*A Journal of Applied Mechanics and Mathematics by DrD, #46*

*Comments on*

*A Calculus Challenge*

I would like to thank all those who took part in the Challenge. My solutions and comments are attached.

]]>A Journal of Applied Mechanics and Mathematics by DrD, #45

(c) DrD, 2018

It has been quite a while since I last posted anything here, but an interesting problem has come to mind that I wanted to share with you. If you really know calculus, this should be straight forward; if you don't know calculus, don't even try!

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THE CHALLENGE IS NOW ENDED. I WILL NOT RESPOND TO FURTHER ANSWERS. i EXPECT TO POST A SOLUTION AND A FEW COMMENTS IN THE NEXT FEW DAYS.

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]]>A Journal of Applied Mechanics and Mathematics by DrD, # 44

Machinery Dynamics Research, 2017

**Mouse Trap / Pendulum Dynamics Challenge - Part I**

**Introduction**

Mice are a problem all over the world, and as a result, I'm sure that there are mouse traps of various sorts found everywhere. It would be utterly amazing if this were not true! In the USA, there is a very common type of mouse trap that I have seen used all my life, the sort of system shown below in Figure 1. I want to spend a few minutes discussing this mouse trap, to be certain that all readers understand how it works, before moving on to the main part of the post.

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A Journal of Applied Mechanics and Mathematics by DrD, #43

(c) Machinery Dynamics Research, 2017

*Four-Bar / Toggle Linkage Mechanism*

**Introduction**

I believe that it would be correct to say that all of the single degree of freedom mechanisms that I have discussed on ME Forums have involved only a single loop. This might lead a reader to conclude that a single degree of freedom implies only a single loop, and vice versa, that a single loop implies only a single degree of freedom. Neither of these statements is true. In this note, I want to discuss a counter example, a mechanism called the four-bar / toggle linkage; it is shown in Figure 1.

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A Journal of Applied Mechanics and Mathematics by DrD

July 31, 2017

**Triple Rocker**

Over at the Kinematics of Machines club, I recently ask if anyone could show me an example of a four-bar linkage that would be classed as a triple rocker. In the terminology of four-bar linkages, a link is classed as either a crank or a rocker:

Crank - can rotate in a complete circle

Rocker - cannot rotate in a complete circle]

Thus my question was for an example of a four-bar linkage where no link is able to rotate around a full circle. My request has not generated any answers, but fortunately, I stumbled onto one.

Since the definition of a rocker is a link that cannot rotate completely, it is evident that the linkage shown is in fact a Triple Rocker. None of the links is able to move through a complete revolution. If we try to rotate the input (left) link further down, it cannot happen without stretching the combination of the coupler and the output (right) links. When the input link (left side) gets to the top, again its motion is stopped by the need to stretch the coupler and output link. Thus, a figure I drew as an illustration for something else turns out to be a Triple Rocker, the item I was looking to find.

In connection with four-bar linkages, some readers will have heard of Grashof's theorem. Let

s = length of shortest link

L = length of the longest link

p, q = lengths of the two intermediate links

Grashof's theorem says that a necessary and sufficient condition for at least one link to be a crank (able to rotate entirely around), it is necessary that

s + L < p + q

This inequality is not satisfied for the four-bar that I drew by chance, so Grashof's theorem says that none of the links can be a crank. That is precisely the condition required for a Triple Rocker (a ground link plus three moving but not fully rotating links). So, there you have it. That is an example of a Triple Rocker, and we now have the criteria for identifying such as a four-bar linkage that does not satisfy Grashof's Theorem.

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*Mechanics Corner*

A Journal of Applied Mechanics and Mathematics by DrD, # 42

© Machinery Dynamics Research, 2017

*Gear Pair Problem*

**Introduction**

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.

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."

Read more at

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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.

]]>A Journal of Applied Mechanics and Mathematics by DrD, #39

(c) Machinery Dynamics Research, 2017

**Comments on a Textbook Theory of Machines
by
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?

A Journal of Applied Mechanics and Mathematics by DrD, #38

Machinery Dynamics Research, 2017

*Rocket Homework Problem*

**Introduction**

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.

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*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.

DrD

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*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*

**Introduction**

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

**Discussion**

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.

]]>A Journal of Applied Mechanics and Mathematics by DrD, #36

*Base Acceleration Problem*

*Introduction*

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.

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A Journal of Applied Mechanics and Mathematics by DrD, # 35

Machinery Dynamics Research, 2017

**Good News --- Bad News
Rolling Disk in a Rolling Ring**

In**troduction**

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 ...

GoodNews-BadNews-DiskInRing.pdf

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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

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”).

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A Journal of Applied Mechanics and Mathematics by DrD, # 34

© Machinery Dynamics Research, 2015

*A Problem in Statics & Dynamics*

**Introduction**

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

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A Journal of Applied Mechanics & Mathematics by DrD, #33

© Machinery Dynamics Research, 2016

**Advanced Polynomial Curve Fitting**

The use of polynomials to fit engineering data is a common engineering practice. In school, we learn that "A data set consisting of n data points ((x_{i},y_{i}), i=1,2,3,…n) can be exactly fitted with a polynomial of degree n-1. Thus three data points can be fitted exactly with a quadratic expression, four data points can be fitted exactly with a cubic expression, and so on. If this approach is pursued much further, something ugly appears: while a polynomial of degree n-1 will pass exactly through n data points, for large values of n, it will oscillate wildly in between the data points. Since one of the most common reason for using a polynomial fit in the first place is for interpolation -- to be able to estimate a function value at locations between the known data points -- this wild oscillation is devastating. It is at this point that least squares fitting is usually introduced to give an approximate fit using a much lower order polynomial. A different approach is employed here.

**Mechanics Corner**

A Journal of Applied Mechanics and Mathematics by DrD, # 32

© Machinery Dynamics Research, LLC, 2015

**Braced Cantilever**

**Introduction**

Anyone who has actually gotten into machine design is familiar with this difficulty. Consider the situation where a project is well advanced, many plans have been made, and it is all based on the assumption of the adequacy of one particular part. When you finally get to the detailed analysis of that part, the calculations show that it is not adequate. What can you do?

To make the problem much more concrete, consider the cantilever beam shown in Figure 1. It supports a weight W at the free end, and when someone finally makes the calculation, the tip deflection, δ, is unacceptably large. The whole system design has been developed on the assumed adequacy of that cantilever, and there is no room to put in a beam with a larger section to give more stiffness. What can be done?

Any of countless machine design texts, mechanics of materials texts, etc., give the formula for the end deflection,

δ=(WL³)/(3EI)

where

E= Young's modulus for the beam material

I= area moment of inertia for the beam cross section

L= length of the beam

W= tip load value

While we can argue that someone should have checked this earlier, finger-pointing does not fix the problem.

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**Mechanics Corner**

A Journal of Applied Mechanics and Mathematics by DrD, # 31

Machinery Dynamics Research, 2016

*ODE Solution --- Fail!!*

**Introduction**

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.

DrD

Saurabh Jain, our host, has identified this location for me, and that is much appreciated.

When I watched the video, I was aghast at all those nearly bare feet in a machine shop! I can appreciate that in Indian culture, the simple sandals are socially quite acceptable, but from a safety perspective, this is an absolute horror. Think of all the opportunities for something to drop on a foot, a tool, a machine part, sparks, etc.

Some years ago (quite a few years ago), I worked in a steel mill. We were required to wear hard hats and steel toed shoes at all times in the mill. And these were not just any old steel toed shoes. These shoes came up ankle high, and had massive steel toes and an additional steel plate, called a metatarsal plate, that came up over the top of the foot almost to the ankle. Each shoe weighed 4 lb, and it was very tiring simply to walk around wearing them. But, .... and this is the key part .... they added much to our safety. Even today, in my advanced old age, I have a pair of steel toed boots (but not metatarsal plates) for when I go into an industrial environment.

What is shown in this video is actually a cautionary tale, a warning of just about everything not to do from a safety perspective. Take heed! Be warned, or you could easily loose all your toes on one foot of the other.

DrD

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I have had some conversations recently with JAG (one of the other writers here at **ME Forums**) regarding the choice of software for 3D modeling and analysis. JAG has made some excellent suggestions, specifically a cloud based program called Onshape. Unfortunately, for reasons that are unclear, my computer cannot run Onshape; I have worked with their help people for several hours, all to no avail. JAG recommends this in part because there is a "free version for the hobbyist" and a relatively inexpensive "full version for the professional." That is pretty attractive, but since I can't run it, I'm stuck.

I gather that virtually all engineering colleges these days are teaching some sort of 3D modeling and analysis software, but that raises a few questions in my mind.

**1. **If your college teaches brandX 3D software, what will you do when you go to work for a small company that cannot afford anything more than 2D drafting (simple CAD), with no analysis capability at all? How will you do your job then? You probably have your own pocket calculator, but will you have your own copy of ANSYS or Pro-E?

**2.** What software does your school teach (every students should have an answer to this question, so I expect lots of replies on this one!)?

**3.** If you have used software extensively for analysis of engineering problems (beam deflections, stress analysis, fluid flow, heat transfer, etc), are you confident that you will be able to work all of those problems if there is no such software available to you on the job?

I might add, as sort of a postscript, most of you know that I am older than dirt (I just had another birthday, so the situation is even worse!), so I tend to look at things from an elderly perspective. One of my great fears as a working engineer was "What will happen when I'm ask to do something that I don't know how to do?" It happened more than once, and it usually resulted in a flurry of intense research to come up to speed on whatever topic was involved. I could usually do that because I have a pretty good library, and I knew how to use a university library as well. But in terms of software, I was always concerned that I had no FEA program, so how could I do problems that others were doing by FEA? I have come up with some interesting work-arounds, including writing my own FEA for some problems, but I never wanted to be dependent on software that I could not afford to own. So, back to my questions about: **How are you going to buy your own copy of ANSYS?**

DrD

]]>A Journal of Applied Mechanics and Mathematics by DrD

© Machinery Dynamics Research, 2016

**Becoming An Expert -- Part 3**

**Introduction**

In the previous article on **Becoming An Expert--Part 2**, I mentioned that there were two big issues for the engineering analysis section at my Houston position, the first being the matter of seismic survivability and the second being torsional vibration. The first item was dealt with in Part 2, and in this article we will take up the second item of concern.

When I joined the engine distributor in Houston in the mid-1970s, the company was about 65 years old, and the torsional vibration problem was not new. This was a problem that they had been dealing with, in one way or another, for many years. There were lots of old torsional vibration analysis reports available to study. I was not at all familiar with torsional vibration of machine trains; I had not studied anything quite like that in school and it had not come up in my previous industrial experience. So I eagerly began reading the old reports, and that is when the problem became acute for me: They did not seem to make any sense. I could not, with any integrity, continue to write reports like that when I thought they were complete nonsense, but I did not know how to analyze the problem correctly. I was in a jam!

There were three major difficulties:

** 1.** The entire crank assembly rotates endlessly, so the stiffness matrix for the system is singular. This results in a zero eigenvalue, something that did not take too long to figure out.

**2.** It is obvious that the system does more than just go round-and-around; it goes up and down as well. I was baffled for a long time about how to deal with the kinematics and their impact on the dynamics.

** 3.** It is apparent that there is a torque acting on the crank, but it is not directly applied to the crank by the combustion process. There is the slider-crank mechanism between the two, and I was at a loss as to how to transfer the cylinder pressure into a crank torque. This is again directly related to the kinematic problem mentioned just above.

**Mechanics Corner**

A Journal of Applied Mechanics and Mathematics by DrD

© Machinery Dynamics Research, 2016

**A Comment Remembered**

Recently, in connection with one of the posts on Becoming An Expert, one of the ** ME Forum** readers made a comment to me, something about seeing everything in terms of differential equations. That comment brought to mind a comment made to me many years ago that I want to pass along to you today.

Most of my college education was at the University of Texas at Austin. It was there that I received BS, MS, and PhD degrees in engineering, and I was there studying for most of a decade. After I finished my PhD, I was asked to stay on the faculty for a year as an Assistant Professor, so that was my first post-graduate teaching position as well.

One of the well known faculty members at UT-Austin was Dr. E.A. Ripperger, a man with a national and international reputation for his work in plastic stress wave propagation. In addition to his teaching responsibilities, Dr. Ripperger directed a laboratory at the Balcones Research Center, a research arm of the University. He had many graduate students working under his direction, and he was riding high in terms of his reputation. He was a rather august figure, somewhat austere and above everyone else.

While I was still a struggling and confused undergraduate, one of Ripperger's graduate students had taught my

The class was fairly difficult, and I was having trouble keeping up with it all. In particular, the solution of the many differential equations just overwhelmed me. Since I was working out at the lab, and Dr. Ripperger was out there from time to time, I thought it might be a good idea to go in to to see him at the lab to discuss the class. I found him at his desk one afternoon, and screwed up my courage to go into talk with him.

I told him that I was finding the class difficult, even though I thought I had a good understanding of dynamics. I told him that my difficulty was particularly with the differential equations, not with dynamics. He listened quietly while I spoke, and then he fixed me with a withering gaze when he spoke, calling me first by name and then saying,

He said no more, and I slunk away to lick my wounds! I don't think I ever spoke to him again.

**Mechanics Corner**

A Journal of Applied Mechanics and Mathematics by DrD

© Machinery Dynamics Research, 2016

Becoming An Expert -- Part 2

**Introduction**

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.

**Mechanics Corner**

A Journal of Applied Mechanics and Mathematics by DrD

© Machinery Dynamics Research, 2016

**How To Become An Expert**

**Introduction**

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.