Mechanical Engineering Community
• entries
76
413
• views
42,780

A Journal of Applied Mechanics and Mathematics by DrD

## Entries in this blog

Mechanics Corner 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. 46 CalcChallengeComments.pdf

## A Calculus Challenge

Mechanics Corner 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! MEForumChallenge.pdf   *********************************************************************************************************** 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. ***********************************************************************************************************

## #44 Mouse Trap / Pendulum Dynamics Challenge

Mechanics Corner
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

Mechanics Corner 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. TogglePress.pdf

## #42 Gear Pair Problem

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

Mechanics Corner A Journal of Applied Mechanics and Mathematics by DrD, #41 (c) Machinery Dynamics Research, July 2017   Modeling Hysteresis 1. Introduction What do you know about hysteresis? Many Mechanical Engineers will associate this term with the magnetization curve of a piece of magnetic material, and quickly conclude, "I don't have to worry about that!" But that would be wrong. While hysteresis does occur in magnetic systems, it happens in many other situations as well, many of them situations of concern to mechanical engineers. Figure 1 Typical Hysteresis Curve   Figure 1 shows a typical hysteresis curve, and it makes no difference as to what physical phenomena are involved. The red curve is the actual hysteresis curve. The blue curve is called the "spine." Read more at 41 Modeling Hysteresis.pdf

## Two Short Math Problems

Mechanics Corner A Journal of Applied Mechanics and Mathematics by DrD, #40 July, 2017   Two Short Math Problems Do you ever read the ads that appear on ME Forum? I try to avoid them as much as possible, but an organization called BRILLIANT has put up some interesting math problems of late that have caught my eye. Two of them are the subject of today's post. The first problem that I want to discuss is actually more recent than the other, but it gives us a good place to start. Following that, we'll go on to the second problem. Along the way, I want to talk about philosophy as well as simply how to solve tow specific problems. The main lessons to be learned here are in regard to how we use mathematics in the practice of Mechanical Engineering. 40 Two Short Math Problems.pdf

## Comments on a Textbook - Khurmi & Gupta

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

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

## Two Balls Rolling On An Incline

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. TwoBallsRollingOnAnIncline.pdf

## Base Acceleration Problem -- #36

Mechanics Corner 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.   BaseAccelerationProblem-36.pdf

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

Mechanics Corner
A Journal of Applied Mechanics and Mathematics by DrD, # 35
Machinery Dynamics Research, 2017
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

Mechanics Corner
A Journal of Applied Mechanics and Mathematics by DrD
Last Post
Time to Hang It Up

## A Problem in Statics & Dynamics, #34

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

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

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

## ODE Solution --- Fail!!

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. ODE_Soln_Fail.pdf

## Where is this?

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. 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

## Becoming An Expert -- Part 3

Mechanics Corner
A Journal of Applied Mechanics and Mathematics by DrD
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

Mechanics Corner
A Journal of Applied Mechanics and Mathematics by DrD
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

Mechanics Corner
A Journal of Applied Mechanics and Mathematics by DrD
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

Mechanics Corner
A Journal of Applied Mechanics and Mathematics by DrD