Dr. B DayalDr. B Dayal

Dr. B DayalDr. B Dayal

RIDERIDE

RIDERIDE

The term “ride” is commonly used in reference to tactile and visual The term “ride” is commonly used in reference to tactile and visual vibrations, while the aural vibrations are categorized as “noise”.vibrations, while the aural vibrations are categorized as “noise”.

Ride – 0 – 25 HzRide – 0 – 25 Hz

Noise – 25 – 20000 HzNoise – 25 – 20000 Hz

25 Hz islower frequency threshold of hearing25 Hz islower frequency threshold of hearing

Noise is usually present when lower frequency vibrations are Noise is usually present when lower frequency vibrations are excited.excited.

Vehicle is a dynamic system, but only exhibits vibrations in Vehicle is a dynamic system, but only exhibits vibrations in response to excitation inputs.response to excitation inputs.

The response properties determine the magnitude and direction The response properties determine the magnitude and direction of vibrations imposed on passenger compartment.of vibrations imposed on passenger compartment.

THE RIDE DYNAMIC SYSTEMTHE RIDE DYNAMIC SYSTEM

Excitation Excitation sourcessourcesRoad Road roughnessroughnessTyre / wheelTyre / wheelDrive lineDrive lineengineengine

Vehicle Vehicle dynamic dynamic responsresponsee

VibrationsVibrations

Ride Ride perceptioperceptio

nn

EXCITATION SOURCESEXCITATION SOURCESROAD ROUGHNESS. Encompasses everything from potholes to ROAD ROUGHNESS. Encompasses everything from potholes to

the ever-present random deviations reflecting the practical limits the ever-present random deviations reflecting the practical limits of precision to which the road surface can be constructed and of precision to which the road surface can be constructed and maintained. Road profile fit the general category of “broad – maintained. Road profile fit the general category of “broad – band random signals” and hence can be described either by band random signals” and hence can be described either by profile itself or its statistical properties. The most useful profile itself or its statistical properties. The most useful representation is Power Spectral Density (PSD) function. A plot of representation is Power Spectral Density (PSD) function. A plot of the amplitude versus spatial frequency is the PSD.the amplitude versus spatial frequency is the PSD.

PSd for average road properties can be represented by:PSd for average road properties can be represented by:GGzz (v) = G (v) = G00 [1 + ( [1 + (νν00/ / νν))22]]

Where,Where, G Gzz (v) = PSD amplitude; (v) = PSD amplitude; νν = wave number; G = wave number; G00 = roughness = roughness magnitude parameter = 1.25 x 10magnitude parameter = 1.25 x 1055 for rough roads and 1.25 x 10 for rough roads and 1.25 x 1066 for smooth roads; for smooth roads; νν00 = cut off number = 0.5 cycle/foot for = cut off number = 0.5 cycle/foot for bituminous road and 0.02 for PCC roads.bituminous road and 0.02 for PCC roads.

ON-BOARD SOURCES: arise from rotating componentsON-BOARD SOURCES: arise from rotating componentsTyre / wheel assembliesTyre / wheel assembliesThe drivelineThe drivelineEngineEngine

TYPICAL SPECTRAL DENSITIES OF TYPICAL SPECTRAL DENSITIES OF ROAD ELEVATION PROFILEROAD ELEVATION PROFILE

ELEVATION, VELOCITY AND ACCELERATION PSDs OF ELEVATION, VELOCITY AND ACCELERATION PSDs OF ROAD ROUGHNESS INPUT IN A VEHICLE TRAVELLING AT ROAD ROUGHNESS INPUT IN A VEHICLE TRAVELLING AT

50 MPH ON A REAL AND AVERAGE ROAD50 MPH ON A REAL AND AVERAGE ROAD

EXCITATION SOURCESEXCITATION SOURCESROAD ROUGHNESS. ROAD ROUGHNESS. At any given temporal frequency the amplitude of the acceleration At any given temporal frequency the amplitude of the acceleration

input will increase with the square of speed.input will increase with the square of speed.ZZrr = A sin (2 = A sin (2πνπνX)X)

Where,Where, ZZrr = profile elevation = profile elevationA = amplitudeA = amplitudeνν = wave number = wave numberX = distance along the roadX = distance along the road

SincSinc X = VtX = Vt ZZrr = A sin (2 = A sin (2πνπνVt)Vt)

Z’’Z’’rr = (2 = (2πνπνV)V)22 A sin (2 A sin (2πνπνVt)Vt) ON-BOARD SOURCES:ON-BOARD SOURCES:Tyre / wheel assemblies. Have got nonuniformities as:Tyre / wheel assemblies. Have got nonuniformities as:

Mass imbalanceMass imbalanceDimensional variationsDimensional variationsStiffness variationsStiffness variations

SPECTRAL DENSITY OF NORMALISED SPECTRAL DENSITY OF NORMALISED ROLL INPUT FOR A TYPICAL ROADROLL INPUT FOR A TYPICAL ROAD

EXCITATION SOURCESEXCITATION SOURCES ON-BOARD SOURCES:ON-BOARD SOURCES:Tyre / wheel assemblies. Have got nonuniformities asTyre / wheel assemblies. Have got nonuniformities as

Mass imbalanceMass imbalanceDimensional variationsDimensional variationsStiffness variationsStiffness variations

These nonuniformities all combine in a tyre / wheel assembly causing it to These nonuniformities all combine in a tyre / wheel assembly causing it to experience variations in the forces and moments at the ground as it experience variations in the forces and moments at the ground as it rolls. These in turn are transmitted to the axle and act as excitation rolls. These in turn are transmitted to the axle and act as excitation sources.sources.

The force variations may be in vertical , longitudinal or in lateral The force variations may be in vertical , longitudinal or in lateral directions.directions.

imbalance force Fimbalance force Fii = mr = mrωω22

Where Where F Fii = imbalance force = imbalance forcemr = the imbalance magnitudemr = the imbalance magnitudeωω = the rotational speed. = the rotational speed.

A non uniform and asymmetric mass distribution along the axis of rotation A non uniform and asymmetric mass distribution along the axis of rotation causes a dynamic imbalance., which creates a rotating torque on the causes a dynamic imbalance., which creates a rotating torque on the wheel, appearing as variations in over turning moments and aligning wheel, appearing as variations in over turning moments and aligning torque at the wheel rotational frequency.torque at the wheel rotational frequency.

TYRE RADIAL SPRING TYRE RADIAL SPRING MODELMODEL

EXCITATION SOURCESEXCITATION SOURCESRadial force Radial force variations take the form as illustrated in figure.variations take the form as illustrated in figure.The peak – to – peak magnitude of force variation is called The peak – to – peak magnitude of force variation is called

“composite force variation”. The force signature may be “composite force variation”. The force signature may be described by the amplitude of harmonics, that is by Fourier described by the amplitude of harmonics, that is by Fourier transform.transform.

The first harmonic of the radial force variation tends to be less The first harmonic of the radial force variation tends to be less than that of composite. And the higher harmonics tend to be of than that of composite. And the higher harmonics tend to be of diminishing magnitude.diminishing magnitude.

The various harmonics are functionally equivalent to The various harmonics are functionally equivalent to imperfection in the shape.imperfection in the shape. Eccentricity. The tyres, wheels and hubs may exhibit radial eccentricity, resulting Eccentricity. The tyres, wheels and hubs may exhibit radial eccentricity, resulting

in first harmonic nonuniformity which produces both radial and tractive excitation in first harmonic nonuniformity which produces both radial and tractive excitation on the axle. The excitation occurs at the rotational speed of wheel (10 – 15 Hz at on the axle. The excitation occurs at the rotational speed of wheel (10 – 15 Hz at normal high way speed).normal high way speed).

Ovality. Radial and tractive force excitation is produced at twice the wheel Ovality. Radial and tractive force excitation is produced at twice the wheel rotational frequency (20 – 30 Hz)rotational frequency (20 – 30 Hz)

Higher order radial variations. Predominantly of importance in the tyre only.Higher order radial variations. Predominantly of importance in the tyre only. Tractive force. Tractive force. Tractive force variations may arise from dimensional Tractive force variations may arise from dimensional

and stiffness nonuniformity as a result of two effects. Magnitude will be and stiffness nonuniformity as a result of two effects. Magnitude will be dependent on load and amount of eccentricity. Independent of speed.dependent on load and amount of eccentricity. Independent of speed.

TYRE RADIAL FORCE VARIATIONSTYRE RADIAL FORCE VARIATIONS

RADIAL NONUNIFORMITIES IN A TYRERADIAL NONUNIFORMITIES IN A TYRE

TRACTIVE FORCE VARIATIONS TRACTIVE FORCE VARIATIONS ARISING FROM AN ECCENTRIC ARISING FROM AN ECCENTRIC

WHEELWHEEL

EXCITATION SOURCESEXCITATION SOURCESLateral force Lateral force variations may arise from nonuniformities of tyre, variations may arise from nonuniformities of tyre,

they tend to be independent of speed. These will affect dynamic they tend to be independent of speed. These will affect dynamic balance of the assembly. Higher order variations are balance of the assembly. Higher order variations are predominantly important in the tyre only. Wheel variations are predominantly important in the tyre only. Wheel variations are absorbed by tyre.absorbed by tyre.

Drive line excitations.Drive line excitations.The drive line consists of drive shaft, gear reduction, differential and The drive line consists of drive shaft, gear reduction, differential and

axle shaft. Drive shaft is the most important for vibrations. axle shaft. Drive shaft is the most important for vibrations. Others are also capable of generating vibrations in the nature of Others are also capable of generating vibrations in the nature of noise. Excitation to the vehicle arise from mass imbalance, and noise. Excitation to the vehicle arise from mass imbalance, and secondary couples, or moments, imposed on the drive shaft.secondary couples, or moments, imposed on the drive shaft.

Mass imbalance. Mass imbalance. An initial imbalance exists as a result of the An initial imbalance exists as a result of the asymmetry given below:asymmetry given below:

Asymmetry of the rotating partsAsymmetry of the rotating parts The shaft may be off center on its supporting flange or end yoke.The shaft may be off center on its supporting flange or end yoke. The shaft may not be straight.The shaft may not be straight. Running clearances may allow the shaft to run off center.Running clearances may allow the shaft to run off center. The shaft is an elastic member and may deflect.The shaft is an elastic member and may deflect.

TYPICAL DRIVELINE TYPICAL DRIVELINE ARRANGEMENTSARRANGEMENTS

TORQUE REACTIONS CAUSING A TORQUE REACTIONS CAUSING A SECONDARY COUPLESECONDARY COUPLE

EXCITATION SOURCESEXCITATION SOURCESDrive line excitations.Drive line excitations.Mass imbalance. Mass imbalance. Magnitude of the excitation force is equivalent to the product of the Magnitude of the excitation force is equivalent to the product of the

imbalance and the speed square.imbalance and the speed square.Secondary couples.Secondary couples.universal joints are direct source oftorque pulsations in the universal joints are direct source oftorque pulsations in the

driveline.driveline.ωω00 / / ωωii = cos = cos θθ / (1 – sin / (1 – sin22 ββ sin sin22 θθ))

Where,Where, ωω00 = output speed = output speed ωωii = input speed = input speed

θθ = angle of the U – joint = angle of the U – jointββ = angle of rotation of driving yoke = angle of rotation of driving yoke

Maximum speed variation changes with joint angle as:Maximum speed variation changes with joint angle as: |ω|ω00 / / ωωii||maxmax = 1 / cos = 1 / cos θθ

The excitation occurs at the second harmonic of the drive line speed.The excitation occurs at the second harmonic of the drive line speed.

SPECTRAL MAP OF VIBRATIONS SPECTRAL MAP OF VIBRATIONS ARISING FROM DRIVELINE AND TYRE / ARISING FROM DRIVELINE AND TYRE /

WHEELWHEEL

EXCITATION SOURCESEXCITATION SOURCESEngine / transmission.Engine / transmission.Engine may be a source for vibration excitation on the vehicle. At Engine may be a source for vibration excitation on the vehicle. At

the crank shaft the torque delivered consists of a series of pulses the crank shaft the torque delivered consists of a series of pulses corresponding to each power stroke.corresponding to each power stroke.

Because of compliance in the engine / transmission mounts, the Because of compliance in the engine / transmission mounts, the system will vibrate in six directions – the three translational system will vibrate in six directions – the three translational directions and three rotations around the translational axis.directions and three rotations around the translational axis.

The most important to vibration is the engine roll direction (about The most important to vibration is the engine roll direction (about the lateral axis of a transverse engine or about the longitudinal the lateral axis of a transverse engine or about the longitudinal axis of an engine mounted in the north – south direction), which is axis of an engine mounted in the north – south direction), which is excited by drive torque oscillations.excited by drive torque oscillations.

TORQUE VARIATIONS AT THE OUTPUT TORQUE VARIATIONS AT THE OUTPUT OF A FOUR STROKE, FOUR CYLINDER OF A FOUR STROKE, FOUR CYLINDER

ENGINEENGINE

TYPICAL TRANSVERSE ENGINE TYPICAL TRANSVERSE ENGINE AND MOUNTING HARDWAREAND MOUNTING HARDWARE

VEHICLE RESPONSE VEHICLE RESPONSE PROPERTIESPROPERTIES

Transmissibility.Transmissibility.Transmissibility is the non-dimensional ratio of response amplitude Transmissibility is the non-dimensional ratio of response amplitude

to excitation amplitude for a system in steady – state forced to excitation amplitude for a system in steady – state forced vibration.vibration.

The ratio of output and input amplitudes represents a gain for the The ratio of output and input amplitudes represents a gain for the dynamic system.dynamic system.

Suspension isolation.Suspension isolation.All vehicles share the “ride isolation” properties common to a All vehicles share the “ride isolation” properties common to a

sprung mass supported by primary suspension systems at each sprung mass supported by primary suspension systems at each wheel. The dynamic behaviour of this system is the first level of wheel. The dynamic behaviour of this system is the first level of isolation from the roughness of the road.isolation from the roughness of the road.

The essential dynamics can be represented by a quarter – car model.The essential dynamics can be represented by a quarter – car model.The sprung mass resting on the suspension and tyre spring is The sprung mass resting on the suspension and tyre spring is

capable of motion in the vertical direction.capable of motion in the vertical direction.““ride rate”. The effective stiffness of the suspension and the tyre ride rate”. The effective stiffness of the suspension and the tyre

springs in series is called the “ride rate”springs in series is called the “ride rate”RR = KRR = Ks s KKt t / K/ Ks s ++ KKtt

QUARTER CAR MODELQUARTER CAR MODEL

VEHICLE RESPONSE VEHICLE RESPONSE PROPERTIESPROPERTIES

Where,Where, RR = ride rateRR = ride rateKKss = suspension stiffness = suspension stiffness

KKtt = tyre stiffness = tyre stiffness

The bounce natural frequency The bounce natural frequency ωωnn = (RR / M) = (RR / M)0.50.5

OrOr ffnn = 0.159 (RR / W/g) = 0.159 (RR / W/g)0.50.5 Where,Where, M – sprung massM – sprung mass

W = Mg = weight of the sprung massW = Mg = weight of the sprung massWhen damping is present, the resonance occurs at the “damped When damping is present, the resonance occurs at the “damped

natural frequency”, natural frequency”, ωωdd..

ωωdd = = ωωnn (1 – (1 – ζζss22))0.50.5

Where,Where, ζζss = damping ratio = C = damping ratio = Css / (4K / (4Kss M) M)0.50.5

CCss = suspension damping coefficient = suspension damping coefficient

W / KW / Kss = static deflection of the suspension due to the weight of the = static deflection of the suspension due to the weight of the vehicle.vehicle.

A static deflection of 10 inches is necessary to 1 Hz natural A static deflection of 10 inches is necessary to 1 Hz natural frequency.frequency.

UNDAMPED NATURAL FREQUENCY UNDAMPED NATURAL FREQUENCY VERSUS STATIC DEFLECTION OF A VERSUS STATIC DEFLECTION OF A

SUSPENSIONSUSPENSION

VEHICLE RESPONSE VEHICLE RESPONSE PROPERTIESPROPERTIES

The dynamic behaviour of the complete quarter car:The dynamic behaviour of the complete quarter car:MZ’’ + CMZ’’ + Css Z’ + K Z’ + Kss Z = C Z = Css Z Zuu’ + K’ + Kss Z Zuu + F + Fbb

mZmZuu’’ + C’’ + Css Z Zuu’ + (K’ + (Kss + K + Ktt) = C) = Css Z’ + K Z’ + Kss Z + K Z + Ktt Z Zrr + F + Fww

Where,Where, Z = sprung mass displacementZ = sprung mass displacementZZuu = unsprung mass displacement = unsprung mass displacement

ZZrr = road displacement = road displacement

FFbb = force on the sprung mass = force on the sprung mass

FFww = force on the unsprung mass = force on the unsprung massThe amplitude ratio for these cases are:The amplitude ratio for these cases are:

Z’’ / ZZ’’ / Zrr’’ = [K’’ = [K11 K K22 + j K + j K11C C ωω] / [] / [χχ ωω44 – (K – (K11 + K + K22 χχ + K + K22) ) ωω22 + K + K11 K K22] ] + j [K+ j [K11 C C ωω – (1 + – (1 + χχ)C)Cωω33]]

Z’’ / FZ’’ / Fww/M = [K/M = [K11 ωω22 + j C + j C ωω33] / [] / [χχ ωω44 – (K – (K11 + K + K22 χχ + K + K22) ) ωω22 + K + K11 K K22] + j ] + j [K[K11 C C ωω – (1 + – (1 + χχ)C)Cωω33]]

Z’’ / FZ’’ / FbbM = [M = [μμ ωω44 – (K – (K1 1 + K+ K22))ωω22 + j C + j C ωω33] / [] / [χχ ωω44 – (K – (K11 + K + K22 χχ + K + K22) ) ωω + + KK11 K K22] + j [K] + j [K11 C C ωω – (1 + – (1 + χχ)C)Cωω33]]

QUARTER CAR RESPONSE TO QUARTER CAR RESPONSE TO ROAD, TYRE/WHEEL AND BODY ROAD, TYRE/WHEEL AND BODY

INPUTSINPUTS

VEHICLE RESPONSE VEHICLE RESPONSE PROPERTIESPROPERTIES

Where,Where, χχ = m / M = ratio of unsprung mass to sprung mass = m / M = ratio of unsprung mass to sprung massC = CC = Css / M / M

KK11 = K = Ktt / M / M

KK22 = K = Kss / M / Mj = complex operatorj = complex operator

The quarter car model is limited to study of dynamic behaviour in The quarter car model is limited to study of dynamic behaviour in the vertical direction only.the vertical direction only.

Yet using above equations, it can be used to examine vibrations Yet using above equations, it can be used to examine vibrations produced on the sprung mass as a result of inputs from road produced on the sprung mass as a result of inputs from road roughness, radial forces or vertical forcesroughness, radial forces or vertical forces

The response properties can be presented by examining the response The response properties can be presented by examining the response gain as a function of frequency.gain as a function of frequency.

For road roughness input, the gain is the ratio of sprung mass For road roughness input, the gain is the ratio of sprung mass motion (acceleration, velocity, or displacement) to the equivalent motion (acceleration, velocity, or displacement) to the equivalent input from the road. At very low frequency, the gain is unity.input from the road. At very low frequency, the gain is unity.

The sprung mass response to tyre / wheel excitation is illustrated The sprung mass response to tyre / wheel excitation is illustrated by choosing an appropriate nondimensional expression for gain ofby choosing an appropriate nondimensional expression for gain of

ISOLATION OF ROAD ISOLATION OF ROAD ACCELERATION BY A QUARTER ACCELERATION BY A QUARTER

VEHICLE MODELVEHICLE MODEL

VEHICLE RESPONSE VEHICLE RESPONSE PROPERTIESPROPERTIES

the system. The input is an excitation force at the axle due to the tyre the system. The input is an excitation force at the axle due to the tyre / wheel. The output – acceleration of the sprung mass – may be / wheel. The output – acceleration of the sprung mass – may be transformed to a force by multiplying by the mass. Thence, the transformed to a force by multiplying by the mass. Thence, the output is the equivalent force on the sprung mass necessary to output is the equivalent force on the sprung mass necessary to produce acceleration. The gain is zero at zero frequency.produce acceleration. The gain is zero at zero frequency.

The response gain for direct force excitation on the sprung mass The response gain for direct force excitation on the sprung mass may be expressed nondimensionally by again using the equivalent may be expressed nondimensionally by again using the equivalent force on the sprung mass as the output. At high frequency the force on the sprung mass as the output. At high frequency the gain approaches unity.gain approaches unity.

VEHICLE RESPONSE VEHICLE RESPONSE PROPERTIESPROPERTIES

The sprung mass acceleration spectrum can be calculated for a The sprung mass acceleration spectrum can be calculated for a linear model by multiplying the road spectrum by the square og linear model by multiplying the road spectrum by the square og the transfer function.the transfer function.

GGzszs (f) = |H (f) = |Hvv (f)| (f)|22 G Gzrzr

Where,Where, GGzszs (f) = acceleration PSD on the sprung mass (f) = acceleration PSD on the sprung mass

HHvv (f) = response gain for road input (f) = response gain for road input

GGzrzr = acceleration PSD for road input = acceleration PSD for road input example problem:example problem:Determine the front and rear suspension ride rates for a 5.0 L Determine the front and rear suspension ride rates for a 5.0 L

Mustang given that the tyre spring rate is 1198 lb/in. the front Mustang given that the tyre spring rate is 1198 lb/in. the front suspension rate is 143 lb/in and rear is 100 lb/in. also estimate the suspension rate is 143 lb/in and rear is 100 lb/in. also estimate the natural frequencies of the two suspensions when the front tyres natural frequencies of the two suspensions when the front tyres are loaded to 957 lb and the rear tyres are at 730 lb each. are loaded to 957 lb and the rear tyres are at 730 lb each.

SUSPENSION STIFFNESSSUSPENSION STIFFNESSBecause the suspension spring is in series with a relatively stiff tyre

spring, the suspension spring predominates in establishing the the ride rate and hence the natural frequency.

Since the road acceleration inputs increase in amplitude at higher frequencies, the best isolation is achieved by keeping the natural frequencies as low as possible. For a vehicle with given weight, it is therefore desirable to use the lowest practical suspension spring rate to minimize the natural frequency.

The effect on acceleration transmitted to the sprung mass can be estimated analytically by approximating the road acceleration input as a function that increases with the square of the frequency. Then the mean – square acceleration can be calculated as a function of frequency.

The lowest acceleration occurs at the natural frequency of 1 hz. At higher values of natural frequency (stiffer suspension springs), the acceleration peak in the 1 to 5 hz range increases, reflecting a greater transmission of road acceleration input, and the mean square acceleration increases by several hundred percent.

ON ROAD ACCELERATION SPECTRA WITH ON ROAD ACCELERATION SPECTRA WITH DIFFERENT SPRUNG MASS NATURAL DIFFERENT SPRUNG MASS NATURAL

FREQUENCIESFREQUENCIES

BOUNCE / PITCH FREQUENCIESBOUNCE / PITCH FREQUENCIESOn most of the vehicles there is a coupling of motions in the vertical On most of the vehicles there is a coupling of motions in the vertical

and pitch directions, such that there is no “pure” bounce and and pitch directions, such that there is no “pure” bounce and pitch modes the behaviour in terms of natural frequenciesand pitch modes the behaviour in terms of natural frequenciesand motion centers, for a vehicle with coupled motions can be readily motion centers, for a vehicle with coupled motions can be readily determined analytically from the differential equations of motion.determined analytically from the differential equations of motion.

Following parameters are defined:Following parameters are defined:αα = (K = (Kff + K + Krr) / M) / M

ββ = (K = (Krrc – Kc – Kffb) / Mb) / M

ϒϒ = (K = (Kffbb22 + K + Krr c c22) / Mk) / Mk22

Where,Where, KKff = front ride rate = front ride rate

KKrr = rear ride rate = rear ride rateb = distance from the front axle to the CGb = distance from the front axle to the CGc = distance from the rear axle to the CGc = distance from the rear axle to the CGIIyy = Mk = Mk22 = pitch moment of inertia = pitch moment of inertiak = radius of gyrationk = radius of gyration

Then the differential equations for bounce Z and pitch Then the differential equations for bounce Z and pitch θθ can be can be

PITCH PLANE MODEL FOR A PITCH PLANE MODEL FOR A MOTOR VEHICLEMOTOR VEHICLE

BOUNCE / PITCH FREQUENCIESBOUNCE / PITCH FREQUENCIESwritten aswritten asZ’’ + Z’’ + ααZ +Z +βθβθ = 0 = 0θθ ‘’ + ‘’ + ββ Z/k Z/k22 + + ϒθϒθ = 0 = 0ΒΒ = coupling coefficient = coupling coefficientWhen When ββ = 0, no coupling occurs and the spring center is at the center = 0, no coupling occurs and the spring center is at the center

of gravity. For this conditioon, a vertical force at the CG produces of gravity. For this conditioon, a vertical force at the CG produces only bounce motion and a pure torque applied to the chassis will only bounce motion and a pure torque applied to the chassis will produce only pitch motion.produce only pitch motion.

The solution to the above differential equations is:The solution to the above differential equations is:Z = Z = Z sin Z sin ωωtt

And thepitch motion will be:And thepitch motion will be:θθ = = θθ sin sin ωωtt

Differentiating twice and putting into equation:Differentiating twice and putting into equation:-Z-Zωω22 sin sin ωωt + t + ααZ sin Z sin ωωt + t + βθβθ sin sin ωωt = 0t = 0((αα – – ωω22)Z + )Z + βθβθ = 0 = 0

OrOr Z / Z / θθ = - = -ββ / ( / (αα – – ωω22))For second equationFor second equation Z / Z / θθ = -k = -k22 ( (ϒϒ – – ωω22) / ) / ββ

THE TWO VIBRATION MODES OF A THE TWO VIBRATION MODES OF A VEHICLE IN THE PITCH PLANEVEHICLE IN THE PITCH PLANE

BOUNCE / PITCH FREQUENCIESBOUNCE / PITCH FREQUENCIESThese equation conditions under which the motions occur.These equation conditions under which the motions occur.The constraint: the ratio of amplitude in bounce and pitch must The constraint: the ratio of amplitude in bounce and pitch must

satisfy both equations.satisfy both equations.Equating right sideEquating right side

((αα – – ωω22) () (ϒϒ – – ωω22) = ) = ββ22 / k / k22

ThenThen ωω44 – ( – (αα + + ϒϒ))ωω22 + + αϒαϒ – – ββ22/k/k22 = 0 = 0The solutions are:The solutions are:

ωω11 = [( = [(αα + + ϒϒ)/2 + (()/2 + ((αα – – ϒϒ))22/4 + /4 + ββ22/k/k22))0.50.5))0.50.5

`̀ ω ω11 = [( = [(αα + + ϒϒ)/2 - (()/2 - ((αα – – ϒϒ))22/4 + /4 + ββ22/k/k22))0.50.5]]0.50.5 These frequencies always lie outside the uncoupled natural These frequencies always lie outside the uncoupled natural

frequencies.frequencies.The oscillation centers can be found using the amplitude ratios with The oscillation centers can be found using the amplitude ratios with

respect to two frequencies respect to two frequencies ωω11 and and ωω22. Z / . Z / θθ((ωω1)1) and Z / and Z / θθ((ωω2)2) will will have opposite signs.have opposite signs.

Z / Z / θθ +ve, oscillation center will be ahead of CG by a distance x = +ve, oscillation center will be ahead of CG by a distance x = Z / Z / θθ

Z / Z / θθ –ve, the oscillation center will be behind the CG. –ve, the oscillation center will be behind the CG.

BOUNCE / PITCH FREQUENCIESBOUNCE / PITCH FREQUENCIES When oscillation center outside the wheel base, the motion is When oscillation center outside the wheel base, the motion is

bounce with bounce frequency.bounce with bounce frequency. When oscillation center inside the wheel base, the motion will be When oscillation center inside the wheel base, the motion will be

pitch with pitch frequency.pitch with pitch frequency. The locations of the motion centers are dependent on the relative The locations of the motion centers are dependent on the relative

values of the natural frequencies of the front and rear values of the natural frequencies of the front and rear frequencies.frequencies.

ffff = (K = (Kffg/Wg/Wff))0.50.5 / 2 / 2ππ

ffrr = (K = (Krrg/Wg/Wrr))0.50.5 / 2 / 2ππ

OILEY’ S CRITERIA FOR FOR OILEY’ S CRITERIA FOR FOR DESIGNING WITH A GOOD RIDEDESIGNING WITH A GOOD RIDEThe front suspension should have a 30% lower ride rate than rear The front suspension should have a 30% lower ride rate than rear

suspension, or the spring center should be at least 6.5% of the suspension, or the spring center should be at least 6.5% of the wheel base behind the CG. Although this does not explicitly wheel base behind the CG. Although this does not explicitly determine the front and rear natural frequencies, since the rear – determine the front and rear natural frequencies, since the rear – front weight distribution on passenger cars is close to 50 – 50, it front weight distribution on passenger cars is close to 50 – 50, it will be generally assure that the rear frequency is greater than the will be generally assure that the rear frequency is greater than the front.front.

The pitch and bounce frequencies should be close together. The The pitch and bounce frequencies should be close together. The bounce frequency should be less than 1.2 x pitch frequency.bounce frequency should be less than 1.2 x pitch frequency.

Neither frequencies should be greater than 1.3 Hz, which means Neither frequencies should be greater than 1.3 Hz, which means that the effective static deflection of the vehicle should exceed that the effective static deflection of the vehicle should exceed roughly 6 inches. The value of keeping natural frequencies below roughly 6 inches. The value of keeping natural frequencies below 1.3 Hz was demonstrated1.3 Hz was demonstrated

The roll frequency should be approximately equal to the pitch The roll frequency should be approximately equal to the pitch and bounce frequencies.and bounce frequencies.

EFFECT OF NATURAL FREQUENCY RATIO ON EFFECT OF NATURAL FREQUENCY RATIO ON POSITION OF MOTION CENTERSPOSITION OF MOTION CENTERS

SPECIAL CASESSPECIAL CASESMost modern vehicles with substential front and rear overhang Most modern vehicles with substential front and rear overhang

have dynamic index close to 1.have dynamic index close to 1.DI = kDI = k22 / bc = 1 / bc = 1

OrOr kk22 = bc = bck = (bc)k = (bc)0.50.5

When equality holds, the front and rear suspensions are located at When equality holds, the front and rear suspensions are located at conjugate centers of percussion (an input at one suspension causes conjugate centers of percussion (an input at one suspension causes no reaction at other). In this case, the oscillation centers are no reaction at other). In this case, the oscillation centers are located at the front and rear axles. There is no interaction in located at the front and rear axles. There is no interaction in front and rear suspensions.front and rear suspensions.

Spring center at CG. The pitch and bounce motions are totally Spring center at CG. The pitch and bounce motions are totally independent.independent.

Dynamic index greater than unity. This occurs when there is Dynamic index greater than unity. This occurs when there is substential overhang at the front / rear axles.substential overhang at the front / rear axles.

Uncoupled motion. (Uncoupled motion. (ββ = 0) and dynamic index = 1 = this results in = 0) and dynamic index = 1 = this results in equal bounce and pitch frequencies.equal bounce and pitch frequencies.

OSCILLATIONS OF A VEHICLE OSCILLATIONS OF A VEHICLE PASSING OVER A ROAD BUMPPASSING OVER A ROAD BUMP

EXAMPLE PROBLEMSEXAMPLE PROBLEMSCalculate the pitch and bounce centers and their frequencies for a Calculate the pitch and bounce centers and their frequencies for a

car with the following characteristics:car with the following characteristics:Front ride rate = 127 lb / inFront ride rate = 127 lb / in front tyre load = 957 lbfront tyre load = 957 lbRear ride rate = 92.3 lb / inRear ride rate = 92.3 lb / in rear tyre load = 730 lbrear tyre load = 730 lbWheel base = 100.6 inWheel base = 100.6 in dynamic index = 1.1dynamic index = 1.1

Find the pitch and bounce centers and their frequencies for a car Find the pitch and bounce centers and their frequencies for a car with following characteristics:with following characteristics:

Front ride rate = 132 lb / inFront ride rate = 132 lb / in front tyre load = 1035 lbfront tyre load = 1035 lbRear ride rate = 93 lb / inRear ride rate = 93 lb / in rear tyre load = varying from rear tyre load = varying from

567 567 to 1000 lbto 1000 lbWheel base = 112 inchesWheel base = 112 inches dynamic index = 1.05dynamic index = 1.05

EXAMPLE PROBLEMSEXAMPLE PROBLEMS

PERCEPTION OF RIDEPERCEPTION OF RIDERide. Ride is a subjective perception normally associated with Ride. Ride is a subjective perception normally associated with

thelevel of comfort experienced when travelling in a vehicle.thelevel of comfort experienced when travelling in a vehicle.Perceived ride is the cumulative product of many factors:Perceived ride is the cumulative product of many factors: The tactile vibrations transmitted to passenger’s body through The tactile vibrations transmitted to passenger’s body through

the seat, and at the hand and feetthe seat, and at the hand and feet Acoustic vibrationsAcoustic vibrations General comfort levelGeneral comfort levelTolerance of seat vibrationsTolerance of seat vibrationsNo universally accepted standard exists for judgment of ride No universally accepted standard exists for judgment of ride

vibrations due to following variables;vibrations due to following variables; Seating positionsSeating positions Influence of hand and foot vibration inputInfluence of hand and foot vibration input Single versus multiple frequency inputSingle versus multiple frequency input Multi direction inputMulti direction input Comfort scalingComfort scaling Duration of exposureDuration of exposure Sound and visual vibration inputsSound and visual vibration inputs

HUMAN TOLERANCE LIMITS FOR HUMAN TOLERANCE LIMITS FOR VERTICAL VIBRATIONSVERTICAL VIBRATIONS

NASA DISCOMFORT CURVES FOR NASA DISCOMFORT CURVES FOR VIBRATION IN TRANSPORT VIBRATION IN TRANSPORT

VEHICLESVEHICLES

HUMAN TOLERANCE LIMITS HUMAN TOLERANCE LIMITS FOR FORE/AFT VIBRATIONSFOR FORE/AFT VIBRATIONS