Vehicle Ride

Dynamic System & Excitations

Vehicle Excitations:

1. Road profile & roughness2. Tire & wheel excitation3. Driveline excitation4. Engine excitation

Road Excitation

• Road excitation is the road profile or the road elevation along the road and includes everything from smooth roads, potholes to “kurangkan laju”

• Road elevation profiles are measured using high speed profilometers

V

X – distance (m)

Roa

d E

leva

tion

(mm

)

Statistical Road Profile

Gz(ν ) = G0[1+(ν0/ν)2]/(2πν)2

Where

Gz(ν) = PSD amplitude (feet2/cycle/foot)

= wave number (cycle/ft)

G0 = roughness parameter

= 1.25 x 105 – rough roads = 1.25 x 106 – smooth roads

ν0 = cut-off wave number

= 0.05 cycle/foot – asphalt road = 0.02 cycle/foot - concrete road

Road Surface Power Spectral Density PSD

Tire&Wheel Assembly Excitation

• Mass imbalance = m r ω2

• Tire/wheel dimensional variation

• Tire radial stiffness variation

Driveline Excitation

• Mass imbalance– Asymmetry of rotating parts– Shaft may be off-center on its supporting flange– Shaft may not be straight– Shaft is not rigid and may deflect

Engine Excitation

• Torque output to the drive shaft from the piston engine is not uniform. It has 2 components– Steady state component– Superimposed torque variations

Ride Isolation

Road roughnessexcitation

Wheel/tire,Driveline excitation

Engine excitation

Suspension Parameters

M – Sprung mass, kg (body, frame, engine, transmission, etc.)m – Unsprung mass, kg (driveline, wheel assembly, chassis, etc.)

Ks – Suspension stiffness, N/mm (spring stiffness)

Kt - Tire Stiffness, N/mm (tire stiffness)

Cs - Suspension damping, N.sec/m (damper)

Z – sprung mass displacement

Zu – unsprung mass displacement

Zr - road elevation

Fb – Force on the sprung mass (engine excitation)

Fw – Force on the unsprung mass (wheel/tire or driveline excitation)

Ride Properties

Ride Rate, RR = Ks*Kt/(Ks + Kt) N/mm

Ride Frequency fn = √RR/M/(2*π) Hz

Damped Frequency, fd = fn √1-ξ2 Hz

Where

ξ = damping ratio = Cs/√4KsM %

Suspension Travel

Static suspension deflection = W/Ks = Mg/Ks (mm)Ride Frequency = 0.159√Ks/M

Hence,

Ride frequency = 0.159√g/static deflection (Hz)

Vehicle ResponseEquations of Motion

M*Z” + Cs*Z’ + Ks*Z = Cs*Z’u + Ks*Zu + Fb --------------------- (1)

m*Z”u + Cs*Z’u +(Ks+Kt)*Zu = Cs*Z’ + Ks*Z + Kt*Zr + Fw- --- (2)

Dynamic Frequency Responses:

Z”/Z”r = Hr(f) = (Ar + j Br)/(D + j E) ---------------------------- (3)

MZ”/Fw = Hw(f) = (Aw + j Bw)/(D + j E) ----------------------- (4)

MZ”/Fb = Hb(f) = (Ab + j Bb)/(D + j E) ----------------------- (5)

Where j = √-1 - complex operator

Vehicle Response

Ar = K1*K2 Br = K1*C*2πf

Aw = K2*(2πf)2 Bw = C*(2πf)3

Ab = μ*(2πf)4 – (K1+K2)*(2πf)2 Bb = C*(2πf)3

D = μ*(2πf)4 – (K1+K2*μ+K2)* (2πf)2 + K1*K2

E = K1*C*(2πf) – (1+μ)*C*(2πf)3

And μ = m/M, C = Cs/M, K1 = Kt/M, K2 = Ks/M

Vehicle Response

|H(f

)|

Observations• At low frequency, gain is unity. Sprung mass moves as the road input• At about 1 Hz, sprung mass resonates on suspension with amplification• Amplitude depends on damping, 1.5 to 3 for cars, up to 5 for trucks• Above resonant frequency, response is attenuated• At 10-12 Hz, un-sprung mass goes into resonance (wheel hop)• Sprung mass response gain to wheel excitation is 0 at 0 frequency as

the force on the axle is absorbed by the tire• Resonance occurs at wheel hop frequency, gain is 1 and axle force

variation is directly transferred to sprung mass• Sprung mass response gain to engine excitation reaches maximum at

sprung mass resonance• At higher frequencies gain becomes unity as displacements become

small, suspension forces do not change and engine force is absorbed by sprung mass acceleration

Isolation of Road Acceleration

Gz(f) = |Hr(f)|2*Gzr(f)

Where: Gz(f) = acceleration PSD of the sprung mass

H(f) = response gain for road input

Gzr(f)= acceleration PSD for the road input

RMS acceleration = sqrt [area under Gz(f) vs f curve]

RMS Acceleration CalculationRoad profile acceleration power spectral density PSD

LOG Gzr(f) = -3.523 when LOG(f) <= 0

LOG Gzr(f) = -3.523 + LOG(f) when LOG(f) >= 0

Frequency Response Function |H(f)|

Sprung mass acceleration power spectral density PSD

Gzs (f) = |H(f)|2 Gzr(f)

RMS acceleration = area under the curve

Gzr

Gzs

|H(f

)|

f

f

f

RMS Acceleration Calculation

Step 1 : Calculate road surface PSD for each frequency from 0.1 Hz to 20 Hz

Step 2 : Frequency response function for each frequency from 0.1 Hz to 20 Hz

Step 3 : Calculate vehicle acceleration PSD for each frequency from 0.1 Hz to 20 Hz

Step 4: Calculate area under the curve found in Step 3.

Step 5: That is RMS acceleration. 99% confidence that the vehicle acceleration will not exceed 3*RMS

Allowable vibration levels

Suspension Stiffness

Acc

eler

atio

n P

SD

Note: softer suspension reduces acceleration level

Suspension Damping

Note: higher damping ratio reduces resonance peak, but increases gain at higher frequencies

Suspension Design

Wheel Hop Resonance

Wheel hop resonant frequency

fa = 0.159√(Kt+Ks)/m

Bounce/Pitch Frequencies

Equations of Motion

Z” + αZ + βθ = 0

θ” + βZ/κ2 + γθ = 0

Where, α = (Kf+Kr)/M

β = (Kr*c-Kf*b)/M

γ = (Kf*b2+Kr*c

2)/Mκ2

Kf = front ride rate

Kr = rear ride rate

b = as shownc = as shown

Iy = pitch inertia

κ = radius of gyration

sqrt(Iy/M)

Bounce/Pitch Frequencies

ω12 = (α+γ)/2 + (α-γ)2/4+ β2κ2

ω22 = (α+γ)/2 - (α-γ)2/4+ β2κ2

f1 = ω1/2π Hz

f2 = ω2/2π Hz

Uncoupled FrequenciesFront Ride Frequency = √Kf/M /(2π) Hz

Rear Ride Frequency = √Kr/M /(2π) Hz

Pitch Frequency = √Kθ/Iy /(2π) Hz

Roll Frequency = √Kφ/Ix /(2π) Hz

Where

Kθ = (Kf*b2+Kr*c

2) = pitch stiffness

Kφ = (Kf+Kr)*t2/2 = roll stiffness

Iy = 0.2154ML2 = pitch moment of inertia

Ix = 0.1475Mt2 = roll moment of inertia

t = tread width and L = wheel base

Olley’s criteria for good ride

• Spring center should be at least 6.5% of the wheelbase behind C.G.• Rear ride frequency should be higher than the front• Pitch and bounce frequencies should be close to each other• Bounce frequency < 1.2 * pitch frequency• Neither frequency should be greater than 1.3 Hz• Roll frequency should be close to bounce and pitch frequencies• Avoid spring center at C.G., poor ride due to uncoupled motion

• DI = κ2/bc >= 1, happens for cars with substantial overhang. Pitch frequency < bounce frequency, front ride frequency < rear ride frequency, good ride

Suspension System Design

Vehicle

•Spring Rate•Tire Rate•Jounce/Rebound Clearance•Shock Rate•Unsprung Mass

Mass, C.G.Roll Inertia

Pitch InertiaWheelbase, Tread

RMS AccelerationRMS Susp TravelFrequenciesOlley’s Criteria

Road PSD