Vehicle Ride

Dynamic System & Excitations Vehicle Excitations: 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 V Elevation (mm) Road 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 (mm) Ride Frequency = 0.159√Ks/M 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) ----------------------- (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 Calculation Road 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 f |H(f)| f Gzs 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 Note: softer suspension reduces acceleration level Acceleration PSD

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 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 Vehicle Road PSD RMS Acceleration RMS Susp Travel Frequencies Olley’s Criteria Spring Rate Tire Rate Jounce/Rebound Clearance Shock Rate Unsprung Mass