Nomograms for the PE exam
Nomograms: Antiquated technology or useful tool?
Many of today’s engineers would not consider using nomograms for the PE exam. It would not surprise me if many of you reading this blog post have no idea what a nomogram (also called a nomograph) is. This blog will quickly introduce them to you to help answer the question… Nomograms: Antiquated technology or useful tool? Though they may be antiquated, you may find that they allow for very quick solutions to some types of problems. Therefore, this blog is an introduction to the idea of using them for the PE exam to help you solve problems faster.
What is a nomogram?
A nomogram is a graphical tool used to perform calculations, and they exist for numerous types of calculations. They are essentially a diagram that gives a quick, approximate solution to a mathematical function. A nomogram will have three or more lines (or curves), each for a different variable in the function. The lines are placed on the nomogram to give the appropriate numerical relationships for the function. If you know two or more of the variable values, you can use a straightedge to line up the values to graphically determine the unknown variable. There will be some error, but it can be a very quick method for doing the calculations!
You can easily search the internet to get more detail on using (or making) nomograms. There are also books teaching the lost art of nomography. You can find books that discuss the history of nomography as well as numerous areas of application. Omer Blodgett has a couple of great books, Design of Welded Structures and Design of Weldments, which have several great nomograms for topics in mechanics of materials.
Here is a quick example
The figure below shows a nomogram used to solve the quadratic equation. Though I did construct this particular nomogram I did not develop the idea (it has been around for a long time). I am also not going to give details on how it was developed, but I will give it only as a representation of how they can be used. The horizontal axis is the value of ‘a’ in the equation, the vertical axis is the value of ‘b’ in the equation, and the circle gives the two roots that solve the equation (values of ‘x’). The dashed line shows an example with a = 1.5 and b = -2. A line is drawn connecting those two points (the dashed line). The roots are determined by the values on the circle where that line crosses. The solutions for that particular example are x = 0.85 and x = -2.35, which are read from the values on the circle.
These nomograms sound obsolete… are they really useful?
OK, I would agree that computers have made nomograms essentially obsolete. However, you cannot use a computer during the PE exam. I would also agree that the power of calculators make nomograms obsolete, but remember that you have a limited selection of acceptable calculators for the PE exam. I am also not suggesting you use nomograms for everyday engineering design (though you could). The real question is whether or not they can save valuable time during the PE exam.
You may not have a need for a nomogram to solve the quadratic equation, but they exist for far more complex problems. Using them for the PE exam could allow for very rapid calculations for some common types of problems. Machine Design Fundamentals by Hindhede, for example, provides a nomogram relating power, torsional stress, and shaft diameter that rapidly determines required shaft diameter. Design of Welded Structures by Blodgett has rather complex nomograms that can be used to determine the required section modulus of beams (or required moment of inertia), which includes information about beam end conditions, beam length, beam loading, and allowable stress. The same text also includes nomograms for deflection of curved beams, fatigue, torsional resistance, column effective lengths, and design aids for plate girders. My textbook, Machine Analysis with Computer Applications (the same figure is in Mechanisms and Dynamics of Machinery by Mabie and Reinholtz), has a nomogram to determine maximum pressure angle in disk cams with roller followers. I have also seen nomograms for fluid flow calculations to aid in pipe sizing. All of these nomograms can save valuable time on the PE exam. You are not limited by nomograms available in books… you can always try making some of your own for different calculations.
Do you think nomograms are useful for the PE exam?
I hope you found this information useful. The PE exam is all about working problems quickly and efficiently. Nomograms can provide quick estimates of the answer (which is usually accurate enough for a multiple choice exam like the PE exam). What are your thoughts? Do you ever use nomograms for design? Do you have recommendations of good nomograms to use for the PE exam (or good books that contain useful nomograms)? Leave a comment letting everyone know and please share the information by tagging the share buttons. Also, take a minute to go over to the Mechanical PE Academy Facebook page to get updates and exclusive content.