Transcript Ride

Vehicle Ride
Dynamic System & Excitations
Vehicle Excitations:
1.
2.
3.
4.
Road profile & roughness
Tire & wheel excitation
Driveline excitation
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
Road
Elevation (mm)
V
X – distance (m)
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
Engine excitation
Wheel/tire,
Driveline excitation
Road roughness
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
Ride Frequency = 0.159√Ks/M
(mm)
Hence,
Ride frequency = 0.159√g/static deflection
(Hz)
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) ----------------------Where j = √-1 - complex operator
(5)
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
|H(f)|
Vehicle Response
Observations
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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 Calculation
Road profile acceleration power spectral density PSD
LOG Gzr(f) = -3.523
|H(f)|
f
when LOG(f) <= 0
LOG Gzr(f) = -3.523 + LOG(f)
when LOG(f) >= 0
Frequency Response Function |H(f)|
f
Gzs
Sprung mass acceleration power spectral density PSD
Gzs (f) = |H(f)|2 Gzr(f)
f
RMS acceleration = area under the curve
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
Acceleration PSD
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*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)
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 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 = 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
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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
Mass, C.G.
Roll Inertia
Pitch Inertia
Wheelbase, Tread
Road PSD
Vehicle
•Spring Rate
•Tire Rate
•Jounce/Rebound Clearance
•Shock Rate
•Unsprung Mass
RMS Acceleration
RMS Susp Travel
Frequencies
Olley’s Criteria