Degree in Aerospace Engineering – Prof. Roberto Lot
The aim of this work is to use the MATLAB® program to create a graphical user interface (GUI) that allows a user to upload data from an Excel file and fit them through the Magic Formula in an intuitive way.
One of the key requirements of the program is the ability to upload any number of data from any folder of the computer, giving the user the possibility of selecting the data and using them for the optimization.
The data are collected through the use of a machinery, called “Mototyremeter”, developed by the “Motorcycle Dynamic Research Group” at the Padova University.
With it, it’s possible to identify the mechanical properties of motorcycle tires. Until now, with this equipment, the following quantities can be measured:
- the lateral force as a function of roll angle,
- the lateral force as a function of the sideslip angle,
- the self aligning torque,
- the twisting torque and the rolling resistance force.
The fitting of the data is obtained through the use of the Pacejka’s Magic Formula, first in the classical formulation modified for motorcycles and, later, in a formulation which takes into account the pressure’s changes. Within the program the user is given the possibility of choosing between the two versions.
2. Mechanics of pneumatic tires
Most of the forces and moments affecting the motion of a ground vehicle are applied through the contact with the ground, which is secured solely by the tires. Acceleration, deceleration, steering and any other movement depends only on the relatively small area of contact is established between the tires and the road.
The tires used in road vehicles are usually required to fulfill the following functions:
• support the weight of the vehicle
• absorb the irregularities of the surface
• provide sufficient traction for driving and braking
• provide adequate steering control and direction stability
Understanding the basic characteristics of the interaction between the tire and the ground is, therefore, essential to the study of performance characteristics, ride quality and handling of a ground vehicle.
2.1 Tire structure
A tire is a torus shaped flexible structure, filled with compressed air. The most important element of the structure is the carcass, which is made up of a series of layers of flexible cords of high modulus of elasticity encased in a rubber compound with low elastic modulus (plies). The cords are made of natural, synthetic or metallic materials and are anchored around the beads built with steel cords of high tensile strength.
Different types of compound, are used to provide the tires with specific properties. Modern tubeless tires have a thin layer of rubber with high air impermeability attached to the inner surface of the carcass.
2.2 Types of construction
The characteristics of the tire are largerly dependent on the design and construction of the carcass. Among the design parameters, the geometrical arrangement of the plies, in particular their direction, has a significant role in the characterization of the behavior of the tire. The direction of the plies is defined by the “crown angle”, which is the angle formed between the direction of the steel cords that make up the plies and the center line of the circumference of the tire, as shown in the next figure.
Depending on the arrangement of the plies, the different existing types of tires can be divided into 3 categories:
• Diagonal (cross) ply
• Radial Ply
In the cross ply tires, the carcass cords extend diagonally from bead to bead with a crown angle of about 40°. The number of plies is variable, from a minimum of 2 (for low-load tires), to 20 (heavy duty). The cords in two adjacent plies run in opposite directions, overlapping and forming a diamond shape pattern (criss-cross). In operation, the diagonal plies flex and rub, elongating the diamond-shaped elements and the matrix rubber, this action produces a bending movement and friction between the tread and the road, which is a major cause of tire wear and rolling resistance.
The radial tire was introduced first by Michelin in 1948 and is now the dominant type of tire on the market; its construction is very different from the cross ply tire. The radial tire has one or more layers of cords in the carcass that extend radially from bead to bead, thus having a “crown angle” of 90 °, as shown in the figure. A belt of several layers of cords with a high modulus of elasticity (usually steel or other high strength materials) is mounted beneath the tread; the belt cords are laid at a low “crown angle”, up to 20 °. Tires for passenger cars are usually made of two radial plies of synthetic materials in the carcass, such as rayon or polyester, and two layers of steel cords and two layers of synthetic material such as nylon in the belt.
In the radial tire, the flexing of the tire creates a very little relative movement of the cords that make up the belt. Being absent this movement of friction between the tires and the road, the power dissipation of the radial tire is lower, by up to 60%, compared to that of cross ply tire under similar conditions, and the life of the radial tire can also come to be twice. Unlike cros s ply tire, the surface pressure, over the entire contact area of a radial tire with the ground, is relatively uniform.
Finally, there are also tires, called bias-belted, constructed by inserting belts in the tread of a tire with cross ply construction. The cords of the belt are made of materials with a higher modulus of elasticity than those in the cross ply. The belt tread offers exceptional rigidity against distortion, and reduces tread wear and rolling resistance compared to conventional tires. Generally belted tires have characteristics midway between those of the other tires considered.
2.3 Tire markings
Information regarding the characteristics of tire construction and its dimensions must necessarily be placed by the manufacturer on the side of each.
Knowing the meaning of the marks is essential to recognize the different types of tires and their characteristics.
For example, in a tire 180/55 ZR17 73W:
• 180 is the nominal width of the tire
• 55 is the aspect ratio: ie the ratio between the nominal height of the shoulder and the nominal width of the tire
• ZR indicates that the tire has a radial construction
• 17 is the rim diameter in inches
• 73 is the load index
• W is the speed index
There are also other marks for which we refer to the following figure:
2.4 Forces and moments
To describe the characteristics of a tire and the forces and moments acting on it, has to be defined an axis system to act as a reference for the definition of various parameters. One axis system commonly used is that recommended by the Society of Automotive Engineers and shown in the below figure:
The origin of the axis system is placed in the center of contact area between the tire and the ground. The X axis is based at the intersection of the wheel plane and the ground plane with the positive direction forward. The Z axis is perpendicular to the ground plane with positive direction downward. The Y-axis is lying on the roadway, and its direction is chosen in order to make the axis system orthogonal and right hand.
There are three forces and three moments acting on the tire by contact with the ground:
• Tractive force (or longitudinal force) Fx: is the component in the X direction of the resultant force exerted on the tire by the road.
• Lateral force Fy: is the component in the Y direction (is the component on which this discussion is focused)
• Normal force Fz: is the component in the Z direction
• Overturning moment Mx: moment about X axis exerted on the tire by the road.
• Rolling resistance moment My: moment obout Y axis
• Aligning torque Mz: moment about Z axis
Thanks to this axis system it is possible to definemany of the performance parameters of the tire, for example, the integration of longitudinal shear stresses over the entire contact path is the braking force.
From the graph we also note the two basic angles, associated with the movement of the tire: the sideslip angle and camber angle. The sideslip angle “α” is the angle between the direction of travel of the wheel and the line of intersection of the plane of the wheel with the road. Camber is the angle between the wheel plane and the XZ plane. The force that develops at the contact between wheel and soil is a function of sideslip angle and camber angle.
In order to study the behavior of the tire and, if necessary, carry out mathematical modelling with formulas that will be introduced later, it is necessary to collect a number of experimental data. In this regard, one useful tool is the so-called Mototyremeter available at the University of Padova.
This machine allows to obtain data on the forces and moments developed by the tire under different operating conditions. There were so created different sets of data, under varying load and pressure conditions, in which the forces (such as lateral force) and moments (such as the moment of self-alignment) are measured sweeping the angle of sideslip (with zero camber angle) and angle of sideslip (with zero sideslip).
These data were then taken into account to carry out the fitting, using the Magic Formula with the MATLAB program.
3. Mathematical models
3.1 The Magic Formula
The study of the dynamics of the vehicle employs a large number of models to describe the properties of forces and moments generated by the tires.
One of the models, by far the most widely used since its conception more than 20 years ago, is the Magic Formula; the model provides a set of mathematical formulas that describe the forces and moments, acting on the pneumatic, under normal load with a sideslip or a camber angle; an extension of the formula describe the case of combined sideslip and camber.
The coefficients appearing in the formula represent typical quantities of the tire. By selecting values for parameters the general curve can serve as a description for the lateral force, the braking force or aligning torque .
In this discussion we will only consider the lateral force.
In the original formulation, developed for cars, the Magic Formula for the lateral force is:
The coefficients appearing in the formula determinate the shape of the curve and depend on the characteristics of the tire.
• Q: is the peak value of the graph and coincides with the friction coefficient
• Stiffness coefficent B: Extends the horizontal axis and therefore influence the slope of the graph at α=0(cornering stiffness BCD)
• C: defines the shape of the curve.
• E: influences the curvature at the peak without affecting the product BCD (cornering stiffness)
The figures following illustrate the influence of different factors on the curve:
3.2 The “Magic Formula” for motorcycles
The Magic Formula was adopted fairly quickly as the standard in the field of tires for passenger cars, however, the formula shown above has some severe limitations if has to be used for motorcycles. In motorcycle driving, the tilt angle of the wheel (camber) generates most of the forces necessary to turn, a sideslip angle occurs when the force generated by the camber is not sufficient. Since the Magic Formula was originally designed to describe the tire forces and moments in a stationary state, with cars as the simulation target, the effect of the camber has been underestimated.
One reason that has contributed significantly to the popularity of the Magic Formula is the fact that the equations of the model have been openly published in the literature. This has allowed, over the years, several developments made to improve the accuracy and extend the capabilities of the model: it has been possible to make changes to the model to make it more suitable for motorcycle simulations and has been developed a special Magic Formula to manage the large camber angles that occur on a motorcycle. The model has been improved to fit with precision large camber angles and to ensure consistent behavior outside the domain of measurements.
A small number of parameters has been kept, which makes the parameter identification more immediate,and has been also maintained a link between the meaning of the parameters and the physical behavior of the tire, making it easy to understand the effect of a variation of the parameter.
In the new model the leading factor D has remained as the maximum lateral force exercisable (μFz). All force contributions appear within the parentheses of the sine function, respecting the limits imposed by the friction coefficient D. To adequately cover the characteristics of the tire, even at high angles of camber, was introduced into the formula a completely independent arctangent function, with its stiffness factor B, factor C and form factor curve E , all of which provides sufficient freedom to describe the characteristics of pure camber accurately. In this way the force due to camber will respect the limit of grip, and the formula will be at the same time characterized by a strong interaction between the contributions of the camber and sideslip.
3.3 The role of pressure
The formulas seen above don’t’ take into account the variations off the tire inflation pressure, so makes it necessary to make separate parameters for different conditions of pressure in addition to requiring an additional measurement effort.
In order to improve the study of the mechanics of the tire, a model that includes the effect of changes of inflation pressure it’s needed, (it is clear that, on many road vehicles, while it is common to mount the same tires on the front and at the rear axes, on the contrary, usually there are different pressures in the tires of the two axes and can often be necessary to adjust the pressure according to the load conditions). This new model eliminates the need for separate set of parameters for different tire pressures and allows to evaluate the behavior of tire at different inflation pressures in the measurement program, thereby reducing the number of measurements required.
In this work,even for the fitting carried out taking into account the variety of pressures, it will be used, for simplicity the Magic Formula for motorcycles, suitably modified.
The formulas showed above contain within them a set of coefficients, which must be determined through procedures of “curve fitting”, from the experimental data obtained using the rotating disc machine. In this work a method of least squares optimization is used, minimizing the sum of squared differences between measured values and those provided by the Magic Formula.
The application of this procedure requires initial values to be assigned to the various coefficients. Since in general the measurements are not extensive enough, it is not possible to obtain experimentally the friction coefficient D, since the force does not reach saturation so we not know the maximum of the curve of the lateral force. Plausible initial values are adopted, giving the user the option to edit them in graphical interface.
4. MATLAB programming
To operate the optimization of the data throught the MATLAB program it will be used a graphical interface to make fitting easy and intuitive, allowing the user quick selection of data and allowing and to change at any time the coefficients of the Magic Formula.
As anticipated the user is given the opportunity of choosing between the normal formula for motorcycles and a formula accounting the changes of pressure.
4.1 MATLAB GUI
The task of the GUI is to make programs easier to use, providing intuitive controls such as buttons, list boxes, sliders or menu. It is important to make the GUI to behave in an understandable and predictable way, so that a user knows what to expect when he perform an action. To accomplish this the design of the GUI must be carefully studied; for example, the labels of each button must indicate the action they start when being pressed, then their position must be consistent, trying to group the buttons that are involved in similar areas in adjacent positions.
The decision to use a graphical interface forces the programmer to prepare the script in order to respond to mouse clicks and keyboard input at any time, as these inputs are known as “events”, a program that responds to a graphical interface is called event-driven.
The three main elements necessary to create a GUI for MATLAB® are:
• Components: Each element of a MATLAB® GUI is a graphical component and the types of components include control elements (buttons, edit boxes, lists, sliders), static elements (frames and text strings), menu and graphics
• Figures: The components of a GUI should be placed inside a window on the computer screen.
• Callback: For each event (such as a click of the mouse on button) the program must respond with the function associated. The code executed in response to an event is known as a callback. There must be a callback for each graphics component on the GUI.
To create a GUI there is a specific tool for MATLAB®, called “guide”, which consists in a “development environment” for the interface.
This tool allows the programmer to create the layout of the GUI, enter the various components in the window and change their properties, such as their name, color, size, font, and displaying text.
When the GUI is saved, a script containing the skeleton of MATLAB® functions is created, the programmer can modify these functions to implement the GUI’s callbacks.
The area within the grid is where the programmer can design its own graphical interface. On the left of the Layout Editor window there is a list of components which can be selected for the GUI; the programmer can create any number of GUI components by first clicking and then dragging the desired component in the layout area. At the top of the window there is a bar with a number of useful tools that allow users to distribute and align the GUI components, change the properties of the components, add menus.
The basic steps needed to create a MATLAB ® GUI are:
1. Decide what elements are necessary for the GUI and what is each function.
2. Use the tool guide (GUI Development Environment) for the layout of the components. The size of the figure, the alignment and spacing of the components on the figure can be adjusted using the tools of “guide”.
3. Use the tool inspector property to give each component a name (“tag”) and to select the characteristics of each component, such as its color, text label.
4. Save the picture in a file. When the figure is saved, two files with the same names but different extensions are created. The. Fig contains the graphical user interface, the M-file contains the code to load the figure and call functions for each element of the GUI.
5. Write code to implement the behavior associated with each callback function.
4.2 The Pacefit program
Now will be detailed the Pacefit program, created to operate the tire fitting of experimental data using the Magic Formula.
An illustration of the operating logic of the graphical interface will be made, in order to create a guide for the use of the program, focusing on points of great interest to understand the work.
For the rest, see the comments in the program.
4.2.1 The user interface
Proceeding clockwise starting from top left:
• Button LOAD: Loads the data for the fitting, selecting, via a dialog box, files in any folder of your computer
• SAVE Button: Saves the results into a. xls file, choosing folder and name
• Listbox for selecting data: Allows the user to select data to be used for the fitting, the data will be plotted in two graphs
• CHECK ALL button: Allows the user to quickly select all the data from the listbox
• Graphs of sideslip: the lateral force is drawn as a function of the angle of sideslip
• Graphic Options Panel: Gives the user the ability to choose whether to draw the real forces or forces normalized, it also gives the opportunity of changing the unit of measure of the abscissa [deg] and [grad]
• Graph of Camber: the lateral force is drawn as a function of the angle of camber
• Table of coefficients: express coefficients to be used for the fitting of the Magic Formula, including the initial values and the maximum and minimum values; are also expressed the coefficients resulting from the optimization. On the right side there is list of buttons which, when checked, let to maintain fixed the initial coefficient with the value showed in the column FITTED VALUES
• Select Button “account pressure changes”: Changes the formula used for the fit, passing from the standard magic formula, (when not selected) to the formula that takes into account the variations of pressure (when selected)
• Table of nominal properties:gives the user the possibility of entering the nominal pressure and the nominal load (not used by the standard formula for motorcycles)
• Button FIT DATA: does the fitting with the selected options and draw graphs of the formula with coefficients rents.
4.2.2 Data files
The program, to run correctly, requires the selected data to be into specific files. Within the file .xls, must be created 2 sheet, the first, called “sideslip” must contain the experimental data measured sweeping the angle of sideslip with zero camber. The second, called “camber” must contain the experimental data measured sweeping the angle of camber with zero sideslip.
4.2.3 Data loading
When LOAD button is pressed, the function “uigetfile opens” a dialog box to allow the user to select the file where are the data to be used.
This feature allows to select files with the .xls extension and provides information such as file names and their paths, knowledge of these two pieces of information will allow to open files in any folder they are
Once selected, datas are all loaded into two variables to be always available quickly at any time without loss of time due execution of “xlsread”.
4.2.4 Data fitting
When the button FIT DATA is pressed the program carries out the curve fitting of data using the coefficients in the table of the GUI and the options chosen.
The subroutine of MATLAB lsqcurvefit, used for the fitting of the data, requires as input the initial values of the coefficients of the Magic Formula, their lower and upper limits and two matrices: the first for the experimental values of the lateral force obtained by the experiment (YDATA), the second to express the conditions under which the data were obtained (XDATA).
YDATA =F( XDATA).
The following figure, easy to understand,in addition to the main operations shows the order in which the execution of a command enables the implementation of the next command.
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