Contact patch, view from above:
2.3 Suspension System
Suspension can mean suspension of wheels (or axles), suspension of sub-frame and drivetrain and suspension of cabin (for heavy trucks). In this compendium, only wheel and axle suspension are
considered. Suspension design is explained in 2.3, but also used in complete vehicle models in 3.4.5- 3.4.7, 4.3.9, and 4.5.3.
Suspension influences road grip and ride comfort, so merely all vehicle motion, Chapters 3..5. The in- fluence is through how vertical forces and camber and steer angles on the wheels changes with body motion (heave, roll, pitch), road unevenness (bumps, potholes, waviness) and wheel forces in ground plane (from Propulsion, Braking and Steering subsystems). Important is also that suspension influences the material stresses (extreme values and fatigue), both in the suspension itself and other in other parts of the vehicle body. Figure 2-52 shows one way to see the suspension systems role.
(sprung) body
road surface
vertical displacement under each wheel
motion of sprung body above each wheel
WhlTorques, AxleSteAngles prop, brk, ste
systems (HW&SW)
forces on vehicle from each tyre to ground contact WhlRotSpeeds,
AxleSteForces driver, environment
(except road surface)
suspension (linkage, elasticities, dampers) wheel & tyre
Fx
Fy
Fz
Fx
Fy
Fz
Figure 2-52: Wheel/axle suspension described as modular sub-model per axle. It may be noted that both wheel model (main geometry such as wheel radius) and tyre model (how and 𝑦 vary with tyre slip and
) is a part of each wheel&tyre sub-model.
The simplest view we can have of a suspension system is that it is an individual suspension between the vehicle body and each wheel, consisting of one linear spring and one linear damper in parallel. Chapter 5 uses this simple view for analysis models, because it facilitates understanding and it is enough for a first order evaluation of the functions studied (comfort, road grip and fatigue load) dur- ing normal driving on normal roads.
The full 3D aspect of suspension is not covered here in 2.3. Instead, a division into 2D is done in 2.3.3 Suspension -- Heave and Pitch and 2.3.4 Suspension -- Heave and Roll, aiming at Longitudinal and Lat- eral dynamics, respectively. The full 3D aspects are briefly addressed in 4.5.3.1.5 and 5.7.2.2.4.
2.3.1 Components in Suspension
Each wheel can rotate in its hub. Each hub can be individually suspended to the body or left and right hub can be mounted on a rigid beam which is suspended to the body. The suspension parts are below grouped in: Linkage, Elasticities and Dampers. One might count in additional parts in the suspension, such as bearings, shafts, brake parts, etc.
2.3.1.1 Linkage
Linkage, which has the purpose to constrain the relative motion between wheel and body via kinemat- ics to one dof (approximately vertical translation), or, for a steered axle, also allow one more dof (ap- proximately yaw rotation) per axle. The linkage defines how longitudinal and lateral tyre forces are brought to the body (sprung mass).
The linkage consists of links (or members) and joints; mainly ball joints, but sometimes others, such as hinge joints. The coordinates (“hard-points”) of these joints are the real design parameters, but the dy- namic behaviour of a complete vehicle model can be expressed in much fewer parameters, namely the “effective pivot points”. These effective points are used in 2.3.3 and 2.3.4.
2.3.1.2 Elasticities or Compliances
Springs are examples of elasticities or compliances. The springs develop forces when the wheels are vertically displaced relative to the body. There is often one spring per wheel but also an anti-roll bar per axle. The anti-roll bar connects left and right wheel to each other to reduce body roll.
Springs often have a rather linear relation between the vertical displacement and force of each wheel, but there are exceptions:
• Anti-roll bars make two wheels dependent of each other (still linear). Anti-roll bars can be used on both individual wheel suspensions and rigid axle suspensions.
• The springs are intentionally designed to be non-linear in the compressed end of their stroke with bump stops. Bump stops at passenger cars are typically designed at (3.5. .4) vertical acceleration when vehicle is fully loaded. (A somewhat opposite non-linearity appears in the rebound end of the stroke, due to wheel lift. Here it is the contact force with ground that is sat- urated to zero, not the spring force. The difference is the damping force.)
• The compliances can be non-linear during the whole stroke, e.g. air-springs and leaf-springs. Air-springs are non-linear due to the nature of compressing gas, e.g. assuming ideal gas: 𝑝 𝑉 = 𝑛 𝑇; ⇒ 𝑜𝑟 𝑒 = 𝑛 𝑇 ( ⁄ 0− 𝐶𝑜 𝑝𝑟𝑒𝑠𝑠 𝑜𝑛);.
• The compliances can be controllable during operation of the vehicle. This can be to change the pre-load level to adjust for varying roads or varying weight of vehicle cargo or to be controlla- ble in a shorter time scale for compensating in each oscillation cycle. The latter is very energy consuming and no such “active suspension” is available on market.
The springs are the main compliance, but also other smaller compliances are present and makes the effective stiffness lower: the links themselves, the bushings in the joints between the links and the brackets where the links are connected to the body. For the complete vehicle model, the tyres vertical compliance adds to the suspension compliance.
2.3.1.3 Dampers
Dampers have the purpose to dissipate energy from any oscillations of the vertical displacement of the wheel relative to the body. The most common design is the hydraulic piston type. Dampers often has a rather linear relation between the vertical deformation speed and force of each wheel, but there are exceptions:
• The dampers are normally intentionally designed to be different in different deformation di- rection. Typical values are about 3 times more damping in rebound (extension) than compres- sion (bump). This can be motivated from that driving over a steep bump requires low damping to reduce upward jerk in vehicle, especially since there is a hard bump stop in the end of the spring stroke. In the other direction, driving over equally steep hole the downward jerk is lim- ited by that the wheel cannot develop pulling forces on the ground; instead it lifts from ground if hole is too steep. So larger damping can be allowed in rebound. A reflection here is that high damping damps oscillations, but high damping also increases the shock transmittance (with this reasoning, the name “shock-absorber” is misleading).
• Damping in leaf springs is non-linear since they work with dry friction.
• The dampers can be designed to be controllable during operation of the vehicle. This can be used to change the damping characteristics to adjust for varying roads or varying weight of ve- hicle cargo or to be controllable in a shorter time scale for compensating in each oscillation cy- cle. The latter is called “semi-active suspension” and is available on some high-end vehicles on market.
2.3.2 Axle and Wheel Rates
All compliances (springs, bushings, etc) contribute to the stiffness between the body and the wheels. The wheels are not independent of each other; we have especially a connection between left and right due to the anti-roll bars. Therefore, we define both axle (compliance) rate and wheel (compliance) rate, see Figure 2-53. We can also call these Effective stiffnesses of the axle and wheel, respectively.