MacAdam 1989 VSD Static Turning Analysis Paper

8/13/2019 MacAdam 1989 VSD Static Turning Analysis Paper

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Vehicle System Dynamics, 18 1989), pp. 345-357 0042-3 14/89/ 1806-0345 $3.00Swets Zeitlinger

St at ic Turning Analys is of Vehicles Subject to Externally AppliedForces - A Momen t Arm Ratio Formulation

C. C. MacADAM*

A formulation for representing the static turning response of a two - axle vehicle due to appliedexternal or c ontrol forces is expressed in terms of a simple r atio of two distances along the vehiclelongitudinal axis. The two distance scoincide with point s on the vehicle at which externally appl.ied/control f orces and their reactive inertial forces act with respect to the vehicle neutral steer point. Theresulting formulat ion is equivalent to the rotatio nal equilibrium equatio n written with respect to th eneutral steer point. T he method allows a s i m l ~ l e

"

visual analysis"

of the steady turn ing process byshowing how key forces and associated m oment ar ms can change with respect to one ano ther du e tovehicle modifications or different operati ngcondit ions, thereby affecting the static turning responseof the vehicle.

1. INTRODUCTION

The intent of this paper is to present a simplified formulation of the staticturning response of a two - axle, pneumatic - tired vehicle subject to arbitrarylateral external forces. The laterally applied forces may have a variety ofsources, such as, the intentional control force introduced from a steered wheel,or, an external disturbance force produced by the aerodynamic shape of thevehicle's body. The basic approach separates forces acting on the vehicle massinto two categories. The first category of forces is termed here control forces andcomprises any external force associated with steering or disturbing the vehicle.The second category of forces is termed motion forces and comprises only thosetire force components associated with sideslip and yawing motion of the vehicle- absent of any steering. Inertial forces acting on the vehicle mass are alsoincluded in this latter category. The conventional analysis that follows helps toidentify these force terms. Linear relationships are assumed so that forces maybe superimposed . The net result is that the static turning motion of a vehicle canbe described simply in terms of a ratio of two distances along the vehiclelongitudinal axis. The two distances are those associated with points on thevehicle axis at which the control force s) and motion forces act with respect tothe neutra l steer point and mass center of the vehicle.

Consequently, knowledge of how the neutral steer point, the mass center, orlocation of the control forces may change due to modifications to a vehicle,allows a sirnple visualization of hour the static turning response of the vehicle

The University of Michigan, Transpor tation Research Institute, Ann Arbor , Michigan48109-2150USA.


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346 C. C. M a c A D A M

will likewise be affected. The development and discussion which follows willarrive at a " formula " which encapsulates this concept. Three examples ofapplying its use to different vehicle contr ol problems completes the paper.

2. MATHEMATICAL ANALYSIS

The linear equations governing the static turning motion of a two - axle vehicleequipped with pneumatic tires are well known [ I , 2 3 4 5 6 ] . However, forpurposes of notation and subsequent referral, the basic lateral force and yawmoment equilibrium equations written with respect to the vehicle mass centerare presented in equations 1) and (2) respectively, using Figure I as a referencediagram.

G f F, , F, - m U r = O 1)

Figure I . Diagram of Forces Acting on Vehicle.

where,

I;; is the total lateral tire force at the front axle6 is the total lateral tire force at the rear axleF is an arbitr ary lateral force (aerodynamic o r otherwise)m is the vehicle massU is the vehicle speed of travel


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STATIC TURNING DUE TO EXTERNALLY APPLIED FORCES 347

is the vehicle yaw rated is a distance forward of the mass center which locates the applied

lateral force Fis the distance of the front axle forward of the mass center

b is the distance of the rear axle behind the mass center L is the vehicle wheelbase a b

The linearized lateral tire forces ar e given by,

where,

f is the front tire cornering stiffness (both tires)C is the rear tire c ornering stiffness (both tires)v is the vehicle sideslip velocity at the mass center

is the vehicle yaw rat eis the steer angle of the front tires

and,6 is the steer angle of the rear tires

Substituting equation s (3) - (4) into equations (1)- (2) an d rearranging, results inan equivalent set of two simultaneous equations for lateral force and yawequilibrium involving sideslip velocity, v, an d yaw rate, r :

The force terms appearing on the left hand side of equation (5) are referred tohere as control forces and include the arbit rary applied force F as well as thosecomponents of the total tire forces resulting from steering of the wheels. Th eforce terms appearing on the right hand side of equation (5) are referred to hereas motion forces a nd include the inertial force acting o n the mass center as wellas those components of the tire forces attributable only to sideslip and yawmotion (tir e forces of the non - steered wheel).

By combining the left hand side control forces int o one total force, FT actingat a distance, e , ahead of the mass center, the equati ons (5) - (6) can be simplifiedto:

where, F Ta n d e are given by,


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348 C. C. M a c D M

FT = F , c,/.SJ+C,,6, 9)and,

e = dF,+aC, /SJ -bC, r6 r ) / F,+C,J6 f+C, rb r ) 10

Solution of equations (7) -(8) results in the following expression f or the rati oof yaw rate to the total control force, FT:

~ F T = [ bGr- G/) + e Cb/+ C r)l/[(a +b)2C4/C,,/U- aCqf- G r ) m u ] 1 1)

By noting that the distance, c, from the neutral steer point forward to the masscenter is given by the expression,

an d following its substitution into equation 1 1 and rearrangement of terms,the following simple expression is obta ined which relates the vehicle yaw rate tothe total control force:

or, in terms of the steady -state lateral acceleration, a, = U r,

Figure 2 supplements equation (13) by illustrating the quantities involved.Equation (13) can be viewed as the yaw moment equilibrium equation writtenwith respect to the neutral steer point. Since moment terms involving fron t an drear tire force components that depend upon sideslip velocity cancel oneanother about the neutral steer point (by definition), no coupling termsinvolving sideslip velocity at the neutral steer point appear in this form of themoment equilibrium equation. Accordingly, only those forces that affect therotational equilibrium of the vehicle about the neutral steer point appear inFigure 2.

In diagram (A) of Figure 2, the distance (c +e) is the moment arm of theapplied lateral control force about the neutral steer point. The distance c is themoment ar m of the inertial force, mUr, acting about the neutral steer point atthe mass center. The distance is equal to the resisting moment of the equal and

opposite non -steered tire force components, Z and 5 generated by a purerotation of the vehicle about the neut ral steer point, normalized by the inertialforce mUr i.e., 5 = tire yaw damping moment abo ut nsp/mUr). Diagram (A)can be simplified by not including the tire damping forces, q and 5 providedthat the inertial force, mUr, is translated forward to the point located by 5,


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or, equivalently:

Figure 2. Diagram of Forces and Moment Arms. (A) lntertial Force Acting at the Mass Center;(B) lntertial Force Located at Distance Ahead of the Mass Center.


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350 C. C. a c A D A M

thereby retaining the tire yaw damping moment , mUr. See diagram (B) ofFigure 2.

Under this latter formulat ion, if the vehicle is viewed as a simple lever, havinga fulcrum or pivot point located at the neutral steer point (nsp), the steadyturn ing process is analogous t o a basic mechanical statics problem. The appliedcontrol force, FT acting at a distance (c e), produces a force of reaction, mUr,acting at a counter - balancing moment arm distance (c 5). Subsequentexamples will utilize diagram (B) of Figure 2 to illustrate the moment armformulation in several applications.

The moment arms c and e depend upon the tire stiffnesses, mass distribution,an d relative magnitudes and location of any steered wheel or addi tional externalforces. The moment arm, 5 is similarly dependent, bu t is also a function of thespeed of the vehicle and is given by the expression,

As seen, the moment arm, 5 which is proportional to the tire yaw dampingmoment, varies as 1/U 2 changing in magnitude from infinity to zero withincreasing vehicle speed. Consequently, changes in vehicle speed stronglyinfluence the moment ar m, 5, but have no effect on the moment arms co r e. Foroversteer vehicles (c < and the nsp lies ahead of the mass center), the classicalcondition fo r directional stability [ I 3 occurs in the diagr am of Figure 2 whenthe moment ar m 5equals the moment arm - c: The forward speed at which thisoccurs is of course the critical speed, LI -L/K)1/2, (where K is the understeergradient) obtained when c. In contr ast, fo r understeer vehicles (c > andthe nsp lies behind the mass center), t he characteristi c speed 161 LI (L/K) /*,

occurs when= c.

3. APPLICATIONS

Diagram (B) of Figure 2 and equation (13) can be used to visualize and predictchanges in the turning response of vehicles when modifications (planned orotherwise) are introduced t hat cause the moment arms c, e, and to becomealtered. The examples that follow help to demonstrate the graphical utility ofthe moment a rm formulation f or explaining different vehicle turn ing responsesderiving from changes in vehicle understeer, speed, or applied forces.

Example I: Aerodynamic Crosswind Response with Fixed Steering.

This example addresses the static t urning response of a vehicle subject to aconstant crosswind and associated lateral aerodynamic force, F . The aero -dynamic force, F acts at the center of pressure located a distance, d, ahead ofthe center of mass. No steering is assumed to occur. See Figure 3. (Vehicledynamicis ts a nd aerodynamicis t shave used this type o f t e s t a s o n e m easu reo f a


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STATIC T U R N IN G D U E TO E X T E RN A L LY A P P L IE D FORCES 5

Figure 3. High speed Aerodynami c Force Example.

vehicle's passive crosswind sensitivity [7,8]. Utilizing equa tions (9) and lo ) ,the total lateral cont rol force, FT, in this case becomes FT=F, , since 6, 6 = 0.Likewise, th e mome nt ar m distance, e , in equation (10) is simply the distance

from the mass center to the center of pressure i.e., e = d) . Equation 13a),relating the static yaw rate response to the applied aerodynamic force, thenbecomes, following these substi tutions,

At very high speeds, where aerodynamic forces have a sizeable influence, themoment arm is normall y quite small [


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Figure 3 would then suggest t hat this influence would increase the baseline yawrate since the ratio (c - A + d)/ c- A + 5- E) would clearly increase (for [ d).In contrast to this, if the same baseline vehicle is operated in a similar or largercrosswind at slower speeds, two primary effects would occur. First, theaerodynamic slip angle would increase, producing a rearward shift in the centerof pressure (due to typical non - linearities in passenger car aerodynamicproperties) and because of the decreased speed, the moment arm 5 wouldincrease. This would then result in a new diagram as shown in Figure 4 withmoment arm, d, locating the center of aerodynamic pressure, with a length lessthan 5 This new set of operating conditions results in a reduced passive windsensitivity (for the same magnitude of crosswind force) because of: (1) thereduced moment arm d, and, (2) the increased tire damping moment arm 5produ ced by the lower speed condition.

Figure 4. Lower Speed Aerodynamic Force Example.

However, under these new baseline conditions, a decrease in understeer forthe vehicle, similar to t hat noted above, can have an opposite effect and producea decrease in the vehicle yaw rate. This is because the ratio of moment a rms seenin equation (15) and Figure 4 is altered oppositely from that seen in Figure 3.This r atio , as before, is given by (c - A d)/ c - A 5- E). Under conditionsin which d is considerably less than 5 the decrease in c to a value c - A willproduce a reduced moment arm rati o, thereby resulting in a decreased yaw ratecompared to its baseline response. (When d is in the vicinity of 5 the picture isless obv ious because E also reduces the denominator .)

Finally, for this latter example, even if the speed is not reduced, b ut the centerof pressure is moved rearward through body aerodynamic styling changesalone, the passive wind sensitivity will still be reduced because of t he shortenedmoment arm d. Obviously, a s d moves rearward toward the neutral steer point,


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STATIC TURNING DUE TO EXTERNALLY APPLIED FORCES 353

the passive wind sensitivity, as measured by the static yaw rat e response t o anaerod ynami c side force, approaches zero i.e., the moment arm d i n equation(15) approaches a value of -c.

Example 2: Turning Response due to a Superelevated Road and GravitationalSide Force.

For non -steered vehicles subject t o a gravitatio nal side force from a laterallyinclined road surface (assuming superelevation levels l ) , the application ofequation (13) is similar to that discussed above for aerodynamic forces.However, in this case, the applied force is acting at the mass center an d thereforethe moment arm distance, e, in equation (13) and Figure 2 is now zero.Consequently, the static yaw rate response due to a consta nt gravitational sideforce, F, W E, is given by,

r / F [m U] I c)/ c + 6 16)

where, W is the weight of the vehicle and E is the superelevation of the roadsurface. Further , the r atio of centrifugal lateral acceleration (in g's) produced byan amount of road superelevation, E, is given by the simple ratio, c/ c 5).Thu s, a t low speeds where is large, the resulting centrifugal accelerationinduced by a r oad surface cross -slope is nil. At very high speeds where is verysmall, the induced centrifugal acceleration (in g's) a pproa ches the am oun t ofroadway superelevation, E. Fo r a neutral steer vehicle c = O ) , no centrifugalacceleration o r path cu rvatu re response is induced from t he gravitational sideforce.

Example.3:

Control Forces from Two - Wheel an d Four - Wheel Steering Vehicles.Control forces attributable to steered wheels can also be introduced and

analyzed in a similar manner using this formulation. The control force termsGJ6,and Gr6,, appea ring on th e left hand side of equations (5) an d 6), are aresult of steering the front an d/o r rear wheels. In the absence of any externalaerodynamic or gravitational forces, the total control force, FT, given inequation (9) for th e front - wheel steering case, is seen t o be,

and the fixed location of this force, e, given by equati on l o ) , becomes after t heaforementioned substitutions:

These equations simply state that for the conventional front - wheel - only steeredvehicle, the cont rol force, cJSJ, acts at the obvious front axle location and islocated at the distance, a, forward of the mass center. See Figure 5.


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Figure 5. Convention al Front-Wheel Steer Vehicle.

The relationship between yaw rate and total lateral control force given by thebasic relationship in equation 13), then becomes for the front -wheel steervehicle:

If this is expressed in terms of the front wheel steer angle, the yaw rate t o steerangle gain is given by,

The same formulation can also be extended to four -wheel steering vehicles.Consider a vehicle having four -wheel steering defined by the simple relation -ship, 6 = k6 / so that the rear wheels are steered in proportion to the frontwheels by an amount k. If k > 0, the wheels are steered in the same direction.Such control strategies are often exercised at mid -range and higher speeds oftravel. Under such schemes in actual practice, k usually takes on a value ofabout 0.3 [9] If no additional lateral forces, F are present, the total controlforce given by equation (9) is seen to be:

and the fixed location of this force, e, given by equation l o ) , becomes after thesubstitution of the above front /rear steering relationship:

indicating t hat the steering control force has now moved considerably rearwardfrom the fron t axle.


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STATIC TURNING DUE TO EXTERNALLY PPLl ED FORCES 355

The re lationship between yaw rate and total lateral control force, given by thebasic relationship in equation 13), then becomes for the four -wheel steerexample:

with the moment arm , e, determined by equation (22). At highway speeds, therelative moment a rm lengths for this four -wheel steering example are providedby the diagram of Figure 6.

In this figure, is approximately L,/lO because of the elevated speed. Themoment arm e is assumed to be about L / 6 (assuming f C,,, a/b 0.6, andk 0.3). Lastly, is approximately L. / 5 reflecting a modest level of understeer.

For cases in which the front - rear steer relationship is more complicated than

a simple gain factor , k, and which may change with vehicle speed and operat ingconditions, the location, e, of the control force, F,,,, is no longer fixed and canmove up and down the longitudinal axis of the vehicle. (Note that this is not thecase with the conventional front -only steering vehicle in which e is fixed at thefront axle position, e a , as noted earlier.)

A good example of this control force " leveraging " is provided in most four -wheel steering systems when counter -steering of the rear wheels (k < 0) occursat low speeds to improve maneuverability in tight quarters. If equation (22) isre-examined with a negative value of k in mind, it is clear tha t the location of thefour wheel steering control force, given by e, is extended out t o a distance wellbeyond the front axle. For example, if k - 0.3 19 101, and assuming equalfront and rear tire corner ing stiffnesses, the length of e becomes,

Consequently, one primary advantage that four -wheel steering vehicles enjoy,by comparison to conventional front -wheel steered vehicles, is the great leewayavailable to them in locating the applied control force (for steering purposes)through intelligent alteration of its associated moment arm. Most four -wheelsteering systems which alter the front - rear steering ratio with speed aresimultaneously adjusting the effective location, e, of the steering control force,I ,, and its magnitude. The product of e and F is the corresponding steeringcontrol moment. Conventional front -wheel steering vehicles maintain aconstant gain relationship between the steering control force and the steeringcontrol input 4,, Cqf af) and a constant gain relationship between the steeringcontrol moment and the steering input (steering moment aCb/Sr) However,most four -wheel steering systems alter these force/moment relationships bychanging k as a function of speed. Ordinar ily, such four -wheel steering systemsproduce larger steering forces and smaller steering moments at high speeds(compared to a comparable front -wheel steering vehicle), and, smaller steeringforces and larger steering moments at low speeds. This can be seen from


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equations (21) and (22) and noting that their product, the steering controlmoment, is given by aC, - kbC;,,)G(,with k typically ranging from - 0.3 to0.3 as speed increases.

An interesting case develops when the rear front steering ratio gain, k, ischosen in such a manner as to maintain e 5 in Figure 6. That is, the location, e,

Figure 6. High Speed - Four -Wheel Steer Example.of the control force varies with forward speed and identically to 5. Examinationof equation (23)shows that this produces a control system that has a constantrelationship between steering control force, F and steady -state lateralacceleration, regardless of speed, since rnUr/ ,, is unity under this scheme. Bysetting e equal to 5 using the relationships provided by equations (14) and ( 2 2 ) ,the resulting speed -dependent expression for k becomes:

As the forward speed, U , approaches zero, the expression for k approaches-C,/ /G,, resulting in low speed counter -steering. At very high speeds, k

approachesa

/bC, , , and producesfront rear

steering of the same polarity./.Interestingly, this concept is very similar to the one proposed by Sano et al. [9]in which their system is designed to produce a zero steady -state sideslipcondition at the mass center for all speeds. The expression given by equation(25) results in a similar zero sideslip condition at low speed, but at the neutralsteer point instead of the mass center.


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MacAdam 1989 VSD Static Turning Analysis Paper

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