Motion Relative to Rotating Axes

Lecture 14

ME 231: Dynamics

2

Question of the Day

ME 231: Dynamics

A wheel rotates 180º in the time it takes a ball dropped from the top of the wheel to reach the ground. What is the motion of the ball observed from a moving reference frame attached to the wheel?

3

Outline for Today

� Question of the day� Vector representation: rotating axes� Time derivatives of unit vectors� Relative velocity� Transformation of a time derivative� Answer your questions!

ME 231: Dynamics

� Question of the day

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Recall: Vector Representation

ME 231: Dynamics

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

� Attach a set of translating (non-rotating) axes x-y to particle B and define the position of A

� Define position of “A relative to B” (“A/B”) in x-y

?�Ar

?�Av

?�Aa

BABA /rrr ��

BABAA /rrrv ��� ���

BABAAA /rrrva ������� ����

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Vector Representation: Rotating Axes

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� ��

6

Outline for Today

� Question of the day� Vector representation: rotating axes� Time derivatives of unit vectors� Relative velocity� Transformation of a time derivative� Answer your questions!

ME 231: Dynamics

� Question of the day� Vector representation: rotating axes

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?�re�

?��e�

Recall: Time Derivativeof Unit Vectors

ME 231: Dynamics

rrdtd

dtd eeee ���

��� ��

�

� ����

�� �� eeee �� ��

� ���dtd

dtd r

r

r � r er

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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 Velocity: Exercise

Aircraft B has a constant speed of 480 mph along an arc with a radius of 9 miles. Aircraft A flies southwest at a constant speed of 360 mph.

Write the vector expression (x-y attached to B) for the velocity of A relative to B.

ME 231: Dynamics

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Transformation of a Time Derivative

ME 231: Dynamics

One of the most important concepts in dynamics!

i�i ��� j�j ���

� � � �jijiV ����yxyx

XY

VVVVdtd

�����

� �

V�VV��

�

� ��

�

� �

xyXY dtd

dtd

i

j

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v � ?

Recall: Time Derivative of a Vector

ME 231: Dynamics

Magnitude changes, but direction constant

r(t)

r(t +��t)

�r

i

irrv 2��� �dtd

ir 2)( tt �

t = 1 s t = 2 st = 0 s

v has

direction of ��rand magnitude |�r/�t|

�r � 4 � 2� � i � 2 i

�r�t

�4 � 2� � i2 �1� � �

2 i1

� 2

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Ý r XY � Ý r xy � � � r

�

Transformation of a Time Derivative: Exercise

ME 231: Dynamics

r(t) ��ri

� 2 i � 2 k � 2t i

t = 1 s t = 2 st = 0 s

x

y

X

Y

� 2 i � 4t j

�� � 2 kir 2)( tt �

k

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Outline for Today

� Question of the day� Vector representation: rotating axes� Time derivatives of unit vectors� Relative velocity� Transformation of a time derivative� Answer your questions!

ME 231: Dynamics

� Question of the day� Vector representation: rotating axes� Time derivatives of unit vectors� Relative velocity� Transformation of a time derivative

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For Next Time…

� Complete Homework #5 due on Wednesday (9/26) at the beginning of class

� Read Chapter 6, Sections 6.3 and 6.4

ME 231: Dynamics