1. Vehicle Ride

2. Dynamic System & Excitations
Vehicle Excitations:
Road profile & roughness
Tire & wheel excitation
Driveline excitation
Engine excitation

3. 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
Elevation (mm)
Road
X – distance (m)

4. 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

5. Road Surface Power Spectral Density PSD

6. Tire&Wheel Assembly Excitation
Mass imbalance = m r ω2
Tire/wheel dimensional variation
Tire radial stiffness variation

7. 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

8. 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

9. Ride Isolation Engine excitation Wheel/tire, Driveline excitation
Road roughness
excitation

10. 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)

11. 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 %

12. 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)

13. Vehicle Response Equations 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

14. Vehicle Response Ar = K1*K2 Br = K1*C*2πf Aw = K2*(2πf)2 Bw = C*(2πf)3
Ab = μ*(2πf)4 – (K1+K2)*(2πf) 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

15. Vehicle Response
|H(f)|

16. 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 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

17. 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]

18. RMS Acceleration Calculation
Road profile acceleration power spectral density PSD
LOG Gzr(f) = when LOG(f) <= 0
LOG Gzr(f) = 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 f
|H(f)| f
Gzs f

19. 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

20. Allowable vibration levels

21. Suspension Stiffness
Note: softer suspension reduces acceleration level
Acceleration PSD

22. Suspension Damping
Note: higher damping ratio reduces resonance peak, but increases
gain at higher frequencies

23. Suspension Design

24. Wheel Hop Resonance
Wheel hop resonant frequency
fa = 0.159√(Kt+Ks)/m

25. 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*c2)/Mκ2
Kf = front ride rate
Kr = rear ride rate
b = as shown
c = as shown
Iy = pitch inertia
κ = radius of gyration
sqrt(Iy/M)

26. 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

27. Uncoupled Frequencies
Front 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*c2) = pitch stiffness
Kφ = (Kf+Kr)*t2/2 = roll stiffness
Iy = ML2 = pitch moment of inertia
Ix = Mt2 = roll moment of inertia
t = tread width and L = wheel base

28. 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

29. Suspension System Design
Mass, C.G.
Roll Inertia
Pitch Inertia
Wheelbase, Tread
Vehicle
Road PSD
RMS Acceleration
RMS Susp Travel
Frequencies
Olley’s Criteria
Spring Rate
Tire Rate
Jounce/Rebound Clearance
Shock Rate
Unsprung Mass