Editor’s note: Guest writer, Alejandro Rivera, Aerospace Engineer at Goddard Space Flight Center,* argues that using dedicated engineering calculations software to ensure accuracy should be part of any engineer’s FEA / CAE Process. Then, he discusses how he used that calculations software to do just that for analyzing the James Webb Space Telescope (JWST), a complex project that Alejandro’s process and software were able to handle.
Why use Mathcad as an essential tool within your FEA / CAE process?
A brief engineering analysis history
Engineering analysis before computers
The leading, competing programs during the space race were two giants of aerospace engineering: on the American side, rocket engineer Dr. Wernher von Braun, and on the Soviet side, astronautics genius Sergei Korolev.
While both represented entirely opposite systems and ideologies, they had one thing in common: both used identical German Albert Nestler A.G. slide rules for their calculations regarding plans and rockets to put a man on the moon. Their slide rules, which today can be found at the National Air and Space Museum in Washington DC, were considered at the time the most accurate and precise.
The slide rule used by engineers on the Mercury, Gemini, and Apollo programs is arguably the computing device that put a man on the moon. That and paper and pencil were NASA engineers’ main tools. Back then, there was no Finite Element Analysis software – as it only became available to a few engineers in the late 1960s and saw a very limited use – and certainly no powerful desktop and laptop computers like we have today.
Finite element analysis software and other modern innovations introduced
It was Thomas G. Butler from NASA Goddard Space Flight Center who championed the creation of a general-purpose finite element analysis (FEA) software.
A contract was awarded to CSC, the Martin Company, and MacNeal-Schwendler Corporation (MSC) who developed NASTRAN (NAsa STRuctural ANalysis) and delivered it to NASA in 1968. In 1972 Hewlett-Packard introduced the first handheld electronic calculator. It was so successful that almost instantly the slide rule became obsolete.
Finite element analysis began to see more widespread use in NASA throughout the 70s during the design of the space shuttle.
Today’s analysis approach
Fifty years later, today’s engineers have high-performance computing and engineering analysis tools on our personal computers that help us perform finite element analysis, dynamics, orbital mechanics analysis, etc. Their goal is to facilitate and speed up the analysis and overall engineering process.
Today’s typical FEA process can be summarized in the figure shown below. Given, for example, a stress analysis related problem, the software’s pre-processor is used to create the model. Typically, this involves importing the geometry, creating the mesh, and specifying boundary conditions, loads, materials, etc. The pre-processor then assembles the data into a format that is suitable for execution by the solver. The solver or processor runs all the computations, generating and then solving the system of equations. The post processor accepts the numeric solution and produces graphic displays of the data that allow the user to interpret the results. Several iterations are usually required to achieve a properly converged solution.
Fig. 1: Standard process for solving an analysis problem
Mathcad Prime: An alternative for better-informed engineering analysis
As part of the FEA / CAE process, I personally try to involve PTC Mathcad Prime as much as possible. As an example, given a dynamics analysis problem, this is the approach I will follow:
- I first draw the free body diagram and then go to the math software and start deriving the differential equations of motion using a Newtonian or Lagrangian approach.
- The equations of motion are then integrated using Mathcad’s solver. The results can then be plotted and interpreted within Mathcad Prime.
- Because I know that eventually I will need to deliver an engineering report, I start the Mathcad Prime worksheet with that end in mind and start writing it concurrently. This saves me considerable time.
- Once I solve the problem in Mathcad Prime, I go to the dynamic analysis software where I build the model and run the analysis.
- Once done, I compare the results using both approaches. If both are within 3%, I know I have reached a good solution. If not, I go back and review both models to see where the discrepancy lies.
Fig. 2: Author's process for solving a dynamics problem using Mathcad Prime and analysis software
By using Mathcad Prime to develop the mathematical model and integrate the differential equations of motion, I can better understand what is occurring, becoming more with engaged with the process.
I feel that this also enhances my analytical skills and allows me to gain a better understanding of the physics involved. By the time the problem reaches the professional software package, I have minimized the potential for modeling errors and can compare the software result to that obtained from Mathcad. That in turn reduces the risk that I could blindly accept faulty software results. Using Mathcad to check fundamental principles such as conservation of momentum and energy is also easy and convenient.
Pain-free reports
A bonus to this approach? No more headaches creating reports, as the worksheet builds them from the beginning. You no longer must merge information from Excel and the analysis software into the word processor, or deal with Excel’s painful equation editor.
Changes and updates to the analysis are automatically captured via Mathcad Prime’s all-encompassing platform. Being able to share the analysis findings without imposing a skillset on the reader such as MATLAB®, FORTRAN®, etc. is an extra plus.
Overall, when using this approach, the engineer regains control of the analysis, and the overall process can even become more gratifying. Personally, I enjoy solving a problem by hand and seeing how all the dynamics involved come into play much more than just pressing a button and letting the software assemble the system of differential equations and solve them for me.
The art of engineering systems modeling
A common argument against solving problems by hand is that “the geometry of my problem or the number of components in my engineering system is way too complex, so I can only do it with the software.”
I argue that the ability to synthesize a complex problem into its key elements to a level that can be solved by hand and obtain a close solution is an “art” that is decreasingly common nowadays, yet one that should be practiced as frequently as possible.
Developing these skills can prove extremely helpful for any engineer. So next time you are ready to press the button to run your analysis model, ask yourself: “Do I fully understand all the engineering principles and mechanics involved in this problem?”
If the answer is no, a simple Mathcad Prime mathematical model could be all you need to feel more comfortable about your modelling approach and solution. Chances are you will enjoy the process more, and along the way, you will become a better engineer.
Can Mathcad truly be used for complex problems?
When I discuss Mathcad Prime with other engineers, I sometimes hear comments to the effect that it can only be used for basic problems. My personal experience, however, is different, as I have been able to use Mathcad to solve rather complex ones. As part of the verification of software results process I described above, Mathcad Prime was used on the analysis of some of the deployments and dynamic events of the James Webb Space Telescope (JWST).
Using Mathcad Prime to become a better-informed engineer
Using Mathcad Prime to become a better-informed engineer
Using Mathcad Prime to become a better-informed engineer
Using Mathcad Prime to become a better-informed engineer
Using Mathcad Prime to become a better-informed engineer
Using Mathcad Prime to become a better-informed engineer
"By using Mathcad Prime to develop the mathematical model and integrate the differential equations of motion, I can better understand what is occurring, becoming more with engaged with the process."
A most interesting example that follows is the dynamic analysis of the separation of the telescope from the Ariane 5 Upper Stage.
Mathcad Prime use for James Webb Space Telescope analysis
The JWST separation
The JWST observatory was launched on a European Space Agency (ESA) Ariane 5 launch vehicle (LV) from the Guiana Space center in Kourou, French Guiana, on Christmas Day 2021. The LV inserted JWST into a direct transfer orbit away from the Sun and towards the Earth-Sun Lagrangian Point L2 approximately 30 minutes after liftoff. Three propulsion mid-course correction (MCC) maneuvers were required for JWST to reach L2. This was because, by design, the LV injection process provides less energy than required, and to account for deviations from the ideal trajectory after the injection and separation processes.
Fig. 3: JWST trajectory to L2
The separation process is one of the most critical moments of the mission. If not done correctly there is a risk of collision between the upper stage and spacecraft; the post-separation spacecraft tip off rates have to be minimized; the relative velocity between the upper stage and the spacecraft has to meet a minimum requirement; and the resulting trajectory vector can’t deviate excessively from the optimal one, or else during the MCC-1a maneuver more propellant than desired would be needed. Because MCC-1a provided the majority of the energy needed to achieve the L2 orbit while MCC-1b and MCC-2 maneuvers provided small corrections to establish the final orbit, even a small error during the LV injection into the direct transfer orbit to L2 or on the separation had large propellant consequences.
Fig. 4: Separation from Ariane 5 Upper Stage
Separation analysis using Mathcad Prime
Earlier, I described in detail how useful it can be to use Mathcad Prime to create mathematical models that can be used to verify the results provided by complex CAE software packages. Following this approach, Mathcad Prime was used to create a dynamic analysis model of the separation event. Six differential equations of motion were derived and then solved simultaneously using Prime’s ODE solver to determine JWST’s post separation linear and angular displacements, velocities, and accelerations, as well as the corresponding linear and angular momenta, for a myriad of scenarios. The effect on the separation dynamics of the propellants’ sloshing inside their tanks was also analyzed via additional differential equations.
The results matched those of the dynamic analysis software with a discrepancy of less than 3%.
The LV injection of JWST into its direct transfer orbit and the separation were so accurate that during the mid-course corrections we ended up using a lot less propellant than expected. Hence, the length of the mission will be extended by additional years, and the Webb telescope will have enough propellant to allow support of science operations in L2 orbit for significantly more than a 10-year science lifetime (~20 years is the latest estimate).
Solving Systems of Equations in Mathcad
Learn how to use Mathcad Prime’s Solve Block to solve linear, non-linear, and differential equations.
Read the Blog
Actual video footage of the separation sequence can be seen on this video courtesy of ESA and Arianespace:
Whenever you face a challenging engineering problem, consider using Mathcad to create a mathematical model of it that can be used to validate the CAE software results. I personally find this approach very effective and rewarding, and I highly recommend it.
*The views, methods and opinions expressed in this article are those of the author and do not necessarily reflect the official policy, procedures or position of NASA Goddard Space Flight Center or the National Aeronautics and Space Administration. NASA does not promote or endorse any products and this article does not represent an endorsement of Mathcad Prime.
Graphics and images courtesy of NASA and ESA.
The author is not affiliated to PTC in any form and received no financial compensation for this article.
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