## Joint venture

Nuclear reentry uncertainties aside, a presentation Brake co-authored at a 2013 ASME conference noted that, in general, stiffness measurements of nominally identical bolted joint hardware can vary by as much as 25 percent, and energy dissipation measurements by up to 300 percent.

“There really is a large amount of variation in these systems,“ Brake says. “We may think we have a model for how joints behave, but this model may not have all the correct physics and thus we are neglecting effects without realizing them.”

One major shortcoming, Brake says, is the inability to accurately model friction. Another ASME presentation he coauthored in 2013 called it a ”grand challenge” for experimental and theoretical mechanics in the 21^{st} century, even though friction has been studied since the 1600s.

The truth is that “the fundamental mechanisms behind the phenomenon of friction are poorly understood,” the presentation added. What happens when two surfaces slide against each other, heating up in the process, actually is an unclear combination of deformation, atomic bond breaking and the generation and damping of stress waves, the authors wrote.

Traditionally, researchers have attempted to model complex joint interactions with fine-mesh finite element analysis. That approximation technique uses differential equations to create three-dimensional, geometric grids divided into discrete parts interconnected at node points and spread throughout the domain being modeled. “You can think of it as taking a car and breaking it up into a whole bunch of Lego blocks,” Brake says.

But there’s a catch. For maximum accuracy, each mathematical block must be extremely small. That data density means joint systems modelers must solve an overwhelming tens-of-millions of equations for each simulation time step. Even short simulations might require millions of time steps producing trillions of equations.

So researchers are exploring Reduced Order Modeling (ROM) in a quest to “find mathematical equivalents that faithfully represents aspects important for analysis while reducing the number of equations that must be solved,” Brake says.

For instance, researchers have found that vibration frequencies above hundreds of kilohertz are too rapidly damped out of structures for proper experimental validation. Eliminating such high vibrations, which can’t be properly accounted for anyway, reduces the number of modeling equations from many millions to tens of thousands per time step, Brake says.

Another ROM tactic called “frequency-based substructuring” evaluates an entire jointed system based on how its segments vibrate instead of how each block is described mathematically. That can reduce by about three orders of magnitude the computational time needed to model jointed structures with interfaces subject to friction, a paper Brake co-wrote says.

Investigators also are trying to broaden ROMs to accommodate nonlinearities – aspects of joint behavior that are difficult or impossible to describe mathematically. Another of Brake’s 21^{st–}century grand challenges is modeling unavoidable uncertainty in joint behavior. He thinks that will require stepping into the realm of chance and probability with tools like Monte Carlo equations.

Why the focus on modeling? To reduce the need for costly experiments, Brake says. “Building prototypes is even more expensive and time consuming than building models. If a test fails and you need to redesign something, it’s a year and a half until you can test the next iteration. By developing a predictive and efficient analysis tool, we can circumvent this lengthy process.

“Our goal is to eventually develop a model that doesn’t necessitate the biggest supercomputer for simulating these systems.”

### About the Author

Monte Basgall is a freelance writer and former reporter for the *Richmond Times-Dispatch, Miami Herald* and* Raleigh News & Observer*. For 17 years he covered the basic sciences, engineering and environmental sciences at Duke University.