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ME 3212: Mechanisms
Course Notebook
Instructor:
Jeremy S. Daily, Ph.D., P.E.
Fall 2013
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Contents
1 Syllabus 6
1.1 Course Bulletin Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Course Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 Course Policies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4.1 Safety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4.2 Text Books . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4.3 Grading Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4.4 Exam Policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.4.5 Computer Usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4.6 Late Submission and Absences . . . . . . . . . . . . . . . . . . . . . . . 9
1.4.7 Class Conduct . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4.8 Academic Misconduct . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4.8.1 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4.8.2 Policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4.8.3 Definition of Academic Misconduct . . . . . . . . . . . . . . 10
1.4.8.4 Prompt Attention . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4.8.5 Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.4.8.6 Sanctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.4.8.7 Appeals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4.9 Center for Student Academic Support . . . . . . . . . . . . . . . . . . . 13
2 Introduction To Mechanisms 14
2.1 Mechanisms Vocabulary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2 Common Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 Kinematic Pairs (a.k.a. Joints) . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3.1 Low Order Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3.2 High Order Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.4 Degrees of Freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.4.1 Mobility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.4.2 Kutzbach Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.5 Grashof’s Law for Four-bar Mechanisms . . . . . . . . . . . . . . . . . . . . . . 20
2.5.1 Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.5.2 Inversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
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2.6 Homework Problem Set 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3 Position Analysis 24
3.1 Loop Closure Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.1.1 Derivation of the Law of Cosines . . . . . . . . . . . . . . . . . . . . . 25
3.1.2 Derivation of the Law of Sines . . . . . . . . . . . . . . . . . . . . . . . 26
3.1.3 Inverted Slider Crank . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.1.4 Offset Slider Crank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.1.5 Four Bar Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.1.5.1 Open Closure . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.1.5.2 Cross Closure . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2 Coupler Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.2.1 Matlab Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2.2 Excel Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2.2.1 Standard Algebraic Solution . . . . . . . . . . . . . . . . . . . 40
3.2.2.2 Use Excel Solver . . . . . . . . . . . . . . . . . . . . . . . . 41
3.2.2.3 SolidWorks Implementation . . . . . . . . . . . . . . . . . . . 473.3 Homework Problem Set 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.4 Newton-Raphson Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.4.1 Rocking Slider Crank . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.5 Homework Problem Set 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.6 Multi Loop Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.7 Toggle and Limit Positions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.8 Transmission Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.9 Homework Problem Set 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4 Mechanism Synthesis 63
4.1 Geometric Constraint Programming . . . . . . . . . . . . . . . . . . . . . . . . 634.2 Homework Problem Set 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5 Velocity Analysis 79
5.1 Vector Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.1.1 Dot Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.1.2 Cross Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.1.3 Derivatives of Vector Products . . . . . . . . . . . . . . . . . . . . . . . 79
5.2 Velocity with a Rotating Reference Frame . . . . . . . . . . . . . . . . . . . . . 80
5.3 Graphical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.3.1 Inverted Slider Crank . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.3.2 Four-Bar Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 835.4 Analytical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.4.1 Inverted Slider Crank . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.4.2 Four Bar Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
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5.5 Homework Problem Set 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
6 Acceleration Analysis 97
6.1 Accelerations in Four-bar Mechanisms . . . . . . . . . . . . . . . . . . . . . . . 97
6.2 Inverted Slider Crank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
6.3 Homework Problem Set 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
7 Cams 103
7.1 Types of Cam Followers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
7.1.1 Flat Faced Radial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
7.1.2 Offset Roller Follower . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
7.1.3 Barrel Cam with Roller Follower . . . . . . . . . . . . . . . . . . . . . . 105
7.1.4 Heavy Truck Brake Cams (S-Cams) . . . . . . . . . . . . . . . . . . . . 106
7.2 Cam Follower Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
7.2.1 Displacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
7.2.2 Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
7.2.3 Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1067.2.4 Jerk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
7.3 Cam Follower Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
7.3.1 Constant Acceleration (Parabolic) . . . . . . . . . . . . . . . . . . . . . 107
7.3.2 Harmonic Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
7.3.3 Cycloidal Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
7.4 Cam Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
7.5 Homework Problem Set 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
8 Gears 1288.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
8.1.1 Cog and Lantern Gears . . . . . . . . . . . . . . . . . . . . . . . . . . . 1288.1.2 Common Types of Gears . . . . . . . . . . . . . . . . . . . . . . . . . . 129
8.2 Fundamental Law of Gearing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
8.3 Conjugate Profiles and Involutometry . . . . . . . . . . . . . . . . . . . . . . . 130
8.3.1 The Involute Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
8.3.2 Cycloidal Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
8.3.3 Gear Sizing and Terminology . . . . . . . . . . . . . . . . . . . . . . . 131
8.4 Homework Problem Set 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
8.5 Gear Train Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
8.5.1 Taxonomy of gear trains: . . . . . . . . . . . . . . . . . . . . . . . . . . 134
8.5.2 Gear Train Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
8.5.3 Idler Gears . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1358.5.4 Compound Gear Trains . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
8.6 Homework Problem Set 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
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8.7 Reverted Gear Train Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
8.7.1 Design Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
8.8 Planetary Gear Trains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
8.8.1 Advantages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
8.8.2 Vector Approach to Planetary Gear Train Analysis . . . . . . . . . . . . 138
8.8.3 Tabular Planetary Gear Train Analysis . . . . . . . . . . . . . . . . . . . 1408.8.4 Compound Planetary Gear Trains . . . . . . . . . . . . . . . . . . . . . 142
8.8.5 Plotting Planetary Gears . . . . . . . . . . . . . . . . . . . . . . . . . . 144
8.9 Differential Gear Trains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
8.9.1 Ackerman Steering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
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1 Syllabus
Instructor: Dr. Jeremy S. Daily
• E-mail: [email protected]
Phone: 918-631-3056
Office: 2080 Stephenson
Office Hours: Tuesday and Thursday 1:30 - 3:00. Otherwise, drop in or schedule an appoint-
ment.
Classroom: U4
Days: Tuesday and Thursday
Time: 12:30 - 1:20 PM
This course notebook is required for the course and costs $20. It includes a 3-ring binder. 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 also be accessed (but not printed) in electronic format at
http://personal.utulsa.edu/~jeremy-daily/ME3212/MechanismsCourseNotebook.
1.1 Course Bulletin Description
Displacement, velocity and acceleration analysis of linkages, cams, and gear trains. An in-
troduction to synthesis. Computer simulation and design of planar mechanisms using modern
engineering software culminating in a design project. Prerequisites: ES 2023 (Dynamics).
This required two-credit hour course is offered once a year, typically at the beginning (fall
semester) of the junior year.
1.2 Objectives
This section of the syllabus is designed to give the student a larger picture of the purpose of
this class. This course is designed to provide the students the analytical skill necessary for
http://personal.utulsa.edu/~jeremy-daily/ME3212/MechanismsCourseNotebook.pdfhttp://personal.utulsa.edu/~jeremy-daily/ME3212/MechanismsCourseNotebook.pdfhttp://personal.utulsa.edu/~jeremy-daily/ME3212/MechanismsCourseNotebook.pdfhttp://personal.utulsa.edu/~jeremy-daily/ME3212/MechanismsCourseNotebook.pdf
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1.3 Course Outline
analyzing mechanism motion analysis. In addition to analysis, student will learn mechanism
synthesis using geometric constraint programming techniques. It is important for the student to
learn to use modern computer tools to aid in mechanism design and analysis. The familiarity
with rigid body analysis in modern computer analysis programs will give students a competitive
advantage in the current marketplace. The course covers position, velocity, and acceleration
analysis of planar linkages and slider crank mechanisms. Grashof’s laws, Kutzbach criterion,and the Newton-Raphson method. Cam profiles and cam follower motion is studied along with
gears and gearing including epicyclic gear trains.
1.3 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 3212)
and can be integrated into your own personal calendaring system. While the calendar can be
updated and changed as the course goes on, the schedule will remain fairly rigid during thesemester. All changes and details concerning specific events and items on the schedule will be
updated through the Google calendar for this course.
The calendar ID is [email protected]
1.4 Course Policies
1.4.1 Safety
Some projects in class may require the use of hand tools to build mechanims. Many mechanisms
are dangerous and may have pinch ponts that can severly injure someone. Always exercise
caution when handling a mechanism or machine. Design mechanisms in such a way to guard or
prevent pinch points.
During the first week of classes, every student must read and sign the rules and safety procedures
to be eligible to work in the laboratory or shop environment. This sheet will be provided the first
lab lesson.
Failure to comply with these rules may result in termination of your privilege to work in the
Mechanical Engineering Shops and Laboratories.
In the event of a minor emergency:
Dial extension 5555 for campus security
http://www.google.com/calendarhttp://www.google.com/calendar
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1.4 Course Policies
In the event of a major emergency:
Dial 9-911 for EMS
1.4.2 Text Books
In addition to this course notebook the following books are useful.
Required: Mechanics of Machines by W. L. Cleghorn, Oxford University Press, 2005, ISBN
0195154525
Reference: Theory of Machines and Mechanisms, 3rd Edition by J.J. Uicker, G.R. Pennock,and J.E. Shigley, Oxford University Press, ISBN 019515598
Other Resources: http://kmoddl.library.cornell.edu/index.php
1.4.3 Grading Procedures
The following table gives the weights of the different aspects of the graded material for this class:
Homework: 15%
Design Project: 35%
Exam 1: 20%
Exam 2: 20%
Final Exam: 20%
90-100 = A, 80-89 = B, 70-79 = C, 60-69 = D, < 60 = F
The instructor reserves the right to lower the minimum requirements for each letter grade.Grades will be kept on WebCT and will be updated on a regular basis.
1.4.4 Exam Policy
Exams are open book and open notes; closed computer.
1.4.5 Computer Usage
Matlab and other specialty software will be used for labs and homework for data analysis and
plotting. These programs are available in the undergraduate computer lab. SolidWorks and Ansys
should be available in the computer labs as well.
SolidWorks is available to all ME students for installation on there personal computer. The media
is available under the TU shared space at S:\ENS\Mechanical Engineering\SolidWorks.
http://kmoddl.library.cornell.edu/index.phphttp://kmoddl.library.cornell.edu/index.php
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1.4 Course Policies
1.4.6 Late Submission and Absences
Late submission of homework will receive no score. Late computer projects will receive no score.
Exams have mandatory attendance. Make-up 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, computer projects, and
exams.
1.4.7 Class Conduct
Please do whatever necessary to maintain a friendly, pleasant and business-like 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.
1.4.8 Academic MisconductAll students are expected to practice and display a high level of personal and professional in-
tegrity. During examinations each student should conduct herself or himself in a way that avoids
even the appearance of cheating. Any homework or computer problem must be entirely the
student’s own work. Consultation with other students is acceptable; however, copying home-
work 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 dismissal. The policy of the University regarding withdrawals
and incompletes will be strictly adhered to. Language in the POLICIES AND PROCEDURES
RELATING TO ACADEMIC MISCONDUCT OF UNDERGRADUATES COLLEGE OF EN-
GINEERING & NATURAL SCIENCES dated August 2013 is included below.
1.4.8.1 Purpose
In keeping with the intellectual ideals and educational mission of the University of Tulsa, all
members of the University of Tulsa community are expected to maintain their intellectual in-
tegrity at all times, to conduct themselves properly in all academic activities, and to adhere to
all academic policies. Cheating, plagiarism and other forms of academic dishonesty violate both
individual honor and the life of the community. The purpose of this document is to encourage
members of the academic community to conduct themselves responsibly toward one another, to
ensure that complaints of academic misconduct are treated fairly and in a timely fashion, and tomaintain the high standards of conduct required at the University of Tulsa.
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1.4 Course Policies
1.4.8.2 Policy
A. This policy prohibits any form of inappropriate conduct that constitutes academic miscon-
duct and applies to all participants in academic courses or programs offered by the College of
Engineering & Natural Sciences.
B. The College of Engineering & Natural Sciences and the University of Tulsa will take appro-priate actions to prevent, correct, and discipline conduct that violates this policy.
C. This policy shall not preclude faculty, academic administrators or a college from proceeding
summarily in appropriate cases.
D. This policy does not preclude anyone from pursuing complaints with any external agency or
other entity, such as other institutions when a student participates in an internship, field place-
ment, academic course or program at such institution; when criminal or civil laws may have been
violated; and other appropriate situations.
1.4.8.3 Definition of Academic Misconduct
A. Academic misconduct includes any conduct pertaining to academic courses or programs that
evidences fraud, deceit, dishonesty, an intent to obtain an unfair advantage over other students, or
violation of the academic standards and policies of the university. It includes, but is not limited to,
plagiarizing; cheating or otherwise violating the procedures for tests and examinations; turning
in counterfeit reports, tests, papers or other work; stealing tests or other academic material;
falsifying academic records or documents; turning in the same work to more than one instructor
without informing the instructors involved; vandalism, unauthorized or inappropriate use of data
files or equipment; violation of proprietary agreements, theft or tampering with the programs and
data of other users; or assisting others in such activities.
B. Academic misconduct also includes any inappropriate behavior that unreasonably interferes
with the educational process and the rights of others to pursue their academic goals. It includes,
but is not limited to, disorderly or disruptive conduct during classroom or other academic activity;
actual or threatened misuse or destruction of equipment or other academic resources; actual or
threatened interference with the right of others to participate fully in academic activities and
failure to respect and adhere to reasonable standards of conduct while participating in academic
activities.
1.4.8.4 Prompt Attention
A. All credible accusations of academic misconduct will be taken seriously and will be investi-
gated promptly, thoroughly and fairly.
B. NOTIFICATION BY INSTRUCTOR TO ASSOCIATE DEAN FOR ACADEMIC AFFAIRS.
All instructors shall notify, in writing and/or email, the Associate Dean for Academic Affairs
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1.4 Course Policies
promptly upon learning, directly or indirectly, about any case of academic misconduct, even in
cases where the instructor intends to investigate and address a complaint directly.
1.4.8.5 Procedures
A. INITIATING A COMPLAINT. A complaint may be initiated by an instructor, administrator,staff member, student or anyone else who has reason to believe that academic misconduct has
occurred.
B. ACTION BY AN INSTRUCTOR. An instructor may investigate and address any complaint
of academic misconduct in the instructor’s course or program.
1. In lieu of addressing a complaint directly, an instructor may choose to refer a complaint to
the associate dean for academic affairs of the college.
2. A decision by an instructor shall be final and binding when the instructor has notified the
student in writing and/or email of that decision. (See Section VII, APPEALS)
C. ACTION BY THE ASSOCIATE DEAN FOR ACADEMIC AFFAIRS. The associate deanfor academic affairs may initiate or pursue any case of academic misconduct in order to enforce
academic policies and to maintain the academic integrity of the college and university.
1. Even when sanctions have been imposed by an instructor for a particular case of academic
misconduct, additional sanctions may be pursued by the associate dean for academic af-
fairs in appropriate cases, such as when a student has committed academic misconduct
previously or when the academic misconduct is serious enough to warrant additional sanc-
tions.
2. In cases where a student has been accused of academic misconduct in a course or program
offered outside of the student’s college of enrollment, action may be initiated and pursued
by either or both the dean of the college in which the academic misconduct occurred andthe dean of the student’s college of enrollment.
3. A decision by the associate dean for academic affairs shall be binding when the associate
dean has notified the student in writing of that decision. (See the APPEALS Section)
1.4.8.6 Sanctions
A. SANCTIONS IMPOSED BY INSTRUCTOR. An instructor may impose sanctions for aca-
demic misconduct that include, but are not limited to, oral and/or written reprimand, counseling,
reduced or failing grades for specific assignments or the entire course or program, additional
assignments or requirements relating to the course or program, or any combination thereof.
B. SANCTIONS IMPOSED BY THE ASSOCIATE DEAN FOR ACADEMIC AFFAIRS OR
COLLEGE COMMITTEE ON ACADEMIC MISCONDUCT. In addition to any sanctions im-
posed by an instructor, the associate dean for academic affairs or college committee may impose
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1.4 Course Policies
sanctions for academic misconduct that include, but are not limited to, oral and/or written rep-
rimand, counseling, reduced or failing grades for a course or program, suspension, probation,
dismissal, notations on a student’s official records and transcript, revocation of academic honors
or degrees, and any other appropriate sanction or combination thereof.
1.4.8.7 Appeals
A. APPEAL TO THE ASSOCIATE DEAN FOR ACADEMIC AFFAIRS OF DECISION OF
AN INSTRUCTOR
1. A student who believes that a decision made by an instructor is unjust may appeal on that
ground in writing to the Associate Dean for Academic Affairs in the College of Engineer-
ing & Natural Sciences.
2. An appeal must be submitted within 7 days after the final decision of an instructor.
3. A decision by the Associate Dean for Academic Affairs shall be binding when the student
is notified in writing and/or email of that decision.B. APPEAL TO THE COLLEGE COMMITTEE ON ACADEMIC MISCONDUCT FROM DE-
CISION OF THE ASSOCIATE DEAN FOR ACADEMIC AFFAIRS
1. A student who believes that the decision made by the associate dean for academic affairs
is unjust may appeal on that ground in writing to the College of Engineering& Natural
Sciences Committee on Academic Misconduct
2. An appeal to the college committee must be submitted within 7 days after the final decision
of the Associate Dean for Academic Affairs.
3. The College of Engineering & Natural Sciences Committee on Academic Misconduct con-
sists of three faculty members elected from the faculty at large and chaired by the AssociateDean for Academic Affairs who sits without voice or vote.
4. Appeals must be in writing and should include all facts and circumstances that have any
bearing on the case, together with all relevant documents, evidence, and names of wit-
nesses.
5. A student shall have the right to request a hearing before the college committee.
6. The Committee on Academic Misconduct shall have the right to conduct a hearing, to
request additional information, and to receive and give such weight to evidence as the
college committee sees fit.
7. A student has the right to present personal testimony and evidence and to have the assis-tance of a friend or other advisor of his or her choosing in the appeal proceedings. Those
providing assistance to the student may only offer advice to the appellant. Advisors &
advocates do not otherwise participate in the proceedings.
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1.4 Course Policies
8. A decision of the College Committee on Academic Misconduct shall be binding when
the college committee has notified the student in writing of that final decision, except as
specifically stated below:
a) If the college committee recommends suspension, probation, dismissal, revocation of
academic honors or degrees, or any combination thereof, such recommendation shall
be forwarded to the Dean of the College for final action.
b) If the college committee is unable to reach a majority decision, the case will be re-
ferred to the Dean of the College for further review and decision.
C. FINAL APPEAL TO THE PROVOST
In the unusual circumstance that the student can make a case that the concept of fundamental
fairness has been violated in the appeal process itself, a final appeal may be made to the Provost,
who may either consider it or decline to do so depending on the Provost’s assessment of the
evidence presented. In all such cases, student appeals on academic issues will be final when a
decision is rendered by the provost.
This policy is not a contract. Policies and interpretation by the administration are subject to
change as circumstances warrant. Please check with the Associate Dean for updates and current
application of any policy.
1.4.9 Center for Student Academic Support
Students with disabilities should contact the Center for Student Academic Support to self-identify
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 StudentAcademic Support such as tutoring, academic counseling, and developing study skills. The
Center for Student Academic Support provides confidential consultations to any student with
academic concerns as well as to students with disabilities.
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2 Introduction To Mechanisms
Mechanisms is a the study of rigid body motion. The concepts in this course are restricted to
analyzing the motion and does not consider the cause of motion. In the field of dynamics, there
are a few broad categories as shown in Fig. 2.1.
Figure 2.1: Hierarchy of Mechanics
2.1 Mechanisms Vocabulary
Fill in the appropriate definitions for the following terms.
Kinematics:
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2.2 Common Mechanisms
Kinetics:
Mechanism:
Machine:
Linkage:
Link:
Joint:
Skeleton Diagram:
2.2 Common Mechanisms
• Slider Crank
– Example: Air compressor
• Four-Bar
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2.3 Kinematic Pairs (a.k.a. Joints)
– Example: Washing Machine Rocker
• Belts and Gears
– Example: Transmission
• Cams
– Example: Internal combustion engine valve train
2.3 Kinematic Pairs (a.k.a. Joints)
2.3.1 Low Order Pairs
• Spherical (G) SeeClegh
Sectio1.4
––
–
–
–
• Revolute (R)
–
–
–
–
–
• Cylindrical (C)
–
–
• Prismatic (P)
–
–
–
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2.3 Kinematic Pairs (a.k.a. Joints)
• Helical or Screw (S)
–
–
–
• Planar or Flat (F)
–
–
–
• Revolute 2
–
–
2.3.2 High Order Pairs
• Rolling-no slip
–
–
–
• Rolling/rotating with slip
– Cams, Link against plane
∗
∗
– Pin-In-Slot
∗
∗
• Rotating Pairs
– Gears, Friction Drives
–
–
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2.4 Degrees of Freedom
• Wrapping Pairs
– Example: Belt on Pulley (sheave)
–
–
2.4 Degrees of Freedom
• The number of inputs needed to get an output.
•
• For planar links there are:
• For spatial links there are:
2.4.1 Mobility
Calculating the number of degrees of freedom for a mechanism is determining its mobility.
2.4.2 Kutzbach Criteria
The formula to calculate mobility is See
Clegh
Sectio
1.5m = 3(n − 1)− 2 j1 − j2
where n. . . j1. . .
j2. . .
if m ≥ 1, then
if m = 0, then
if m ≤ −1, then
Remember, rolling pairs count as a 2 d.o.f. joint.
Example: Consider the planar slider crank mechanism shown in Fig. 2.2a. Determine the mobil-ity using the Kutzbach Criteria. Determine the number of links, the number of single degree of
freedom joints, and the number of 2 d.o.f. joints.
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2.4 Degrees of Freedom
x
y
A
BO2 θ 3
θ 2
(a) Planar Slider Crank Mechanism
(b) Four-bar slider
(c) Four-bar linkage
Figure 2.2: Determine the mobility of the mechanisms shown above.
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2.5 Grashof’s Law for Four-bar Mechanisms
2.5 Grashof’s Law for Four-bar Mechanisms
Grashof’s law is a method to categorize four-bar mechanisms based on the ability for a link to
make a complete rotation compared to the other links. In a four-bar mechanism there are four
possible link lengths:
• s. . .
• l. . .
• p. . .
• q. . .
s+ l < p +q (2.1)
If Eq. 2.1 is satisfied, then
If s is the input link, then
If s is the frame (base) link, thenIf s is the coupling link, then
Example: Can the shortest link make a full revolution in a four-bar mechanism where s = 4,l = 9, p = 6, and q = 6?
2.5.1 Special Cases
Change point mechanism
Parallelogram four-bar mechanisms
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2.5 Grashof’s Law for Four-bar Mechanisms
2.5.2 Inversions
• Every mechanism has a “ground” or “base” or “frame” link that is fixed. SeeClegh
Sectio
1.7
•
•
Let’s do an example to find the inversions for a four-bar mechanism where s = 1, l = 8, p = 6,and q = 6 using the grids shown in Fig. 2.3.
Check Grashof’s Criteria:
(a) Crank Rocker (b) Crank Rocker
(c) Double Crank (d) Double Rocker
Figure 2.3: Inversions of a four-bar mechanism.
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2.6 Homework Problem Set 1
2.6 Homework Problem Set 1
1. The link lengths of a planar four-bar linkage are 1, 3, 5, and 5 inches.
a) Assemble the links in all possible combinations and sketch (with a ruler and compass
on engineering graph paper) the four inversions of each.
b) Describe each inversion by name (e.g., crank-rocker, drag-link, rocker-rocker, change-
point)
c) Do these linkages follow Grashof’s Law?
2. Book Problem P1.1.
3. Book Problem P1.5.
4. Complete the “Introduction to SolidWorks” Tutorial. The tutorial can be found by access-
ing the SolidWorks Help Menu. Under the Getting Started Tutorial Category, perform the
Introduction to SolidWorks Tutorial. Turn in a print of the drawing you create.
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2.6 Homework Problem Set 1
Due on: _____________________________
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3 Position Analysis
The goal of a position analysis is to describe any position of any point in any mechanism config-
uration. The mechanical engineering skill from learning how to do a position analysis is to learn
the following concepts:
• Vector analysis
• Computer programming and mathematical modeling
• Numerical methods for root finding
• Develop the skill of abstracting physical objects using mathematics
A locus is defined as
3.1 Loop Closure Equations
The loop closure equations are fundamental to modeling mechanisms. The vectors that describe
the components must add to zero when the links form a loop: See
Clegh
Sectio
4.2
n
∑i=1
Ri = 0
where n. . .
Consider a loop of three fixed lengths: A, B, and C with angles α , β , and γ .
C A
Bα
β
γ
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3.1 Loop Closure Equations
Let’s define the x-axis to be along length B and the vectors defining the loop to go in a clockwise
direction.
C A
Bα
β
γ x y
Breaking these vectors into components gives the following six expressions. Keep in mind all
angles are defined from the positive x-axis and are positive when measure in a counterclockwise
direction. A x :
A y:
B x :
B y :
C x :
C y :
Now all the components in the x direction can be summed to zero and all the y components can
be summed to zero:
x : A x + B x +C x = 0
y : A y + B y +C y = 0
or
x :
y :
3.1.1 Derivation of the Law of Cosines
In Section 3.1 the concept of the loop closure equations was presented. If two sides, say C and
B, are known and the angle between them α is known, then the loop closure equations can besolved for the length of A. The following procedures demonstrate solution techniques for the
loop closure equations.
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3.1 Loop Closure Equations
1. Rewrite each component equation to move the unknown and unneeded angle as a single
term on one side:
x :
y :
2. Square each component equation and add them together. Invoking the relationship
sin2θ + cos2θ = 1
x :
y :
Sum:
3. Now the unknown angle γ has been eliminated and only the remaining length A is un-known. Expand the binomial terms:
4. Simplify and solve for A
A =
C 2 + B2 − 2 BC cosα
Similarly
C =
A2 + B2 − 2 AB cosγ
and
B =
A2 +C 2 − 2 AC cosβ
3.1.2 Derivation of the Law of Sines
If only one length is known and two angles are known, then the solution technique of the loop
closure equations requires a slightly different approach. This time, an unknown length needs to
be eliminated from the loop closure equations. The following procedure will derive the Law of
Sines from the loop closure equations. Let’s say we know α , γ and B. If two angles are knownin a triangle, then the other is also known because the sum of all the angles must be π radians(180 degrees).
1. Start by arranging the loop closure equations to eliminate C by moving all terms with C on
the left hand side.
x :
y :
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3.1 Loop Closure Equations
2. Divide the x equation into the y equation: y
x:
3. Eliminate the unknown length and cross multiply:
4. Arrange so B sinα is the only term on the left hand side:
5. Recall that β = π −α −γ . Take the sine of both angles and recall a trigonometric identity:
sinβ = sin(π −α − γ )
sinβ = sin(π )cos(α + γ )− cos(π )sin(α + γ )
sinβ = 0 − (−1) sin(α + γ )
sinβ = sin(α )cos(γ )+cos(α ) sin(γ )
6. Notice the right hand side of the equation matches the term from the loop closure equations.
Make the appropriate substitution:
B sinα = A sinβ
This is the law of sines. A similar formulation will give the familiar ratios:
sinα
A=
sinβ
B=
sin γ
C (3.1)
3.1.3 Inverted Slider Crank
An Inverted Slider-Crank is a mechanism that most used as an actuator. For example, a hydraulic
or pneumatic cylinder can be modeled as an Inverted Slider Crank. Fig. shows a photo of an
electric linear actuator and how it is modeled as an inverted Slider-Crank.
This section describes a simple mechanism to understand the concepts of position analysis.
Many texts will label the axis as real and imaginary instead of x and y. Either is valid. See
Clegh
Sectio
4.3.2
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3.1 Loop Closure Equations
(a) Retracted (b) Extended
Figure 3.1: A pneumatic cylinder is an example of an inverted slider-crank where the stroke and
angle can change.
x
y
A
D
BC
θ 3
θ 2
r
d
Figure 3.2: Inverted slider crank. Let a be the fixed length from A to C (the crank length).
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3.1 Loop Closure Equations
What is the mobility of the inverted slider crank in Fig. 3.2?
Another way to set up the loop closure equations is to find two different paths to the same point
in the mechanism. This formulation may be a little easier depending on how the angles are
defined. Consider two different loop closure formulations shown in Fig. 3.3 on the next page.
Both formulations describe the same physical system, so they should ultimately produce the
same solution.
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3.1 Loop Closure Equations
x
y
a
θ 3
θ 2
r
d
(a) a+ r + d = 0
x
y
a
θ 3θ 2
r
d
(b) a = d + r
Figure 3.3: Loop closure formulations for the inverted slider crank shown in Fig. 3.2.
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3.1 Loop Closure Equations
For the inverted slider crank shown in Fig. 3.2, the following variables are known: a = 0.15 m,d = 0.20 m and θ 2 = 35
◦. The goal of the position analysis is to find r and θ 3.
1. Break the vectors into components
a) For the loop closure equation from Fig. 3.3a on the preceding page
x : y :
b) For the loop closure equation from Fig. 3.3b on the previous page
x : y :
2. Rearrange the loop closure equations to have all terms of θ 3 on one side. Notice that either
formulation will give the same results.
x :
y :
3. Square each equation and add them together to eliminate θ 3:
4. Expand the squared terms:
5. Solve for r :
6. Divide the equations in Step 2 to eliminate r :
7. Compute the arctangent and adjust for the correct quadrant. Use the atan2 function.
The above example is also implemented in Mathematica and the Mathematica notebook can be
downloaded from
http://personal.utulsa.edu/~jeremy-daily/ME3212/InvertedSliderCrank.
nb
http://personal.utulsa.edu/~jeremy-daily/ME3212/InvertedSliderCrank.nbhttp://personal.utulsa.edu/~jeremy-daily/ME3212/InvertedSliderCrank.nbhttp://personal.utulsa.edu/~jeremy-daily/ME3212/InvertedSliderCrank.nbhttp://personal.utulsa.edu/~jeremy-daily/ME3212/InvertedSliderCrank.nb
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3.1 Loop Closure Equations
3.1.4 Offset Slider Crank
Given: r 2 =r 3 =
e =θ 2 =
Find: θ 3 and x B
1. Write the Loop equations:
2. Break into components
a) x :
b) y :
3. Rearrange y equation to solve for θ 3
4. Substitute into x expression and solve for x B
Note: For a standard slider crank. . .
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3.1 Loop Closure Equations
3.1.5 Four Bar Mechanism
The four bar mechanism is a very versatile device and is the building block for many other
mechanisms. A four bar linkage in conjunction with a slider crank is frequently employed in
lifting mechanisms and construction equipment. See
Clegh
Sectio
4.3.3
Consider the crank rocker in Fig. 3.4 where r 1 = 10 cm, r 2 = 4 cm, r 3 = 12 cm, r 4 = 8 cm, andθ 2 = 120
◦. The goal is to find θ 3, θ 4, and the transmission angle.
x
y
r 3
r 2
r 4
A
B
O2 O4
θ 4θ 2
r 1
θ 3
Figure 3.4: Crank rocker four-bar mechanism.
1. Consider the triangles formed by drawing a line from A to O4 and label it s.
2. Use the Law of Cosines (LOC) to determine s
3. Determine β :
4. Solve for ψ using LOC
5. Solve for λ using LOC
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3.1 Loop Closure Equations
6. Solve for θ 3
7. Solve for θ 4
8. Determine the transmission angle, γ using LOC.
3.1.5.1 Open Closure
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3.1 Loop Closure Equations
3.1.5.2 Cross Closure
See course website for a Mathematica notebook with the solutions to this example.
http://personal.utulsa.edu/~jeremy-daily/ME3212/4barPositions.nb
http://personal.utulsa.edu/~jeremy-daily/ME3212/4barPositions.nbhttp://personal.utulsa.edu/~jeremy-daily/ME3212/4barPositions.nb
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3.2 Coupler Curves
3.2 Coupler Curves
The goal of a coupler curve analysis is to determine the locus of some arbitrary point P on a
mechanism.
Physically a mechanism can have many different shapes for a single skeleton diagram.
Let’s add onto the example of Section 3.1.4:
Point P is. . .
Analysis Steps: See
Clegh
Sectio
4.2
1. Write down loop closure equations and solve for θ 3.
2. Write the equations for the vector describing point P.
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3.2 Coupler Curves
3.2.1 Matlab Implementation
A computer is useful to plot the curves with θ 2 as the independent variable. Make sure both axishave the same scale to get the true shape of the coupler curve. Below is some example Matlab
code to generate the coupler curve.
1 %ME 3 2 1 2 : M e c h a n i sm s%In −c l a s s E xa mp le
3 %O f f s e t s l i d e r c r a n k −d i r e c t s o l u t i o n
5 c l c %c l e a r s c r e e n
c l e a r a l l %c l e a r memo ry
7 c l o s e a l l %c l o s e a l l f i g u r e wi n d o ws
9 %i n p u t known v a l u e s
r 2 = 0 . 5
11 r 3 = 1 . 2
r 4 = 1 . 3
13 e = 0. 3
15 %d e f i n e t h e t a 2 f r o m 0 t o 360 i n 5 d eg re e i n c r e m e n t s u s i n g r a d i a n s
t h e t a 2 = [ 0 : p i / 3 6 : 2 ∗ p i ] ;17
%s o l v e f o r t h e t a 3
19 t h e t a 3 = a s i n ( ( e + r 2 ∗ s i n ( t h e t a 2 ) ) / r 3 ) ;
21 %s o l v e f o r xB
xB=r2 ∗ c o s ( t h e t a 2 )+ r 3 ∗ c o s ( t h e t a 3 ) ;23 yB=z e r o s ( s i z e ( xB ) ) ; %T h i s i s n ee d ed f o r t h e c o up l er c u r v e
25 %D e t er m i n e C o up l er c u r v e
xp = r 2 ∗ c o s ( t h e t a 2 ) − r 4 ∗ c o s ( t h e t a 3 ) ;27 yp = e + r 2 ∗ s i n ( t h e t a 2 ) + r 4 ∗ s i n ( t h e t a 3 ) ;
29 f o r n = 1 : l e n g t h ( t h e t a 2 ) %I t e r a t e t hr o u gh t h e v a l u e s o f t h e t a 2
31 %d e f i n e k e y p o i n t s on mec h a n is m i n a
%f a s h i o n t o p l o t t h e c o n f i g u r a t i o n
33 p t s x = [ 0 , r 2 ∗ c o s ( t h e t a 2 ( n ) ) , xB ( n ) , x p ( n ) ] ;p t s y = [ e , e + r 2 ∗ s i n ( t h e t a 2 ( n ) ) , 0 , y p ( n ) ] ;
35
%p l o t t h e c o u p l e r c u r v e p o i n t s and o n e m e c h a n i sm o r i e n t i a t i o n
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3.2 Coupler Curves
−1.5 −1 −0.5 0 0.5 1 1.5
−0.5
0
0.5
1
1.5
x−position (m)
y − p o s i t i o n ( m )
Coupler Curve: Slider crank shown with 60 degree input
(a)
0 50 100 150 200 250 300 350
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
Crank Angle, θ2 (deg)
S l i d e D i s t a n c e , x B (
m )
Slide position as a function of input angle
(b)
Figure 3.5: Output graph from Matlab code for the coupler curves of an Offset slider crank mech-
anism.
3.2.2 Excel Implementation
Consider the following four-bar mechanism where
r 1 = ,r 2 = , r 3 = , r 4 = , r 5 = , r 31 = , θ 2 = .
Coupler Curve Equations: x p =
y p =
Trigonometric Identities:
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3.2 Coupler Curves
3.2.2.1 Standard Algebraic Solution
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3.2 Coupler Curves
3.2.2.2 Use Excel Solver
Open Closure
1
23
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
3334
A B C D E F G
Parameter Value Units Vertex X Y
r1 10 in 1 0 0
r2 3 in 2 1.5000 2.5981
r3 12 in 3 4.98362767 12.6957
r4 8 in
Radians Degrees Vertex X Y
theta2 2.094 120.000 3* 6.6708 7.2744
theta3 1.000 57.296 4 10.0000 0.0000
theta4 2.000 114.592
Difference 1.6872E+00 5.4213E+00
residual 5.6778E+00
Lower Leg
Upper Leg
ME 3212: Mechanisms
FourBar
Linkage
Analysis
Plot Coordinates
0
2
4
6
8
10
12
14
4 2 0 2 4 6 8 10 12
Upper Leg
Lower Leg
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3.2 Coupler Curves
1
2
3
4
56
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
2627
28
29
30
31
32
33
34
A B C D E F G
Parameter Value Units Vertex X Y
r1 10 in 1 0 0
r2 3 in 2
1.5000 2.5981r3 12 in 3 9.233836209 7.9632
r4 8 in
Radians Degrees Vertex X Y
theta2 2.094 120.000 3* 9.2338 7.9632
theta3 0.464 26.557 4 10.0000 0.0000
theta4 1.667 95.496
Difference 3.7786E06 7.8999E06
residual 8.7571E06
Lower Leg
Upper Leg
ME 3212: Mechanisms
FourBar Linkage Analysis
Plot Coordinates
0
1
2
3
4
5
6
7
8
9
4 2 0 2 4 6 8 10 12
Upper Leg
Lower Leg
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3.2 Coupler Curves
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
1617
18
19
20
21
22
23
24
25
26
2728
29
30
31
32
33
34
A B C D E F G
Parameter Value Units Vertex X Y
r1 10 in 1 0 0
r2 3 in 2 =B5*COS(B10) =B5*SIN(B10)
r3 12 in 3 =F5+B6*COS(B11) =G5+B6*SIN(B11)
r4 8 in
Radians Degrees Vertex X Y
theta2 =2*PI()/3 =DEGREES(B10) 3* =F11+B7*COS(B12) =B7*SIN(B12)
theta3 0.463515306 =DEGREES(B11) 4 =B4 0
theta4 1.666714283 =DEGREES(B12)
Difference =F6F10 =G6G10
residual =SQRT(F13^2+G13^2)
Lower Leg
Upper Leg
ME 3212: Mechanisms
FourBar Linkage Analysis
Plot Coordinates
0
1
2
3
4
5
6
7
8
9
4 2 0 2 4 6 8 10 12
Upper Leg
Lower Leg
Cross Closure
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3.2 Coupler Curves
1
2
3
4
56
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
2627
28
29
30
31
32
33
34
A B C D E F G
Parameter Value Units Vertex X Y
r1 10 in 1 0 0
r2 3 in 2
1.5000 2.5981r3 12 in 3 9.030990743 3.1550
r4 8 in
Radians Degrees Vertex X Y
theta2 2.094 120.000 3* 14.3224 6.7318
theta3 0.500 28.648 4 10.0000 0.0000
theta4 1.000 57.296
Difference 5.2914E+00 3.5767E+00
residual 6.3869E+00
Lower Leg
Upper Leg
ME 3212: Mechanisms
FourBar Linkage Analysis
Plot Coordinates
8
6
4
2
0
2
4
5 0 5 10 15 20
Upper Leg
Lower Leg
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3.2 Coupler Curves
1
2
3
4
56
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
2627
28
29
30
31
32
33
34
A B C D E F G
Parameter Value Units Vertex X Y
r1 10 in 1 0 0
r2 3 in 2
1.5000 2.5981r3 12 in 3 5.884873205 6.8604
r4 8 in
Radians Degrees Vertex X Y
theta2 2.094 120.000 3* 5.8849 6.8604
theta3 0.908 52.019 4 10.0000 0.0000
theta4 2.111 120.957
Difference 1.8606E06 5.1288E07
residual 1.9300E06
Lower Leg
Upper Leg
ME 3212: Mechanisms
FourBar Linkage Analysis
Plot Coordinates
8
6
4
2
0
2
4
4 2 0 2 4 6 8 10 12
Upper Leg
Lower Leg
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3.2 Coupler Curves
3.2.2.3 SolidWorks Implementation
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3.3 Homework Problem Set 2
3.3 Homework Problem Set 2
1. An offset slider crank mechanism is driven by rotating the crank. The axis of the slider is
1 inch below the x-axis, the crank is 2.5 inches, and the connecting rod is 7 inches. Solve
for the position of the slider as a function of the crank angle θ 2. Write a Matlab program
that plots the position of the slider for a complete revolution of the crank. Turn in yourhand analysis, Matlab program (the .m file) and a properly labeled plot.
http://personal.utulsa.edu/~jeremy-daily/ME3212/HW2Prob1.avi
2. Using the figure of book problem P4.10, plot the locus of Point C for a complete revolution
of link 2. Also, plot on the same graph the configuration of the mechanism when θ 2 = 10◦.
For now, ignore the questions regarding velocity and acceleration. Be sure to use the com-
mand axis equal to plot the configuration of the mechanisms to scale.
http://personal.utulsa.edu/~jeremy-daily/ME3212/HW2Prob2.avi
3. In book problem P4.18a, plot the locus of point C for a complete revolution of link 2
for the open closure configuration. Also, plot on the same graph the configuration of the
mechanism when θ 2 = 30◦. Draw the coupler as a triangle (△ BCD). For now, ignorethe questions regarding velocity and acceleration. A movie showing the positions of the
mechanism is available at:
http://personal.utulsa.edu/~jeremy-daily/ME3212/HW2Prob3.avi
4. Perform the “Assembly Mates” SolidWorks Tutorial under the Building Models Tutorial.
Turn in a print of your completed part.
http://personal.utulsa.edu/~jeremy-daily/ME3212/HW2Prob1.avihttp://personal.utulsa.edu/~jeremy-daily/ME3212/HW2Prob2.avihttp://personal.utulsa.edu/~jeremy-daily/ME3212/HW2Prob3.avihttp://personal.utulsa.edu/~jeremy-daily/ME3212/HW2Prob3.avihttp://personal.utulsa.edu/~jeremy-daily/ME3212/HW2Prob2.avihttp://personal.utulsa.edu/~jeremy-daily/ME3212/HW2Prob1.avi
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3.4 Newton-Raphson Method
3.4 Newton-Raphson Method
What if you can’t solve for the positions by trigonometry and algebra?
3.4.1 Rocking Slider Crank
Consider the following example:
Given: r 2, xo, R, θ 2
Find: r and θ 3Write down the Loop closure equations:
x :
y:
Rearrange to eliminate r :
This equation has just 1 unknown θ 3 but no analytical solution. Therefore, we’ll solve using theNewton-Raphson method.
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3.4 Newton-Raphson Method
In Matlab, the algorithm for the example looks like the following:
1 %ME 3 2 1 2 : M e c h a n i sm s
%Newton− R a h p so n Meth od f o r s o l v i n g l o o p c l o s u r e e q u a t i o n s3 %Dr . J e r e my D a i l y
%
5 c l c ; c l e a r a l l ; c l o s e a l l
7 % k n o w n s:
r 2 = 1 . 2 % m e t e r s
9 x0 = 1 .8
R = . 4
11
t h e t a 2 = [ 0 : p i / 1 0 1 : 4 . ∗ p i ] ; %Two c y c l e s o f t h e c ra nk 13
t h e t a 3 = 0; %g u e ss a s o l u t i o n f o r f i r s t i t e r a t i o n
15 f o r n = 1 : l e n g t h ( t h e t a 2 )
d e l t a t h e t a 3 = 1; %s e t t h e l o op c o n d i t i o n a l
17 %g ue ss t h e same s o l u t i o n a s t he t i m e b e f o re
19 wh i l e norm ( d e l t a t h e t a 3 ) > 0 .0 0 01 7 5 %0 . 0 1 d e g r e e s
f = t a n ( t h e t a 3 ( n ) ) . ∗ ( x0 +R∗ t h e t a 3 ( n )− r 2 ∗ c o s ( th e t a 2 ( n ) ) ) . . .21 −r 2 ∗ s i n ( t h e t a 2 ( n ) ) ;
d f d t h e t a 3 = ( 1 . / ( c o s ( t h e t a 3 ( n ) ) . ̂ 2 ) ) . ∗ ( x0 +R∗ t h e t a 3 ( n ) . . .23 −r 2 ∗ c o s ( t h e t a 2 ( n ) ) ) + R∗ c o s ( t h e t a 3 ( n ) ) ;
d e l t a t h e t a 3 =−f / d f d t h e t a 3 ;25 %u pd at e t h e s o l u t i o n u n t i l i t i s w i t h i n t o l e r a n c e
t h e t a 3 ( n ) = t h e t a 3 ( n ) + d e l t a t h e t a 3 ;27 en d
t h e t a 3 ( n + 1 ) = t h e t a 3 ( n ) ;
29
en d
31%S i n c e t h e l a s t command i n t h e l oo p c r e a t e d an e x t r a t h e t a 3 e n t r y
33 %we m u s t r e mo ve i t b y a s s i g n i n g i t t o t h e e m p t y s e t
t h e t a 3 ( n + 1 ) = [ ] ;
35
%S o l v e f o r t h e r e ma i ni n g u n k o w n s
37 r _ s i n e s = r2 ∗ s i n ( t h e t a 2 ) . / s i n ( t h e t a 3 ) ; %l a w o f s i n e sr _ c o s i n e s = s q r t ( r 2 . ^ 2 + ( x 0 +R∗ t h e t a 3 ) . ^ 2 . . .
39 − 2∗ r 2 . ∗ ( x0+R. ∗ t h e t a 3 ) . ∗ c o s ( t h e t a 2 ) ) ; %l a w o f c o s i n e sx=R∗ t h e t a 3 ;
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3.4 Newton-Raphson Method
41 p l o t ( t h e t a 2 , t h e t a 3 , ’ : ’ , t h e t a 2 , r _ s i n e s , ’− ’ , t h e t a 2 , r _ c o s i n e s , . . .’−− ’ , th et a2 , x , ’ −. ’ )
43 x l a b e l ( ’ \ t h e t a _ 2 ( r a d i a n s ) ’ )
y l a b e l ( ’ P o s i t i o n v a r i a b l e s ’ )
45 l e g e n d ( ’ \ t h e t a _ 3 ( r a d ) ’ , ’ r : s i n e s (m) ’ , ’ r : c o s i n e s (m) ’ , ’ x (m) ’ )
The graph generated by the code is shown in Fig. 3.6.
0 2 4 6 8 10 12 14−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
3
3.5
θ2 (radians)
P o s i t i o n v a r i a b l e s
θ3 (rad)
r:sines (m)
r:cosines (m)
x (m)
Figure 3.6: Output of Newton-Raphson example. What are the spikes from?
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3.4 Newton-Raphson Method
How does Newton-Raphson work?
The N-R method requires a good first guess or:
•
•
How to make a good initial guess:•
•
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3.5 Homework Problem Set 3
3.5 Homework Problem Set 3
1. Use the Newton-Raphson method to solve for θ 3 by hand for the four-bar linkage shownin Fig. 3.4 on page 33. Compare the result from the N-R method to the analytical solution
for θ 2 = 2π /3 radians (120 degrees). Then, create a Matlab program that implements the
N-R method and plots θ 3 against θ 2 for every 5 degrees of a complete revolution of thecrank. Plot the solution from the N-R method with a dot and the solution from the analyti-
cal method with a square. Generate two plots: one for the open closure configuration and
one for the cross closure configuration. The graphs you generate should look like the ones
in Fig. 3.7 on page 58. The Matlab code used to plot the results is as follows:
1 F= f i g u r e ( 1 ) ;
p l o t ( t h e t a 2 ∗ 1 8 0 / pi , t h e t a 3 ∗ 1 8 0 / p i , ’ . ’ , . . .3 t h e t a 2 ∗ 1 8 0 / pi , t h e t a 3 _ o p e n ∗ 1 8 0 / p i , ’ s ’ )
%N o t e : t h e t a 3 _ o p e n m u s t b e d e f i n e d i n o r d e r f o r t h i s t o work
5 l e g e n d ( ’Newton−R a p h so n ’ , ’ A n a l y t i c a l ’ , ’ L o c a t i o n ’ , ’ N o r t h W e s t ’ )x l a b e l ( ’ I n p u t C ra nk A ng le , \ t h e t a _ 2 [ d eg ] ’ )7 y l a b e l ( ’ C o u p l e r A ng le , \ t h e t a _ 3 [ d e g ] ’ )
a x i s t i g h t
9 s e t ( gca , ’ XTick ’ , [ 0 : 3 0 : 3 6 0 ] )
g r i d on
11 t i t l e ( ’ C ou pl er a n g le i n t h e open c l o s u r e c o n f i g u r a t i o n ’ )
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3.5 Homework Problem Set 3
2. Eccentric Cam Analysis (See figure below):
a) Plot the locus of point P for a complete turn of the cam when r 1 = 100 mm, r 2 = 150mm. e = 20 mm, and R = 40 mm. Hint: use N-R to determine θ 2.
b) Plot the y position of point P as a function of θ 3.
c) Determine the maximum and minimum of θ 2 by hand.
An animation of the motion of this mechanism is available at
http://personal.utulsa.edu/~jeremy-daily/ME3212/HW3Prob2.avi
Hints: r and R form a right angle.
R and e do not form a right angle (the angle changes).
Due on:___________________
P
x
y
e
R
θ 3θ 2
r 1
r
r 2
O2
O3
3. Perform the “Advanced Design” SolidWorks Tutorial. Turn in a print of your completed part.
http://personal.utulsa.edu/~jeremy-daily/ME3212/HW3Prob2.avihttp://personal.utulsa.edu/~jeremy-daily/ME3212/HW3Prob2.avi
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3.5 Homework Problem Set 3
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3.5 Homework Problem Set 3
0 30 60 90 120 150 180 210 240 270 300 330 360
20
25
30
35
40
45
50
55
60
65
Input Crank Angle, θ2 [deg]
C o u p l e r A n g l e ,
θ 3
[ d e g ]
Coupler angle in the open closure configuration
Newton−Raphson
Analytical
(a) Open Closure Configuration
0 30 60 90 120 150 180 210 240 270 300 330 360−65
−60
−55
−50
−45
−40
−35
−30
−25
−20
Input Crank Angle, θ2 [deg]
C o u p l e r A n g l e ,
θ 3
[ d
e g ]
Coupler angle in the cross closure configuration
Newton−Raphson
Analytical
(b) Cross Closure Configuration
Figure 3.7: Graphs for the Newton-Raphson and analytical solution for θ 3
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3.6 Multi Loop Mechanisms
3.6 Multi Loop Mechanisms
Multi loop mechanisms are analyzed by constructing the loop closure equations for all the ele-
mentary loops. Open chains can also be considered.
Consider the following example: See
CleghSectio
4.2
Calculate the mobility using the Kutzbach Criteria:
Determine the known and unknown variables:
There are four equations after breaking up the loop closure equations:
x1 :
y1 : x2 :
y2 :
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3.7 Toggle and Limit Positions
3.7 Toggle and Limit Positions
Mechanical Advantage See
Clegh
Sectio
2.6
Consider an offset slider crank in the “dead” center positions.
Top center:
Bottom center:
The criteria to find a toggle position are:
Loop Closure Equations:
x :
y :Substitute θ 2 = θ 3 into the y equation:
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3.8 Transmission Angle
Substitute θ 2 = θ 3 +π into the y equation:
3.8 Transmission Angle
For a slider crank, the transmission angle is γ . See
Clegh
Sectio
2.7
γ =
Goal: find the maximum and minimum γ and corresponding θ 2.
Rewrite Loop equations:
x :
y:
but,
and,
so
x :
y :
To find extrema, find values of θ 2 when d γ
d θ 2= 0. Use implicit differentiation:
x′ :
y′ :
Consider only the numerator:
θ 2 =
So the transmission angles are:
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3.9 Homework Problem Set 4
3.9 Homework Problem Set 4
1. Using the offset slider-crank mechanism shown in Fig. 4.4 of Cleghorn’s text, determine
the extreme values of the transmission angle, γ . Hint: Write an expression for γ in terms
of θ 2, differentiate with respect to θ 2, and let d γ /d θ 2 = 0 for extremes.
2. For a four bar mechanism where r 1 = 400 mm, r 2 = 200 mm, r 3 = 500 mm, and r 4 = 400mm,
a) Determine θ 2, θ 3, θ 4, and γ for both limit positions.
b) Draw the mechanisms in each limit position to scale.
c) Determine the total rocking angle (∆θ 4). Ans: 78 ◦
3. Perform the “Animation” SolidWorks Tutorial. Turn in a print of your completed part.
Due on: ________________
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4 Mechanism Synthesis
The design or creation of a mechanism to achieve the desired motion.
Type Synthesis:
Number Synthesis:
Dimensional Synthesis:
Classical Analysis:
4.1 Geometric Constraint Programming
Everyone should download their own copy of:
Kinzel et al., “Kinematic synthesis for finitely separated positions using geometric constraint
programming.” Journal of Mechanical Design (2006)
Meet in the Computer Lab (L1) to see the following demonstration:
Goal: pick up an object with a scoop and dump it at a point 3” higher and 4” over.
Approach: design a four bar mechanism that has a coupler that will follow 5 precision points and
directions.
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4.1 Geometric Constraint Programming
To synthesize a mechanism that satisfies these constraints, follow these instructions:
1. Create a new Part in SolidWorks.
2. Change the document properties to the IPS system. This is found under the Tools, Options,
Document Properties menu.
3. Under the System Options tab (Tools ⊲ Options), click on the Relations/Snaps branch andmake sure the Automatic Relations box is unchecked.
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4.1 Geometric Constraint Programming
4. Create a new sketch on any plane.
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4.1 Geometric Constraint Programming
5. Create 5 3-point arcs. They do not have to be the same size yet and placement is arbitrary
at this time. These arcs will represent the scoop.
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4.1 Geometric Constraint Programming
6. Make all the arcs equal by selecting them while holding the Shift key. Then click on the
Equal button under the Add Relations Pane.
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4.1 Geometric Constraint Programming
7. Create centerlines an connect each end of the arc.
8. Make the center of the arc coincident with the line.
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4.1 Geometric Constraint Programming
10. Draw congruent triangles from each of the arc endpoints. To ensure congruency, make all
corresponding legs of the triangle equal. The tip of the triangle will represent the connec-
tion to one of the links.
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4.1 Geometric Constraint Programming
11. Draw a perimeter circle through three of the five tips of the triangles.
12. Make the points coincident with the circle.
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4.1 Geometric Constraint Programming
13. Drag the other two points close to the circle and make them coincident. All five points
should be coincident with the circle.
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4.1 Geometric Constraint Programming
14. Create another set of congruent triangles to represent the other connecting point for the
coupler.
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4.1 Geometric Constraint Programming
15. Draw the other circle and maneuver the locations of the arc and the triangles to get a com-
pact package. This may be difficult and you may have to delete some fixed constraints.
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4.1 Geometric Constraint Programming
16. Once the position is where you like, Select All and make a Block (Tools ⊲ Make Block).
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4.1 Geometric Constraint Programming
17. Draw the frame (link 1) by connecting the two circle centers. Fix the ends. The draw
moving links 2 and 4. Trace the coupler. Add dimensions to keep the lengths fixed.
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5 Velocity Analysis
5.1 Vector Operations
5.1.1 Dot Product
5.1.2 Cross Product
5.1.3 Derivatives of Vector Products
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5.2 Velocity with a Rotating Reference Frame
5.2 Velocity with a Rotating Reference Frame
Draw two vectors. The first vector, r 1, describes the position of a point at time t 1. The secondvector, r 2, describes the position at time t 2. Draw tangent and normal unit vectors at each position.Also, draw ∆r .
To determine velocity, take the limit:
Relate inertial unit vectors to rotating unit vectors:
Determine the velocity based on the product rule:
But r can also be written in terms of a tangent-normal system:
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5.2 Velocity with a Rotating Reference Frame
Take the derivative with respect to time to get the velocity:
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5.3 Graphical Analysis
4. Measure Magnitudes
reference x
0v
Goal: Determine the angular velocity of link 4.
5.3.2 Four-Bar Mechanism
Example of a four-bar mechanism: Determine the angular velocities of link 3, link 4, and thevelocity of point C. See
Clegh
Sectio
3.3
where r 1 = 600 mm, r 2 = O2 A = 140 mm, r 3 = 690 mm, r 4 = 400 mm, r 5 = 200 mm, r 6 = 200mm, θ 2 = 240
◦, θ 3 = 44◦, θ 4 = 116
◦, θ̇ 2 = ω 2 = 50 rad/sec (constant).
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5.3 Graphical Analysis
r 6
r 3
r 4
A
BC
O2
O4
x
y
θ 4θ 2
θ̇ 2
r 1
r 5
θ 3
Figure 5.1: Four-Bar Mechanism Example
Use the space allocated in Fig. 5.2 to construct the velocity polygon for this example. Recall
velocity equivalence:
v B = v A + v B/ A = vO4 + v B/O4
1. Determine the angles for the lines of action.
2. Draw v A
3. Draw a construction line at the angle of v B through Ov.
4. Construct a line through v A perpendicular to r 3 to show the line of action of v B/ A
5. Find the intersection of the v Bconstruction line and the v B/ A construction line. Draw thevector representing v B/ A and measure its length.
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5.4 Analytical Analysis
reference
x
0v
Figure 5.2: Workspace for graphical velocity determination for
the four-bar mechanism in Fig. 5.1.
Notes on velocity images:
• The velocity image has a triangle...
• The ratio...
• If no angular velocity, then...
• The point 0v...
• Determining absolute velocity ...
5.4 Analytical Analysis
Once the position vectors are known, then they can be differentiated with time to get the velocity
equations. A couple examples show this technique.
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5.4 Analytical Analysis
5.4.1 Inverted Slider Crank
Determine θ̇ 4 and ṙ in the following inverted slider crank mechanism:
r 2
A
O4
C
O2 x
y
θ 2θ 4
θ̇ 2
r
r 1
where r 1 =r 2 =r =θ 2=θ 4 =θ̇ 2=
1. Loop Equations:
a) x :
b) y :
2. Differentiate with respect to time:
a)
b)
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5.4 Analytical Analysis
3. Arrange in matrix form:
4. Solve the linear equation.
a) On TI-89:
b) In Matlab:
c) By Hand:
Compare graphical solution to analytical solution:
Variable Graphical Analytical
We can write a computer program to solve the above system for various angles of θ 2 and a
constant angular velocity. An example in Matlab is as follows:
1 %ME 3 2 1 2 : M e c h a n i sm s
%I n v e r t e d S l i d e r Cra nk
3 %F i nd i ng t h e v e l o c i t i e s o f l i n k s%Dr . J e r e my D a i l y
5%c l o s e a n i n i t i a l i z e s y s t e m
7 c l c
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5.4 Analytical Analysis
c l e a r a l l
9 c l o s e a l l
11 %I n v e r t e d s l i d e r c ra n k me c h a n is m
% k n o w n s:
13 r 1 = 45r 2 = 2 5
15 omega2=20
17 t h e t a 2 = l i n s p a c e ( 0 , 2 ∗ p i , 1 5 0 ) ;
19 %p o s i t i o n s o l u t i o n :
r = s q r t ( r 1 ^ 2+ r 2 ^2−2∗ r 1 ∗ r 2 ∗ c o s ( p i−t h e t a 2 ) ) ;21 t h e t a 4 = a c o s ( ( r . ^ 2 + r 1 . ^ 2 − r 2 . ^ 2 ) . / ( 2 ∗ r ∗ r 1 ) ) ;
t h e t a 4 = a t a n 2 ( r 2 ∗ s i n ( t h e t a 2 ) , r 1 + r 2 ∗ c o s ( t h e t a 2 ) ) ;23
%v e l o c i t y a n a l y s i s25 f o r i =1: l e n g t h ( t h e t a 2 )
A= [ c o s ( t h e t a 4 ( i ) ) −r ∗ s i n ( th e t a 4 ( i ) ) ;27 s i n ( t h e t a 4 ( i ) ) r ∗ c o s ( th e t a 4 ( i ) ) ] ;
C= [ −r 2 ∗ omega2 ∗ s i n ( t h e t a 2 ( i ) ) r 2 ∗ omega2 ∗ c o s ( t h e t a 2 ( i ) ) ] ’ ;29 v=A\ C ;
r d o t ( i ) = v ( 1 ) ;
31 omega4 ( i ) = v ( 2 ) ;
en d
33
%P l o t t he r e s u l t s
35 F= f i g u r e ( 1 )
s u b p l o t ( 2 , 2 , 1 )
37 p l o t ( t h e t a 2 ∗ 1 8 0 / p i , t h e t a 4 )x l a b e l ( ’ \ t h e t a _ 2 ( d e g ) ’ )
39 y l a b e l ( ’ \ t h e t a _ 4 ( r a d ) ’ )
g r i d on
41 a x i s t i g h t
%s e t s t he c u r r e n t a x i s t o h a v e t i c k s e v e ry 45 d e g
43 s e t ( gca , ’ X t i c k ’ , [ 0 : 4 5 : 3 6 0 ] )
45 s u b p l o t ( 2 , 2 , 2 )p l o t ( t h e t a 2 ∗ 1 8 0 / p i , r )
47 x l a b e l ( ’ \ t h e t a _ 2 ( d e g ) ’ )
y l a b e l ( ’ s l i d e r p o s i t i o n (mm) ’ )
49 g r i d on
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5.4 Analytical Analysis
a x i s t i g h t
51 s e t s t he c u r r e n t a x i s t o h a v e t i c k s e ve ry 45 deg
s e t ( gca , ’ X t i c k ’ , [ 0 : 4 5 : 3 6 0 ] )
53
s u b p l o t ( 2 , 2 , 3 )
55 p l o t
( t h e t a 2 ∗ 1 8 0 / p i
, omeg a4 )x l a b e l ( ’ \ t h e t a _ 2 ( d e g ) ’ )
57 y l a b e l ( ’ \ omega_4 ( rad / s ) ’ )
g r i d on
59 a x i s t i g h t
s e t ( gca , ’ X t i c k ’ , [ 0 : 4 5 : 3 6 0 ] )
61
s u b p l o t ( 2 , 2 , 4 )
63 p l o t ( t h e t a 2 ∗ 1 8 0 / p i , r d o t )x l a b e l ( ’ \ t h e t a _ 2 ( d e g ) ’ )
65 y l a b e l ( ’ s l i d e r v e l o c i t y (mm/ s ) ’ )
g r i d on67 a x i s t i g h t
s e t ( gca , ’ X t i c k ’ , [ 0 : 4 5 : 3 6 0 ] )
69
s a v e a s ( F , ’ I n v e r t e d S l i d e r C r a n k V e l o c i t y . p d f ’ )
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5.4 Analytical Analysis
0 45 90 135 180 225 270 315 360
−0.5
0
0.5
θ2 (deg)
θ 4
( r a d )
0 45 90 135 180 225 270 315 360
30
40
50
60
70
θ2 (deg)
s l i d e r
p o s i t i o n ( m m )
0 45 90 135 180 225 270 315 360
−6
−4
−20
2
4
6
θ2 (deg)
ω 4 ( r a d / s )
0 45 90 135 180 225 270 315 360
−400
−200
0
200
400
θ2 (deg)
s l i d e r v e l o
c i t y ( m m / s )
Figure 5.3: Output of the velocity analysis program of an inverted slider crank.
5.4.2 Four Bar Mechanism
Recall the four-bar mechanism from Section 5.3.2 on page 83.
1. To find θ̇ 3 and θ̇ 4 by starting with the loop closure equations:
a) x :
b) y :
2. Differentiate each equation with respect to time:
a) ˙ x :
b) ˙ y
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5.4 Analytical Analysis
3. Arrange in matrix form:
4. Solve the linear equation.
a) On TI-89:
b) In Matlab:
c) By Hand:
For the velocity of point C:
5. Write a vector equation from O2 to C :
6. Break in to x and y components.
a) x :
b) y :
7. Differentiate with respect to time.
vc, x =
vc, y =
8. Substitute the appropriate values.
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5.4 Analytical Analysis
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5.5 Homework Problem Set 6
r 5
r 6
r 3
r 2
r 4
A
B
C
O2O4 x
y
θ 4θ 2θ̇ 2
r 1
θ 3 φ
Ans: θ̇ 4 = 45.51 rad/s, |vc| = 9.025 m/s and ∠vc = 137.9◦
3. Determine the velocities of points B, C, and D along with the angular velocity of the con-
necting rod (△ ABC ) in a double slider crank where r 2 = 2 in., r 3 = 10 in, CA = 4 in,CD = 8 in., θ 2 = −120
◦, θ̇ 2 = ω 2 = 42 rad/s cw, and O2 is at (−3,0).
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5.5 Homework Problem Set 6
Ans: |vc| = 67.9 in/s and ∠vc = 154◦
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6 Acceleration Analysis
Graphical techniques are discussed in Cleghorn’s Chapter 3. This section is focused on analytical
techniques. Simply put, the acceleration is the time derivative of the velocity equations.
Accelerations are needed to determine forces: F = m a
6.1 Accelerations in Four-bar Mechanisms
Start with velocity equations from the example in Section 5.4.2 on page 91. SeeClegh