optimizing tire vertical stiffness based on ride, handling, performance and fuel consumption...

Amir SoltaniDepartment of Mechanics,

Faculty of Engineering,

Islamic Azad University,

Hamedan Branch,

Hamedan 65181-15743, Iran

e-mail: [email protected]

Avesta Goodarzi1Department of Mechanical

and Mechatronics Engineering,

University of Waterloo,

Waterloo, ON N2L 3G1, Canada


Mohamad HasanShojaeefard

Automotive Engineering Department,

Iran University of Science & Technology,

Tehran 16846-13114, Iran


Khodabakhsh SaeediDepartment of Mechanical

and Mechatronics Engineering,

University of Waterloo,

Waterloo, ON N2L 3G1, Canada


Optimizing Tire VerticalStiffness Based on Ride,Handling, Performance,and Fuel Consumption CriteriaResearchers mostly focus on the role of suspension system characteristics on vehicledynamics. However tire characteristics are also influential on the vehicle dynamicsbehavior. In this paper, the effects of tire vertical stiffness on the ride, handling, acceler-ating/braking performance, and fuel consumption of a vehicle are analytically investi-gated. Furthermore, a method for determining the optimum vertical stiffness of tires ispresented. For these purposes, first an appropriate mathematical criterion for the ride,handling, accelerating/braking performance, and fuel consumption is developed. Next, toachieve the optimum tire characteristic, a performance index, which contains all of theabove-mentioned criteria, is defined and optimized. In the proposed performance index,the tire vertical stiffness is a design variable and its optimization provides a compromiseamong ride, handling, accelerating/braking performance, and fuel consumption of thevehicle. Last, the analytical optimization results are confirmed by performing precisenumerical simulations. [DOI: 10.1115/1.4031459]

1 Introduction

In numerous articles, the contradictory effects of the suspensionsystem characteristics on vehicle ride and handling have beeninvestigated. It is well known that the spring and damper charac-teristics requirement for good handling are not the same as thosefor good ride comfort. Any choice between the spring and dampercharacteristic is, therefore, a compromise between ride comfortand handling [1–3].

Tire characteristics are also crucial in ride, handling, accelerating/braking performance, fuel consumption, and generally thedynamics behavior of vehicles. They are the only contact betweenthe road and the vehicle. The tire forces and moments are theresult of the interaction between the wheel and the road. Hence,tire characteristics strongly affect the vehicle handling anddynamics performance. Another role of tires is to cushion the ve-hicle and the passengers from road irregularities. The road surfaceexcitations pass through the tires and reach to the main body ofthe automobile, therefore, they have a great influence on the qual-ity of vehicles ride comfort [4]. Furthermore, the tire has a stronginfluence on the vehicle fuel consumption through their rollingresistance properties [5].

One of the most important tire characteristics, which affects allof the above-mentioned performances, is tire vertical stiffness.The vertical stiffness of the tire depends on the size, construction,and the inflation pressure, but the main parameter that affects thetire vertical stiffness is the inflation pressure [6].

The idea of changing the tire’s vertical stiffness through controlof its inflation pressure goes back to World War II. Since thistime, the central tire inflation system (CTIS) was used as standardequipment on most wheeled military vehicles to improve the

vehicle ride, handling, and accelerating/braking performance [7].In order to enhance the vehicle dynamic behavior on rough ter-rains, CTIS decreases the vertical stiffness of the tires by reducingthe inflation pressure to provide better ground contact, as well as amore comfortable ride. The military demonstrated that use ofCTIS provides a better ride and better mobility for tacticalvehicles, especially in off-highway conditions [8].

As mentioned above, the effects of the tire’s vertical stiffnesson the vehicle dynamics behavior are conceptually known, how-ever, there are few published analytical research papers in thisfield and most of the existing research has focused on the tire’seffects on ride comfort and/or the research methods are qualitativerather than quantitative [7,8]. The effects of the vertical stiffnesson the tire’s rolling resistance and the vehicle’s fuel consumptionhave been individually studied by a number of researchers [5,9].

The main object of this research is to analytically investigate theeffects of tire vertical stiffness on the behavior of vehicles in termsof four different aspects: ride, handling, accelerating/braking per-formance, and fuel consumption. In this research, for each per-formance aspect a suitable mathematical criterion is introduced;afterward, using those criteria, the effects of the tire vertical stiff-ness on the vehicle behavior in each case are individually investi-gated. In the next step, by combining these criteria, a generalizedperformance index is formed which its maximization leads to theoptimum tire vertical stiffness for ride, handling, accelerating/braking performance, and fuel consumption concurrently. Last, byusing precise numerical simulation in the Carsim

VR

software envi-ronment, the performance of the generally optimized tire in differ-ent situations is studied and compared with those tires which areexclusively optimized for only ride, handling, accelerating/braking, or fuel consumption performances.

2 Ride Comfort Analysis

The passenger comfort and the driver feeling in a moving vehi-cle are introduced as ride quality. Ride comfort is defined in terms

1Corresponding author.Contributed by the Dynamic Systems Division of ASME for publication in the

JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript receivedFebruary 20, 2014; final manuscript received August 3, 2015; published onlineSeptember 22, 2015. Assoc. Editor: Junmin Wang.

Journal of Dynamic Systems, Measurement, and Control DECEMBER 2015, Vol. 137 / 121004-1Copyright VC 2015 by ASME

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of human response to the vehicle vibration. Transmitted vibrationto the vehicle originates from different sources, such as roadunevenness, engine, and transmission. However, road irregular-ities are the major source of vehicle vibration.

To study the vehicle ride comfort, one needs to quantify itbased on the measurable variables. Most of the ride comfort stand-ards such as International Organization for Standardization (ISO)2631, consider the vertical acceleration as the best variable forevaluating the ride comfort of a vehicle [10]. Hence, the ride crite-rion can be defined based on the magnitude of average verticalacceleration of the vehicle body, azs. For calculation of the verti-cal acceleration, the standard road input is a crucial issue. It canbe a simple sinusoidal road with a fixed excitation frequency, afrequency sweep over a defined frequency range, or a real roadwith a specific given power spectrum density (PSD). In this study,frequency sweep in a specific frequency range ðxi;xf Þ has beenselected as the way to generate the road excitation input. Accord-ingly, the average value of azs can be found using its root meansquare (RMS):

RMS azsð Þ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1

xf � xi

ðxf

xi

azs2dx

s(1)

Road surface irregularities in the range from 0 to 30 Hz representthe most intensive source of input energy to the vibration systemof the vehicle [11]. Hence, in this paper, the ride comfort analysisis performed by analyzing the vertical behavior of a quarter-carmodel in the low (0–10 Hz) and high frequency (10–30 Hz)domains. In the quarter-car model, as illustrated in Fig. 1, the ver-tical motions of the sprung and unsprung masses are two degrees-of-freedom. The quarter-car is an analytical model, and can pro-vide useful insight into trends. Moreover, it has been proven byprevious research that regardless of the model simplicity, thequarter-car model offers a quite reasonable accuracy for simulat-ing the bounce motion of the sprung mass [12].

The parameters of the quarter-car model, as seen in Fig. 1, aresprung mass (ms), suspension stiffness (ks), suspension dampingcoefficient (cs), tire/wheel mass (i.e., unsprung mass) (mu), andtire vertical stiffness (kt). The tire vertical damping coefficient is

usually very small and to simplify the model can be neglected[13].

As aforementioned, the system behavior is studied in thefrequency-domain. Frequency-domain analysis considers thebehavior of the vehicle in terms of its response at any given fre-quency of stimulus. In order to find the frequency response, theroad excitation is assumed as

y ¼ Y sin ðxtÞ (2)

where Y is the magnitude of the sinusoidal road and x is the roadexcitation frequency, which depends on the road wavelength andlongitudinal speed of the vehicle. By solving the quarter-carequations of motion for the given road excitation, the magnitudeof the vertical acceleration of sprung mass, azs is calculated asfollow [13]:

azs ¼ Yx2ð Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4n2 x=xsð Þ2 þ 1

Z21 þ Z2

2

s(3)

where

Z1 ¼ ðx=xsÞ2 � ½ðx=xsÞ2a2 � 1� þ 1� ð1þ eÞðx=xsÞ2a2 (4)

Z2 ¼ 2 � f � ðx=xsÞ � ½1� ð1þ eÞðx=xsÞ2a2� (5)

The other parameters are described in Table 1According to Eq. (3), one can define the vertical acceleration

transmissibility function Az as the ratio of vertical acceleration azs

to the road excitation magnitude Y

AZ ¼ azs=Y (6)

Now, using this new variable, without any change in the inher-ent of problem, we can represent Eq. (1) in the following form:

RMS Azð Þ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1

xf � xi

ðxf

xi

Az2dx

s(7)

The root mean square value of the Az, which is used for percep-tion of the vehicle ride comfort, is called passenger discomfort(PD), because lower RMSðAzÞ causes a more comfortable ride.Hence, the ride comfort criterion is introduced as

CR ¼1

RMS Azð Þ

¼ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

xf�xi

ðxf

xi

x2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4n2 x=xsð Þ2 þ 1

� �= Z2

1 þ Z22

� �r" #28<:

9=;dx

vuuut(8)

Figure 2 shows the variation of Az versus the excitation fre-quency x for different values of tire vertical stiffness. The valuesof the main system parameters are listed in Table 2. Note that

Fig. 1 Quarter-car model configuration

Table 1 Quarter-car model parameters and equations

Symbol Definition Relationship

xs Natural frequency of the sprung mass xs ¼ffiffiffiffiffiffiffiffiffiffiffiffiks=ms

pxus Natural frequency of the unsprung mass xus ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffikt=mus

pa Frequency ratio of sprung to unsprung mass a ¼ xs=xus

n Damping ratio n ¼ cs=ð2msxsÞe Mass ratio e ¼ ms

mus

121004-2 / Vol. 137, DECEMBER 2015 Transactions of the ASME


resonance happens when the frequency of excitation coincideswith one of the natural frequencies. The resonance of theunsprung mass is usually referred to as “wheel hop” resonance. Itcan be seen that in the low frequency range, the change in tire ver-tical stiffness does not have considerable effect on the sprungmass vibration, however, when excitation frequency is close tothe natural frequency of the unsprung mass (around 10 Hz), astiffer tire intensifies wheel hop resonance strongly. In fact, in thehigh frequency range, the stiffer tires cause a significant worseride quality.

The above-mentioned finding can be proven again by using theplot of RMS (Az). It has been illustrated in Fig. 3 in two differentfrequency ranges; low frequency range from 0 to 10 Hz and highfrequency range from 10 to 30 Hz. Based on Fig. 3(a), change inthe vertical stiffness of tire has insignificant influence on thevibration of the sprung mass in the low frequency range, as theRMS value of Az can be approximated with a constant value of ar.While, according to Fig. 3(b), a less-stiff tire provides bettervibration isolation in the mid to high frequency ranges.

To formulate the ride comfort criterion in high frequency-domain based on Fig. 3(b), the passenger discomfort can beapproximated as

RMSðAzsÞ ffi �ar0 þ ar1kt (9)

where ar0 and ar1 are constant coefficients. The coefficients ar0

and ar1 for our case study vehicle equipped with a 195/65R15 tireare tabulated in Table 3. Finally, by combining Eqs. (8) and (9),the ride comfort criterion can be obtained

CR ¼1

�ar0 þ ar1kt(10)

3 Handling Analysis

The term handling is often used to describe the cornering,maneuverability, and directional response of a road vehicle. Eval-uation of vehicle handling is almost subjective (due to the varietyand the different experience of those who drive vehicles) [14,15].However, as a general rule, the handling characteristic of tires isin connection with their capacity to generate lateral force when a

vehicle drives through a corner [15,16]. Therefore, to quantify thehandling of vehicles, a performance index is required to determinethe potential of vehicles’ tires in producing the lateral force in dif-ferent conditions. Generally, the lateral force of a tire can be for-mulated as

Fy ¼ CFaa (11)

where a and CFa are the side slip angle and the cornering coeffi-cient of the tire, respectively. Based on Eq. (11), it is clear that agreater cornering coefficient increases the potential of a tire inproducing the lateral force. So the handling criterion can besimply defined as

CH ¼ CFa (12)

Fig. 2 Acceleration transmissibility versus excitation fre-quency for different tire stiffness

Table 2 Quarter-car model parameters value

Parameter Symbol Value Unit

Sprung mass ms 350 kgUnsprung mass mu 35 kgDamping ratio n 0.3 —Suspension stiffness ks 27 N/mm

Fig. 3 RMS of sprung mass vertical acceleration: (a) Lowfrequency range and (b) high frequency range

Table 3 Ride, handling, and accelerating/braking performancecriteria coefficients

Parameters Value Parameters Value

ar 255 ah3 1.837� 10�11

ar0 64.27 ah4 �1.895� 10�17

ar1 2.16� 10�3 ap0 9216ah0 16188 ap1 0.769ah1 0.985 ap2 2.291� 10�6

ah2 �6.622� 10�6

Journal of Dynamic Systems, Measurement, and Control DECEMBER 2015, Vol. 137 / 121004-3


Maximizing the presented handling criterion reduces the side-slip angle, which is crucial for improving the stability of vehicles.

To evaluate the effect of tire characteristics on the handling ofa vehicle based on the handling criterion in Eq. (12), we shoulddevelop relationships between tire cornering coefficient and tirevertical stiffness. Since more than 80% of the total tire stiffness isdue to inflation pressure [6], we can utilize relationships betweenlateral stiffness and inflation pressure approximately.

To include the effects of the inflation pressure in the tire corner-ing coefficient, CFa’s relationship of magic formula (MF) 6.1 tiremodel [17–19] is used. Relative to original form of MF, two linearexpressions depending on normalized change in inflation pressure(dp) are added

CFa ¼ PKY1 1þPKY1dpð ÞFZ0 sin 2 tan�1 Fz

PKY2 1þPKY2dpð ÞFZ0

� � (13)

where the Ps are MF parameters, which for our case study 195/65R15 tire have been listed in Table 4, Fz and Fz0 are the actualand the nominal vertical load, respectively, and dp is

dp ¼ p

p0

� 1 (14)

where p and p0 are the existing and nominal inflation pressures,respectively.

The next step is to describe the inflation pressure and the verti-cal stiffness relationship. Based on the method that is presentedby Schmeitz et al. [17], the vertical stiffness can be obtained fromthe following empirical equation:

kt ¼ ð1þ qFz3dpÞðqFz1 þ 2qFz3qÞ (15)

where the qFz s are fitting parameters, their typical value for thecase study tire have been listed in Table 4, and q is tire deflection.Practically, the qFz1 is much bigger than 2qFz3q, and then the sec-ond one can be ignored. In other words, it can be assumed that therelation between the vertical stiffness and the inflation pressure isapproximately linear [15,16]. Hence, by solving Eq. (15) for dp,the expression for the normalized change in inflation pressure isderived as follows:

dp ffi 1

qFz3

kt

qFz1

� 1

� �(16)

By substituting Eq. (16) in Eq. (13), one can find a relationshipbetween tire cornering stiffness and tire vertical stiffness. Thisrelationship is graphically presented for the 195/65R15 tire inFig. 4. For this specific tire, the maximum value of corneringcoefficient occurs at kt¼ 140 N/mm. Relative to this point, anyincrease/decrease in tire vertical stiffness causes a reduction in thetire cornering coefficient. Consequently, the potential of the tirefor producing lateral forces decreases. As a physical explanationof this tire characteristic, one should note that the value of the

cornering coefficient is proportional to the tire/road contact patcharea and the lateral stiffness of the tire. Besides, the tire verticalstiffness is directly related to the tire lateral stiffness andindirectly related to the contact patch area. So any change in thevertical stiffness from the optimum point, decreases one of theabove-mentioned influential factors and results in a reduction ofthe cornering coefficient.

In order to use a simple equation for the handling criterion,using the least square method, a fourth-order polynomial is fittedto the graph of CFa versus kt. The simplified handling criterion isobtained in the following form:

CH ffiX4

i¼0

ahikit (17)

where the constant coefficients ahis are tabulated for the 195/65R15 tire in Table 3.

4 Accelerating/Braking Analysis

The vehicle’s accelerating/braking performance correlates withthe longitudinal forces that a tire generates during braking oraccelerating maneuvers. Consider the longitudinal force formula-tion of tires as

Fx ¼ CFj j (18)

where CFj is the longitudinal slip stiffness and j is the longitudi-nal slip of the tire. Similar to the approach in Sec. 3 to decreasethe longitudinal slip as a way to improve the vehicle stability [16]and to increase the potential of generating longitudinal force, thefollowing performance criterion is introduced:

CP ¼ CFj (19)

Considering the goal of this research, the introduced accelerat-ing/braking performance criterion should be presented as a func-tion of the tire vertical stiffness. For this purpose, the improvedMF 6.1 tire model [17–19] can be used. Based on the MF modellongitudinal slip stiffness CFj relates to inflation pressure of thetire as

CFj ¼ FzðPKX1 þ PKX2dfzÞePKX3dfzð1þ PPX3dpþ PPX4dp2Þ (20)

where the Ps are MF parameters, which for the case study tirehave been listed in Table 4, and dfz is the normalized change invertical load and is defined based on the actual and nominal verti-cal loads

Table 4 195/65R15 Tire magic formula parameters value[17–19]

Parameter Value Parameter Value

P0 2.2 PPY2 0.89Fz0 4000 PKX1 18.886qFz1 200,000 PKX2 �3.988qFz3 0.9166 PKX3 0.21542PKY1 19.797 PPX3 �0.38PKY2 1.7999 PPX4 �1.08PPY1 0.33 frr0 0.015966

Fig. 4 Cornering coefficient versus tire vertical stiffness



dfz ¼Fz � FZ0

FZ0

(21)

As can be seen in Eq. (20), the first part is a function of dfz andbecause it is a constant value, the first part is supposed to be con-stant. Besides, the second part of CFj equation includes a second-order algebraic form of normalized change in inflation pressure(dp) which, as aforementioned, is a function of the tire verticalstiffness. Finally, combining Eqs. (20), (21), and (14), a relation-ship between the tire longitudinal slip stiffness and the tire verticalstiffness is formed as follow:

CP ¼ ap0 þ ap1kt � ap2k2t (22)

where ap0, ap1, and ap2 are constant coefficients which have beentabulated for the 195/65R15 tire in Table 3.

Figure 5 shows the variation of the longitudinal slip stiffness of195/65R15 tire versus vertical stiffness. The maximum pointoccurs at kt¼ 170 N/mm. It is clear that after or before the maxi-mum point, any change in the tire vertical stiffness causes a reduc-tion in the tire longitudinal slip stiffness. The similar physicalexplanation with what was addressed in Sec. 3 for relationshipbetween the cornering stiffness and the vertical stiffness can bestated here again.

5 Fuel Consumption Analysis

When a tire is rolling, due to deflection of the carcass in theground contact area, the center of normal pressure is shifted in thedirection of rolling. This shift produces a negative moment aboutthe rotation axis of the tire, which is called the rolling resistancemoment. The rolling resistance is simply the manifestation of allof the energy losses associated with the rolling of a tire under theload. In a vehicle, approximately 5–15% of the fuel is consumedto overcome the rolling resistance [20]. At low speeds, fuel con-sumption is determined up to 40% by the tire rolling resistance[21]. It is clear that less-stiff tires (low inflation pressure tires)increases the rolling resistance coefficient leads to more fuel con-sumption. Therefore, the fuel consumption criterion is definedbased on the rolling resistance coefficient. Because the higherrolling resistance of a tire causes more fuel consumption, we canconsider the reverse of rolling resistance coefficient (frr) as fuelconsumption criterion

CFC ¼1

frr(23)

Usually, the following general form is used to formulate therolling resistance based on ISO rolling resistance test [22]:

frr ¼ frr0

p

po

� ��a FZ

FZ0

� �b

(24)

where frr0 coefficient for case study tire has been listed in Table 4and the following values are applicable for the other coefficientsof Eq. (24) [22]:

� For a passenger cars, a¼ 0.4 and b¼ 0.9� For a trucks, a¼ 0.2 and b¼ 0.9

According to Eqs. (14) and (16), the p/p0 term in Eq. (24) canbe replaced by

p

p0

ffi kt

qFz3 � qFz1

� 1� qFz3

qFz3

� �(25)

By substituting Eq. (25) in Eq. (24), a relationship between therolling resistance and the tire vertical stiffness is formed

frr ¼ Krr �kt þ qFz4

qFz1

� ��a

(26)

where the constants Krr and qFz4 are defined as follows:

Krr ¼ frr0:qFz3a:

FZ

FZ0

� �b

qFz4 ¼ qFz1 qFz3 � 1ð Þ (27)

By combining Eqs. (23) and (26), the final form of fuel con-sumption criterion is written as follows:

CFC ¼1

Krr� kt þ qFz4

qFz1

� �a

(28)

Figure 6 shows the effect of tire vertical stiffness variation onthe rolling resistance coefficient of the case study 195/65R15 tire.Based on this figure, it is clear that by increasing the tire verticalstiffness, the rolling resistance coefficient dramatically decreases,leads to less fuel consumption. Hence, the optimum value of thetire stiffness that provides the least fuel consumption is chosenequal to 300 N/mm. It is the biggest practically possible value.

Fig. 5 Longitudinal slip stiffness versus tire vertical stiffnessFig. 6 Rolling resistance coefficient as a function of tire verti-cal stiffness



6 Optimization

Based on the results that were discussed in Secs. 2–5, there isno unique optimum tire’s vertical stiffness which satisfies ride,handling, accelerating/braking performance, and fuel consumptionrequirements simultaneously. For example, a soft tire providesgood ride quality, while the fuel consumption of a vehicle withhard tires is better. In this section, with the combination of eachindividual criterion discussed earlier, an evaluating function isintroduced which can provide a compromise among different ve-hicle characteristics in various situations. This evaluation functioncan be presented as

EF ¼ kwRðksRCRÞ þ kwHðksHCHÞ þ kwPðksPCPÞ þ kwFCðksFCCFCÞ(29)

where kwR, kwH, kwP, kwFC and ksR, ksH, ksP, ksFC are the weightingfactors and scaling factors, respectively. Weighting factors deter-mine the relative importance of different terms, which in thiscase, the same importance is considered for ride, handling,accelerating/braking performance, and fuel consumption. There-fore, the entire weighting factors are equal to one. On the otherhand, because the different terms of Eq. (29) do not have the sameorder of magnitude, the scaling factor is needed to equalize thenumerical values of them. Considering Eqs. (10), (17), (22), and(28), for the case study tire, the average values of ride, handling,accelerating/braking performance, and fuel consumption criteriain the interval ðktmin; ktmaxÞ are 0.003, 64,300, 63,700, and 117,respectively. The scaling factors of different criteria are selectedin such a manner that can compensate these numerical differences.In other words, the scaling factors are defined as follows:

ksi ¼1ðkt:max

kt:min

Cidkt

kt:max � kt:min

264

375

i ¼ R;H;P & FC (30)

which for the 195/65R15 case study tire, coefficients ksR, ksH, ksP,and ksFC are 321, 0.0000155, 0.0000157, and 0.008, respectively.

Finally, in Eq. (29), the ride, handling, accelerating/brakingperformance, and fuel consumption criteria can be substitutedfrom Eqs. (10), (17), (22), and (28), respectively, and the finalform of the evaluating function is found as follows:

EF¼ ksRkwR

�ar0þ ar1ktþ ksHkwH ah0þ ah1kt þ ah2k2

t þ ah3k3t þ ah4k4

t

� �þkspkwp ap0þ ap1kt � ap2k2

t

� �þ ksFCkwFC

Krr� kt þ qFz4

qFz1

� �a

(31)

Maximizing Eq. (31) in the interval ðktmin; ktmaxÞ leads us to theoptimum value of vertical stiffness. For this purpose the criticalpoint of the evaluating function, in the interval ðktmin; ktmaxÞshould be determined. The critical point yield to a local maxi-mum, if the evaluating function is concave downward at this point

d

d ktEFð Þ ¼ 0 and

d2

d kt2

EFð Þ < 0 (32)

If the evaluation functions value is larger than the function val-ues at each endpoint of ktmin and ktmax, this point is the optimumtire’s vertical stiffness. The optimum vertical stiffness of the tireis found through the solving of Eq. (32) by using numerical tech-niques. As shown in Fig. 7, the optimum value of the verticalstiffness for the case study 195/65R15 tire is about 195 N/mm.

A radar chart of the different performances of the generallyoptimized tire for all ride, handling, accelerating/braking and fuelconsumption, in comparison with those of the individuallyoptimized tires (optimized only for one of the above-mentioned

performances) is plotted in Fig. 8. The radii represent the fourabove-mentioned criteria that are normalized relative to theirgreatest value. According to the figure, while the individuallyoptimized tires get the best rank (score 1) in their incorporatingaspect, they do not get an acceptable score in the others. Forinstance, in the case of kt¼ 300 N/mm which provides the bestfuel economy, it gets scores of 0.3, 0.7, and 0.45 in ride, handling,and accelerating/braking performance, respectively, whereas, thegenerally optimized tire gets the acceptable scores from differentperspectives.

7 Simulation

The results of analyses in Secs. 2–6 are based on simplifiedmodels. In order to evaluate the effect of tire characteristics on thevehicle behavior in an approximately real world situation, precisesimulations by using Carsim

VR

software, which is a well-knownaccurate vehicle dynamic simulation package [23], have beenperformed.

For simulation studies, a compact, hatchback vehicle has beenconsidered as the case study vehicle. The main specifications ofthe car are listed in the Table 5. This vehicle is equipped with dif-ferent tires with same size of 195/65R15 but with different verti-cal stiffness as follows:

� An individually optimized tire for ride with kt¼ 100 N/mm� An individually optimized tire for handling with kt¼ 140 N/mm� An individually optimized tire for accelerating/braking per-

formance with kt¼ 170 N/mm� An individually optimized tire for fuel consumption with

kt¼ 300 N/mm� A generally optimized tire which is optimized for all ride,

handling, performance and fuel consumption, concurrently,with kt¼ 195 N/mm.

For these tires, the values of ride, handling, and other met-rics, as well as the evaluation functions are tabulated in Table6. The characteristic curves of the lateral and longitudinalforces of the five above-mentioned tires are obtained based onthe MF 6.1 tire model [19]. The curves have been illustrated inFigs. 9 and 10 versus side slip angle and longitudinal slip ratio,respectively.

In Secs. 7.1–7.4 based on standard test procedures, the vehicleperformances in terms of ride, handling, braking performance,and fuel consumption for different tires are simulated anddiscussed.

Fig. 7 The evaluation function (EF) versus tire verticalstiffness



7.1 Ride. In order to simulate the effects of the tire verticalstiffness on the ride comfort at low and high frequencies, a stand-ard test has been simulated on a straight sinusoidal road. For thelow frequency test, the vehicles with the speed of 20 m/s. passover the sinusoidal road. The spectral wavelength of the road isselected in such way that the exciting frequency of 1Hz is pro-vided. For the high frequency case, the speed of vehicles andexciting frequency are 30 m/s. and 15 Hz, respectively. The rela-tionship between the exciting frequency fR (Hz) and vehicle speedu(m/s.) and road’s spectral wavelength kR (m) is as follow:

fR ¼u

kR(33)

The simulation results are shown in Fig. 11. At the low fre-quency ride test, there is no significant difference between thevehicles center of gravity’s vertical acceleration. However, in thehigh frequency test, the softest tire (kt¼ 100 N/mm) contributesthe best ride comfort. The maximum of the vertical acceleration

for the hardest tire is about 0.7 g, while this value for the vehiclewith the softest tire is 50% less. These results confirm the illus-trated results of Figs. 2 and 3.

7.2 Handling. To evaluate the handling of the case studyvehicle with different tires, a double lane change (DLC) test isselected. This test determines the characteristics of vehicles’ han-dling in a highly transient situation. DLCs are obstacle avoidancemaneuvers that frequently occur in the real world. The targetedspeed in this maneuver is 33 m/s. (120 km/hr.). The vehicle pathof the five mentioned vehicles during the test has been illustratedin Fig. 12. As illustrated by this figure, by using the same drivermodel, the target path is approximately achieved for all fivevehicles.

Also, Fig. 13 shows the yaw velocity, lateral acceleration, andslip angle responses of the vehicles versus time. Because thevehicles follow the same path and speed approximately, the lateralacceleration and yaw rate responses of vehicles with different tiresare very similar. However, according Fig. 13(c), the slip angleresponse of the vehicles are not the same. The maximum value ofslip angle for the vehicle with optimized for handling tire is about1.5 deg, showing that it is in the safe zone completely. However,for the vehicle with the hardest tire, this value is more than doubleand reaches about 3.5 deg. For the vehicle equipped with generallyoptimized tires (kt¼ 195 N/mm), the maximum side slip angle isabout 2 deg. Although this value is a little more than 1.5 deg, itstill is in the safe zone [24,25].

7.3 Braking Performance. A severe straight-line brakingwith initial speed of 27.7 m/s. (100 km/hr.) is utilized to evaluate

Fig. 8 Radar chart of the optimum tire vertical stiffness

Table 5 Main specifications of the case study vehicle

Parameter Value Unit

Vehicle sprung mass 1410 kgVehicle sprung mass 1274 kgWheelbase 2578 mmFront track 1539 mmRear track 1539 mmCG height 540 mm

Table 6 The criteria’s value of 195/65R15 tire with different vertical stiffness

Criterion

Optimizedtire for ride

kt¼ 100 N/mm

Optimized tire forhandling

kt¼ 140 N/mm

Optimized tire forperformance

kt¼ 170 N/mm

Optimized tire for fuelconsumption

kt¼ 300 N/mm

Optimum tire ingeneral

kt¼ 195 N/mm

Ride 0.00657 0.00419 0.00329 0.00171 0.00279Handling 64,922 67,855 66,706 58,680 65,561Performance 63,175 71,927 73,679 33,614 71,989Fuel consumption 48 57 63 78 66Evaluation function 3.46 3.52 3.58 3.25 3.61



the braking performance of the vehicles. This test is simulated ona dry level road with the coefficient of friction of one. The reduc-tion in vehicles speed versus time is shown in Fig. 14. This figureshows that the maximum deceleration belongs to the vehicle withkt¼ 170 N/mm (optimized tire for accelerating/braking perform-ance). As can be seen in Fig. 15, the stopping distance of this ve-hicle is about 68 m, while for the vehicle with the hardest tire, thestopping distance is more than 75 m.

7.4 Fuel Consumption. As mentioned before, the rollingresistance of tires affects fuel economy of vehicles. Hence, toinvestigate the effect of tire vertical stiffness on fuel consumption,we can compare their rolling resistances. The simplest way forevaluating rolling resistance of tires is the coast down test method.In this method, an engine-disengaged vehicle from a certain low

speed is released. The distance that the vehicle travels before itstops is a measure of the vehicle rolling resistance. A longer stop-ping distance means a lower rolling resistance.

For this purpose, a simulation is done in which five coast-downvehicles are released with the initial speed of 5.6 m/s (20 km/hr).Each vehicle is equipped with a different set of tires as mentionedearlier. Except the rolling resistance force, other resistance forcesfor all the vehicles are the same. The longitudinal velocity of thesevehicles versus traveled distance from origin has been plotted inFig. 16, respectively. It is clear that the vehicle equipped with thehardest tire kt¼ 300 N/mm (optimized tire for fuel consumption)has the longest stopping distance, about 220 m, whereas for thevehicle with the softest tire this value reduces by more than 40%.This value decreases just by 15% for the vehicle with generallyoptimized tire.

Fig. 9 Lateral force versus side slip angle for different tires

Fig. 10 Longitudinal force versus slip ratio for different tires



8 Conclusion

In this paper, the influences of the tire vertical stiffness on theride comfort, handling, accelerating/braking performances, andfuel economy of vehicles have been analytically investigated. Fur-thermore, a new method for determining the optimum tire verticalstiffness has been developed. The proposed method is formedbased on definition of analytical performance criteria as functionsof tire vertical stiffness for the above-mentioned vehicle perform-ances. Reverse of acceleration transmissibility function, tire cor-nering coefficient, tire longitudinal slip stiffness, and reverse of

Fig. 11 Simulation results for ride test on the sinusoidal road:(a) Low frequency and (b) high frequency

Fig. 12 Path of the vehicles during double lane change test

Fig. 13 Vehicle handling responses during double lanechange test: (a) Yaw velocity, (b) lateral acceleration, and (c)Vehicle slip angle



rolling resistance coefficient have been chosen as ride, handling,accelerating/braking performance, and fuel consumption criterion,respectively. Optimizing each individual criterion has determinedthe optimum tire vertical stiffness from that perspective. At theend, in order to compromise between ride, handling, accelerating/braking performance, and fuel consumption of a vehicle, an evalu-ating function has been formed by combining the individual crite-ria. Maximizing the evaluating function determines thegeneralized optimum tire vertical stiffness. Precise simulations byusing Carsim

VR

software show that the generally optimized tire forall aspects has been able to provide good performances in termsof ride, handling, accelerating/braking, and fuel consumption,while the individually optimized tires just provide good result intheir incorporated aspect and not in the others.

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Fig. 14 Vehicle longitudinal velocity versus time during brak-ing test

Fig. 15 Vehicle longitudinal velocity versus time during brak-ing test

Fig. 16 Longitudinal velocity of vehicles versus time duringcoast down test



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http://www.mate.tue.nl/mate/pdfs/8727.pdf

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