A Journal of Applied Mechanics and Mathematics by DrD, #46
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!
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.
A Journal of Applied Mechanics and Mathematics by DrD, # 44
Machinery Dynamics Research, 2017
Mouse Trap / Pendulum Dynamics Challenge - Part I
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.
A Journal of Applied Mechanics and Mathematics by DrD, #43
(c) Machinery Dynamics Research, 2017
Four-Bar / Toggle Linkage Mechanism
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.
A Journal of Applied Mechanics and Mathematics by DrD
July 31, 2017
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.
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
A Journal of Applied Mechanics and Mathematics by DrD, #41
(c) Machinery Dynamics Research, July 2017
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
41 Modeling Hysteresis.pdf
A Journal of Applied Mechanics and Mathematics by DrD, #40
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
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
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
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.
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.
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.
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.
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.
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 ...
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.
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.
Does anyone recognize where this video is shot? Is it a group of students at a school (what school?), or is it an industrial site (what company)? I am anxious for someone to locate this for me, please.
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.
Many of you have asked me various questions, so now it is my turn. Let me lay a bit of background first, and then the questions.
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 is a retired Professor of Mechanical Engineering in the USA. He can be reached for comments, questions, or requests via the ME Forum message system. Be sure to check back soon at www.http://mechanical-engineering.in/forum/blog/206-mechanics-corner/ for more articles.