Abstract:
A method of modeling a tire is provided. The tire model is used in simulating vehicle response to a simulated ground profile. The tire has an undeformed envelope that defines the outer circumference when no forces are exerted on the tire. The method includes the steps of mathematically characterizing the tire undeformed envelope and a simulated ground profile. Integrating the characterized simulated ground profile and the characterized tire undeformed envelope with respect to the horizontal direction such that a tire deformed area is determined. Calculating the magnitude of the resultant force vector from the deformed area. Determining the direction of the resultant force vector.

Description:
BACKGROUND AND SUMMARY OF THE INVENTION 
     The present invention relates generally to vehicle simulations for ride, handling, and road load prediction. In particular, the present invention relates to an analytical tire model that is used for vehicle simulations. 
     The use of simulations has become an important design tool of the automotive industry for predicting vehicle ride and load characteristics. A critical component of these simulations is the tire model used to characterize the interaction between the tire and the ground. Due to the complexity of the tire structure and composition, a range of tire models are used depending on the particular simulation application. The tire models can be divided into three general categories; finite element models, lumped mass models, and analytical tire models. 
     Finite element tire models can provide highly accurate results for simulations of tire and vehicle interactions. However, finite element models require an excessive amount of computation time as well as requiring a complex and time consuming set-up. In addition, some nonlinear finite element models may not be stable for all operating conditions of the simulation, causing additional lost time determining the source of the instability. 
     Lumped mass tire models are particularly suited to simulating tire tread bend. This model type can be used to simulate tire local resonance and its interaction with the vehicle. In some cases a lumped mass tire model can be used to directly define the contact between the tread bend and road surface. Unfortunately, the lumped mass tire models are very difficult and costly to create, often requiring special tire tests, programs, and experience to create an accurate model of the physical tire. 
     Analytical tire models do not require the detailed modeling of the physical tire that the finite element and lumped mass tire models both require. Instead of providing a detailed model of the physical tire, an analytical tire model uses global tire parameters such as tire radial stiffness, radial damping, and tire radius to model a tire. Using global tire parameters simplifies the creation of the tire model and leads to much faster computation times. However, since the physical tire is not modeled, mathematical assumptions must be made in order to simulate the interaction of the tire and the road surface. Determining what assumptions to use and how to mathematically implement those assumptions is generally determinative of the tire model accuracy and computation speed. 
     The first analytic tire model was created during the &#39;60s. Since then, numerous types of analytic tire models have been created, each based on different assumptions and having varying limitations in the usage of the particular model. Currently, various mathematical implementations of the radial spring tire model are the most widely used conventional analytical tire models because of the simplicity and accuracy compared with other analytical models. Referring to FIGS. 1A and 1B, radial spring tire models in general rely on the assumption that a tire is formed by a series of radial springs emanating from the center of the tire. At every time step as the tire progresses, the deformed area between the road profile and the tire undeformed envelope is calculated from the deformation of the radial springs. One method that many of the radial spring models employ to determine the magnitude of the resultant force is by hypothetically pressing the tire on a flat surface to deform the same area. Likewise, the angle of the resultant force is determined by summing all of the radial spring forces. Generally speaking, for gradually changing road surfaces conventional radial spring models provide reasonably accurate results. However, suddenly changing surfaces such as a step or pothole road input, may cause conventional radial spring models to provide inaccurate results. Due to the assumptions that were made in mathematically describing conventional radial spring models, the models do not always converge when a step input or pothole input is used as the ground profile for the simulation. In addition, for ground profile inputs similar to a step input, the models generally require excessive computation time in order to complete the numerical iterations that are necessitated for convergence. 
     Therefore, it is an object of the invention to provide an analytical tire model that can be used in vehicle simulations to accurately predict the tire and vehicle interaction when subjected to a predetermined ground profile. Also, it is desirable for the tire model to minimize the computation time required for the simulation. Additionally, it is an object to provide a tire model that is easily created. Also, it is desirable for the tire model to provide accurate simulation results when subjected to a ground profile describing a sudden change such as a step input. 
     To achieve the foregoing objectives an analytical tire model is provided for modeling a tire for use in simulating vehicle response to a simulated ground profile. The tire is described by an undeformed envelope that is mathematically characterized. The difference between the tire undeformed envelope and the ground profile is integrated with respect to the horizontal direction in order to determine the tire deformed area. The magnitude of a resultant force vector acting on the tire is calculated from the deformed area. The direction of the resultant force vector is combined with the magnitude of the resultant force vector. 
     The above described system is only an example. Systems in accordance with the present invention may be implemented in a variety of ways. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     These and other objects of the present invention, as well as the advantages thereof over other analytical tire models will become apparent to those skilled in the art from the following detailed description in conjunction with the attached drawings. 
     FIGS. 1A and 1B illustrate a radial spring tire model; 
     FIG. 2 is a block diagram of a presently preferred embodiment of a simulator including a tire model in accordance with the principles of the invention; 
     FIG. 3 illustrates a presently preferred embodiment of a tire model in accordance with principles of the invention; 
     FIG. 4 illustrates a presently preferred embodiment of a method of simulating the interaction between the vehicle and a ground profile; and 
     FIGS. 5A and 5B illustrate a simulation diagrams of a presently preferred embodiment for calculating vertical and horizontal spindle forces. 
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT 
     Referring to FIG. 2, a system model  10  for simulating the interaction between a vehicle  12  and a tire  14  when subjected to an actual ground profile  16  is illustrated. In the presently preferred embodiment, the system model  10  is incorporated in a Simulink simulation, however the scope of the invention includes using other simulation programs including mathematical simulation programs such as Matlab, as well as using multi-body dynamic simulation programs such as ADAMS and DADS. The system model  10  includes a quarter vehicle model  18  representing spindle-coupled sprung and unsprung mass. Those skilled in the art will readily recognize that there are numerous means of representing the vehicle model  18  other than the present implementation. 
     A tire model  20  in accordance with the principles of the invention is coupled to the vehicle model  18 . The tire model  20  describes the tire characteristics so that the interaction between the tire  14  and a ground profile  16  may be simulated analytically without detailed modeling of the physical tire. Referring to FIG. 3, the relationship between the ground profile  22  and the tire undeformed envelope  24  is illustrated. Located at approximately the center of the tire undeformed envelope  24  is the spindle  25  about which the tire  14  rotates. A resultant force  26  is exerted upon the spindle  25  due to the interaction of the tire  14  and the ground profile  22 . To determine the magnitude of the resultant force  26 , the tire deformed area lying between the road profile  22  and the tire undeformed envelope  24  is calculated. The invention recognizes that integrating with respect to the horizontal axis provides an accurate determination of the tire deformed area, from which the magnitude of the resultant force  26  may be determined. 
     Referring to FIG. 4, a presently preferred method of modeling a tire in conformance with the principles of the invention is illustrated. At steps  30  and  32 , the tire undeformed envelope  24  and the ground profile  22  are mathematically characterized. At step  34 , an integration range over which the horizontal integration is to be conducted is determined. In the presently preferred embodiment, the integration range is determined to be an integration range ratio that is selected after evaluating the intersection of the ground profile  22  with the tire undeformed envelope  24 . With additional reference to FIG. 3, construction lines  36  are projected from the tire center to the intersection of the ground profile  22  and the tire undeformed envelope  24 . The integration range ratio is then selected based on the angle of the construction lines  36  relative to the horizontal. By limiting the range over which the integration is performed, the duration of the simulation time is further decreased. For simulations in which the ground profile  22  describes potholes and curb strikes, the integration range is set somewhat broader relative to a flat surface to ensure that the entire deformed area is included within the range. Although, in the presently preferred embodiment the magnitude of the positive and negative integration range limits are equivalent, it is within the scope of the invention for the magnitudes to differ. In addition, it is within the scope of the invention to select fixed values for the integration range such as “+r” and “−r”. At step  40 , horizontal integration is used for determining the deformed area, DA.        DA   =       ∫     -   ar     ar            (       q                   (   x   )       -   y   +         r   2     -     x   2           )                        x                                
     where; 
     
       
           q ( x )− y+{square root over (( r   2   −x   2 ))}= 0 when  q ( x )− y+{square root over (r 2   −x   2 )} &lt;0 
       
     
     “q(x)” is the ground profile, and “a” is the integration range ratio. 
     At step  42 , the magnitude of the resultant force  26  is determined from the spindle deformed displacement, DD. The deformed displacement is the distance the spindle moves when the tire  16  is hypothetically pressed on a flat surface to deform the same area as the deformed area. The deformed displacement may also be calculated by determining the length of a perpendicular bisector extended from the deformed tire envelope to the undeformed tire envelope. One method of mathematically calculating the resultant force magnitude is as follows: 
     
       
         
           |Fr|=K * DD 
         
       
     
     where; “K” is the radial stiffness of the tire, and 
     “DD” is a function of the deformed area. 
     The variable “K” is generally obtained from manufacturer&#39;s data that illustrates the relationship between tire loading and deformed displacement, DD, for a tire. Following is a method of calculating the deformed displacement: 
     
       
           DD=r· (1−cos(θ/2)) 
       
     
     
       
         
           
             
               θ 
               - 
               
                 sin 
                  
                 
                     
                 
                  
                 θ 
               
             
             = 
             
               
                 2 
                 · 
                 DA 
               
               
                 r 
                 2 
               
             
           
         
                 
         
             
         
      
     
     where: “r” is the radius of the undeformed envelope, and “θ” is the angle formed by extending a perpendicular bisector from the envelope of the deformed tire to the undeformed area envelope. 
     In the presently preferred embodiment of the invention, the direction of the resultant force  26  is determined by connecting a hypothetical line from the deformed area centroid to the tire center. However, it is within the scope of the invention to calculate the resultant force direction by using other analytical tire models such as the circumferential integration radial spring tire model. Also, although the presently preferred embodiment uses horizontal integration to determine the centroid, the scope of the invention encompasses using other methods to calculate the centroid of the deformed area. A presently preferred method for calculating the deformed area centroid is described by the following equations, step  44 :          X   c     =       (       ∫     -   ar     ar            (       q                   (   x   )       -   y   +         r   2     -     x   2           )                   x                      x         )     /   A               Y   c     =       (       ∫     -   ar     ar            1   2                     (       q                   (   x   )       -   y   +         r   2     -     x   2           )                     (       q                   (   x   )       +   y   -         r   2     -     x   2           )             x         )     /   A                            
     where; 
     
       
           q ( x )− y+{square root over (( r   2   −x   2 ))}= 0 when  q ( x )− y+{square root over (r 2   −x   2 )} &lt;0 
       
     
     “q(x)” is the ground profile, and 
     “a” is the integration range ratio. 
     At step  46 , the direction of the resultant force is determined by constructing a line from the centroid of the deformed area to the center of the tire. The direction of the resultant force is approximately in the same direction as the construction line. This aspect of the invention recognizes that the resultant force direction can be approximated by a line extending from the deformed area centroid to the tire center. Approximating the resultant force direction in this manner additionally reduces the computation time required for the simulation. At step  48 , the spindle horizontal and vertical forces are determined from the resultant force. 
     Referring to FIGS. 5A and 5B, presently preferred Simulink diagrams in accordance with the teachings of the invention illustrate the models used for calculating deformed area, area centroid direction, vertical spindle force, and horizontal spindle force of a vehicle and ground profile interaction. The tire model used in the Simulink diagrams uses horizontal integration to determine the deformed area and additionally approximates the determination of the resultant force direction by constructing a line from the deformed area centroid to the tire center. Although the presently preferred embodiment includes using both horizontal integration and the above described approximation for resultant force direction, it is within the scope of the invention to include only horizontal integration or only the resultant force direction approximation. 
     Although certain preferred embodiments of the invention have been herein described in order to afford an enlightened understanding of the invention, and to describe its principles, it should be understood that the present invention is susceptible to modification, variation, innovation and alteration without departing or deviating from the scope, fair meaning, and basic principles of the subjoined claims.