In this lecture, we are going to take a closer look into the transmission mechanisms of the vertical force of a tyre and physical effects that we saw thereof. A tyre can only transmit forces and torques when it is in contact with the road surface. In contrast to horizontal tyre forces, which are friction and shear deformation-based forces, the vertical force is a pure deformation-based or structural force and depends on properties such as the tyre's construction and the inflation pressure. Simply fact, the tyre can be seen as some kind of air spring. So before dealing with horizontal tyre forces, let's start with the tyre road contact area and take a closer look into the basic mechanisms of vertical tyre force transmission by using simple physical considerations. When a tyre has pressed to the ground by a certain wheel out offset, it is deformed in vertically direction. Let's take a look at the simple substitute model shown on the left, where the tyre belt is considered to be elastically attached to the rim. Here you can see two deflections in radial direction of the loaded tyre. The deflection delta z b of the tyre belt and the deflection delta z f of the tyre flank. These two together form the overall vertical tyre deflection delta z, which can also be determined by the difference between the unloaded tyre radius r0 and the loaded tyre radius rl in a more convenient way. The vertical tyre contact force, which counteracts the wheel load, is a function of this deflection. Simplify it can be described as the product of the overall vertical stiffness of the tyre cz and the deflection delta z. Let's talk a bit more about this vertical stiffness of the tyre. The resistance of the tyre against the deformation in vertical direction is described by this vertical stiffness, which determines the relationship between vertical force and deflection. Here you can see examples of measured vertical force deflection characteristics. Exampleally they are shown on the left side for a passenger tyre and on the right side for a track tyre. With increasing vertical load, the vertical stiffness increases, leading to a non-linear progressive force deflection behavior. This means that the vertical stiffness of a tyre is a function of the vertical deflection 2. To give you a better sense, simplify the stiffness of a typical passenger tyre is in the range of 200 kN per meter and for a track tyre it is in the range of 1000 kN per meter. The overall vertical stiffness of a tyre depends on two main factors. First, on the structural stiffness of the carcass, which depends on the tyre's construction. Second, on the inflation pressure, as you can see here. Of the influences on the overall vertical stiffness, the internal inflation pressure is the predominant one. Considering a passenger car tyre, the structural rigidity contributes about 15% or less to the overall vertical stiffness. The remainder of about 85% is due to the internal inflation pressure. Thus a tyre's ability to carry load depends significantly on the inflation pressure. During driving the vertical force, acting on a tyre is usually not constant and will deviate from the static vertical force of payload and standstill. Such a change in vertical force can also occur on even road surfaces. For example during braking, the vertical shift from the tyres on the rear axle to the ones on the front axle. At this point we have already talked about the vertical stiffness of a tyre. Now we will take a look on the relations of the vertical force to the size of the contact area and the pressure distribution. As you will see in later lectures, these two have a significant effect on the horizontal tyre force transmission. Let's go back to the simple model from earlier and take a look at what tyre deflection means for the contact area between tyre and road, which is called contact patch. The resulting contact patch length L can be approximated by means of a simple geometric approach. Assuming the undefirmed belt is a circle and using the geometric relations shown on the right, we can derive a relationship between the half of the contact length L, the belt deflection delta set B and the unloaded tyre radius R0. We assume that the flank deformations are in the same range as the belt deformations and that delta set B is much smaller than R0, which holds in normal driving situations. In this way an approximated relation between contact patch length L and overall vertical deflection delta set can be derived. The complete derivation can be found under the following link. To give you a better sense of the dimensions of a typical contact patch, we have created an example for you. On the left side, there is the approximated relation between the contact patch length and the influencing factors unloaded tyre radius R0, vertical tyre stiffness C set and the vertical tyre load F set. This approximated behaviour of the contact length L with respect to the deflection is shown in the figure below. Now calculate the contact patch length of a typical passenger car tyre by means of this approximation with the following parameters. The vertical force is 3.5 kN, the vertical stiffness is 200 kN per meter and the unloaded tyre radius is 0.3 m. This is the right solution. A description of the width of the contact patch is more complex than for the length. Apart from additional influences such as camber angle, the roundness of the tyre structure plays an important role. The cross section of a tyre is somewhere between a rectangle and a circle. To highlight this, we show you two extreme examples. On the right side, there is a tyre with an almost rectangular shape, as often seen for passenger cars or truck tyres. Motorcycle tyres, on the other hand, typically have a round cross section. The main reason for this is that they can and usually will be driven at significantly higher camber angles than passenger or truck tyres. Compared to the rectangular shape, the width of the contact patch changes more for a round tyre with a changing vertical force. Let's take a look at two so-called footprints of a typical passenger car tyre. The left one shows the measured shape of the contact patch at the low vertical load and the right one at the high load. What can be seen is the transition from a more elliptical shape at low loads to a more rectangular one at high loads. Most tyres are profiled tyres, this means that only the positive part of the tread is in contact with the road. The contact area can be distinguished between positive and negative areas. The main task of the negative areas is to drain water from the contact area on wet roads. As rule of thumb, about 70% of the overall area is positive, which means that only 70% of this already rather small areas really in contact with the road. For a certain tyre, the exact value of the positive area significantly depends on its individual tread design. So keep in mind, all forces acting on a tyre transmitted via a contact area that is smaller than the contact patch itself, which has the size of a few centimeters in length and width. Think about that the next time you sit in the car. Finally in the last part of this lecture, let's take a closer look at the vertical force transmission mechanisms within the contact patch. In general, when an on-rotating tyre without camber is loaded and pressed a flat surface, the vertical load of set produces a vertical pressure distribution cooset in the contact patch, as shown here on the left. The vertical pressure distribution in the contact area shows a symmetric shape. It is symmetric both in the longitudinal plane, as shown here and in the lateral plane of the tyre. For many vehicle dynamics applications, the vertical pressure distribution can be simplified to a description by a resulting contact force, which in this case acts in the wheel contact point W. The simplification is shown on the right figure. That's it for now. We will continue in a second part.