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Download PDF: Efficient Large Displacement/Large Rotation Dynamic Simulations Using Nonlinear Dynamic Substructures

Utilizing reduced-order dynamic math models (DMM) in linear system-level dynamic analyses is a well-known practice that enables extreme computational efficiencies. But what about nonlinear system dynamics? Reduced-order DMMs have found their way into contact dynamics. The engineer must look no further than the Henkel-Mar pad separation analysis methodology to verify this fact. More sophisticated applications of DMMs in contact dynamics are possible when certain repetitive geometry pattens are present. For example, Figure 1 shows a type of pipe known as a “flexible” pipe used by the subsea industry. This design features four layers of helically wound steel wires that provide the pipe with its stick/slip behavior during bending, thereby enabling a longer fatigue life in harsh ocean environments. With these helically wound armor layers presenting a repetitive contact topology, contact surfaces can be constructed and tracked enabling the friction logic to operate resulting in the friction hysteretic moment-curvature plot provided in Figure 1 (top). 

As seen from Figure 1, the pipe was subjected to many bending cycles and executed in essentially a real-time computation. A single bending cycle of the same pipe in full finite element model (FEM) resolution (i.e., no use of DMMs) would require 48 hours of computation on 36 central processing units (CPUs) running in parallel given the very large order of the FEM.   

What about utilizing DMMs for computationally efficient nonlinear dynamics involving large displacements and rotations? Before addressing this question, the residual flexibility mixed boundary transformation (RFMB¹) must be defined. The RFMB coordinate transformation is given as follows: 

The transformation is a mix of the following submatrices: constraint modes (ψ) due to unit displacements on the b-set boundary degrees of freedom (DoFs) that remain fixed during the eigenvalue problem, residual flexibility (g) due to unit forces at the c-set boundary DoFs that remain free during the eigenvalue problem, and a truncated set of normal modes (φ) computed with the b-set DoFs constrained. It can be shown that the transformation retains full flexibility at the DMM physical DoFs and retains the full dynamics of the FEM up to the user-selected truncation frequency for the normal modes. The reduction of DoFs, and hence the computational efficiency, arises from the number of kept modes (k) being significantly less than the number of interior FEM DoFs. 

To enable DMM large displacements/rotations, four coordinates are added to the above RFMB to track large rotations. These quaternions replace the rigid-body modes that are only valid for infinitesimal rotations. With this process, the RFMB is transformed into a nonlinear dynamic substructure (NDS). Solution algorithms need to be modified accordingly as well to allow for equilibrium iterations since the problem now is highly nonlinear. As an example, consider the undeformed cantilever beam model (Figure 2) composed of 20 DMMs (single DMM of a beam composed of 5 CBAR elements repeated 20x).   

A moment is applied at the free end (right end) of Figure 2. While small displacement theory is limited and breaks down after a few degrees of rotation, the cantilever beam can be completely rolled up using NDS (see Figure 3) in a highly nonlinear dynamic simulation. Also note that the entire nonlinear dynamic simulation was executed in seconds on a laptop and included all dynamic effects. Similarly, the beam can be bent into a “catenary-like²” shape by turning on gravity and enforcing displacements at each end to the required coupling location (see Figure 4). 

One application for this large displacement/rotation NDS capability has been to include umbilical models in the coupled loads analysis (CLA) framework. Figure 5 shows the Interim Cryogenic Propulsion Stage (ICPS) umbilical that was integrated into the Space Launch System (SLS) CLA. The SLS CLA is an integrated assembly of various component DMMs (boosters, core stage, mobile launcher (ML), upper stage, etc.) to which the ICPS umbilical (ICPSU) and its hoses as NDS DMMs can now be added. For each hose, one end connects to the SLS vehicle and the other end to the ML structure. As an example, Figure 6 shows the evolution of the deformations of the forward vent hose (modeled with 20 NDS DMMs) as it goes from the undeformed geometry (straight line) into its prelaunch geometry during the initial condition setup in the CLA. 

As the timed command for umbilical separation is given, the vehicle-side ground plate separates (using the Henkel-Mar contact/separation algorithm) and the ML gantry rotates the separating umbilical away from the already lifting vehicle (the gantry was brought into the CLA as a NDS capable of large rotations). Figure 7 captures the post-separation forward vent hose dynamics (extracted from the CLA). From this, 100  ICPSU hose clearances to the lifting vehicle can be computed. 

The power of the reduced-order models does not end with linear dynamics. It is possible to introduce large displacements and rotations into reduced-order models to enable seamless integration into large substructured integrated system dynamic analyses such as a CLA. For the specific case of the SLS, this capability allowed us to integrate umbilicals into the CLA to more accurately capture the impact of system flexibilities, dynamic response to forcing functions, pad separation “twang” effects, ML dynamics, and gantry/umbilical timings on clearances.  

For information, contact Dr. Dexter Johnson.  dexter.johnson@nasa.gov 

Winter winds lofted clouds of dust from the Sahara Desert, carrying it north toward the Mediterranean and dispersing it widely across Europe in March 2026. When the dust combined with moisture-laden weather systems, a dirty rain fell in parts of Spain, France, and the United Kingdom.

This animation highlights the concentration and movement of dust throughout the region from March 1 to March 9. It depicts dust column mass density—a measure of the amount of dust contained in a column of air—produced with a version of the GEOS (Goddard Earth Observing System) model. The model integrates satellite data with mathematical equations that represent physical processes in the atmosphere.

The animation shows dust plumes originating in northwestern Africa being blown both to the west across the Atlantic Ocean and north toward the Mediterranean. As plumes spread throughout Western Europe over several days, people observed hazy skies from southern England, where sunrises and sunsets took on an eerie glow, to the Alps in Switzerland and Italy, where a dust layer encroached on the Matterhorn.

Not all of the dust remained aloft. Storms encountered some of the dust, causing particles to fall to the ground with rain and coat surfaces with a brownish residue. A low-pressure system, named Storm Regina by Portugal’s weather service, moved across the Iberian Peninsula and brought so-called blood rain to southern and eastern Spain, along with parts of France and the southern UK in early March, according to news reports.

Over the Mediterranean, areas of “dusty cirrus” clouds developed higher in the atmosphere, where dust particles can act as condensation nuclei for ice crystals, according to MeteoSwiss, Switzerland’s Federal Office for Meteorology and Climatology. Scientists are studying these clouds to better understand their formation and how they affect weather, climate, and even solar power generation.

In a new analysis, researchers used NASA’s MERRA-2 (Modern-Era Retrospective Analysis for Research and Applications, Version 2), observations from MODIS (Moderate Resolution Imaging Spectroradiometer), and other satellite products to parse the effect of airborne Saharan dust on solar power in Hungary. They found that photovoltaic performance dropped to 46 percent on high-dust days, compared with 75 percent or more on low-dust days. They determined the greatest losses occurred because dust enhanced the presence and reflectance of cirrus clouds and reduced the amount of radiation that reached solar panels.

Some research suggests more frequent and intense wintertime dust events have affected Europe in recent years. Researchers have proposed several factors contributing to these outbreaks, including drier-than-normal conditions in northwestern Africa and weather patterns more often driving winds north from the Sahara.

NASA Earth Observatory animation by Lauren Dauphin, using GEOS-FP data from the Global Modeling and Assimilation Office at NASA GSFC. Story by Lindsey Doermann.

References & Resources

• Barcelona Dust Regional Center (2026, March) Daily Dust Products. Accessed March 11, 2026.
• FOX Weather (2026, March 9) Blood rain, a rare weather phenomenon, falls across southern Europe. Accessed March 11, 2026.
• IQAir (2026, March 6) Southwest Europe Air Quality Alert: Southwest Europe Dust. Accessed March 11, 2026.
• Met Office (2026, March 4) What is ‘blood rain’ and will we see it this week? Accessed March 11, 2026.
• MeteoSwiss (2026, March 4) He’s here again, the visitor from North Africa. Accessed March 11, 2026.
• NASA Earth Observatory (2022, March 31) Dusty Storm Clouds Over Europe. Accessed March 11, 2026.
• NASA Earthdata (2026) Dust/Ash/Smoke. Accessed March 11, 2026.
• Seifert, A., et al. (2023) Aerosol–cloud–radiation interaction during Saharan dust episodes: the dusty cirrus puzzle. Atmospheric Chemistry and Physics, 23, 6409–6430.
• Varga, G., et al. (2026) Saharan dust and cirrus clouds: Dominating indirect impact of dust events on photovoltaic energy generation in Hungary (2019–2024). Solar Energy, 307.

By USGS Landsat Missions 

The William T. Pecora Award is presented annually to individuals or teams using satellite or aerial remote sensing that make outstanding contributions toward understanding the Earth (land, oceans, and air), educating the next generation of scientists, informing decision-makers, or supporting natural or human-induced disaster response. Both national and international nominations are welcome.

The award is sponsored jointly by the U.S. Department of the Interior and the National Aeronautics and Space Administration and was established in 1974 to honor the memory of Dr. William T. Pecora, former Director of the U.S. Geological Survey and Under Secretary, Department of the Interior.  

Dr. Pecora was a motivating force behind the establishment of a program for civil remote sensing of the Earth from space.  His early vision and support helped establish what we know today as the Landsat satellite program. 

Nominations for the 2026 award will be accepted until May 29, 2026.

Visit the William T. Pecora Awards webpage for eligibility requirements and the nomination process.  

NASA’s University Innovation (UI) project funds university-led innovation to address the agency’s Aeronautics Research Mission Directorate’s system-level challenges via independent, NASA-alternate-path, multi-disciplinary awards.

Strategic Goals

The UI portfolio’s strategic goals in descending order of importance are:

1.    Assist in achieving aviation outcomes defined in the ARMD Strategic Implementation Plan through NASA-complementary research.
2.    Transition research results to an appropriate range of stakeholders that leads to a continuation of the research.
3.    Provide broad opportunities for students at different levels, including graduate and undergraduate, to participate in aeronautics research.

Portfolio Elements

The UI project’s strategic goals are achieved through two opportunities that are available through NASA Research Announcement awards.

University Leadership Initiative (ULI)
ULI provides the opportunity for university teams to exercise technical and organizational leadership in proposing unique technical challenges, defining interdisciplinary solutions, establishing peer review mechanisms, and applying innovative teaming strategies to strengthen the research impact. By addressing the most complex challenges associated with ARMD’s strategic thrusts, universities will accelerate progress toward achievement of high impact outcomes while leveraging their capability to bring together the best and brightest minds across many disciplines. To transition their research, principal investigators are expected to actively explore transition opportunities and pursue follow-on funding from stakeholders and industrial partners during the course of the award.

University Students Research Challenge (USRC)
USRC seeks to develop novel concepts with the potential to create new capabilities in aeronautics by stimulating aeronautics research in the U.S. student community. USRC provides students, from accredited U.S. colleges or universities, with grants for aeronautics projects that also raise cost sharing funds using crowdfunding platforms. By including the process of creating and preparing a crowdfunding campaign, USRC can act as a teaching accelerator to help students develop entrepreneurial skills.

Gateways To Blue Skies
Gateways to Blue Skies expands engagement between universities and NASA’s University Innovation Project, industry, and government partners by providing an opportunity for multi-disciplinary teams of students from all academic levels (i.e., freshman, sophomore, junior, senior, and graduate) to tackle significant challenges and opportunities for the aviation industry through a new project theme each year. The competition is guided by a push toward new technologies as well as environmentally and socially conscious aviation.

UI Project Page, University Innovation (UI) Tech Talks

This March 3, 2026, image combines views from ESA’s (European Space Agency) Euclid and NASA’s Hubble Space Telescope to feature one of the most visually intricate remnants of a dying star: the Cat’s Eye Nebula, also known as NGC 6543. This extraordinary planetary nebula lies 4,400 light-years away in the constellation Draco and has captivated astronomers for decades with its elaborate and multilayered structure.

See what this observation reveals about this planetary nebula.

Image credit: ESA/Hubble & NASA, ESA Euclid/Euclid Consortium/NASA/Q1-2025, J.-C. Cuillandre & E. Bertin (CEA Paris-Saclay), Z. Tsvetanov

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