Combining Multiple-Body Dynamics (MBD) with Finite Element Analysis (FEA)

Multi-Body Dynamics (MBD) can help to reduce model complexity by lumping masses, stiffnesses and damping into parameterized elements connected via linkages and contacts. The technique is particularly useful for kinematic evaluation of mechanisms and studying dynamics where a full Finite Element Analysis (FEA) would be expensive.

While the subjects discussed in this article can apply to any mechanical system, I will focus on a passenger car as a working example that most folks can relate to. A car consists of a tire that contacts the road. The tire is coupled to a rim, which is coupled to a spindle, suspension links, steering links and a sway bar. Everything that moves with the tire rim is termed "unsprung mass," which isn't technically correct, because the tire itself has stiffness and damping. The vehicle body is connected to the unsprung mass via four spring/damper assemblies and is termed the "sprung mass." All of these items can be modeled simply using a 2 degree-of-freedom (2 DoF) system, represented by the spring of the tire and the spring of the suspension.

Nowadays, we can be a lot more sophisticated than that. Software packages such as MSC Adams and Dassault Systemes Simpack have libraries of components that are commonly used for automotive applications, so that the entire suspension and steering assembly can be modeled accurately. However, modeling components on a macro-level can only go so far. For example, modeling a tire as a simple set of springs and dampers has proven to be inaccurate, and the suspension itself includes components (such as control arms) that can not be considered infinitely stiff, not to mention the car's body (termed the Body in White or BIW).

Tire Modeling

It is important to include an accurate tire model, so that the virtual vehicle can be driven over a virtual road surface. There are several options for the tire, depending on the objectives. One is to model the tire structure as a composite using FEA (tools exist in Abaqus to do this). This can be very expensive and time consuming because every material used in the tire needs to be accurately characterized, and the model takes time to build and solve. FEA tire models are typically used for NVH applications, where dynamics at high frequencies become important. At the opposite end of the spectrum, there are formulae for tires called "magic formulae" which are used primarily for ride and handling work. For durability work, which is the world I live in, we use what I call "gray box" options. You may know the concept of a "black box," where you don't care what is inside the box, you can characterize its behavior by simply building equations that relate its output to its input. These equations, or transfer functions are usually linear. Since a tire is very non-linear, a black box approach will not work, you have to have at least some knowledge about it's characteristic behavior. Models exist for gray boxing, where the tire's response is measured in various predetermined conditions to parameterize it. One example of this, and probably the most popular is Ftire by Cosin. Companies around the world specialize in calibrating tires for FTire so that parameters can be plugged into their algorithm to model the tire's behavior. The thing that sets Ftire apart, is its ability to predict the tire's behavior for road inputs that are smaller than the tire patch, and so it is particularly suited to the types of events we are concerned about for durability testing.

Linear Flexible Bodies

In cases where the engineer is interested in evaluating the fatigue or dynamics of flexible components, it becomes necessary to integrate FEA into the MBD. If we are only interested in looking at linear behavior, there is a well established method to extrapolate stress in a meshed component when integrated into MBD. It uses the principles of Modal Analysis. First, an FEA model is built of the flexible body, and the solver is used to extract the modes of vibration of the structure (Eigen Values). Since the FEA algorithm will generate modes that are not relevant, it is important to distill the results into the primary modes of vibration. This is a process of orthogonalization. It essentially extracts the independent modes. This is a very important step and may be missed if you are using a mixed-vendor environment. MSC uses their Nastran solver to perform an additional step that they call orthonormalization that handles anomalies at boundary conditions, where a mode shape can be mistakenly be identified between two nodes. Think of it as spacial aliasing. Abaqus can do this too.

Once this step is complete, the results are integrated into the MBD model. When the model is solved, it generates modal coordinates, which are essentially the displacements of each mode resulting from the dynamic interaction with the other elements in the model as it is driven over the virtual road. These modal coordinates can then be used to calculate stress at each node. If you are using nCode's Design Life, it can accept the original FEA results and the Modal Coordinates file directly. However note that nCode does not support every FEA/MBD vendor combination. Abaqus/Simpack and Nastran/Adams are safe choices.

Non Linear Flexible Bodies

Obviously the method described above only works if the flexible body is perfectly linear, which is often not the case. Factors such as internal joint friction, local plasticity, geometry, etc. can introduce errors that would not be accounted for in this technique. To solve a non-linear flexible body in an MBD environment, we would use a co-simulation approach. Abaqus can used as the FEA solver, with Adams or SimPack as the MBD solver. Forces are passed one way and accelerations or velocities are passed the other, and the two models communicate on a point-by-point basis. There is an open source library which most vendors support called the Functional Mockup Interface (FMI) that provides for a communication protocol between models, or alternatively, Fraunhofer sells a server-based product called MpCCI. You may hear about other non-linear options, such as MaxFlex or Marc, but these have their limitations and so use these with caution.

Need Help?

If all this seems confusing, and you need help with building flexible bodies in MBD, or need help picking the right tools, Re:Test, Inc. can build and solve the models for you, or provide you with the help and guidance you need to do it yourself.