Lecture 39

ME 231: Dynamics

Final Review

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-Energy Equation: Usage

ME 231: Dynamics

VTU ����21TU ��21

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)

Final Exam General Breakdown

ME 231: Dynamics

For Next Time…

� Review Chapters 1, 2, 3, 4, 5, 6, 7, & 8� Review Lectures slides

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

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

� Final Exam on Monday (12/10) @ 8:00am here (in Min Kao 524)

ME 231: Dynamics