Using the Aircraft Basic Tire Model
The Aircraft Basic Tire Model is comprised of the Adams Tire Fiala tire model, with modifications that are necessary for aircraft landing gear analysis.
This section contains information for using the Aircraft Basic Tire Model:
Overview
Assumptions
■Single contact point with the road profile.
■Disk representation of wheel and tire.
■No lateral, longitudinal, and twist deformation effects on tire center moments.
■First-order lag on longitudinal and lateral slip.
Inputs
The inputs to the Aircraft Basic Tire Model come from two sources:
■Input parameters from the tire property file (.tir), such as tire undeflected radius, that the tire references.
■Tire states, given through the tire interface with the solver, such as slip angle (
).
). The following table summarizes the input data from the tire property file (.tir) that the Aircraft Basic Tire Model requires.
Table 1 Aircraft Basic Tire Model Input Data
Parameters: | Description: |
|---|---|
[UNITS] block: LENGTH | Units of length for all tire property file values that involve length units. Valid entries: 'inch', 'cm', 'centimeter', 'foot', 'ft', 'kilometer', 'km', 'm', 'meter', 'mile', 'millimeter', 'mm'. |
[UNITS] block: FORCE | Units of force for all tire property file values that involve force units. Valid entries: 'dyne', 'kg_force', 'kilogram_force', 'knewton', 'kpound_force', 'lbf', 'millinewton', 'newton', 'ounce_force', 'pound_force'. |
[UNITS] block: MASS | Units of mass for all tire property file values that involve mass units. Valid entries: 'gram', 'kg', 'kilogram', 'kpound_mass', 'lbm', 'megagram', 'ounce_mass', 'pound_mass', 'slug'. |
[UNITS] block: ANGLE | Units of angle for all tire property file values that involve angle units. Valid entries: 'angular_minutes', 'am', 'angular_seconds', 'as', 'degree', 'deg', 'radian', 'rad'. |
[UNITS] block: TIME | Units of time for all tire property file values that involve time units. Valid entries: 'hour', 'millisecond', 'ms', 'minute', 'second', 'sec'. |
PROPERTY_FILE_FORMAT | Must be 'AIR_BASIC'. |
FUNCTION_NAME | Must be 'TYR1500'. |
HANDLING_MODE | 1 = don't compute handling forces (zero) 2 = Fiala-based handling force computations |
FRICTION_MODE | 1 = slip ratio-based friction coeff. model 2 = slip velocity-based friction coeff model A 3 = slip velocity-based friction coeff model B 4 = user-input custom Mu versus slip ratio See Friction Models. |
UNLOADED RADIUS | Tire's outer radius under zero loading. (Units: length.) |
WIDTH | Tire's maximum undeflected (or unloaded) width. In simple geometry graphics, WIDTH represents the tread width, for visualization purposes only. In computations, however, WIDTH represents the tire's maximum undeflected width. (Units: length.) |
ASPECT_RATIO | Ratio of "rim-to-tread distance" to WIDTH. Used only for tire geometry graphics. (Units: none.) |
BOTTOMING_RADIUS (optional) | Wheel bottoming radius. (Units: length.) See Wheel Bottoming. |
VERTICAL_DAMPING | Vertical tire damping force coefficient. (Units: force/(length/time).) |
RELAXATION_LENGTH | Relaxation length. (Units: length.) See Lagged Longitudinal and Lateral Slip Quantities (transient tire behaviour). |
LOW_SPEED_DAMPING (optional) | The low speed damping rate when transient tire modelling is used (relaxation length not equal to zero). (Units: none.) See Lagged Longitudinal and Lateral Slip Quantities (transient tire behaviour). |
LOW_SPEED_THRESHOLD (optional) | The speed below which the low speed damping will be applied. (Units: length/time.) If not specified in the tire property file the value for this parameter is 4 m/s. See Lagged Longitudinal and Lateral Slip Quantities (transient tire behaviour). |
ROLLING_RESISTANCE | Rolling resistance moment coefficient, which represents the longitudinal shift in the vertical center of pressure, during pure rolling. (Units: length.) |
CSLIP | Tire’s longitudinal stiffness, partial derivative of longitudinal force (Fx) with respect to longitudinal slip ratio (S) at zero longitudinal slip. (Units: force/(non-dimensional slip ratio).) |
CALPHA | Tire's cornering stiffness. Partial derivative of lateral force (Fy) with respect to slip angle (  ) at zero slip angle. (Units: force/angle.) |
UMAX | Coefficient of friction at zero slip. (Units: none.) See Friction Models. |
UMIN | Coefficient of friction when tire is sliding. Not used if FRICTION_MODE = 2. (Units: none.) See Friction Models. |
V_UREF | Reference velocity for friction coefficient determination. Used only if FRICTION_MODE = 2 or 3. (Units: length/time.) See Friction Models. |
RR_DEFL_FACTOR | Factor used in the calculation of unbraked, unyawed tire rolling radius. (Units: none.) |
[AIR_CURVE] block: pen | Column of tire/road penetration (deflection) values, corresponding to the adjacent tire radial force value. (Units: length.) |
[AIR_CURVE] block: fz | Column of tire radial force values, corresponding to the adjacent tire/road penetration (deflection) value. (Units: force.) |
[SHAPE] block: radial (optional) | Column of tire radial scale values, corresponding to the adjacent tire width station value. This value is multiplied with UNLOADED RADIUS. (Units: none.) See Tire Carcass Shape. |
[SHAPE] block: width (optional) | Column of tire width station values, corresponding to the adjacent radial scale value. 0.0 represents the tire centerline tread station and 1.0 represents the outermost tire tread station. Symmetry about the tire centerline is assumed. (Units: none.) See Tire Carcass Shape. |
[BOTTOMING_CURVE] block: pen (optional) | Column of rim/road penetration (deflection) values, corresponding to the adjacent rim radial force value. (Units: length.) See Wheel Bottoming. |
[BOTTOMING_CURVE] block: fz (optional) | Column of rim radial force values, corresponding to the adjacent rim/road penetration (deflection) value. (Units: force.) See Wheel Bottoming. |
Tire Property File Format Example
The following file, located in the shared database, is an example of the Aircraft Basic Tire Model tire property file:
install_dir/acar/shared_car_database.cdb/tires.tbl/AA_small_basic_relax.tir
where install_dir represents the location of the Adams installation directory.
Contact Methods
The Aircraft Basic Tire model supports all Adams Tire contact methods.
■One Point Follower Contact, used by default for 2D Road, 3D Spline Road, OpenCRG road and RGR Road.
■3D Equivalent Volume Contact, used by default for 3D Shell Road.
■3D Enveloping Contact, can be used with all road types when the keyword CONTACT_MODEL = '3D_ENVELOPING' is specified in the [MODEL] section of the tire property file.
The contact method supplies the tire model with the (effective) road height and road plane for the tire deflection/penetration calculation.
Wheel Bottoming
You can optionally supply a wheel bottoming deflection - load curve in the tire property file in the [BOTTOMING_CURVE] block. If the deflection of the wheel is so large that the rim will be hit (defined by the BOTTOMING_RADIUS parameter in the [DIMENSION] section of the tire property file), the tire vertical load will be increased according to the load curve defined in this section.
Note: | The rim-to-road contact algorithm is a simple penetration method (such as the 2D contact) based upon the tire-to-road contact calculation, which is strictly valid only for rather smooth road surfaces (the length of obstacles should have a wavelength longer than the tire circumference). The rim-to-road contact algorithm is not based on the 3D volume penetration method, but can be used in combination with the 3D Contact (that takes into account the volume penetration of the tire itself). If you omit the [BOTTOMING_CURVE] block from a tire property file, no force due to rim road contact will be added to the tire vertical force. |
The BOTTOMING_RADIUS may be chosen larger than the rim radius to account for the tire's material left in between the rim and the road, while the bottoming load-deflection curve may be adjusted for the change in stiffness.
If (Pentire- (Rtire - Rbottom) - ½ width · | tan(
) |) < 0 the left or right side of the rim has contact with the road. Then the rim deflection Penrim can be calculated with:
) |) < 0 the left or right side of the rim has contact with the road. Then the rim deflection Penrim can be calculated with:■
= max(0 , ½width · | tan(
) | ) + Pentire- (Rtire - Rbottom) )
= max(0 , ½width · | tan() | ) + Pentire- (Rtire - Rbottom) )■Penrim= 
/(2 · width · | tan(
) |)
/(2 · width · | tan() |)■Srim= ½width - max(width , 
| tan(
) |)/3
| tan() |)/3with Srim the lateral offset of the force with respect to the wheel plane.
If the full rim has contact with the road, the rim deflection is
■Penrim = Pentire- (Rtire - Rbottum)
■Srim= width2 · | tan(
) | · /(12· Penrim)
) | · /(12· Penrim) Using the load - deflection curve defined in the [BOTTOMING_CURVE] section of the tire property file, the additional vertical force due to the bottoming is calculated, while Srim multiplied by the sign of the inclination 
is used to calculate the contribution of the bottoming force to the overturning moment. Further, the increase of the total wheel load Fz due to the bottoming (Fzrim) will not be taken into account in the calculation for Fx, Fy, My and Mz. The Fzrim will only contribute to the overtuning moment Mx using the Fzrim· Srim.
is used to calculate the contribution of the bottoming force to the overturning moment. Further, the increase of the total wheel load Fz due to the bottoming (Fzrim) will not be taken into account in the calculation for Fx, Fy, My and Mz. The Fzrim will only contribute to the overtuning moment Mx using the Fzrim· Srim.Note: | Rtire is equal to the unloaded tire radius, Pentire is similar to effpen. |
Normal Force of Road on Tire
The normal force of a road on a tire at the contact patch in the SAE coordinates (+Z downward) is always negative (directed upward). The normal force is:
Fz = min (0.0, {Fzk + Fzc}) + min (0.0, Fzrim)
where
■Fzk is the normal force due to the tire radial load-deflection curve
♦Fzk = - f (effpen, tire load-deflection spline)
■Fzc is the normal force due to tire vertical damping
♦Fzc = - VERTICAL_DAMPING x Vpen
■Fzrim is the normal force due to bottoming of the wheel
The normal penetration (effpen, or Δ) and penetration velocity (Vpen) are obtained from the appropriate road contact model.
Handling Forces of Road on Tire
The following topics are included:
Basic Tire Kinematics
All tire kinematic values are in the tire contact patch (SAE) reference system.
Figure 2 Unbraked, Unyawed, Effective Rolling Radius
Unloaded (and Ungrown) Radius
Ro = UNLOADED_RADIUS
Geometric Deflected Radius
R = UNLOADED_RADIUS - (effpen)
Effective Unbraked Rolling Radius
Re,o = UNLOADED_RADIUS - (effpen x RR_DEFL_FACTOR)
And RR_DEFL_FACTOR is usually set to 1/3.
Wheel Carrier Translational Velocity
Vx, Vy, Vz
Total Rotational Velocity of Spinning Tire and Rotating Wheel Carrier
Contact Patch Rubber Velocity
Vxc = X-component of 
where
is the vertical radius vector of the scalar Re,o.
is the vertical radius vector of the scalar Re,o. Vyc = Y-component of 
where 
is the vertical radius vector of the scalar R.
is the vertical radius vector of the scalar R. Contact Patch Rubber Slip (or Scrub) Velocity
Vsx = Vx + Vxc
Vsy = Vy + Vyc
Definition of Tire Slip Quantities
Figure 3 Slip Quantities at Combined Cornering and Braking/Traction
The longitudinal slip velocity Vsx in the SAE-axis system is defined using the longitudinal speed Vx, the wheel rotational velocity 
, and the effective rolling radius Re:
, and the effective rolling radius Re:
The lateral slip velocity is equal to the lateral speed in the contact point with respect to the road plane:
The practical slip quantities 
(longitudinal slip) and 
(slip angle) are calculated with these slip velocities in the contact point with:
(longitudinal slip) and (slip angle) are calculated with these slip velocities in the contact point with:
The rolling speed Vr is determined using the effective rolling radius Re:
Note that for realistic tire forces the slip angle 
is limited to 45 deg and the longitudinal slip 
in between -1 (locked wheel) and 1.
is limited to 45 deg and the longitudinal slip in between -1 (locked wheel) and 1.Zero Handling Force Model
If this option is selected in the tire property file, friction and slip parameters are not used, and all handling forces will be zero:
Longitudinal Force
Fx = 0
Lateral Force
Fy = 0
Oversteering Moment
Tx = 0
Rolling Resistance Moment
Ty = 0
Aligning Moment
Tz = 0
Fiala Handling Force Model — Basic Tire
The Aircraft Basic Tire Model's Fiala Handling Force model is an extended Fiala model (Fiala, E., "Seitenkrafte am rollenden Luftreifen," VDI-Zeitschrift 96, 973 (1964)). This model provides reasonable results for simple maneuvers where inclination angle is not a major factor and where longitudinal and lateral slip effects may be considered unrelated.
Modifications are included to make the Fiala model more general and more appropriate for use in Adams.
Additional Parameters
Before calculating the current maximum available friction coefficient, the Fiala tire model requires the evaluation of some additional variables. First is the comprehensive slip S*s
:
:S*s
= (S2s + tan2(
))1/2
= (S2s + tan2())1/2The truncated comprehensive slip (Ss
):
): S*s
= min(1, S*s
)
= min(1, S*s)Friction Models
You can choose from four friction models. The friction mode parameter within the tire property file is used to select the friction model. The friction model ultimately computes the maximum available comprehensive friction coefficient.
Slip Ratio-based Friction Model A (Linear U-Slip)
The resultant friction coefficient between the tire tread base and the terrain surface is determined as a function of the resultant comprehensive, truncated slip ratio (Ss
) and friction parameters (Umax and Umin). The friction parameters are experimentally obtained data representing the kinematic property between the surfaces of tire tread and the terrain.
) and friction parameters (Umax and Umin). The friction parameters are experimentally obtained data representing the kinematic property between the surfaces of tire tread and the terrain. A linear relationship between Ss
and U(
), the corresponding maximum available road-tire friction coefficient, is assumed. The following figure shows this relationship.
and U(), the corresponding maximum available road-tire friction coefficient, is assumed. The following figure shows this relationship. 
Figure 4 Linear Tire-Terrain Friction Model
Therefore, the current value coefficient of friction (U):
U = Umax + (Umin-Umax) x Ss
Note: | The figure illustrates that the available friction coefficient, U, varies linearly with slip ratio, but the actual curve of Fx/Fz is not linear, as described in Longitudinal Force. |
Slip (or Scrub) Velocity-based Friction Decay Model A
The resultant friction coefficient between the tire tread base and the terrain surface is determined as a function of the total planar slip (or scrubbing) velocity Vsxy, maximum friction parameter (Umax), and the friction coefficient reference velocity parameter V_UREF from the tire property file. The friction parameters are experimentally obtained data representing the kinematic property between the surfaces of tire tread and the terrain.
A decay relationship between Vsxy and U (
), the corresponding maximum available road-tire friction coefficient, is assumed. The following figure shows this relationship.
), the corresponding maximum available road-tire friction coefficient, is assumed. The following figure shows this relationship. 
Figure 5 Friction Decay Model A
Therefore, the current value coefficient of friction (U):
Notice that Umin is not used in this friction model. Also, notice the effect of V_UREF upon the decay of the available friction coefficient with total slip (or scrub) velocity Vsxy:
♦If Vsxy = 0, then U = Umax
♦If Vsxy = V_UREF/2, then U = 60.7% of Umax
♦If Vsxy = V_UREF, then U = 36.78% of Umax
Note: | The figure illustrates the available friction coefficient, U, as it varies with slip ratio. The actual curve of Fx/Fz deviates from this curve, as described in Longitudinal Force. |
Slip (or Scrub) Velocity-based Friction Decay Model B
The resultant friction coefficient between the tire tread base and the terrain surface is determined as a function of the total planar slip (or scrubbing) velocity Vsxy, friction parameters (Umax and Umin), and the friction coefficient reference velocity parameter V_UREF from the tire property file. The friction parameters are experimentally obtained data representing the kinematic property between the surfaces of tire tread and the terrain.
A decay relationship between Vsxy and U (
), the corresponding maximum available road-tire friction coefficient, is assumed (AGARD-R-800 "The Design, Qualification and Maintenance of Vibration-Free Landing Gear": Denti, E., Fanteria D., "Analysis and Control of the Flexible Dynamics of Landing Gear in the Presence of Antiskid Control Systems" (1996)). The following figure shows this relationship.
), the corresponding maximum available road-tire friction coefficient, is assumed (AGARD-R-800 "The Design, Qualification and Maintenance of Vibration-Free Landing Gear": Denti, E., Fanteria D., "Analysis and Control of the Flexible Dynamics of Landing Gear in the Presence of Antiskid Control Systems" (1996)). The following figure shows this relationship. 
Figure 6 Friction Decay Model B
Therefore, the current value coefficient of friction (U):
Notice the effect of V_UREF on the decay of the available friction coefficient with total slip (or scrub) velocity Vsxy:
■If Vsxy = 0, then U = Umax
■If Vsxy = V_UREF, then U = Average of Umax and Umin
■If Vsxy =
, then U = Umin
, then U = Umin Note: | The figure illustrates the available friction coefficient, U, as it varies with slip ratio. The actual curve of Fx/Fz deviates from this curve, as described in Longitudinal Force. |
Slip Ratio based Model B (User-Defined Mu-Slip)
The resultant friction coefficient between the tire tread base and the terrain surface is determined as a function of the resultant comprehensive, truncated slip ratio (Ss
) and a user-defined table of U (
). The tabular data are experimentally obtained and represent the kinematic property between the surfaces of tire tread and the terrain.
) and a user-defined table of U (). The tabular data are experimentally obtained and represent the kinematic property between the surfaces of tire tread and the terrain. The following figure shows the relationship between Ss
and U (
), the corresponding maximum available road-tire friction coefficient.
and U (), the corresponding maximum available road-tire friction coefficient. 
Figure 7 User-Defined Fiala Tire-Terrain Friction Model
Therefore, the current value coefficient of friction (U):
U = a cubic spline interpolation of U versus Ss
curve
curveAdd a section called [MU_SLIP_CURVE] to the tire property file, in a similar way as for example, [RIMPACT_CURVE] shown in the example tire property file acar/shared_car_database.cdb/tires.tbl/AA_small_basic_relax.tir, to represent the mu versus slip ratio data.
Note: | The figure illustrates the available friction coefficient, U, as it varies with slip ratio. The actual curve of Fx/Fz deviates from this curve, as described in Longitudinal Force. |
Handling Force Evaluation
Now that the current maximum available total friction coefficient U is known, the Fiala handling forces can be calculated.
Longitudinal Force
The longitudinal force depends on the vertical force (Fz), the current maximum available total coefficient of friction (U), and the longitudinal slip ratio (Ss).
Fiala defines a critical longitudinal slip (S_critical):
S_critical = 
This is the value of longitudinal slip beyond which the tire is sliding.
Case 1. Elastic Deformation State:|Ss| < S_critical
Case 2. Complete Sliding State: |Ss| > S_critical
Fx = -sign(Ss)(Fx1-Fx2)
where:
■Fx1 = U x Fz
■
The calculations of Fx can be used to calculate Fx/Fz, which can be contrasted to the available total coefficient of friction (U) curves shown above. All of the above figures are plots of U, but they are not the plots of Fx/Fz. The U curves show the maximum possible friction coefficient, but the actual longitudinal force, while based on U, is modified by the rolling characteristics of the tire.
For example, consider the plot of Linear Tire-Terrain Friction Model. The coefficient of friction is a straight line. Consider next the following figure based on the equations for Fx shown in Case 1 and Case 2 above. The following figure, created using arbitrarily chosen parameters, illustrates that Fx/Fz is less than the value of U at every value of slip, Ss
. The actual Fx/Fz curve is a function of the U curve, CSLIP, and tire vertical force, Fz.
. The actual Fx/Fz curve is a function of the U curve, CSLIP, and tire vertical force, Fz. 
This type of difference between the chosen U curve and Fx/Fz affects all four friction models. You should keep this in mind when creating your tire property file. Also, after you run a simulation, such as a braking or wheel test simulation, you can plot Fx/Fz to determine whether the friction values are what you require.
Lateral Force
Like the longitudinal force, the lateral force depends on the vertical force (Fz) and the current coefficient of friction (U). And similar to the longitudinal force calculation, Fiala defines a critical lateral slip (Alpha_critical):
Alpha_critical = arctan 
The lateral force peaks at a value equal to U x |Fz| when the slip angle (Alpha) equals the critical slip angle (Alpha_critical).
Case 1. Elastic Deformation State: 
Fy = - U x |Fz| x (1-H3) x sign(Alpha)
where: 
Case 2. Sliding State: |Alpha| > Alpha_critical
Fy = -U|Fz|sign(Alpha)
Oversteering Moment
Tx = 0
Rolling Resistance Moment
When the tire is rolling forward: Ty = -ROLLING_RESISTANCE * Fz
When the tire is rolling backward: Ty = ROLLING_RESISTANCE * Fz
Aligning Moment
Case 1. Elastic Deformation State: 
Case 2. Complete Sliding State: |Alpha| > Alpha_critical
Tz= 0.0
Force Reducer
In a balancing simulation, you can switch on the force reducer by using the tire user array. If the first element reads the value 1500 and the second 1, the force reducer is switched on. Except for the vertical load Fz, all tire forces and moments are reduced drastically to reach airplane equilibrium in a more efficient way.
Fx = Fx * FORCE_REDUCER_X
Fy = Fy * FORCE_REDUCER_Y
Mx = Mx * FORCE_REDUCER_Y
My = My * FORCE_REDUCER_X
Mx = Mz* FORCE_REDUCER_Y
FORCE_REDUCER_X = 0.01
FORCE_REDUCER_Y = 0.0
Fy = Fy * FORCE_REDUCER_Y
Mx = Mx * FORCE_REDUCER_Y
My = My * FORCE_REDUCER_X
Mx = Mz* FORCE_REDUCER_Y
FORCE_REDUCER_X = 0.01
FORCE_REDUCER_Y = 0.0
Transient Behavior
In the upper sections described force calculations are valid for the 'so-called' steady-state tire response, in other words tire dynamics is not taken into account. However, in general, the tire will be exposed to changes of input in terms of vertical load and longitudinal and lateral slip continuously.
For estimating transient tire behavior, a linear transient model is used as described in [1].
In the linear transient model the tire contact point S' is suspended to the wheel-rim plane with a longitudinal and lateral spring, with respectively stiffness's CFx and CFy. In the figure below a top view of the tire with the single contact point S' and the longitudinal (u) and lateral (v) carcass deflections is shown.
The contact point may move with respect to the wheel-rim plane and road. Movements relative to the road will result in tire-road interaction forces. Differences in slip velocities at point S and point S' will result in the tire carcass to deflect. The change of the longitudinal deflection u can be defined as:
and the lateral deflection v as:
For small values of slip the side force Fy can be calculated using the cornering stiffness CFα as follows:
While the lateral force on the carcass reads:
When introducing the lateral relaxation length σα as:
the differential equation for the lateral deflection can be written as follows:
For linear small slip we can define the practical slip quantity α' as:
With α' the equation for the lateral deflection becomes:
Similar the differential equation for longitudinal direction with the longitudinal relaxation length σκ can be derived:
with the practical slip quantity 
These practical slip quantities 
and 
are used instead of the usual 
and 
definitions for steady-state tire behavior.
and are used instead of the usual and definitions for steady-state tire behavior. 
and 
optional the damping rates that can be applied to achieve more damping at low speeds (below the LOW_SPEED_THRESHOLD value). The LOW_SPEED_DAMPING parameter in the tire property file yields:
= 100 · 
= LOW_SPEED_DAMPINGThe longitudinal and lateral relaxation length are read from the tire property file, see Tire Property File Format Example.
Note that in transient mode the tire model is able to deal with zero speed (stand-still), because this linear transient model works with tire deflections instead of slip velocities. The effective lateral compliance of the tire at stand-still in transient mode is:
And similar in longitudinal direction the compliance is:
Note: | If the tire property file's REL_LEN_LON or REL_LEN_LAT = 0, then steady-state tire behavior is calculated. |
Reference
1. H.B. Pacejka, Tyre and Vehicle Dynamics, 2002, Butterworth-Heinemann, ISBN 0 7506 5141.