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A Journal of Applied Mechanics and Mathematics by DrD

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    Mechanics Corner
    A Journal of Applied Mechanics and Mathematics by DrD, # 16
    © Machinery Dynamics Research, 2015

Vibrations -- Part III
Effects of Damping


    In the previous discussions of the simple, displacement driven mechanical oscillator, damping has been explicitly omitted. In this part, viscous damping is taken into account. Viscous damping is the effect of forces that (1) always oppose existing relative motion, and (2) are proportional to the relative velocity between surfaces. It always removed energy from a system, converting mechanical energy into heat energy that is then lost by conduction, convection, or radiation. Note that, with no motion there is no viscous damping force. Viscous action is associated with surfaces separated by a film of oil, grease, or other viscous substance (such as animal fat, glycerin, or any other viscous fluid).
    There are many other types of energy dissipation mechanisms, including dry friction (Coulomb damping), aerodynamic drag (proportional to the square of velocity), and various other nonlinear damping mechanisms. In many cases, these other models are closer to reality, but there is a reason why the viscous damping is often the preferred modeling technique instead; it is mathematically tractable, which is to say, that it lends itself to mathematical solution far more readily than most other models. (Tractable is simply a big word, popular with mathematicians, meaning that something is easy to work with for a solution.) In this part of the series, the same system considered previous is again presented in slightly modified form; a viscous damper is added as shown in Figure 1.
    [Figure 1  Damped, Displacement Driven Mechanical Oscillator.]





    Mechanics Corner
    A Journal of Applied Mechanics and Mathematics by DrD, # 15
    © Machinery Dynamics Research, 2015

Introduction & Review

    In Part I of this discussion, the single degree of freedom oscillator was introduced with displacement excitation. The system is shown in Figure 1.

    The impressed support motion is s(t) and the response of the mass is measured by x(t). As shown in the upper figure, s(0)=0, x(0)=0, and there is no strain in the spring. The elongation (or compression) of the spring is s(t)-x(t), so that the force acting to the right on the mass is F=K[s(t)-x(t)]. Applying Newton's Second Law gives the system equation of motion


    which may be re-written as


    ω_{n}²=K/M= square of the natural frequency of vibration. The quantity ω_{n} (or ω_{n}²) is a fundamental system property.
    The driving displacement was specified as


    and the solution was eventually determined, subject to the provision ω_{n}²≠Ω². In this part, the objective is to investigate what happens when the excitation is near or at the natural frequency, Ω≈ω_{n}. The frequency response plot, shown in Part I, leads us to think that the oscillations will be catastrophically large near resonance, but that is not the whole picture.




Mechanics Corner
    A Journal of Applied Mechanics and Mathematics by DrD, # 14
    © Machinery Dynamics Research, LLC, 2015

SDOF Vibrations -- Part I
Undamped System With
Non-Resonant Conditions



    The undamped single degree of freedom oscillator is a suitable model for countless physical systems. It might represent a wheel on a vehicle (without shock absorbers), a machine on an elastic foundation, a sensitive instrument mounted on a shaky support, or any number of other possible real situations. A simple schematic diagram for such a system is shown in Figure 1 below.

    [Figure 1  Undamped SDOF Oscillator with Base Excitation.]

    The impressed support motion is s(t) and the response of the mass is measured by x(t). As shown in the upper figure, s(0)=0, x(0)=0, and there is no strain in the spring.

SDOF Vibs1.pdf




    Mechanics Corner

    A Journal of Applied Mechanics and Mathematics by DrD, #13

    © Machinery Dynamics Research, LLC, 2015



Numerical Methods

Roots & Solutions of Equations




    When we write a mathematical expression of the form y=f(x), a value of x for which y=0 is called a root of the equation. The root is a value of x at which the ordinate is zero and the curve passes through the x-axis.

    Further, we are often called upon to consider two equations of the form y=f₁(x) and y=f₂(x), each of which defines a curve in the x-y plane. To find a solution for this system means to find a pair of numbers, (x,y), such that the point represented by that pair is simultaneously on both of the curves.

    Everyone who aspires to become an engineer learns in their pre-college mathematics about finding roots of linear and quadratic equations and about solving systems of linear simultaneous equations. We learn that finding roots of equations of first and second degree (linear and quadratic equations) is a fairly simple matter, but that roots of equations of higher degree is difficult to impossible, depending on the exact situation. We learn that solving a system of two simultaneous linear equations is a simple matter, a system of three simultaneous linear equations is still manageable, but more than three equations becomes very laborious. Systems of nonlinear equations are rarely dealt with at all at the introductory level.

    As has been demonstrated in previous articles on Kinematics and Statics by Virtual Work, these problems lead to some equations and systems of equations that are at times extremely difficult, or impossible, to solve by traditional algebraic methods. This points to the need for a numerical approach to the solution of these systems.




Mechanics Corner

A Journal of Applied Mechanics and Mathematics by DrD, #12

© Machinery Dynamics Research, LLC, 2015



Numerical Solution of

Ordinary Differential Equations




When we take the mathematics course titled Differential Equations, we learn a bag of tricks for the solution of many different types of differential equations. We also learn that every different type seems to require a completely new approach. There is no general approach to differential equations as a whole. This is rather disheartening news. We also learn that for linear ordinary differential equations, there are a number of fairly general methods, and hence much of our study tends to focus on systems described by this class of equations. But what are we to do when we need to understand systems not described by linear differential equations?

One of the earlier approaches that enjoyed some considerable success was electronic analog computation (there were also mechanical analog computers used earlier, such as the "ball and disk integrator" of the Norden bomb sight). In its electronic embodiment, analog computation involved the construction of a DC circuit that obeyed the same differential equation as the original system of interest. Imagine that we are interested in a spring--mass oscillator, subject to velocity squared damping. This system obeys the differential equation




Mechanics Corner

A Journal of Applied Mechanics and Mathematics by DrD, # 11

© Machinery Dynamics Research, LLC, 2015



Eksergian's Equation


Motion for SDOF






In undergraduate engineering education, when someone says "equation of motion," it is almost reflexive to think "Newton's Second Law of motion." Recall that Newton's Second Law says


F=m a


where both the left and right sides of the equation are vector expressions. Vector are very powerful, but they are also very demanding for proper handling. Energy quantities, which are scalars, are much easier to work with, by comparison. In most undergraduate work, and much graduate work as well, Newton's Second Law is the first, last, and only word to be said about equations of motion. But in fact, there is more, much more!

The most commonly discussed part of "more" is what is called the Lagrange equation of motion, an energy based approach to obtaining the equations of motion (as opposed to a vector approach) that originated with J.L. Lagrange (1736-1813). This approach has great applicability and will be discussed in detail in a later article.

A much less well known part of the "more" is what is called the Eksergian equation of motion, an energy based approach to the equation of motion for single degree of freedom systems. Since it only applies to SDOF systems, one might ask, "Why bother? Why not just use the Lagrange equations?" The answer is two fold: (1) the Eksergian approach is slightly easier than the Lagrange equation in application, and (2) the Eksergian approach offers more insight into the meaning of terms.

The Eksergian approach first appeared in print in English with Eksergian's 15 part paper titled "Dynamical Analysis of Machines," appearing over the years 1930 -- 1931, although there are hints that something similar may have appeared earlier in German. This series of papers was extracted from Eksergian's doctoral dissertation at Clark University, 1928. In many ways, Eksergian's work was ahead of its time; it is well suited to digital computation which was virtually nonexistent at the time that this work appeared, but it is too labor intensive for hand computation. This is probably why it is relatively obscure. Eksergian's equation is particularly useful for systems that are kinematically complicated.

 Eksergian's Equation of Motion for SDOF.pdf.467597bd8f63b9f0057f49487b2f4bac


An Integrity Problem


The article referenced below points to a serious integrity problem, one that should be a matter of concern to all engineers. Is their no honor among these people? Is this how India expects to advance? By fraud?

It involves blatant cheating on exams. Most of us do not enjoy taking exams, but we recognize that they are necessary to evaluate who is competent and who is not. The determination of who is competent to practice any profession, be it law, medicine, engineering, etc., is a matter of concern to all of society. It is damaging to society as a whole when those who are not competent, for whatever reason, are allowed to practice, putting society at risk of inferior work.

The Daily Mail article includes pictures showing friends and family members passing notes to those taking their final exams. The whole matter appears to be very poorly policed. Why is this possible? Why is not the exam site secured? It makes a joke out of the entire examination process, and renders it meaningless. This degrades the work of those who are diligent and work hard, right along with those who take the lazy way. Shame!!


The following is a verbal description of a Doonesbury cartoon of unknown date by Garry Trudeau. Doonesbury has long been one of America’s major cartoon strips, with a very dry wit and a decidedly left-of-center outlook. I found this today in going through some old files.

SCENE: A college classroom, the teacher lecturing in a rather absent minded fashion, the students silently bent over, taking notes and keeping their heads down.

TEACHER: Of course, in his deliberations on American capitalism, Hamilton could not have foreseen the awesome private fortunes that would be amassed at the expense of the common good.

TEACHER: Take the modern example of the inventor of the radar detector. In less than ten years, he made $175 million selling a device whose sole purpose is to help millions of people break the law.

TEACHER: In other words ...

STUDENT (suddenly sitting up and interjecting): Maybe the fuzz buster is a form of Libertarian civil disobedience, man. You know, like a blow for individual freedom.

TEACHER: I ... I don’t believe it!

STUDENT: Believe what, man?

TEACHER (smiling in happy elation!): A Response! I finally got a thinking response from one of you. And I thought you were all stenographers! I have a student! A student LIVES!

TEACHER (kneeling down, hand extended like one might approach a shy animal): Who are you lad? Where did you come from? Don’t be frightened ...

STUDENT: (looking around himself): What’s the deal here? Am I in trouble?

The above all appeared in print many years ago, but it is an apt description of Mechanics Corner.



Mechanics Corner

A Journal of Applied Mechanics and Mathematics by DrD, #10

© Machinery Dynamics Research, LLC, 2015



Virtual Work -- Part II (Revised)




The main ideas related to virtual displacements and virtual work were introduced in a previous article titled Virtual Work -- Part I; understanding of that article is essential background for this article.

Everything that is really necessary to be said about virtual work has been said previously; there is really no need to say anything more. Having read that, the reader is entitled to ask, "Then why are we spending time on a second article on the same topic?," an entirely valid question. The answer in a single word is convenience. While all that is necessary, that is essential, has been said, there remains more to be said that will make the ideas of virtual work more powerful and convenient to use. The purpose for this article is to bring to bear the idea of potential energy which forms a natural extension to virtual work.




Mechanics Corner

A Journal of Applied Mechanics and Mathematics by DrD, #9

© Machinery Dynamics Research, LLC, 2015



Virtual Work -- Part I




The whole topic of virtual work is one that is usually not well handled at the undergraduate level (and frequently not well done at the graduate level, either!). It is, however, a very powerful concept, and critical to the application of many of the most powerful tools available for both statics and dynamics.

The term virtual is an old word (preceding the current usage in computer related matters by several hundred years), meaning something proposed for consideration and discussion as opposed to something that actually happens. It is an adjective, and is used frequently to modify such nouns as displacement and work. Thus a virtual displacement is a possible displacement of the system under consideration, not an actual displacement of the system (the distinction is initially not easy to grasp, but it will become clear if you bear with it!). A virtual displacement is an infinitesimal displacement, of the most general sort possible consistent with the system constraints.

What does that mean? Consider a block sitting on a solid plane surface. The block can be moved parallel to the plane, so a virtual displacement must include any and all possible infinitesimal displacements parallel to the plane. On the other hand, the block cannot be moved downward through the plane; the solid surface is a constraint. Thus virtual displacements of the block do not included downward displacements.



Mechanics Corner

A Journal of Applied Mechanics and Mathematics by DrD, # 8

© Machinery Dynamics Research, LLC, 2015



Rolling Constraints






The previous discussion of Constraints, Part IV, dealt mostly with holonomic constraints representing a rigid link between two points in a mechanism. That is a very useful concept, but it neglects another important constraint type, the rolling constraint. This article will deal with the latter.

Rather than make a very formal, comprehensive presentation, this topic is addressed simply by a sequence of examples. These will cover most of the common situations and indicate how such problems may be handled.


Rolling Along a Track


As an initial example, consider a steam locomotive operating on a straight, level track. For a fine animation of this, see Mechanizmalar at

There the locomotive is going forward; here, increasing angle θ moves the locomotive to the rear. This is the situation shown in Fig. 1.



Engineering Philosophy


Engineering Philosophy


You have probably heard of Engineering Econ, Engineering Management, etc., but what about Engineering Philosophy?


Most of the time, in engineering discussions we talk about “hard information,” that is, facts, ideas, and methods for doing various engineering activities. But behind all that, guiding it, there needs to be a correct philosophy, an approach to life. Late at night, when you are alone in your bed, do you ever think about, “What does it mean that I am an engineer? What does this mean about the way I approach my daily work? What does it mean about the way I think about politics? What does it mean about the way I live my life?”


A recent comment on one of my previous articles points indirectly toward these questions. I do not wish to criticize the person making the comment in the least, but rather to use his comment as a spring-board to get into my reflection on philosophy. The comment read, in part, as follows:


...but we learn in in india from our regular course, book contents are diferent from differnet author, so we have problem which one is correct. for example my project is analysisi of drum brake, the book I preferred for making the project and for formulation is wrong according to some people.


This person is pointing to a misunderstanding of engineering education. He wants an authority that is always correct, so there will be no conflict between one authority and the next. He wants to be able to say that a particular approach is correct because this author said so. What is he missing?


To try to get to that question, let us first look a bit deeper at his complaint. We have engineer W complaining that he wants to execute his project following the methods described by author X, but various others are saying that he should follow the approach given by author Y or the one set forth by author Z. What is engineer W supposed to do?


Assuming that authors X, Y, and Z each describe a different approach to his problem, how should he know which one to follow? Is it possible that each of the authors, X, Y, and Z are correct even though they are different? Well, maybe.


1. It may be the case that each of the authors describes an approach of different accuracy. Perhaps X is describing a simple, “back of the envelope” hand calculation that is only a rough approximation but requires very little effort. At the same time, Y is describing an approach incorporated into a widely accepted standard, and Z is describing a very thorough, detailed analysis (FEA, CFD, elaborate simulation, etc.). Which one is correct? Well, they all are “correct” in their own context. Then the question becomes, what does the project justify? Is it worth spending the resources (time, manpower, etc) for the very detailed approach given by Z, or is a very rough approximation sufficient for the present purposes, something like the hand calculation described by X? In this situation, the “correct choice” is the one that is appropriate to the purposes of the project.


2. Another possibility may be related to the technology involved. For example, imagine that author X is describing a graphical solution while author Y is describing a simple but effective computer solution and Z is again proposing a very involved computer solution requiring massive computing power and software resources. The graphical solution may offer a good bit of insight, but it is necessarily relatively slow and has only limited accuracy. Repeated applications of the graphical methods mean many clean sheets of paper to start again and again. The simple computer solution is likely to be much more accurate and lends itself to repeated application as needed to iterate toward a final solution. The very detailed computer solution can also be used to iterate, but often at very high costs in terms of time and resources. In this case, the simple computer solution described by author Y is likely the optimum approach.


3. There is also the unhappy possibility that the method recommended by author X may simply be in error. We may assume that he would not have published it if he thought it was incorrect, but the fact that it is in a book is no guarantee that it is correct. If this is the situation, then it is clear that the method of author X is to be avoided.


So where does this leave engineer W? The answer is really pretty simple: He should not use any tool that he does not fully understand, both in terms of how it works and what it costs. And this brings us back toward the original question, “What does it mean that I am an engineer?” An engineer is one who applies knowledge. He does not simply use tools without understanding what they do.


This last statement leads into another discussion, that regarding just how fully do you need to understand your tools. We have to recognize that no one can know everything that is known; it is simply too much. That said, it is clear also that the person who tries to drive a screw with a hammer does not understand his tool. One place where we can easily get into difficulties is with the application of commercial computer programs for FEA, CFD, etc. These are highly sophisticated programs, and the internal technical details of their workings is clearly the realm of the specialist. On the other side, the misapplication of these programs have produced some stunningly bad results, leading to catastrophic failures and false alarms. How is an engineer supposed to deal with these?


The first guiding principal is “Do Not Accept Anything Blindly Just Because the Computer Said So.” We must remember that the computer has no brain at all. It is simply a very fast way to make calculations, calculations which can be made correctly or in error. The computer is able to generate errors far faster than anything you could possibly do by hand! You must always, without exception, check computer results for reasonableness. You should also run test cases, cases for which solutions are known by other means, that incorporate as many of the features of you actual problem as possible.


Let us look at a specific case to expand upon this. Suppose that our engineer needs to design a shaft. The shaft is required to transmit certain amounts of power, it has various lateral loads, and in all likelihood has numerous steps along the length (steps are useful for locating items such as wheels, bearings, disks, blades, etc., mounted on the shaft). Our engineer may very well choose to use a finite element program or a specialty shaft design program to assist him in his design work. How will he be able to judge the correctness of the computer results?


In college, he took a course in Mechanics of Materials (sometimes called Strength of Materials) in which he studied shaft and beam deflections and stresses. The problems in that course were, without exception, based on simple geometries. The beam/shaft is usually uniform along the length, and the supports are either simple supports (knife edges) at the ends or built-in ends. The course will usually consider a variety of discrete and distributed loads. How does this relate to the design problem he faces on the job?


1. He should have learned that the boundary conditions are critical to a correct model. Changing from a simple support to a fixed (built-in) support can make a great difference.


2. He should have learned that the corners near a step along the length are essentially dead material; they do not carry much stress or strain at all because of having a free surface on two sides.


3. He should have learned how to combine the effects of different load types by superposition. This will allow him to consider one load, or one group of loads, at a time if that is useful.


4. He should have learned how to combine stresses in order to compute principal stresses and the von Mises stress at a point for failure evaluation.


5. He should have learned about stress concentration at corners, such as a step, and the means to mitigate this problem.


The list above is only a beginning, but the point is, the information from the Mechanics of Materials course is going to be fundamental in his shaft design problem on the job. He may also need to draw on information from Dynamics, Vibrations, and Material Science courses to deal with the whole problem. For this reason, it is absolutely essential that he actually learned the content of these courses. They are not there simply as hurdles to be jumped on the way to a degree; they are the basis for professional engineering work.


When I say “learn the content of these courses,” I mean just that. Really learn, understand, and make the content of these courses your own. This means learn all of the mathematics required to work through and fully understand all of the derivations. Again these are not simply hurdles to be passed; they are the foundations for your career.


Sadly, I have to admit that there are a great many people in engineering positions who have not done this. While I was still an undergraduate myself, I had an opportunity to visit with the director of the state highway department, the man ultimately responsible for the design of all the roads and bridges in the State of Texas. I asked him how much he used calculus on a day-to-day basis. He laughed at me, and told me that he never used calculus. He could not remember any of it. That puzzled me at the time, but I understand it now. This man was no longer really an engineer; he was not competent to design anything at all. He was a paper-pusher, and executive, but he was not an engineer. You can only be an engineer if you know what you are doing!


That brings me back to the comment that appeared on this blog, the complaint about different authorities in conflict. The final answer to that question is simply this: You, the engineer, must be able to correctly identify the proper approach and justify your decision. You can do this if you really know what you are doing, but you will never be able to do it if you don’t know what you are about. You, the engineer, must ultimately be your own authority!



Mechanics Corner

A Journal of Applied Mechanics and Mathematics by DrD, # 7

© Machinery Dynamics Research, LLC, 2015

Degrees of Freedom &  Constraints




The term "Degrees of Freedom" (often abbreviated as DOF) has been carefully avoided for the most part in these presentations up to this point, although it has crept in unavoidably a time or two. In this article, we attempted to face the matter squarely and deal with it fully. It is an important concept, one that is very widely confused, and is critical to correct understanding of countless mechanics problems. There are several other concepts that must be discussed along with degrees of freedom including the idea of a particle or point mass and the idea of various types of constraints.

This article is different from those that went before in that there is (almost) no calculation involved. It is almost entirely focused on matters of philosophy, a perspective or point of view, that has proven useful for countless generations of workers in the field of mechanics.



Mechanics Corner

A Journal of Applied Mechanics & Mathematics by DrD, #6

© Machinery Dynamics Research, LLC, 2015

AC Power in Real Variables Only



Most mechanical engineers get a pretty good understanding of DC circuits, and this carries over fairly well into single phase AC circuits. The difficulties come when we get into industry and discover that almost everything is powered by three phase AC circuits. This is where it starts getting sticky!

In the discussion of three phase AC electrical power, it is almost universal to use complex notation, otherwise known as phasor notation. For most purposes, the results might just as well be simply pulled out of the blue for all the understanding that complex mathematics gives, because everyone knows that the quantities involved -- voltage and current -- are fundamentally real, physical variables. These real quantities are not described by complex numbers, but rather by real numbers. The customary mantra says, "... we are considering the real part ...," but that really does not explain things very well because all of the mathematics being done is using complex algebra which considerably obscures the picture. Complex variables are, to use a colloquial term, "unreal." What is needed is a simple, straight--forward presentation of the problem in terms of real variables. We will give that a shot here.



Puzzled (A Poll)

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

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

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

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

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



Mechanics Corner

A Journal of Applied Mechanics and Mathematics by DrD, # 5

© Machinery Dynamics Research, LLC, 2015



Vector Loop Kinematics -- Part III

Acceleration Analysis



In the first article in this series, titled "Vector Loop Kinematics - Part I/Position Analysis" the idea of using closed vector loops for the position analysis of mechanisms and machines was introduced. A second article, "Vector Loop Kinematics - Part II/Velocity Analysis" extended the process to include mechanism velocity analysis. In this, the third article in the series, the process is extended further to cover the analysis of acclerations.

For each mechanism considered, we have first identified a single variable as the input, a variable to be assigned at will over some range representing the full motion of the system. (In so doing, we are limiting the discussion to Single Degree of Freedom systems, although this term has not yet been defined in this series.) It happens that in both examples used, the primary variable has been called θ, but there is no real significance to this naming. The position loop equations have then been written in terms of this primary variable and such other secondary variables as might be needed (secondary variables have been denoted as A, B, and x in the examples). The first step is always completion of the position solution, determining values for the secondary variables for any values of the primary varible of interest.



Mechanics Corner

A Journal of Applied Mechanics and Mathematics by DrD, # 4

© Machinery Dynamics Research, LLC, 2015



Vector Loop Kinematics -- Part II

Velocity Analysis




In the previous article in this series, titled "Vector Loop Kinematics - Part I/Position Analysis" the idea of using closed vector loops for the position analysis of mechanisms and machines was introduced. This is an extremely powerful method; I have never found a kinematics problem that was beyond its scope (now watch someone challenge me with such a problem!). As we left it at the end of that article, the technique of finding out all of the position information was at hand, but we had done nothing at all about discussing velocities or accelerations. This article will introduce the extension of this method to velocity analysis, but accelerations are differed until a later article.

This article is built upon the previous article, even to the extent of using some of the same example problems. If you have forgotten the content of the previous article, you might want to review it before getting to far into the present article.




The use of desktop, laptop, tablet, and other computers has become routine these days for engineering work. Along with this, there has been an ever-increasing number of software options for engineering calculations. It would be interesting to know just what software the readership here uses in their daily work and/or study.




Position Analysis



Many years ago, when I first began to study mechanics, the "conventional wisdom," expressed by both teachers and fellow students, was this: "Statics is easy, Dynamics is hard, and Kinematics -- who bothers to actually study kinematics? Kinematic relations, when needed, simply drop from the sky like rain, but nobody seriously studies kinematics." I eventually found the truth to be a bit more subtle: Statics of structures is generally easy, while the statics of mechanisms and machines may, or may not, be easy, depending a lot on the kinematics. Further, I found that the key to most dynamics problems is having a good tool to deal with the necessary kinematics.

The purpose for this article is to present the most powerful tool I have ever found for dealing with mechanism and machine kinematics, the vector loop method. This will be demonstrated in the context of two simple problems.



#2 -- Box Tipping

Mechanics Corner

A Journal of Applied Mechanics and Mathematics by DrD, No. 2

© Machinery Dynamics Research, LLC, 2015

It is a common practice for manufacturers to ship their products in packing crates that are strapped down on pallets for handling. There is often concern about the stability of this package as it is handled in transit to the purchaser. For this problem, we understand that the manufacturer wants to perform a simple test on each package shipped to assure that it will not tip over in transit. The test will consist of tipping the package slightly to the left and placing a block under the right edge of the pallet. The block is then quickly pulled out and the question is whether or not the package will fall over to the right. The answer depends upon the amount of the initial tip to the left and the location of the center of mass of the combined packing crate and pallet.


It is clear that the falling box impacts the floor, causing an impulsive distributed load to act on the bottom of the package. This will apply both an impulsive upward force and an impulsive moment to act on the box. Since the actual distribution of the force is unknown (and unknowable), an impulse--momentum approach to this problem is not likely to get very far. There is, however, a much simpler energy analysis available. Go on over to the attached PDF for more details.



#1 -- Introductions

Welcome to the first installment of Mechanics Corner, a feature that we hope will become a regular blog item on Mechanical Engineering Forum. The intent is that every week we will have a new article on some aspect of Applied Mechanics and Mathematics, things of broad interest to mechanical engineers. Some of these articles will be fairly elementary, while others will be considerably more advanced, but the idea is to have something for everyone. We will hope to amuse, entertain, and most importantly, to inform you with each article. We may even have a little bit of engineering humour from time to time!



Most of these articles will involve the use of pictures, diagrams, and mathematics, all things that are fairly difficult to accomplish in a blog post. For this reason, and beginning today, the bulk of the post will be in an attached PDF file. It turns out that there are fairly simple ways to have all the necessary tools available in a PDF file even though they are not available directly on the Internet. Thus I hope that each of you will click on over to the attached PDF file to read the rest of today's article.


Mechanics Corner is written by DrD, which of course raises the question, "Who is DrD?" Well, that’s me, but that doesnt tell you very much, does it? My intention is to say very little about myself in most of the articles, but since today is the day for introductions, for the blog, for myself, and a few other matters, it seems appropriate to tell you a little bit more about who I am.


I am an elderly man, a semiretired engineer and Mechanical Engineering Professor, living in Texas, USA. All of my engineering degrees are from The University of Texas at Austin, and I have professional engineering registrations in both Texas and Wisconsin (in the USA, engineers are licensed by the several states to prevent incompetents from holding themselves out as engineers and thus endangering the public safety and welfare). As I mentioned, I am old, many would say "older than dirt." I have had a long and very interesting career as an engineer working in a number of different industries, as an engineering faculty member, and as a consulting engineer (I continue to do some consulting yet today). In my engineering career, I have worked in the automotive, aerospace, naval, offshore, gas compression, steel, and electric power generation industries. I have worked on diesel and natural gas engines, steam turbines, gas turbines, large electric motors, generators and host of other machine types. If it moves, it is likely that I have worked on it; if I have not, I sure would like to work on it!


Being as old as I am, I have seen a lot of changes in my life, a few of which I wanted to touch on here. One of the most profound changes has been the shift of manufacturing industry away from the USA to India, China, Mexico, Brazil, and Southeast Asia. When I was young, the USA was arguably the greatest manufacturing power in the world, but that is no longer true today. We talk about being an "information society" (although Im not sure what that is) but we have very little to do with machinery and similar things today. But here I am, and this is one of the reasons why I feel a need to talk with many of you in the developing countries.


In October, 1957, Russia put the Sputnik satellite in orbit. It was a tiny thing, about the size of a soccer ball, but it shook the world. At that time I was in my last year in high school, preparing to go off to study engineering in college. Sputnik caused a great shake-up in American engineering education, with many warning cries that we were "behind" and had to "catch up." This meant many changes in education, but one in particular: now everything had to be done in vector notation, something that had not been done much before. In my freshman year in college, I took the introductory Mechanics course in physics, and fell in love with the subject matter. As a result, I have been studying mechanics, in one form or another, for well over half a century. Interestingly, although I initially learned everything in vector notation, I have come to the conclusion that I prefer to use scalars wherever possible. In particular this means the use of energy methods whenever they are suitable.


We all go off to college to study mechanical engineering; this is how we enter the profession. We are constantly told that we must never stop learning, but how many really believe that? Do you still hit the books every night? Are you still doing homework problems? I want to tell you a story about learning after school has ended.


About 12 years after I completed a Ph.D., I was on the faculty at Texas A&M University, one of the great engineering schools of America. I was assigned to teach Theory of Machines, and I figured that I could handle it, even though I had never had such a course in my own education. I selected a textbook that was somewhat unorthodox but I thought it looked attractive. It was a very good textbook, and it proved to be one of the greatest learning experiences in my whole life (there is nothing like trying to teach a course to be sure that you learn the course). I struggled to stay a few days ahead of the students, but that book brought me many new and powerful ideas that I had never seen previously. At the end of the semester, I asked the students what they thought of the book. They hated it! Their complaints really came down to two things: (1) the book was too big, too long, nearly 700 pages, and (2) the author had some really awkward notations. A few years later, when I set out to write my own textbook on Theory of Machines, I kept these two objections in mind and was able to produce what I think was a much better text. The point of the story is this: here I was, supposedly educated and having industrial experience, and yet I had the greatest learning experience of my life. It profoundly changed the way I work all kinds of problems to this day. The moral of the story is that we are never too old to learn, unless we think we are.


One of the great changes that I have seen, and you have seen it also, is the profound impact of the Internet. Thirty years ago there is no way that I would have been writing for you, and no possibility that you would have been reading the words of an old man in Texas. But all of that has changed now. Sadly, there is much trash on the Internet. On the good side, there is also much of a value. Among those good things, I would like to direct you to the site of a friend who has done some most excellent work in mechanism animations. When you see his animations you simply cannot help but have a better feel for how these machines work.


The URL is: By all means take a look at them and see for yourself! I hope to see all of you and many more next week when we will go on to things of a more technical nature. Please check back here at Mechanical Engineering Forum for the next article.