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Questions of the Day
What is the most important concept in Chapter 2?
ME 231: Dynamics
Time Derivative of a Vector
What is the most important concept in Chapter 6? Transformation of a Time Derivative
What is the most important concept in mechanics?Free Body Diagram
What is the most important concept in dynamics? Equations of Motion
Outline for Today� Questions of the day� Concept and calculation of dynamics?� Velocity and acceleration of a point� Absolute- and relative-motion analysis� Locating the instantaneous center� Velocity and acceleration of a body� Inverse vs. forward dynamics� Kinetics: cause of motion� Direct application of Newton’s 2nd Law� Impulse-momentum principles� Equations, equations, equations…� Principle of work and kinetic energy� Work vs. potential energy and
work-energy equation� Work of forces and couples & energy of
rigid bodies� Answer your questions!
ME 231: Dynamics
� Questions of the day
Kinematics(Exam 1)
Kinetics(Exam 2)
Work-Energy(new stuff)
Kinematics Kinetics Dynamics
Concept: What is dynamics?
ME 231: Dynamics
Chapters 1, 2, 6 Chapters 3, 4, 5, 7, 8
Relationship among position, velocity, and acceleration
Relationship among forces and acceleration
F = m a
Calculation: How Do We Use Dynamics?
ME 231: Dynamics
Acceleration. Velocity rate of change with
respect to time
Newton’s 2nd Law
Force. A push or pull exerted on a body, characterized by:�magnitude�direction�point of application
Mass. Measure of the resistance of a body to linear acceleration.
Lecture Velocity Acceleration
2. Rectilinear
3. Curvilinear
4. Normal & Tangential
5. Polar Coordinates
7. Relative Motion
Velocity and Acceleration of a Point
ME 231: Dynamics
BABAA /rrrv ��� ��� BABAAA /rrrva ������� ����
�� eev r �� rr �� � � � � ���� eea r 2 2 ������� rrrr ����
tn eea 2
vv���
�v � v et
jirv yx ��� ��� jirva yx ������� ����
2
2
0lim
dtsd
dtdv
tva
t��
��dt
dstsv
t�
�� 0
lim
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
==
Relative-Motion Analysis: Acceleration
ME 231: Dynamics
aA � aB � aA / B� �t � aA / B� �n
Translational portion
Rotationalportion
Ba
BaBA /a
Aa
� �nBA /a� �tBA /a
� � � �r��a ���nBA /
� � r�a ��tBA /
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
���
�
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
Lecture Velocity Acceleration
9. Rotation
10. Absolute Motion
11. Relative Velocity
13. Relative Acceleration
14. Rotating Axes
Velocity and Acceleration of a Body
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 �
Outline for Today� Questions of the day� Concept and calculation of dynamics?� Velocity and acceleration of a point� Absolute- and relative-motion analysis� Locating the instantaneous center� Velocity and acceleration of a body� Inverse vs. forward dynamics� Kinetics: cause of motion� Direct application of Newton’s 2nd Law� Impulse-momentum principles� Equations, equations, equations…� Principle of work and kinetic energy� Work vs. potential energy and
work-energy equation� Work of forces and couples & energy of
rigid bodies� Answer your questions!
ME 231: Dynamics
Kinematics(Exam 1)
Kinetics(Exam 2)
Work-Energy(new stuff)
� Questions of the day� Concept and calculation of dynamics?� Velocity and acceleration of a point� Absolute- and relative-motion analysis� Locating the instantaneous center� Velocity and acceleration of a body
Kinematics(Exam 1)
Inverse vs. Forward Dynamics
ME 231: Dynamics
aF m�inverse
dtd
dtd
Positions
Velocities
Forces
aF m�
ME 231: Dyna
forward
� � Positions
Velocities
Forces
Statics Kinematics Kinetics
Kinetics: Cause of Motion?
ME 231: Dynamics
Chapters 1, 2, 6 Chapters 3, 4, 5, 7, 8
Relationship among position, velocity, and acceleration
Relationship among forces(and moments)andacceleration
Concept: What is kinetics?
ME 202
Relationship among forces (and moments)and equilibrium
Direct Application of Newton’s 2nd Law
ME 231: Dynamics
rF ��m�� aF m��or
iim rfF �������iim rfffFFF ���� �������� 321321
xx amF ��
yy amF ��
zz amF ��
� Integrate to describe the effect of the resultant force ��F on linear momentum over a finite period of time
ME 231: Dynamics
dtdtt
t
t
t 2
1
2
1�� �� GF �
21 2
1
GFG ��� � dtt
t
GF ��� � � � �xt
t xx vmdtFvm 21 2
1
��� �� � � �y
t
t yy vmdtFvm 21 2
1
��� �� � � �z
t
t zz vmdtFvm 21 2
1
��� �impulse-momentum diagram
Impulse-Momentum Principle: Linear vG m�
Impulse-Momentum Principle: Angular
� Integrate to describe the effect of the angular impulse ��MO*t on angular momentum HOof m about O over a finite period of time
ME 231: Dynamics
dtdtt
t O
t
t O 2
1
2
1�� �� HM �
� � � �21 2
1O
t
t OO dt HMH ��� �
� � 21 )( )( 2
1
zvyvmdtzvyvm yz
t
t xOyz ����� � M
� � 21 )( )( 2
1
xvzvmdtxvzvm zx
t
t yOzx ����� � M
� � 21 )( )( 2
1
yvxvmdtyvxvm xy
t
t zOxy ����� � M
vrH mO �� OO HM ���
Lecture Equations
18. Newton 2nd Law19. Eqs. of Motion20. Rectilinear
21. Curvilinear
27. Lin. Imp. Mom.
28. Ang. Imp. Mom.
29. Sys. Imp. Mom.
Equations, Equations, Equations…
ME 231: Dynamics
aF m��xmmaF xx �����
ymmaF yy �����zmmaF zz �����
rr maF ��
�� maF ��nn maF ��
tt maF ��
GF ���vG m� 21 2
1
GFG ��� � dtt
t0G �
vrH mO ��
OO HM ���� � � �21 2
1O
t
t OO dt HMH ��� �0H � O
vG m�� �iiiO m vrH ���
v�HH mGP ���� �iiiG m ��H ����
GG HM ��� a�HM mGP ���� �
Lecture Equations
18. Newton 2nd Law22. Gen. Eqs. Mot. I23. Gen. Eqs. Mot. II
24. Fixed-Axis Rot.
25. Gen. Plane Mot. I
26. Gen. Plane Mot. II
31. Body Imp. Mom.
Equations, Equations, Equations…
ME 231: Dynamics
GG HM ���iiG F�H ����
mad�IM GP ���PPP mI a��M ����
2mrII GO ��aF m�� �M GG I�� �M OO I��
aF m�� �M GG I��mad�IM GP ���PPP mI a��M ����
vG m�
GF ���
21 2
1
GFG ��� � dtt
t
GG HM ���
� � � �21 2
1G
t
t GG dt HMH ��� �
�H GG I� mvd�IH GP ��
PP HM ��� OO HM ���
�IH OO �
2OO mkI �
Outline for Today� Questions of the day� Concept and calculation of dynamics?� Velocity and acceleration of a point� Absolute- and relative-motion analysis� Locating the instantaneous center� Velocity and acceleration of a body� Inverse vs. forward dynamics� Kinetics: cause of motion� Direct application of Newton’s 2nd Law� Impulse-momentum principles� Equations, equations, equations…� Principle of work and kinetic energy� Work vs. potential energy and
work-energy equation� Work of forces and couples & energy of
rigid bodies� Answer your questions!
ME 231: Dynamics
Kinematics(Exam 1)
Kinetics(Exam 2)
Work-Energy(new stuff)
� Questions of the day� Concept and calculation of dynamics?� Velocity and acceleration of a point� Absolute- and relative-motion analysis� Locating the instantaneous center� Velocity and acceleration of a body� I f d d iInverse vs. forward dynamics� Kinetics: cause of motion� Direct application of Newton’s 2nd Law� Impulse-momentum principles� Equations, equations, equations…
Kinematics(Exam 1)
Kinetics(Exam 2)
Principle of Work and Kinetic Energy
� The kinetic energy T of a particle is
� Work done to bring it a particle from velocity v1to a velocity v2
ME 231: Dynamics
T �12
mv 2
U1�2 � T2 �T1 � T
U1�2 �12
m v22 � v1
2� �
T1 �U1�2 � T2
(work-energy eq.)
Work vs. Potential Energy
� Recall: Work is done by the weight mg over an altitude change (y2-y1)
� Potential energy is simply the opposite sign (-work) because of its potential to be converted into energy
ME 231: Dynamics
U � �mg dyy1
y2� � �mg y2 � y1� �
Work-Energy Equation
� The work of all external forces other than gravitational and spring forces is
ME 231: Dynamics
VTU ����21
222111 VTUVT ����� �
(work-energy eq.)
21��U
Work of Forces and Couples
ME 231: Dynamics
dU � F� dr
dU � Ft dsdU � F ds cos
dU � M d�
dU � F b d�� �
Kinetic Energy for Rigid Bodies
ME 231: Dynamics
translation fixed-axis rotation general plane motion
T �12
mv 2
T � �12
miv2
T �12
IO�2
T � �12
mi ri�� �2
T �12
mvG2 �
12
IG�2
T � �12
mi vG2 � �i
2� 2� �
Work-Energy Equation for Rigid Bodies
ME 231: Dynamics
Express weight and springs as doing work
� Work done to bring a rigid body from kinetic energy T1 to a kinetic energy T2
� The work of all external forces other thangravitational and spring forces is
T1 �U1�2 � T2
T1 �V1 � �U 1�2 � T2 �V2
Express weight and springs by means of potential energy
Outline for Today� Questions of the day� Concept and calculation of dynamics?� Velocity and acceleration of a point� Absolute- and relative-motion analysis� Locating the instantaneous center� Velocity and acceleration of a body� Inverse vs. forward dynamics� Kinetics: cause of motion� Direct application of Newton’s 2nd Law� Impulse-momentum principles� Equations, equations, equations…� Principle of work and kinetic energy� Work vs. potential energy and
work-energy equation� Work of forces and couples & energy of
rigid bodies� Answer your questions!
ME 231: Dynamics
Kinematics(Exam 1)
Kinetics(Exam 2)
Work-Energy(new stuff)
� Questions of the day� Concept and calculation of dynamics?� Velocity and acceleration of a point� Absolute- and relative-motion analysis� Locating the instantaneous center� Velocity and acceleration of a body� I f d d iInverse vs. forward dynamics� Kinetics: cause of motion� Direct application of Newton’s 2nd Law� Impulse-momentum principles� Equations, equations, equations…� Principle of work and kinetic energy� Work vs. potential energy and
work-energy equation� Work of forces and couples & energy of
rigid bodiesA ti !
Kinematics(Exam 1)
Kinetics(Exam 2)
Work-Energy(new stuff)