MECH3100Engineering Design

Lecture 6Roll centers for independent suspensions

Suspension Dynamics - simple modelsEstimating Design loads

In the last lecture we ...

• … looked at the roles the control arms serve in acceleration and braking.

• … developed design guidelines for control arm geometry based on anti-squat and anti-dive and saw there’s a design compromise to make - you can’t simultaneously realize both.

This final lecture on suspections

• … looks at how the suspension geometry determines suspension roll centers and the vehicle roll-axis.

• … introduces simple models for the dynamics of suspensions.

• … looks at how we might use Newton’s 2nd Law to estimate loads acting on the suspension

Vehicle roll axis

• Roll axis: instantaneous axis is about which the unsprung mass rotates with respect to the sprung mass when a pure couple is applied to the unsprung mass.

Based on Fig 7.14 of T. Gillespie,Fundamentals of vehicle dynamics, SAE Press,1992, pp 258.

Suspension roll centers

• Suspension roll center: point in the transverse vertical plane through the wheel centers at which lateral forces may be applied to the sprung mass without producing suspension roll.

• Both front and rear suspensions will have role centres

Role center for a solid axle

• In a plan view of the suspension, find the linkages that react the side force. Then find where the projection of the linkages crosses the vehicle centerline.

• In a side view, find the same two points and connect them. That’s the suspension roll axis.

• The roll center is the point where the roll axis is over the wheel center.

Note that this is an instantaneous roll center.It changes as the body rolls.

Solid Axle Roll CenterRef. T. Gillespie, Fundamentals of Vehicle Dynamics, SAE Press, pp 260

Plan view

Side view

Roll center analysis of a four-link rear suspension.

SLA suspension - equivalent four-bar mechanism

A

B C

D

E

ICDE

Instantaneous point about which C, D, & E rotaterelative to the body of the car

Instantaneous Center

• Fictitious point about which the wheel (instantaneously) rotates under constraints provided control links

Independent suspension

AR

BRCR

DR

ER

AL

BLCL

DL

EL

IL IR

CL

Independent Suspension Roll Centers

• In a front view, locate the point about which the wheel rotates (Virtual Reaction point or Instantaneous Center). For double-A arms, this is the intersection of the projection of the arms.

• Draw a line from the tire-ground contact point to the virtual reaction pt. (A)

• Where the line crosses the centerline of the body is the suspension roll center.

Independent Suspension

Positive swing arm independent suspension.Called (+) because roll ctr is above ground

Ref. T. Gillespie, Fundamentals of Vehicle Dynamics, SAE Press

Recap - vehicle roll axis

• We construct the roll axis by drawing a line through the front and rear suspension roll centres.

Based on Fig 7.14 of T. Gillespie,Fundamentals of vehicle dynamics, SAE Press,1992, pp 258.

AR

BRCR

DR

ER

AL

BLCL

DL

EL

IL IR

Relationship between suspension motion and chassismotion

As wheelmoves down

This point moves up and chassis rolls

Imagine this link is fixed

CL

AR

BRCR

DR

ER

AL

BLCL

DL

EL

IL IR

Suspension motion from cornering

COG2m rω

Right tyre move down and point inwards

Left tyre moves up and out

Suspension geometry chassis roll

• The suspension geometry controls the vehicles roll axis.

• Desire your suspension keep tyres close to vertical during cornering

• Some designs seek to have the roll axis pass as near as possible to the centre of gravity of the vehicle to minimize the roll moment.

• But this is not a universal design objective!

Negative roll centre

(-) because roll ctr lies below ground

Negative swing arm independent suspension.

Zero roll centre

Parallel horizontal link independent suspension.

Positive roll centre

Inclined parallel link independent suspension.

Independent Suspension

• Roll center can change if there is body roll

no body roll

with body roll

Ref. Milliken &Milliken, Race Car Vehicle Dynamics, SAE Press

Dynamics: 1/4 Car Suspension• First approximation: consider the 1/4

car, single DOF system:independent suspensionmCar body

(sprung mass)z

k b

M = 1/4 of total car massk = combined tyre and suspension stiffness b = combined tyre and suspension damping

Ride comfort• One function of the suspension is to isolate the

chassis from the road. • Ride comfort is a measure of this and is affected

by – high frequency vibrations – body roll and pitch– vertical spring action

• Ride quality normally associated with the vehicles response to bumps is a function of the bounce and rebound movements of the suspension.

• Following a bump the undamped vehicle with experience oscillations that cycle at the natural frequency of the ride.

Ride comfort• Ride is perceived as most comfortable when the

natural frequency is in range of 1 to 1.5 Hz. • A high performance car will typically have a stiffer

suspension with a natural frequency of 2 to 2.5 Hz. • Sensitivity to frequency was at one time thought to

be associated with the natural oscillations of the body during walking (70 to 90 steps per minute with 5 cm vertical oscillation.)

• Early suspension design tried to mimic this.

Human sensitivity to vibration

• 0.5 to 0.8 Hz produces motion sickness• 5 to 6 Hz adversely affects the visceral

regions• 18 to 20 Hz is bad for the head and neck. • Humans are most uncomfortable with

longitudinal vibrations in range 1-2 Hz. The frequency at which most comfortable with vertical vibrations

Vibrational Characteristics• The unforced equations of motion of this system

are

• Which has natural frequency and damping

0=++ kzzbzm ⋅

2nk b

m mkω ζ= =

Vibrational Characteristics of Suspensions

• For the suspension that we’re modeling, the input usually comes from a road disturbance, not a force on the car body.

m Zc sprung massdisplacement

b kZr road displacement

Vibrational Characteristics of Suspensions

• We can look at the relative motion of the car with respect to the the road for a range of frequencies.

transmissibilityplot

System Dynamics• What do the system dynamics tell us?• If we known m we can choose k and b to satisfy

various goals based on requirements for natural frequency and damping.

• Natural frequency is 2 to 2.5 Hz. • ζ ≈ .2 to .4 for most cars.• This analysis is much simplified

– Damping in jounce and rebound are not usually equal.

– Usual to have lower damping on jounce than rebound.

½ Car Model and Coupling Effect

forward velocity

ms, I θzsr

za b

mur muf

zsf

ksr bsr ksf bsf

zur zuf

ktfktr zrr zrf

Equations of Motion

rfs ffzm +=

bfafI rf ⋅−⋅=θ

)( rfuftffufuf zzkfzm −−−=

)( rrurtrrurur zzkfzm −−−=

Bounce and Pitch

• If the road wavelength is equal to the vehicle wheelbase, or has an integer multiple equal to the wheelbase, then the ½ car model will experience pure bounce.

• If the road wavelength is equal to twice the wheelbase, or has an odd integer multiple equal to twice the wheelbase, then the ½ car will experience pitch.

Bounce and Pitch

⇒

Estimating design loads• We’ve been concentrating on roles of a suspension • Remember the suspension acts as the interface

between car and road and sees various loads– React the drive and braking forces– React cornering and other lateral loads– React vertical loads, e.g. those due to static

weight and those generated during acceleration.• Suspension has to be strong enough to withstand

these loads - I.e. in addition to basic kinematic issues discussed suspension should be designed for structural integrity under static loads and fatigue

How do we estimate design loads?• Intimately linked with suspension geometry. • In practices advanced modelling packages are used to

estimate loads, e.g. ADAMS • Suggest you make simple estimates using basic physics,

e.g. Newton’s second law.• A strategy might be to identify component ‘loads’ at

wheels under steady state conditions– Static weight– Cornering and other lateral forces– Motive forces– etc.

• Use these with appropriate multi-axis stress failure predictor (Von Mises) and factors of safety.

Estimating design loads • Vertical forces

– Static weight - mass of vehicle and centre of mass

– Acceleration forces - How quickly can vehicle accelerate/decelerate

• Identify component of reaction in each link

Lb a

h

V

FzrFzf

Fx

Force decreasesby m ax h/LForce increases

by m ax h/L

Estimating design loads • Drive forces

– What is the maximum drive torque at the wheels?

– What are the braking torques. – What forces do these generate at the tyres?

• Identify component of reaction in each link? L

b a

V

h

Fx Fz Fzr f

Estimating design loads • Lateral forces, e.g. those due to cornering

– What is the ‘tightest corner the car will take’– How fast will it take turn.– Note interactions with other loads, e.g vertical

loads on tyres• Identify component of reaction in each link

c

Force decreasesby mw2rh/c

Force increasesby mw2rh/c

c

F = mw2r

h