ME 231: Dynamics

Kinematics of Rigid Bodies (Ch. 6) ReviewLecture 17

2

Question of the Day

What is the most important concept in Chapter 6?

ME 231: Dynamics

Transformation of a Time Derivative

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Outline for Today

� Question of the day� Absolute-motion analysis� Relative-motion analysis� Locating the instantaneous center� Rotating coordinate systems� Velocity and acceleration� Exam 1 breakdown (kinematics of rigid

bodies)

ME 231: Dynamics

� Question of the day

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Absolute-Motion Analysis

� The method relates the position of a point, A, on a rigid body to the angular position, ��, of a linecontained in the body

� The velocity and acceleration of point Aare obtained in terms of the angular velocity, �, and angular acceleration, �, of the rigid body

ME 231: Dynamics

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Relative-Motion Analysis: Velocity

ME 231: Dynamics

BABA /vvv ��

Translational portion

Rotationalportion

r�v ��BA /Bv

Bv

BA /v

Av

90º

6

==

Relative-Motion Analysis: Acceleration

ME 231: Dynamics

aA � aB � aA /B t � aA /B n

Translational portion

Rotationalportion

Ba

BaBA /aAa

nBA /a tBA /a

r��a ���nBA /

r�a ��tBA /

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Solution of Relative-Acceleration Eq.: Exercise

Calculate the angular acceleration of the plate, where AO has a constant angular velocity ��OA = 4 rad/s and � = 60º for both links.

ME 231: Dynamics

8

Outline for Today

� Question of the day� Absolute-motion analysis� Relative-motion analysis� Locating the instantaneous center� Rotating coordinate systems� Velocity and acceleration� Exam 1 breakdown (kinematics of rigid

bodies)

ME 231: Dynamics

� Question of the day� Absolute-motion analysis� Relative-motion analysis

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Locating the Instantaneous Center: Case #1

ME 231: Dynamics

� Directions of absolute velocities for A and B are known (and not parallel)

� Point A has circular motion about some point on the lineperpendicular to velocity vA

� Point B has a similar motion

� Point C is the instantaneous center of zero velocity (may lie on or off the body)

B

A

Bv

Av

C

Ar

Br

B

B

A

A

rv

rv

���

�

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Locating the Instantaneous Center: Case #2

ME 231: Dynamics

� Directions of absolute velocities for A and B are known AND parallel

� The line joining the points is perpendicular to velocity vAand vB

� Instantaneous center found by direct proportions

B

A

Bv

Av

C

A

d

B

A

Bv

Av

C

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Outline for Today

� Question of the day� Absolute-motion analysis� Relative-motion analysis� Locating the instantaneous center� Rotating coordinate systems� Velocity and acceleration� Exam 1 breakdown (kinematics of rigid

bodies)

ME 231: Dynamics

� Question of the day� Absolute-motion analysis� Relative-motion analysis� Locating the instantaneous center

12

Rotating Coordinate System

ME 231: Dynamics

� Absolute position of B is defined in an inertial coordinate system X-Y

� Moving reference frame x-y has its origin at B and rotates with angular velocity ��

� Define “A relative to B” using unit vectors in x-y

BABA /rrr ��

jirr yxBA ���

i

j

k� ��

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Time Derivatives of Unit Vectors

ME 231: Dynamics

i

j

ME 231 D i

d��d��

Infinitesimal change during dt

dtd �� �

jirr yxBA ���

k� ��

ji ��� ij ����

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Relative Velocity

ME 231: Dynamics

i

j

ji ��� ij ����

i�i ��� j�j ���

jirr yxBA ���

k� �� jirr yx

dtd

BA ��� ��

relBA vr�vv ����

jijirr yxyxBA ������ ����� jij�i�rr yxyxBA ���� �������

jiji�rr yxyxBA ���� ������

jjjj

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Relative Acceleration

ME 231: Dynamics

i

j

jirr yxBA ���

k� ��relBA vr�vv ����

relBA vr�r�aa ��� ������

i�i ���

j�j ���

relvr�r ����

relrelBA vvr��r�aa �� ��������

relrelBA vv�r��r�aa �� ���������

relrelBA av�r��r�aa ��������� 2�relrelrel av�v ����

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Relative Acceleration: Exercise

A disk with the radial slot is rotating about Owith constant �� = 4 rad/s. The slider A is positioned at x = 0.25 m and moves with constant speed vrel = 0.5 m/s relative to the slot.

ME 231: Dynamics

Determine the absolute acceleration of A for this position.

relrelOA av�r��r�aa ��������� 2�

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Outline for Today

� Question of the day� Absolute-motion analysis� Relative-motion analysis� Locating the instantaneous center� Rotating coordinate systems� Velocity and acceleration� Exam 1 breakdown (kinematics of rigid

bodies)

ME 231: Dynamics

� Question of the day� Absolute-motion analysis� Relative-motion analysis� Locating the instantaneous center� Rotating coordinate systems

18

Lecture Velocity Acceleration

9. Rotation

10. Absolute Motion

11. Relative Velocity

13. Relative Acceleration

14. Rotating Axes

Velocity and Acceleration

ME 231: Dynamics

�� �� ��� ��� ���rv � 2�ran ��rat �

relBA vr�vv ����

r�v �� r��a ���nr�a ��t�� �� ��� ��� ���rv � 2�ran ��rat �r�v �� r��a ���nr�a ��t

vA � vB � vA /B

vA /B � �� � r

aA � aB � aA /B t � aA /B n r��a ���nBA / r�a ��tBA /

relrel

BA

av�r��r�aa

���������

2 �

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Outline for Today

� Question of the day� Absolute-motion analysis� Relative-motion analysis� Locating the instantaneous center� Rotating coordinate systems� Velocity and acceleration� Exam 1 breakdown (kinematics of rigid

bodies)

ME 231: Dynamics

� Question of the day� Absolute-motion analysis� Relative-motion analysis� Locating the instantaneous center� Rotating coordinate systems� Velocity and acceleration

20

Exam 1 Breakdown (kinematics of rigid bodies)

ME 231: Dynamics

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For Next Time…

� Complete Homework #6 due on Tuesday (10/2)

� Review Lectures slides– http://connect.mcgraw-hill.com/class/me231

� Review Chapters 2 & 6� Review Lectures slides

– http://rrg.utk.edu/resources/ME231/lectures.html

� Review Examples from class– http://rrg.utk.edu/resources/ME231/examples.html

� Exam #1 on Wednesday (10/3)ME 231: Dynamics