Rollover Stability Index Including Effects of Suspension Design

SAE TECHNICAL PAPER SERIES2002-01-0965 Rollover Stability Index Including Effects of Suspension Design Aleksander Hac Delphi Automotive Systems Reprinted From: Vehicle Dynamics and Simulation 2002 (SP–1656) SAE 2002 World Congress Detroit, Michigan March 4-7, 2002 400 Commonwealth Drive, Warrendale, PA 15096-0001 U.S.A. Tel: (724) 776-4841 Fax: (724) 776-5760 222 Rosewood Drive. 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. Positions and opinions advanced in this paper are those of the author(s) and not necessarily those of SAE. This consent is given on the condition.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. for advertising or promotional purposes. 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For permission to publish this paper in full or in part. without the prior written permission of the publisher. To request permission to reprint a technical paper or permission to use copyrighted SAE publications in other works. or for resale. Inc. ISSN 0148-7191 Copyright © 2002 Society of Automotive Engineers. standards. Danvers. for creating new collective works. however. 1999) on the other. involving a variety of factors. 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. rollover is the primary cause of fatalities in accidents involving sport utility vehicles (SUVs). The results of simulations correlate well with the predictions based on the proposed model. There exists a variety of models. . ABSTRACT In this paper a simple yet insightful model to predict vehicle propensity to rollover is proposed. For these reasons.. For vehicles with low static stability factors. it is not easy to separate the primary influences from the higher order effects. The lateral accelerations at the rollover threshold predicted by the model are compared to the results of simulations. which may have broad statistical distributions. Inc. It provides a measure of rollover resistance not only in the untripped. This measure of rollover propensity reflects only the most fundamental relationship. usually known at the design stage.. which includes the effects of suspension and tire compliance. In reality vehicle suspension allows for significant movements of wheels with respect to the body. Real-world rollovers are complex events. maneuver induced rollovers. inputs or environment can significantly affect vehicle behavior. during rollover vehicle experiences a loss of stability. such as tripping on soft soil. road and shoulder inclination angles. since certain minimal lateral force is necessary do initiate rollover. type of shoulder. presence or absence of obstacles on the vehicle path. etc. type of road surface. and pose problems in interpreting the results. Computer models require a large number of parameters. Design recommendations for passive suspensions. There is a . 1999). and some of these factors may be beyond control of vehicle designers. the cumulative effect of secondary factors may be sufficient to reduce the lateral acceleration threshold to the value achievable during emergency handling maneuvers. The model uses only a few parameters. Even though rollovers constitute a small percentage of all accidents. because of complexities of the model. in which vehicles with the same static stability factor. Computer models do not provide clear guidance regarding changes that could improve the design if the results are not satisfactory. they have unproportionally large contribution to severe and fatal injuries.2002-01-0965 Rollover Stability Index Including Effects of Suspension Design Aleksander Hac Delphi Automotive Systems Copyright © 2002 Society of Automotive Engineers. INTRODUCTION In recent years rollover has became an important safety issue for a large class of vehicles. It is obtained under the assumption that a vehicle is a rigid body and ignores all higher order effects. resulting in changes in halftrack width and position of vehicle center of gravity during large lateral acceleration. For example. They range from a simplistic static stability factor ground (Garrot et al. One commonly used indicator of rollover propensity is the lateral acceleration at the rollover threshold. which predict vehicle rollover propensity and in particular lateral acceleration at the rollover threshold. Such factors include driver steering patterns. which would increase rollover stability are discussed. a condition in which small changes in vehicle parameters. Static stability factor is a ratio of half-track width to the height of vehicle center of gravity above ground (Garrot et al. which may not be readily available. coefficient of friction. but different suspension characteristics and payloads are subjected to rollinducing handling maneuvers. in particular the effects of suspension and tire compliance. unless other insights are available. but also in some types of tripped rollovers. need to develop models based on sound physical principles that would help the vehicle designer in predicting and reducing rollover risk. 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. In particular. None of these extremes provides significant insights into the effects of vehicle suspension design on rollover propensity. 1999) at one side of the spectrum to complex computer models (Allen et al. In addition. existence of drop off in transition from road to shoulder. Fzi. In this paper the major secondary effects resulting from suspension and tire compliance affecting rollover propensity are discussed. It can be viewed as an extension of the models proposed earlier. which in general do not cancel out and may elevate vehicle center of gravity. Bernard et al. but different suspension characteristics and payloads. The rollover stability threshold predicted by the proposed factor is compared to the results of simulations. Therefore these link forces have vertical components. (1989) provide relationships for rollover thresholds of varying complexity with the static stability factor being the simplest.There exists a number of analytical formulas for rollover threshold. g gravity acceleration. which include higher order effects to varying degrees. This lateral movement is determined primarily by suspension kinematics. Taking moments about the center of contact patches for the outside tires results: ∑ TA = Mgtw/2 – Mayh0 . The results of simulations correlate well with the predictions based on the simple model. in which a full vehicle model is subjected to roll-inducing handling maneuvers. The results of this paper are limited to vehicles with passive suspensions and without stability enhancement systems. where deflections of tires and suspension are neglected. which in general are not parallel to the ground. lateral tire distortion and the effect of overshoot in roll angle during dynamic maneuvers. Rigid Vehicle Model At the limit cornering condition (rollover threshold) the normal load.Fzitw (1) where M is vehicle mass. which in addition to the effect of body roll. This is illustrated in Figure 1. During cornering the lateral tire forces on the ground level (not shown) counterbalance the lateral inertial force acting at vehicle center of gravity. tw vehicle track width (assumed the same front and rear). resulting in a roll .e. rotation about the longitudinal axis) of the wheels. An analytical formula for the optimal height of roll centers from the viewpoint of rollover resistance is derived. the effects of suspension kinematics and of lateral compliance of tires. At the same time lateral forces of the outside tires cause lateral deformation of the tires and camber (i. which acts to stabilize the vehicle. ay is vehicle lateral acceleration. the effects of damping. which become more common in modern SUVs and afford the designer more freedom in selecting suspension kinematics than the dependent suspensions. This moment is counterbalanced by the moments of vertical forces. Dixon (1996) gives a more refined formula. the lateral forces are transmitted between the body and the wheels by rigid suspension arms. which may have a profound effect on vehicle propensity to rollover. Neglecting the compliances of suspensions and tires leads to overestimation of rollover threshold. Most of these effects are illustrated in Figure 2. which would increase the rollover stability. which reflects these secondary influences is proposed. which includes the effect of lateral movement of vehicle center of gravity due to body roll. At the same time. The focus is on independent suspensions. vertical movement of the wheel with respect to the body is usually accompanied by the lateral movement. payload and gyroscopic forces. The more complex model includes the effects of lateral movement of vehicle center of gravity. Design recommendations for passive suspensions. reaches zero. reflects the influences of tire lateral distortion and the gyroscopic moments due to wheel rotation. are discussed. and Fzi the total normal load on the inside tires. the effects of jacking forces in suspension (which may elevate the body center of gravity). During cornering vehicle body rolls about the roll axis. A quasi-dynamic stability factor. All these factors contribute to the reduction of the moment arm of the gravity force. They include the effects of the lateral movement of vehicle center of gravity due to body roll during cornering. Hence at the rollover threshold the lateral acceleration is aylim = gtw/(2h0) = gSSF where SSF = tw/(2h0) is the static stability factor. The vehicle has the same static stability factor. which can change the half-track width. Gillespie (1992) provides an expression for the rollover threshold. Figure 1. resulting in the lateral shift of vehicle center of gravity towards outside of turn. h0 the height of vehicle center of gravity above ground. The effects of active systems are discussed in a separate paper. moment. In addition. or other active control systems. The final result is the reduction of the effective half-track width and usually increase in height of vehicle center of (2) ROLLOVER MODEL Static stability factor is obtained by considering the balance of forces acting on a rigid vehicle in steady-state cornering. usually reduced. The amount of overshoot depends on the type of maneuver. hroll is the height of vehicle body center of gravity above the roll axis and κφ is the total roll stiffness of vehicle suspension and tires. contributing to the moment equation as a destabilizing moment. each effect is discussed separately and simplified equations are provided which describe their impact on rollover threshold as function of known vehicle parameters. The term Mshroll / κφ may be determined experimentally as a roll rate. 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. both of which reduce the rollover threshold. This results in gyroscopic moments. In what follows.gravity. during cornering. the analysis presented here can be conducted separately for front and rear axle. Mgsinφ. during cornering vehicle wheels rotate about the lateral axis (axis of their rotation) and concurrently about the vertical axis (the axis of vehicle turn). CHANGE IN HALF-TRACK WIDTH Due to lateral compliance of tires and suspension. which can approximately be determined as φ = Tφ / κφ = Mshrollay / κφ (3) where Tφ is the roll moment acting on vehicle body. and then results can be combined. Finally. the gravity force Mg can be decomposed into two components as shown in Figure 2 with the lateral where kyt is the total lateral stiffness of both outside tires. In addition. 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. the body roll reduces the stabilizing moment. Vehicle Model with Deformable Suspension and Tires VEHICLE BODY ROLL ANGLE Under the influence of lateral inertial forces in steady state cornering. and tire compliance (in vertical direction) contributes to the roll angle. especially for vehicles with independent suspensions. which may be interpreted as having an effect of reducing the effective half-track width. which in turn is approximately equal to the product of lateral acceleration and vehicle mass. For most vehicles this simplification can be used without introducing unacceptable errors. the roll angle of vehicle body may exceed (overshoot) the steady-state value. Figure 3. In more accurate analysis body roll should be separated from axis roll. in dynamic maneuvers. vehicle body rolls about the roll axis by an angle φ. In this paper this distance is defined as a half-track width. which contribute to the moment equation. Assuming a linear tire model. since the body roll angle is measured with respect to the road. 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. the distance in lateral direction between the centerline of vehicle and the tire contact patches is changed. For SUVs their contribution may exceed 1 degree of roll angle at maximum lateral acceleration. Ms is vehicle sprung mass. This is a reasonable simplification since the sprung mass is large as compared to the unsprung mass. as well as changes in wheel lateral location due to suspension kinematics and changes in camber angle. that is the body roll angle per unit lateral acceleration. component. It is noted that the roll stiffness includes the tire stiffness. The body roll angle results in a lateral shift of vehicle center of gravity. In any case. 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. Change in Track Width Resulting from Suspension Kinematics . Thus the reduction in half-track width due to tire compliance is ∆tw1 = May/kyt (4) Figure 2. When geometry and compliance of front and rear suspensions are significantly different. Alternatively. or more precisely the moment arm of gravity force. This method was selected in this paper. Since Iw = mwρw2. often referred to as “jacking” forces. During suspension deflection. 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. 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). which tend to lift the vehicle center of gravity above the static location. equation (11) can be written as Tx = 4mwρw2ay/rd (12) Thus the increase in half-track width due to suspension kinematics is proportional to lateral acceleration. which in turn is proportional to lateral acceleration (equation 3). Suspension stiffness characteristics are usually progressive. This effect is highly dependent on the particular stiffness characteristic. which is approximately parallel to the axis of vehicle roll. Thus the increase in the half-track width resulting from suspension compression of ∆z is ∆tw2 = ∆ztanγ = ∆z2hrollc/tw (5) Tx = Iϖy×ϖz (8) and the moment vector is along axis x perpendicular to both y and z. The total reduction in half track width resulting from tire lateral compliance and suspension kinematics is: ∆tw = ∆tw1. progressive characteristic of suspension permits smaller deflection in compression of the outside suspension than deflection in extension of the inside suspension. so it would decrease the value of ∆tw2.∆tw2 (7) Thus the gyroscopic moment is proportional to lateral acceleration. This results in wheel displacement in lateral direction and a change of the wheel camber angle with respect to the body. Thus the total gyroscopic moment about the x axis. 1989) . height of vehicle center of gravity increases. If the roll center is located below ground level. It will generally act to reduce the half-track width. The wheels also rotate (with the entire vehicle) with angular velocity ωz = v/R (10) 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. This yields ∆tw2 = Mshrollhrollcay/κφ (6) where R is the radius of curvature of the vehicle path. Additional change caused by the lateral compliance of suspension elements can also be factored in. The first effect is illustrated in Figure 3. 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. EFFECT OF JACKING FORCES During cornering maneuvers on smooth roads vehicle body is usually subjected to vertical forces. The symbol I denotes the moment of inertia of the body about the axis of rotation. respectively. and × is a vector product. y. then the moment necessary to rotate this body about another axis. where ∆tw1 and ∆tw2 are given by equations (5) and (6). 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. so it is difficult to capture in a general approach. where mw is the wheel mass and ρw denotes the wheel radius of gyration. If a body rotates about its own axis of rotation. y. As a result. that is stiffness increases with suspension deflection in order to maintain good ride properties with a full load.Additional change of half-track width occurs primarily because of suspension kinematics and secondarily due to lateral compliance of suspension elements. During a cornering maneuver vehicle wheels are spinning with the angular velocity ωy = v/rd (9) where v is vehicle speed and rd the tire radius. the compression of outside suspension is approximately a linear function of roll angle (∆z = φtw/2). It is neglected in the present analysis. with an angular velocity ϖy. z. The negative sign in front of the second term appears because ∆tw2 is an increase in half-track width. the distance hrollc is negative and the half-track width is reduced during suspension compression. and in the steady state cornering ay = v2/R. since it is proportional to the lateral force and therefore lateral acceleration. In a first approximation. this path is perpendicular to the line AC. with velocity ϖz is (Hibbeler. During cornering. is Tx = 4Iwv2/(rdR) (11) The symbol Iw denotes the moment of inertia of each wheel about the axis of rotation. During cornering maneuvers. This is illustrated in Figure 4 for a double A arm suspension. Msg/kst. It tends to increase as the height of roll center increases. Thus the vertical component of the link force. In order to simplify subsequent equations. It is known (Gillespie. such that tan γ = 2hrollc/tw (13) gSSF = ------------------------------------------------------------------[1+ ∆h/h0 + Msghroll(1-hrollc/ h0)/κφ + Mg/(kyth0) + 4mwρw2/(Mh0rd)] (19) The incremental change in the height of vehicle center of gravity. Using a small roll angle assumption. which usually do not cancel out. In general these members are not parallel to the ground. ∆h/h0 is the effect of the increase in the height of center of gravity resulting from jacking forces. it is 20% below the threshold computed from the static stability factor): ay = 0. which pushes the body up. In the limit steady state cornering maneuver. 1993. Lateral forces generated during cornering maneuvers are transmitted between the body and the wheels through relatively rigid suspension links. is given by equation (17). It is small for suspensions with roll centers close to the ground and nearly linear stiffness characteristics. F.g.The second jacking effect is a result of forces in suspension links. Fz. according to equation (17). threshold is approximately equal to 0.8gtw/(2h0) (16) This simplification is justified because it has an effect only on higher order terms in subsequent analysis. Taking moments about the center of contact patches of the outside wheel for the compliant vehicle model shown in Figure 2. is Fz = Fy tan γ = 2Mshrollcay/tw (14) The jacking force results in the vertical displacement of the body center of gravity equal to ∆h = Fz/kst = 2Mshrollcay/(twkst) (15) The symbol kst denotes the total stiffness of the suspension in vertical direction. at which lateral forces applied to the sprung mass do not produce suspension roll. is a static deflection of suspension. Equations (15) and (16) yield ∆h = 0. it can be assumed that for an average SUV the lateral acceleration at the rollover . is inclined under an angle γ to the horizontal plane. Roll Center and Jacking Force The resultant force of two reactions in the links. yields the following expression for the lateral acceleration at the rollover threshold: gtw/(2h0) ay = ---------------------------------------------------------------. one obtains at the rollover threshold ∑TA = M(h0+∆h)ay . It is seen that the lateral acceleration at the rollover threshold is lower than that computed from the static stability factor. resulting in a vertical net force. therefore the reaction forces in these elements have vertical components. ∆h.8gSSF (e. substituting ∆tw from equations (4) through (7) and ignoring higher order terms.8(hrollc/h0)(Msg/kst) (17) The last term. which reacts along the line from the tire contact patch to the roll center of suspension. it may contribute up to 5% to the reduction in the lateral acceleration threshold. 1996) that forces transmitted between the vehicle body and a wheel through lateral arms are dynamically equivalent to a single force. The roll center is by definition the point in the transverse vertical plane. the total lateral force in the links Fy = Msay.Mgcosφ(tw/2 – ∆tw) + Mgsinφh0cosφ + 4mwρw2ay/rd = 0 (18) The last term on the left-hand side represents the gyroscopic moment according to equation (12). The terms contributing to the reduction in lateral acceleration threshold along with the typical range of values for an SUV are listed below. Reimpell and Stoll. 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 where hrollc is the height of the roll center above ground and tw is the track width.= [1 + ∆h/h0 + Msghroll(1-hrollc/h0)/κφ + Mg/(kyth0) + 4mwρw2/(Mh0rd)] Figure 4. the maximum value of the roll angle during transient exceeds the steady state value. It contributes up to 5% increase in lateral acceleration threshold and may partially offset the effects of tire lateral compliance. hrollc. results in reduced damping ratio and increased overshoot. or any steering input on uniform surface. This is discussed in more detail in the simulation section. Nearly ideal unit step in lateral acceleration can be achieved when vehicle is sliding from low friction surface onto the high-friction one. which increases the moment of inertia. The moment equation (18) can now be modified to include the effect of roll angle overshoot in transient maneuvers. Tφ is the moment of external loads with respect to the roll axis. The simplest model. It is therefore more realistic to consider a step function. Vehicle Body Roll Model In the above analysis steady-state cornering was considered. It depends primarily on the height of the roll center above ground. The maximum roll angle under dynamic conditions is φmax = φss(1 + ∆os) (21) where ∆os is the degree of overshoot above the steadystate value. which reduces the distance hroll. This effect depends on roll rate of vehicle. followed by the effects of suspension kinematics and change in height of vehicle center of gravity. For the model described by equation (20) the degree of overshoot in response to a step function can be determined analytically as ∆os = exp[-ζπ/(1-ζ2)1/2] (22) where ζ is the non-dimensional roll damping ratio. especially when the roll mode is heavily underdamped. All the secondary factors combined can reduce the lateral acceleration at the rollover threshold by as much as 2025% for a typical SUV. that is when the roll damping decreases relatively to the roll stiffness and the moment of inertia. without increasing damping. model shown in Figure 5. It is described by the following equation: Isd2φ/dt2 + cφ dφ/dt + κφφ = Tφ (20) where Is is the moment of inertia of vehicle body with respect to the roll axis. 4mwρw2/(Mh0rd) is the effect of gyroscopic forces. it usually exhibits an overshoot in roll response to suddenly applied lateral acceleration. This yields the following equation for lateral acceleration at the rollover threshold: Figure 5.5%. Since a vehicle is a dynamic system. Thus increasing the vehicle payload. to the critical damping cφcr = 2(Isκφ)1/2. then Tφ = T01(t) and the steady state roll angle is φss = T0/κφ. Msghrollhrollc/(κφh0) is the effect of increase in half-track width as the result of suspension kinematics. Mshroll/κφ. cφ is the total roll damping of front and rear suspensions and κφ is the total roll stiffness of front and rear suspensions. That is. which contributes only 1 to 1. With the exception of tire lateral compliance. in which the lateral acceleration increases linearly within a finite period time. each one of these factors can be significantly influenced by suspension design. It is the only factor that increases vehicle stability if roll center is above ground. The effect of gyroscopic forces is very small and can be neglected. The degree of overshoot depends on a particular type of maneuver. cφ. compliant tires). as in a ramp function. in roll motion exited by lateral acceleration Tφ = -Msayhroll. Mg/(kyh0) is the effect of reduction in half-track width due to lateral compliance of tires. because lateral acceleration does not build up instantaneously. including springs and roll bars. which captures this phenomenon is the second order roll . Thus the damping ratio can be expressed in terms of vehicle parameters as ζ = cφ / [2(Isκφ)1/2] (23) The degree of overshoot. that is the ratio of the actual roll damping. it contributes 3 to 8% (again this value tends to be larger for SUVs because of high profile.travel necessary for off road use). If the lateral acceleration input is a unit step. but it is impossible to achieve with a step steer. It decreases with increasing roll stiffness of suspension and with increasing height of roll centers. Thus the actual overshoot in a step steer maneuver will usually be substantially less than the one identified by equation (22). increases as the damping ratio decreases. ∆os. It decreases with the increasing lateral stiffness of tires. in which the steady-state value of roll angle was assumed. The largest contributing factors are the lateral displacement of vehicle center of gravity and the lateral compliance of tires. Substituting these values into equation (27) yields the optimal value of 0.201 m. the value obtained from equation (27) would be subject to limitations resulting from other design constraints. the effect of height of roll center is not so transparent. The vehicle used in simulation is a midsize sport utility vehicle with all independent suspensions and a marginal static stability factor of only 1. 1996).652 m. In order to make the vehicle easier to roll over during severe handling maneuvers. κφ = 66.09 in unladen state. the lateral acceleration can be maximized by minimizing the denominator with respect to the roll center height. the steering patterns illustrated in Figure 6 were used. It permits one to approximately determine the effects of various design parameters on rollover propensity. kst. and contribute to the incremental increase in half-track width due to suspension kinematics. such as allowable changes in camber angle (limited for example by tire wear) and limitation of variations in track width. In most vehicles the roll center of the front suspension is lower than that of the rear. high roll centers tend to increase the jacking forces and hence increase the height of vehicle center of mass during cornering. significantly exceeding the angle corresponding to two-wheel lift off condition. Increasing the roll center height has both positive and negative influences on vehicle rollover stability. In an attempt to induce the rollover by aggressive steering maneuvers. Large variations of track width with suspension travel. the total vertical stiffness of suspension kst = 74. In practice. this yields the following optimal value: hrollcopt = 2h0 – 0. and especially the dynamic. κφ. b = 2Msg/κφ .8κφ/(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.000 N/m and the total roll stiffness. but it should be accompanied by an increase in roll damping.where a. and thereby vehicle resistance to rollover. In each case the steering rate . It is often the case that when a design variable exerts influences acting in opposite directions. Since the numerator is a constant. vehicle simulations were conducted using a full-car 16degree of freedom vehicle model. which may reduce the benefit of higher roll stiffness in dynamic maneuvers. Let us neglect the effect of dynamic overshoot and consider the lateral acceleration threshold given by equation (19). For the vehicle parameters used in this study (a midsize SUV) the height of center of gravity. hrollc: f(hrollc) = ahrollc2 – bhrollc + c (25) RESULTS OF SIMULATIONS In order to verify the accuracy of the proposed model. would reduce this value to 0. However. the lateral acceleration capability of the vehicle was slightly increased by assuming more aggressive than standard tires. as well as contribute to reduction of half-track width due to tire compliance.087 m. For example.8 Msg/(ksth02) (26) c = 1 + Msg h0/κφ + Mg/(k yth0) + 4m wrw2/(Mh0rd) Function f(hrollc) reaches minimum when the variable hrollc = b/2a. They represent a J-turn maneuver and a fishhook maneuver. roll angle. While the influences of suspension roll stiffness or damping are obvious. It may also increase the height of vehicle center of gravity. 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. which is slightly higher than typically used. The model permits simulation of vehicle dynamics under large roll angles.500 Nm/rad. ∆tw2. the optimal value is quite sensitive to the ratio κφ/kst. 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. h0 = 0. a = Msg/(κφh0) . The location of roll center is no exception to this general rule. there exists an optimal value for this variable. As expected. Bearing in mind that hroll = h0-hrollc and substituting the value of ∆h from equation (17). to the total vertical stiffness of suspension. affect straight line stability during driving on rough roads (Reimpell and Stoll. especially when occuring on the front axle. In terms of vehicle parameters.0. denominator of equation (19) can be expressed as the following quadratic function of roll center height. On one hand. Otherwise the damping ratio and the overshoot will increase. increasing the roll stiffness by 10% by employing stiffer roll bars. For example. b and c are the following constants: gtw/(2h0) ay = -------------------------------------------------------------------[1+ ∆h/h0 + Msghroll(1-hrollc/ h0)(1+∆os)/κφ + Mg/(kyth0) + 4mwρw2/(Mh0rd)] (24) Equation (24) provides a simple model to determine rollover threshold using only a few parameters. increasing roll stiffness will improve rollover resistance due to reduction of body roll angle (and associated lateral displacement of the center of gravity). On the other hand. they reduce the roll angle of vehicle and the associated lateral displacement of vehicle center of mass. which was validated against vehicle test data. The next series of simulations is designed to study the effects of changes in payload and selected suspension design parameters on the rollover propensity. The roll angle and lateral acceleration responses are illustrated in Figure 8 for both steering patterns and the nominal level of damping. respectively.261.5 seconds. In Figure 7 the degree of overshoot computed from equation (22) is plotted as a function of damping ratio. Figure 8. however. the dotted curve in figure (7) illustrates the overshoot calculated when the lateral acceleration rises linearly from zero to the final value in 0. For very firm damping.at the steering wheel is reduced to about 1000 deg/s.4ζ (28) Figure 6. but with increasing amplitudes A of steering angle up to the point when either the maximum steering angle of 540 degrees was reached. which in turn causes the roll angle to overshoot. which corresponds to the maximum rates that can be generated by human drivers. 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. All maneuvers were performed with the entry speed of 25 m/s (about 56 mph). The overshoot in lateral acceleration is caused primarily by dynamic increase of tire normal forces due to transient body roll and heave. For example. The amplitudes of the steering angle were 54 and 27 degrees for the J-turn and fishhook steering patterns. For this purpose vehicles with the following parameter variations were considered: 1) Vehicle 1 is a baseline vehicle with all nominal parameters and without payload. This line is much closer to the simulation test data. which increases lateral forces and consequently lateral acceleration. Within the range of damping usually encountered in SUVs. This occurs because in full vehicle simulations the lateral acceleration rises above its steady-state value after rapid changes in steering angle.35 – 0. or the vehicle rolled over. Degree of Overshoot as Function of Damping Ratio . the simple model over-predicts the overshoot when damping is low. the model underestimates the degree of overshoot. ζ for the roll mode. Figure 7. 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. the degree of overshoot can be approximated by the following linear function ∆os = 0. 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. Vehicle Lateral Acceleration and Roll Angle Responses in J-turn and Fishhook Maneuvers Since the lateral acceleration does not build up immediately after a step in steering angle. The nominal value of ζ is 0. Responses of Vehicle 1 and 3 in a Fishhook Maneuver with the Steering Angle Amplitude of 90 Degrees The amplitudes of the steering angle are 90 degrees in both cases. In the case of fishhook maneuver both the nominal vehicle 1 and the vehicle 2 with payload rolled over. 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. The nominal vehicle did not roll over in the J-turn maneuver regardless of the amplitude of the steering wheel angle.78 m/s2. but with front and rear roll bar stiffness double the nominal values and the damping coefficients increased by 25%. but the vehicle with payload did at the lower level of lateral acceleration and at a lower steering angle. larger reduction in half track width due to lateral compliance of tires and suspension under increased lateral forces.. It is seen that both vehicles 3 and 4 remain stable. however. which increases jacking forces. respectively. while the baseline vehicle rolls over.3 degrees). Traces of lateral acceleration and roll angle for both vehicles in a J-turn with 90 degrees steering input are compared in Figure 9. Figure 10. The payload. 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. These improvements of resistance to maneuver induced rollover for vehicle 3 and 4 were predicted by the simplified model. The damping was increased in order to maintain approximately the same damping ratio in the roll mode despite increase in roll stiffness. Oversteer is known to be a contributing factor in rollovers (Marine et al. a shift of the center to the rear. but with full payload it rolled over for the steering angle of 70 degrees and at a rather low lateral acceleration of 6. Thus relatively minor changes in suspension design can improve rollover resistance of vehicle with marginal static stability factor. and shift of center of mass to . Nevertheless. 3) Vehicle 3 is the same as the baseline. which promotes tendency to oversteer. Figure 9. the steering angle amplitude required to rollover these vehicles is larger than for the base vehicle. Responses of each vehicle to both steering patterns with increasing amplitudes were simulated from an initial speed of 25 m/s (56 mph). slightly higher center of gravity. tendency of suspension to bottom in heavy cornering. 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). This corresponds to the increase of total roll stiffness by 41%. the rear. 1999) since peak lateral forces on tires of both axles are developed at relatively large side slip angles. in addition to changing inertial properties. 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. 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. and deflection of suspension (which shifts its operating point towards higher stiffness and facilitates bottoming of suspension during cornering). It should be noted. causes a slight increase in the height of vehicle center of mass. but it carries an additional payload of 500 kg.2) Vehicle 2 has the same parameters as the baseline. 4) Vehicle 4 is the base vehicle with both roll center heights increased to the optimal value calculated from equation (27). 47 Aymax(Eqs.25 REFERENCES 1... “Computer Simulation Analysis of Light Vehicle Lateral/Directional Dynamic Stability”. 24. The effect of gyroscopic moments was neglected in the analytical models. 24. 891991. Gillespie. of gyroscopic forces. Figure 11. J. Howe. the effects of tire lateral compliance. G. without changing the static stability factor. and the effects of dynamic overshoot in the roll angle. Garrott... R. Warrendale.58 8. The area of possible future improvements include modeling of nonlinearities of suspension stiffness characteristics. S. Thomas. T. A simplified formula was derived for the lateral acceleration at the rollover threshold. 4. 5.. W. Design guidelines for suspension parameters to improve rollover resistance were discussed. “An Experimental Examination of Selected Maneuvers that May Induce On-Road Untripped. 1990. Bernard. It is most likely due to the fact that the simple model does not take suspension nonlinearities into account. such as SUVs.65 8. the effects of changes in track width due to suspension kinematics. H. T. J.18 6.04 7. 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). SAE paper No.CONCLUSION In this paper the effects of some design parameters of passive independent suspensions on rollover propensity of vehicles with high center of gravity. Table1 Aymax(Eqs. J. Rosenthal... Equation (24) includes the effect of dynamic overshoot. SAE paper No. 1999. M. J. Inc. R. the effects of suspension jacking forces. D. The analytical results obtained are supported by the results of simulations.39 8. Allen. in which vehicle did not roll over. 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. PA 15096-0001. 2. In particular. Inc. which increases the height of vehicle center of gravity and lateral tire forces under dynamic conditions. G. J. stiffness in the heave mode and locations of roll centers. Since in some cases there are peaks of extremely short time duration. 1996. 1989. “Testing and Analysis of Vehicle Rollover Behavior”. N. The model includes the effects of lateral movement of vehicle center of gravity during body roll.37 7.33 The vehicle with payload (vehicle 2) is an exception – the model tends to overestimate the lateral acceleration at the rollover threshold.. 19) 8. the variations in suspension parameters can change the character of vehicle response from unstable to stable in typical dynamic rollover tests considered by NHTSA.. an analytical expression for the optimal roll center height from the viewpoint of rollover resistance was developed. “Fundamentals of vehicle Dynamics”. R. Klyde. 22) 8. the lateral acceleration was low-pass filtered. Warrendale. SAE. Dixon.35 8.) 8. The value of lateral acceleration derived from simulation is the maximum value of lateral acceleration in any of the considered maneuvers. . were examined. 1999.16 8.45 8. D. and Forkenbrock. J.. and Vanderploeg. “Tires. 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). “Vehicle Rollover on Smooth Surfaces”. not sufficient to cause rollover. T. Cooperrider.. 28) 8. such as roll stiffness and damping.36 8. 1992. W. The results of analysis and simulations also indicate that for a marginally stable SUVs.19 7. Shannon. in particular the effect of bottoming of suspension. SAE paper 1999-01-0124.37 Aymax (sim. which includes the effects of suspension design parameters. and Hammond. while in the fourth column the quasi-empirical equation (28) was utilized. Suspension and Handling”.C.78 8. Light Vehicle Rollover – Phase II of NHTSA’s 19971998 Vehicle Rollover Research Program” 6. A model derived from simple physical principles was proposed to evaluate vehicle propensity to rollover. Vehicle 1 2 3 4 Aymax (Eq.. 900366. which play especially important role in modeling rollovers of fully loaded vehicles. and Hogue. There is a good agreement between the results of simulations and those obtained from the analytical models. while equation (19) does not. 3. PA 15096-0001. In the third column the dynamic overshoot computed from the theoretical formula (22) was used.. SAE. 1989. “The Automobile Chassis. 1999. SAE.. Inc. M. J. Engineering Principles”. Hibbeler... SAE paper 1999-01-0122. Dynamics”. Warrendale. “Characteristics of On-Road Rollovers”. J. M. 9. Reimpell. and Stoll. C. New York. PA 15096-0001... Wirth. . and Thomas T.. “Engineering Mechanics. L. R. Inc. MacMillan. C. 1996. 8. Marine.7. H.