ME 4024:Machine Dynamics
Course Notebook
Instructor:
Jeremy S.Daily,Ph.D.,P.E.
Spring 2013
Contents
I Lecture
21
1 Dynamics ProblemSolving
11
1.1 Dynamics Vocabulary
.................................11
1.2 ProblemSolving
...................................12
1.3 Vector Operations
...................................13
1.3.1 Cartesian Coordinates
............................13
1.3.2 Dot Product
..................................13
1.3.3 Cross Product
.................................14
1.3.4 Product Rule
.................................14
1.3.5 Partial and Total Derivatives
.........................14
1.3.6 Time Derivative of a Vector
.........................15
1.4 Kinematic Relationships for Rectilinear Motion
...................16
1.4.1 Velocity,distance,and time
.........................16
1.4.2 Acceleration,velocity and time
.......................16
1.4.3 Displacement,velocity and acceleration
...................16
1.4.4 Special Cases for Constant Acceleration
...................16
1.5 Newton's Laws of Motion
..............................16
1.6 Inertial Reference Frames
...............................17
1.7 Force,Mass and Weight
................................18
1.8 Example
........................................19
1.9 Tangent and Normal Coordinates
...........................111
1.10 Radial and Transverse Coordinates
..........................112
1.11 Homework ProblemSet 1
...............................113
1.12 Momentum
......................................116
1.12.1 Concept of Impulse
..............................118
3
Contents
1.12.2 SystemMomenta
...............................119
1.12.3 Moment of Inertia
..............................119
1.13 Collisions in 2D
...................................123
1.13.1 Center of Percussion
.............................123
1.13.2 Collision Reconstruction Example Using The Conservation of Linear Mo
mentum
....................................124
1.13.3 Collision Reconstruction Example Using Angular Momentum
.......129
1.14 Homework ProblemSet 2
...............................134
1.15 Work and Energy
...................................136
1.15.1 Kinetic Energy
................................136
1.15.2 Potential Energy
...............................137
1.15.3 Work
.....................................137
1.15.3.1 Conservative Forces
........................137
1.15.3.2 NonConservative Forces
.....................137
1.15.3.3 Workless Forces
.........................137
1.15.4 The WorkEnergy Theorem
.........................138
1.15.5 The Conservation of Mechanical Energy
...................138
1.16 Homework ProblemSet 3
...............................141
1.17 Principle of Virtual Work
...............................143
1.18 Lagrange Equations
..................................144
1.19 Simulation
.......................................145
1.19.1 StateSpace Form
...............................146
1.19.2 Euler's Method
................................147
1.19.3 RungeKutta
.................................148
1.20 Homework ProblemSet 4
...............................149
2 Vibration
21
2.1 Free Vibration
.....................................21
2.1.1 Undamped Systems
..............................21
2.1.2 Systems with Damping
............................22
2.2 Homework ProblemSet 5
...............................23
2.3 Forced Vibration
...................................26
2.3.1 Harmonic Forcing
..............................26
2.3.2 Forcing by Rotating Unbalance
.......................27
2.3.3 Base Excitation
................................28
4
Contents
2.3.4 Frequency Response Functions
........................29
2.4 Homework ProblemSet 6
...............................210
2.5 Critical Shaft Speed
..................................211
2.5.1 Single Degree of FreedomModel
......................211
2.5.2 Rayleigh's Method
..............................212
2.6 Homework ProblemSet 7
...............................213
3 Dynamic Force Analysis
31
3.1 D'Alembert's Principle
................................31
3.2.1 Example
...................................32
3.3 Dynamics of a 4Bar Mechanism
...........................37
3.3.1 Velocity Analysis (ME3212)
.........................38
3.3.2 Free Body Diagrams
.............................39
3.3.3 Equations of Motion
.............................311
3.3.4 Solution Technique
..............................312
3.4 Homework ProblemSet 8
...............................315
3.5 Dynamics of an Inverted SliderCrank Mechanism
..................317
3.5.1 Kinematic Analysis
..............................317
4 Balancing
41
4.1 Balancing Rotating Masses
..............................41
4.1.1 Graphical Solution
..............................42
4.1.2 Analytical Solution
..............................43
4.1.3 Static Balance
.................................44
4.2 Dynamic Balancing
..................................47
4.3 Homework ProblemSet 9
...............................48
4.4 Field Balancing
....................................49
4.5 Balancing Reciprocating Masses
...........................410
4.5.1 Single Cylinder Engines
...........................410
4.5.2 Multi Cylinder Inline Engines
........................411
4.6 Homework ProblemSet 10
..............................412
5
Contents
II Reference Material
415
5 Standards for Measurement
51
5.1 Measurement
.....................................51
5.2 Physical Quantities and Units of Measure
......................51
5.2.1 Fundamental Units
..............................53
5.2.1.1 Standard of Length
........................53
5.2.1.2 Standard of Mass
.........................54
5.2.1.3 Standard of Time
.........................55
5.2.1.4 Standard of Electric Current
....................57
5.2.1.5 Standard of Temperature
.....................57
5.2.1.6 Standard of Amount of Substance
.................58
5.2.1.7 Standard of Intensity of Light
...................58
5.2.2 Derived Units
.................................58
5.2.2.1 Velocity and Speed
........................58
5.2.2.2 Acceleration
............................59
5.2.2.3 Force
................................59
5.2.2.4 Angles
...............................511
5.2.2.5 Pressure
..............................512
5.2.2.6 Density
...............................513
5.2.2.7 Work and Energy
.........................513
5.2.2.8 Torque and Moments
.......................514
5.2.2.9 Power
...............................515
5.2.3 Factors and Ratios
..............................516
5.2.4 Dimensional Analysis
............................516
5.3 Accuracy and Precision
................................518
5.4 Numerical Precision
..................................520
5.4.1 Signicant Figures
..............................521
5.4.2 Rounding
...................................522
5.4.3 Truncating
..................................523
5.5 Quantifying Uncertainty
...............................524
6 Vectors
61
6.1 Vector and Scalar Quantities
.............................61
6.2 Vector Basics and Drawing Vectors
..........................63
6
Contents
6.3 Length of a Vector
..................................65
6.4 Vector Addition
....................................67
6.4.1 Vector Addition Example
...........................68
6.5 Vector Subtraction
..................................610
6.6 Vector Multiplication
.................................610
6.6.1 VectorScalar Multiplication
.........................611
6.6.2 Dot Product
..................................611
6.6.3 Cross Product
.................................613
6.7 Converting Polar Coordinates to Rectangular Coordinates
..............616
6.8 Resolving a Vector into Components
.........................617
6.8.1 Example
...................................618
7 Motion In One Dimension
71
7.1 The Study of Motion:Kinematics
..........................71
7.2 Rectilinear Motion
..................................73
7.2.1 Constant Velocity
...............................74
7.2.2 Constant Acceleration
............................77
7.3 Uniformly Accelerated Motion
............................79
7.3.1 Relationship Between Acceleration and Velocity
..............79
7.3.2 Relationship between Acceleration,Time,and Displacement
........710
7.3.3 Relationship between Acceleration,Velocity,and Displacement
......712
7.4 Variable Acceleration
.................................715
7.4.1 Kinematic Relations Using Calculus Concepts
...............716
7.4.2 Dealing with Variable Acceleration without Calculus
............724
7.5 Relative Motion
....................................728
7.5.1 Example
...................................729
8 Motion in More than One Dimension
81
8.1 Degrees of Freedom
..................................81
8.2 Motion in Space
....................................81
8.3 Displacement
.....................................83
8.4 Speed and Velocity
..................................84
8.4.1 Average Velocity
...............................84
8.4.2 Instantaneous Velocity
............................86
7
Contents
8.5 Acceleration
......................................87
8.5.1 Average Acceleration
.............................87
8.5.2 Instantaneous Acceleration
..........................88
8.6 UniformProjectile Motion
..............................88
9 Friction and Acceleration Factors
91
9.1 Coefcient of Friction
.................................91
9.2 Acceleration (Drag) Factor
..............................92
9.3 Determining Drag Factors
..............................94
9.4 Effects of Uneven Braking on Drag Factor
......................98
9.4.1 Denition of Percentage of Braking
.....................98
9.4.2 Application of Percentage of Braking
....................99
9.5 Gathering Road Friction Data
.............................910
9.5.1 Tables
.....................................910
9.5.2 Drag Sleds
..................................910
9.5.3 Tests with Skidding Vehicles
.........................912
9.5.3.1 Measured Test Skids
........................912
9.5.3.2 Test Skids with Shot Markers (Bumper Guns)
..........914
9.5.3.3 Test Skids with Recording Radar
.................915
9.5.3.4 Test Skids with Accelerometers
..................916
9.6 Determining Drag Factors fromTest Data
......................919
9.7 The Friction Circle and Lateral Friction
.......................920
10 Dynamics and Newton's Laws of Motion
101
10.1 Newton's First Law
..................................101
10.2 Newton's Second Law
................................102
10.3 The Concepts of Mass and Weight
..........................103
10.4 Newton's Third Law
.................................104
10.5 The Concept of Friction
................................104
10.6 Free Body Diagrams
.................................106
10.6.1 Free Body DiagramExample
.........................106
10.7 The Concept of Torque
................................107
10.7.1 Torque Example
...............................109
10.8 The Concept of Equilibrium
.............................1010
10.8.1 EquillibriumExample
............................1011
8
Contents
10.9 Center of Mass
....................................1016
10.9.1 Determining Center of Mass
.........................1017
10.9.2 Center of Mass Height
............................1018
10.9.3 Center of Mass with Cargo
..........................1023
10.9.4 Center of Mass with Cargo Example
.....................1024
10.10Dynamic Weight Shift
................................1026
11 Linear Momentum
111
11.1 Linear Momentumand Impulse
............................111
11.2 Conservation of Linear Momentum
..........................118
11.3 InLine Momentum
..................................1110
11.4 Elastic and Inelastic Collisions
............................1115
11.5 The Presence of External Forces
...........................1121
12 Collision Analysis Using Conservation of Linear Momentum
121
12.1 Introduction
......................................121
12.2 Collision Types and Congurations
..........................122
12.2.1 Collinear,Central Collisions
.........................122
12.2.2 Collinear,NonCentral Collisions
......................122
12.2.3 TwoDimensional,Central Collisions
....................123
12.2.4 TwoDimensional,NonCentral Collisions
..................123
12.3 Collision Analysis Examples in One Dimension
...................123
12.3.1 A Moving Vehicle into a Stopped Vehicle
..................123
12.3.2 Vehicles Colliding in the Same Direction
..................126
12.3.3 Vehicles Colliding in Opposite Directions
..................129
12.4 Collision Analysis in Two Dimensions
........................1210
12.4.1 Graphical Analysis
..............................1211
12.4.1.1 Coordinate System
.........................1212
12.4.1.2 Vector Addition
..........................1212
12.4.1.3 Collision Analysis Using Vector Diagrams
............1216
12.4.1.4 Constructing a Vector Diagram
..................1221
12.4.2 Mathematical Analysis
............................1226
12.4.2.1 Derivation of Basic Equations
...................1226
12.4.2.2 Derivation of the
v and PDOF Equations
............1228
9
Contents
12.5 Evidence Required for COLMAnalysis
.......................1234
12.5.1 Determining Pre and PostImpact Directions Using the Impact Circle
...1234
12.5.2 Determining Vehicle Weights
........................1237
12.5.3 Determining PostImpact Speeds
.......................1237
12.6 Special Considerations and Limitations
.......................1238
12.6.1 Multiple Departure Analysis
.........................1238
12.6.2 Multiple Collisions
..............................1239
12.6.3 Low Speeds
..................................1241
12.6.4 Small Approach Angles
...........................1241
12.6.5 Large Weight Differences
..........................1242
12.6.6 Large MomentumRatios
...........................1245
12.7 Summary
.......................................1246
13 Work and Energy
131
13.1 Work
..........................................131
13.2 Mechanical Energy
..................................136
13.2.1 Kinetic Energy
................................136
13.2.2 Potential Energy
...............................138
13.3 Conservation of Energy
................................1310
13.3.1 Systems
....................................1310
13.3.2 Conservative Forces
.............................1311
13.3.3 NonConservative Forces
...........................1312
13.4 WorkEnergy Theorem
................................1314
13.5 Derivation of the Kinetic Energy Formula
......................1315
13.6 Power
.........................................1317
14 Rotational Mechanics
141
14.1 UniformCircular Motion
...............................141
14.2 Lateral Acceleration
..................................143
14.3 Rotational Motion
...................................147
14.3.1 Angular Displacement
............................148
14.3.2 Angular Velocity
...............................148
14.3.3 Angular Acceleration
.............................148
14.4 Mass Moment of Inertia
................................149
14.4.1 The Parallel Axis Theorem
..........................1412
10
Contents
14.4.2 Radius of Gyration
..............................1414
14.5 Newton's Second Law for Rotation
..........................1414
14.6 Changing Torque and Gear Ratios
..........................1415
14.7 Rotational Kinetic Energy
..............................1419
14.8 Angular Momentum
..................................1421
14.9 Eccentric Collision Analysis Using Rotational Mechanics
..............1431
14.9.1 Derivation of v fromRotational Mechanics Concepts
...........1432
14.9.2 The WorkEnergy Theoremfor Rotation
..................1434
14.9.3 Computing Impact Speed from v
......................1437
III Laboratory Exercises
1447
15 Laboratory Policies
151
15.1 Safety
.........................................151
15.2 Laboratory Schedule
.................................151
16 Motor Control and Speed Sensing
161
16.1 Introduction
......................................161
16.2 Assignment
......................................161
16.3 Proceedure
......................................162
16.4 Reporting Requirements
...............................168
17 Determining Inertial Properties
171
17.1 ProblemStatement
..................................171
17.2 ProblemSolving Strategy
...............................171
17.3 Reporting Requirements
...............................172
18 Jumping Impulse
181
18.1 Objective
.......................................181
18.1.1 Calibration Using Multiple Regression
...................181
18.1.2 Measure Force and Determine Impulse
...................181
18.2 Theory
.........................................181
18.2.1 Determining Jump Height Based on Impulse
................181
18.2.2 Jump Height fromHang Time
........................184
18.2.3 Multiple Regression for Calibration
.....................185
11
Contents
18.3 Procedure
.......................................186
18.3.1 Set Up LabVIEWfor Data Acquisition
...................187
18.3.2 Determine Gain Constants
..........................187
18.3.3 Measure Impulse and Height
.........................189
18.3.4 Data Analysis
.................................1811
18.4 Reporting Requirements
...............................1811
19 Balancing Rotating Masses
191
19.1 Objective
.......................................191
19.2 Theory
.........................................191
19.3 Procedure
.......................................193
19.3.1 Establish Smooth Baseline
..........................193
19.3.2 Establish Unbalanced Baseline
........................194
19.3.3 Force Balance
.................................194
19.3.4 Moment Balance
...............................194
19.4 Reporting Requirements
...............................195
20 Free Vibration
201
20.1 Objective
.......................................201
20.2 Theory
.........................................201
20.2.1 Differential Equation of Motion
.......................201
20.2.2 Log Decrement
................................204
20.2.3 Indirect Weighing
...............................205
20.3 Procedure
.......................................205
20.4 Reporting Requirements
...............................207
21 Forced Vibration
211
21.1 Objective
.......................................211
21.2 Theory
.........................................211
21.3 Procedure
.......................................213
21.4 Reporting Requirements
...............................213
22 Small Engine Rebuild
221
22.1 Objective
.......................................221
22.2 Tasks
.........................................221
22.3 Dissassembly
.....................................222
12
Contents
22.4 Reporting Sheet for Engine Rebuild Lab
.......................2210
23 Crash Analysis and Deposition Exercise
231
23.1 Objective
.......................................231
23.2 The Crash Scenario
..................................231
23.3 Supporting Documents
................................235
23.3.1 Computer Generated Drawings
........................235
23.3.2 Friction Determination
............................237
23.3.3 Vehicle Information
.............................238
23.4 Assignment
......................................239
23.5 Ethics for Accident Investigation and Reconstruction
................239
23.6 Reporting Requirements
...............................2310
24 Ansys Workbench Exercise
241
24.1 Objective
.......................................241
24.2 Theory and Hand Calculations
............................241
24.3 Ansys Exercise
....................................2418
25 Rocket Lab Practical
251
25.1 Objective
.......................................251
25.2 Background and Theory
...............................252
25.2.1 Thrust Calculation
..............................252
25.2.2 Mass Moment of Inertia
...........................255
25.2.3 Angular Velocity Prediction
.........................257
25.3 Example Numerical Integration for a Simple Pendulum
...............258
25.4 LabVIEWData Acquisition
.............................258
25.5 Instrument Hookup and Calibration
.........................259
25.5.1 Accelerometer
................................259
25.5.2 Pressure Transducer
.............................2511
25.5.3 LabVIEWProgramming
...........................2511
25.5.4 Quadrature Encoder
.............................2511
25.6 Procedure
.......................................2513
13
Contents
IV Supplemental Material
2515
A Grading Sheet for Technical Reports
A1
B Participation Survey
B1
C Formal Letter Template
C1
Bibliography
C2
14
Syllabus
Instructor:Dr.Jeremy S.Daily
Email:jeremydaily@utulsa.edu
Phone:9186313056
Ofce:2080 Stephenson
Ofce Hours:Right after class on M and W.Otherwise,drop in or schedule an appointment
(email or phone)
Classroom:U1
Laboratory Stephenson 1085
Lecture Time:2:003:15 PM,Mondays and Wednesdays
Lab Time:2:005:00 PM,Tuesdays and Thursdays
This course notebook is required for the course and can only be purchased from Dr.Daily.While
the pages in here are designed to help you take notes,additional writing space will be required.
Therefore,loose leaf paper is recommended to augment the notebook.
This notebook can be accessed in electronic format
http://personal.utulsa.edu/∼jeremydaily/ME4024/MachineDynamicsCourseNotebook.pdf
A website dedicated to Machine Dynamics is located at
http://personal.utulsa.edu/∼jeremydaily/ME4024/ME4024Syllabus.html
Engineering graph paper can be printed from
http://personal.utulsa.edu/∼jeremydaily/downloads/EngineeringPaper.pdf
15
Contents
Course Bulletin Description
Kinematic and force analysis of machines and mechanisms.Mechanical vibrations,balancing,
and critical speed.Dynamic measurement using transducers and data acquisition systems,analysis
and interpretation of data,lab report writing.Introduction to multibody simulation using modern
engineering software.Written laboratory reports.Three hours lecture and three hours laboratory
per week.Prerequisite:ES 2023  Dynamics
This required fourcredit hour course is offered once a year,typically at the end (spring semester)
of the junior year.
Dynamic Course Outline
The following is a dynamically updated schedule for the class.It is a shared Google calendar
http://www.google.com/calendar
(Search for ME 4024)
and can be integrated into your own personal calendaring system.While the calendar can be up
dated and changed as the course goes on,the schedule will remain fairly rigid during the semester.
All changes and details concerning specic events and items on the schedule will be updated
through the Google calendar for this course.
The calendar ID is i26vok4v83gtp2ckn7b64em95o@group.calendar.google.com
Course Policies
Text Books
Reference (not required)
Fundamentals of Trafc Crash Reconstruction by J.G.Daily,N.Shigemura,and J.S.Daily,
Institute of Police Technology and Management,2006,ISBN 1884566634
Engineering Vibration,3rd Edition by D.J.Inman,Prentice Hall,2008,ISBN 0132281732
Mechanics of Machines by W.L.Cleghorn,Oxford University Press,2005,ISBN 019
5154525
16
Contents
Theory of Machines and Mechanisms,4th Edition by J.J.Uicker,G.R.Pennock,and J.E.
Shigley,Oxford University Press,2011,ISBN 0195371239
Any sophomore level dynamics book.
Grading Procedures
A separate sheet will be provided to explain grade allocations.
90100 = A,8089 = B,7079 = C,6069 = D,< 60 = F
The instructor reserves the right to lower the minimumrequirements for each letter grade.
ExamPolicy
Exams are open book and open notes;closed computer.
Computer Usage
Matlab,Mathematica,SolidWorks,LabVIEW,ANSYS and other specialty software will be used
for labs and homework.These programs are available in the Shared Undergraduate Computer lab
and the Machine Dynamics Lab.
Late Submission and Absences
Late submission of homework will receive no score.Late computer projects will receive no score.
Exams have mandatory attendance.Makeup exams will be offered only under very exceptional
circumstances provided prior permission from the instructor is obtained.Neatness and clarity of
presentation will be given due consideration while grading homework and exams.
Class Conduct
Please do whatever necessary to maintain a friendly,pleasant and businesslike environment so that
it will be a positive learning experience for everyone.Please turn off all cell phone ringers or any
other device that could spontaneously make noise.
17
Contents
Academic Misconduct
All students are expected to practice and display a high level of personal and professional integrity.
During examinations each student should conduct himself in a way that avoids even the appearance
of cheating.Any homework or computer problem must be entirely the students'own work.Con
sultation with other students is acceptable;however copying homework from one another will be
considered academic misconduct.Any academic misconduct will be dealt with under the policies
of the College of Engineering and Natural Sciences.This could mean a failing grade and/or dis
missal.The policy of the University regarding withdrawals and incompletes will be strictly adhered
to.
Center for Student Academic Support
Students with disabilities should contact the Center for Student Academic Support to selfidentify
their needs in order to facilitate their rights under the Americans with Disabilities Act.The center
for Student Academic Support is located in Lorton Hall,Room 210.All students are encouraged
to familiarize themselves with and take advantage of services provided by the Center for Student
Academic Support such as tutoring,academic counseling,and developing study skills.The Center
for Student Academic Support provides condential consult ations to any student with academic
concerns as well as to students with disabilities.
The University of Tulsa Mission
The University of Tulsa is a private,independent,doctoraldegreegranting institution whose
mission reects these core values:excellence in scholarsh ip,dedication to free inquiry,integrity of
character,and commitment to humanity.The university achieves its mission by educating men and
women of diverse backgrounds and cultures to become literate in the sciences,humanities,and
arts;think critically,and write and speak clearly;succeed in their professions and careers;behave
ethically in all aspects of their lives;welcome the responsibility of citizenship and service in a
changing world;and acquire the skills and appetite for lifelong learning.
While one course cannot accomplish the mission of the University experience,ME4024 does em
phasize the following aspects of the University's mission:
18
Contents
Clear writing is practiced by submitting technical laboratory reports following the laboratory
experiment.Also,neat and clean homework assignments must be submitted in a timely
manner.
The understanding of dynamic processes is a fundamental skill required for success in an
engineering career.Also,the ability to interact with modern measurement systems is a nec
essary skill for success.
The objective reporting of experimental data is paramount and fundamental to engineering
ethics.
The ability to learn howto use newsoftware and hardware systems is critical to being able to
maintain the appetite for lifelong learning in a dynamic engineering environment.
19
Part I
Lecture
1 Dynamics ProblemSolving
Machine Dynamics is a full and rich subject for study.Concepts in machine dynamics apply to all
industries that employ moving parts.The goal of this course is to develop the skills and problem
solving strategies to understand these concepts in both industry and further study.This chapter
should solidify techniques and methods taught in previous dynamics courses and develop funda
mental concepts in problemsolving with dynamic systems.
1.1 Dynamics Vocabulary
Fill in the appropriate denitions for the following terms.
Mechanics:
Statics:
Dynamics:
Kinematics:
Kinetics:
Machines:
Control:
11
1 DynamicsProblemSolving
1.2 ProblemSolving
The following steps are necessary when trying to solve machine dynamics problems
1.Read the problemcarefully and understand what it is asking.Take note of units.
2.Draw large diagrams and carefully tabulate data.This includes a Free Body Diagram (FBD)
and an Inertial Response Diagram(IRD).
3.Establish the appropriate coordinate systems and a transformation between them.
4.Solve the problem using symbols as far as possible using relevant kinetic and kinematic
principles.
5.Write a well commented computer programto obtain numerical and graphical results.
6.Check to see if the the solution makes sense.
7.Solve the problemusing another technique and try to get the same answer.
All work should be on engineering paper.The header should contain the course number/course
name,your name,the project/assignment,and page numbers.No markings should be outside the
work area.All diagrams and sketches should be centered,make use of a large portion of the page,
and no work should be shown on the side of the drawings.All work must be neat and orderly with
written explanations for the procedures.The nal answer sh ould be boxed with the appropriate
signicant gures.
12
1.3 VectorOperations
1.3 Vector Operations
Vectors are a fundamental mathematical tool that are used to describe physical pheneomenon.
1.3.1 Cartesian Coordinates
~
A =
i,
j,
k...
Magnitude of
~
A:
Direction of
~
A:
Direction Cosines:
1.3.2 Dot Product
~
A
~
B =
or
~
A
~
B =
or
~
C =
~
A×
~
B =
~
A
~
B =
13
1 DynamicsProblemSolving
1.3.3 Cross Product
~
C =
~
A×
~
B =
Expand the determinate to compute
~
C
1.3.4 Product Rule
d(a
~
B)
dt
=
d(
~
A
~
B)
dt
=
d(
~
A×
~
B)
dt
=
1.3.5 Partial and Total Derivatives
Given a function of n variables that are functions of time:
f = f (x
1
,x
2
, ,x
n
,x
1
,x
2
, ,x
n
,t)
where each x
i
=x
i
(t).
14
1.3 VectorOperations
d f
dt
=
if t does not explicitly appear in f,then
1.3.6 Time Derivative of a Vector
d
dt
(
~
A) =
~
A =
15
1 DynamicsProblemSolving
1.4 Kinematic Relationships for Rectilinear Motion
1.4.1 Velocity,distance,and time
1.4.2 Acceleration,velocity and time
1.4.3 Displacement,velocity and acceleration
1.4.4 Special Cases for Constant Acceleration
1.
2.
3.
Drag Factor is a ratio of accelerations when slowing:
f =−
a
g
Acceleration Factor is a ratio of accelerations when speeding up:
f =
a
g
1.5 Newton's Laws of Motion
1.
16
1.6 InertialReferenceFrames
2.
3.
1.6 Inertial Reference Frames
17
1 DynamicsProblemSolving
1.7 Force,Mass and Weight
Force:
1.
2.
3.
Mass:
Weight:
1.
2.
3.
How much does a 4200lb car weigh in the SI system?
SI Units
US Units
18
1.8 Example
1.8 Example
Two masses A and B are hung from a pulley and released from rest.Determine the velocity of A
after 2 seconds.Also,determine the velocity of A after 8 feet.Mass A weighs 250 lb and mass B
weighs 150 lb.
Ignore the moment of inertia of the pulley,the friction in the bearings,the mass of the cable,and
the extension in the cable.
Determine the values of mass.
Draw Kinetic Diagrams (Inertial Response Diagrams):
Write the Governing Equations:
A:
B:
19
1 DynamicsProblemSolving
Now what if w
A
=1250 lb and w
B
=1150 lb?
Solve using work and energy.
110
1.9 TangentandNormalCoordinates
1.9 Tangent and Normal Coordinates
111
1 DynamicsProblemSolving
1.10 Radial and Transverse Coordinates
112
1.11 HomeworkProblemSet1
1.11 Homework ProblemSet 1
1.A vehicle begins skidding on a surface with a drag factor of 0.72.It skids for 24 m and then
skids 19 macross another surface with a drag factor of 0.50.It then impacts a tree at 48 kph
and stops.The PerceptionResponse Time (PRT) of the driver/vehicle is determined to be
about 1.8 seconds.
a) What is the initial speed of the vehicle?95.42kph
b) What is the PRT distance?47.71m
c) What is the total distance to impact,starting at the perceptionresponse (PR) point?90.71m
d) What is the distance fromthe PR point to the beginning of the second skid?71.71m
e) If there is no impact with the tree,what is the speed at the beginning of the last skid?49.12kph
f) What is the maximum speed the vehicle could be going to stop at the tree without
impact,using the initial PR point as a reference?(Hint:The distance of the rst skid is
now unknown.) 87.01kph
g) What is the skid distance across the rst surface?28.20m
2.The two weights (A and B) are initially at rest.Determine the angular velocity (magnitude
and direction) of the pulley 1 second after letting go.The inner diameter is 15 cm and the
outside diameter is 25 cm.Weight A is 10 kg and weight B is 8 kg.Assume the chord is
inextensible and has negligible mass.
a) Solve neglecting the moment of inertia of the pulley.(Ans:
= 31.68
rad/sec)
b) Solve including the moment of inertia of the pulley when its radius of gyration is 20 cm
and it weighs 30.6 kg.
(Ans:
=
4.48 rad/sec)
113
1 DynamicsProblemSolving
B
A
114
1.11 HomeworkProblemSet1
3.Do the following problemfromdynamics.
4.Do the following problemfromdynamics.
115
1 DynamicsProblemSolving
1.12 Momentum
Newton's 2nd Law of Motion for Rigid Bodies
Translation:
Rotation:
where
~
F...
~
L...
~
H
G
...
~
H
G
=
{
}
=
[
]
{
}
Angular MomentumVector =
So,what does this mean?
1.
2.
116
1.12 Momentum
x
y
z
117
1 DynamicsProblemSolving
1.12.1 Concept of Impulse
118
1.12 Momentum
1.12.2 SystemMomenta
In Words:
X:
Y:
:
1.12.3 Moment of Inertia
FromWikipedia:
Abaseball bat is a smooth wooden or metal club used in the game of baseball to hit the
ball after the ball is thrown by the pitcher.It is no more than 2.75 inches in diameter at
the thickest part and no more than 42 inches (1,100 mm) long.It typically weighs no
more than 33 ounces (0.94 kg),but it can be different fromplayer to player.
Baseball bat swing is easier when the bat is backwards,because
The following Solidworks model was downloaded from
http://grabcad.com/home
Let's see what Solidworks thinks the mass properties are.Cl ick on the Tools > Mass Properties
menu.
119
1 DynamicsProblemSolving
The dialog box shows
120
1.12 Momentum
The print dialog gives:
121
1 DynamicsProblemSolving
How can we verify these numbers.Information is cheap and easy with the internet.Evalating
information is what an engineer should be able to do.
1.Verify weight using a measurement
2.Doublecheck dimensions (e.g solid vs hollowtubes)
3.Adjust denisty so
=
4.Calculate moment of inertia of simple shapes.
122
1.13 Collisionsin2D
5.Use Parallel Axis Theorem
Inertia is a tensor
Radius of Gyration
Vibration Measurements
1.13 Collisions in 2D
1.13.1 Center of Percussion
Impacts at the centerofpercussion result in zero net force at the pivot point,this location has long
been identied with the sweet spot.
123
1 DynamicsProblemSolving
1.13.2 Collision Reconstruction Example Using The Conservation of
Linear Momentum
124
1.13 Collisionsin2D
125
1 DynamicsProblemSolving
126
1.13 Collisionsin2D
127
1 DynamicsProblemSolving
128
1.13 Collisionsin2D
1.13.3 Collision Reconstruction Example Using Angular Momentum
129
1 DynamicsProblemSolving
130
1.13 Collisionsin2D
131
1 DynamicsProblemSolving
132
1.13 Collisionsin2D
133
1 DynamicsProblemSolving
1.14 Homework ProblemSet 2
1.Do the following problem from dynamics.The radius of the disk A is 0.2 m.Report your
results in slugft
2
.
2.A 40g bullet is red with a horizontal velocity of 600 m/s i nto the lower end of a slender
7kg bar of length L =600 mm.Knowing that h =240 mmand that the bar is initially at rest,
determine
a) the angular velocity of the bar immediately after the bullet becomes embedded,
b) the impulsive reaction at C,assuming that the bullet becomes embedded in 0.001 sec
onds.
134
1.14 HomeworkProblemSet2
3.A Chevrolet Caprice weighing 3600 lb and a Ford Crown Victoria weighing 3800 lb are
involved in a collision.The Chevrolet may be assumed to be on the xaxis with an approach
of 0°.The approach of the Ford is at 80°.The departure of the Chevy is 30° and the departure
of the Ford is 45°.The departure speed of the Chevy is 30 mph and the departure speed of
the Ford is 35 mph.Assume the friction forces are small compared to the collision forces
(i.e.assume no external impulse).
a) What is the speed of the Ford at impact?
b) What is the speed of the Chevy at impact?
c) What is the v of the Ford in the collision?
d) What is the v of the Chevy in the collision?
e) Determine the value of the collision impulse assuming the duration of the impact is
0.150 seconds.
f) Nowtest your assumption.If the coefcient of friction is 0.7 and the duration of the im
pact is 0.150 seconds,estimate the impulse fromthe friction force during the collision.
Compare this value to the collision impulse.
g) What is the angle of the Principal Direction Of Force (PDOF) for the Ford?
h) What is the angle of the PDOF for the Chevy?
i) Are the PDOF angles opposite in direction?If so,why?If not,why not?
j) How much energy is lost in the collision phase of this accident?
135
1 DynamicsProblemSolving
1.15 Work and Energy
1.15.1 Kinetic Energy
Consider a mass translating and rotating in a plane.
Kinetic Energy fromTranslation:
T =
Kinetic Energy fromRotation:
T =
Kinetic Energy about point O:
136
1.15 WorkandEnergy
1.15.2 Potential Energy
Gravity
Elastic Strain Energy
1.15.3 Work
1.15.3.1 Conservative Forces
Work of a Weight
Work of a spring
1.15.3.2 NonConservative Forces
Friction
In General
1.15.3.3 Workless Forces
Constraints:
137
1 DynamicsProblemSolving
1.15.4 The WorkEnergy Theorem
Energy is the ability to do work.
In General:
The work fromfriction:
1.15.5 The Conservation of Mechanical Energy
For Conservative Forces we get the Conservation of Energy:
Example:Charpy Impact Machine
138
1.15 WorkandEnergy
Example:A 10kg rod has its ends constrained to move in a vertical or horizontal slot.A spring
is attached to the end in the vertical slot and has a stiffness of 800 N/m.It is not stretched when
the rod is horizontal.The rod is 0.4 m long and its center of gravity is in the middle.Find static
equilibriumand nd the angular velocity when the rod is rele ased from 30 degrees.Use Newton's
Laws to develop the equation of motion.
139
1 DynamicsProblemSolving
140
1.16 HomeworkProblemSet3
1.16 Homework ProblemSet 3
1.Using the WorkEnergy Theorem,show that the speed calculation based on the distance of
a locked wheel skid is independent of the vehicle weight.This will require you to derive a
slide to stop equation based on work done by friction and the kinetic energy of the vehicle.
The speed equation is
S =
p
30d f
where S is in mph,d is in ft,and f is the drag factor.
2.A30lb uniformrod is released fromrest when it is in the near vertical position.It is allowed
to fall freely.Determine the angle at which the bottom end starts to lift off the ground.
Neglect friction at the bottomand the rod is 10 ft long.
3.A ball is dropped from Point A to plate B and bounces to point C.For
=20
◦
and a coef
cient of restitution of 0.40,determine the distance,d,as a function of the height,h.
141
1 DynamicsProblemSolving
4.The system is released from rest.Knowing the energy dissipated in the axle friction is 10 J
and the inertia of the pulley is negligible,determine
a) the velocity of B as it hits the ground and
b) the tension of the cable on each block.
Nowconsider the pulley has a moment of inertia about its center of 4 kgm
2
and a radius
of 1m,determine
c) the velocity of B as it hits the ground and
d) the force fromof the cable on each block.
142
1.17 PrincipleofVirtualWork
1.17 Principle of Virtual Work
143
1 DynamicsProblemSolving
1.18 Lagrange Equations
Example:The Equations of Motion of a Pendulum
144
1.19 Simulation
1.19 Simulation
145
1 DynamicsProblemSolving
1.19.1 StateSpace Form
146
1.19 Simulation
1.19.2 Euler's Method
147
1 DynamicsProblemSolving
1.19.3 RungeKutta
148
1.20 HomeworkProblemSet4
1.20 Homework ProblemSet 4
Simulate 15 seconds of a double rod pendulumfalling froma horizontal position.Assume friction
less bearings and the mass and moment of intertia each have a value of 1.You will need to derive
the equations of motion rst,then put the equations in state space form.Once in state space form,
you will need to write a computer program to simulate the motion of the pendulum.Turn in your
computer code and a plot of the locus of the trace point,P on an XY plot with equal axes.
149
2 Vibration
2.1 Free Vibration
2.1.1 Undamped Systems
21
2 Vibration
2.1.2 Systems with Damping
22
2.2 HomeworkProblemSet5
2.2 Homework ProblemSet 5
1.Show that the two systems below have the same equation of motion (in the presence of
gravity).
K
C
+x
m
K
C
+y
m
23
2 Vibration
2.Evaluate A,
n
,and
and plot the responses of at least 3 periods with a computer.
x(t) =Acos(
n
t +
)
a) m=5 kg,K =2000 N/m,x(0) =5 cm,v(0) =0
b) m=10 kg,K =1000 N/m,x(0) =5 cm,v(0) =0
c) m=2 kg,K =1000 N/m,x(0) =5 cm,v(0) =2 cm/s
K
+x
m
3.Evaluate A,
d
,C,and
for the system below.Plot the displacement responses with a
computer with at least 4 periods.Compute the logdecrement,
,based on
and compare
that value to the logdecrement measured measured fromthe response.
a) m=5 kg,K =2000 N/m,
=0.4,x(0) =0 cm,v(0) =5 cm/s
b) m=5 kg,K =2000 N/m,
=0.2,x(0) =2 cm,v(0) =0
K
C
+x(t)
m
24
2.2 HomeworkProblemSet5
4.Evaluate
,
d
,and
n
,for the following system when I
o
=1.6 kgm
2
(about the point of
rotation).
K =850 N/m
C =50 Ns/m
L
d
=0.3 m
L
s
=0.6 m
L
p
=0.7 m
5.Fromthe stripchart data below,Determine
d
,f
d
,T,
,
,
n
,m,andC when K =200 N/m.
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
4
3
2
1
0
1
2
3
4
5
t (sec)
x(t)
25
2 Vibration
2.3 Forced Vibration
2.3.1 Harmonic Forcing
26
2.3 ForcedVibration
2.3.2 Forcing by Rotating Unbalance
27
2 Vibration
2.3.3 Base Excitation
28
2.3 ForcedVibration
2.3.4 Frequency Response Functions
29
2 Vibration
2.4 Homework ProblemSet 6
1.Produce normalized graphs of the magnitude ratio and phase for the response of a system
subject to a base excitation.
2.Produce normalized graphs of the magnitude ratio and phase for the response of a system
subject to a rotating unbalance.
3.Consider the following steadystate forced vibration problem:
K
C
m
where:m=5 kg,k =2000 N/m,c =30 Ns/m,F =20 N,and
=15 rad/s.
a) Write down the equation for the steady state response of the system,x(t).
b) Plot exactly 3 periods of the steadystate solution.
c) Locate
and H(
) on the plots fromproblem1.Compare the results fromthe graphs
to your answer in part a.
210
2.5 CriticalShaftSpeed
4.Plot the amplitude of the response and the phase of the response based on of a rotating un
balance from0 <
<60 rad/sec.M=5 kg,K =2000 N/m,
=0.1,l =4 cm,and m=0.2
kg.
K
C
M
+
The small dot represents a hole with a length from the center of rotation of l and a missing
mass of m.
2.5 Critical Shaft Speed
2.5.1 Single Degree of FreedomModel
211
2 Vibration
2.5.2 Rayleigh's Method
212
2.6 HomeworkProblemSet7
2.6 Homework ProblemSet 7
1.Determine the eigenvalues and eigenvectors of the following matrix:
a)
"
8 −12
−12 1
#
b)
1 0 0 0 0
0 49 0 0 0
0 0 16 0 0
0 0 0 144 0
0 0 0 0 3
c) What is the highest systemfrequency for the multiple degree of freedomsystemwhose
eigenvalues are determined in part b.
2.Determine the critical speed in RPMof an electric motor.The steel (E = 30e6 psi) shaft is 0.5
inches in diameter and the distance between the bearings is 12 inches.Consider the rotating
element as being a single disk with a weight of 25 lb located midway between the bearings.
Neglect the mass of the shaft and assume simple supports for the bearings.Is 1200 RPM a
safe operating speed?
3.For the gure below,consult a text book for the static dee ction and determine the critical
speed in radians per second neglecting the mass of the shaft
a) if the bearings act as simple supports
b) the bearings act as cantilever supports (i.e.do not allowrotation)
c) What do you suppose the actual critical speed of a real system would be?In other
words,what if the previous boundary conditions are not exact.
213
2 Vibration
4.The static deection curve for a shaft supported in three b earings is shown below.The de
ections and the corresponding weights are given.Find the l owest critical shaft speed using
Rayleigh's Method.
5.Using Rayleigh's Method,show that the critical shaft spe ed for a simply supported uniform
shaft is
n
=9.874
r
EIg
WL
3
where W is the total weight of the bar.The static deection curve for a uniform horizontal
shaft is
y =
mgx
24EI
(L
3
−2Lx
2
+x
3
)
where m is the mass per unit length.
214
3 Dynamic Force Analysis
In machines that move...
Quantication...
Recall,
Kinematics:
Kinetics:
For Plane Motion:
3.1 D'Alembert's Principle
Newton's 2nd Law:
31
3 DynamicForceAnalysis
3.2
3.2.1 Example
x
y
Relative Velocity Analysis
Absolute Velocity Analysis
32
3.2
Accelerations
33
3 DynamicForceAnalysis
Use D'Alembert's Principle:
x
y
34
3.2
Now,Let's work out the problemwithout an inertial couple.
h =
Apply the 4 criteria:
x
y
Finally,use an Energy Method:
35
3 DynamicForceAnalysis
36
3.3 Dynamicsofa4BarMechanism
3.3 Dynamics of a 4Bar Mechanism
Find all pin reactions and the torque applied to crank,r
2
.Gravity is in the z direction.
x
y
Data froma Kinematic Analysis:
r
1
=
2
=
2
=
¨
2
=
r
2
=
3
=
3
=
¨
3
=
r
3
=
4
=
4
=
¨
4
=
r
4
=
Data for Inertial Properties:
O
2
G
2
=
2
=
m
2
=
I
G2
=
AG
3
=
3
=
m
3
=
I
G3
=
O
4
G
4
=
4
=
m
4
=
I
G4
=
37
3 DynamicForceAnalysis
3.3.1 Velocity Analysis (ME3212)
38
3.3 Dynamicsofa4BarMechanism
3.3.2 Free Body Diagrams
x
y
Force and Moment Balance:
F
x
=
F
y
=
M
O
2
=
39
3 DynamicForceAnalysis
Include internal forces by developing a free body diagramfor each link:
Links 2 and 4:
x
y
Link 3:
x
y
310
3.3 Dynamicsofa4BarMechanism
3.3.3 Equations of Motion
Link 2:
F
x
=
F
y
=
M
O
2
=
Link 3:
F
x
=
F
y
=
M
A
=
Link 4:
F
x
=
F
y
=
311
3 DynamicForceAnalysis
M
O
4
=
Unknowns:
3.3.4 Solution Technique
Put the coupled equations in matrix form:
[
]
{
}
=
{
}
where
Inertia
3
=
Inertia
4
=
Substitute numbers:
312
3.3 Dynamicsofa4BarMechanism
A =
[
]
A =
[
]
Inertia
3
=
Inertia
4
=
m
3
a
3x
=
m
3
a
3y
=
So the inertia vector is
313
3 DynamicForceAnalysis
C =
{
}
Solve the linear systemof equations:
A
−1
C =
{
}
=
{
}
Complete the solution:
R
2x
=
R
2y
=
R
4x
=
R
4y
=
314
3.4 HomeworkProblemSet8
3.4 Homework ProblemSet 8
1.In the following problem,gravity is acting downward.
315
3 DynamicForceAnalysis
2.Assume the following mechanismis mounted sideways and is not inuenced by gravity.
Verify your torque answer using the Lagrange Equations.
316
3.5 DynamicsofanInvertedSliderCrankMechanism
3.5 Dynamics of an Inverted SliderCrank Mechanism
Goal:Determine the input torque and pin reactions
x
y
Given Data:
r
1
=
r
2
=
O
3
B =
BC =
O
2
G
2
=
2
=
m
2
=
I
G2
=
O
3
G
3
=
3
=
m
3
=
I
G3
=
2
=
2
=
m
4
=
I
G4
=
P =
3.5.1 Kinematic Analysis
Loop Closure Equations:
x)
317
3 DynamicForceAnalysis
y)
Square both equations and add to eliminate
3
:
r
2
3
=
Expand:
r
2
3
=
Which gives the law of Cosines:
r
3
=
Similarly,the law of sines gives:
3
=
To determine velocities,differentiate the loop closure equations with respect to time.
x)
y)
Cast in matrix formand solve:
[
]{
}
=
{
}
318
3.5 DynamicsofanInvertedSliderCrankMechanism
With numbers:
[
]{
}
=
{
}
r
3
=
3
=
To determine accelerations,differentiate once more with respect to time:
x)
y)
Note the Coriolis term:
Write in matrix form:
[
]{
}
=
{
}
With numbers:
[
]{
}
=
{
}
319
3 DynamicForceAnalysis
Given the kinematic results fromthe loop closure equations,we can determine the accelerations of
the mass centers for link 2.
320
3.5 DynamicsofanInvertedSliderCrankMechanism
Now determine the accelerations of the mass centers for link 3.
Once the complete kinematic results are know,we can setup and solve the kinetic problem.
Free body diagramof Link 2:
321
3 DynamicForceAnalysis
Equations:
F
x
=
F
y
=
M
O
2
=
Free body diagramof Link 3:
Equations:
F
x
=
322
3.5 DynamicsofanInvertedSliderCrankMechanism
F
y
=
M
O
3
=
Free body diagramof Link 4:
Equations:
F
x
=
F
y
=
M
O
3
=
Number of Equations:
Number of Unknowns:
323
3 DynamicForceAnalysis
The last equations come fromslider friction.In tangential and normal coordinates:
324
3.5 DynamicsofanInvertedSliderCrankMechanism
F
t
=
F
x
=
F
y
=
If frictionless,then...
Set up the systemof equations:
[A]{f } ={B}
where
325
3 DynamicForceAnalysis
326
3.5 DynamicsofanInvertedSliderCrankMechanism
B =
With Numbers:
327
3 DynamicForceAnalysis
B =
f =
Thought Questions:
328
3.5 DynamicsofanInvertedSliderCrankMechanism
What is the effect of including a larger ywheel to the crank?
How does friction inuence the solution?
Why are the forces applied to the slider not equal and opposi te?
What is the effect of including gravity?
How does this apply if the slider is a linear actuator?
A Mathematica notebook also accompanies this example.It can be downloaded from:
http://personal.utulsa.edu/∼jeremydaily/ME4024/InvertedSliderCrankDynamics.nb
The PDF version of the notebook can be downloaded from
http://personal.utulsa.edu/∼jeremydaily/ME4024/InvertedSliderCrankDynamics.pdf
329
4 Balancing
4.1 Balancing Rotating Masses
41
4 Balancing
Balance Inertial Forces
4.1.1 Graphical Solution
Draw a Force Polygon:
42
4.1 BalancingRotatingMasses
4.1.2 Analytical Solution
Set up a table:
Number
Weight
Radius
Angle
w
i
r
i
sin
i
w
i
r
i
sin
i
1
2
3
Total
Balance
43
4 Balancing
4.1.3 Static Balance
x
z
y
M
y−axis
=
M
x−axis
=
What along balancing along the shaft?
Example:
z
y
x
44
4.1 BalancingRotatingMasses
Build a table for moments:
Moments about A
Around Y
Around X
Number
w
i
r
i
a
i
i
w
i
r
i
a
i
cos
i
w
i
r
i
a
i
sin
i
1
2
3
Total
B
Balance
Solve for magnitude and angle:
45
4 Balancing
Force Balance
X
Y
Number
w
i
r
i
i
w
i
r
i
cos
i
w
i
r
i
sin
i
1
2
3
B
Total
A
Balance
Solve for magnitude and angle:
Why is balancing important?
46
4.2 DynamicBalancing
4.2 Dynamic Balancing
47
4 Balancing
4.3 Homework ProblemSet 9
1.A rigid rotor is to be balanced by the addition of a fourth mass at a 178mm radius.Three
masses already exist and they are as follows:M
1
=1.81 kg,r
1
=381 mm,
1
=120
◦
,M
2
=
2.27 kg,r
2
=254 mm,
2
=225
◦
,M
3
=0.907 kg,r
3
=305 mm,
3
=330
◦
.Determine the
mass and angular position of the balancing mass using both graphical and analytical methods.
Draw a scale diagram of the rotor using solid lines for the known masses and dashed lines
for the balancing mass.
2.Determine the amounts and angular positions of two masses which,if added at a 51mm
radius in planes L and R,will balance the rotor.
z
y
x
3.Determine the bearing reaction forces for the unbalanced rotor in the previous problemwhen
the rotor is spinning at 1000 rpm.The shaft is mounted vertically so gravity is not in effect.
48
4.4 FieldBalancing
4.4 Field Balancing
49
4 Balancing
4.5 Balancing Reciprocating Masses
4.5.1 Single Cylinder Engines
410
4.5 BalancingReciprocatingMasses
4.5.2 Multi Cylinder Inline Engines
411
4 Balancing
4.6 Homework ProblemSet 10
1.The following data are given for a single cylinder internalcombustion engine with the piston
translating in the horizontal direction:speed = 1500 rpm,stroke = 204 mm,mass of crank
and crankpin = 3.63 kg,mass of piston and piston pin = 3.18 kg,distance from crankshaft
axis to the center of mass of the crankshaft = 63.5 mm,length of the connecting rod = 408
mm,mass of the connecting rod = 3.63 kg,distance fromthe center of mass of the connecting
rod to the crank pin = 102 mm.
a) Determine the magnitude and direction of the shaking force for a crank angle of
=
150
◦
if no counterbalance is used.
b) Determine the magnitude and direction of the shaking force for a crank angle of
=
150
◦
if a counterbalance force equal to the crank inertia plus 60% of the maximum
primary inertia force is used.
c) Write a programto plot magnitude of the shaking force for the above two scenarios.
2.A proposed engine conguration is shown below.The cylind ers are equally spaced with a
distance a between their centerlines.
z
y
x
a) Construct an engine balancing table and determine if any components are unbalanced.
b) Write an expression for the shaking force in terms of
1
(the rotation of the rst cylinder
bank).
412
4.6 HomeworkProblemSet10
c) Write an expression for the shaking moment in terms of
1
.
d) Write an expression for the location of the shaking force in terms of
1
.
3.Show that an inline 6 cylinder engine with is completely balanced in terms of primary and
secondary forces and moments.The crank angles are as follows:1 & 6 are at 0
◦
,3 & 4 are
at 120
◦
,2 &5 are at 240
◦
.Construct a balancing table to show this.
413
Part II
Reference Material
5 Standards for Measurement
5.1 Measurement
Crash reconstructionists are interested in measuring things that happened in the past.Since the nal
results of a reconstruction are measurements of some kind,we need to understand the denitions
and standards for measurements.
If we say a skid mark is measured to be 5 what does that mean?The number 5 does not mean
anything unless some units are used because,for example,5 inches is much different than 5 miles.
Developing standards for measurements was no easy task.Measurements in past times were very
often based on the dimensions of human or animal body parts.For example,kings of old were so
selfcentered that they made lengths and masses of their own bodies and possessions the standard
for the whole land.The length of the king's foot became his ki ngdom's standard unit of measure for
the foot.Or,the inch was the width of his thumb.Kings could be replaced and so could the unit of
measure.Furthermore,neighboring kingdoms had different standards.These practices,of course,
made scientic progress difcult.As scientic inquiry,co mmerce,and colonization grew,the world
began to adopt standards for measurement.Finally,the units of measure were standardized and they
are still used today.
5.2 Physical Quantities and Units of Measure
The laws of physics are expressed in terms of what are known as physical quantities.Examples
of physical quantities are force,time,velocity,temperature,and electric current.Physical quan
tities are categorized into fundamental quantities and derived quantities.Fundamental quantities
are those quantities that are based on known standards.They are not dened in terms of other
physical quantities.Derived quantities are those quantities that are based on other physical quanti
51
5 StandardsforMeasurement
Prex Abbreviation Exponent Multiplier
Tera T 10
12
1,000,000,000,000
Giga G 10
9
1,000,000,000
Mega M 10
6
1,000,000
Kilo k 10
3
1000
Hecta H 10
2
100
Deca D 10
1
10
deci d 10
−1
0.1
centi c 10
−2
0.01
milli m 10
−3
0.001
micro
or u 10
−6
0.000 001
nano n 10
−9
0.000 000 001
pico p 10
−12
0.000 000 000 001
fempto f 10
−15
0.000 000 000 000 001
Table 5.1:Denition of prexes for factor of ten relationsh ips
ties.The units of measure associated with fundamental quantities and derived quantities are called
fundamental units and derived units,respectively.
There are two major systems of measurement used in the world today.The rst,used mainly in
America,is the US
∗
system.The second is the SI system(metric system),which has been adopted
by the rest of the world,including Canada.SI is the abbreviation for International System of Units
(from the French Le Système International d'Unités).Both systems are used throughout this text,
with the majority of the examples worked in the US system.
The metric system uses fundamental units that are related by factors of ten.
This is convenient
because only the fundamental units need to be known.For example,if we know the length of a
meter,then a kilometer is 1000 times the length of a meter.Table
5.1
lists the different names that
are prepended to the fundamental unit in order to change it.Some of these will be familiar from
our experience with consumer electronics.Outside the US the metric system has been adopted
∗
Sometimes referred to as the English or Imperial system.
A factor of ten is also known as an order of magnitude.
52
5.2 PhysicalQuantitiesandUnitsofMeasure
by most countries and the scientic community for their tech nical publications.However,typical
Americans (juries) can relate much easier to the US system.This book presents concepts using
both systems of measurement.
5.2.1 Fundamental Units
There are seven fundamental units,also known as base units,that dene the SI system.The seven
base units are:
1.Length
2.Mass
3.Time
4.Electric Current
5.Temperature
6.Amount of Substance
7.Intensity of Light
The denitions of the seven base units are established throu gh the General Conference on Weights
and Measures.The rst three,length,mass,and time,are fam iliar to crash investigators and are
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