Jump to content
Mechanical Engineering Community
DrD

The New Way or the Old Way?

Recommended Posts

Long, long ago, in a land far away, I was an undergraduate engineering student. I knew some of my fellow students who studied a course called "Kinematics," and that puzzled me. Kinematics had been one of the parts of my first course in dynamics, so I wondered what else there was to the matter; I really did not know much at all. I borrowed a textbook for the Kinematics course, and it looked interesting, particularly the aspect that seemed to deal with real machines. I liked that. But then I asked about the course and I learned that it was a lot like the drafting course that we were all required to take. Specifically, the solutions were all obtained by scale drawings. With that bit of information, my interest in Kinematics cooled completely.

Now way back then, I was a pretty good draftsman, and I certainly was not put off by that in itself. I enjoyed drafting, and I had begun doing it for fun back in junior high school. But for kinematics, to major difficulties were immediately evident:

(1) No matter how sharp you make your pencil point, drawn lines have width (without width, we could not see them). This very substantially limits the accuracy of a drawn solution. Jumping ahead some years, when I worked for Hamilton Watch Company, our draftsmen would make greatly oversized drawings to graphically check clearances, and other mechanism features. The common scale was 40:1, meaning that the drawing was 40 times the actual size. This was before CAD had really taken hold, so these huge drawings would be spread out on drafting tables and everything was carefully drawn to scale.

(2) A single drawing can only show one mechanism position. Since most mechanisms operate through a considerable range of motion, the graphical analysis in one position was of little value for understanding velocities and accelerations in a second position. This could lead to the need to make many drawings, one for each position of interest. I was rapidly losing interst at the prospect of repeating essentially the same drawing, in slightly shifted positions, time and again!

The long and short of it is that, I did not seriously study kinematics of machines until some 15 years after I finished my PhD. It happened when i was called upon to teach Theory of Machines at Texas A&M University. As I began reviewing textbooks in preparation for this course, I found that much had changed in the intervening time. There were still some books focused on the old graphical techniques. They claimed that this gave the student a feel for the motion, an intuitive understanding. It also gave him at most 3 digit accuracy. On the other hand, there were books out there that presented new methods, particularly based on the idea of a closed vector loop. While it is useful to have a sketch to identify the variables in this method, there are no graphical constructions involved in obtaining the results. It is all mathematical, and can be carried out to whatever level of precision the user wants. Usually 8 decimal digits is common, but 14 to 20 digits are readily available. No more worrying about how sharp your pencil point is!!

I immediately adopted the mathematics based vector loop method for my own work and for teaching, and I have done so ever since that. The vector loop method has enabled me to answer many vexing questions that I would have never resolved by the graphical method. I went so far as to write a Theory of Machines textbook based on vector loop kinematics, and cannot imagine doing these problems any other way.

An then, in the last few years, I have come to ME Forum, where I encounter folks who are still solving kinematics problems graphically. I know that they have access to computers; that is how I encounter them, at an Internet website. Thus I have to ask, Why are you still using graphical solutions? I simply do not understand. I hope that there will be some comments that will shed light on this question.

DrD

Share this post


Link to post
Share on other sites

Dr D, I can't answer your question directly but I have an example of something I did recently. I was reviewing literature about rigging for lifting objects. An example had a two legged cable of different lengths. The unequal lengths were to compensate for a center of gravity not centered on the object being lifted. The example provided two trigonometric equations for determining the length of the short and long cables. Almost out of instinct I recreated it graphically vs. working through the trig to convince myself that the equations provided were correct.

I think the approach one takes has to do with comfort level.  You did not come upon the vector loop until post PhD. You would have had a lot more math experience than most at that point. If I were to venture into vector loop today, I would likely continue with a graphical approach as a sanity check. After many applications I may begin to develop a comfort level as I have with algebra. I often can tell an answer is wrong before I know why it is wrong.  

Math can be like a black box. When we add 2 + 2 we may not consciously think 2 objects + 2 objects as when teaching a child but I think something similar is going on in our mind. When I look at a complicated equation I have to take it on faith it is correct or work from first principles to verify it is correct. If I can't do that with confidence, or not at all, I will not use the equation unless it is published in multiple independent sources. With the trigonometric example above I could have worked through the trig, but the graphical method was faster and easier to trust.

For a cable length the accuracy is not as critical as a cam dimension in an engine, so my example is not the best, but your question got me thinking. You and I have discussed differential equations. Having had little opportunity to use what was presented in class years ago; my confidence using them is low. I would have to use another approach or two to arrive at the answer. Fortunately none of the work I do requires using differential equations.

And for the trigonometric equations in the example I mentioned, there was a glaring error I did not see at first. The author repeated the same equation for both lengths. The graphical vs. the equation results did not agree. I instinctively measured the other leg and got the correct answer. I then realized the problem. I have yet to work through the trig to get correct equation for the other leg.

Joe 

Share this post


Link to post
Share on other sites



×
×
  • Create New...