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A Journal of Applied Mechanics and Mathematics by DrD

## #17 -- MDOF Vibrations / Part IV

Mechanics Corner
A Journal of Applied Mechanics and Mathematics by DrD, # 17
© Machinery Dynamics Research, 2015   Vibrations -- Part IV
Multiple Degrees of Freedom Introduction     The first three parts of this series have involved only a single mass moving in a single direction. Such a system constitutes a single degree of freedom. In this part, a second mass is added which changes the required approach considerably. The new system is shown in Figure 1.

Fig. 1  Two Degree of Freedom System     In the upper part of the figure, (a), the system is shown at equilibrium rest with no strain in either spring. In the lower figure, (b), the system is displaced x₁ and x₂ under the action of the applied external forces F₁(t) and F₂(t). The problem to be addressed in this part is the determination of the dynamic response of the system. VibsIV-MDOF.pdf

## #16 -- Vibration Part III/Effects of Damping

Mechanics Corner
A Journal of Applied Mechanics and Mathematics by DrD, # 16
Vibrations -- Part III
Effects of Damping
Introduction     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.]   VibsIII-Damping-Rev.pdf

## #15 -- SDOF Near & At Resonance, Part II

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     Mx=K(s-x)     which may be re-written as     x+ω_{n}²x=ω_{n}²s     where
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     s(t)=Ssin(Ωt)     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. VibsII-Resonance.pdf

## #14 -- SDOF Vibrations Part I

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      Introduction     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

## #13 -- Numerical Methods/Roots & Solutions of Equations

Mechanics Corner     A Journal of Applied Mechanics and Mathematics by DrD, #13     © Machinery Dynamics Research, LLC, 2015     Numerical Methods Roots & Solutions of Equations   Introduction       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.   RootsAndSolutions-Pt1.pdf

## #12 -- Numerical Solution of Ordinary Differential Equations

Mechanics Corner A Journal of Applied Mechanics and Mathematics by DrD, #12 © Machinery Dynamics Research, LLC, 2015     Numerical Solution of Ordinary Differential Equations   Introduction   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    NumericalSolnOfODE.pdf.57a35d58f014bf08467ba76cc1ca36f5

## #11 -- Eksergian's Equation of Motion for SDOF

Mechanics Corner A Journal of Applied Mechanics and Mathematics by DrD, # 11 © Machinery Dynamics Research, LLC, 2015     Eksergian's Equation of Motion for SDOF       Introduction   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!! http://www.dailymail.co.uk/news/article-3004353/Remarkable-pictures-dozens-people-scaling-four-storey-building-pass-cheating-relatives-notes-high-pressure-school-leaving-exams-India.html

## Verbal Description of a Doonesbury Cartoon

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.

## #9 -- Virtual Work / Part I

Mechanics Corner A Journal of Applied Mechanics and Mathematics by DrD, #9 © Machinery Dynamics Research, LLC, 2015     Virtual Work -- Part I   Introduction   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.  VirtualWork-Part1.pdf.bff10ab7622992387af105e83220675a

## #8 -- Kinematics Part V / Rolling Constraints

Mechanics Corner A Journal of Applied Mechanics and Mathematics by DrD, # 8 © Machinery Dynamics Research, LLC, 2015     Rolling Constraints       Introduction   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.  Kinematics-V.pdf.f197539bc46c566e6a63f7ae09714729

## #7 -- Kinematics Part IV / Degrees of Freedom & Constraints

Mechanics Corner A Journal of Applied Mechanics and Mathematics by DrD, # 7 © Machinery Dynamics Research, LLC, 2015 Degrees of Freedom &  Constraints Introduction   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. DOF-Constraints.pdf

## #6 -- AC Power in Real Variables Only

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. ACPowerInRealVariables.pdf

## Puzzled (Part II, Continuation of the Poll)

This is simply a continuation of the poll started just previously.

## #5 -- Kinematics Part III / Accelerations

Mechanics Corner A Journal of Applied Mechanics and Mathematics by DrD, # 5 © Machinery Dynamics Research, LLC, 2015     Vector Loop Kinematics -- Part III Acceleration Analysis Introduction 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. Kinematics-III.pdf

## #4 -- Kinematics Part II / Velocity Analysis

Mechanics Corner A Journal of Applied Mechanics and Mathematics by DrD, # 4 © Machinery Dynamics Research, LLC, 2015     Vector Loop Kinematics -- Part II Velocity Analysis   Introduction 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. Kinematics-II.pdf

## How Do You Compute?? -- A Poll

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.

## #3 -- Kinematics Part I / Position Analysis

Position Analysis Introduction 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. Kinematics-I.pdf

## #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. BoxTipping.pdf