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Published byStephany Parsons Modified about 1 year ago

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Vehicle Ride

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Dynamic System & Excitations Vehicle Excitations: 1.Road profile & roughness 2.Tire & wheel excitation 3.Driveline excitation 4.Engine excitation

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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) Road Elevation (mm)

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Statistical Road Profile G z (ν ) = G 0 [1+(ν 0 /ν) 2 ]/(2πν) 2 Where G z (ν) = PSD amplitude (feet2/cycle/foot) = wave number (cycle/ft) G 0 = roughness parameter = 1.25 x 105 – rough roads = 1.25 x 106 – smooth roads ν 0 = cut-off wave number = 0.05 cycle/foot – asphalt road = 0.02 cycle/foot - concrete road

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Road Surface Power Spectral Density PSD

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Tire&Wheel Assembly Excitation Mass imbalance = m r ω 2 Tire/wheel dimensional variation Tire radial stiffness variation

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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

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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

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

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Suspension Parameters M – Sprung mass, kg (body, frame, engine, transmission, etc.) m – Unsprung mass, kg (driveline, wheel assembly, chassis, etc.) K s – Suspension stiffness, N/mm (spring stiffness) K t - Tire Stiffness, N/mm (tire stiffness) C s - Suspension damping, N.sec/m (damper) Z – sprung mass displacement Z u – unsprung mass displacement Z r - road elevation F b – Force on the sprung mass (engine excitation) F w – Force on the unsprung mass (wheel/tire or driveline excitation)

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Ride Properties Ride Rate, RR = K s *K t /(K s + K t ) N/mm Ride Frequency f n = √RR/M/(2*π) Hz Damped Frequency, f d = f n √1-ξ 2 Hz Where ξ = damping ratio = C s /√4K s M %

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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)

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Vehicle Response Equations of Motion M*Z” + C s *Z’ + K s *Z = C s *Z’ u + K s *Z u + F b --------------------- (1) m*Z” u + C s *Z’ u +(K s +K t )*Z u = C s *Z’ + K s *Z + K t *Z r + F w- --- (2) Dynamic Frequency Responses: Z”/Z” r = H r (f) = (A r + j B r )/(D + j E) ---------------------------- (3) MZ”/F w = H w (f) = (A w + j B w )/(D + j E) ----------------------- (4) MZ”/F b = H b (f) = (A b + j B b )/(D + j E) ----------------------- (5) Where j = √-1 - complex operator

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Vehicle Response A r = K 1 *K 2 B r = K 1 *C*2πf A w = K 2 *(2πf) 2 B w = C*(2πf) 3 A b = μ*(2πf) 4 – (K 1 +K 2 )*(2πf) 2 B b = C*(2πf) 3 D = μ*(2πf) 4 – (K 1 +K 2 *μ+K 2 )* (2πf) 2 + K 1 *K 2 E = K 1 *C*(2πf) – (1+μ)*C*(2πf) 3 And μ = m/M, C = C s /M, K 1 = K t /M, K 2 = K s /M

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Vehicle Response |H(f)|

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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

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Isolation of Road Acceleration G z (f) = |H r (f)| 2 *G zr (f) Where: G z (f) = acceleration PSD of the sprung mass H(f) = response gain for road input G zr (f)= acceleration PSD for the road input RMS acceleration = sqrt [area under G z (f) vs f curve]

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RMS Acceleration Calculation Road profile acceleration power spectral density PSD LOG G zr (f) = -3.523 when LOG(f) <= 0 LOG G zr (f) = -3.523 + LOG(f) when LOG(f) >= 0 Frequency Response Function |H(f)| Sprung mass acceleration power spectral density PSD G zs (f) = |H(f)| 2 G zr (f) RMS acceleration = area under the curve G zr Gzs |H(f)| f f f

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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

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Allowable vibration levels

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Suspension Stiffness Acceleration PSD Note: softer suspension reduces acceleration level

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

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Suspension Design

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Wheel Hop Resonance Wheel hop resonant frequency f a = 0.159√(K t +K s )/m

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Bounce/Pitch Frequencies Equations of Motion Z” + αZ + βθ = 0 θ” + βZ/κ 2 + γθ = 0 Where, α = (K f +K r )/M β = (K r *c-K f *b)/M γ = (K f *b 2 +K r *c 2 )/Mκ 2 K f = front ride rate K r = rear ride rate b = as shown c = as shown I y = pitch inertia κ = radius of gyration sqrt(I y /M)

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Bounce/Pitch Frequencies ω 1 2 = (α+γ)/2 + (α-γ) 2 /4+ β 2 κ 2 ω 2 2 = (α+γ)/2 - (α-γ) 2 /4+ β 2 κ 2 f 1 = ω 1 /2π Hz f 2 = ω 2 /2π Hz

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Uncoupled Frequencies Front Ride Frequency = √K f /M /(2π) Hz Rear Ride Frequency = √K r /M /(2π) Hz Pitch Frequency = √K θ /I y /(2π) Hz Roll Frequency = √K φ /I x /(2π) Hz Where K θ = (K f *b 2 +K r *c 2 ) = pitch stiffness K φ = (K f +K r )*t 2 /2 = roll stiffness I y = 0.2154ML 2 = pitch moment of inertia I x = 0.1475Mt 2 = roll moment of inertia t = tread width and L = wheel base

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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 = κ 2 /bc >= 1, happens for cars with substantial overhang. Pitch frequency < bounce frequency, front ride frequency < rear ride frequency, good ride

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

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