ + đđ Ă đđđđ + ÔŠđđđđCoriolis Acceleration
Rotating frame: đ â 0
https://www.youtube.com/watch?v=Wda7azMvabE
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ÔŠđđŽ = ÔŠđđ” + ÔŠđŒ Ă ÔŠđđđđ âđ2 ÔŠđđđđ + đđ Ă đđđđ + ÔŠđđđđCoriolis Acceleration
Rotating frame: đ â 0
https://www.youtube.com/watch?v=Wda7azMvabE
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ÔŠđŁđŽ = ÔŠđŁđ” +đ Ă ÔŠđđđđ + ÔŠđŁđđđInterpretation 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.
ÔŠđŁđŽ = ÔŠđŁđ” +đ Ă ÔŠđđđđ + ÔŠđŁđđđÔŠđŁđŽ = ÔŠđŁđ” + ÔŠđŁđ/đ” + ÔŠđŁđŽ/đ
ÔŠđŁđŽ = ÔŠđŁđ + ÔŠđŁđŽ/đ
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ÔŠđđŽ = ÔŠđđ” + ÔŠđŒ Ă ÔŠđđđđ âđ2 ÔŠđđđđ + 2đ Ă ÔŠđŁđđđ + ÔŠđđđđInterpretation 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.ÔŠđđŽ = ÔŠđđ” + ÔŠđŒ Ă ÔŠđđđđ âđ2 ÔŠđđđđ + 2đ Ă ÔŠđŁđđđ + ÔŠđđđđ
ÔŠđđŽ = ÔŠđđ” + ÔŠđđ/đ” + ÔŠđđŽ/đ
ÔŠđđŽ = ÔŠđđ + ÔŠđđŽ/đ
ÔŠđđŽ = ÔŠđđ” + ÔŠđđŽ/đ”
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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¶đ =
125 mm/s and á·đ = 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.
ÔŠđŁđŽ = ÔŠđŁđ + ÔŠđŁđŽ/đSince point P fixed on the rotating disk it makes fixed axis
rotation about O,
ÔŠđŁđ = đ Ă ÔŠđđ = 4đ Ă 0.15 Æžđ = 0.600 Æžđ đ/đ
ÔŠđŁđŽ/đ = á¶ÔŠđ = 0.125 Æžđ đ/đ
ÔŠđŁđŽ = 0.6 Æžđ + 0.125 Æžđ = 0.1250 Æžđ + 0.600 Æžđ đ/đ ÔŠđđŽ = ÔŠđđ + 2đ Ă ÔŠđŁđđđ + ÔŠđđđđÔŠđđ = ÔŠđŒ Ă ÔŠđđđđ â đ2 ÔŠđđđđ = â10đ Ă 0.15 Æžđ â 420.15 ÆžđÔŠđđ = 2.40 Æžđ â 1.500 Æžđ đ/đ 2
ÔŠđđŽ = 2.40 Æžđ â 1.500 Æžđ + 2 â 4đ Ă 0.125 Æžđ + 2.025 ÆžđÔŠđđŽ = â0.375 Æžđ â 0.500 Æžđ đ/đ 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¶đŠ = â600 mm/s and á·đŠ = 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.
ÔŠđŁđŽ = ÔŠđŁđ + ÔŠđŁđŽ/đSince point P on the disk makes fixed axis rotation about O,
ÔŠđŁđ = đ Ă ÔŠđđ = 5đ Ă â0.15 Æžđ + 0.25 Æžđ = â1.25 Æžđ â 0.75 Æžđ đ/đ ÔŠđŁđŽ/đ = á¶đŠ Æžđ = â0.6 Æžđ đ/đ
ÔŠđŁđŽ = â1.25 Æžđ â 0.75 Æžđ â 0.6 Æžđ = â1.250 Æžđ â 1.350 Æžđ đ/đ ÔŠđđŽ = ÔŠđđ + 2đ Ă ÔŠđŁđđđ + ÔŠđđđđÔŠđđ = ÔŠđŒ Ă ÔŠđđđđ â đ2 ÔŠđđđđÔŠđđ = â3đ Ă â0.15 Æžđ + 0.25 Æžđ â 52 â0.15 Æžđ + 0.25 ÆžđÔŠđđ = 4.50 Æžđ â 5.80 Æžđ đ/đ 2
ÔŠđđŽ = 4.5 Æžđ â 5.8 Æžđ + 2 â 5đ Ă â0.6 Æžđ + 0.75 ÆžđÔŠđđŽ = â10.50 Æžđ â 5.05 Æžđ đ/đ 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.
đŁđŽ = 72 đđ/â ⥠20 đ/đ đŁđ” = 54 đđ/â ⥠15 đ/đ When motion is observed through B using x-y coordinates it is rotating
about the center of curvature of the road therefore
đ =đŁđ”đ=
15
100= 0.1500 đđđ/đ (đ¶đ¶đ)
ÔŠđŁđŽ = ÔŠđŁđ” + đ Ă ÔŠđđđđ + ÔŠđŁđđđÔŠđđđđ = ÔŠđđŽ/đ” = â40 Æžđ đ
20 Æžđ = 15 Æžđ + 0.15đ Ă â40 Æžđ + ÔŠđŁđđđÔŠđŁđđđ = 20.0 Æžđ â 9.00 Æžđ đ/đ Also consider car A at the center of curvature where ÔŠđđđđ = ÔŠđđŽ/đ” =
â 100 Æžđ đ and about to hit car B where ÔŠđđđđ = ÔŠđđŽ/đ” = 0
For center of curvature:
20 Æžđ = 15 Æžđ + 0.15đ Ă â100 Æžđ + ÔŠđŁđđđÔŠđŁđđđ = 20.0 Æžđ â 0.00 Æžđ đ/đ At the instant of hit:
20 Æžđ = 15 Æžđ + 0.15đ Ă 0 + ÔŠđŁđđđÔŠđŁđđđ = 20.0 Æžđ â 15.00 Æžđ đ/đ
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ÔŠđđŽ = ÔŠđđ” + ÔŠđŒ Ă ÔŠđđđđ âđ2 ÔŠđđđđ + 2đ Ă ÔŠđŁđđđ + ÔŠđđđđÔŠđđŽ = 0ÔŠđđ” = đ2 ÔŠđ = â0.152 â 100 Æžđ = â2.25 Æžđ đ/đ 2
ÔŠđŒ = 0, đ = đđđđ đĄ0 = â2.25 Æžđ + 0 â 0.152 â â40 Æžđ + 2 â 0.15đ Ă 20.0 Æžđ â 9.00 Æžđ + ÔŠđđđđÔŠđđđđ = 1.350 Æžđ â 6.00 Æžđ đ/đ 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).
![Page 25: ME 208 DYNAMICS · 2021. 7. 20. · đÔŠ =đÔŠ +đŒÔŠĂ ÔŠ âđ2 ÔŠ +2đĂđŁÔŠ +đÔŠ InterpretationofTerms Assume the moving coordinate system x-y be formed of a plate](https://reader036.vdocument.in/reader036/viewer/2022081623/6146f8c3f4263007b13584aa/html5/thumbnails/25.jpg)
One may assume a point P fixed on the disk and instantly coincident with A
ÔŠđđŽ = ÔŠđđ + 2đ Ă ÔŠđŁđđđ + ÔŠđđđđÔŠđđđđ = 0.15đĄđđđ Æžđ + 0.15 Æžđ = 0.0866 Æžđ + 0.15 Æžđ đÔŠđđ = ÔŠđŒđđđ đ Ă ÔŠđđđđ â đđđđ đ
2 ÔŠđđđđÔŠđđ = 0 Ă 0.0866 Æžđ + 0.15 Æžđ â 32 0.0866 Æžđ + 0.15 ÆžđÔŠđđ = â0.779 Æžđ â 1.350 Æžđ đ/đ 2
ÔŠđŁđđđ =á¶ÔŠđđđđ = á¶đ„ Æžđ =
đ
đđĄ0.15đĄđđđ Æžđ + 0.15 Æžđ
ÔŠđŁđđđ = 0.15 á¶đđ đđ2đ Æžđ = 0.15 â 2 â đ đđ230° Æžđ = 0.400 Æžđ đ/đ
ÔŠđđđđ =á¶ÔŠđŁđđđ = á·đ„ Æžđ =
đ
đđĄ0.15 á¶đđ đđ2đ Æžđ = 2 â 0.15 á¶đđ đđ2đđĄđđđ = 0.923 Æžđ đ/đ for á·đ = 0
ÔŠđđŽ = â0.779 Æžđ â 1.350 Æžđ + 2 â2đ Ă 0.4 Æžđ + 0.923 Æžđ = 0.1230 Æžđ â 1.050 Æžđ đ/đ 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.
![Page 27: ME 208 DYNAMICS · 2021. 7. 20. · đÔŠ =đÔŠ +đŒÔŠĂ ÔŠ âđ2 ÔŠ +2đĂđŁÔŠ +đÔŠ InterpretationofTerms Assume the moving coordinate system x-y be formed of a plate](https://reader036.vdocument.in/reader036/viewer/2022081623/6146f8c3f4263007b13584aa/html5/thumbnails/27.jpg)
Assume a point P fixed on body ODE instantly
coincident with A.
ÔŠđŁđ = ÔŠđŁđŽ + ÔŠđŁđ/đŽÔŠđŁđ = đđđ·đž Ă ÔŠđđ/đ = đđđ·đž
đ Ă 0.12 Æžđ = 0.12đđđ·đž Æžđ
ÔŠđŁđŽ = đđ Ă ÔŠđđŽ/đ¶ = 4đ Ă â0.12 Æžđ = 0.48 Æžđ đ/đ
ÔŠđŁđ/đŽ = đŁđ/đŽ đđđ 45° Æžđ + đ đđ45° Æžđ
0.12đđđ·đž Æžđ = 0.48 Æžđ + đŁđ/đŽ đđđ 45° Æžđ + đ đđ45° Æžđ
Æžđ: 0 = 0.48 + đŁđ/đŽđđđ 45°, đŁđ/đŽ = â0.679 đ/đ
Æžđ: 0.12đđđ·đž = đŁđ/đŽđ đđ45°, đđđ·đž = â4 đđđ/đ
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Assume a point P fixed on body ODE instantly
coincident with A.ÔŠđđ = ÔŠđđŽ + ÔŠđđ/đŽÔŠđđ = ÔŠđđđĄ + ÔŠđđđÔŠđđ = ÔŠđŒđđ·đž Ă ÔŠđđ/đ â đđđ·đž
2 ÔŠđđ/đÔŠđđ = đŒđđ·đž đ Ă 0.12 Æžđ â 42 â 0.12 ÆžđÔŠđđ = â1.920 Æžđ + 0.12đŒđđ·đž Æžđ đ/đ 2
ÔŠđđŽ = ÔŠđđŽđĄ + ÔŠđđŽđ = ÔŠđŒđ¶đŽ Ă ÔŠđđŽ/đ¶ âđ2 ÔŠđđŽ/đ¶
ÔŠđđŽ = 0 Ă â0.12 Æžđ â 42 â â0.12 Æžđ = 1.920 Æžđ đ/đ 2
ÔŠđđ/đŽ = 2đđđ·đž Ă ÔŠđŁđ/đŽ + ÔŠđđđđÔŠđđ/đŽ = 2 â â4đ Ă â0.679 đđđ 45° Æžđ + đ đđ45° Æžđ + đđđđ đđđ 45° Æžđ + đ đđ45° Æžđ
ÔŠđđ/đŽ = 3.84 â Æžđ + Æžđ + đđđđ đđđ 45° Æžđ + đ đđ45° Æžđâ1.92 Æžđ + 0.12đŒđđ·đž Æžđ = 1.92 Æžđ + 3.84 â Æžđ + Æžđ + đđđđ đđđ 45° Æžđ + đ đđ45° Æžđ
Æžđ: â1.92 = â3.84 + đđđđđđđ 45°đđđđ = 2.72 đ/đ 2
Æžđ: 0.12đŒđđ·đž = 1.92 + 3.84 + 2.72đ đđ45°đŒđđ·đž = 64.0 đđđ/đ 2