Asymmetric Roll Centers Author(s): Wm. C. Mitchell Source: SAE Transactions , 1998, Vol. 107, SECTION 6: JOURNAL OF PASSENGER CARS (1998), pp. 2632-2639 Published by: SAE International Stable URL: https://www.jstor.org/stable/44741226 JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at https://about.jstor.org/terms SAE International is collaborating with JSTOR to digitize, preserve and extend access to SAE Transactions This content downloaded from 13.245.201.252 on Mon, 21 Mar 2022 13:03:03 UTC All use subject to https://about.jstor.org/terms 983085 Asymmetric Roll Centers Wm. C. Mitchell Wm. C. Mitchell Software Copyright © 1998 Society of Automotive Engineers, Inc. many applications they produce the same result. But ABSTRACT when we apply the definitions to asymmetric suspensions the differences can be critical. Indeed, not all authors The roll center is an important analysis tool for vehicle dynamics. But most analysis of the roll center is agree basedon the definition. We must understand the assumptions that underlie the definitions before we on production cars, which usually have symmetric susthem. pensions and a center of gravity near the centerlineextend of the vehicle. Racing cars, particularly oval track stock cars, The best discussion of roll centers I have found is a paper often have asymmetric suspensions and usually have a by J.C. Dixon (1) titled "The roll-centre concept in vehicle weight bias. For a car that is only going to turn left there handling dynamics."" This appeared in the Procedings of is no reason for the left side front suspension to be anythe Institute of Mechanical Engineers, volume 20, numthing like the right side. Oval track cars usually have ber 1. as much weight on the inside as the rules allow. THE KINEMATIC ROLL CENTER Analytical tools adapted from the standard industry texts or production car use do not properly address asymmetThis is the most common definition of a roll center, but ric suspensions. This paper will analyze the asymmetric suspension and discuss the role of the roll center. relies It will upon some assumptions that are dubious for production begin with a theoretical analysis of the roll center and the cars and impossible for oval track racing cars. underlying assumptions. It will show that the roll center is Ellis (2) has the following definition: a clever device for calculating forces where you do not know how lateral force is distributed between the inside "Traditionally the vehicle has been assumed to roll and outside tires. about a 'roll-axis' which has been defined as an The paper will include an analysis of the relationship centres' ... between lateral movement of the roll center as a result of roll and vertical movement of the roll center as a result of axis joining two imaginary points, the 'roll The roll centres themselves have been taken as bump movement. The paper will prove that a symmetric suspension which exhibits no lateral movement of the roll kinematic centres of rotation of the suspension assuming that the wheels are rigid and do no center as a result of chassis roll will also have a roll cen- move sideways on the road surface. ter which moves vertically with the center of gravity when the chassis exhibits vertical movement. This desirable Dixon (1) does a good job of discussing the underlyi assumptions of the kinematic roll center. These assum property preserves the moment arm and contributes to tions are severely violated in the motorsports arena. stability. It also explains why lateral movement of the roll Assumption #1 : The contact point between tire and ro center, which is calculated by popular software programs, remains fixed even as the tire changes camber. can be used as a design criteria. This analysis will then be extended to asymmetric sus- This also means there is no lateral deflection of the pensions and will derive stability criteria for suspensionswheel or tire. Photographs and under-car TV camthat do not have the roll center on the centerline of the eras show this is not true. This definition depends upon a pin-joint between tire and road. With a wide car. Techniques will be developed which allow the pneumatic tire there is no simple description of the designer to place the roll center wherever he wants and contact between tire and road. keep it near there as the vehicle moves. Assumption #2: There is no track change. WHAT IS A ROLL CENTER? This assumption would restrict the vehicle to one ride The formal definitions of roll center distinguish between aheight for each roll angle. But the vehicle goes through ride height changes (heave) as it moves kinematic roll center and a force-based roll center. For 2632 This content downloaded from 13.245.201.252 on Mon, 21 Mar 2022 13:03:03 UTC All use subject to https://about.jstor.org/terms THE SPRING CENTER around the race track. This is particularly true of banked tracks where the lateral force is applied with a normal force resulting from the banking or aerody- If the chassis truly did rotate about the 'roll center' the namic downforce. we could use the lateral location of the roll center to determine how much one spring compresses and ho Assumption #3: The roll center is really an instantaneous center because it can migrate as the car moves. much the other extends. But this kinematic definition ignores the springs entirely. The springs define a 'spring This is why many industry texts restrict the definition center' which relates to the relative stiffness of the to low lateral forces. Race cars are designed for springs. Oval track cars often have different springs the inside and outside of the car. If you consider a c large lateral forces. with a massive spring on the outside and a light spring The kinematic roll center concept also depends upon the inside, that car is going to roll about the massi spring. This example contradicts the kinematic approach. rigid links without bushings. In this case race cars fit the assumption better than street cars. m ^ l' _[ III J m r > 'ns'an' ^ ļ l' _[ Center ^ Fi i I / Roll '' Center ' Contaci Patch Contact Patch Figure 2. The Roll Center is located at the intersection o with the tire Contact Patch. For a symmetric vehicle i 2633 This content downloaded from 13.245.201.252 on Mon, 21 Mar 2022 13:03:03 UTC All use subject to https://about.jstor.org/terms i i r~r~] Figure 3. In Figure FORCE-BASED ROLL CENTER vertical plane through any pair of wheel centres at which lateral forces may be applied to the sprung The force-based roll center relies upon the same projec-mass without producing suspension roll." tion of A-arms to an instant center. But some definitions specify a roll center where these lines intersect the centerline of the car. Applying the forces at a point beneath the CG makes it easier to understand jacking forces. Gillespie (3) states: "A lateral force in the contact patch of the left wheel reacts along the line from the contact point to the Dixon goes on to say: "This does not call for the roll-centre to be in the centre-plane and is therefore ambiguous, although it is usually taken to be there." If you assume the forces from the tire contact patches are applied at the intersection of the lines from tire contact patch to IC, then you can combine the forces from the left and right tires into one force without knowing how the latcenter plane of the vehicle establishes the roll cen- eral force is distributed between the tires. The force vecter R." tor from the outside tire adds to the force vector from the pivot point ... Its elevation where it crosses the Others use the intersection of the two lines. For a sym- inside tire giving a combined force with magnitude equal to the total lateral force. This allows you to define a metric car the lines intersect on the centerline of the car. height of the roll center. Dixon (2) quotes the SAE definition: TotalForce = p * G * Force + (1 - p) * G * Force = G * "The roll centre is defined in SAE J670c Vehicle Dynamics Terminology in the following way: 'TheForce irregardless of proportion p of force from outside tire [S.A.E.] roll-centre is the point in the transverse 2634 This content downloaded from 13.245.201.252 on Mon, 21 Mar 2022 13:03:03 UTC All use subject to https://about.jstor.org/terms THE LATERAL ROLL CENTER LOCATION For determining the jacking effect, it makes more sense to apply these forces at a point beneath the CG of the vehicle. Then the lateral force can be translated into a None of these definitions really deal with the lateral loca jacking force (or de-jacking force from the inside tire) tion of the roll center. However a belief has develope without creating a moment. But this construction that if the roll center does not move laterally when th requires a knowledge of how the lateral force is distribvehicle rolls then the car will be "stable". There is no par uted. In this case the two forces do not add; they subticular reason stated for this, but it seems to have prove tract. Designers often assume that most of the lateral out in many practical examples. In the next portion of th force comes from the outside tire and thus ignore the depaper we will describe the relationship between later of the roll center and the moment arm jacking effect from the inside tire. This assumption movement is questionable on high-banked tracks where the inside tire between RC height and CG. It turns out that if the ro may contribute a significant amount to the lateral force. center does not move laterally in roll then the momen arm remains constant. That is, if the vehicle moves dow Milliken and Milliken (4) have a good explanation of this in one mm (bump) then the RC height also moves dow their chapter on Suspension Geometry, which was writone mm and preserves the length of the moment arm. ten by Terry Satchell. This means the vehicle remains at a constant roll angle rather than trying to change roll angle. This gives a fee WHY A ROLL CENTER? ing of stability to the driver. this roll center remains fixed in the vehicle frame of refThe roll center is used in two types of analysis.IfThe height of the roll center relates to the amount of lateral erence then the forces acting through the suspension load transfer through the suspension links. It also reflects links to not attempt to change the roll angle of the vehicle. allows the car to "take a set" and gives a feeling of the amount of jacking force exerted on the chassisThis from lateral force. The higher the roll center the more jacking stability. force. The jacking force raises the chassis, which has the ROLL CENTER CONTROL detrimental effect of raising the center of gravity ACHIEVING of the vehicle and also decreasing the aerodynamic forces. This is an argument for having a roll center near, or even Designing a suspension with the desired roll center below, the ground. migration properties can be difficult. The roll center is the intersection of two lines which connect the contact pat Swing-axle cars are the classic example of this situawith the instant center, which is also an intersection o tion. It explains why swing-axles are not used on modern two geometric lines. This type of problem does not easi race cars. lead to a closed-form equation that can be solved. It The distance between the height of the roll center possibleand to develop analytical techniques that addre the height of the CG defines a moment arm.this Thequestion, longer but the easiest way to explain them is t the moment arm the more the car will try to look roll at inthe reacproblem from a different vantage point. tion to lateral forces. A tendency to roll may force the use Instead of using the road as a frame of reference a having the car move, consider the problem from t of heavier springs or anti-roll bars to restrict the roll. This is an argument for a roll center height near the CG. frame of reference of the vehicle. The body will rema These two conflicting consequences of roll center height stationary while the road moves under it. Or the body make it impossible to specify the best roll center. But stationary roll remains while the suspension moves up a center height does remain a useful tuning tool. down. Now the problem of finding a roll center that do not move laterally in roll and preserves the moment arm /K Figure 5. In the road frame of reference the read remains fixed while the chassis moves in ride and roll. 2635 This content downloaded from 13.245.201.252 on Mon, 21 Mar 2022 13:03:03 UTC All use subject to https://about.jstor.org/terms under ride changes becomes a problem of having a fixed area by designing a suspension that projects through a roll center. What we want is a suspension design that desired point in this frame of reference. keeps the line from instant center to contact point moving We are not guaranteed a solution, or a practical solution. through a specific point on the chassis. This point may be on the centerline of the vehicle or not. If the suspension from either side keeps the force vector line going through the desired point then we will have a stationary The desired suspension lengths may place the pickup points inside the engine block. In this case the location of the Instant Center may have to be moved to achieve a stable roll center. point. And the solution is really an instantaneous solution. It will be very accurate for small movements but the roll center will begin to migrate as the movements increase. But the RC envelope will still be smaller than with an uncontrolled suspension. This formulation has several nice properties. First of all, it produces a fixed target rather than a moving target. This is always easier to hit. Secondly, it splits the problem into two parts. We can design each side of the suspension to achieve the desired characteristics rather than designing two suspensions that interact in the desired way. It is always easier to solve two distinct linear equations with one unknown rather than one pair of equations This effect can work against designers converting production cars to race cars. A production car with a sixinch ride height may be designed with excellent roll cen- with two unknowns. ter control. If the car is to be raced it will often be lowered an inch or more to lower the center of gravity. It may be impossible to achieve roll center control at this ride height without changing pickup points, which may be against the The way to achieve roll center stability while maintaining the instant center (IC) location, and thus the first-order kinematic effects, is to change the lengths of the suspension arms. We can shorten or lengthen the arms without changing the IC location. This preserves our first-order ride-camber, roll-camber, and scrub-change effects. But the changes will effect roll center stability. You can even rules. Many designers of race cars are clearly using roll center control as a design criteria. I have seen Indy cars with roll centers that do not move even 0.1 inches with one degree of roll. maintain the length of one arm and change the other until the desired stability is achieved. This is a single-valued THE INCLINE RATIO function of one variable. The variable is the length of one arm. The result is the change in intersection height at the desired point. We desire a length which gives a change of zero. This formula can be solved many ways. It can be done graphically or with numerical techniques which The best technique for designing a suspension with roll center control is the incline ratio. This is the ratio of ride height change to change in the point where the line provide clever guesses of the next approximation. Once the proper lengths have been determined for each side, we can put the suspension together and return to the road as a frame of reference. You may quibble that a fixed point in one frame of reference is not fixed in the other frame. This is true, but we have devised a tech- nique that produces a roll center which is constrained within a limited area. We can even adapt the same technique to the road frame of reference and derive a suspension that will hold the roll center within a very limited between tire contact point and Instant Center crosses a reference line. This is based on the angle of the line from the tire contact patch to the Instant Center. Both kinematic and force-based definitions agree on the location of this Instant Center. This dimension could be expressed as an angle, but I find it easier to express it as a distance along a vertical line. The reference line may be the centerline of the vehicle, or a line through the CG, or a line through the roll center when the car is measured, or any other line. We seek an incline ratio of 1 .0, meaning each unit of change in ride height moves the incline point one unit. '|/ Figure moves '|/ 6. In the Ve up and down 2636 This content downloaded from 13.245.201.252 on Mon, 21 Mar 2022 13:03:03 UTC All use subject to https://about.jstor.org/terms M - The Incline Ratio can be calculated like the motion ratio. sponds, to the same amount of movement in the intersecThe easiest way to calculate the motion ratio is to contion point as in ride height. sider two ride heights perhaps 0.01 inch apart. Take the difference in spring movement and divide by 0.01 . This EXAMPLE gives a motion ratio calculated as a difference rather than a true derivative. The amount of the difference can be To demonstrate the method with an asymmetric suspe reduced to better approximate the true differential but sion, we take the previous suspension and raise the rig round-off errors become significant if we use too small a upper inner pickup point one inch. This creates an side difference. asymmetric suspension with the roll center located 10.363 inches to the left of the centerline and 1.775 The Incline Ratio is calculated based upon the same 0.01 inches above ground. Since we did not change the difference. We are seeking a ratio of 1 .0, which correside we can use 6.316 lower arm pickup point which us a stable symmetric roll center. '/ '/ Figure center Table 1 Lower Inner Incline - Roll Center Lateral - RC Height Pickup lateral Ratio 0.1 deg | 1.0 deg | 2.0 deg ãõÕ 1.3791 -0.274 -2.738 -5.431 1 .228 ÕÕÕ 1.3105 -0.224 -2.237 -4.435 1.296 8Í5Õ 1.2459 -0.177 -1.763 -3.494 1.361 ÉTÕÕ 1.1846 -0.132 -1.314 -2.603 1.422 7ÜÖ 1.1264 -0.089 -0.887 -1.757 1.480 7ÃÕ 1.0712 -0.048 -0.483 -0.954 1.535 Õ5Õ 1.0186 -0.010 -0.098 -0.189 1.587 &ÕÕ 0.9686 0.027 0.269 0.539 1.636 ÕlÕ 0.9784 0.020 0.197 0.396 1.627 Õ20 0.9883 0.013 0.125 0.252 1.617 Õ30 0.9885 0.005 0.051 0.106 1.607 ēāī 0.9994 0.004 0.044 0.091 1 .606 Õ32 1.0004 0.004 0.037 0.077 1.605 Õ33 1.0015 0.003 0.029 0.062 1.604 Õ34 1 .0025 0.002 0.022 0.047 1.603 Õ35 1.0035 0.002 0.014 0.033 1.602 036 1.0044 0.001 0.007 0.018 1.601 Õ37 1.0055 0.000 0.000 0.003 1 .600 Õ38 1.0064 -0.001 -0.008 -0.011 1.599 Õ39 1.0075 -0.001 -0.015 -0.026 1.598 Õ4Õ 1.0085 -0.002 -0.023 -0.041 1 .597 2637 This content downloaded from 13.245.201.252 on Mon, 21 Mar 2022 13:03:03 UTC All use subject to https://about.jstor.org/terms Table 2. Verification of Roll Center movement with Lower Inner Pickup Lateral of 6.31 6 Roll Angle RC Lateral RC Height Change in CG-RC Moment Arm ÕÕÕ ÕÕÕÕ 2^587 0Ī0 0.004 2.588 ÕÕÕÕ 0.001 0.20 0.008 2.588 0.001 0.50 0.020 2.589 0.002 TÕÕ ÕÕ4Õ 2J593 2.00 0.083 2.610 0.023 Ž689 0.102 3ÃÕ 0.133 0.006 Ride Height RC Height Change in CG-RC Moment Arm ^0 TĪ44 0.244 TÕ 066Ī ÕÕ74 -1.0 1.605 0.018 0.0 2.587 0.000 1.0 3.607 0.020 2X> 4.663 0.076 Tab Lower Pickup Inner Incline lateral Ratio ÕIÕ 1.3791 9JÕÕ 1.3105 ÍTÕÕ 1.2459 7XK) 1.1170 ãÕÕ 1.0049 ŠT0 0.9030 5^90 0.9943 5.95 0.9994 Table 4. Verification of Roll Pickup lateral at 5.95 Roll Angle RC Lateral RC Height Change in CG-RC Moment Arm ÕÕÕ -10.363 1.775 0.000 0Ī0 -10.359 1.775 0.000 Õ2Õ -10.355 1.775 0.000 Õ5Õ -10.338 1.776 0.001 TÕÕ -10.296 1.781 0.006 2M -10.155 ãÕÕ -9.935 1.800 0.025 1.834 0.059 Ride Height RC Lateral RC Height Change in moment arm 23.088 -0.436 ^5 -444.580 -8.114 TÕ -20.520 0.607 TŠ -13.800 1.216 ÕÕ -10.355 1.775 05 -8.259 2.318 Ū) -6.850 2.859 Ž0 -5.074 3.947 3X) -4.003 5.058 2638 This content downloaded from 13.245.201.252 on Mon, 21 Mar 2022 13:03:03 UTC All use subject to https://about.jstor.org/terms 0.789 -9.889 0.168 0.059 0.000 0.043 0.084 0.172 0.283 Table 4 shows that the roll center is contained within a fairly small area. For 3 degrees of roll the roll center moves 0.4 inches laterally and less than a tenth of an inch vertically. For ride height changes the roll center moves laterally a bit but the moment arm is constrained within 0.3 inches except for the area where one instant center moves below ground and the roll center is outside the track of the vehicle. This is a singularity of the way the roll center is constructed. When one IC is above ground and the other below ground the roll center must be outside of the track of the vehicle. When both instant centers move below ground then the roll center returns to a position between the tires. This suspension was chosen using a stability criteria that constrained the inclination of the force vector on the centerline of the vehicle. Had we chosen an Incline Ratio designed to optimize the asymmetric roll center location we could have reduced the roll center movement farther. CONCLUSIONS The incline ratio is a technique that can be added to suspension geometry software to better understand a double A-arm suspension. If you want to constrain the roll center within a limited area then the incline ratio allows the user to construct a suspension more quickly. It also lets you quickly understand how you need to move pickup points. It is far easier than the trial-and-error methods required previously. REFERENCES: 1. Dixon, J.C. 'The roll-centre concept in vehicle handling dynamics", Procedings of the Institute of Mechanical Engineers, volume 201 , number D1 , 1 987. 2. Dixon, J.C. Tvres. Suspension and Handling. Cambridge University Press, 1 991 . 3. Gillespie, Thomas D., Fundamentals of Vehicle Dynamics. SAE Publications Group, Warrendale, PA, 1 992. 4. Milliken, William F., and Millikem, Douglas R., Race Car Vehicle Dynamics. SAE Publications Group, Warrendale, PA, 1995. 2639 This content downloaded from 13.245.201.252 on Mon, 21 Mar 2022 13:03:03 UTC All use subject to https://about.jstor.org/terms