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A Journal of Applied Mechanics and Mathematics by DrD

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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.



Puzzled (A Poll)

The response here at Mechanical Engineering Forums, or should I say, the lack of response, has left me puzzled. There was a modest response (as indicated by comments) to my first post, but the number of comments has dropped to almost nothing since then. I am only aware of one person who has actually worked on one of the challenge problems that I have posed (but I hope that there are more who have). In the previous poll, there have been a good number of views, but an extremely small number of people have answered the poll. What does this all mean?

There are a lot of possible interpretations. Is the material too difficult? Is the material too simple? Are the topics boring? Are the topics too general? Is the application of these ideas not evident? Do you want to see more problems carried through to numerical answers?

Would you like to see more articles on other topics, such as (1) vibrations, (2) stress and deflection analysis, (3) gears, (4) cams, (5) electromechanics, (6) applied mathematics, (7) computer methods?

Would you like to see more articles on specific applications, with the presumption that you already know all of the necessary back ground? I’m thinking, for example, of an article I intend to write eventually about a vibration attenuation system. In order to understand that article, the reader is going to need to have a general background in multi-degree of freedom linear vibrations. Are all readers ready to simply jump into that subject?

Please answer the following poll questions, and add your comments on this topic at the bottom. I am most interested in your feedback.



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, # 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.




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.




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.



#2 -- Box Tipping

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.



#1 -- Introductions

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.