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A Journal of Applied Mechanics and Mathematics by DrD
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A Calculus Challenge
#44 Mouse Trap / Pendulum Dynamics Challenge
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. MouseTrapPendulumDynamics-1.pdf
#43 Four-Bar / Toggle Linkage Mechanism
Triple Rocker
#42 Gear Pair Problem
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. 42 Gear Pair Problem.pdf
Modeling Hysteresis
Two Short Math Problems
Comments on a Textbook - Khurmi & Gupta
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? Comments on Textbook - Khurmi.pdf
Rocket Homework Problem
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. RocketHWProblem.pdf 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?
Two Balls Rolling On An Incline
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. TwoBallsRollingOnAnIncline.pdf
Base Acceleration Problem -- #36
Good News -- Bad News -- Rolling Disk in a Rolling Ring
A Journal of Applied Mechanics and Mathematics by DrD, # 35
Machinery Dynamics Research, 2017
Good News --- Bad News
Rolling Disk in a Rolling Ring Introduction 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
Last Post -- Time to Hang It Up
A Journal of Applied Mechanics and Mathematics by DrD
© Machinery Dynamics Research, 2017
Last Post
Time to Hang It Up
This will be the final post of Mechanics Corner here on Mechanical Engineering Forums. It has run almost exactly two years, and there have been ups and downs along the way. In this final post, I want to reflect a bit on my original goals for the blog, and also on what has actually happened. When our host first proposed to me that I might write a blog for ME Forums, I was excited about it. About half of my career had been spent in engineering education, and I always loved working with students. It seemed like a way to get back to something that I had long enjoyed, and so I accepted his suggestion. A long time ago, back when I was about 14 or 15 years old, in Junior High School, my shop teacher mentioned, in an off-hand way in class, that various curves could be described mathematically. I’d never heard that before, but I thought immediately, “This has interesting possibilities.” Moving ahead a few years, I discovered that I wanted to study and build my career around was the area known as Applied Mechanics, although it was a time before I first heard that term. In my freshman physics class, I discovered the laws of motion, and thought to myself, “This is great stuff! I can use math to describe how things move!” All of that happened back in the 1950s, and I’m still doing the same thing today (some might say I am in a rut!). As a teacher, I taught mostly undergraduate engineering courses, although I taught my share of graduate courses as well. It was the undergraduate courses that I liked most, because I firmly believe that the economy of a nation is strongly dependent on the quality of the baccalaureate level engineers produced in that nation. Engineers with graduate degrees are valuable as well, but the vast majority of the national engineering workload falls to BS level engineers. Thus, I envisioned Mechanics Corner as a sort of continuation of the several undergraduate courses I most enjoyed teaching — kinematics, dynamics of machines, vibrations, and mechanics of materials. For the most part, I have stuck to the plan, so that most of the technical posts I have made have dealt with problems that I considered suitable for undergraduate engineering students, say perhaps, junior level. I have posted a few topics from my industrial experience, but those have been situations that baccalaureate level engineers would be expected to handle. Now I knew it would not be exactly like continuing to teach my classes. In particular, you would not have any homework or tests, and I would not have any grading to do – a win-win, or so I thought. I did hope, that even with no assigned homework, readers would take an interest in the problems discussed, even to the point of working through the details for themselves (I was terribly naive, apparently!). I knew from my own experience that the only way I ever really learned a new idea was to get in and work with it, work some problems, make some numbers, plot some curves, until I really understood what it was all about. I’ll venture to say that nobody ever learned any technical material simply by reading only. In actual fact, in the early days, I had one or two folks say that they would in fact work through the problems, so I was encouraged. What I was not prepared for, however, was the fact that the vast majority never seemed to even read very carefully, much less work through the problems! The questions that have come, and there have been a few, have largely been about matters totally unrelated to the posts. The most common question has been, “Suggest a topic for my final project,” which relates to not a single post. Needless to say, that aspect of my vision was totally unfulfilled. But there is another side. I ventured to write a few “philosophical” articles, items dealing with academic integrity and cheating, with how to ask for help, with how to write a report or a paper, and various other matters. I really thought all of this would be considered obvious and trivial, so I was completely unprepared for the excitement that some of these articles generated. There were, in some cases, many, many comments, and people seemed to really be interested. I’m left to wonder: why? Are these ideas foreign to the culture of India and SE Asia? Are these things not all taught at home and in the public schools? I don’t know, but there was a lot of interest in these matters. But Mechanics Corner was intended to be primarily a technical blog, and there, it just did not excite the interest of the readership. As time passed, there was less and less interest. First, the comments dropped off to just about zero, and later, there were fewer and fewer who even bothered to “like” the articles. Finally, the number of reads has dropped to almost nothing (there may be no one left to read this final note). Well, there could hardly be any more clear indication that it is time to stop. I asked for opinions about this from some of the administrators, and was told that the blog was just over the heads of the readership. That makes me sad; that was never the intent. If it is true, I do not see how engineering has a very bright future among this readership. Even so, I wish all of you the best for your careers. I hope that you are able to find rewarding and beneficial work in which you will be happy and make a real contribution to your societies. To use an old cowboy metaphor perhaps familiar to many of you from Bollywood, “It is time to hang up the bridle and saddle, and say, ‘Adios’ (Adios is literally, “to God”).
A Problem in Statics & Dynamics, #34
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 StatDynProb.pdf
Advanced Polynomial Curve Fitting
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. AdvancedPolynomialCurveFitting.pdf
Braced Cantilever
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. BracedCantilever.pdf
ODE Solution --- Fail!!
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. ODE_Soln_Fail.pdf
Where is this?
A Question for Readers
Becoming An Expert -- Part 3
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. BecomingAnExpert--Part3.pdf
A Comment Remembered
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 Mechanics of Materials course, and I had done well in that class. This fellow liked me, and when a job opening came up out at the lab, he let me know about it and helped me to land it. Thus I was working a few hours a week as a lab assistant for this particular graduate student who was himself working under Dr. Ripperger. Before long, I signed up to take a class in Intermediate Dynamics, and Dr. Ripperger was the assigned teacher. Truth to tell, he was only mediocre teacher, nothing to get excited about.
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, "Did you think there was anything else besides the differential equations?"
He said no more, and I slunk away to lick my wounds! I don't think I ever spoke to him again.
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.
Becoming An Expert -- Part 2
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. BecomingAnExpert--Part2.pdf
How To Become An Expert
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. HowToBecomeAnExpert.pdf