Welcome to the second installment in our series on dirt late model vehicle dynamics. In part 1 of the series, we talked a little about some of the technical resources available on race car vehicle dynamics and the differences between the engineering involved when looking at a pavement race car as compared to dirt track race car. We also talked about the differences between analytical tools used to evaluate a race car, and conceptual tools used to visualize the dynamic characteristics of a car. In part 2, we are going to start looking at the analytical tools in more detail.
Before we start looking at the equations that govern the dynamics of a dirt late model, we need to make sure we have a good general understanding of some basic engineering analytical techniques. We won’t try to cram an entire year of statics and dynamics courses into one paragraph, but we will touch on the necessary basics to understand how our system of equations work.
First, let’s look at the concepts of vectors and coordinate systems. In general, there are two types of quantities that we are interested in when developing a mathematical model. One is a “scalar” quantity, and the second is a “vector” quantity. A scalar quantity only gives you information about the magnitude of something. For example, the length of the left upper four-bar link is a scalar quantity. It is the same length no matter how it is oriented in 3D space. Seventeen inches is seventeen inches, no matter how you look at it. A vector quantity, on the other hand, gives you information about the magnitude and direction of a quantity. This directional information is determined by breaking the vector down into its component quantities which are relative to some frame of reference coordinate system. This coordinate system is made up of three axes which are the “x-axis”, the “y-axis”, and the “z-axis”. Each component quantity of the vector will also be a vector itself and be parallel with the associated axis of the coordinate system, so the x component of the vector will be parallel with the x-axis, etc., etc. Looking at our four-bar example, the left upper four bar link will have a magnitude of some value, say seventeen inches, and a direction in 3D space that is defined by the three components of the vector. The components of the vector are defined by the locations of the tail and head points of the vector in the 3D space of the reference coordinate system. We will not go into great detail describing this here as it is a little more complicated for our discussion; however, it can be found in any appropriate text book on engineering and/or mathematics. The point here is to understand that a vector has magnitude and direction, and it is relative to some frame of reference.
Second, let’s look at how systems of equations are derived. In mechanical engineering, it is common practice to develop systems of equations by summing forces along given directions, and/or summing moments (or torques) about given points. Forces along a given coordinate axis are typically summed and made equal to some known quantity. If a system is static, or not moving, they are set to zero. If the system is dynamic, or accelerating in some direction, they are set to some known force. Moments are rotational forces taken about an axis of a given coordinate system relative to some point. Moments can also be static or dynamic. For example, we may take a moment about point A around the x-axis. In vector mechanics, we typically use the cross product to simplify this process. We sum forces and moments until we have a system of equations to define our unknown quantities. In order to solve for a number of unknown quantities, a system of equations must contain at least one independent equation for each unknown quantity. This means if you have five things that you are trying to define, you will need to have five independent equations that define the unknowns. If you do not have much experience in mathematics, some of this may not make much since. That’s okay, it will make more sense as we move along. The key thing to remember is that we sum forces and moments to derive equations, and we need an equation for each unknown quantity we are trying to define. We can then solve the system of equations to find values for our unknowns. It should also be pointed out that forces and moments are always vector quantities.
With this basic understanding of vectors, coordinate systems, and equation derivation, we can now start looking at how to develop the equations that govern dirt late models (or any other race car for that matter). We will cover other mathematical concepts as they arise.
The first step in developing our system of equations is to define our coordinate systems. The first coordinate system is what we will call the world coordinate system. This is the system relative to the flat surface of Earth. In this coordinate system, gravity always points down. The second coordinate system is what we will call the track coordinate system. This coordinate system will move with the flat surface, so as the track surface banks, this coordinate system will move with it. For example, if there is no banking, gravity will point straight down, but if there is 15° banking, the gravity vector will have a vertical component and a component pointing to the left or right depending on the direction of banking. The third coordinate system is what we will call the chassis coordinate system. The chassis coordinate system is relative to the chassis frame. As the chassis moves around in roll, pitch, and yaw, the chassis coordinate system moves with it. You can think of the driver as having a perspective from the chassis coordinate system, provided he/she doesn’t move around inside the cockpit. We will be developing all of our equations in the chassis coordinate system. This means that we will use our chassis as a fixed object, and we will move the world around the chassis instead of moving the chassis around in the world and/or track coordinate systems. This may seem odd at first, but doing it this way simplifies the mathematics. In addition, this method also makes incorporating data acquisition information into our models much easier. DAQ boxes are generally mounted, in some known orientation, to the chassis frame and will give accelerations along all three axes relative to the accelerometer mounted in the DAQ box. The acceleration values can simply be input into the model, which is conveniently already in the chassis coordinate system. If data is wanted in the world coordinate system or track coordinate system, it is easy to do a coordinate transformation of the model results using the roll, pitch, and squat of the car to transform the data into a perspective that we as observers see the race car in. This would be done to obtain things like camber, caster, bird cage indexing, etc. that we would physically measure when setting up a race car in the real world. We also need to define directions within our coordinate systems. For our model, we will define the x-axis as running longitudinally with the chassis with positive x pointing towards the front. We will define the y-axis as running laterally with the car with positive y pointing to the left of the driver. The z-axis will be defined as up and down with the positive direction pointing up from the drivers perspective. As we define point locations within our coordinate system, we will define them by where they lie relative to each of the three coordinate axis. Any given point will have an “x” location, a “y” location, and a “z” location. We denote this by putting the xyz locations in brackets like [x, y, z]. The given xyz locations can be negative or positive depending on where they lie on the given axis. For example, a point in space defined by the 3D coordinates of [25”, -34”, 18”] will lie a distance of 25 inches on the positive side of the origin along the x axis, 34 inches on the negative side of the origin along the y axis, and 18 inches on the positive side of the origin along the z axis. Along with directions, we also need to define where the origin is located. The origin is the center point of the coordinate system and has the coordinates of [0, 0, 0]. When measuring points on a chassis, the head of the positional vector will be at the point [x, y, z] and the tail will be at the origin. In theory, you could choose to place the origin anywhere in space, and as long as you can accurately measure the coordinates of a point in question, the math model will work out. In reality, when working with a race car, it is good to place the origin somewhere in the center of the car near ground level. It is worth noting here that as you begin to understand how a coordinate system is laid out, you can start to see why things like roll center locations are not very useful as mentioned in part 1 of this series. When talking about roll center location, you often hear people say “put the roll center so many inches to one side of center and so far off the ground”. Well, where exactly is center, especially on an asymmetric car like a dirt late model? Depending on where you place the origin, three different people may place the origin in three different locations. If the three different people set their roll centers “so many inches” from their center, they will have three completely different set ups, but set up to the same roll center location target!
The second step in developing our system of equations is to determine what things we want to solve for. For our model, we have eight different things that we are interested in solving for. The first four unknowns are the vertical loads at each of the four tire contact patch centers. Note that when we refer to vertical in this case, it is vertical in the chassis coordinate system, not vertical with respect to the ground. In a more technical sense, we are looking for the z component of the force vector at the tire contact patch center in the chassis coordinate system. We already have an idea of what the lateral component (i.e. the y component), and the longitudinal component (i.e. the x component) of this force vector is based off of either DAQ system accelerometer outputs, or from track geometry and throttle/brake inputs. Note the use of the words “an idea of”. Although we may have an idea of what the lateral and longitudinal components are, we don’t yet know how they are distributed among the four tire contact patch centers. We will get into that later. We do not, however, have a good idea of the vertical component because of the weight transfer that takes place as a result of applying the lateral and longitudinal forces. The vertical loadings on the tires are a combination of the gravitational force, centripetal force from banking, and lateral & longitudinal load transfer forces. The last four unknowns are the vertical displacements of the four contact patch centers from some initial starting position. This starting position is simply some initial condition that is used to give the simulation process somewhere to start from. We will get into this more later, but for now you can think of it as a point where the chassis is setting somewhere just above the static ride height. Don’t worry, this will make sense later. Again, this vertical displacement is in the chassis coordinate system and is the z component of the displacement vector of the tire contact patch. We can translate the force and displacement vectors into the world or track coordinate system later as we see fit in order to give more “real world” useful results. The vertical loads and displacements give us a total of eight unknown quantities that we will build a system of equations to solve; therefore, we will need to develop eight equations to solve for the eight unknown quantities.
The eight equations can be broken down into three subgroups. The first group consist of equations one through three and are the “load transfer” equations. The second group consist of equations four through seven and are the “load distribution” equations. The third group consist of the single last equation eight which is a compatibility equation that ties everything together. In part three of this series, we will start to look at these equations in detail.
As always, if you find this blog helpful, then you can help me in return by thinking of Bartlett Motorsport Engineering next time you need to buy parts for your race car.