vehicle ride. dynamic system & excitations vehicle excitations: 1.road profile & roughness...
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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
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