The Aircraft Enhanced Tire Model is comprised of the Adams Tire Fiala and UA (University of Arizona) tire models, with modifications that are necessary for aircraft landing gear analysis in Adams

This section contains information for using the Aircraft Enhanced Tire Model:

The inputs to the Aircraft Enhanced Tire Model come from two sources:

The following table summarizes the input data from the tire property file (.tir) that theAircraft Enhanced Tire Model requires.

The following file, located in the shared database, is an example of the Aircraft Enhanced Tire Model tire property file:

where install_dir represents the location of the Adams installation directory.

The Aircraft Enhanced Tire model supports all Adams Tire contact methods.

The contact method supplies the tire model with the (effective) road height and road plane for the tire deflection/penetration calculation.

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.

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:

with 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

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.

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:

where

The normal penetration (effpen, or Δ) and penetration velocity (Vpen) are obtained from the appropriate road contact model.

The following topics are included:

All tire kinematic values are in the tire contact patch (SAE) reference system.

The current cornering stiffness is a function of the current vertical load:

C = f(abs(Fz)), tire cornering stiffness versus vertical load spline

The current longitudinal stiffness is a function of the current vertical:

Klon = f(abs(Fz)), tire longitudinal stiffness versus vertical load spline

The current lateral stiffness is a function of the current vertical:

Klat = f(abs(Fz)), tire lateral stiffness versus vertical load spline

The current longitudinal stiffness is a function of the current vertical:

where e,o is the vertical radius vector of the scalar Re,o.

where is the vertical radius vector of the scalar R.

Vsx = Vx + Vxc

Vsy = Vy + Vyc if LAT_SLIP_MODE = 0

Vsy = Vy + Vyc + Vzc*tan() if LAT_SLIP_MODE = 1

The component Vzc* tan() takes into account the lateral slip due to a vertical movement of the tire if the roll inclination with the road is not zero. The default for LAT_SLIP_MODE is zero.

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:

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:

and

When the UA Tire is used for the force calculation the slip quantities during positive Vsx (driving) are defined as:

and

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.

If this option is selected in the tire property file, friction and slip parameters are not used, and all handling forces will be zero:

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.

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 :

The truncated comprehensive slip (Ss):

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.

A linear relationship between Ss and U(), the corresponding maximum available road-tire friction coefficient, is assumed. The following figure shows this relationship.

Therefore, the current value coefficient of friction (U):

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.

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:

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.

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:

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.

The following figure shows the relationship between Ss and U (), the corresponding maximum available road-tire friction coefficient.

Therefore, the current value coefficient of friction (U):

Add 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.

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):

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

where:

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 Fiala 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.

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:

where:

Case 2. Sliding State: |Alpha| > Alpha_critical

Fy = -U|Fz|sign(Alpha)

Oversteering Moment

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

The Aircraft Enhanced Tire Model's UA Tire Handling Force model is an extension of the University of Arizona tire model, recently developed by Drs. P.E. Nikravesh and G. Gim (Gim, Gwanghun, "Vehicle Dynamic Simulation with a Comprehensive Model for Pneumatic Tires," Ph.D. Thesis, The University of Arizona (1988)). This analytical model has been extensively tested and verified against experimental data and other analytical models. Please note that the current implementation of the University of Arizona tire model uses a friction circle rather than the original friction ellipse. The ellipse caused undesirable results under some circumstances due to its dependence on the integration time step.

Modifications are included to make the UA Tire model more general and more appropriate for use in Adams.

The UATire tire model requires the evaluation of some additional variables:

Lateral Slip Ratios

The lateral slip ratio due to slip angle, , may then be defined as:

whether traction or braking

The lateral slip ratio due to inclination angle, , is defined as:

A combined lateral slip ratio due to slip and inclination angles, , is defined as:

whether traction or braking

where is the length of the contact patch.

A comprehensive slip ratio due to slip ratio, slip angle, and inclination angle may be defined as:

Note that:

Now a slip velocity directional angle may be defined as:

Slip properties and slip ratio relationships are shown in the following figure (a), (b), and (c):

You can choose from three 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

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 () and friction parameters (U0 and U1). The friction parameters are experimentally obtained data representing the kinematic property between the surfaces of tire tread and the terrain.

A linear relationship between and , the corresponding maximum available road-tire friction coefficient, is assumed. The following figure shows this relationship.

Therefore, the current value coefficient of friction (U) is:

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 , the corresponding maximum available road-tire friction coefficient, is assumed. The following figure shows this relationship.

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:

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 , the corresponding maximum available road-tire friction coefficient, is assume, from (Denti, E., Fanteria D., "Analysis and Control of the Flexible Dynamics of Landing Gear in the Presence of Antiskid Control Systems" AGARD-R-800 "The Design, Qualification and Maintenance of Vibration-Free Landing Gear" (1996)). The following figure shows this relationship.

Therefore, the current value coefficient of friction (U) is:

Notice the effect of V_UREF upon the decay of the available friction coefficient with total slip (or scrub) velocity Vsxy:

In the UATire model, the friction circle concept allows for different values of longitudinal and lateral friction coefficients ( and ) but limits the maximum value for both coefficients to . The relationship that defines the friction circle follows:

or

and

where

Note that and in the following figure depend on the untruncated value .

To compute longitudinal force, lateral force, and self-aligning torque in the SAE coordinate system, you must perform a test to determine the precise operating conditions. The conditions of interest are:

If then a camber stiffness is estimated using:

with Clen the contact length is estimated by:

The lateral force Fh can be decomposed into two components: Fha and Fhg. The two components are in the same direction if and in opposite direction if .

Case 1.

Before computing the longitudinal force, the lateral force, and the self-aligning torque, some slip parameters and a modified lateral friction coefficient should be determined. If a slip ratio due to the critical inclination angle is denoted by , then it can be evaluated as:

If Ssc represents a slip ratio due to the critical (longitudinal) slip ratio, then it can be evaluated as:

If a slip ratio due to the critical slip angle is denoted by , then it can be determined as

when Ss < Ssc.

The term "critical" stands for the maximum value which allows an elastic deformation of a tire during pure slip due to pure slip ratio, slip angle, or inclination angle. Whenever any slip ratio becomes greater than its corresponding critical value, an elastic deformation no longer exists, but instead complete sliding state represents the contact condition between the tire tread base and the terrain surface.

A nondimensional slip ratio Sn is determined as:

where:

A nondimensional contact patch length is determined as:

A modified lateral friction coefficient is evaluated as:

where is the available friction as determined by the friction circle.

To determine the longitudinal force, the lateral force, and the self-aligning torque, consider two subcases separately. The first case is for the elastic deformation state, while the other is for the complete sliding state without any elastic deformation of a tire. These two subcases are distinguished by slip ratios caused by the critical values of the slip ratio, the slip angle, and the inclination angle. Specifically, if all of slip ratios are smaller than those of their corresponding critical values, then there exists an elastic deformation state, otherwise there exists only complete sliding state between the tire tread base and the terrain surface.

Same as in Case 1, a slip ratio due to the critical value of the slip ratio can be obtained as:

A slip ratio due to the critical value of the slip angle can be found as:

when Ss < Ssc.

The nondimensional slip ratio Sn, is determined as:

where:

The nondimensional contact patch length ln is found from the equation ln = 1 - Sn, and the modified lateral friction coefficient is expressed as:

For the longitudinal force, the lateral force, and the self-aligning torque two subcases should also be considered separately. A slip ratio due to the critical value of the inclination angle is not needed here since the required condition for Case 2, CαSα > CγSγ, replaces the critical condition of the inclination angle.

Similar to cases 1 and 2, slip ratios due to the critical values of the inclination angle and the slip ratio are obtained as:

The nondimensional slip ratio Sn, is expressed as:

where:

For the longitudinal force, the lateral force, and the self-aligning torque, two subcases should also be considered similar to Cases 1 and 2. A slip ratio due to the critical value of the slip angle is not needed here since the required condition for Case 3, CαSα< CγSγ, replaces the critical condition of the slip angle.

For the general formulations of the longitudinal force Fx, lateral force Fy, and self-aligning torque Mz, in the SAE coordinate system (see the figure, SAE Tire Coordinate System), the three possible combinations of the slip ratio, the slip angle, and the inclination angle are also considered.

Where Cr = ROLLING_RESISTANCE.

An adjustment to the tire moments is conducted to capture the effects of the longitudinal and lateral shifting of the approximate contact patch vertical center of pressure and center of longitudinal shear pressure, in the presence of longitudinal and lateral tire forces.

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.

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:

For Air Basic Tire, the relaxation σ is equal to the relaxation as specified in the tire property file.

For Air Enhance Tire, the relaxation σ is corrected with the footprint length:

The relaxation for Air Enhance is then:

With the relaxation value specified in the tire property file.

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 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_DAMPING

The 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: