Engineers and designers often solve equations in the course of their work.
Problems they encounter might be as simple as finding the maximum stress in a statically loaded simple beam, or as complex as determining the transient response of a flexing aircraft wing subjected to transient loads. In any case, there are often lots of numbers to crunch.
In the past, the sheer volume of calculations often meant resorting to approximations and best guesses. When spreadsheet software came along, the technical community quickly realized it could be used for engineering calculations. But spreadsheets are mostly geared to accounting and finance and lack support for complex engineering calculations.
That’s why over 25 years ago we started using Maple mathematical software. In earlier releases, it was primarily a calculation engine. Users invoked suitable commands, provided input, and the software delivered answers. A later version introduced the Document mode. It combines the math capabilities of Maple with the text capabilities of a word processor. Workflows are similar to those of paper and pencil. Users employ standard mathematical expressions, make annotations with text notes to explain what they are doing and why, and results appear in the document. Everything is associative and parametric, so any changes reflect through the entire document. For a simple example, if x = 3 and y = 7, then x + y = 10. Change x to 5, and the total immediately changes to 12.
The later version also included a handwriting-recognition system so that when you couldn’t remember which menu contains a specific symbol, you sketched it with the mouse, and the system would attempt to find the symbol. Maple 11, the latest release, expands on this to include formulas and equations instead of just single characters. The developer emphasizes that the function is experimental.
Although the idea is clever, a downside is it’s difficult to draw characters and symbols with a mouse. Stylus-operated systems for recognizing characters are approaching 100% hit rates, but I could never get much above 20% with the mouse-driven system.
That said, Maple 11 boasts an incredible arsenal of mathematical weaponry. It has more than 4,000 builtin math functions and 1,000 math symbols. Conversion factors for over 250 different simple and compound units let users mix and match units. The software helps users here by “sanity checking.” It displays an error message when users try to enter a volume unit when it expects a time unit, or similar incongruities. Version 11 also includes over new 100 improvements and additions.
For example, many functions in the program can be altered by numeric inputs, pushbutton selections, or sliders, producing 2D or 3D graphical outputs that change accordingly. If a user varies the parameters of a complex spring-mass-damper system, the response graph immediately updates.
An interesting feature of the interface is the new self-documenting context menus. Consider the following calculation:
Creating this equation is simple. Just type 2/3, press the right arrow key, type 5/7, and finally press Ctrl=. Maple 11 formats the equation as shown and calculates the result. Now for the self-documenting feature. Right-click on the answer and then chose, for example (Approximate 20) from the list of 5, 10, 20, 50, or 100 that appears in the context menu to have the answer rounded to 20 digits.
Maple 11 converts the fraction to a decimal, rounds it the selected amount, and adds the arrow and annotation to indicate what it had done, as shown:
|2||+||5||=||29||at 20 digits 1.380952380952390952|
To further assist with calculations, the software sports a new Back-Solver Assistant. It produces the value for any variable in an equation when you supply values for the other variables. This trims the time usually spent in rearranging a formula to solve for a specific variable.
A nifty capability lets users turn a Maple 11 document into a slide show. Better yet, slides do not have to be static. Value inputs, buttons, and sliders are still active. A presenter might, for example, move a slider to show how frequency response varies with damping.
Users can also cross reference equations between documents. For example, suppose a main equation is stored on a file server. Other documents can point to it, instead of users recreating the formula in each document. Better yet, any change to the structure or values of the master reflects back to the referencing documents.
The computation engine has also been improved. The developer says the engine is the fastest polynomial real-root finder in the world and that it will find the real roots of any polynomial. In addition, Maple 11 now imports from and exports to Excel spreadsheets. The program has also added a physics package and a differential-geometry package. Maple 11 comes from Maplesoft, 615 Kumpf Dr., Waterloo, Ont., Canada N2V 1KB, maplesoft.com.
— Bill Fane, B.A.Sc, P.Eng, ATC
Bill Fane was a product engineer and then product engineering manager for Weiser Lock in Vancouver, B.C., for 27 years and has taught mechanical design since 1996 at the British Columbia Institute of Technology.