kinematics rigid body 1

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Planar Kinematics of a Rigid Body

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Page 1: Kinematics Rigid Body 1

Planar Kinematics of a Rigid Body

Page 2: Kinematics Rigid Body 1

Planar Motion of Rigid Body

• Rigid Body

– Distance between any two given points is constant.

– No Deformation

• Planar Motion

– Distances of the paths are constant from a fixed plane.

03/07/53 ME212 ดร. พภัิทร 2

Page 3: Kinematics Rigid Body 1

Types of Planar Motion

• Translation

• Rotation about a fixed axis

• General Motion

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Page 4: Kinematics Rigid Body 1

Translation

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Position

ABAB /rrr

Page 5: Kinematics Rigid Body 1

Translation

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Velocity

AB vv 0B/Ardt

d

Acceleration

AB aa

Page 6: Kinematics Rigid Body 1

Rotation about a fixed axis

• Position is defined by θ.

• Displacement = ∆θ or dθ

• Angular Velocity

• Angular Acceleration

03/07/53 ME212 ดร. พภัิทร 6

dt

d

2

2

dt

d

dt

d dd

Page 7: Kinematics Rigid Body 1

Rotation about a fixed axis

• For a constant α=αc

03/07/53 ME212 ดร. พภัิทร 7

)(2

2

1

020

2

200

0

c

c

c

tt

t

( +)

( +)

( +)

Page 8: Kinematics Rigid Body 1

Motion of point P

03/07/53 ME212 ดร. พภัิทร 8

Position

r

Velocity

rv

rvr

Page 9: Kinematics Rigid Body 1

Motion of point P

03/07/53 ME212 ดร. พภัิทร 9

Acceleration

dt

rdr

dt

d

dt

vda

rrrra 2)(

na

ta

Special Case rr

,

nt urura ˆˆ 2

Page 10: Kinematics Rigid Body 1

HB 16-3

The angular velocity of the disk is defined by ω=(5t2+2) rad/s, where t is in seconds. Determine the magnitudes of the velocity and acceleration of point A on the disk when t = 0.5 s

03/07/53 ME212 ดร. พภัิทร 10

Page 11: Kinematics Rigid Body 1

HB 16-5

If gear A starts from rest and has a constant angular acceleration of αA = 2 rad/s2, determine the time needed for gear B to attain an angular velocity of ωB = 50 rad/s.

03/07/53 ME212 ดร. พภัิทร 11

Page 12: Kinematics Rigid Body 1

Absolute Motion Analysis

• Observed by a stationary observer.

• Analyzed in a fixed frame of reference.

• Translation and rotation motion.

03/07/53 ME212 ดร. พภัิทร 12

Page 13: Kinematics Rigid Body 1

Procedure for analysis

• หาจดุ P โดยใช้พิกดัต าแหน่ง S โดยวดัจากจดุก าเนิดถงึจดุ P ในแนวการเคลื่อนท่ีของ P

• วดัต าแหน่งเชิงมมุ θ จากแนวอ้างอิงถงึแนวในตวัของ

• หาความสมัพนัธ์ระหว่าง S และ θ โดยใช้ เรขาคณิตหรือตรีโกณมิติ

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Page 14: Kinematics Rigid Body 1

Absolute Motion Analysis

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cos222 abbas

cos2

cos2

2

122 abba

abv

Page 15: Kinematics Rigid Body 1

Example 1

The large window is opened using a hydraulic cylinder AB. If the cylinder extends at a constant rate of 0.5 m/s, determine the angular velocity and angular acceleration of the window at the instant θ = 30°

03/07/53 ME212 ดร. พภัิทร 15

Page 16: Kinematics Rigid Body 1

HB 16-36

Determine the angular velocity of rod AB when θ = 30⁰. The shaft and the center of the roller C move forward at a constant rate v = 5 m/s.

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Page 17: Kinematics Rigid Body 1

Relative Motion Analysis

• The general plane motion of a rigid body can be described as a combination of translation and rotation. Both are separately analyzed.

• The x, y coordinate system is fixed and measures the absolute position of two points A and B on the body.

• The origin of the x’, y’ coordinate system will be attached to the selected “base point” A, which generally has a known motion.

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Page 18: Kinematics Rigid Body 1

Relative Motion Analysis

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Position.

• The position vector rA specifies the location of the “base point” A, and the relative-position vector rB/A

locates point B with respect to point A.

ABAB /rrr

Page 19: Kinematics Rigid Body 1

Relative Motion Analysis

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Displacement.

ABAB ddd /rrr

due to rotation about A

due to translation about A

due to translation and rotation

Page 20: Kinematics Rigid Body 1

Relative Motion Analysis

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ABAB /vvv

ABAB /rvv

Page 21: Kinematics Rigid Body 1

Procedure for Analysis

• Establish the directions of the fixed x, ycoordinates and draw a kinematics diagram of the body. Indicate on it the velocities vA, vB of points A and B, the angular velocity ω, and the relative-position vector rB/A

03/07/53 ME212 ดร. พภัิทร 21

Page 22: Kinematics Rigid Body 1

Example 2

The link is guided by two block A and B, which move in the fixed slots. If the velocity of A is 2 m/s downward, determine the velocity of B at the instant θ = 45°.

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Page 23: Kinematics Rigid Body 1

HB 16-66

The planetary gear system is used in an automatic transmission for an automobile. By locking or releasing certain gears, it has the advantage of operating the car at different speeds. Consider the case where the ring gear R is rotating at ωR = 3 rad/s, and the sun gear S is held fixed, ωS = 0. Determine the angular velocity of each of the planet gear P and shaft A.

03/07/53 ME212 ดร. พภัิทร 23

Page 24: Kinematics Rigid Body 1

HB 16-70

At the instant shown, the truck is traveling to the right at 12 m/s, If the pipe does not slip at B, determine its angular velocity if its mass center G appears to an observer on the ground to be moving to the right at 3 m/s.

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Page 25: Kinematics Rigid Body 1

Instantaneous Center of Zero Velocity

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ABAB /vvv

ICBB r /v

0 ICA vv

Page 26: Kinematics Rigid Body 1

Location of IC

The velocity of a point on the body is always perpendicular to the relative position vector

extending from the IC to the point.

ความเร็วของจดุใดๆ บนตวัวตัถ ุตัง้ฉากกบั

เวกเตอร์สมัพทัธ์แสดงต าแหน่งระหว่าง IC และจดุนัน้เสมอ

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Page 27: Kinematics Rigid Body 1

Location of IC

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Page 28: Kinematics Rigid Body 1

Example 3

Show how to determine the location of the instantaneous center of zero velocity for (a) member BC shown in (a); and (b) the link CB shown in (b).

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Page 29: Kinematics Rigid Body 1

Example 3

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Page 30: Kinematics Rigid Body 1

Relative Motion Analysis (Cont’)

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ABABAB

nABtABAB

ABAB

dt

d

dt

d

dt

d

/

2

/

//

/

rωrαaa

aaaa

vvv

where aA denotes the acceleration of base point AaB denotes the acceleration of point Bα denotes the angular acceleration of the bodyω denotes the angular velocity of the body

(aB/A)t denotes the relative tangential acceleration of B with respect to A(aB/A)n denotes the relative normal acceleration of B with respect to ArAB denotes the relative position vector from A to B

Page 31: Kinematics Rigid Body 1

Procedure for Analysis

• Determine the angular velocity ω of the body by using a velocity analysis as discuss in previous section. Also determine the velocities vA and vB of points A and B if these points moves along curved paths.

• Establish the directions of the fixed x, y coordinates and draw the kinematics diagram of the body. Indicate on it aA, aB, ω, α, rB/A.

03/07/53 ME212 ดร. พภัิทร 31

Page 32: Kinematics Rigid Body 1

HB 16-110

Determine the angular acceleration of link AB at the instant θ=90⁰ if the collar C has a velocity of vc = 4 m/s and deceleration of ac = 3 m/s2

03/07/53 ME212 ดร. พภัิทร 32