Rollover Stability Index Including Effects Of Suspension Design

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SAE TECHNICAL PAPER SERIES  2002-01-0965 Rollover Stability Index Including Effects of Suspension Design Aleksander Hac Delphi Automotive Systems Reprinted From: Vehicle Vehicle Dynamics and Simulation 2002 (SP–1656) SAE 2002 World Congress Detroit, Michigan March 4-7, 2002 400 Comm Commonw onwealt ealth h Drive, Drive, Warre Warrenda ndale, le, PA PA 1509615096-000 0001 1 U.S.A. U.S.A. Tel: Tel: (724) (724) 776-4 776-4841 841 Fax: Fax: (724) (724) 776-57 776-5760 60 The appearance of this ISSN code at the bottom of this page indicates SAE’s consent that copies of the paper may be made for personal or internal use of specific clients. This consent is given on the condition, however, that the copier pay a per article copy fee through the Copyright Clearance Center, Inc. Operations Center, 222 Rosewood Drive, Danvers, MA 01923 for copying beyond that permitted by Sections 107 or 108 of the U.S. Copyright Law. 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A process is available by which discussions will be printed with the paper if it is published in SAE Transactions. For permission to publish this paper in full or in part, contact the SAE Publications Group. Persons wishing to submit papers to be considered for presentation or publication through SAE should send the manuscript or a 300 word abstract of a proposed manuscript to: Secretary, Engineering Meetings Board, SAE. Printed in USA 2002-01-0965 Rollover Stability Index Including Effects of Suspension Design Aleksander Hac Delphi Automotive Systems Copyright © 2002 Society of Automotive Engineers, Inc. ABSTRACT In this paper a simple yet insightful model to predict vehicle propensity to rollover is proposed, which includes the effects of suspension and tire compliance. The model uses only a few parameters, usually known at the design stage. The lateral accelerations at the rollover threshold predicted by the model are compared to the results of simulations, in which vehicles with the same static stability factor, but different suspension characteristics and payloads are subjected to rollinducing handling maneuvers. The results of simulations correlate well with the predictions based on the proposed model. Design recommendations for passive suspensions, which would increase rollover stability are discussed. INTRODUCTION In recent years rollover has became an important safety issue for a large class of vehicles. Even though rollovers constitute a small percentage of all accidents, they have unproportionally large contribution to severe and fatal injuries. For example, rollover is the primary cause of fatalities in accidents involving sport utility vehicles (SUVs). There is an urgent need to develop both analytical and experimental tools to predict rollover propensity of vehicles and to improve their design from the viewpoint of rollover resistance. Real-world rollovers are complex events, involving a variety of factors, which may have broad statistical distributions, and some of these factors may be beyond control of vehicle designers. Such factors include driver steering patterns, type of road surface, type of shoulder, road and shoulder inclination angles, existence of drop off in transition from road to shoulder, coefficient of friction, presence or absence of obstacles on the vehicle path, etc. In addition, during rollover vehicle experiences a loss of stability, a condition in which small changes in vehicle parameters, inputs or environment can significantly affect vehicle behavior. For these reasons, it is nearly impossible to device a simple test or a method that would reflect a majority of real-world rollover scenarios and reliably determine rollover propensity. There is a need to develop models based on sound physical principles that would help the vehicle designer in predicting and reducing rollover risk. One commonly used indicator of rollover propensity is the lateral acceleration at the rollover threshold. It provides a measure of rollover resistance not only in the untripped, maneuver induced rollovers, but also in some types of tripped rollovers, such as tripping on soft soil, since certain minimal lateral force is necessary do initiate rollover. There exists a variety of models, which predict vehicle rollover propensity and in particular lateral acceleration at the rollover threshold. They range from a simplistic static stability factor ground (Garrot et al., 1999) at one side of the spectrum to complex computer models (Allen et al. , 1999) on the other. None of these extremes provides significant insights into the effects of vehicle suspension design on rollover propensity. Computer models require a large number of parameters, which may not be readily available, and pose problems in interpreting the results. In particular, because of complexities of the model, it is not easy to separate the primary influences from the higher order effects, unless other insights are available. Computer models do not provide clear guidance regarding changes that could improve the design if the results are not satisfactory. Static stability factor is a ratio of half-track width to the height of vehicle center of gravity above ground (Garrot et al., 1999). This measure of rollover propensity reflects only the most fundamental relationship. It is obtained under the assumption that a vehicle is a rigid body and ignores all higher order effects, in particular the effects of suspension and tire compliance. In reality vehicle suspension allows for significant movements of wheels with respect to the body, resulting in changes in halftrack width and position of vehicle center of gravity during large lateral acceleration. For vehicles with low static stability factors, the cumulative effect of secondary factors may be sufficient to reduce the lateral acceleration threshold to the value achievable during emergency handling maneuvers. There exists a number of analytical formulas for rollover threshold, which include higher order effects to varying degrees. Gillespie (1992) provides an expression for the rollover threshold, which includes the effect of lateral movement of vehicle center of gravity due to body roll. Dixon (1996) gives a more refined formula, which in addition to the effect of body roll, reflects the influences of tire lateral distortion and the gyroscopic moments due to wheel rotation. Bernard et al. (1989) provide relationships for rollover thresholds of varying complexity with the static stability factor being the simplest. The more complex model includes the effects of lateral movement of vehicle center of gravity, lateral tire distortion and the effect of overshoot in roll angle during dynamic maneuvers. In this paper the major secondary effects resulting from suspension and tire compliance affecting rollover propensity are discussed. They include the effects of the lateral movement of vehicle center of gravity due to body roll during cornering, the effects of suspension kinematics and of lateral compliance of tires, the effects of jacking forces in suspension (which may elevate the body center of gravity), the effects of damping, payload and gyroscopic forces. A quasi-dynamic stability factor, which reflects these secondary influences is proposed. It can be viewed as an extension of the models proposed earlier. The rollover stability threshold predicted by the proposed factor is compared to the results of simulations, in which a full vehicle model is subjected to roll-inducing handling maneuvers. The vehicle has the same static stability factor, but different suspension characteristics and payloads. The results of simulations correlate well with the predictions based on the simple model. Design recommendations for passive suspensions, which would increase the rollover stability, are discussed. An analytical formula for the optimal height of roll centers from the viewpoint of rollover resistance is derived. The focus is on independent suspensions, which become more common in modern SUVs and afford the designer more freedom in selecting suspension kinematics than the dependent suspensions. The results of this paper are limited to vehicles with passive suspensions and without stability enhancement systems, or other active control systems, which may have a profound effect on vehicle propensity to rollover. The effects of active systems are discussed in a separate paper. ROLLOVER MODEL Static stability factor is obtained by considering the balance of forces acting on a rigid vehicle in steady-state cornering. This is illustrated in Figure 1, where deflections of tires and suspension are neglected. During cornering the lateral tire forces on the ground level (not shown) counterbalance the lateral inertial force acting at vehicle center of gravity, resulting in a roll moment. This moment is counterbalanced by the moments of vertical forces. Taking moments about the center of contact patches for the outside tires results: ∑ TA = Mgtw /2 – Ma yh0 - Fzitw (1) where M is vehicle mass, g gravity acceleration, t w vehicle track width (assumed the same front and rear), ay is vehicle lateral acceleration, h 0 the height of vehicle center of gravity above ground, and F zi the total normal load on the inside tires. Figure 1. Rigid Vehicle Model At the limit cornering condition (rollover threshold) the normal load, F zi, reaches zero. Hence at the rollover threshold the lateral acceleration is aylim = gtw /(2h0) = gSSF (2) where SSF = t w /(2h0) is the static stability factor. Neglecting the compliances of suspensions and tires leads to overestimation of rollover threshold. During cornering vehicle body rolls about the roll axis, resulting in the lateral shift of vehicle center of gravity towards outside of turn. At the same time lateral forces of the outside tires cause lateral deformation of the tires and camber (i.e. rotation about the longitudinal axis) of the wheels. All these factors contribute to the reduction of the moment arm of the gravity force, which acts to stabilize the vehicle. At the same time, vertical movement of the wheel with respect to the body is usually accompanied by the lateral movement, which can change the half-track width. This lateral movement is determined primarily by suspension kinematics. In addition, the lateral forces are transmitted between the body and the wheels by rigid suspension arms, which in general are not parallel to the ground. Therefore these link forces have vertical components, which in general do not cancel out and may elevate vehicle center of gravity. Most of these effects are illustrated in Figure 2. The final result is the reduction of the effective half-track width and usually increase in height of vehicle center of gravity, both of which reduce the rollover threshold. In addition, during cornering vehicle wheels rotate about the lateral axis (axis of their rotation) and concurrently about the vertical axis (the axis of vehicle turn). This results in gyroscopic moments, which contribute to the moment equation. Finally, in dynamic maneuvers, the roll angle of vehicle body may exceed (overshoot) the steady-state value. The amount of overshoot depends on the type of maneuver, but for a given maneuver it is related to the roll damping of suspension as well as suspension stiffness and the body moment of inertia about the roll axis. In what follows, each effect is discussed separately and simplified equations are provided which describe their impact on rollover threshold as function of known vehicle parameters. Figure 2. Vehicle Model with Deformable Suspension and Tires VEHICLE BODY ROLL ANGLE Under the influence of lateral inertial forces in steady state cornering, vehicle body rolls about the roll axis by an angle φ, which can approximately be determined as φ = Tφ / κ φ = Mshrollay / κ φ (3) where Tφ is the roll moment acting on vehicle body, M s is vehicle sprung mass, h roll is the height of vehicle body center of gravity above the roll axis and κ φ is the total roll stiffness of vehicle suspension and tires. The term M shroll  / κ φ may be determined experimentally as a roll rate, that is the body roll angle per unit lateral acceleration. It is noted that the roll stiffness includes the tire stiffness, since the body roll angle is measured with respect to the road, and tire compliance (in vertical direction) contributes to the roll angle. For SUVs their contribution may exceed 1 degree of roll angle at maximum lateral acceleration. The body roll angle results in a lateral shift of vehicle center of gravity, which may be interpreted as having an effect of reducing the effective half-track width, or more precisely the moment arm of gravity force. Alternatively, the gravity force Mg can be decomposed into two components as shown in Figure 2 with the lateral component, Mgsin φ, contributing to the moment equation as a destabilizing moment. This method was selected in this paper. In any case, the body roll reduces the stabilizing moment. It should be noted that in the simplified analysis the centers of gravity of vehicle and that of the body are assumed to be collocated. This is a reasonable simplification since the sprung mass is large as compared to the unsprung mass, especially for vehicles with independent suspensions. In more accurate analysis body roll should be separated from axis roll. CHANGE IN HALF-TRACK WIDTH Due to lateral compliance of tires and suspension, as well as changes in wheel lateral location due to suspension kinematics and changes in camber angle, the distance in lateral direction between the centerline of vehicle and the tire contact patches is changed, usually reduced, during cornering. In this paper this distance is defined as a half-track width. The approximate analysis of the change in half-track width under dynamic conditions presented here is conducted using average values for front and rear suspensions and tire parameters. For most vehicles this simplification can be used without introducing unacceptable errors. When geometry and compliance of front and rear suspensions are significantly different, the analysis presented here can be conducted separately for front and rear axle, and then results can be combined. Assuming a linear tire model, the lateral displacements of the tire contact patches with respect to the body resulting from lateral distortion of tires is proportional to the lateral force, which in turn is approximately equal to the product of lateral acceleration and vehicle mass. Thus the reduction  in half-track width due to tire compliance is ∆tw1 = Ma /k y yt (4) where kyt is the total lateral stiffness of both outside tires. Figure 3. Change in Track Width Resulting from Suspension Kinematics Additional change of half-track width occurs primarily because of suspension kinematics and secondarily due to lateral compliance of suspension elements. The first effect is illustrated in Figure 3. During suspension deflection, the wheel rotates with respect to the body about the instantaneous center of rotation (point C) located on the line connecting the tire contact patch with the roll center (point R). This results in wheel displacement in lateral direction and a change of the wheel camber angle with respect to the body. A cumulative effect of both can be analyzed by tracking the path of the contact point A between the tire and the road during suspension deflection. In a first approximation, this path is perpendicular to the line AC. Thus the increase  in the half-track width resulting from suspension compression of ∆z is ∆tw2 = ∆ztanγ  = ∆z2hrollc /tw (5) Here γ  is the inclination angle of line AC with respect to a horizontal line and hrollc is the height of roll center above ground. If the roll center is located below ground level, the distance h rollc is negative and the half-track width is reduced during suspension compression. During cornering, the compression of outside suspension is approximately a linear function of roll angle ( ∆z = φtw /2), which in turn is proportional to lateral acceleration (equation 3). This yields ∆tw2 = Mshrollhrollca /    y κφ (6) Thus the increase in half-track width due to suspension kinematics is proportional to lateral acceleration. Additional change caused by the lateral compliance of suspension elements can also be factored in, since it is proportional to the lateral force and therefore lateral acceleration. It will generally act to reduce the half-track width, so it would decrease the value of ∆tw2. The total reduction  in half track width resulting from tire lateral compliance and suspension kinematics is: ∆tw = ∆tw1- ∆tw2 (7) where ∆tw1 and ∆tw2 are given by equations (5) and (6), respectively. The negative sign in front of the second term appears because ∆tw2 is an increase in half-track width. EFFECT OF GYROSCOPIC FORCES DUE TO WHEEL ROTATION Any rigid body rotating about one axis (usually an axis of symmetry) tends to resist rotation about another axis perpendicular to the axis of rotation. If a body rotates about its own axis of rotation, y, with an angular velocity ϖy, then the moment necessary to rotate this body about another axis, z, with velocity ϖz is (Hibbeler, 1989) Tx = Iϖy×ϖz (8) and the moment vector is along axis x perpendicular to both y and z. The symbol I denotes the moment of inertia of the body about the axis of rotation, y, and × is a vector product. During a cornering maneuver vehicle wheels are spinning with the angular velocity ωy = v/rd (9) where v is vehicle speed and r d the tire radius. The wheels also rotate (with the entire vehicle) with angular velocity ωz = v/R (10) where R is the radius of curvature of the vehicle path. Thus the total gyroscopic moment about the x axis, which is approximately parallel to the axis of vehicle roll, is 2 Tx = 4Iwv /(rdR) (11) The symbol Iw denotes the moment of inertia of each 2 wheel about the axis of rotation. Since I w = m wρw , where mw is the wheel mass and ρw denotes the wheel radius of 2 gyration, and in the steady state cornering a y = v /R, equation (11) can be written as 2 Tx = 4mwρw a /r y d (12) Thus the gyroscopic moment is proportional to lateral acceleration. EFFECT OF JACKING FORCES During cornering maneuvers on smooth roads vehicle body is usually subjected to vertical forces, often referred to as “jacking” forces, which tend to lift the vehicle center of gravity above the static location. In steady-state cornering there are primarily two sources of   jacking forces: nonlinearities in suspension stiffness characteristics and vertical components of forces transmitted by suspension links. Suspension stiffness characteristics are usually progressive, that is stiffness increases with suspension deflection in order to maintain good ride properties with a full load. During cornering maneuvers, progressive characteristic of suspension permits smaller deflection in compression of the outside suspension than deflection in extension of the inside suspension. As a result, height of vehicle center of gravity increases. This effect is highly dependent on the particular stiffness characteristic, so it is difficult to capture in a general approach. It is neglected in the present analysis. The second jacking effect is a result of forces in suspension links. Lateral forces generated during cornering maneuvers are transmitted between the body and the wheels through relatively rigid suspension links. In general these members are not parallel to the ground; therefore the reaction forces in these elements have vertical components, which usually do not cancel out, resulting in a vertical net force, which pushes the body up. threshold is approximately equal to 0.8gSSF (e.g. it is 20% below the threshold computed from the static stability factor): It is known (Gillespie, 1993; Reimpell and Stoll, 1996) that forces transmitted between the vehicle body and a wheel through lateral arms are dynamically equivalent to a single force, which reacts along the line from the tire contact patch to the roll center of suspension. The roll center is by definition the point in the transverse vertical plane, at which lateral forces applied to the sprung mass do not produce suspension roll. This is illustrated in Figure 4 for a double A arm suspension. ∆h = 0.8(h rollc /h0)(Msg/kst) ay = 0.8gtw /(2h0) (16) This simplification is justified because it has an effect only on higher order terms in subsequent analysis. Equations (15) and (16) yield (17) The last term, M sg/kst, is a static deflection of suspension. Taking moments about the center of contact patches of the outside wheel for the compliant vehicle model shown in Figure 2, one obtains at the rollover threshold ∑TA = M(h0+∆h)ay - Mgcosφ(tw /2 –  ∆tw) 2 + Mgsinφh0cosφ + 4mwρw a /r =0 y d (18) The last term on the left-hand side represents the gyroscopic moment according to equation (12). Using a small roll angle assumption, substituting ∆tw from equations (4) through (7) and ignoring higher order terms, yields the following expression for the lateral acceleration at the rollover threshold: gtw /(2h0) ay = ---------------------------------------------------------------- = [1 + ∆h/h0 + Msghroll(1-hrollc /h0)/ κφ  + Mg/(kyth0) 2 + 4mwρw /(Mh0rd)] Figure 4. Roll Center and Jacking Force The resultant force of two reactions in the links, F, is inclined under an angle γ  to the horizontal plane, such that tan γ  = 2hrollc /tw (13) where hrollc is the height of the roll center above ground and tw is the track width. In the limit steady state cornering maneuver, the total lateral force in the links F y = M say. Thus the vertical component of the link force, F z, is Fz = Fy tan γ  = 2Mshrollca /t y w (14) The jacking force results in the vertical displacement of the body center of gravity equal to = 2Mshrollca /(t k) ∆h = F /k z st y w st (15) The symbol k st denotes the total stiffness of the suspension in vertical direction. In order to simplify subsequent equations, it can be assumed that for an average SUV the lateral acceleration at the rollover gSSF = ------------------------------------------------------------------[1+ ∆h/h0 + Msghroll(1-hrollc / h0)/ κφ  + Mg/(k yth0) 2 + 4mwρw /(Mh0rd)] (19) The incremental change in the height of vehicle center of gravity, ∆h, is given by equation (17). It is seen that the lateral acceleration at the rollover threshold is lower than that computed from the static stability factor. The terms contributing to the reduction in lateral acceleration threshold along with the typical range of values for an SUV are listed below. ∆h/h0 is the effect of the increase in the height of center of gravity resulting from jacking forces; it may contribute up to 5% to the reduction in the lateral acceleration threshold. It is small for suspensions with roll centers close to the ground and nearly linear stiffness characteristics. It tends to increase as the height of roll center increases, according to equation (17). Msghroll / κφ  is the effect of lateral displacement of vehicle center of gravity due to body roll and may contribute 5 to 12% (SUVs tend to roll more than passenger cars because of high center of gravity and large suspension travel necessary for off road use). This effect depends on roll rate of vehicle, M shroll / κφ  . It decreases with increasing roll stiffness of suspension and with increasing height of roll centers, which reduces the distance hroll. Msghrollhrollc /(κ φh0) is the effect of increase in half-track width as the result of suspension kinematics. It depends primarily on the height of the roll center above ground, hrollc. It is the only factor that increases vehicle stability if roll center is above ground. It contributes up to 5% increase in lateral acceleration threshold and may partially offset the effects of tire lateral compliance. Mg/(kyh0) is the effect of reduction in half-track width due to lateral compliance of tires; it contributes 3 to 8% (again this value tends to be larger for SUVs because of high profile, compliant tires). It decreases with the increasing lateral stiffness of tires. 2 4mwρw /(Mh0rd) is the effect of gyroscopic forces, which contributes only 1 to 1.5%. The effect of gyroscopic forces is very small and can be neglected. The largest contributing factors are the lateral displacement of vehicle center of gravity and the lateral compliance of tires, followed by the effects of suspension kinematics and change in height of vehicle center of gravity. With the exception of tire lateral compliance, each one of these factors can be significantly influenced by suspension design. All the secondary factors combined can reduce the lateral acceleration at the rollover threshold by as much as 2025% for a typical SUV. Figure 5. Vehicle Body Roll Model In the above analysis steady-state cornering was considered, in which the steady-state value of roll angle was assumed. Since a vehicle is a dynamic system, it usually exhibits an overshoot in roll response to suddenly applied lateral acceleration. That is, the maximum value of the roll angle during transient exceeds the steady state value. The simplest model, which captures this phenomenon is the second order roll model shown in Figure 5. It is described by the following equation: 2 2 Isd φ /dt + cφ dφ /dt + κ φφ = Tφ (20) where Is is the moment of inertia of vehicle body with respect to the roll axis, c φ is the total roll damping of front and rear suspensions and κ φ is the total roll stiffness of front and rear suspensions, including springs and roll bars. Tφ is the moment of external loads with respect to the roll axis; in roll motion exited by lateral acceleration Tφ = -Msayhroll. If the lateral acceleration input is a unit step, then T φ = T01(t) and the steady state roll angle is φss = T /    . The maximum roll angle under dynamic 0 κφ conditions is φmax = φss(1 + ∆os) (21) where ∆os is the degree of overshoot above the steadystate value. For the model described by equation (20) the degree of overshoot in response to a step function can be determined analytically as 2 1/2 ∆os = exp[-ζπ /(1-ζ ) ] (22) where ζ is the non-dimensional roll damping ratio, that is the ratio of the actual roll damping, c φ, to the critical 1/2 damping cφcr = 2(Isκ φ) . Thus the damping ratio can be expressed in terms of vehicle parameters as 1/2 ζ = cφ / [2(I sκ φ) ] (23) The degree of overshoot, ∆os, increases as the damping ratio decreases, that is when the roll damping decreases relatively to the roll stiffness and the moment of inertia. Thus increasing the vehicle payload, which increases the moment of inertia, without increasing damping, results in reduced damping ratio and increased overshoot. The degree of overshoot depends on a particular type of maneuver. Nearly ideal unit step in lateral acceleration can be achieved when vehicle is sliding from low friction surface onto the high-friction one, but it is impossible to achieve with a step steer, or any steering input on uniform surface, because lateral acceleration does not build up instantaneously. Thus the actual overshoot in a step steer maneuver will usually be substantially less than the one identified by equation (22), especially when the roll mode is heavily underdamped. It is therefore more realistic to consider a step function, in which the lateral acceleration increases linearly within a finite period time, as in a ramp function. This is discussed in more detail in the simulation section. The moment equation (18) can now be modified to include the effect of roll angle overshoot in transient maneuvers. This yields the following equation for lateral acceleration at the rollover threshold: where a, b and c are the following constants: gtw /(2h0) ay = -------------------------------------------------------------------[1+ ∆h/h0 + Msghroll(1-hrollc / h0)(1+∆os)/ κφ  + Mg/(kyth0) 2 + 4mwρw /(Mh0rd)] (24) Equation (24) provides a simple model to determine rollover threshold using only a few parameters. It permits one to approximately determine the effects of various design parameters on rollover propensity. For example, it is seen that vehicle payload will usually increase vehicle tendency to rollover because of increased mass and moment of inertia which tend to increase the steadystate, and especially the dynamic, roll angle, as well as contribute to reduction of half-track width due to tire compliance. It may also increase the height of vehicle center of gravity. As expected, increasing roll stiffness will improve rollover resistance due to reduction of body roll angle (and associated lateral displacement of the center of gravity), but it should be accompanied by an increase in roll damping. Otherwise the damping ratio and the overshoot will increase, which may reduce the benefit of higher roll stiffness in dynamic maneuvers. EFFECT OF SUSPENSION KINEMATICS ON ROLLOVER Equations (19) and (24) indicate how various vehicle and suspension design parameters can affect the lateral acceleration at the rollover threshold, and thereby vehicle resistance to rollover. While the influences of suspension roll stiffness or damping are obvious, the effect of height of roll center is not so transparent. Increasing the roll center height has both positive and negative influences on vehicle rollover stability. On one hand, high roll centers tend to increase the jacking forces and hence increase the height of vehicle center of mass during cornering. On the other hand, they reduce the roll angle of vehicle and the associated lateral displacement of vehicle center of mass, and contribute to the incremental increase in half-track width due to suspension kinematics, ∆tw2. It is often the case that when a design variable exerts influences acting in opposite directions, there exists an optimal value for this variable. The location of roll center is no exception to this general rule. Let us neglect the effect of dynamic overshoot and consider the lateral acceleration threshold given by equation (19). Since the numerator is a constant, the lateral acceleration can be maximized by minimizing the denominator with respect to the roll center height. Bearing in mind that h roll = h0-hrollc and substituting the value of ∆h from equation (17), denominator of equation (19) can be expressed as the following quadratic function of roll center height, h rollc: 2 f(hrollc) = ah rollc – bhrollc + c (25) a = Msg/(κ φh0) , 2 b = 2M sg/ κφ  - 0.8 M sg/(ksth0 ) 2 c = 1 + M sg h0/ κφ  + Mg/(k yth 0) + 4m wr w /(Mh0rd) (26) Function f(hrollc) reaches minimum when the variable h rollc = b/2a. In terms of vehicle parameters, this yields the following optimal value: opt hrollc = 2h0 – 0.8κ φ /(ksth0) (27) The optimal height of roll center from the point of view of rollover resistance depends on the nominal height of vehicle center of gravity and the ratio of suspension roll stiffness, κ φ, to the total vertical stiffness of suspension, kst. In practice, the value obtained from equation (27) would be subject to limitations resulting from other design constraints, such as allowable changes in camber angle (limited for example by tire wear) and limitation of variations in track width. Large variations of track width with suspension travel, especially when occuring on the front axle, affect straight line stability during driving on rough roads (Reimpell and Stoll, 1996). For the vehicle parameters used in this study (a midsize SUV) the height of center of gravity, h 0 = 0.652 m, the total vertical stiffness of suspension k st = 74,000 N/m and the total roll stiffness, κ φ = 66,500 Nm/rad. Substituting these values into equation (27) yields the optimal value of 0.201 m, which is slightly higher than typically used. However, the optimal value is quite sensitive to the ratio κ φ /kst. For example, increasing the roll stiffness by 10% by employing stiffer roll bars, would reduce this value to 0.087 m. In most vehicles the roll center of the front suspension is lower than that of the rear. RESULTS OF SIMULATIONS In order to verify the accuracy of the proposed model, vehicle simulations were conducted using a full-car 16degree of freedom vehicle model, which was validated against vehicle test data. The model permits simulation of vehicle dynamics under large roll angles, significantly exceeding the angle corresponding to two-wheel lift off condition. The vehicle used in simulation is a midsize sport utility vehicle with all independent suspensions and a marginal static stability factor of only 1.09 in unladen state. In order to make the vehicle easier to roll over during severe handling maneuvers, the lateral acceleration capability of the vehicle was slightly increased by assuming more aggressive than standard tires. In an attempt to induce the rollover by aggressive steering maneuvers, the steering patterns illustrated in Figure 6 were used. They represent a J-turn maneuver and a fishhook maneuver. In each case the steering rate at the steering wheel is reduced to about 1000 deg/s, which corresponds to the maximum rates that can be generated by human drivers. All maneuvers were performed with the entry speed of 25 m/s (about 56 mph), but with increasing amplitudes A of steering angle up to the point when either the maximum steering angle of 540 degrees was reached, or the vehicle rolled over. Figure 8. Vehicle Lateral Acceleration and Roll Angle Responses in J-turn and Fishhook Maneuvers Figure 6. Steering Patterns Used in Simulations The first series of simulations was conducted to evaluate the degree of overshoot in handling maneuvers with rapid changes in steering angle. In Figure 7 the degree of overshoot computed from equation (22) is plotted as a function of damping ratio, ζ for the roll mode. The nominal value of ζ is 0.261. The degrees of overshoot obtained from full car simulations of both maneuvers are also shown in Figure 7 for several damping levels ranging from half to double the nominal value. The roll angle and lateral acceleration responses are illustrated in Figure 8 for both steering patterns and the nominal level of damping. The amplitudes of the steering angle were 54 and 27 degrees for the J-turn and fishhook steering patterns, respectively. Since the lateral acceleration does not build up immediately after a step in steering angle, the simple model over-predicts the overshoot when damping is low. For very firm damping, however, the model underestimates the degree of overshoot. This occurs because in full vehicle simulations the lateral acceleration rises above its steady-state value after rapid changes in steering angle, which in turn causes the roll angle to overshoot. The overshoot in lateral acceleration is caused primarily by dynamic increase of tire normal forces due to transient body roll and heave, which increases lateral forces and consequently lateral acceleration. A better match between the degree of overshoot obtained from simulations and from the analytical formula can be obtained if a more realistic ramp function in lateral acceleration instead of a step function is used. For example, the dotted curve in figure (7) illustrates the overshoot calculated when the lateral acceleration rises linearly from zero to the final value in 0.5 seconds. This line is much closer to the simulation test data. Within the range of damping usually encountered in SUVs, the degree of overshoot can be approximated by the following linear function ∆os = 0.35 – 0.4 ζ (28) The next series of simulations is designed to study the effects of changes in payload and selected suspension design parameters on the rollover propensity. For this purpose vehicles with the following parameter variations were considered: Figure 7. Degree of Overshoot as Function of Damping Ratio 1) Vehicle 1 is a baseline vehicle with all nominal parameters and without payload. 2) Vehicle 2 has the same parameters as the baseline, but it carries an additional payload of 500 kg. The payload, in addition to changing inertial properties, causes a slight increase in the height of vehicle center of mass, a shift of the center to the rear, and deflection of suspension (which shifts its operating point towards higher stiffness and facilitates bottoming of suspension during cornering). 3) Vehicle 3 is the same as the baseline, but with front and rear roll bar stiffness double the nominal values and the damping coefficients increased by 25%. This corresponds to the increase of total roll stiffness by 41%. The damping was increased in order to maintain approximately the same damping ratio in the roll mode despite increase in roll stiffness. 4) Vehicle 4 is the base vehicle with both roll center heights increased to the optimal value calculated from equation (27). Responses of each vehicle to both steering patterns with increasing amplitudes were simulated from an initial speed of 25 m/s (56 mph). The nominal vehicle did not roll over in the J-turn maneuver regardless of the amplitude of the steering wheel angle, but with full payload it rolled over for the steering angle of 70 degrees and at a rather low lateral acceleration of 6.78 2 m/s . Traces of lateral acceleration and roll angle for both vehicles in a J-turn with 90 degrees steering input are compared in Figure 9. the rear, which promotes tendency to oversteer. Oversteer is known to be a contributing factor in rollovers (Marine et al., 1999) since peak lateral forces on tires of both axles are developed at relatively large side slip angles. In the case of fishhook maneuver both the nominal vehicle 1 and the vehicle 2 with payload rolled over, but the vehicle with payload did at the lower level of lateral acceleration and at a lower steering angle. Simulations performed for vehicle 3 (with increased roll stiffness and roll damping) and vehicle 4 (with modified suspension geometry) indicate that neither of them can be rolled over in J-turn or fishhook maneuvers regardless of the amplitude of steering angle. Thus relatively minor changes in suspension design can improve rollover resistance of vehicle with marginal static stability factor. These improvements of resistance to maneuver induced rollover for vehicle 3 and 4 were predicted by the simplified model. The results of simulations in the case of fishhook maneuver for vehicles 3 and 4 as compared to the baseline vehicle 1 are shown in Figures 10 and 11, respectively. Figure 10. Responses of Vehicle 1 and 3 in a Fishhook Maneuver with the Steering Angle Amplitude of 90 Degrees Figure 9. Responses of Vehicle 1 and 2 in a J-turn Maneuver with the Steering Angle of 90 Degrees Simulation performed for vehicle 2 was terminated when the body roll angle reached 1 radian (57.3 degrees). The vehicle with payload rolls over much easier because of several factors: larger roll angle in handling maneuvers due to increased inertia (which increases both the steady state value and the dynamic overshoot), tendency of suspension to bottom in heavy cornering, which increases jacking forces, larger reduction in half track width due to lateral compliance of tires and suspension under increased lateral forces, slightly higher center of gravity, and shift of center of mass to The amplitudes of the steering angle are 90 degrees in both cases. It is seen that both vehicles 3 and 4 remain stable, while the baseline vehicle rolls over. It should be noted, however, that the individual changes in suspension parameters as defined by vehicle 3 and 4 are not sufficient to prevent the vehicle with full payload from rolling over. Nevertheless, the steering angle amplitude required to rollover these vehicles is larger than for the base vehicle. CONCLUSION Figure 11. Responses of Vehicle 1 and 4 in a Fishhook Maneuver with the Steering Angle Amplitude of 90 Degrees In Table 1 the results of simulations in terms of the accelerations at the rollover threshold are compared to the values obtained from the equations (19) and (24). The effect of gyroscopic moments was neglected in the analytical models. Equation (24) includes the effect of dynamic overshoot, while equation (19) does not. In the third column the dynamic overshoot computed from the theoretical formula (22) was used, while in the fourth column the quasi-empirical equation (28) was utilized. The value of lateral acceleration derived from simulation is the maximum value of lateral acceleration in any of the considered maneuvers, in which vehicle did not roll over. Since in some cases there are peaks of extremely short time duration, not sufficient to cause rollover, the lateral acceleration was low-pass filtered. There is a good agreement between the results of simulations and those obtained from the analytical models, with the value of lateral acceleration at the rollover threshold derived from simulations being usually and very close to the analytical result obtained from equation (24) with the overshoot modeled by equation (28). Vehicle 1 2 3 4 max Ay (Eq. 19) 8.37 7.65 8.58 8.47 Table1 Ay (Eqs. 24, 22) 8.04 7.16 8.35 8.25 max max Ay (Eqs. 24, 28) 8.19 7.36 8.45 8.37 max Ay (sim.) 8.18 6.78 8.39 8.33 The vehicle with payload (vehicle 2) is an exception –  the model tends to overestimate the lateral acceleration at the rollover threshold. It is most likely due to the fact that the simple model does not take suspension nonlinearities into account, in particular the effect of bottoming of suspension, which increases the height of vehicle center of gravity and lateral tire forces under dynamic conditions. In this paper the effects of some design parameters of passive independent suspensions on rollover propensity of vehicles with high center of gravity, such as SUVs, were examined. A model derived from simple physical principles was proposed to evaluate vehicle propensity to rollover. The model includes the effects of lateral movement of vehicle center of gravity during body roll, the effects of suspension jacking forces, the effects of changes in track width due to suspension kinematics, the effects of tire lateral compliance, of gyroscopic forces, and the effects of dynamic overshoot in the roll angle. A simplified formula was derived for the lateral acceleration at the rollover threshold, which includes the effects of suspension design parameters, such as roll stiffness and damping, stiffness in the heave mode and locations of roll centers. Design guidelines for suspension parameters to improve rollover resistance were discussed. In particular, an analytical expression for the optimal roll center height from the viewpoint of rollover resistance was developed. The analytical results obtained are supported by the results of simulations, which show that the lateral accelerations at the rollover threshold predicted by the model are in a reasonably good agreement with the results of simulations. The area of possible future improvements include modeling of nonlinearities of suspension stiffness characteristics, which play especially important role in modeling rollovers of fully loaded vehicles. The results of analysis and simulations also indicate that for a marginally stable SUVs, the variations in suspension parameters can change the character of vehicle response from unstable to stable in typical dynamic rollover tests considered by NHTSA, without changing the static stability factor. REFERENCES 1. Allen, R. W., Rosenthal, T. J., Klyde, D. H. and Hogue, J. 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