Machine Design

# High-end math package complements engineering work

Mathcad 13 is easy-to-use yet powerful software for creating, calculating, and documenting engineering and scientific analyses.

— Ted Diehl

The program combines data, text, images, illustrations, equations, and 2D and 3D graphs in a readable, electronic calculation "worksheet" that also handles a mix of numbers and strings, arrays, and nested arrays. MC13 complements such other widely used engineering and business applications such as spreadsheets, finite-element-software, and graphic or presentation programs.

Matchcad has a broad appeal. It lets users perform everything from simple calculations to those that are large, complex, and highly nonlinear. For example, the software has many numerical and symbolic calculation features, and it performs both matrix and complex math. MC13 also solves ordinary differential and linear, nonlinear, and transcendental equations and computes derivatives and integrals (symbolically and numerically). Some of the many tools include programming, curve fitting, statistical functions, and data interpolation by means of splines. A few examples illustrate simple and more-advanced calculations.

In its simplest use, Mathcad is superior to the ubiquitous spreadsheet in writing and displaying clear equations that compute and include units. Users can create simple equations and functions that can be combined with internal Mathcad functions like integration. Users can also create high-level function-programs that include looping, logic, and other advanced features. Un-like a spreadsheet, these equations and functions look natural, similar to common engineering notation. If your document becomes too cluttered, you can hide details by collapsing areas or highlight regions of special interest.

Mathcad also works with mixed units, doing the conversions for you. Say, for example, you are calculating Force from Mass and Acceleration (F= MA). In Mathcad you can define the variables for M and A in any units you like (kg and m/sec 2 , lbm and ips 2 , or a mixed set such as kg and fps 2 ). Regardless of the units, the formula is still the same (F = MA). Mathcad internally does the conversions and then allows the user to select units for the result (for example, Newtons, lbf, ton). This is useful for global industries that deal with metric and English units. With a spreadsheet, this type of flexibility simply does not exist.

However, there are a few cases in which MC13's units-conversion feature could produce undesirable results. Vibration problems, for example, involve frequency, which is typically defined with units of Hertz or radian/second. For mechanical engineers, 1 Hz is classically defined to represent 1 cycle/sec, where 1 cycle is equivalent to 2 radian. These users consider Hertz and radian/second to be two different unit measures of the same quantity, frequency, just differing by the scaling value of 2 (much like radians and degrees both represent angles or inch and meter represent length).

Unfortunately, the ISO standards that Mathcad follows, deviate from the original definition and define the unit of Hertz more generically as 1/sec, causing a potential unit conversion error of 2 in a vibration problem. The software lets users address this shortcoming in several ways, including explicitly placing the 2 factor in equations as needed or redefining Hertz to equal 2 radian/sec.

For more advanced tasks, Mathcad works well with other engineering tools, such as FEA software. In a recent application, I used both Abaqus/Standard and Abaqus/Explicit to analyze the failure of a metal T-bracket as it is pulled apart. I output the load versus displacement results from the FEA simulations to ASCII files and read them into MC13 for further analysis. The resulting "Raw Data" graph showed that the highly noisy explicit transient-dynamic solution did not correlate well with the smoother implicit solution produced by Abaqus/Standard. Applying special DSP-based smoothing filters from the Cornucopia add-on toolbox to remove noise improved the interpretation, accurately showing the two solution methods yield similar results. During these calculations, the MC13 worksheet read-in large x-y data files (over 100,000 lines each) and computed in less than 2 seconds (although it did take 40 seconds to initially open).

Users can also develop their own toolboxes. These are just worksheets that include a variety of functions and function programs. This capability and the software's clear visual layout makes short work of developing and using programs which, for instance, compute derivatives of numerical data using the coefficients of a spline of the same data. And the "Insert Reference" feature simplifies creating libraries of worksheets and then referencing them into a document as needed.

Minor gripes: There should be a scheme for numbering and referencing images, figures, equations, and sections and headings. And publication-quality graphing (2D and 3D) needs improvement. Also, for many "power users," the ability to compile user-created worksheets and functions is long overdue. It would let users directly create linkable libraries for use with other applications that use languages such as C and Fortran.

On balance, considering its pluses and minuses, Mathcad 13 is a unique and useful tool for the engineering and scientific computing arena.

Mathcad 13 comes from Mathsoft Engineering & Education Inc., (800) 628-4223), mathcad.com

 Calculations for Stretching a Rod shows how Mathcad provides a clear visual display of text, sketches, plots, equations, integrals, and results.

Ted Diehl is president of Bodie Technology Inc., (610) 368-3638, bodietechnology.com, which provides expertise in the use of DSP and analysis software and in deciphering complex problems in non-linear structural mechanics. 