Mechanisms Course Notebook

8/9/2019 Mechanisms Course Notebook

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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


3.6 Multi Loop Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.7 Toggle and Limit Positions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.8 Transmission Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61


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


5.3.2 Four-Bar Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 835.4 Analytical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86


5.4.2 Four Bar Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91


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6 Acceleration Analysis 97

6.1 Accelerations in Four-bar Mechanisms . . . . . . . . . . . . . . . . . . . . . . . 97

6.2 Inverted Slider Crank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98


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


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


8.5 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.

pdf

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


2.3.1 Low Order Pairs

• Spherical (G) SeeClegh

Sectio1.4

––

–

–

–

• Revolute (R)

–

–

–

–

–

• Cylindrical (C)

–

–

• Prismatic (P)

–

–

–


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• 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)

–

–


• 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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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


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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2.5.2 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


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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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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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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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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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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(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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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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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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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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3.1.4 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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3.1.5 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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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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3.1.5.2 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


r1 10 in 1 0 0

r2 3 in 2

1.5000 2.5981r3 12 in 3 9.233836209 7.9632

r4 8 in


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


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


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


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


r1 10 in 1 0 0

r2 3 in 2

1.5000 2.5981r3 12 in 3 9.030990743 3.1550

r4 8 in


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


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


r1 10 in 1 0 0

r2 3 in 2

1.5000 2.5981r3 12 in 3 5.884873205 6.8604

r4 8 in


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


Plot Coordinates

8

6

4

2

0

2

4

4 2 0 2 4 6 8 10 12

Upper Leg

Lower Leg


47/147

3.2 Coupler Curves

3.2.2.3 SolidWorks Implementation


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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.


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:


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


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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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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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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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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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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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


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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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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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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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. Create a new sketch on any plane.


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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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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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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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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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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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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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14. Create another set of congruent triangles to represent the other connecting point for the

coupler.


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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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16. Once the position is where you like, Select All and make a Block (Tools ⊲ Make Block).


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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


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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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 ...


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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5.4.1 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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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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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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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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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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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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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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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

Documents

Mechanisms Course Notebook