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1. Suspension systems

1.1 Introduction to suspension design

Ride quality and handling are two of the most important issues related to vehiclerefinement. Together they produce some design conflicts that have to be resolved bycompromise. Also, the wide range of operating conditions experienced by a vehicleaffect both ride and handling and must be taken into account by the chassis designer.This all adds up to some very challenging tasks for the chassis designer.

While the chassis designer has a checklist of functional requirements for a givendesign, there are also a number of other constraints that have to be met. These includecost, weight and packaging space limitations, together with requirements forrobustness and reliability, ease of manufacture, assembly and maintenance.

Suspension design like other forms of vehicle design are affected by developmenttimes dictated by market forces. This means that for new vehicles, refined suspensionsneed to be designed quickly with a minimum of rig and vehicle testing prior to launch.Consequently, considerable emphasis is placed on computer-aided design. Thisrequires the use of sophisticated mathematical models and computer software thatenables a variety of “what-if” scenarios to be tested quickly and avoids the need for alot of prototype testing.

In order to understand the issues facing the suspension designer it is necessary to haveknowledge of: The requirements for steering, handling and stability The ride requirements related to the isolation of the vehicle body from road

irregularities and other sources of vibration and noise How tyre forces are generated as a result of braking, accelerating and cornering The needs for body attitude control Suspension loading and its influence on the size and strength of suspension

members.

This module addresses these issues. Further details can be found in a number oftextbooks, [1] to [6], with [1] being particularly appropriate for the course.

1.1.1 The role of a vehicle suspension

The principle requirements for a suspension are: To provide good ride and handling performance - this requires the suspension to

have vertical compliance to provide chassis isolation while ensuring that thewheels follow the road profile with very little tyre load fluctuation.

To ensure that steering control is maintained during manoeuvring, requiring thewheels to be maintained in the proper positional attitude with respect to the roadsurface.

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To ensure that the vehicle responds favourably to control forces produced by thetyres as a result of longitudinal braking and accelerating forces, lateral corneringforces and braking and accelerating torques - this requires the suspension geometryto be designed to resist squat, dive and roll of the vehicle body.

To provide isolation from high frequency vibration arising from tyre excitation -this requires appropriate isolation in the suspension joints to prevent thetransmission of “road noise” to the vehicle body.

To provide the structural strength to resist the loads imposed on the suspension.

It will be seen that these requirements are very difficult to achieve simultaneouslyparticularly when the added constraints of cost, packaging space, robustness and otherfactors are also taken into account. This leads to some design compromises that oftenresult in less than perfect performance for some of the desired outcomes.

1.1.2 Definitions and terminology

There is a lot of terminology associated with suspension design that may appear novelfor a student meeting the subject for the first time. Most of this will be described as itarises. The student can find a very useful summary of the most common vehicledynamics definitions in reference [2]. Some care is needed here however as there aresome differences between American and European terminology.

Figure 1.1 illustrates whole vehicle reference axes together with terms to describerotation about these axes. These axes have their origin at the centre of gravity of thevehicle. Numerous axis systems are used in vehicle dynamics analysis.

Figure 1.1 Whole-vehicle reference axes

Roll

Pitch

Yaw

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1.1.3 What is a vehicle suspension?

The isolation of a vehicle body from the road undulations that are fed into the tyre of avehicle at the road / wheel interface requires relative movement between the wheeland the body. This motion is, in general, controlled by some form of linkagemechanism. It is this mechanism that is termed a suspension.

1.1.4 Suspension classifications

In general suspensions can be broadly classified as dependent, independent or semi-dependent types.

With dependent suspensions the motion of a wheel on one side of the vehicle isdependent on the motion of its partner on the other side. That is when a wheel on oneside of the vehicle strikes a pothole, the effect of it is transmitted directly to its partneron the other side. Generally this has a detrimental effect on the ride and handling ofthe vehicle.

As a result of the trend to greater vehicle refinement, this type of suspension is nolonger common on passenger cars. However, they are still commonly used oncommercial and off-road vehicles. Their advantages are simple construction andalmost complete elimination of camber change with body roll (resulting in low tyrewear). They are also commonly used at the rear of front-wheel drive light commercialand off-road vehicles with rear driven axles (live axles) and on commercial and off-highway vehicles. They are occasionally used in conjunction with non-driven axles(dead axles) at the front of some commercial vehicles.

With independent suspensions the motion of wheel pairs is independent, so that adisturbance at one wheel is not directly transmitted to its partner. This leads to betterride and handling capabilities. This form of suspension usually has benefits inpackaging and gives greater design flexibility when compared to dependent systems.Some of the most common forms of front and rear designs will be considered later.Both McPherson struts; double wishbones and multi-link systems are employed forfront and rear wheel applications. Trailing arm, semi-trailing arm and swing axlesystems tend to be used predominantly for rear wheel applications.

There is also a group of suspensions that fall some way between dependent andindependent ones and are consequently called semi-dependent. With this form ofsuspension, the rigid connection between pairs of wheels is replaced by a compliantlink. This usually takes the form of a beam that can twist providing both positionalcontrol of the wheel carrier as well as compliance. Such systems tend to be simple inconstruction while having scope for design flexibility when used in conjunction withcompliant supporting bushes.

1.1.5 Defining wheel position

Since one of the most important functions of a suspension system is to control theposition of the road wheels, it is important to understand the definitions relating towheel location. The location of the wheels relative to both the road and the vehicle

suspension are important and will in general be affected by suspension deflection andtyre loading. In this sub-section we consider additionally the effect of wheel positionon handling behaviour.

Camber angle

This is the angle between the wheel plane and the vertical - taken to be positive whenthe wheel leans outwards from the vehicle (Figure 1.2).

Figure 1.2 Wheel position (viewed in x-direction, i.e. from the front of the vehicle)

Camber is affected by vehicle loading and by cornering. A slight positive camber(0.1) gives more even wear and low rolling resistance. However, on passenger carsthe setting is often negative (even when the vehicle is empty). Front axle values rangefrom 0 to -1 20’. This improves lateral tyre grip on bends and improves handlingbecause of camber thrust effects (these are a consequence of tyre characteristics).

A disadvantage of an independent suspension is that the wheels incline relative to thevehicle body on a bend (Figure 1.3). This tends to produce increased negative camberon the inner wheels and increased positive camber on the outer wheels. The outerwheels then try to roll towards the outside of the bend tending to produce anundersteer1 condition. Note: the change of camber (relative to the road) is acombination of changes due to suspension movement (rebound on the inside, bump onthe outside) and body roll.

Kingpin inclination (steered axles)

Sometimes called swivel pin inclination, KPI is the angle between the kingpin axisand the vertical (Figure 1.2). It has the effect of causing the vehicle to rise when thewheels are turned and produces a noticeable self-centring effect for swivel pininclinations greater than 15. When the wheels are turned, the magnitude of the self-aligning moment is dependent on the kingpin angle, the kingpin offset and the casterangle.

1 Understeer – relates to a handling stability condition – see sub-section 1.8.1

Kingpin ground offset, dg

Kingpininclination,

Kingpin hub offset, dh

Camberangle,

i

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Roll angle,

o i

Figure 1.3 Camber angles when cornering with an independent suspension

The effects of KPI are as follows: Tyre wear - induced camber changes can create wear particularly at high lock

angles Returnability2 - this is improved with increased values of KPI due to work done in

lifting the vehicle from the straight ahead to the turned position. Torque steer3 - this is related to the drive shaft angles. Lift-off / traction steer4 on bends - increasing KPI may result in an increased

vehicle reaction to mid-bend traction changes. Steering effort - KPI causes lift of the vehicle as the steering angle is increased.

This results in an increasing amount of work done at the steering system to achievea given wheel angle. However an increase of KPI from zero to 15 is unlikely toincrease the steering effort by more than 10%

Tyre clearance - changes in KPI affect wheel swept envelope and hence tyreclearance.

Typical kingpin angles for passenger cars are between 11 and 15.5.

Kingpin offset

This is the distance between the centre of the tyre contact patch and the intersection ofthe swivel pin axis at the ground plane (Figure 1.2) - taken to be positive when theintersection point is at the inner side of the wheel. For a given suspension, KPO canbe changed by changing tyre width. Increasing KPO improves the returnability of thesteering (see below). The disadvantage is that longitudinal forces at the tyre contactpatch due to braking, or striking a bump or pothole is transmitted through the steeringmechanism to the steering wheel.

2 Returnability – the tendency for the steering system to return to the “straight-ahead” position after an

induced steering change.3 Torque-steer – the tendency for a drive shaft to induce a change in the steering angle4 Lift-off / traction steer – the tendency for sudden accelerator change to produce a steer effect

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The effects of kingpin offset are thus: Returnability - positive increases of KPO will increase returnability due to the

increased lift with steering angle Brake stability (outboard brakes) - with a diagonal split braking system, negative

KPO will assist in counteracting the steer effect of a failed system. A similar effectoccurs with split-5braking where one wheel locks or with unbalanced frontbrakes. For braking on a bend, negative KPO produces toe-in on the outer wheelscreating a tendency to oversteer. (With inboard brakes the hub offset (see below) isthe critical dimension).

Torque steer - with a conventional non-biased differential ensuring equal torque toeach shaft, if one wheel loses traction, large values of KPO will cause steeringfight.

Steering effort - it is generally assumed that either positive or negative KPO willreduce static steering effort by allowing some rolling of the tyre on the road surfacereducing tyre scrub. In practice there appears to be little effect even with largevalues of KPO

In practice the KPO varies from small positive to small negative values.

Kingpin hub offsetThis is defined as the perpendicular distance from the king pin axis to the intersectionof the hub axis and the tyre centreline as shown in Figure 1.2. Hub offset is defined aspositive when the tyre centre line lies outside the king pin axis at hub centre height.

Castor angle Castor angle is the inclination of the swivel pin axis projected into the fore-aft

plane through the wheel centre - positive in the direction shown (Figure 1.4). Thecastor angle produces a self-aligning torque for non-driven wheels. It is dependenton suspension deflection and for steered wheels it influences the camber angle as afunction of steering angle

Castor angle,

Castor trail, lt

Kingpin axis

Castor offset, loff

Figure 1.4 Defining caster angle and mechanical trail

The effects of castor angle are as follows:

5 Split- – different coefficients of friction between tyres and road on the right / leftsides of the vehicle

Front ofVehicle

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Tyre wear - induced camber changes can create shoulder wear particularly at highlock angles.

Steer response - a positive castor angle will generate negative camber on a outsidewheel thus enhancing turn-in response.

Castor trailCastor trail is the longitudinal distance from the point of intersection of the swivel pinaxis and the ground to the centre of the tyre contact patch - Figure 1.4. It influencesthe magnitude of the self-righting moment. On some front wheel drive cars, there is anincreased self-righting moment during cornering due to the offset of the tractive forceand lateral force. This is undesirable in that the understeer effect is too great and thesteering is unduly influenced by rough road surfaces.

The effects of castor trail on vehicle behaviour are as follows: Straight line stability - improved with higher castor trail Returnability - stronger with higher castor trail Braking stability - castor angle will generally reduce with braking (due to hub

wind-up and vehicle pitch) resulting in reduced castor trail leading to a degradationin braking stability, i.e. braking stability is enhanced with higher levels of castortrail.

Steering effort - castor trail has very little effect on steering effort at stationary or atvery low speeds, but otherwise steering effort increases with increasing castor trail.Steering feel at high lateral accelerations (with impending loss of grip) is alsoinfluenced by castor trail.

Castor offsetCastor offset is the longitudinal distance from the vertical centre line through thewheel centre to the intersection of the longitudinal axis through the wheel centre andthe king pin axis. It is positive when in front of the wheel centre as shown in Figure1.4. It is possible to use a negative offset to reduce mechanical trail.

Toe-in / Toe-outThis is the difference between the front and rear distances separating the centre planeof a pair of wheels (quoted at static ride height and measured to the inner rims of thewheels). Toe-in is when the wheel centre planes converge towards the front of thevehicle as shown in Figure 1.5. Braking and rolling resistance forces tend to produce atoe-out effect, while tractive forces (front wheel drive vehicles) tend to produce theopposite effect. This leads to front wheels being set to toe-in for undriven and drivenfront axles. In the later case this is to ensure driving stability in those cases when thedriver suddenly takes his foot off the accelerator. With independent front suspensionsbody roll can produce changes of toe and hence roll steer6. This is discussed later.

6 Roll steer – a tendency for the wheels to steer as result of suspension articulation arising from bodyroll

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Figure 1.5 Toe-in definition (plane view)

1.1.6 Tyre loadsSuspension loads result from the set of forces and moments that are generated at thetyre / road interface during vehicle motion. These forces depend on the static loadsproduced by masses of the vehicle, occupants and payload, together with the dynamicforces arising from accelerating, braking, cornering, tyre rolling resistance andaerodynamic effects. These combined forces affect the handling ability of the vehicleand it is the chassis designer’s task to ensure that their effects are satisfactorilycontrolled.

The tyre contact patchDue to tyre compliance, the tyres deform to produce a contact area over which tyreload is distributed. This is called the tyre contact patch. For static loading conditionsthis is a symmetrical area as shown in Figure 1.6a. The centre of pressure in this caselies vertically below the axle centre. During motion of the vehicle, the tyre contactpatch becomes distorted depending on whether the vehicle is accelerating, braking,cornering etc. These actions produce a distributed set of normal and shear forces in thetyre contact patch.

Vertical forcesUnder static conditions, the load distribution in a vehicle determines the vertical loadsupported by each wheel. These forces are altered by a process called load transfer.During acceleration the vertical load increases on the rear wheels and decreases on thefront wheels. The reverse happens during braking. During cornering there are lateral(centrifugal force) forces exerted on the vehicle through its mass centres. Since theselie above the ground plane they increase the vertical load at the outer wheels anddecrease them at the inner wheel. Aerodynamic forces also affect the vertical tyreloads. They become significant for passenger car speeds above 80 km/h and aredependent on vehicle type and body design. Since the forces acting on a vehicle areabove the ground plane they produce moments that cause pitch and roll. These in turnproduce load transfer and hence affect tyre loading.

Front of vehicle

Toe is the differencein these dimensions

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a) Static loading b) Effect of cornering force

Figure 1.6 Tyre contact patch

Longitudinal tyre forcesLongitudinal tyre forces are produced by braking and accelerating, rolling resistanceand aerodynamic forces. They are affected by both wheel-slip and vertical loading.

Lateral force and slip angleAs a result of the cornering force described above there is also shear force produced inthe plane of the tyre contact patch. This produces distortion of the contact patchresulting in the direction of wheel motion differing from the wheel heading by anangle call the slip angle. Figure 1.6b shows this for the case of an unsteered wheel.This clearly this has implications for directional control and handling stability of avehicle. Aerodynamic forces and camber angle also affect lateral tyre forces.

Pneumatic trail and self-aligning momentAs a result of distortion of the tyre contact patch the resultant lateral tyre force Fy actsat a distance tp (termed the pneumatic trail), behind the vertical centre line through thewheel centre (Figure 1.6b). This produces a self-aligning moment, Mz, of magnitudetpFy about an axis normal to the road surface.

1.2 Vehicle suspensions

In this section we examine the factors that influence the selection of suspension types,discuss the kinematic requirements for suspensions and examine a range ofsuspension configurations.

Having selected a suspension mechanism for a given vehicle a number of issues needto be considered. These include how the wheel movement changes relative to both thevehicle body and the road surface over the range of suspension travel and whether the

Distortedcontact patch

Lateralforce

Wheelheading

Directionof travel

Slipangle

Symmetricalcontact patch

tp

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suspension members are able to cope with the variety of loads imposed on them. Ofparticular concern are changes in camber and toe angles together with changes in track(this is related to tyre scrub and hence tyre wear). The loading in some suspensionmechanisms produces a need for additional support members. In general these do notaffect the kinematic behaviour of the suspension mechanism.

Initially, we will assume that suspension mechanisms undergo rigid body motion, withno compliance at the joints between members. However, most modern suspensions doincorporate some compliance at certain joints to facilitate subtle wheel / bodymovements such as compliance steer to enhance handling performance. Also we willassume initially that the motion of the wheel relative to the body is 2-dimensional, i.e.occurs in a plane perpendicular to the longitudinal axis of the vehicle. In reality,because of additional design requirements, e.g. pitch control of the vehicle body, thewheel motion needs to be 3-dimensional.

The complexity of the issues facing the chassis designer can be appreciated when onealso considers how various operational factors affect chassis performance. These arediscussed in [1].

1.2.1 Factors influencing suspension selection

Choice of suspension is primarily determined by the engine-drive combination, whichis in turn determined by the following: The functional requirements of the vehicle (use, performance capabilities and cargo

space). The need to place a high proportion of the vehicle mass over the driven axles to aid

traction.

The most common combinations are outlined as follows. The advantages anddisadvantages of these configurations are more fully discussed in [1].

Front mounted engine and rear wheel drive

In this case there is not the same restriction on engine length as there is with the frontengine, front wheel drive combination. For this reason it is adopted for more powerfulpassenger cars and estate cars. Under full load most of the vehicle mass is on thedriven axle. However, when lightly loaded, e.g. in the 2-up condition, it leads to poortraction on wet and wintry roads. This problem can be overcome with the use oftraction control.

With this configuration, the implications for the choice of a rear suspension are that ithas to accommodate the differential and drive shafts while still providing anacceptable boot space.

Front mounted engine and front-wheel drive

In this case the engine and transmission form one unit which sits in front of, over, orjust behind the front axle. This leads to a very compact arrangement compared to theprevious case and this is no doubt responsible for its adoption on small to medium

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sized saloons (up to 2-litre capacity). The main advantages include a significant loadon the driven and steered wheels for normal vehicle loads providing good traction androad holding on wet and icy roads. With a fully laden vehicle this advantage tends tobe lost.

The implications for front suspension design are that provision must be made for driveshafts, and steering. In general the packaging space for the suspension is limited by thesize of the engine-transmission unit. This engine-drive configuration favoursMcPherson strut and double wishbone types of suspensions. A variety of rearsuspensions are possible for this drive configuration. The type of vehicle and thedegree of refinement required usually influence the choice.

Four wheel drive

In this case it is possible to have all the wheels driven continuously or one of the pairof wheels is always connected to the engine while the other pair are selected manuallyor automatically. The aim is of course to improve traction for all road conditionsespecially wet and wintry conditions. All wheel drive is particularly advantageous foroff-road conditions, and to improve climbing ability regardless of loading conditions.

From the point of view of suspension selection, the requirements for the frontsuspension is similar to that on a vehicle with front mounted engine and front wheeldrive. The rear suspension requirement is similar to that for a rear suspension on avehicle with a front mounted engine and rear wheel drive.

Suspensions have been broadly categorised in section 1.1.4 as dependent, independentor semi-dependent. Solid axles have been categorised as dependent systems, but otherconnections across the vehicle, such as anti-roll bars, are not classed as dependent,because although they influence suspension forces, they do not provide a geometricconstraint.

Other factors

The mass of the components associated with the wheel hub, brake components(outboard brakes), and parts of the other masses connected to the wheel hub, e.g.suspension links, suspension spring and damper, create what is know as the unsprungmass. This produces an additional resonant frequency when the vehicle system issubjected to road induced vibration (a 2 degree of freedom system). The effect is toincrease the dynamic tyre load and cause degradation in ride. The aim should thereforebe to keep the unsprung mass to a minimum (reduce the system to 1 degree offreedom – the vehicle body).

Another factor to be considered is roll centre (and hence roll axis) location. Thisaffects such things as the amount of body roll in cornering, the amount of lateral loadtransfer and the loads in the suspension links. It will be discussed in detail later. There

is a roll centre in the vertical transverse plane passing through each axle. Suspensiontype and geometry determine its location. Some suspension types allow much greaterflexibility in the choice of roll centre location.

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1.2.2 Kinematic requirements for suspensions

The main objective in suspension design is to isolate the vehicle body from roadsurface undulations. Ideally, this requires that the wheels of the vehicle perfectlyfollow the road surface undulations with no vertical movement of the vehicle body. Inanalysing this scenario, it is possible to assume the vehicle body is stationary while weconsider the motion of the wheels.

The motion requirements of the wheels relative to the vehicle body can be satisfied ina number of different ways. If each wheel can move independently of its partner onthe other side of the vehicle (i.e. an independent suspension, as defined in section1.1.4), this can be achieved kinematically as shown in Figure 1.7. In the first example,the wheel moves vertically relative to the vehicle body via a sliding contact. In thesecond example the required vertical motion is accompanied by rotation about a pivotattached to the body, while in the third example the wheel and hub are attached to oneof the links in a 4-bar mechanism. This provides the required vertical wheelmovement accompanied by a small rotation.

All of these examples have some practical disadvantages. These will be discussedlater when we have a look at some applications. In an independent suspension themechanism coupling the wheel to the body of the vehicle is said to have a singledegree of freedom, i.e. the motion of the wheel relative the body can be described by asingle coordinate.

(a) (b) (c)Figure 1.7 Examples of possible independent suspension mechanisms

If it is necessary to couple a pair of wheels together on a single axle (as in a dependentsuspension), the mechanism required must have two degrees of freedom to allowvertical movement of each wheel while at the same time having a rigid connectionbetween the two wheels. An example of how this can be achieved is shown in Figure1.8.

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Figure 1.8 A possible dependent suspension mechanism

1.2.3 Examples of dependent types of suspensions

These may be categorised into three sub-groups: Drive axles – sometimes these are called “live” axles – they carry the driven

wheels and differential. Non-driven axles usually called “dead” axles - in which a solid beam simply

connects the two undriven wheels which may be steerable. De Dion axles - which connect the two driven wheels but do not carry the

differential and drive shafts.

Solid axles are used rarely in passenger vehicle design. They are however almostuniversally used for trucks and commercial vehicles and for many off-highwayvehicles because of their simplicity (relative to independent systems). They virtuallyeliminate wheel camber (except for the small amount arising from different tyredeflections on the inner and outer wheels) and wheel alignment is easy to maintain,providing good tyre wear properties.

Many methods of mounting solid axles have been and still are used in vehicle design.Only the main types and associated principles are covered here. Further descriptionsof suspensions are covered in [1], [4] and [5].

Hotchkiss driveThe most familiar form of mounting a solid drive axle is on a pair of semi-elliptic leafsprings; this is known as the Hotchkiss arrangement (Figure 1.9). It is a remarkablysimple suspension and produces satisfactory location of the wheels with the minimumnumber of components. Its simplicity derives from the leaf spring properties, i.e.compliant in the vertical direction but relatively stiff laterally and longitudinally.

Developments in leaf springs over the years have overcome problems with inter-leaffriction. Single and taper leaf designs now tend to be used. However, for the lowspring rates required for good ride performance, single leaf springs are generallyinadequate to resist lateral loads and driving and braking torques on modern vehicles.In such cases, additional strengthening links such as Panhard rods are used.

The detailed design of the leaf spring itself and the associated shackle mountinggeometry (Figure 1.10) which accommodates changes in effective spring length is not

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straightforward. Design guidelines are published in the SAE Spring Design Manual[7].

Four-link arrangements

The requirements for the axle to have two degrees of freedom and also withstand alltypes of loading can be satisfied by a number of four-link suspension mechanisms.

In the mechanism shown in Figure 1.11, two upper and two lower links - usuallytrailing (but not always), react the driving / braking torques while providing therequired vertical and roll degrees of freedom. Lateral control may be provided byangling the upper links (as shown in Figure 1.11), triangulating the upper links, or byproviding an additional Panhard rod. Any springing medium can be used, but coil orair springs are the most common. The advantages of this suspension design over theHotchkiss arrangement are greater design flexibility in arranging the roll centre, theanti-dive/squat geometry and controlling roll steer.

Figure 1.9 Hotchkiss suspension system

Figure 1.10 Leaf spring kinematics

Bump stopLeaf spring

Shackle

C

Leaf spring with shackle

Equivalent linkage torepresent leaf spring

C

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De Dion system

In this suspension design the drive is separated from the solid axle (Figure 1.12).Hence, the differential is chassis mounted and the simple axle is typically located byone of many four-link variations. Its main advantage is that the unsprung mass isreduced dramatically compared with a live axle. However, it does not have significantcost or performance advantages over an independent system and hence is generally notused these days.

1.2.4 Examples of independent front suspensions

Almost all passenger cars, and increasing numbers of light trucks, use independentsuspensions at the front. The gradual shift towards independent suspensions hasresulted from their benefits in packaging, (i.e. with front wheel drive vehicles they arebetter able to fit into the limited space available around the engine- transmissionsystem). They also permit greater suspension design flexibility and overcomeproblems

Figure 1.11 Four link suspension

Figure 1.12 De Dion suspension system

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of steering shimmy7 vibrations often associated with beam axle designs. Olley, one ofthe early influential automotive engineers, is credited with proposing some of the firstdesigns for independent suspensions. His paper [9] makes interesting reading eventoday. Over the years a bewildering number of designs have been proposed and thedescriptions here are restricted to the generic types.

McPherson strut suspensions

Great simplicity is the main benefit of this design with the wheel controlled by a lowerarm in combination with a sliding strut (Figure 1.13). The configuration shown in thediagram is well able to withstand the longitudinal and lateral loads that will arise forvarious operating conditions. Advantages of the design are good packaging andsimplicity, whilst its disadvantages are a high installation height (which may conflictwith the bonnet line) and wheel loads that produce a moment which must be reactedby the strut. This produces friction problems in the strut. Angling the axes of thespring and strut can reduce this problem.

Figure 1.13 McPherson strut suspension

Wishbone suspensions

Upper and lower wishbones - or arms as they are called in the United States - arecombined to form a classic four-bar linkage mechanism when viewed from the front.(Figures 1.7c and 1.14) The wishbones are nearly always unequal in length; the upperarm is invariably shorter than the lower one to meet the space limitations in front-engined vehicles.

7 Steering shimmy – a coupled bounce and steer oscillation found on some earlyvehicles

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1.2.5 Examples of independent rear suspensions

Differences in rear suspension design are dictated by whether the axle is driven orundriven. In the case of rear-wheel drive, one must consider the packaging space(affect by both differential and a need to minimise intrusion into the boot space) andalso torque-steer effects. This later point is particularly relevant for larger, morepowerful, saloon cars.

Trailing arm

A simple single trailing arm (Figure 1.15) can be used with a torsion bar spring (e.g.as in the Renault 5) or with a rubber or hydro-elastic one (e.g. the early design of theMini). Again, the wheel camber is the same as the body roll angle.

Swing axle

This is also a simple way of achieving an independent rear suspension (Figure 1.16) -made famous on the VW Beetle design. The length of the swing axle controls thecamber behaviour and this has to be relatively short to accommodate the differential.Camber changes (and consequently scrub) are significant during normal wheel travel.

Figure 1.14 Wishbone suspension

Figure 1.15 Rear trailing arm suspension

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Also this design is particularly prone to the problem of jacking8 (vertical translation ofthe vehicle) and is little used nowadays.

Semi-trailing arm systems

This design is a cross between a pure trailing arm and a swing axle (Figure 1.17). Theadditional flexibility in the design allows a compromise in the control of camber andjacking. However care must be taken with the geometry to control the amount of steerwhich results from the inclination of the pivot axis. More recent variants of this typeof design try to exploit small amounts of rear steering to improve overall vehiclehandling qualities (e.g. Vauxhall Omega).

Multilink designs

Multi-link suspensions are now a common feature on passenger cars from mid-rangemodels upward. They are usually based on three, four, or five links per wheel stationand are designed to give more complete control over wheel positioning [10]. Thiscomes at a price due to the added number of components, and the extra effort requiredin setting them up. The Mercedes Benz design (Figure 1.18) is typical of such designs.Because of the number of links (5 in this case), the suspension mechanism is over-

8 Jacking –in certain suspensions, lateral tyre forces produce progressive trackchanges and body lift leading to instability

Figure 1.16 Swing axle suspension

Figure 1.17 Semi-trailing arm suspension

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constrained kinematically. Suspension movement is only possible, because ofcompliant suspension bushes at the ends of some of the links.

Figure 1.18 Mercedes Benz five link suspension

1.2.6 Semi-independent rear suspensionsThis arrangement has become common at the rear of small front-wheel drive vehiclesdue to its relative simplicity and low cost. It is a cross between a fully independenttrailing arm arrangement and a dependent beam axle arrangement. The axle isreplaced by a crossbeam that is allowed to twist by an amount controlled by thegeometry of the suspension and the structural properties of all the members.

An example is shown in Figure 1.19. Such designs are known by various names, e.g.trailing twist-axles and semi-dependent suspensions. The cross member is effectivelyrigid in bending but has torsional compliance and this performs a function similar tothat of an anti- roll bar.

Figure 1.19 Trailing twist axle

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The support bushes can be angled and designed to have different stiffnesses along andperpendicular to their principal axes (Figure 1.20). This can produce a shear centrebehind the wheel axis to give a compliance steer in the correct direction consistentwith understeer.

Figure 1.20 Trailing twist axle with lateral force understeer

The shape of the open section cross-member can be varied to alter the position of itsshear centre (a feature of open sectioned members in torsion). This permits someflexibility in the choice of roll centre location. This will be discussed later.

1.3 Suspension components

Suspension components, particularly springs and dampers, have a profound effect onride and handling performance. In addition to the constraints imposed by suspensionperformance requirements, component designers have a range of other constraints toconsider. These include cost, packaging, durability and maintenance. Because of thehostile environment in which suspension components operate and the high fluctuatingloads (and hence stresses) involved, fatigue life is another of the designer’s primeconcerns.

1.3.1 Springs - types and characteristics

Suspension systems require a variety of compliances to ensure good ride, handling andNVH performance. The need for compliance between the unsprung and sprung massesto provide good vibration isolation has long been recognised. In essence, a suspensionspring fitted between the wheel and body of a vehicle allows the wheel to move upand down with the road surface undulations without causing similar movements of thebody. For good isolation of the body (and hence good ride), the springs should be assoft as possible consistent with providing uniform tyre loading to ensure satisfactoryhandling performance.

The relatively soft springing required for ride requirements is normally inadequate forresisting body roll in cornering, therefore it is usual for a suspension system to alsoinclude additional roll stiffening in the form of anti-roll bars. Furthermore, there is the

Angled bushAngled bush

Lateral tyreforce

Lateral tyreforceShear centre

of bushes

Directionof turn

Axlerotation

1-21

possibility of the suspension hitting its stops at the limits of its travel as a result ofabnormal ground inputs (e.g. as a result of striking a pothole). It is then necessary toensure that the minimum of shock loading is transmitted to the sprung mass. Thisrequires the use of additional springs in the form of bump stops to decelerate thesuspension at its limits of travel.

Finally, there is also a requirement to prevent the transmission of high frequencyvibration (>20 Hz) from the road surface, via the suspension to connection points onthe chassis. This is achieved by using rubber bush connections between suspensionmembers.

The following compliant elements are thus required in suspension systems:suspension springs, anti-roll bars, bump-stops and rubber bushes. In this section ourattention is concentrated on suspension springs and anti-roll bars. Further details ofsuspension components can be found in [3] to [5], [11] and [12].

Suspension springs

The main types of suspension spring are: Steel springs (leaf springs, coil springs and torsion bars) Hydropneumatic springs

a) Leaf springs

Sometimes called semi-elliptic springs, these have been used since the earliestdevelopments in motor vehicle technology. They rely on beam bending principles toprovide their compliance and are a simple and robust form of suspension spring stillwidely used in heavy commercial applications, e.g. lorries and vans. In somesuspensions (e.g. the Hotchkiss type) they are used to provide both verticalcompliance and lateral constraint for the wheel travel. Size and weight are amongtheir disadvantages.

Leaf springs can be of single or multi-leaf construction (Figure 1.21). In the latter case(Figure 1.21 (b)) interleaf friction (which can affect their performance) can be reducedwith the use of interleaf plastic inserts. Rebound clips are used to bind the leavestogether for rebound motion. The swinging shackle accommodates the change inlength of the spring produced by bump loading. The main leaf of the spring is formedat each end into an eye shape and attached to the sprung mass via rubber bushes.Suspension travel is limited by a rubber bump stop attached to the central reboundclip. Structurally, leaf springs are designed to produce constant stress along theirlength when loaded.

Spring loading can be determined by considering the forces (see section 1.5) acting onthe spring and shackle as a result of wheel loading (Figure 1.22). The spring is a threeforce member with FA, FW and FC at A, B and C respectively. The wheel load FW, isvertical and the direction of FC is parallel to the shackle (a two-force member). Thedirection of FA must pass through the intersection of the forces FW and FC (point P) forthe link to be in equilibrium. Knowing the magnitude of the wheel load enables theother two forces to be determined. The number, length, width and thickness of the

1-22

leaves determine the stiffness (rate) of the spring. See [7] and [13] for stiffness andstress formulae. Angling of the shackle link can be used to give a variable rate [7].When the angle < 90 (Figure 1.22), the spring rate will increase (i.e. have a risingrate) with bump loading.

(a) Single leaf type (b) Multi-leaf type

Figure 1.21 Examples of leaf spring designs

(a) Wheel load (b) Force on the spring member

Figure 1.22 Leaf spring loading

b) Coil springs

This type of spring provides a light and compact form of compliance that is animportant feature in terms of weight and packaging constraints. It requires littlemaintenance and provides the opportunity for co-axial mounting with a damper. Itsdisadvantages are that because of low levels of structural damping, there is apossibility of surging (resonance along the length of coils) and the spring as a wholedoes not provide any lateral support for guiding the wheel motion.

Most suspension coil springs are of the open coil variety. This means that the coilcross-sections are subjected to a combination of torsion, bending and shear loads.Spring rate is related to the wire and coil diameters, the number of coils and the shearmodulus of the spring material. Cylindrical springs with a uniform pitch produce alinear rate. Variable rate springs are produced either by varying the coil diameterand/or pitch of the coils along its length. In the case of variable pitch springs, the coils

FC

FW

FA

FW

P

A CB

Bump stop

Shackle

Single taperedleaf spring

Bump stop

Shackle

Multi-leafspring

1-23

are designed to “bottom-out” as the spring is loaded, thereby increasing stiffness withload.

Coil spring design is well covered in the literature. In addition to Wahl’s classic text[14], the reader is also recommended to consult [7] and chapters in, [14] and [15].

c) Torsion bars

These are a very simple types of springs and consequently very cheap to manufacture.They are both wear and maintenance free. Despite their simplicity they cannot easilybe adopted for some of the more popular forms of suspension.

The principle of operation (Figure 1.23) is to convert the applied load FW into a torqueFW x R producing twist in the bar. A circular cross-section bar gives the lowest springweight for a given stiffness. In this case simple torsion of shafts theory can be used todetermine the stiffness of the spring and the stresses in it. As the lever-arm rotatesunder load, the moment-arm changes, requiring twist angle corrections (for largerotations) to be made in design calculations [14] and [7]. In general, stiffness is relatedto diameter and length of the torsion bar and the torsion modulus of the material. Abending moment FW x L is induced in the twist section of the member and supportsshould be positioned to minimise this.

Figure 1.23 Principle of operation of a torsion bar spring

d) Hydropneumatic springs

In this case the spring is produced by a constant mass of gas (typically nitrogen) in avariable volume enclosure. The principle of operation of a basic diaphragmaccumulator spring is shown in Figure 1.24. As the wheel deflects in bump, the pistonmoves upwards transmitting the motion to the fluid and compressing the gas via theflexible diaphragm. The gas pressure increases as its volume decreases to produce ahardening spring characteristic.

The principle was exploited in the Moulton-Dunlop hydrogas suspension wheredamping was incorporated in the hydropneumatic units. Front and rear units wereconnected to give pitch control. See [11] for a detailed description of the system.

A further development of the principle has been the system developed by Citroen.This incorporates a hydraulic pump to supply pressurised fluid to four hydropneumaticstruts (one at each wheel-station). Height correction of the vehicle body is achieved

LTwist section

Fixing plane

RFW

1-24

with regulator valves that are adjusted by roll-bar movement or manual adjustment bythe driver. The system and its operation can be found in [11].

In general, hydropneumatic systems are complex (and expensive) and maintenancecan also be a problem in the long term. Their cost can, however, be offset by goodperformance. Patents cover the two systems discussed in the preceding paragraphs, butthere is still scope for development of alternative hydropneumatic systems. Some ofthese are incorporated into controllable suspensions.

Figure 1.24 Principles of a hydropneumatic suspension spring

Anti-roll bars (roll stabilisers)These are used to reduce body roll and have an influence on a vehicle’s corneringcharacteristics (in terms of understeer and oversteer). Figure 1.25a shows how atypical roll bar is connected to a pair of wheels. The ends of the U-shaped bar areconnected to the wheel supports and the central length of the bar is attached to thebody of the vehicle. Attachment points need to be selected to ensure that bar issubjected to torsional loading without bending.

If one of the wheels is lifted relative to the other, half the total anti-roll stiffness actsdownwards on the wheel and the reaction on the vehicle body tends to resist body roll.If both wheels are lifted by the same amount, the bar is not twisted and there is notransfer of load to the vehicle body. If the displacements of the wheels are mutuallyopposed (one wheel up and the other down by the same amount), the full effect of theanti-roll stiffness is produced.

(a) Anti-roll bar layout (b) Roll bar contribution to total roll moment

Figure 1.25 Anti-roll bar geometry and the effect on roll stiffness

Gas Spherical container

LiquidDiaphragm

Piston

Wheel load

RollMoment

Bump stopeffect

Anti-roll bar

Roll bar

Roll angle

Chassis fixingpoints

Total rollmoment

Suspension

1-25

Roll stiffness is equal to the rate of change of roll moment with roll angle. Figure1.25(b) shows the contributions to the total roll moment made by suspension stiffnessand anti-roll bar stiffness.

1.3.2 Dampers - types and characteristics [8]

Frequently called shock absorbers, dampers are the main energy dissipaters in avehicle suspension. They are required to dampen vibration after a wheel strikes a pot-hole and to provide a good compromise between low sprung mass acceleration(related to ride) and adequate control of the unsprung mass to provide good roadholding.

Suspension dampers are telescopic devices containing hydraulic fluid. They areconnected between the sprung and unsprung masses and produce a damping force thatis related to the relative velocity across their ends. The features of the two mostcommon types of damper are shown in Figure 1.26. Figure 1.26(a) shows a dual tubedamper in which the inner tube is the working cylinder while the outer cylinder is usedas a fluid reservoir. The latter is necessary to store the surplus fluid that results from adifference in volumes on either side of the piston. This is a result of the variable rodvolume in the inner tube. In the monotube damper (Figure 1.26(b)) the surplus fluid isaccommodated by a gas-pressurised free piston. An alternative form of monotubedamper (not shown in Figure 1.26) uses a gas/liquid mixture as the working fluid toabsorb the volume differences. Comparing the two types of damper shown in Figure1.26, the dual tube design offers better protection against stones thrown up by thewheels and is also a shorter unit making it easier to package. On the other hand, themonotube strut dissipates heat more readily.

(a) Dual tube damper (b) Free-piston monotube damperFigure 1.26 Damper types

In dealing with road surface undulations in the bump direction (the damper incompression) relatively low levels of damping are required when compared with therebound motion (the damper extended). This is because the damping force produced

Rod seal

Gas

Piston rod

Piston andvalves

Floatingpiston

ValvesGas

1-26

in bump tends to aid the acceleration of the sprung mass, while in rebound anincreased level of damping is required to dissipate the energy stored in the suspensionspring. These requirements lead to damper characteristics that are asymmetrical whenplotted on force-velocity axes. The damping rate (or coefficient) is the slope of thecharacteristic. Ratios of 3:1 for rebound to bump are quite common (Figure 1.27).However, one manufacturer has discarded this reasoning and used linear dampers onone of its vehicles with some success [16].

Figure 1.27 Non-linear damper characteristics

The characteristics called for in damper designs are achieved by a combination oforifice flow and flows through spring-loaded one-way valves. These provide a lot ofscope for shaping and fine-tuning of damper characteristics9. Figure 1.28 illustrateshow the force-velocity characteristics can be shaped by a combined use of dampervalves. At low relative velocities damping is by orifice control until the fluid pressureis sufficient to open the pre-loaded flow-control valves. Hence the shape of thecombined characteristic.

Figure 1.28 Shaping of damper characteristics

A driver operated adjustment mechanism can be used to obtain several dampingcharacteristics from one unit. Typical curves for a three position adjustable damper areshown in Figure 1.29. A continuous electronically controlled adjustment forms thebasis of one type of controllable suspension designed to improve both ride andhandling.

9 For a more detailed account of damper operation and characteristics see [8]

Relative velocity

Valve control

Orifice control

Dampingforce

Dampingforce

Relative velocity(bounce)

1-27

Figure 1.29 Different operating modes for adjustable dampers

1.4 The kinematics of suspensions

Kinematics is the study of the geometry of motion, without reference to the forcesinvolved. In the case of vehicle suspensions, which are simply examples of the manylinkages and mechanisms encountered in mechanical engineering, study of theirkinematic properties involve some important assumptions in relation to their practicaloperation. For example, elastic deformations are excluded from the analysis (sinceforces are excluded). This means that bush compliances cannot be included. Hence thesuspension must be treated as if all the link connections are perfect pivots, sliders orball joints.

A kinematic analysis of a proposed suspension is invariably the first analysisperformed by the designer in order to: Check that the mechanism actually works over the full bump/rebound range and

does not interfere with the body Check the way in which the wheel geometry is controlled over the suspension

working range Calculate the effective ratios involved between spring and damper travel relative

to wheel travel.

1.4.1 Kinematic analysis of suspension mechanisms

Suspensions are really three-dimensional mechanisms so that analyses of them shouldbe three-dimensional. However, the largest displacement components occur in atransverse plane perpendicular to the longitudinal axis of the vehicle. It follows that alot of useful information can be obtained from a two-dimensional analysis.

There are three ways in which suspension kinematics can be analysed: Analytically using vector-based methods Using a computer package with a mechanisms analysis capability Graphically (only really relevant for a two-dimensional analysis).

It is the third approach that we are going to use in the following example. Readersunfamiliar with graphical kinematic analysis should consult an undergraduate textsuch as [17].

Dampingforce

Relative velocity

Sport

Standard

Comfort

1-28

Example 1.1

A scaled drawing of a double wishbone suspension is shown diagrammatically inFigure E 1.1

The damper is mounted between E and F and E can be assumed to be on the line AD.The centre of the tyre contact patch is at W and WV is the centre-line of the wheel.

Use a velocity diagram to determine:

a) The scrub to bump derivative at the tyre-road interface (ratio of wheel horizontalto vertical velocities)

b) The damper rate to bump rate (ratio of relative velocity across the damper to thevertical velocity of the wheel)

c) The suspension ratio (the inverse of the damper rate to bump rate)

Figure E1.1

SolutionBegin by drawing the mechanism to scale as shown in Figure E1.2 and assume thechassis fixing points to be stationary. Apply a unit angular velocity of =1 rad/s tolink AD (the absolute magnitude is not important as the things to be determined are allratios). The magnitude of the velocity of D is vD = 360 x 1 = 360 mm/s and the vectorvD is perpendicular to AD as shown.The velocity of C is given by: vC = vD + vCD (a vector addition, where a symbolunderlined denotes a vector). vC is perpendicular to BC – its magnitude is unknown atthis stage. Because CD can be considered as a rigid link, vCD is perpendicular to CD –its magnitude is unknown at this stage. The directions of these vectors are alsounknown (see dotted lines in Figure E1.2). Also draw a line from D to W – it will beassumed this imaginary link is rigid and form part of a link CDW.

A

ED

B

CF

V

W Scale 1 : 6

Dimensions in mm

AE = ED 180CD = 315WD = 159

1-29

Construction of the velocity diagram (Figure E1.3) begins by first locating the pole Ov

Vectors drawn from here are “absolute” velocities.

1. Draw vD to some scale and locate D at its end.2. Draw lines representing the directions of vC and vCD. Note that vC is an absolute

velocity and hence is drawn from Ov

3. Locate C at their intersection and add arrows to satisfy the vector equation above.The magnitudes of vC and vCD can now be determined by scaling the diagram.

4. There should be a velocity image CDW in the velocity diagram of similar shape toCDW in Figure E1.1. Determine W by proportion from FigureE1.1 and hence addthe vector vWD as shown in Figure E1.3

vD

A

ED

B

CF

V

W

Figure E1.2

Velocity diagram(to some scale)

Figure E1.3

vWD

vWvert

Ov

vD

D

vC

C

vCD

vWhor

vE

vEr

vEt

W

E

1-30

5. The (absolute) vertical and horizontal velocity components of W, vWhor and vW

vert

can be added to the diagram as shown in Figure E1.3

6. The (absolute) velocity of E, vE, is half that of D and can now be added to thediagram. There are components parallel and perpendicular to EF which are relatedrespectively to relative velocity across the damper and its angular velocity. Addthe component vectors vE

r and vEt.

7. Relevant values can now be scaled from the diagram. These are: vWvert = 336

mm/s; vWhor = 114 mm/s and vE

r = 168 mm/s

Answers

a) Scrub to bump derivative:

vWhor / vW

vert = 114 / 366 = 0.393

b) Damper rate : bump rate

vEr / vW

vert = 168 / 366 = 0.459

c) Suspension ratio

vWvert / vE

r = 366 / 168 = 2.18

Bump to scrub ratio is related track change and tyre wear. Damper rate to bump rateand suspension ratio are both related to modelling for ride analysis and thedetermination of component parameters.

The facility to perform kinematic analysis of three-dimensional mechanisms isavailable on many CAD systems. Alternatively, independent packages have beenwritten specifically to analyse three-dimensional suspension kinematics, for example,the one embodied in VDAS, [18]. The industry standard is ADAMS [19].

Some typical output from the VDAS package for a three-dimensional analysis ofdouble wishbone suspension is shown in Figure 1.30. Note that the results have beencalculated over the full range of suspension travel and are for a particular steeringangle. The package also permits the evaluation of driveshaft data such as plunge atCV joints and CV joint angles.

1.4.2 Roll centres and roll axis [20]

Roll centre and roll axis concepts are important aids for studying vehicle handling andare also used in the calculation of lateral load transfer for cornering operations.

There are two definitions of roll centre. One is based on forces and the other onkinematics. The first of these (the SAE definition) states that: the roll centre is a pointin the transverse plane through any pair of wheels at which a transverse force may beapplied to the sprung mass without causing it to roll. The second is a kinematic

1-31

definition that states that: the roll centre is the point about which the body can rollwithout producing any lateral movement at either of the wheel contact areas.

Figure 1.30 Examples of results from a 3-D kinematic analysis

In general each roll centre lies on the vertical line produced by the intersection of thelongitudinal centre plane of the vehicle and the vertical transverse plane through a pairof wheel centres. The roll centre heights in the front and rear wheel planes tend to bedifferent as shown in Figure 1.31. The line joining the centres is called the roll axis,with the implication that a transverse force applied to the sprung mass at any point onthis axis will not cause body roll.

Figure 1.31 Roll centres and roll axis

1.4.3 Roll centre determinationFor a given front or rear suspension, the roll centre can be determined from thekinematic definition by using the Aronhold-Kennedy theorem of three centres. Thisstates: when three bodies move relative to one another they have three instantaneouscentres all of which lie on the same straight line.

To illustrate the determination of roll centre by this method consider the doublewishbone suspension shown in Figure 1.32. Consider the three bodies capable ofrelative motion as being the sprung mass, the left-hand wheel and the ground. Theinstantaneous centre of the wheel relative to the sprung mass Iwb, lies at theintersection of the upper and lower wishbones, while that of the wheel relative to theground lies at Iwg. The instantaneous centre of the sprung mass relative to the ground(the roll centre) Ibg, must (according to the above theorem), lie in the centre plane ofthe vehicle and on the line joining Iwb and Iwg, as shown in the diagram.

Rear RC

Front RC

Roll axis

1-32

Figure 1.32 Roll centre determination for a double wishbone suspension

Iwg always lies in the position shown. Changing the inclination of the upper and lowerwishbones can vary Iwb, thereby altering the location of Ibg and ultimately the loadtransfer between inner and outer wheels during cornering. The importance of rollcentre location is discussed later. Figures 1.33 to 1.36 illustrate the locations of rollcentres for a variety of independent suspensions. In the case of the McPherson strutsuspension (Figure 1.33) the upper line defining Iwb is perpendicular to the strut axis.

Figure 1.33 Roll centre location for a McPherson strut

For the trailing arm suspension (Figure 1.35), the trailing arm pivots about atransverse axis (forward of the wheel centre). In the front view (Figure 1.35(c)), thewheel is constrained to move in a vertical plane (with no transverse movement) andhence Iwb lies at infinity along the pivot axis (to the right). The roll centre thereforelies in the ground plane on the centre-line of the vehicle.

Figure 1.34 Roll centre location for a swing axle suspension

Iwb

Ibg (RC)

Iwg

Iwb

Ibg (RC)

Iwg

Iwb

Ibg (RC)Iwg

1-33

(b) Side view (c) Front view

Figure 1.35 Roll centre location for a trailing-arm suspension

For the semi-trailing arm suspension (Figure 1.36) the pivot axis is inclined andintersects the vertical lateral plane through the wheel centre at Iwb a distance L fromthe centre plane of the wheel. The roll centre Ibg lies on the line connecting Iwb withthe instantaneous centre of the wheel relative to the ground Iwg.

Figure 1.36 Roll centre location for semi-trailing arm suspension

Figure 1.37 shows the roll centre determination for a four-link rigid axle suspension.In this case the wheels and axle can be assumed to move as a rigid body. The upperand lower control arms produce instant centres at A and B respectively (projected to

Iw

Iw

Ibg

Ibg

Iw

(a) Plan view

Iw

(a) Plan view

Vehiclec.l.

L

L

(c) Front(b) Side view

1-34

the vehicle centreline). Connecting these together produces a roll axis for thesuspension. The intersection of this axis with the transverse wheel plane defines theroll centre.

Figure 1.37 Roll centre for a four link rigid axle suspension

Our final example illustrating roll centre determination is the Hotchkiss rearsuspension shown in Figure 1.38. The analysis in this case is somewhat different tothe previous examples. Lateral forces are transmitted to the sprung mass at A and B.The roll centre height is at the intersection of the line joining these points and thevertical transverse plane through the wheel centres. The roll centre is of course at thisheight in the centre plane of the vehicle.

Figure 1.38 Roll centre location for a Hotchkiss suspension

1.4.4 Roll centre migrationWhen a suspension deflects, the roll centre locations move. For modest suspensiondeflections this migration of the roll centre can be quite significant, (as shown inFigure 1.39), arising in this case from the vertical deflection of the vehicle body.Large roll centre migrations can have undesirable effects on vehicle handling. Henceattempts are made at the design stage to limit them for the full range of suspension

(a) Plan view

(b) Side view

ARC

B

B

A

Roll centreheight

RC

1-35

travel. Some suspension types are more amenable than others for controlling rollcentre migration.

Figure 1.39 Effect of suspension deflection on roll centre migration

1.4.5 Roll centre effects

A high roll centre produces low lateral load transfer (good anti-roll) but produces highlevels of half-track changes. The high roll centre is typically closer to the centre ofgravity and hence there is a reduced lever arm and lower roll moment. Lateral forcesgenerated under cornering therefore produce less roll. However, the suspensiongeometry to give a high roll centre typically means that the wheel scrub is significantover the suspension travel, hence the larger half-track changes.

A low roll centre produces small half-track changes but poor anti-roll properties.

The migration of the roll centre needs to be limited to minimise the steering balancevariation under different driving conditions.

Steering ResponseThe balance between steady-state understeer / oversteer is largely dictated by the frontto rear lateral load transfer ratio which is dependent on roll centre heights. This can bemodified by changes in roll stiffness usually with anti-roll bars. Migration of rollcentres can create unwanted variations in steering balance.

Straight Line StabilityIncreasing front to rear lateral load transfer ratio produces an understeer tendency.Unsymmetrical road undulations induce half-track changes producing lateral forcevariations at the tyre contact patches with resultant steer effects. A positive half trackchange produces a force towards the vehicle centreline and this effect, when combinedwith bump steer, will affect vehicle stability.

Braking StabilityRoll centre migration due to induced pitch changes can cause plunging front andrising rear roll centres resulting in an undesirable oversteer tendency

I’w

b

Iwb

Ibg (RC)

I’bg

Iwg

1-36

Lift-Off / Traction Steer on Bends

Roll centre migration due to induced pitch changes can cause variations in steeringbalance (see Steer Response above).

Ride

Half-track changes produce resisting forces having a vertical component. In somecases this may be large enough to affect ride quality. On asymmetric undulations, thelateral forces due to half-track changes may excite roll-rock. Vehicle roll isfundamentally linked to roll centre height, but can be modified by the use of anti-rollbars.

Jacking

Vehicle jacking is affected by roll centre height.

1-37

1.5 Suspension Force Analysis

A thorough analysis of all the forces in a suspension system is a complex problemrequiring the use of computer packages. FE packages can be used for static analysesand stress distribution within members while multi-body system (MBS) packages arerequired for the analysis of three-dimensional dynamic loading. Analysing staticforces is straightforward, but superimposed on these are dynamic conditions due tovertical and longitudinal road irregularities, braking, accelerating, cornering etc. Manyof these dynamic loads are of a transient nature and are very difficult to model. Forthis reason they tend to be modelled by dynamic load factors.

In this section we will discuss how dynamic loads can be quantified and analysegraphically some typical loading situations. The relationship between the suspensionspring stiffness (spring rate) and the equivalent stiffness in the vertical plane of thewheel (wheel rate) are also examined. This relationship is important in terms ofdetermining suspension forces and the natural frequencies of vibration of the vehicle.

A generalised schematic of wheel load against wheel vertical displacement is givenbelow. The figure identifies the critical points where the response changes, e.g. due toengagement of a bump stop. Whilst the main suspension spring and its associatedspring rate may be linear (or intentionally designed to be non-linear) the wheel rate isalmost always non-linear owing to the suspension geometry which defines how thespring rate translates to give the wheel rate.

A – unladen conditionB – fully loaded conditionC – bump stop engagementD - bump stop fully engagedE – bump stop engagementF – zero vertical wheel forceG – bump stop fully engaged

1-38

1.5.1 Suspension loading

a) Longitudinal loads due to braking and accelerating

Vertical, longitudinal and lateral tyre loads have been discussed briefly in section1.1.6. It is useful to examine in a little more detail the generation of longitudinal loadsarising from control actions such as braking and accelerating.

For these cases the loads are dependent on how braking and drive-torques are reacted.They in turn depend upon whether the vehicle has inboard or outboard brakes andwhether the drive is to a live axle or an independent suspension.

Braking - outboard brakes

In this case the braking torque is reacted at the hub assembly. Consider the free-bodydiagram of the wheel in Figure 1.40a. The corresponding force and moment on thehub carrier are shown in Figure 1.40b. These provide the suspension loading. Anequivalent suspension loading is shown in Figure 1.40c.

Figure 1.40 Loads due to braking with outboard brakes

Motion

Moment exerted bybrake pads Fx r

Fx Tyre force

a) Wheel forces and moment

Force exertedby

suspensionon hub, Fx

Moment exerted bythe wheel on thebrake pads, Fx r

Force exerted byhub on

suspension, Fx

b) Hub force and moment

Fx

c) Equivalent force exerted on suspension

r

Braking - inboard brakesIn this case the braking torque is reacted inboard at the vehicle body. The free-bodydiagram of the wheel is the same as in Figure 1.40a. However, in this case there is nomoment exerted on the hub carrier. The suspension load is simply equal to thelongitudinal tyre force acting through the centre of the hub, Figure 1.41a. Anequivalent (resultant) force system is shown in Figure 1.41b.

Figure 1.41 Loads due to bra

Drive to a live axleIn this case the drive torque is reacted outbothe case of outboard brakes. The equivalent f

Figure 1.42 Suspension load resu

Drive to independent suspension

In this case the drive torque is reacted inboainboard brakes. The equivalent force systeFigure 1.43.

Figure 1.43 Suspension load resulting fromturn (for speeds b

Force exerted by thehub on suspension, Fx

a) Force exerted on suspension

Motion

moment, Fx r

Compensating

king with inboard brakes

ard at the hubs. The analysis is similar toorce is shown in Figure 1.42

lting from drive to a live axle

rd. The analysis is similar to the case ofm acting on the suspension is shown in

moment, Fx r

Fx

b) Equivalent force system on the suspension

Fx

Compensating

1-39

drive to an independent suspension ofelow critical).

Fx

1-40

b) Modelling transient loads

In order to safeguard the robustness of a suspension it is particularly important that theeffects of worst-case loading scenarios are analysed. These tend to arise from thetransient (shock) loads due to wheels striking potholes and kerbs and panic braking.

Because every set of events is likely to be different, designers tend to cover worst-casescenarios by using load factors based on the static wheel load. Different manufacturerstend to adopt their own set of design factors. Alternatively, the manufacturers work toa set of worst-case accelerations along the different vehicle axes; a typical example isshown in Table 1.1. The force loadings can then be obtained using an appropriatemass for the wheel and unsprung mass.

1.5.2 Wheel-rate required for constant natural frequency

Wheel-rate is the vertical stiffness of the suspension in a wheel plane. One of themany problems facing the suspension designer is that as the body mass of a vehiclechanges from the unladen to the laden condition, the natural frequency of the mass canchange. This is undesirable and happens when the wheel rate k, is constant over therange of suspension travel. The problem can be overcome by arranging a wheel-ratethat increases with suspension travel. The following analysis shows that it is possibleto determine how the wheel-rate needs to vary in order to provide a constant naturalfrequency as the payload changes.

Load case Load factorLongitudinal Transverse Vertical

Front / rearpothole bump

3g at the wheelaffected

0 4g at the wheelaffected, 1g atother wheels

Bump duringcornering

0 0 3.5g at wheelaffected, 1g atother wheels

Lateral kerbstrike

0 4g at frontand rear

wheels onside affected

1g at all wheels

Panic braking 2g at frontwheels, 0.4g at

rear wheels

0 2g at frontwheels, 0.8g at

rear wheels

Table 1.1 Typical worst case accelerations for dynamic suspension loads..A vehicle body mass supported on its suspension can be modelled most simply as amass on a spring for which the natural frequency is given by:

n

k

m

1

2

where k = wheel rate and m = effective sprung mass.

(1-1)

1-41

Equation (1-1) may first of all be written in terms of an effective static deflection, s,which is defined by

s

mg

k

Hence, substituting in Equation (1-1) gives

n

s

g

( )

1

2

Note the units. When g is in m/s2 and is in m, the units of n are rad/s.

A typical load -deflection curve for a rising-rate spring is shown in Figure 1.44. Theeffective rate is the local slope of this curve at a particular point. To maintain aconstant value of natural frequency requires a particular relationship between the rateand the load i.e.

Load

Rateconstant

W

dW dxs constant

Equation (1-5) now needs to be re-arranged and integrated to obtain an expression forW as a function of x. Re-arranging gives,

dW

W

dx

s

Hence by integration,

es

Wx

clog

(1-2)

(1-3)

(1-4)

(1-5)

Figure 1.44 Rising rate spring characteristics

(1-6)

(1-7)

1-42

The integration constant, c, is given by substituting values at the static load conditionas indicated in Figure 1.48.

W W s sx x when

c Wx

e ss

s

log

Hence equation (1-7) becomes

es

s

s

W

W

x xlog

( )

1.5.3 The relationship between spring-rate and wheel-rateThe rising-rate stiffness of the suspension in the wheel-plane will dictate the requiredcharacteristic for the suspension spring. The relationship between wheel-rate andspring rate will depend on the kinematics of the suspension. In general, thedisplacement relationships between wheel travel and deflection across the suspensionspring will be nonlinear. Some examples will illustrate how the relationship can beobtained.

A simple exampleThis example is based on a swing axle suspension, a design that is not commonly usednowadays because it is prone to the suspension-jacking problem mentioned earlier.However, it serves as a useful example to introduce spring/wheel-rate calculations.For the case shown in Figure 1.45, the wheel-rate may be calculated as follows.Assume the suspension spring has stiffness ks.

(1-8)

(1-9)

(1-10)

Figure 1

A small displacement, w, at the wh

sc w

d

Hence the spring force is

)d

wc(kF ss

Taking moments about the pivot at

w sF d F c

Then expressing the wheel force in

wkF ww

Substituting in equation (1-12) then

w sk kc

d ( )2

In this case the rates are related byonly for small displacements adisplacements the relationship is no

Spring

w

FS

FW

c

s A(a)

1-43

.45 Swing axle example

eel results in a displacement at the spring of,

A (Figure 1.49b) gives,

terms of an effective wheel rate, kw, gives,

gives

the lever arm ratio squared. This is strictly validround the mean position shown. For largern-linear

(1-11)

(1-12)

(1-13)

(1-14)

(1-15)

d

c

d

FA

(b)

1-44

A general example – suspension with a coil spring

A more general relationship between spring rate and wheel rate can be derived usingFigure 1.46. The suspension ratio is defined as

RS

F

The spring stiffness is

skdS

dx

d

dxRF ( )

sk RdF

dX

dX

dxF

dR

dX

dX

dx

Also by virtual work,FdXSdx

This leads to an alternative definition of the suspension ratio,

RS

F

dX

dx

Noting that the wheel rate is

wkdF

dX

Equation (1-18) may be rewritten as

s wk k R SdR

dX 2 ( )

1.5.4 Analysis of forces in suspension members

Despite all the complexities associated with the practical situations, it is neverthelesspossible to carry out some simple design calculations. These can be carried out

(1-16)

(1-17)

(1-18)

Figure 1.46 Calculation of suspension ratio

(1-19)

(1-20)

(1-21)

(1-22)

X

1-45

graphically for a given load application and enables the loads in the suspensionmembers and at body mounting points to be determined. Inevitably these calculationsinvolve some simplifying assumptions. Some examples will illustrate the procedure.

a) Vertical loading

Double wishbone suspension example

A diagram of a double wishbone suspension in which the spring acts on the lower armis shown in Figure 1.47. The following assumptions are made. Loads due to the masses themselves are ignored on the basis that they will be

insignificant compared with loads arising from the static wheel load, Fw . All joints are treated as simple pin joints so that effects of friction, compliance in

bushes etc. are ignored.The problem can be treated in two dimensions.

The static wheel load, FW, would typically be calculated from the vehicle data. Theproblem then is to calculate the forces transmitted to the body mounting points.Clearly this is a statics problem since all the members are in equilibrium and thesimplest approach is a graphical one. The free body diagrams and associated forcevector diagrams are shown in Figure 1.47.

Note the following features in obtaining these results. The force in the upper arm DA must be along the link since it is pin jointed at both

ends. Consider the wheel plus kingpin assembly first and construct the FBD. The

directions of FW and FD are known so the direction of FC can be found (it mustpass through G for equilibrium)

Next, the FBD for the lower arm can be drawn.

A full numerical solution to this problem requires that (i) the position diagram of thelinkage is drawn to scale to define all the geometry and (ii) the force vector diagramsare either drawn to scale or the forces calculated using the geometry defined by theforce vector diagrams.

1-46

In order to analyse the suspension forces over the full range of suspension travel, theprocedure can be repeated at suitable increments between full bump and reboundpositions. To do this some data about either the spring rate or the wheel rate must beknown or assumed. For example, if the load-deflection curve for the spring is known,then the load-deflection curve for the wheel can be plotted and the effective wheel ratecalculated (Figure 1.48). Note that because of geometrical effects, the relationshipbetween wheel deflection and spring deflection is not linear and so even a linearspring will result in a non-linear wheel rate. An alternative calculation is to assume aparticular wheel rate and hence calculate the spring load-deflection curve (Figure1.49). This defines the spring characteristics necessary to achieve a desired wheel-rate, which may be linear or non-linear.

Force vector diagrams

Free Body Diagramfor wheel/kingpin assembly

Free Body Diagramfor lower arm

Figure 1.47 Force analysis of a double wishbone suspension

1-47

McPherson strut example

A diagram of a McPherson strut suspension is shown in Figure 1.50. The sameassumptions as in the previous section are used and in addition the strut is treated as apiston which is free to slide within a cylinder. The free body and force vectordiagrams are constructed as follows. The force in the arm, CB, is along its longitudinal axis since it is pin-jointed at

both ends and there are no loads applied along its length. For the wheel-axle-strut assembly, DCE, the resultant force at D must pass

through point G for equilibrium The force at D is reacted by two components: the spring force in the direction of

AC and a side force on the cylinder. Note that this is only true if the spring isconcentric with the strut AC. The side force is undesirable since it producesfriction and wear on the piston in the strut.. Mounting the spring more nearly inline with AG can reduce this side force and this is commonly done in McPhersonstrut designs.

Slope = spring rate

Static deflection

Spring deflection

Spring Load

FS

Slope = wheel rate

Static loadposition

Wheel deflection

Wheel Load

FW

Rebound Bump

Max

Min

Figure 1.48 Typical load vs. Deflection characteristics for spring and wheel

Slope = spring rate

Static load position

Spring length

Spring Force

FS

Full bump

Full Rebound

max

min

Figure 1.49 Spring rate calculation

1-48

A

BC

D

E

G

FD

Fc

Fw

Fspring

FslideFD

FD

Fc

Fw Fspring

Fslide

FD

Free Body Diagramfor wheel/strut assembly

Force vector diagrams

Free Body Diagramfor piston slider

b) Lateral, Longitudinal and Mixed Loads

When a lateral tyre force is applied in addition to a vertical load, the analysis for theexample in Figure 1.47, is modified to that shown in Figure 1.51. The resultant wheelforce is first calculated from the vector combination of FW and FY. This then defines anew point for G and the force acting on the lower link is changed in both magnitudeand direction compared with the previous analysis.

For loads in the fore-aft direction, the situation is shown in Figure 1.52. A fore-aftload may be applied at either points E or O (i.e. at right angles to the plane of thediagram.) A force at the tyre-ground contact point, E, would arise from braking withconventional outboard brakes. The braking torque is transmitted through the stubaxle/kingpin assembly. This may be treated as a single rigid member. If the fore-aftforce is due to drag, shock loading, braking with inboard brakes or traction, then itseffective point of application is at the wheel centre, O.

Figure 1.50 Analysis of a McPherson strut suspension

1-49

Figure 1.51 Combined lateral and vertical loads on double wishbone suspension

In general any force at a given point may be treated as the same force at another pointplus a couple around that point. This is a convenient way of analysing the effect ofthese fore-aft forces. If the fore-aft force is moved to the kingpin axis then a momentof either Fd1 or Fd2 acts around the kingpin axis. This moment is, in practice, reactedby the steering arm and track rod so that the force in the track rod is either:

FY

FW FC

FDG

Resultant wheel force

Figure 1.52 Defining the loading points on a suspension

1-50

1

3

Fd

d

or2

3

Fd

d

d3 is the effective moment arm of the steering arm around the kingpin axis. In the sideview (Figure 1.53), this longitudinal force is reacted by the upper and lower arms. Itmay then be possible to proceed to analyse all the forces on the upper and lower arms.However, the analysis is now starting to become complicated. Remember also we arenot taking proper account of all the three dimensional effects. Despite this, our simpleanalysis does provide a good indication of the primary forces in the suspension. Thesubtle effects arising from a full three-dimensional treatment are of secondaryimportance.

Figure 1.53 Longitudinal forces reacted by suspension arms

Suspension travel is limited by bump stops. Normally these are made of rubber andproduce a rapidly increasing spring-rate with deflection. This obviously preventsdamage from metal to metal contact and provides a rapidly increasing force to reactthe extreme motion of the wheel. There are two philosophies involved in bump stopdesign. The first, and perhaps more conventional approach is to make them relativelyshort and stiff; this means that contact is relatively infrequent and restricted to extremeevents. The forces involved are high. The second is to make them relatively long andcompliant; this means that they come into contact much more often, but much lowerforces are involved and they may be viewed as a secondary spring. This effectivelyhelps to stiffen the suspension for large deflections. Whichever approach is adoptedcontact with the bump stops changes the force system in the suspension members.

An example is shown in Figure 1.54. One method of tackling this problem is bysuperposition, i.e. by separating the forces arising from the main suspension spring atfull deflection from those due to the bump stop spring. Both sets of suspension loadsare then added together using the principle of superposition.

The procedure is as follows;

(1-23)

(1-24)

Force on lower arm

F

E D

CD

1

Force on upper arm

F

E D

CD

1

Upper Arm

1-51

Draw the suspension in the required position (in contact with the bumpstop). Ignoring the force due to the bump stop compression, determine the forces and

wheel load F’w.

Calculate the forces as described previously - this will give a nominal loads F’c

and F’s. Calculate the excess wheel load carried by the bump-stop,

'wwex F-FF

Ignoring the spring force, draw the FBD of the upper arm and determine thesuspension forces arising from the bump-stop load F’w.

Determine the total suspension forces by adding the results from steps (c) and (e).

A

B

C

D

Fex

H

Fb

F

FA

D

Fb

FA

FD

excess load

FBD for upperarm

Force Diagram

Figure 1.54 Effect of bump stop on suspension force analysis

1-52

1.6 Suspension Geometry to Combat Squat and Dive

Squat (nose up) and dive (nose down) are changes in vehicle body attitude arisingrespectively from acceleration and braking. It is possible to design a suspension thatwill combat squat or dive, however, the geometry required for anti-squat will bedifferent to that for anti-dive. Again, this is one of those areas of suspension designthat requires compromises to be made. The design requirements to combat divedepend on whether a vehicle has outboard or inboard brakes, while anti-squat designdepends on whether the vehicle has dependent or independent suspension. In the lattercase suspension stiffness also plays a part.

1.6.1 Anti-Dive Geometry

The free body diagram for a car during braking is shown in Figure 1.55. Note thepseudo (D’Alembert)-force, mf, in which f is the deceleration (m/s2). Braking forces,Bf and Br, are applied at the front and rear wheel pairs and the relationship betweenthem is defined by the fixed braking ratio,

kB

B B

f

f r

Under these conditions, the vertical loads on the axles are different from their staticvalues, and can be calculated as follows.

Taking moments about the rear tyre contact patch,

fN l mfh mgb 0

fNmgb

l

mfh

l

(1-25)

(1-26)

Figure 1.55 Free Body Diagram of vehicle during braking

(1-27)

mgBf Br

NrNf mf

a b

l

h

1-53

The first term is the static load and the second term is the load transfer due to braking.Hence, the load at the rear is,

rNmga

l

mfh

l

This load transfer term will cause an increased deflection at the front axle and areduced deflection at the rear axle. Hence the change of attitude of the vehicle body.

Outboard brakes

Consider now the diagram of the front suspension shown in Figure 1.56. The axis ofthe link pivots are inclined such that the effective centre of rotation of the wheel inside view is at Of.

Figure 1.56 Free Body Diagram for front suspension

The spring force may be expressed as static load plus a perturbation due to braking,

f f fS S S

Under static conditions (f = 0) the static spring load is

fSmgb

l

For the braking condition, taking moments about Of gives,

f f fN d S d B e 0

dmgb

l

mfh

ld S S B ef f f( ) ( ) 0

Removing the static load and setting Sf to zero to represent the case in which thesuspension load does not change gives

mfhd

lB ef 0

Substituting for the braking force

(1-28)

(1-29)

(1-30)

(1-31)

(1-32)

(1-33)

Bf

dNf

e

Sf Spring force

Of

1-54

fB mkf

This results ine

d

h

kl tan

Hence, if the instantaneous centre of the suspension, Of, lies anywhere along the linedefined by equation (1-35) then the condition for no front suspension deflection issatisfied. If the instantaneous centre is designed to lie below this line at an angle of say, then the percentage anti-dive is defined as

(tan

tan)

100%

A similar analysis can be applied at the rear (Figure 1.57) to give

)k1(l

h

d

etan

(1.37)

Again, if the instantaneous centre lies along the line defined by equations (1-36) and(1-37), then will be no change in spring force and hence no lift will occur. Therefore,the condition for 100% anti-dive is that the front and rear instantaneous centres liealong the two lines shown in Figure 1.58.

It is common practice to use only around 50% anti-dive. The reasons are: Zero pitch braking appears to be undesirable from a subjective viewpoint, Full anti-dive conflicts with anti-squat so that some compromise must be reached,.

(1-34)

(1-35)

(1-36)

Figure 1.57 Free Body Diagram for rear suspension

Figure 1.58 Condition for 100% anti-dive

Br

dNr

e

Sr Spring force

h

kl (1-k)l

Or

1-55

Full anti-dive can cause large caster angle changes because all the brake torque isreacted through the suspension links. Such caster changes may tend to make thesteering unacceptably heavy during braking

Inboard Brakes

All the foregoing analysis applies to the most common case in which the brakes aremounted outboard, (i.e. at the wheel). Inboard brake systems differ in that the braketorque is reacted through the driveshaft. (Section 1.5.1) The situation for say a frontwheel is shown in Figure 1.59. The additional torque applied by the driveshaft to thewheel unit must be included in the freebody diagram. Taking moments about thewheel centre,

f fM B r mfkr

Now, taking moments about Of gives

f f fN d S d B e mfkr 0

Comparing this with equation (1-31) it can be seen that an additional term is nowincluded. Using the same analysis as before leads to,

( )tan

e r

d

h

kl

Note from the above expression that the angle is now defined from the wheel centrerather than the ground level. The analogous situation occurs at the rear to give

tan)k1(l

h

d

)re(

1.6.2 Anti-squat geometry

The analysis for anti-squat or anti-pitch design during acceleration follows a verysimilar process to that adopted in the previous section. In Figure 1.60, the pseudoforce is now reversed in direction where f is now defined as acceleration. Tractiveforces may be applied at either axle, or both in the special case of four-wheel drive.

(1-38)

Figure 1.59 FBD for front suspension with inboard brakes

(1-39)

(1-40)

(1-41)

eMf

Nf

Bf

d

Sf Spring force Of

1-56

There are two general cases to consider; solid axle drive or independent suspension /driveshaft designs.

Solid rear drive axle

This configuration is shown in Figure 1.61 with the axle assumed to be mounted ontwin trailing arms, with an instantaneous centre of rotation at Or. This point issometimes referred to as the virtual reaction point (i.e. the point at which the forces inthe suspension arms can be resolved and assumed to act on the vehicle body). Notethat this treatment of the suspension is exactly the same as if a trailing arm of length,d, connected rigidly to the rear axle acting about a pivot at Or. Note that the drivetorque is reacted through the axle casing and the suspension members (thus makingthe analysis analogous to the case of outboard brakes). The analysis assumes that theaxle mass is small relative to the vehicle mass. The vertical wheel load is,

rNmga

l

mfh

l

mfh

lis the load transfer onto the rear axle.

The tractive force is,rT mf

Taking moments about Or gives

Figure 1.60 FBD for an accelerating car (all-wheel drive)

(1-42)

e

O

Sr

TrNrd

Figure 1.61 Rear axle mounted on twin trailing arms

(1-43)

mgTf Tr

NrNf mf

a bl

h

Or

1-57

r r rT e S d N d 0

Substituting for all the forces and eliminating the static spring force gives

r r rS mfh

l

e

dk ( )

where r = rear suspension deflection, kr = rear spring rate

The front suspension must also deflect under these conditions due to the load transferterm, mfh l , reacted there. Note that it is not possible to design the front suspension

to directly accommodate this. It was possible to do this for the braking case, becausepart of the braking force was reacted there. Hence, the spring force change at the frontis,

f f fS kmfh

l

The pitch angle, , of the vehicle is simply,

( )r f

l

mf

k

h

l

e

d

mfh

l k

l

r f

( )

)(d

ehh

l

mf

klklk rrf

Hence the condition for zero pitch may be expressed as,

e

d

k

kh

l

r

f

( )1

Independent suspension – rear wheel drive

The freebody diagram is now altered to include the moment provided by the driveshaft(Figure 1.62). Note that this is similar to the case of the inboard mounted brakes.Thus equation (1-48) becomes,

r r r rT e S d N d M 0

where Mr = Trr

(1-44)

(1-45)

(1-46)

(1-47)

(1-48)

(1-49)

(1-50)

(1-51)

1-58

Carrying this small difference through the analysis results in the condition for zeropitch,

( )( )

e r

d

k

kh

l

r

f

1

Example E2A multi-purpose vehicle has a live rear axle with outboard brakes. The pivot centre of the rearsuspension is located at Or, which is positioned relative to the underside of the wheels by thedimensions d and e, as shown in Figure E2.1

Relevant vehicle data is as follows:Vehicle mass 2.1 tonnesWheelbase 2.8 mVehicle c.g. height above road 0.85 mFront spring-rate (both sides) 52.5 kN/mRear spring-rate (both sides) 35.2 kN/mBraking ratio (front: rear) 70: 30

a) Determine the ratio e/d for zero pitch during acceleration.

b) What then is the lift at the rear suspension when the vehicle brakes with a constantdeceleration of 0.3g.

Figure E2.1

e

O

Sr

TrNrd

Mr

Torque appliedby driveshaft

Figure 1.62 Independent rear suspension

(1-52)

Or

d

e

Or

1-59

Solution

a) The free body diagram of wheel and axle during acceleration is shown in Figure E2.2:

e

O

Sr

TrNrd

Figure E2.2

It has been shown that for zero pitch during acceleration:

e

d

k

kh

l

r

f

( )1

………………..(E-1)

1-60

Using the vehicle data: 507.08.2

85.05.52

2.351

d

e

b) For the lift at the rear suspension due to 0.3g deceleration, note that the vehicle hasoutboard brakes

In this case, a braking force replaces the traction force in the above diagram andacts in the opposite direction.

The moments equation about Or is: –Br e + Sr d – Nr d = 0…………….(E-2)

l

hfm

l

agmN r , rrr SSS , fmk1B r

Substituting into E-2 and cancelling out terms gives:

l

h

d

ek1fm

l

hfm

d

ek1fmSr

Change of spring force

9368.2

85.0507.0x3.081.9x3.0x2100Sr

N

The negative sign indicates a force reduction

Lift at rear suspension:

mm6.26mm/N2.35

N936

k

S

r

rr

The negative sign indicates lift at the rear axle.

.

1.7 Vehicle ride analysis

Ride comfort is one of the most important characteristics defining the quality of avehicle. It is principally related to vehicle body vibration, the dominant source ofwhich is due to road surface irregularities. In order to design vehicles that have goodride properties it is essential to be able to model ride performance in the early stagesof vehicle development. This requires an understanding of road surface characteristics,human response to vibration and vehicle modelling principles. These are introduced inthis section.

1.7.1 Road surface roughness and vehicle excitation

When a vehicle traverses over a road profile, the irregularities in the profile areconverted into time-varying vertical displacement excitation inputs at each tyre

1-61

contact patch. In general, road surfaces have random profiles, which means that theexcitations that they generate are also of a random nature.

The spatial random profile of a road surface can be frequency analysed and shown tocomprise a set of harmonic components having different amplitudes and wavelengthsas shown in Figure 1.63. In general the components having the longest wavelengthshave the largest amplitudes. While only three components are shown in Figure 1.63,there are in reality an infinite number. For a given frequency component it is possibleto define spatial10 frequency n as the number of cycles per metre. It can then be shown[21] that the frequency characteristics of the profile are described as a function of n,by the spatial power spectral density S(n). The units of S are m3 / cycle.

Figure 1.63 Harmonic components of a random road surface profile

From large amounts of measured road data it has been established [22], that S and nare related approximately by:

S n n 2 5.

is the road roughness coefficient.

When a vehicle travels at a velocity V m/s over a random road profile described byS(n), the resulting random excitation to the vehicle is described by a temporal11

spectral density S(f), where f = n V is the excitation frequency in Hz.

It may be shown [21] that

S fS n

V( )

( )

10 Spatial – position related11 Temporal – time related

(1-53)

(1-54)

1-62

It follows that S f V f( ) . . 1 5 2 5

The units for S(f) are m2 / Hz. Typical values of for a motorway, a principal roadand a minor road are 0.25.(10)-6, 4.(10)-6 and 15.(10)-6 m2/(cycle/m)1.5 respectively for0.01<n<10 cycle/m [22]. The variation of S(f) for a vehicle traversing a poor minorroad at 20 m/s is shown in Figure 1.64. It is important to note the distribution offrequency components in this spectrum in relation to the natural frequencies of asuspended vehicle.

1 10 1001 10

7

1 106

1 105

1 104

0.001

0.01

Frequency, Hz

Figure 1.64 Spectral density of a road input as a function of vehicle speed (poorminor road)

1.7.2 Human perception of ride

Ride is dominated by the level of road induced vibration in a vehicle body. Inassessing ride performance of a vehicle it is necessary to take into account thesensitivity of the human body to vibration.

The human body is sensitive to vibration and has a number of natural frequencies. Itfollows that when the human body is subjected to excitation at any of thesefrequencies resonance and discomfort will result. From the last section it obvious thatroad induced vibration will be broadband and hence some of the components of theexcitation will tend to produce human body resonance. From an understanding ofhuman body resonances ([23], [24]) it is possible to weight the vehicle body responsein order to obtain a single figure assessment of ride performance.

Human response to whole body vibration

From a vibration point of view the human body can be considered to be a complexassemblage of mass, elastic and damping elements that result in a number of naturalfrequencies in the range from 1 to 900 Hz. From a ride point of view we are concernedwith whole-body vibration of a seated person in the range from 0.5 to 15 Hz. Thereare two whole-body natural frequencies in this range. They are associated with thehead-neck (1-2 Hz) and the thorax-abdomen (4-8 Hz). Head-neck resonance can beexcited by vehicle pitching and rolling motions, while resonance of the thorax-abdomen can be produced by vehicle bounce motion.

(1-55)

SpectralDensitym2 / Hz

1-63

In general the tolerance to whole-body vibration decreases with time, so a high levelof vibration for a short time can produce the same level of discomfort as a low levelfor a long time. Equal discomfort curves have been produced in standards, e.g.International Standard ISO 2631 [25] and British Standard BS. 6841:1987 [26]. Someexamples are given in Figure 1.65 for vertical and longitudinal / lateral excitation. Itshould be noticed that the human body is acceleration-sensitive and the minimumvalues of the graphs correspond to the resonances discussed above.

1 10 1000.1

1

10

Frequency, Hz

R.m

.s.

acce

lera

tio

naz

,m

/s^2

1 10 1000.01

0.1

1

10

Frequency, Hz

R.m

.s.a

ccel

erat

ion

sax

and

ay,

m/s

^2

(a) Vertical vibration (b) Longitudinal and lateral directionsFigure 1.65 Equal discomfort curves

The shapes of the graphs are used to establish weighting curves that can be applied tovehicle body accelerations determined from simulations of vehicle response to roadinduced vibration. Thus enabling a single figure estimate of human response to beobtained. A typical weighting curve for vertical vibration is shown in Figure 1.66.

Figure 1.66 Weighting curve for vertical whole-body acceleration

1.7.3 Vehicle models

Ride performance is assessed at the design stage by simulation of vehicle response toroad excitation. This requires the development of a vehicle model and analysis of itsresponse.

284

8h

8h

4h4h

Weighting,dB

-12

--8

--4

-0-0

+4

0.5 1 2 4 16

1-64

Models of varying complexity are used. For a passenger car, the most comprehensiveof these has seven degrees of freedom (DOFs) - Figure 1.67. These comprise threeDOFs for the vehicle body (pitch, bounce and roll) and a further vertical degree offreedom at each of the four unsprung masses. This model allows the pitch, bounce androll performance of the vehicle to be studied.

Figure 1.67 Full vehicle model

The suspension stiffness and damping rates in the model are derived from theindividual spring and damping units using the approach discussed in Section 1.3.Various tyre models have been proposed [27]. The simplest of these uses a point-contact model to represent the elasticity and damping in the tyre with a simple springand viscous damper. Since tyre damping is several orders of magnitude less thansuspension damping it can be neglected in basic vehicle models.

Much useful information can be derived from simpler vehicle models. Because for anormal road surface profile the components having the longer wavelength componentsare in phase (coherent) across the left and right tracks of a vehicle, there is notendency to excite body roll. This then justifies the use of a half vehicle model asshown in Figure 1.68. This has four DOFs; a body mass translation and rotation, plusa translation for each of the unsprung masses.

Figure 1.68 Half-vehicle model

The half vehicle model can be simplified still further to a quarter vehicle model, if thebody mass satisfies certain conditions. Let the sprung mass in Figure 1.68 be denoted

Sprungmass

Unsprungmasses

a b

L

G

1-65

by M and its moment of inertia about its centre of gravity G by I. G is located bydistances a and b. The wheelbase is L. For the sprung mass in Figure 1.69 to bedynamically equivalent to that in Figure 1.68 the following conditions have to besatisfied.

Equal mass: MMMM Grf (1.56)

Mass centre in the same position: bMaM rf (1.57)

The same moment of inertia: IbMaM 2r

2f (1.58)

Figure 1.69 Half-vehicle model with dynamically equivalent sprung mass

From equation 1.57 and 1.58

Ibb

aMaM 2

f2

f

Re-arranging IbaaM f

But Lba

HenceLa

IM f (1.59)

It follows thatLb

IM r (1.60)

Substituting into equation 1.56:

ba

IM

Lba

baIM

Lb

I

La

IMMG

Denoting I by M rG2, where rG is the radius of gyration of the sprung mass about

lateral axis through G, it follows that:

Unsprungmasses

a b

L

Mf MG Mr

G

1-66

ba

r1MM

2G

G (1.61)

If the inertia coupling ratio 1ba

r2G , MG = 012. Then the dynamically equivalent

system comprises a sprung mass Mf at the front suspension and a sprung mass Mr atthe rear suspension and the motions at the front and rear suspensions are uncoupled.For this case a quarter vehicle model as shown in Figure 1.70 can represent either thefront or rear suspension.

Figure 1.70 Quarter vehicle model

1.7.4 Vibration analysis of the quarter vehicle model13

The model shown in Figure 1.70 consists of linear elements, i.e. the springs anddampers have linear characteristics. These are valid approximations for smalloscillations about the static equilibrium position of the system. Let x0, z1 and z2 betime varying displacements from this equilibrium position.

a) Vibration characteristics of a quarter vehicle model (see Appendix A1)

For this case we assume Cs = 0 and examine the free vibration of the model (x0 = 0).

The equations of motion are

0

0

z

z

KK

KKK

z

z

M0

0M

2

1

ss

sst

2

1

s

u

(1.62)

Assume solutions z1 = A1 sin t and z2 = A2 sin t. Substitute these expressions and theirsecond differentials into (1.62) and cancel throughout by sin t:

12 The inertia coupling ratio ranges typically from 0.8 for sports cars to 1.2 for somefront wheel drive cars.

13 For further details of vibration analysis see [28]

Sprungmass, Ms

Unsprungmass, Mu

CsKs

Kt

z1

z1

x0

1-67

0

0

A

A

MKK

KMKK

2

12

sss

s2

ust (1.63)

The non-trivial solution of (1.63) is

0

MKK

KMKK2

sss

s2

ust

(1.64)

This is called the frequency or characteristic determinant, i.e. describes the vibrationcharacteristics of the system.

Consider a typical set of data for the rear of a mid-range saloon car:Sprung mass Ms = 317.5 kgUnsprung mass Mu = 45.4 kgSuspension stiffness (wheel-rate) Ks = 22 kN/mTyre stiffness Kt = 192 kN/m

Substitute the into (1.64), expand and then simplifying the result gives:

010x2934786 324

The solution to this quadratic equation in 2 is:

2

10x293x447864786,

3222

21

From which the natural frequencies are 1 = 7.87 rad/s (f1 = 1.25 Hz) and 2 = 68.73 rad/s (f2

= 10.9 Hz). These results are very typical of what one would expect for a car suspension.Corresponding to each natural frequency there is a mode of vibration. These describe therelative amplitudes of free vibration of the masses at each of the natural frequencies. Theyare determined by substituting first 1

2 and then 22 into either of the equations in (1.63).

First mode of vibration

Substituting 12 into the second equation and add a second suffix 1 to A1 and A2 to denote

that they are the amplitudes at the first frequency:

0AMKAK 2121ss11s (1.65)

For the numerical data: 0122.0A

A

12

1

This indicates that since the result is positive, both masses move in the same direction at anyinstant and the amplitude at the unsprung mass is 0.0122 that of the sprung mass. This modeof vibration is called the body mode and can be represented graphically as shown in Figure1.71a. Note amplitudes are plotted horizontally for clarity.

1-68

Second mode of vibration

Substituting 22 into the second equation and add a second suffix 2 to A1 and A2 to denote

that they are the amplitudes at the second frequency:

0AMKAK 2222ss12s (1.66)

For the numerical data: 7.7A

A

22

1

This indicates that since the result is negative, the masses are moving in opposite directionsat any instant and the amplitude at the unsprung mass is 7.7 times that of the sprung mass.This mode of vibration is called the wheel-hop mode and can be represented graphically asshown in Figure 1.71b.

(a) Body mode (b) Wheel-hop modeFigure 1.71 Vibration modes for a quarter-vehicle suspension.

From the above analysis it is apparent that for the body mode the amplitude of the unsprungmass is very small compared to that of the sprung mass, so in this case the system could beapproximated as shown in Figure 1.72a. Also, for the wheel-hop mode the amplitude of thesprung mass is very small in comparison to that of the unsprung mass. For this case thesystem could be approximated as shown in Figure 1.72b. The corresponding approximatenatural frequencies and damping ratios are as shown in the diagrams.

(a) Body mode

(b) Wheel-hop modeFigure 1.72 Approximate models for body mode and wheel-hop mode

A22

7.7

-11

0.0122A21

A11

f1 = 1.25 Hz

Ks

Mu

Kt

Ms

Ks

Cs

Cs

s

s1

M

K

2

1f

ss

s1

KM2

C

u

ts2

M

KK

2

1f

tsu

s2

KKM2

C

A12

f2 = 10.9 Hz

A22

1-69

b) Frequency response of the quarter vehicle model (see Appendix A1)

In this case we are concerned with the complete quarter vehicle model shown in Figure 1.70and begin with the equations of motion for the system:

0

xK

z

z

KK

KKK

z

z

CC

CC

z

z

M0

0M 0t

2

1

ss

sst

2

1

ss

ss

2

1

s

u

(1.67)

Assume

i) A harmonic input at the base of the tyre spring, tieXtXx 000 Imsin .

ii) Steady state solutions for the response of the masses as tieZz 11 Im and

tieZz 22 Im . The quantities Z1 and Z2 are in general complex quantities where a

complex exponent indicates a phase change with respect to the input waveform.

For the purpose of solving the equations of motion, the exponential form of the input andoutput responses is used. The square root of the sum of the squares of the resulting complexexpression give the amplitude response and the inverse tangent of the imaginary part dividedby the real part give the phase change with respect to the input waveform.

When these expressions (and their first and second order differentials) are substituted into(1.67) it is possible to cancel throughout by eit. After some manipulation the result is:

0

XK

Z

Z

MKiCKiC

KiCMKKiC 0t

2

12

sssss

ss2

usts (1.68)

Note that Z1 and Z2 are complex in nature due to the inclusion of damping, i.e. the vibrationof the masses lags behind the excitation. Solving this pair of non-homogeneous equations inZ1 and Z2 produces the following expressions:

2sssss

ss2

usts

2ssst

0

1

MKiCKiC

KiCMKKiC

MKiCK

X

Z

(1.69)

and

2

sssss

ss2

usts

sst

0

2

MKiCKiC

KiCMKKiC

KiCK

X

Z

(1.70)

Replacing with 2 f (where f is the excitation frequency in Hz), in these expressions leadsto frequency response functions H1(f) and H2(f) as follows:

)f(X

ZfH

0

11 …(1.71) and )f(

X

ZfH

0

22 ..(1.72)

H1(f) and H2(f) are complex of the form H1r(f) + i H1i(f) and H2r(f) + i H2i(f)respectively. It follows that the amplitude responses are given respectively by:

1-70

2i12

r11 fHfHfH (1.73)

and 2i22

r22 fHfHfH (1.74)

where ‘r’ denotes the real part and ‘i’ the imaginary part of each function. H1(f) andH2(f) can be thought of as transfer functions between the respective parts of thesystem.

1.7.5 Response of a linear system to random excitation [28]

Consider a general linear structural system as shown in Figure 1.73. If A is thelocation of a random excitation having a psd SA(f), and B is the location of the outputfrom the system, having a psd SB(f), it can be shown [22] that:

fSfHfS A2

B , (1.75)

where H(f) is the frequency response function relating A to B. Furthermore it can beshown that the r.m.s. value of a stationary random signal is equal to the area under thepsd graph. This requires lower and upper frequency limits to be specified. Forsuspension performance analysis these can be taken to be between 0.5 and 15 Hz.

Figure 1.73 Input and output spectral densities

1.7.6. Parameters used in assessing suspension performance

There are three key parameters as follows.

a) Discomfort Parameter ACC

This is defined as the r.m.s. sprung-mass acceleration weighted for human response.The FRF for the weighted sprung-mass acceleration is given by:

acc2hrwACC fHfHfH (1.76)

Hhrw(f) = FRF for the human response weighting for vertical vibration

fHf2fH 22

acc2 = FRF for the sprung mass acceleration

ACC is then given by:

15

5.02

ACC dffSfHACC (1.77)

S(f) = power spectral density associated with the displacement input x0.

ioInput PSD, Si(f)

Output PSD, So(f)

System

1-71

b) Dynamic Tyre Load Parameter DTL

This is defined by the r.m.s. tyre-load fluctuation. Since the tyre load variation fromstatic is given by Kt(x0 –z1), it follows that the FRF for tyre load fluctuation is givenby:

fH1KX

ZXKfH 1t

0

10tDTL

(1.78)

DTL is then given by:

15

5.02

DTL dffSfHDTL (1.79)

c) Suspension Working Space Parameter SWS

This is defined as the r.m.s. relative displacement between sprung and unsprungmasses. The relative displacement is random and Gaussian. If the bump-stop to bump-stop displacement is some multiple of SWS, there will be some bump-stop contact. Iffor example the unsprung mass is equidistant between the bump-stops in the staticposition and the maximum suspension travel is n x SWS, the percentage of time thatbump-stop contact occurs is given in Table 1.2.

n 1 2 3Contact time, % 31.7 4.6 0.03

Table 1.2 Bump-stop contact related to suspension travel and SWS

One of the design aims is to minimise bump-stop contact, therefore a value of n = 3would be a realistic choice.

The FRF for relative movement between unsprung and sprung masses is: fHfHfH SWS 21 (1.80)

SWS is then given by:

15

5.02

SWS dffSfHSWS (1.81)

1.7.7 Quarter vehicle model results and design guidelines

The equations given above can be used to investigate the effect of suspension stiffnessand damping for a given road surface and vehicle speed. Some typical results areshown in Figure 1.74.

The objective from a design point of view is to have low body acceleration and lowdynamic tyre load. For a given SWS of 30 mm it is seen (upper diagram) that lowdamping requires higher suspension stiffness than would be required by highdamping. In terms of ACC (middle diagram), high damping produces higher bodyacceleration than low damping. From the bottom diagram, DTL is relativelyinsensitive to change in stiffness, while over the design range DTL favours higher

1-72

damping. In general, damping should be between 0.3 and 0.5 of critical - at the lowerend for “optimum comfort” and the higher end for “optimum handling”.

Figure 1.74 Typical quarter vehicle model results

The conflicting issues regarding stiffness and damping selection can be clarified if afull set of results (for a number of different damping values), are plotted on a conflictdiagram for a selected SWS (Figure 1.75). Note f1 and 1 are given by the equations inFigure 1.72a.

Figure 1.79 Conflict diagram

Figure 1.75 Conflict diagram

DTL, kN

ACC, m/s2

Point f1 1

Conventionaldesign range

Stiffnessincreasing

1-73

The optimum points on the conflict diagram (to minimise ACC and DTL) are 2 and 3.However, from a practical point of view, this will tend to make the suspensionstiffness too low. The result will be large amounts of body roll during cornering. Thiscan be counteracted with very stiff anti-roll bars. Unfortunately these can degrade rideon roads having significantly different characteristics across the road lane width. Abetter selection of parameters would be based on point 1.

1.8 The Influence of Chassis Design on Vehicle Handling

1.8.1 Steady State Cornering and Stability

Appendix A2 outlines the dynamics of a simple handling model. Note theassumptions and the notation. When forward speed U and steering input f areconstant, v and r are constant for steady state conditions representing the lateralvelocity and yaw velocity (an angular term) respectively (i.e. 0 rv ). Under theseconditions equation A2.11 reduces to:

Denoting r as rss (steady-state yaw rate) the motion is described in Figure 1.76 and thesimultaneous equations 1.82 can be solved to give:

where L = a + b = wheelbase

Figure 1.76 A steady-state turn

Steady-state yaw rate can be related to path curvature:

82.1.............Ca

C

r

v

U

CbCa

U

CbCa

U

CbCaUm

U

CC

ff

f

r2

f2

rf

rfrf

83.1....................

CaCbUmCCL

CCLUr

fr2

rf2

rf

f

ss

Tyre force

Tyre force

Front slip angle

Rear slip angleYaw rate rss

vssCentre of turn

Radius of turn

Constant forwardspeed U

f

1-74

84.1....................UR

1Ur ssss

Substitute into 1.83 and re-arrange to give:

Where:

In this relationship b Cr – a Cf is called the Stability Margin. Its magnitude (positive,zero or negative) determines the behaviour of the vehicle. Possible effects as afunction of speed U are shown in Figure 1.77.

Figure 1.77 Effect of speed on ss / f for different K

Consider the three possibilities.

K = 0 (b Cr = a Cf) Neutral Steer condition

L

1

f

ss

. This represents a pure rolling (Ackermann) condition. According to Figure

1.78,R

Lf

K > 0, (b Cr > A Cf), Understeer condition

The vehicle is always stable and tends to “run wide”. f needs to be increased tomaintain ss.

K < 0, b (Cr < a Cf), Oversteer condition

85.1..........................UKL

12

f

ss

86.1....................

CCL

CaCbmK

rf

fr

K < 0, Oversteer

K < 0, Oversteer

Forward

speed UK > 0, Understeer

K = 0, Neutral steer

Critical speed

Ucrit

ss / f

1-75

There is a critical speed at which ss tends to . This occurs at a speed Ucrit given by:

Figure 1.78 “Ackermann” vehicle in a steady turn

U < Ucrit

ss increases with speed. At a given speed (< Ucrit) it is necessary to decreasef to maintain ss.

U > Ucrit

Need to apply negative steering lock to maintain ss – requires special skill.

1.8.2 The Influence of Chassis Design on Handling

Despite the many limitations in the analysis of the basic handling model, a number ofimportant deductions can be made as to how chassis design can influence handlingstability. In effect we envisage ways of modifying cornering stiffness producingrevised values C’f and C’r. We limit our attention to the following:

a) Load transferb) Camber anglec) Compliance steer.

Load Transfer

When a vehicle corners load is transferred from the inside to the outside wheels. In thecase of an unsprung vehicle (Figure 1.79) with a static axle load Fz0 = mg / 2, thelateral load transfer is:

f

Centre

of turn

a

b

L

R

f

tan fL

R

87.1........................K

LUcrit

1-76

88.1..................t2

hamF

yz

Figure 1.79 Load transfer – unsprung vehicle

Figure 1.80 shows a typical tyre characteristic for Fy as a function of Fz. The graphshows the vertical tyre load for zero acceleration Fz0 and the corresponding lateral tyreforce capability Fy0. For a lateral load transfer Fz the lateral tyre force capability atinner and outer wheels are Fy,inner and Fy,outer,.

Figuer 1.80 Tyre characteristics and lateral load transfer

The following conclusions can be drawn regarding the total lateral tyre forcecapability Fy,total (= Fy,inner + Fy,outer) as a function of Fz:

1. Fy,total is less than 2Fy0.2. Fy,total decreases as Fz increases, i.e. Fy,total decreases with increasing acceleration.3. Fy,total is related to the effective cornering stiffness C’ (C’ decreases in proportion

to Fytotal), so by inducing more load transfer Fz, C’ can be reduced.4. Since bC’r needs to be greater than aC’f for an understeer vehicle, understeer can

be promoted by making C’f smaller. This can be achieved by inducing more loadtransfer at the front axle.

On a sprung vehicle, load transfer is reacted by the roll stiffness at front and rear axles- the higher the roll stiffness at a given axle the more the load transfer. The balance of

t t

Fzo -Fz

h

may

mg

Fzo +Fz

Vertical tyre

load, Fz

Lateral tyre force, Fy

Outer wheel

Inner wheel

Static wheel load

Outer wheelInner wheel

Fz0

Fz Fz

Fy,outer

Fy,inner

Fy0

1-77

load transfer at front and rear axles can be changed by altering the roll stiffness atthese axles. It follows that by increasing the roll stiffness at the front axle, thecornering stiffness can be decreased there and vehicle understeer can be made morepronounced. In practice this can be achieved by adding or increasing anti-roll barstiffness. It is then possible for the vehicle to be stable up to its cornering limit.

Camber Angle

Wheel camber produces a lateral tyre fiorce called camber thrust. Its magnitude is afunction of camber angle and vertical load. Denoting camber angle (see Fig 1.2),camber stiffness can be defined as:

When a vehicle is stationary is small, but with suspension movement can change,depending on the type of suspension. For small camber angles camber thrust is givenby:

90.1..............CF camber,y

Fy,camber is a second order force compared to Fy due to a slip angle and together theyproduces an effective cornering stiffness C’. Camber thrust can be used to make minoradjustments to handling balance. Figure 1.81 shows how a net reduction in corneringforce can be achieved due to camber thrust. Front positive camber used with negativerear camber can be used to promote understeer.

Figure 1.81 Reduction of cornering force due to camber thrust

Compliance steer

Cornering force can induce steer due to compliance in the suspension system. This canbe deliberately used to “fine-tune” and improve the handling of a vehicle. It is afeature exploited currently by many vehicle manufacturers.

89.1.................F

C

0

y

m ay

Lateral tyre forcedue to camber

Lateral tyre forcedue to slip angles

Parallel links result inequal body roll andcamber angles

1-78

Assume that the cornering forces produce a small compliance steer r,comp at the rearaxle. If this is in the same direction as the steering input at the front axle f, it movesthe centre of curvature away from the vehicle, thereby increasing the understeertendancy – Figure 1.82. It can be viewed as an increase in the rear cornering stiffnessC’r, thereby increasing the margin of stability.

Figure 1.82 The effect of rear compliance steer

f

f

r

r

To centreof turn

1-79

References 1

1. Reimpell, J; Stoll, H and Betzler, JM. The Automotive Chassis: EngineeringPrinciples, 2nd Ed; Butterworth-Heinemann 2001.

2. Gillespie, T.D. Fundamentals of Vehicle Dynamics. SAE. 1992

3. Dixon, J.C. Tyres, Suspension and Handling, (2nd Ed). Edward Arnold.(distributed by SAE), 1996.

4. Milliken, W.F. and Milliken, D.L. Chassis Design: Principles and Analysis,Professional Engineering Publishing, 2002.

5. Bastow, D and Howard, G. Car Suspension and Handling, (3rd Ed). PentechPress, London. (distributed by SAE, Warrendale, USA.) 1993.

6. Happian-Smith, J (Ed). An Introduction to Modern Vehicle Design, Butterworth-Heinemann, 2002.

7. Anon. Spring Design Manual. 2nd Ed; SAE. 1999.8. Dixon, JC. The Shock Absorber Handbook, SAE. 1999

9. Olley, M. Independent wheel suspensions - its whys and wherefores. SAEJournal, Vol 34, pp73-81, 1934.

10. Enke, K. Improvement of the ride and handling compromise by progress in theelasto-kinematic system of wheel suspension. Conf on Road Vehicle Handling,Nuneaton. Proc I MechE pp91-96, 1983.

11. Heisler, H. Advanced Vehicle Technology, Edward Arnold, 1989

12. Hillier, V.A.W. Fundamental of Motor Vehicle Technology. 4th Ed. Thornes.1991

13. Wahl, A.M. Mechanical Springs, 2nd Ed. McGraw-Hill 1963

14. Shigley, J.E. and Mitchell, L.D. Mechanical Engineering Design. McGraw-Hill.1983.

15. Mott, R.L. Machine Elements in Design. Charles E Merrill. 1985.

16. Bosworth, R. “Rover’s System Approach to Achieving First Class Ride Comfortfor the New Rover 400”. Automotive Refinement (Selected papers fromAutotech95), pp113-122, IMechE, 1996.

17. Grosjean, J. Principles of Dynamics. Stanley Thornes. 1986.

1-80

18. Horton, D. VDAS - Vehicle Dynamics Analysis Software, Users Manual. Version3.4. University of Leeds, Feb 1992.

19. Ryan, R. Multibody Systems Analysis Software, Multibody Systems Handbook(Ed W Schielen), Springer Verlag, 1990

20. Dixon, J.C. The roll centre concept in vehicle dynamics. Proc IMechE, Vol 201,No D1, pp69-78, 1987.

21. Wong, J.Y. Theory of Ground Vehicles. (2nd Ed) SAE. 1993

22. Dodds, C.J. and Robson, J.D. “The description of road surface roughness” JSound and Vib. 31 (2), pp175-183, 1973.

23. Butkunas, A.A. “Power spectral density and ride evaluation”. Sound andVibration, 1 (12), pp25-30, 1967.

24. Reason, J. Man in Motion: the Psychology of Travel. Weidenfeld and Nicholson.London. 1974.

25. ISO 2631-1 “Guide for the evaluation of human exposure to whole-bodyvibration”. International Standards Organisation. International StandardsOrganisation. 1978.

26. B.S. 6841:1987 “Guide to the measurement and evaluation of human exposure towhole-body mechanical vibration and repeated shock”. British StandardsInstitution, 1987.

27. Bohm, F and Wilhemeit, H-P (Eds). Tyre Models for Vehicle Dynamic Analysis.Proc 2nd Int Colloquium on Tyre models for vehicle dynamic analysis, Berlin1997. Supplement to Vehicle System Dynamics, vol 27. Swets and Zeitlinger,1997.

28. Thompson, W.T. Theory of Vibration with Applications, 3rd Ed. Unwin Hyman.1989.

29. Meirovitz, L. Elements of Vibration Analysis, 2nd Ed; McGraw-Hill, 1986.

30. Rao, S.S; Mechanical Vibrations, 3rd Ed, Addison-Wesley, 1995.

31. Dimaragonas, A.D and Haddad, S; Vibration for Engineers, Prentice-HallInternational, 1992.

32. Schwarzenbach, J and Gill, K.F. System Modelling and Control. Arnold, 1984

1-81

Appendix A1: Summary of vibration fundamentals [29,30,31]

The general approach to vibration analysis is to: Develop a mathematical model of the system and formulate the equations of

motion Analyse the free vibration characteristics (natural frequencies and modes) Analyse the forced vibration response to prescribed disturbances Investigate methods for controlling undesirable vibration levels if they arise.

Mathematical models

These provide the basis of all vibration studies at the design stage. The aim is torepresent the dynamics of a system by one or more differential equations. The mostcommon approach is to represent the distribution of mass, elasticity and damping in asystem by a set of discrete elements and assign a set of coordinates to the masses.

Elasticity and damping elements are assumed massless – there is a need to knowthe constituent equation describing the character of these elements

The number of degrees-of-freedom (DOFs) of the system is determined by thenumber of coordinates, e.g. the model of a simply-support beam in Figure A1.1

Each coordinate z1, z2.. etc is a function of time t The number of elements chosen dictates accuracy - aim to have just sufficient

elements to ensure that an adequate number of natural vibration modes andfrequencies can be determined while avoiding unnecessary computing effort

An n-DOF system will be described by n-second order differential equations andhave n-natural frequencies and modes.

The simplest model has one DOF

Formulating equations of motion

Equation(s) of motion can be determined by drawing a free-body diagram (FBD)of each mass including all relevant forces and applying Newton’s 2nd law

Each equation of motion is a second order differential equation For a multi-DOF system the equations can be assembled in matrix form Figure A1.2 shows the FBD’s of the quarter vehicle model discussed in Section

1.7.4 z1(t) and z2(t) are the generalised coordinates (measured from the static

equilibrium position). x0(t) represents the dynamic displacement input from theterrain

zn(t)z3(t)z2(t)z1(t)

Figure A1.1 A lumped-parameter model of a simply-supported beam

1-82

Applying Newton’s second law to the unsprung mass gives:

1u21s21s10t zMzzCzzKzxK

0t1t1t21s21s1u xKzKzKzzKzzCzM

Applying Newton’s second law to the sprung mass gives:

2s21s21s zMzzCzzK

0zzKzzCzM 21s21s2s

These equations can be re-expressed in matrix form as:

Ks (z1 – z2) Cs 21 zz

Kt (x0 – z1)

Mu111 ,, zzz

Ks (z1 – z2) Cs 21 zz

Ms 222 ,, zzz

z2(t)

z1(t)

x0(t)

(b) Free body diagrams

Figure A1.2 The quarter vehicle model and free-body diagramdiagrams

(a) Model

Ms

KsCs

Mu

Kt

1-83

0

xK

z

z

KK

KKK

z

z

CC

CC

z

z

M0

0M 0t

2

1

ss

sst

2

1

ss

ss

2

1

s

u

The equations can be written concisely as: tFzKzCzM

[M], [C] and [K} are the mass / inertia, damping and stiffness matrices andz,z z are displacement, velocity and acceleration vectors

{F(t)} is the excitation vector

Single degree of freedom systems

Knowledge of SDOF behaviour provides an understanding of more complexsystems.

For the SDOF system in Figure A1.3 the equation of motion is: tFzkzczm

The system characteristics are obtained from the free-vibration behaviour (F(t) =0)

With c = 0, there is simple harmonic motion tcosZz n ,

Z (the amplitude) and are arbitrary constants (determined from the conditions at t= 0) and n is the undamped natural frequency given by:

m

kn

Figure A1.3 Notation for SDOF system

When 0c , the level of damping can be described in terms the damping ratio defined as:

cc

c , mk2cc

When disturbed, the mass either returns to its equilibrium position with or withoutoscillation

Two possible characteristics: < 1, the system is under damped - decaying oscillation

m

k

F(t)

z(t)

c

1-84

1 , the system is over damped - the mass returns to its equilibrium without

oscillation When = 1 the system is critically damped

For an under damped system: tcoseZz dtn , where Z and are

arbitrary constants and d is the damped natural frequency given by:2

nd 1

If F(t) 0 the solution to the equation of motion has two components: the

Complementary Function (CF) and the Particular Integral (PI). The CF is identical to the free vibration solution and quickly dies away with

realistic levels of damping14 to leave z = PI. If F t F t 0 sin , the steady-state response of the mass (after the CF has become

zero) is given by: tsinAz

A() is the steady-state amplitude and is the phase lag - both are dependenton .

It may be shown that

222

0

200

cmk

F

icmk

1FHFA

H() is called the frequency response function (complex) - relates the input(excitation) to the output (response) in the frequency domain

The amplitude response can be presented in dimensionless form in terms of thedynamic magnifier 0FAkD and frequency ratio r n

It may be shown [30] that

2

n

2

n

21

1D

The variation of D for two different values of are shown in Figure A1.4:

0 1 2 3 40

1

2

3

4

5

6

Frequency ratio, r

Dynam

icm

agnif

ier,

D

Figure A1.4 Amplitude response in terms of D

Conclude from Figure A1.4 that:a) Maximum amplitude occurs at resonance - when n

14 In practice ranges from approximately 0.02 for low damping elastomers to 0.5 forvehicle suspensions.

= 0.1

= 0.3

1-85

a) The amplitude is strongly dependent on the damping in the system when n

b) When and n are very dissimilar, damping has very little effect on responseamplitude – an important point when considering the use of damping tocontrol vibration levels.

Multi-degree of freedom systems (MDOF-systems)

Equations of motion in matrix form: tFzKzCzM

Characteristics (F(t)=0), given by the solution of: 0zKzCzM

For negligible damping:([C] = [0]) and equations of motion are: 0zKzM

Assuming solutions of the form steAz leads to a set of homogeneous

equations: 0AKsM 2

The non-trivial solution can be written in the form of a characteristic equation (or

frequency determinant): 0KsM 2 or eigenvalue equation:

uKuM

This leads to a set of real roots, typically 2ii

2is , where i is the i-th

eigenvalue and i the i-th natural frequency. For each eigenvalue i, there is an eigenvector {u}i ,which relates the relative

amplitudes at each of the degrees of freedom. Vibration in the i-th mode then can be described by: iiii tsinAuz

i

The general free vibration of the systems is a combination of all modes:

tquzzn

1ii

, where: n21 u......uuu is called the modal

matrix and

q t

A t

A tn n n

( )

sin

sin

1 1 1

is a vector of modal (principal)

coordinates. [u] represents a linear transformation between generalised coordinates {z} and

modal coordinates {q} For a lightly damped system ( 0C ), the frequency determinant is:

0KsCsM 2

This equation has a set of complex conjugate roots having negative real parts -provides information about the frequency and damping associated with each modeof vibration

For harmonic excitation of damped MDOF systema) The system vibrates (in its steady state) at the same frequency as the excitationb) The dynamic displacement at each DOF lags behind the excitation.c) The displacement amplitudes at each of the degrees of freedom are dependent

on the frequency of excitation

1-86

In MDOF systems the excitation can be applied simultaneously at any of theDOFs.

For a linear system the response at any of the DOFs is the sum of the responsesdue to each excitation force.

When the excitation is applied simultaneously at all DOFs, F(t) can be written as:{F(t)} = {F}f(t) = {F}sin(t), then )t(fFzKzCzM

Take Laplace transforms of both sides with zero initial conditions, replace s with

i [32] and pre-multiply both sides by 12 KCiM

gives:

FHHx

{Hz()} is a vector of frequency responses at the DOFs

nn1n

n22221

n11211

HH

HHH

HHH

H

is a matrix of frequency response functions such

that Hij = frequency response at i due to unit amplitude excitation at j. The frequency response at DOF i is:

j

n

1jijiin22i11iiz FHFHFHFHH

The response at a particular DOF is made up of contributions from excitations atthe various DOFs in the systems.

FRFs and hence responses are complex (when damping is included) – amplitudeand phase at a given DOF is given by the modulus and argument of the complexresponse

Appendix A2: The Basic Handling Model and its Dynamics

Steady state analysis of a basic vehicle-handling model leads to some importantconclusions. First of all consider the following modelling assumptions:

1. The vehicle moves on a flat horizontal plane2. The forward velocity is constant3. There is no suspension compliance (rigid suspension, no body roll)4. No compliance in the steering mechanism5. Input to the model is a small steering angle f at the front axle6. No wheel camber7. No lateral load transfer8. No aerodynamic forces

The model is shown in Figure A2.1. The origin of the moving axes is at A, the vehiclec.g; and the following notation is used:

u = forward velocity (= U, the constant forward speed)v = lateral velocityr = yaw velocitym = vehicle mass

1-87

I = moment of inertia about the z-axisa,b = distance from A to front and rear axles

Figure A2.1 Basic handling model

Lateral forces at the front and rear axles can be lumped together to give a bicyclemodel – Figure A2.2. Longitudinal tyre forces are negligible for constant forwardvelocity, so two equations of motion are sufficient to describe the dynamics of thevehicle. In the lateral direction:

yryf FFR

Uvm

2

where R is the radius of the curve. Noting that RrU (linear velocity, U, equals theproduct of the radius, R, and the angular velocity, r):

1.2A..................yryf FFUrvm

2.2A..........................FbFarI yryf

Figure A2.2 Bicycle model

Ground-fixed

axis system G

X

Y

f

x

y

u

vr

Vehicle-fixed

axis system A Steering angle

x

y

U

vr

X

Y

G

AFyf

Fxf

Fyr

Fxr

a

b

1-88

Equations for the tyre forces can be derived from the definition of slip angle and tyrecharacteristics. Slip angle is given by:

3.2A............u

vtan

w

w

Relevant tyre characteristics are shown in Figure A2.3 where lateral tyre force Fy isplotted as a function of , for a range of vertical loads Fz, Cornering stiffness C is theslope of the appropriate curve at the origin.

Figure A2.3 Lateral tyre force as a function of slip angle

For small slip angles ( < 6)

4.2A............................CFy

The negative sign is required to make C positive consistent with the sign conventionused in the analysis. C is the cornering stiffness for the whole axle and depends on thevertical load. Fy is assumed to act in the y direction (based on the assumption that issmall).

It can be shown that the slip angles at the front and rear axles f and r are:

5.2A.......................U

ravff

6.2A...............................U

rbvr

Increasingvertical tyre load

Fz

Lateral tyreforce Fy, N

0 1 2 3 4 5 60

1000

2000

3000

4000

Corneringstiffness

C, is slopeat origin

Slip angle, deg

1-89

Lateral forces Fyf and Fyr are then:

7.2A..............U

ravCF ffyf

8.2A........................U

rbvCF ryr

Cf and Cr are cornering stiffness at front and rear axles. Fyf and Fyr can be substitutedinto equations A2.1 and A2.2. After some manipulation the following equationsresult.

These two first order differential equations can be written in matrix form:

With constant steering angle δf and forward speed U, and considering the steady stateresponse under these conditions, v, r are constant hence, 0rv , hence:

f

f

f

rfrf

rfrf

aC

C

r

v

U

CbCa

U

bCaC

U

bCaCmU

U

CC

22

Solving for r, which is denoted as rss to represent the steady state yaw rate, andrepresenting the terms of the 2x2 matrix on the left hand side as a11, a12 etc:

ff

ff

aC

C

r

v

aa

aa

2221

1211

ff

ff

aC

C

aa

aa

aaaar

v

1121

1222

12212211

1

10.2A........CrU

CbCaUmv

U

CCvm ff

rfrf

9.2A.......CarU

CbCav

U

CbCarI ff

r2

f2

rf

11.2A...Ca

C

r

v

U

CbCa

U

CbCa

U

CbCaUm

U

CC

r

v

I0

0mf

f

f

r2

f2

rf

rfrf

1-90

Hence r maybe found as:

ffff aCaCaaaaa

r 1121

12212211

1

,

Which simplifies to:

frrf

rf

aCbCmULCC

LCCUr

22

Considering that r is the yaw rate (an angular velocity) and in this steady statecondition, r = rss,

ssss UR

Ur

where R is the radius of the corner, and ss is the curvature.Hence:

frrf

rf

f

ss

aCbCmUCCL

LCC

22

2

1

KUL

with

LCC

aCbCmK

rf

fr

marginstability