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A Journal of Applied Mechanics and Mathematics by DrD

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#42 Gear Pair Problem


Mechanics Corner
    A Journal of Applied Mechanics and Mathematics by DrD, # 42
    © Machinery Dynamics Research, 2017

Gear Pair Problem


    In this post, I want to discuss a seemingly simple problem currently being discussed at Physics Forums (PF). The original question, posed by someone, perhaps a student but perhaps not, is quoted below:


So, we have a pinion and a gear. I give an input torque Tp in the clockwise direction. Therefore, the pinion will rotate with ωp angular velocity in clockwise and the gear ωg in counter-clockwise. There is a load TL against the gear motion. The bearing friction both in pinion and gear are considered by means of linearly-viscous damping coefficients cp and cg for pinion and gear, respectively. The friction between the gear mesh is neglected at this point. The moments of inertia of the pinion and the gear are Ip and Ig, respectively. Moreover, the radii of the pinion and the gear are rp and rg, respectively. My question is what the output torque To is because I want to find the efficiency of this gear pair.
I have tried four options for To and simulated them in MATLAB, but I have not found the correct results yet. Followings are the explanation of each option I tried for To.

    The sketch in Figure 1 and the two paragraphs following are exactly as posted by the original questioner. There follows on PF a long sequence of responses and more questions, but he still seems no closer to understanding what is going on. Let us see what we can do to help him.
    Before someone thinks badly of me for not helping him, let me say that I did give several hints, but the rules of PF forbid me to actually post an analysis. I have been severely scolded in the past for doing just that.

42 Gear Pair Problem.pdf


Mechanics Corner

A Journal of Applied Mechanics and Mathematics by DrD, #41

(c) Machinery Dynamics Research, July 2017


Modeling Hysteresis

1. Introduction

What do you know about hysteresis? Many Mechanical Engineers will associate this term with the magnetization curve of a piece of magnetic material, and quickly conclude, "I don't have to worry about that!" But that would be wrong. While hysteresis does occur in magnetic systems, it happens in many other situations as well, many of them situations of concern to mechanical engineers.


Figure 1 Typical Hysteresis Curve


Figure 1 shows a typical hysteresis curve, and it makes no difference as to what physical phenomena are involved. The red curve is the actual hysteresis curve. The blue curve is called the "spine."


41 Modeling Hysteresis.pdf



Mechanics Corner

A Journal of Applied Mechanics and Mathematics by DrD, #40

July, 2017


Two Short Math Problems

Do you ever read the ads that appear on ME Forum? I try to avoid them as much as possible, but an organization called BRILLIANT has put up some interesting math problems of late that have caught my eye. Two of them are the subject of today's post.

The first problem that I want to discuss is actually more recent than the other, but it gives us a good place to start. Following that, we'll go on to the second problem. Along the way, I want to talk about philosophy as well as simply how to solve tow specific problems. The main lessons to be learned here are in regard to how we use mathematics in the practice of Mechanical Engineering.

40 Two Short Math Problems.pdf


    Mechanics Corner
    A Journal of Applied Mechanics and Mathematics by DrD, #38
      Machinery Dynamics Research, 2017


Rocket Homework Problem


    Most engineers find problems involving rockets to be exciting. There is something about a rocket that fires our imagination, whether we think of going to the moon or one of the planets, or simply of shooting down an incoming missile. The subject of this post involves a rocket on a mobile launcher. The rocket is intended to be transported in a horizontal position, but it must be elevated in order to be fired. Both positions are shown in the accompanying figure.

Read the attached PDF for more on this problem.



Addendum:  One reader has posted a proposed solution for this problem as a comment. It was not my intent that solutions be posted in the comments at all. I only want solutions sent to me by the personal message system. DO NOT POST YOUR SOLUTION IN THE COMMENTS!!

Regarding the solution that has been posted, let me say the following:

1. Some of the answers are correct, while others are not. Do not be misled into following this solution because there are errors therein.

2. Even where the results are correct, there are a number of methods that I would not recommend using. Thus again, I say to all other readers, do not follow this solution, but work it out for yourself.

3. Be sure to document your solution, so that if someone else were to ask how you obtained a particular result, you would be able to explain it in a clear and reasonable manner.




Mechanics Corner
A Journal of Applied Mechanics and Mathematics by DrD, #39
(c) Machinery Dynamics Research, 2017

Comments on a Textbook
Theory of Machines
R.S. Khurmi & J.K. Gupta

1 Introduction
Recently, through the wonders of the Internet, I have come across a copy of the textbook Theory of Machines by R.S. Khurmi and J.K. Gupta (S.Chand & Co., Ltd., 2005). Since theory of machines has been my primary technical interest since the early 1980s, I was interested to see what would be in this book, particularly in view of the many favorable comments posted in regard to it. Many people seem to think that this is a most excellent book, and I’m always interested to see what brings forth comments of that sort.

As I looked through the Table of Contents, I saw that one of the last chapters was given to the topic of Torsional Vibrations (Ch. 24). Since the area of torsional vibrations has been a topic of intense personal interest for 40+ years, I was naturally drawn to this chapter. The comments that follow are based on what I found in that chapter; I have not reviewed the remainder of the book at all. In my comments below, I will refer to the authors, Khurmi and Gupta, simply as K&G to avoid writing their names out repeatedly.

One of the things I think is necessary in a textbook is that it should be directed toward teaching students to solve real problems, not simply textbook examples. Certainly, textbook examples should be simple so that they can be easily understood, but they should also be as general as possible. Where they involve special, limiting assumptions that may likely not be true in actual practice, this should be made clear. Failure to do that marks an author as one who has never actually done engineering in the real world. If the assumptions are not made clear, there is a tendency for students to later want to simply apply directly the results from the textbook problem, not realizing that they may not apply at all. So, what did I find?
Comments on Textbook - Khurmi.pdf
Mechanics Corner
Where Would You Publish It?
Since long before my time, there has been a desire to have important results published where they become accessible to many others. Some of the great names, such as Newton, Euler, Bernoulli, and others, we know primarily because of what they published. Their work formed the fundamentals upon which modern engineering and science is built. Publication of research results has long been particularly important to faculty members; it is often taken as a measure of just how intelligent and useful they are (there is a lot of doubt about the validity of this measurement, but that has not prevented it use). When I was a young faculty member (many, many years ago), there was the mantra "Publish or Perish." This referred to the idea that those faculty members that did not publish research work would not receive tenure, and would be out of employment after several years. Agencies that funded research were eager to see publication of results that they had funded; it was considered evidence of the importance of the work supported by the agency. This was particularly true of the National Science Foundation (NSF) and other governmental funding sources in the USA.
It was not too long before publication was replaced as the measure of academic value, to be replaced by funding. A faculty member was expected to write research grant proposals, and the Dean's Office expected a significant cut of the proceeds, ostensibly for their role in "supervision." In practical terms, Dean's Offices almost never contributed anything of value to research efforts, but this was a form of graft to assure their cooperation. But publication remained essential as well. Any research that could not be published in a reputable journal was considered to be unworthy, a waste of time. So the criteria for success became, get money and publish, a tougher goal that simply publishing.
More recently, the goal posts have been moved again. Today the big cry is for "undergraduate research." To my mind, this is the height of absurdity. For folks who are just beginning to learn a profession, how can anyone think that they are capable of fundamental new discoveries? For undergraduates that are still struggling with Mechanics of Materials, do we really expect them to discover new understanding of fatigue or fracture mechanics? For a student laboring to understand dynamics, do we really expect them to come up with breakthroughs in orbital mechanics, seismic shock resistance, or multidegree of freedom models for gear box noise? But, rest assure, there is no place more insane than a university!! The utterly absurd is treated as absolutely essential!!
Thus far, I've talked a lot about academia, but we must not neglect industry. Publication is important to industrial firms as well, although for different reasons. Published research, done by your firm, is a way of establishing the technical excellence of your company. If you want to be known as an industry leader in your area, you want your employees to publish work that makes the company look like it is on the cutting edge of new technology. Often industry imposes constraints on what can be published; they do not want proprietary information to be put into the public domain. But they really like to have results published that make them look sophisticated, ahead of the pack, so to speak.
For consulting engineers, publication can be important as a means to establish your expertise in an area. If you publish a lot in a particular subject area, people begin to think you kow something about the area and come to you when they have problems. New work is the life blood of consulting engineers, so this can be very important. You will also be asked to review the work of others and to sit on panel discussions and other public appearances that can upgrade your image and bring in more work.
I hope that it is evident that most engineers will need to publish some work at some point in their career. It may be a central matter of those in more research oriented areas, or it may be only occasional for those in less cutting edge business sectors, but everyone will eventually need to publish something. So, back to the original question: Where Would You Publish It?
Most professional societies publish research work, and there are also a vast number of trade magazines. Fifty years ago, when the volume of "research" was much less, it was not too difficult to publish through any number of venues. I have published articles through the various Transactions of the American Society of Mechanical Engineers (ASME), through the Transactions of the Society of Automotive Engineers (SAE), and the Journal of Mechanism and Machine Theory. I have also published through some much less well known venues such as Machine Design magazine, and most recently through IPTEK Journal, a small journal headquartered in Indonesia (that was an experience!) and other places. But the game is ever changing!
When I first began to publish papers back in the 1960s, it was a fairly simple process. You wrote up your text, with figures and equations, and mailed it to the editor in type written form (this before the days of word processing). After a few months, you would get something back from the editor. It might be an outright acceptance (rare), a conditional acceptance which meant that the paper would be accepted with certain modifications/corrections that were described in the letter (fairly common), or it might be a flat rejection (not extremely uncommon). If you got a conditional acceptance, you made the revisions, and about 6 months later, it would be published in whatever journal you were dealing with. The classier the journal, the higher the standards were, but all worked about the same.
Many of these organizations that publish papers also hold meetings, and they want people to come to the meetings. I have presented papers at the ASME Winter Annual Meeting (always in New York), at various SAE meetings, etc. But, there is a problem. It is expensive to go to these meetings. There is the travel expense (transportation, hotel, food, etc), and there is usually an admission fee (you have to pay money to present your own paper, an absurdity, but very real). Often the papers is only accepted for publication if you agree to come to the meeting to present it and pay the admission fee. Now if your paper is the result of funded research, or if your employer will pay the expenses, this is usually not a personal burden. If neither of these apply, the burden of the costs fall of the individual, and it is often prohibitive, often approaching $1000. The publisher then sell your work for a subscription fee, usually several hundred dollars per year. Libraries are the principal subscribers (university, municipal, and industrial libraries), along with a few individual.
In recent years, there has been a glut of material offered for publication, and everybody thinks that their paper is extremely important for the world to see. The volume of publications have increased drastically, but so has the cost. Who will pay for all the paper, printing, etc.? For years, it has been common to impose what are called "page charges," typically around $100 per page, to publish in most journals. Funded research usually included a line item for page charges, so that paid those bill. In the past, any unfunded research, if it was accepted, would usually be published with the page charges waived. Today, that is not longer true, and page charges are usually mandatory. But it gets worse.
We all know the Internet is a wonderful thing, but it does have some downsides as well. One of those downsides is in the area of publication. There is a relatively recent trend in publication called "Open Access," and it is particularly popular with a number of on-line journals. These journals are free to all on the internet, but the journals charge the authors a very steep price to publish their work. Thus you, as an author, must prepare the article according some very demanding rules about formatting, style, etc, then you must pay several thousand dollars, just so the world can see your work. It means that your work becomes available to all for free (which is a good thing), but it means that you the author must bear the full cost of supporting the publishing operation. I know that I, as an individual, cannot afford this, and thus it is almost impossible for me to publish anything now. It means that those with money will get their work published, and those without money will not. The quality of the published work is virtually certain to decline, but that is modern life. What can you do?
As a closing note, I'm currently writing another technical paper that I would like to publish, preferably where folks who work with IC engines will read it. I think I have something of real value to present, but I have no idea where I will publish it, or if I will be able to find a publisher at all. If any readers have a suggestion for an appropriate journal, I would certainly appreciate a suggestion in the comments.
Mechanics Corner
    A Journal of Applied Mechanics and Mathematics by DrD, # 37
    29 April 2017

Two Balls Rolling On An Incline
A Problem Where I Learned Something New

    In previous articles, I have mentioned another web site called Physics Forums (PF) where people post problems for which they need help. In this note, I want to present to you one such problem and it solution, along with a new insight that came from another commenter at PF, one of the advisory folk on that site. At first, I thought the adviser was wrong, but it turns out that he was correct and had something new that I had never seen before. Here is the problem.
Problem Statement
    A thin wall spherical shell with a mass of 0.605 kg and a radius of 0.0402 m is released from rest at the top of an incline. The spherical shell rolls down the incline without slipping. The spherical shell takes 7.49 s to get to the bottom of the incline.
    A solid sphere with mass of 0.127 kg and a radius of 0.1123 m is released from rest at the top of the same incline. The solid sphere rolls down the incline without slipping. How much time does it take for the solid sphere to reach the bottom of the incline.
    Note that ---
Thin spherical shell        I=(2/3)MR^2
Solid sphere        I=(2/5)MR^2

    The original problem statement is above. Note what is given, and perhaps more importantly, what is not given. In particular, we are not given
1.The time for the solid sphere to reach the bottom -- this is the item to be determined;
2.The angle of the incline;
3.The length of the incline;
4.The local value of g, the acceleration of gravity.
    The last three items are things that we might expect to have given in such a problem, but here they are not. This is the major difficulty in this problem, and the solution must find a way to work around this missing information.
    Mechanics Corner
    A Journal of Applied Mechanics and Mathematics by DrD, #20
    © Machinery Dynamics Research, 2015

A Question of Stability
    The word stability in its several forms is widely used in nontechnical communication. A person whose life it highly consistent from day to day is said to have a stable life. When the political situation in a particular area appears to be unlikely to change, it is said to be stable. A person who is well balanced and unlikely to be easily provoked to anger is said to be a stable person. When the medical condition of a sick or injured person ceases to get worse, the person is said to be stabilized. A company on the verge of bankruptcy is said to be an unstable company. But what does the word stability mean in a technical context? Each of the foregoing examples hints at the technical meaning without really being explicit about it.

A factor g = accel of gravity was missing in the potential energy expression. That is now corrected.
Mechanics Corner
    A Journal of Applied Mechanics and Mathematics by DrD
    © Machinery Dynamics Research, 2016

Becoming An Expert -- Part 2
    In the previous article on How To Become An Expert, I covered a lot of points in generalities with some short anecdotes from my own experience. In this article and the next, I will describe in considerably more detail a critical period in my own formation, an time of considerable professional embarrassment which was a real spur to learning.
    In the summer of 1974, I took a position as the head of the engineering analysis section with a large diesel engine distributor in Houston, TX. This company purchased diesel engines, mostly from General Motors (GM) and packaged them on a skid with some driven machine such as a generator, a pump, air compressor, or other driven machine, along with the required controls. For me, it was a fascinating place to be as I had always been intrigued by diesel engines. I soon found out how little I actually knew about the whole matter.
    The analysis section consisted of three other engineers (two men from India and a lady from Turkey) and myself. The men were there before I came, and I hired the lady. They were all good workers, but they were best at following directions. They did not ask "Why?" very often. If this is the way it had been done previously and nobody objected, they would repeat that same pattern over and over without wondering why we do it that way. More about that aspect later.
    This was a time of great activity in terms of nuclear power plant construction in the USA, and the company was building a lot of very large engine-generator sets to serve as standby power in nuclear power plants. In a nuke plant, pumps continuously supply cooling water to the core to take away the heat and as a means to move heat to the steam generators. If those pumps fail for any reason, the core can over heat and meltdown, a major catastrophe. The great fear was that the pumps would lose power from their regular supply, in which case the standby generators would need to start up and provide power to the pumps. The proposed cause of loss of power was an earthquake, and that meant that the standby generator set must survive the earthquake and be able to start and run.
The following is a verbal description of a Doonesbury cartoon of unknown date by Garry Trudeau. Doonesbury has long been one of America’s major cartoon strips, with a very dry wit and a decidedly left-of-center outlook. I found this today in going through some old files.

SCENE: A college classroom, the teacher lecturing in a rather absent minded fashion, the students silently bent over, taking notes and keeping their heads down.

TEACHER: Of course, in his deliberations on American capitalism, Hamilton could not have foreseen the awesome private fortunes that would be amassed at the expense of the common good.

TEACHER: Take the modern example of the inventor of the radar detector. In less than ten years, he made $175 million selling a device whose sole purpose is to help millions of people break the law.

TEACHER: In other words ...

STUDENT (suddenly sitting up and interjecting): Maybe the fuzz buster is a form of Libertarian civil disobedience, man. You know, like a blow for individual freedom.

TEACHER: I ... I don’t believe it!

STUDENT: Believe what, man?

TEACHER (smiling in happy elation!): A Response! I finally got a thinking response from one of you. And I thought you were all stenographers! I have a student! A student LIVES!

TEACHER (kneeling down, hand extended like one might approach a shy animal): Who are you lad? Where did you come from? Don’t be frightened ...

STUDENT: (looking around himself): What’s the deal here? Am I in trouble?

The above all appeared in print many years ago, but it is an apt description of Mechanics Corner.
    Mechanics Corner
    A Journal of Applied Mechanics and Mathematics by DrD, # 34
    © Machinery Dynamics Research, 2015

A Problem in Statics & Dynamics
        A problem was recently posted on this Forum, requesting help, that has led me to consider a somewhat more general problem for this post. The scope of this post will include the original problem, although not by the method required there, but will also go beyond to a more general geometry. We begin here by stating the present problem; interested readers are invited to search back for the original problem posted 19 December, 2016, by iivii.

Assembly Drawing, with Dimensions
Mechanics Corner
A Journal of Applied Mechanics and Mathematics by DrD, # 4
© Machinery Dynamics Research, LLC, 2015
Vector Loop Kinematics -- Part II
Velocity Analysis

In the previous article in this series, titled "Vector Loop Kinematics - Part I/Position Analysis" the idea of using closed vector loops for the position analysis of mechanisms and machines was introduced. This is an extremely powerful method; I have never found a kinematics problem that was beyond its scope (now watch someone challenge me with such a problem!). As we left it at the end of that article, the technique of finding out all of the position information was at hand, but we had done nothing at all about discussing velocities or accelerations. This article will introduce the extension of this method to velocity analysis, but accelerations are differed until a later article.
This article is built upon the previous article, even to the extent of using some of the same example problems. If you have forgotten the content of the previous article, you might want to review it before getting to far into the present article.
Position Analysis

Many years ago, when I first began to study mechanics, the "conventional wisdom," expressed by both teachers and fellow students, was this: "Statics is easy, Dynamics is hard, and Kinematics -- who bothers to actually study kinematics? Kinematic relations, when needed, simply drop from the sky like rain, but nobody seriously studies kinematics." I eventually found the truth to be a bit more subtle: Statics of structures is generally easy, while the statics of mechanisms and machines may, or may not, be easy, depending a lot on the kinematics. Further, I found that the key to most dynamics problems is having a good tool to deal with the necessary kinematics.
The purpose for this article is to present the most powerful tool I have ever found for dealing with mechanism and machine kinematics, the vector loop method. This will be demonstrated in the context of two simple problems.
Welcome to the first installment of Mechanics Corner, a feature that we hope will become a regular blog item on Mechanical Engineering Forum. The intent is that every week we will have a new article on some aspect of Applied Mechanics and Mathematics, things of broad interest to mechanical engineers. Some of these articles will be fairly elementary, while others will be considerably more advanced, but the idea is to have something for everyone. We will hope to amuse, entertain, and most importantly, to inform you with each article. We may even have a little bit of engineering humour from time to time!

Most of these articles will involve the use of pictures, diagrams, and mathematics, all things that are fairly difficult to accomplish in a blog post. For this reason, and beginning today, the bulk of the post will be in an attached PDF file. It turns out that there are fairly simple ways to have all the necessary tools available in a PDF file even though they are not available directly on the Internet. Thus I hope that each of you will click on over to the attached PDF file to read the rest of today's article.
Mechanics Corner is written by DrD, which of course raises the question, "Who is DrD?" Well, that’s me, but that doesnt tell you very much, does it? My intention is to say very little about myself in most of the articles, but since today is the day for introductions, for the blog, for myself, and a few other matters, it seems appropriate to tell you a little bit more about who I am.
I am an elderly man, a semiretired engineer and Mechanical Engineering Professor, living in Texas, USA. All of my engineering degrees are from The University of Texas at Austin, and I have professional engineering registrations in both Texas and Wisconsin (in the USA, engineers are licensed by the several states to prevent incompetents from holding themselves out as engineers and thus endangering the public safety and welfare). As I mentioned, I am old, many would say "older than dirt." I have had a long and very interesting career as an engineer working in a number of different industries, as an engineering faculty member, and as a consulting engineer (I continue to do some consulting yet today). In my engineering career, I have worked in the automotive, aerospace, naval, offshore, gas compression, steel, and electric power generation industries. I have worked on diesel and natural gas engines, steam turbines, gas turbines, large electric motors, generators and host of other machine types. If it moves, it is likely that I have worked on it; if I have not, I sure would like to work on it!
Being as old as I am, I have seen a lot of changes in my life, a few of which I wanted to touch on here. One of the most profound changes has been the shift of manufacturing industry away from the USA to India, China, Mexico, Brazil, and Southeast Asia. When I was young, the USA was arguably the greatest manufacturing power in the world, but that is no longer true today. We talk about being an "information society" (although Im not sure what that is) but we have very little to do with machinery and similar things today. But here I am, and this is one of the reasons why I feel a need to talk with many of you in the developing countries.
In October, 1957, Russia put the Sputnik satellite in orbit. It was a tiny thing, about the size of a soccer ball, but it shook the world. At that time I was in my last year in high school, preparing to go off to study engineering in college. Sputnik caused a great shake-up in American engineering education, with many warning cries that we were "behind" and had to "catch up." This meant many changes in education, but one in particular: now everything had to be done in vector notation, something that had not been done much before. In my freshman year in college, I took the introductory Mechanics course in physics, and fell in love with the subject matter. As a result, I have been studying mechanics, in one form or another, for well over half a century. Interestingly, although I initially learned everything in vector notation, I have come to the conclusion that I prefer to use scalars wherever possible. In particular this means the use of energy methods whenever they are suitable.
We all go off to college to study mechanical engineering; this is how we enter the profession. We are constantly told that we must never stop learning, but how many really believe that? Do you still hit the books every night? Are you still doing homework problems? I want to tell you a story about learning after school has ended.
About 12 years after I completed a Ph.D., I was on the faculty at Texas A&M University, one of the great engineering schools of America. I was assigned to teach Theory of Machines, and I figured that I could handle it, even though I had never had such a course in my own education. I selected a textbook that was somewhat unorthodox but I thought it looked attractive. It was a very good textbook, and it proved to be one of the greatest learning experiences in my whole life (there is nothing like trying to teach a course to be sure that you learn the course). I struggled to stay a few days ahead of the students, but that book brought me many new and powerful ideas that I had never seen previously. At the end of the semester, I asked the students what they thought of the book. They hated it! Their complaints really came down to two things: (1) the book was too big, too long, nearly 700 pages, and (2) the author had some really awkward notations. A few years later, when I set out to write my own textbook on Theory of Machines, I kept these two objections in mind and was able to produce what I think was a much better text. The point of the story is this: here I was, supposedly educated and having industrial experience, and yet I had the greatest learning experience of my life. It profoundly changed the way I work all kinds of problems to this day. The moral of the story is that we are never too old to learn, unless we think we are.
One of the great changes that I have seen, and you have seen it also, is the profound impact of the Internet. Thirty years ago there is no way that I would have been writing for you, and no possibility that you would have been reading the words of an old man in Texas. But all of that has changed now. Sadly, there is much trash on the Internet. On the good side, there is also much of a value. Among those good things, I would like to direct you to the site of a friend who has done some most excellent work in mechanism animations. When you see his animations you simply cannot help but have a better feel for how these machines work.
The URL is: http://www.mekanizmalar.com/ By all means take a look at them and see for yourself! I hope to see all of you and many more next week when we will go on to things of a more technical nature. Please check back here at Mechanical Engineering Forum for the next article.
Mechanics Corner
    A Journal of Applied Mechanics and Mathematics by DrD, # 31
      Machinery Dynamics Research, 2016

ODE Solution --- Fail!!
    Digital computation has become a major tool for engineers, and it is a great benefit. It can also lead to many pitfalls for the unwary. This note is about the latter, a potential pitfall that many engineers risk on a daily basis, most of them with little awareness of the danger.
    Early in the development of digital computation, every problem required that the user write a program specific to the problem at hand. If speed was a very important issue, the programs were written in machine language, so that they would execute as fast as possible. If speed was a little less critical, programs were written in so-called "high level languages." This included FORTRAN, BASIC, ALGOL, C, C++, and a host of other such names. But even with a high level language, there was the problem of generating a program for the solution of the specific problem at hand.
    As things have continued to evolve, it was soon evident that a lot of the work in writing each program was the same from one problem to the next. The major mathematical operations, such things as numerical integration, matrix operations and the solution of systems of linear equations, plotting, and many other steps were re-usable from one problem to the next. It was natural that this would eventually lead to the development of general purpose programs, able to solve broad classes of problems. This group includes programs like Mathematica, Maple, MatLab, SciLab, Maxima, TKSolver, and numerous others. Most of those just mentioned have built-in capability to solve ordinary differential equations, in some cases by analytical means, and in practically all cases, by numerical means. This has taken the sting out of working with differential equations
from many engineering problems, and we must all be grateful for that.
    At the same time, we must also be somewhat skeptical about any general purpose solver when applied to a particular problem. How do we know that the solution generated is correct? How do we even know if it is reasonable? Most of the time, when engineers resort to numerical solutions, it is because there is no readily available analytical solution. Thus, when faced with a problem that cannot be solved in closed form, how can we know when to trust the numerical solution? This is a very serious question, one that all must consider. It you blindly trust a numerical solution, the old excuse, "The computer said it was OK" will not get you very far. The computer cannot be fined, fired, or (in extreme cases) possibly sent to prison, but all of these things can happen to an engineer!
    So, what can the engineer do when the differential equation has no known solution? Well, there are several options.
    (1) He can resort to any physical principles that apply to the situation. For example, if the system is such that energy should be conserved, then he can add code to calculate the total system energy at every instant. Just verifying that energy is conserved does not "prove" that the solution is correct, but if energy is not conserved when it should be, you can be sure there is an error in the solution.
    (2) He can try various approximations that may apply to see if they are in reasonable agreement with the computed solution.
    (3) He can verify the solution code by applying it to a similar problem for which there is a known solution. It is this last approach that I want to talk about in this post.
Mechanics Corner
A Journal of Applied Mechanics and Mathematics by DrD, # 5
© Machinery Dynamics Research, LLC, 2015
Vector Loop Kinematics -- Part III
Acceleration Analysis

In the first article in this series, titled "Vector Loop Kinematics - Part I/Position Analysis" the idea of using closed vector loops for the position analysis of mechanisms and machines was introduced. A second article, "Vector Loop Kinematics - Part II/Velocity Analysis" extended the process to include mechanism velocity analysis. In this, the third article in the series, the process is extended further to cover the analysis of acclerations.

For each mechanism considered, we have first identified a single variable as the input, a variable to be assigned at will over some range representing the full motion of the system. (In so doing, we are limiting the discussion to Single Degree of Freedom systems, although this term has not yet been defined in this series.) It happens that in both examples used, the primary variable has been called θ, but there is no real significance to this naming. The position loop equations have then been written in terms of this primary variable and such other secondary variables as might be needed (secondary variables have been denoted as A, B, and x in the examples). The first step is always completion of the position solution, determining values for the secondary variables for any values of the primary varible of interest.
Mechanics Corner
A Journal of Applied Mechanics and Mathematics by DrD, #36
Base Acceleration Problem
In a recent post (#35) I mentioned that I often participate in another forum called Physics Forums (PF). The problem that I want to discuss here is an elaboration on a problem that recently appeared at PF. I'm going to add a little bit of complexity to the problem (the problme as stated at PF was extremely simple) in order to make a particular point.

The system of interest is shown in Figure 1, a body with a single wing attached to one side. You might consider this to be one side of an airplane, or perhaps a stirring paddle used to mix paint. The mass of the wing is M, and the center of mass for the wing is at the point marked CM, a known distance u from the main body. We are told that the main body has an acceleration a sub z in the z-direction, and that the whole system is immersed in a viscous liquid such that the drag force is proportional to the square of the velocity in the z-direction. Our concern is with the connection between the wing and the main body. We need to determine the shear and bending moment on that connection due to z-direction motion.
    Mechanics Corner
    A Journal of Applied Mechanics and Mathematics by DrD, # 35
      Machinery Dynamics Research, 2017

Good News --- Bad News
Rolling Disk in a Rolling Ring
    Well, it looks like Mechanics Corner is back, at least in terms of an occasional post. It will probably be less frequent than previously, but there are just too many interesting things to talk about to remain entirely silent! The title for this post may leave you wondering what is the Good News, and what is the Bad News? Why is there both? Well, let me tell you about it ...

Mechanics Corner
    A Journal of Applied Mechanics and Mathematics by DrD
    © Machinery Dynamics Research, 2016

How To Become An Expert
    This is going to be another of those personal experience/opinion pieces, so if these bore you, be warned! This may be the time to click on something else.
    A reader recently wrote to me asking how to become an expert. I have to tell you, I don't spend much time thinking about being an expert, but I suppose on some reflection, the shoe probably fits. (Most of the time, I see myself as simply a tired old man, still enjoying the things I have done almost all my working life.) In the discussion below, I will describe a few events and observations that seem to relate to the question at hand.
Find Your Place
    Nobody can hope to be an expert on everything, there is simply too much to know. You have to find the area that excites you, the area that really makes you want to dig in more. If you do not really enjoy it, you will never be an expert!
    I was very fortunate in this regard. When I was in High School, I was rather good in Mathematics, and my school advisers all told me, "You should become an engineer." Sadly, I really had no idea what that meant, and neither did they. The town where I grew up had rather little industry, and no one in my family knew an engineer of any sort. I did a little bit of research on engineering (this was thousands of years before the Internet), and it sounded interesting in a very vague way; there seemed to be little specific information available to me. But I went off to college, intending to study mechanical engineering, whatever that was.
    In my first semester of college, I took a Physics course in classical mechanics, and I really enjoyed it. This was exactly what I wanted to do, I just did not know the right name for it. I thought Newton's Second Law was the greatest thing ever discovered, and when implemented with Calculus, it was really fun. I was astounded at the power of it all, the questions that could be answered. If I could just get a job doing mechanics problems, I was sure I would be happy.
    Mechanics Corner
    A Journal of Applied Mechanics and Mathematics by DrD
    © Machinery Dynamics Research, 2017

Last Post
Time to Hang It Up
This will be the final post of Mechanics Corner here on Mechanical Engineering Forums. It has run almost exactly two years, and there have been ups and downs along the way. In this final post, I want to reflect a bit on my original goals for the blog, and also on what has actually happened.
When our host first proposed to me that I might write a blog for ME Forums, I was excited about it. About half of my career had been spent in engineering education, and I always loved working with students. It seemed like a way to get back to something that I had long enjoyed, and so I accepted his suggestion.
A long time ago, back when I was about 14 or 15 years old, in Junior High School, my shop teacher mentioned, in an off-hand way in class, that various curves could be described mathematically. I’d never heard that before, but I thought immediately, “This has interesting possibilities.” Moving ahead a few years, I discovered that I wanted to study and build my career around was the area known as Applied Mechanics, although it was a time before I first heard that term. In my freshman physics class, I discovered the laws of motion, and thought to myself, “This is great stuff! I can use math to describe how things move!” All of that happened back in the 1950s, and I’m still doing the same thing today (some might say I am in a rut!).
As a teacher, I taught mostly undergraduate engineering courses, although I taught my share of graduate courses as well. It was the undergraduate courses that I liked most, because I firmly believe that the economy of a nation is strongly dependent on the quality of the baccalaureate level engineers produced in that nation. Engineers with graduate degrees are valuable as well, but the vast majority of the national engineering workload falls to BS level engineers.    Thus, I envisioned Mechanics Corner as a sort of continuation of the several undergraduate courses I most enjoyed teaching — kinematics, dynamics of machines, vibrations, and mechanics of materials. For the most part, I have stuck to the plan, so that most of the technical posts I have made have dealt with problems that I considered suitable for undergraduate engineering students, say perhaps, junior level. I have posted a few topics from my industrial experience, but those have been situations that baccalaureate level engineers would be expected to handle.
Now I knew it would not be exactly like continuing to teach my classes. In particular, you would not have any homework or tests, and I would not have any grading to do – a win-win, or so I thought. I did hope, that even with no assigned homework, readers would take an interest in the problems discussed, even to the point of working through the details for themselves (I was terribly naive, apparently!). I knew from my own experience that the only way I ever really learned a new idea was to get in and work with it, work some problems, make some numbers, plot some curves, until I really understood what it was all about. I’ll venture to say that nobody ever learned any technical material simply by reading only.
In actual fact, in the early days, I had one or two folks say that they would in fact work through the problems, so I was encouraged. What I was not prepared for, however, was the fact that the vast majority never seemed to even read very carefully, much less work through the problems! The questions that have come, and there have been a few, have largely been about matters totally unrelated to the posts. The most common question has been, “Suggest a topic for my final project,” which relates to not a single post. Needless to say, that aspect of my vision was totally unfulfilled.
But there is another side. I ventured to write a few “philosophical” articles, items dealing with academic integrity and cheating, with how to ask for help, with how to write a report or a paper, and various other matters. I really thought all of this would be considered obvious and trivial, so I was completely unprepared for the excitement that some of these articles generated. There were, in some cases, many, many comments, and people seemed to really be interested. I’m left to wonder: why? Are these ideas foreign to the culture of India and SE Asia? Are these things not all taught at home and in the public schools? I don’t know, but there was a lot of interest in these matters.
But Mechanics Corner was intended to be primarily a technical blog, and there, it just did not excite the interest of the readership. As time passed, there was less and less interest. First, the comments dropped off to just about zero, and later, there were fewer and fewer who even bothered to “like” the articles. Finally, the number of reads has dropped to almost nothing (there may be no one left to read this final note). Well, there could hardly be any more clear indication that it is time to stop.
I asked for opinions about this from some of the administrators, and was told that the blog was just over the heads of the readership. That makes me sad; that was never the intent. If it is true, I do not see how engineering has a very bright future among this readership. Even so, I wish all of you the best for your careers. I hope that you are able to find rewarding and beneficial work in which you will be happy and make a real contribution to your societies.
To use an old cowboy metaphor perhaps familiar to many of you from Bollywood, “It is time to hang up the bridle and saddle, and say, ‘Adios’ (Adios is literally, “to God”).
Mechanics Corner
A Journal of Applied Mechanics and Mathematics by DrD, # 7
© Machinery Dynamics Research, LLC, 2015
Degrees of Freedom &  Constraints

The term "Degrees of Freedom" (often abbreviated as DOF) has been carefully avoided for the most part in these presentations up to this point, although it has crept in unavoidably a time or two. In this article, we attempted to face the matter squarely and deal with it fully. It is an important concept, one that is very widely confused, and is critical to correct understanding of countless mechanics problems. There are several other concepts that must be discussed along with degrees of freedom including the idea of a particle or point mass and the idea of various types of constraints.
This article is different from those that went before in that there is (almost) no calculation involved. It is almost entirely focused on matters of philosophy, a perspective or point of view, that has proven useful for countless generations of workers in the field of mechanics.
Mechanics Corner
A Journal of Applied Mechanics & Mathematics by DrD, #6
© Machinery Dynamics Research, LLC, 2015
AC Power in Real Variables Only

Most mechanical engineers get a pretty good understanding of DC circuits, and this carries over fairly well into single phase AC circuits. The difficulties come when we get into industry and discover that almost everything is powered by three phase AC circuits. This is where it starts getting sticky!
In the discussion of three phase AC electrical power, it is almost universal to use complex notation, otherwise known as phasor notation. For most purposes, the results might just as well be simply pulled out of the blue for all the understanding that complex mathematics gives, because everyone knows that the quantities involved -- voltage and current -- are fundamentally real, physical variables. These real quantities are not described by complex numbers, but rather by real numbers. The customary mantra says, "... we are considering the real part ...," but that really does not explain things very well because all of the mathematics being done is using complex algebra which considerably obscures the picture. Complex variables are, to use a colloquial term, "unreal." What is needed is a simple, straight--forward presentation of the problem in terms of real variables. We will give that a shot here.
The use of desktop, laptop, tablet, and other computers has become routine these days for engineering work. Along with this, there has been an ever-increasing number of software options for engineering calculations. It would be interesting to know just what software the readership here uses in their daily work and/or study.

Mechanics Corner
A Journal of Applied Mechanics and Mathematics by DrD, No. 2
© Machinery Dynamics Research, LLC, 2015
It is a common practice for manufacturers to ship their products in packing crates that are strapped down on pallets for handling. There is often concern about the stability of this package as it is handled in transit to the purchaser. For this problem, we understand that the manufacturer wants to perform a simple test on each package shipped to assure that it will not tip over in transit. The test will consist of tipping the package slightly to the left and placing a block under the right edge of the pallet. The block is then quickly pulled out and the question is whether or not the package will fall over to the right. The answer depends upon the amount of the initial tip to the left and the location of the center of mass of the combined packing crate and pallet.

It is clear that the falling box impacts the floor, causing an impulsive distributed load to act on the bottom of the package. This will apply both an impulsive upward force and an impulsive moment to act on the box. Since the actual distribution of the force is unknown (and unknowable), an impulse--momentum approach to this problem is not likely to get very far. There is, however, a much simpler energy analysis available. Go on over to the attached PDF for more details.