DISCUSSION: Cornering Stiffness

When a tire is orientated at an angle not equal to its direction of motion, a side force acts perpendicular to the plane of the wheel. The relation is nearly linear for small slip angles. The relationship that defines side force as a function of slip angle makes use of the tire cornering stiffness. The SMAC program calculates tire side forces based on a nondimensional side force function whereby the small-angle properties of the tires become progressively "saturated" at larger angles (Figure 6). The "friction circle" concept (Figure 7) is used to approximate interaction between side and circumferential (braking and tractive) tire forces. This concept  is based on the assumption that the maximum value of the resultant tire friction force is independent of its direction relative to the wheel plane. The general approach to the modeling of tire properties is patterned after that of Reference 12. The cornering stiffnesses of the individual wheels are input separately in the SMAC program.

 

The program requires the user to input a cornering stiffness for small slip angles on M-SMAC cards 6 and 7. (Inputs for original units, lbs/rad, for metric units, N/deg, and for Default units (EDSMAC compatible), lbs/deg). Typical values for the cornering stiffness in lbs. per degree of slip angle are generally in the range of 16-17% of the load on the tire. Inputs defined on this basis for each of the tires on a vehicle will produce cornering forces which correspond to a "neutral steer" vehicle.

Figure 5 Automobile Tire Cornering Stiffness from Reference 27

 

The steering characteristics of vehicles in cornering maneuvers, can be classified as being either oversteer, neutral steer or understeer.

Figure 6 Tire cornering stiffness

 

Figure 7 Friction Circle

For most automobiles, the relationship between the distribution of tire cornering stiffnesses and the vehicle weight distribution is designed to produce understeer. A simple means of approximating the general properties of a vehicle in a cornering maneuver is to calculate the vehicle Static Margin (SM). The vehicle static margin is defined by the relationship between the front and rear cornering stiffnesses and the weight distribution of the vehicle.

As previously mentioned, the original SMAC input instructions produced inputs corresponding to the condition of neutral steer (i.e., a zero value for the static margin). The use of a Static Margin=0.0 reflects that the emphasis for original SMAC was on collision  responses as opposed to vehicle handling. The inputs for tire cornering stiffnesses can be made more realistic for the case of  prolonged pre-impact maneuvers by means of the specification of an appropriate non-zero value for the static margin.

 

The static margin (SM) may be defined as

 

                  

 

          where C_F, C_R =        cornering stiffnesses per tire at front and rear respectively, LB/RAD.

 

                    A, B    =        longitudinal distances from the complete-vehicle   center of gravity to the front and rear wheel                                      centers, respectively, INCHES.

 

For most automobiles, the SM is positive (i.e., understeer) with values ranging from approximately 0.05 to 0.07 (e.g., Reference 13).

 

In the m-edit environment, when you enter a field on m-smac input cards 6 or 7, a popup dialog appears adjacent to the field entitled the “Static margin Calculator”

          

Figure 8 Input Card 6 or 7 produces popup Static margin dialog

A close up of the pop is as follows:

 

Figure 9   Closeup of the Static Margin Calculator

The Static Margin is calculated as follows:

 

For the example above:

 

SM = -5158/(-7123-5158) - 37.83/(37.83+59.17) = 0.03

 

You can also use the Static Margin dialog box  to determine:

 

•      the tire cornering stiffness properties required to change the Static Margin to a specified value

or

•      the Static Margin which would result in a change to the Cornering Stiffness.

 

For example if you change the value for the static margin to 0.0 and press the <Enter> Key, the Cornering Stiffness for the Rear wheels will change to values that will produce the selected Static Margin.

 

Likewise, if you change the value(s) of any of the individual Tire cornering stiffness, and Press <Enter>, the Static Margin will be recalculated.

 

Please note that the requirement of a Key Press event is to signal to the Static Margin Procedure the basis on which you wish to calculate. The default action is to calculate the Static Margin for the current values of the Tire Cornering Stiffness.

 

 

 

The Default recommended procedure to calculate the Cornering Stiffness for the tires in SMAC is to approximate the cornering Stiffness as 16.5% of the weight on the tire.  The default value for the SM is 0.00.

The procedure is as follows:

 

Temp1 = A + (SM) * (A+B)

Temp2 = B - (SM) * (A+B)

        W_RF = W_LF = (W/2) *B/(A+B)        LBS.

 

For Original Units:

 

                C_RF = C_LF = -9.455 * W₁      LB/RAD

                C_RR = C_LR = C_RF * (TEMP1/TEMP2) 

 

For Default compatible Units:

 

                C_RF = C_LF = -.165 * W_RF             LB/DEG

                C_RR = C_LR = C_RF * (TEMP1/TEMP2) 

 

For Metric Units:

 

                C_RF = C_LF = -42. * W_RF             N/DEG

                C_RR = C_LR = C_RF * (TEMP1/TEMP2) 

 

Note: the sign of the value for the cornering stiffness is ignored in the m-smac program.

 

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