Welcome to the fourth installment in our series on dirt late model vehicle dynamics. In part three, we looked at the development of the first three equations in our dynamic system. We are referring to the first three equations as the “load transfer equations”. In part four, we are going to take a closer look at how engine torque reactions are incorporated into the second and third equations.
The first three equations in our system focus on how the vehicle reacts to external inputs of force such as gravity, lateral acceleration, and longitudinal acceleration. When looking at the three sources, we have to understand that they are the reactions to some change in state of the system that cause the accelerations. Newton laws are in effect here. First, we have gravity which is the result of a natural source of acceleration driven by the relative mass between two objects (a topic far beyond the scope of this blog). Second, we have lateral acceleration which is the result of the changing direction a body takes as it traverses a curve, and is a function of the bodies speed and the curve radius. Third, we have longitudinal acceleration which is the reaction to the torque generated by the engine. This torque is the result of a conversion of “external” chemical energy, in the form of liquid fuel, to mechanical energy, in the form of rotational torque. The power generated by the engine can be thought of as an external source of energy acting on the vehicle.
In each of the scenarios noted above, the sources of external force must eventually be traced back to a “fixed ground” for each of them to “push” against. You can visualize this by thinking of a hydraulic cylinder. One end of the cylinder needs to be grounded in order for the other end to generate force and motion to move or support a load. Gravity is a source of energy that is constant and does not need to “push” against anything solid to be generated. It is, in a sense, its own source of solid foundation to push against. The change of direction that generates the lateral acceleration is from the slip angle forces generated in the front tires as a result of steering input. In this case, the Earth is the ground that the tire pushes against to change vehicle direction. This force transmits up through the suspension to the chassis to cause yaw rotation. We will talk more about this later. The torque from the engine is slightly different from the other two scenarios. The “fixed ground” in this case is the chassis itself. This means that the reaction from the external energy is “grounded” within the very system we are looking at; therefore, we need to account for it in our dynamic equations.
The engine torque has an action and a reaction. We can think of the engine as the “action” side and the axle housing as the “reaction” side. In reality, both the engine and axle housing responses will be reactions to the explosion in the combustion chamber, just in opposite directions, but we will call the axle side the “reaction” and the engine side the “action” to keep things separated verbally. We can actually think of the entire chassis, minus the rear axle, as being on the “action” side since the engine is bolted directly to the chassis frame.
Let’s take a look at the “reaction” at the axle. As engine torque is transmitted to the wheel, it follows a path along the driveshaft, makes a 90° turn at the axle ring & pinion, and splits down each axle shaft until it finally reaches each of the wheels. Since there are two paths, the axle housing will react in two directions. The first reaction will be along the axis of the axle pinion shaft, which is roughly in line with the x-axis of our coordinate system. This reaction is in response to the drive shaft torque and is countered and distributed to the rear tire contact patches through the axle housing, and conversely to the front contact patches through the frame and front suspension. The second reaction is in response to the pinion gear in the rear end trying to “climb” the ring gear. This reaction will be along the axis of the axle shafts, which are roughly in line with the y-axis of our coordinate system. This reaction will be countered by a fifth-arm device, a pull-arm device, or directly through the suspension links depending on what type of rear suspension configuration is used. A fifth-arm device is assumed in this example. We will evaluate each of the reactions with respect to the x-axis and to the y-axis and incorporate them into equations two and three.
In order to integrate the reactions into our system of equations, we will need to know what type of forces and moments we are dealing with. We can establish this information in various ways. Since we are utilizing information from a data acquisition system, we can determine what our longitudinal accelerations are at various points. We can use this information along with the loaded radius of the tire/wheel assemblies and the gear ratio of the ring and pinion in the rear end to determine what the moments are along the axle and pinion axes. Once we know the moments, we can use them, along with our known suspension geometry points from a positional analysis, to establish what the reaction forces are and where they are applied. They can then be integrated into equations two and three accordingly.
Illustration #1 below shows a side planar view of how the fifth-arm forces would be applied when looking at the moments. This is for demonstration purposes only. In reality, we would use the cross product method described in part 3 to ensure that we are accounting for moments about both the x and the y axes. The blue arrows indicate the two ends of the fifth arm. One end is attached to the axle and pushes down on the axle housing, while the other end is attached to a “fifth arm shock” coil over assembly and pushes up on the chassis frame. This action generates what is commonly referred to as anti-squat. If the position of the fifth arm shock is positioned so that the rear chassis lifts up under acceleration, we have greater than 100% anti-squat. If it is positioned so that the rear goes down under acceleration, we have less than 100%. Perfect 100% anti-squat would result in no upward or downward movement under forward acceleration. Note that fully decoupled rear suspensions typical on most dirt late models have other mechanisms that contribute to anti-squat. The left and right four-bar links, along with the J-bar, all contribute to anti-squat when conditions are right. When using a decoupled rear suspension, we typically always have significantly more than 100% anti-squat. The amount of rear lift is controlled by the use of a limiter chain that controls the amount of separation between the frame and the axle. This is a good time for the reader to take a minute and think about what is happening here. What will happen if you put a longer fifth-arm on the car? What will happen if you put a shorter arm on and move the limiter chain left or right along the axle tube? For reference, test conducted by Bartlett Motorsport Engineering show that forces in a fifth-arm spring can be upwards of 800 to 900 lbs. when using a 36 to 38 inch fifth-arm.
Illustration #2 shows the reaction forces with respect to the x-axis. Since most domestic engines rotate CCW as viewed from the rear, the “action” on the frame will be to rotate CW. This, of course, assumes that the crankshaft axis is parallel to the x-axis of our coordinate system. This is seldom the case in real life, and can be corrected for, but to simplify things in our case, we are going to assume that it is parallel. As the engine transmits torque to the rear-end through the drive shaft, the entire axle housing will try to rotate with the drive shaft causing it to rotate CCW. As this action-reaction occurs, force #1 will decrease, force #2 will increase, force #3 will increase, and force #4 will decrease. This can be validated anytime you spin the right rear tire of a car with a standard differential. As engine power is applied, the chassis rotates CW while the axle housing rotates CCW. The imbalance in roll couple between the front and rear causes the axle housing to rotate more than the chassis; consequently, the right rear tire unloads. As a result, power is diverted to the path of least resistance (i.e. the right rear), and the tire spins. The dashed lines, intentionally drawn opposite direction, indicate the input forces from the reaction moments. They can be found using the known geometry of the four tire contact patches and the known reaction moments to develop two equations with two unknowns. For the front, use the “action” from the engine and take a moment about contact patch one for the first equation, then about contact patch two for the second equation. Repeat for the rear accordingly. In reality, if the chassis were symmetrical, you could probably get away without even doing this, and the forces would fall out in equations to be developed in later blog post; however, with an asymmetric chassis, it is best to break them out and define them here for added detail.
In part five, we will start to look at each of the four corners of the car and determine the path that the forces follow through the suspension, and how much each corner of the suspension will move.
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.