# Modeling Connected Moving Objects

## Introduction

Computing the motion of connected objects is a complex, but common, problem in engineering. It occurs in many practical scenarios, from a tugboat moving a barge to mechanisms for measuring the fuel depth in a car. In FLOW-3D, these sorts of problems are handled with the general moving objects (GMO) model. While the GMO model does not have a specific "linked motion" option, there are three approaches for modeling the motion of linked moving objects in the GMO model. The first approach is prescribing the motion of the objects, the second approach takes advantage of the collision feature in the GMO model, and the third approach uses the spring and rope feature of the GMO model to connect the objects. The pros and cons of each of these approaches are discussed in the next section.

## Modeling Approaches

### Prescribed Motion

Simple cases, where the connection between the objects is designed to force some known, regular motion are often better modeled by prescribing the behavior of each object because this requires less computational effort and results in less uncertainty in the motion profiles than if they were computed quantities. Some examples where prescribing the motion of the objects is preferred include simulations of camshaft and valve motion in an engine and the motion of gears in a transmission. This approach can also be used to model situations that would otherwise be intractable in FLOW-3D, like the motion of a timing belt, by prescribing the motion of small pieces of a larger object.

### Collision Model

With this approach, the actual geometry of the connector region is resolved by the mesh and the collision feature in the GMO model is used to allow the different pieces of the link to interact directly. This makes it the most intuitive of the three methods, though there are some difficulties that arise because the collision model was developed to compute the effects of a brief, impulsive collision between two objects. This leads to two important restrictions when using the collision model to simulate linked objects: (1) the interacting objects cannot be touching at time t=0, requiring that the small gap between the components is resolved by the mesh and (2) only two components can collide at a given time. While relatively minor, these restrictions limit what can be modeled with a practical mesh to relatively simple situations. For example, a three-bar linkage could not be modeled with this approach because only two of the components could interact.

To illustrate the use of the collision model, consider the following example of a double pendulum. The geometry of each pendulum is simple, consisting of a bar with a hole at each end that allows it to be connected to another pendulum with a pin. In FLOW-3D, the connecting pin is defined as part of one of the pendulum components to bypass the restriction described in (2). The gap between the connecting pin and the pendulums has been artificially increased to allow for practical modeling. This setup is shown in Figure 1. From the image, it is clear that resolving the gap between the pin and the pendulums will require cells that are small in comparison to both the length of the pendulums and the expected range of motion. This will lead to a large cell count and long runtimes. Even so, the simulation gives good results, as shown below.

### Springs and Ropes

The third approach uses the spring and rope feature of the GMO model. Springs and ropes are special objects in FLOW-3D that can be used to apply forces on moving objects, making them well suited for linking moving objects together. The ends of the springs (or rope) can be attached to moving objects or fixed to a point in space, applying a force on the objects that follows Hooke's law. Modeling a pivot with a spring requires creating a spring with each end attached to a different moving component at the pivot location as shown in Figure 2. Then, as the components move apart, the model will apply a force pulling the two points back towards their initial locations. This approach bypasses the restrictions associated with the collision model, allowing an arbitrary number of components to be linked together and using a coarser mesh. For example, using springs to link the components of a double pendulum together allows the cells to be four times larger than would be required when using the collision model.

When using a spring to connect two components, the time-step size must be limited to ensure that the computed forces do not result in any objects moving more than a single cell in a time step. A rough guideline for the maximum allowable time-step size comes from vibration analysis where the natural frequencies are used to characterize a system. In this case, a simple criterion can be derived by neglecting all external forces except for the spring force. The resulting criterion suggests that the time step should be much less than the period of the first natural mode of oscillation to obtain quality results; mathematically, . From this it is clear that the time-step size becomes small when the spring stiffness (k) is much larger than smallest mass in the system (m), so springs that are rigid in comparison to the mass will result longer runtimes. It should also be noted that this criterion may not be sufficient in cases where the spring force is not the dominant force in the system. Even so, it is often possible to get a good approximation of the behavior of rigidly connected objects with this approach, as shown below.

## Summary

Prescribing the motion of the interacting objects and using the spring model to connect objects are the most commonly used approaches of the three methods outlined above. Prescribing the motion of the interacting objects is often a simple, accurate, and efficient method for modeling the appropriate motion. It avoids the difficulties that arise when computing the motion of the objects, but it is limited in the scope of problems that can be reasonably solved with this approach. A pragmatic approach to modeling scenarios where the motion of the objects cannot be prescribed is to connect the objects with springs. This method easily avoids the restrictions associated with the collision model (small cells and only two interacting objects) without introducing any major drawbacks. The main challenge with this method is managing the parameters to maintain a reasonable time-step size while simultaneously restraining the motion of the objects to an acceptable range.