Japan J. Indust. Appl. Math., 17 (2000), 403-426
Area (3)
Application of Genetic Algorithms for Optimization of Tire Pitch Sequences Yukio NAKAJIMA and Akihiko ABE Bridgestone Corporation, 3-1-1, Ogawahigashi-Cho, Kodaira-Shi, Tokyo 187-8531, Japan Received October 27, 1998 Revised July 9, 1999 A simple genetic algorithms (GAs) has been applied to generate the optimum pitch sequence. Though a simple GAs worked properly, there was the problem of the premature convergence. To solve this problem, we introduced the new operator named the growth and combined it with a simple GAs. The growth operator, which is a kind of the hillclimbing technique, has the function to get the local optimum in a small CPU time. The GA with growth generated better sequence than a simple GAs. The GA with growth was verified not to have the premature convergence even in the smaller population size. The optimum pitch sequence generated by the GA with growth improved the noise performance such as pass-by noise compared with the current pitch sequence.
Key words: tire, optimization, genetic algorithm, pitch sequence
1. Introduction The noise nuisance problem has been increased due to the increase of traffic volume and is becoming a community noise environment problem. In order to improve the noise problem, the regulations have become strict and manufacturers have invested for the research of the noise reduction. The vehicle manufacturers have been particularly successful for significant noise reduction such as engine, exhaustion and transmission systems. The tire manufacturers have also studied noise reduction of a tire [1]. A tire, however, is becoming a dominant noise source in a vehicle system and a tire causes 70% of total vehicle noise in a steady rolling and 20% of total noise in acceleration [2]. The tire noise is generated from the tread and sidewall vibration due to tread pattern and an irregularity of road surface. In order to improve the tire noise performance, the pattern, contour, material and construction of a tire have been studied. One of tire noise researches is the tire pitch sequence. Pitch refers to the distance between one groove on a tire's tread and corresponding point of the groove adjacent to it along the tire's circumference. Use of the same pitch length along the entire circumference of the tire leads to a constant noise frequency which results in a high and jarring noise. Therefore, pitches of different lengths are usually allocated at random, in order to disperse the noise frequency and maintain lower noise levels. Current technology seeks to lower noise levels by analyzing numerous, randomly generated pitch sequences and choosing the quietest from among them, but it needs a lot of computer resources or it will not mathematically assure an optimal pitch sequence [3]—[7]. In order to solve the problem, a simple genetic algo-
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rithms (GAs) was applied for the optimization of the pitch sequence by Nakajima et al. [8]. Though the pseudo optimum sequence was obtained by a simple GAs, the large population size and many iteration were necessary for the optimization. Hence, in this paper the idea of growth is combined with a simple GAs, in order to obtain better sequence by the small population size as well as faster convergence. 2. Theory of Pitch Noise Figure 1 shows the model for the pitch noise of a tire. The pitch noise is mainly generated at the leading and the trailing edges of a tire. The wave form of one groove impact is measured at the leading edge and the trailing edge on a three-meter diameter drum tester installed in an anechoic room. Though the wave form can be expressed by a sinusoidal function with exponential decay, we simply approximated it by the Dirac 8-function [9]. Figure 2 is the comparison of tire noise spectrum between the measurement at the leading edge and the noise model. The velocity is 100 km/h and the number of pitches is 60. The noise model is a good predictor of the measurement up to 1k Hz. Since we are interested in the pitch noise which frequency range is less than 1k Hz, the approximation of 8-function input is validated. Since the sequences of grooves generate the sequences of the wave form, we approximate them by the sequences of the following Dirac 8-function: f(t) = 1 t = ti f(t)=0 t # ti
Fig. 1. Model for tire pitch noise.
(1)
f(t)
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Fig. 2. Comparison between measurement and noise model. LZ
(2)
= V
ti
where L i is the length measured from a reference point in the circumferential direction and V is the velocity. If we need to express the measured wave form, we can approximate it by a sinusoidal function with exponential decay [9]: f(t)=0
t < tz
(3)
f (t) = e at sin bt t > t i —
where a = 316.0 and b = 27r x 10 3 which are determined by curve fit to the experiment. If we apply a Fourier series expansion to the equation (1), we can obtain the tire noise spectrum in the frequency domain: _
^
ao
00
/2 -ir n
2Tn
+L , a,^ cos ^, t+ b,^sin ^, t
(4)
i=i
where a n, and bn are expressed by the following equations:
an = T
bn, = T
f j
T
T
f(t) cos T 2 tdt
(5)
2 tdt(6) f (t) sin T
where T is a cycle of rotation. Substituting equation (1) to the equations (5) and (6), the following equations are obtained:
1 N2irn
a n = ^ cos ^ cost
(7)
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=
N27rn T1 sin T ti
(8)
i=1
where N is the number of pitches. The amplitude of nth harmonic number is expressed by the following equation: C^ = an + b 2(9)
The example output is shown in Figure 3 in which the spectrums without the pitch variation and with the pitch variation are compared. The peak amplitude of tire noise spectrum is reduced by a pitch variation treatment. The spectrum with pitch variation treatment is widely distributed in frequency range like the white noise which has the same spectrum amplitude in every frequency. The tonal noise, hence, can be improved by a pitch variation treatment.
Fig. 3. Spectrum with pitch variation and without pitch variation.
3. Optimization of Tire Pitch Sequence 3.1. Difficulty of optimization of tire pitch sequence The pitch sequence is defined by a series of integers. For example, 1233233311233332212... is a pitch sequence, when the number of pitch sizes is three. Each integer value indicates the different pitch size in the circumferential direction. When the number of pitches is 60, the number of pitch sizes is 3, and the number of pitches in each pitch size is 20, the number of combination of pitch sequences becomes an astronomical number: Combination =
60 i!
2 • (20!)33 x 10 26 .
(10)
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If one thousandth second CPU time is necessary to evaluate one pitch sequence, it will take three hundred trillion years to obtain an optimal sequence. Hence, we cannot overcome the so called "the cruse of dimensionality", if we search the whole design space of the tire pitch sequence. Hence, we cannot obtain the optimum by the random search which has been used in the tire industry [4]—[7]. Moreover, the mathematical programming, which is the most popular technique in the optimization, cannot be used for the integer type problem in which the design variable is expressed by integer [10], [11], [8]. Hence, we applied a GAs (genetic algorithms) for the optimization of the pitch sequence. Since a GAs is a kind of optimization technique which is not based on the sensitivity analysis, a GAs can be applied for the integer type problem. A GAs is search algorithms based on the mechanics of natural selection and natural genetics. The biological processes, reproduction, crossover, selection and mutation are replaced by the simple mathematical models, and the models are used as the optimization techniques. In a GAs, the evolutionary process of living things is grasped, the process of survival of the fittest. After rearrangement of the gene, the existing living thing is regarded as the pseudo-optimum. 3.2. General idea of optimization Optimization technique has been studied by many researchers for many years. Since the optimization requires a lot of CPU time in the sensitivity calculation, a research could not be pursued. Great strides, however, have been made in computer hardware, corresponding to the formidable computational requirements of large scale structures and the optimization researches have been advanced. We can write constrained optimization problem mathematically as follows [12]: (a) Objective function
(b)
Minimize : f (X)
(11)
g^(X) < 0 j = 1, m
(12)
hk(X)=0 k=1,l
(13)
Xi < X , < XZ i = 1, N
(14)
Constraints i. inequality constraints
ii.
equality constraints
iii. side constraints ;
(c) Design variables X = {Xl, X2, X3, ... , XN}
(15)
where a vector X is referred to as a vector of design variables and XZ and X2 are a lower and upper bound constraint respectively.
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3.3. Design variables The design variable Xi is the pitch length in the circumferential direction and ti in the equation (7) and (8) is defined by the following equation:
1
2—I
t 2 = VXk
(16)
k=1
where V is a rolling velocity of a tire. Since the pitch noise is periodic, a reference point can be selected in any place. So we chose t I as a reference point such as ti = 0.
(17)
3.4. Objective functions The objective function must be clearly defined in the optimization process. We can settle a few kinds of objective functions such as the spectrum amplitude or the spectrum deviation. The objective function of the spectrum amplitude is expressed as a min-max problem by the following equation: Minimize : maxc i(i = nmin
— nmax
)
(18)
where n m j n and n max are the minimum and the maximum of harmonic number to evaluate the objective function. The spectrum deviation is expressed by two ways; a deviation from mean spectrum amplitude and an information entropy. If the spectrum deviation is minimized, we will obtain the same spectrum amplitude in each harmonic number. So we can obtain the white noise by minimizing the spectrum deviation. One way to express the spectrum deviation is to use the following equation: (ci — c) 2(19)
f(X) = 2= n-in
where c is a mean amplitude of c n, expressed by c=
1n.nax E ci . nmax — nmiri + 1
(20)
=fl,,,.,.
The other way to express the spectrum deviation is to use an information entropy. The information entropy H is defined by the following equation: M
H = — ^pi logpi
(21)
i=I
where pi is the probability in which an event i is occurred, and M is the maximum harmonic number to be analyzed in this case. The p z possesses the next relation: M
pi = 1. I
(22)
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is satisfied, H becomes H has a specific relation proved easily: when p i = a maximum. Suppose that pi is related to an amplitude of ith harmonic number. We can prove easily that pi have the same values, when H has a maximum. Hence, if H is bigger, the spectrum deviation becomes smaller. Considering the equation (22), p i can be related to the spectrum amplitude ci by the following equation: pi = Mci(23) Ek=1 Ck
Substituting the equation (23) to the equation (21), M
H = Iog
Ck
k=1
1 M — Mci Lik=1 e k i=1
1og ci(24)
is obtained. In this case H becomes the objective function. 3.5. Constraints The constraints are required not to sacrifice other performances. When a small pitch, for example, is adjacent to a large pitch, an irregular wear will be occurred due to the imbalance of shear forces between a small and a large pitches. In order to circumvent an irregular wear, it is necessary to control a ratio of the adjacent pitches. For example the constraints are expressed by the following equation: 1 <
Xj+1
r X^
< r j= 1— N— 1
(25)
N —<---
(26)
r
X1
-
where r is the ratio of a large pitch and a small pitch in the adjacent pitches. The equations (25) and (26) will be easily transformed to the form of equation (12). 4. Genetic Algorithms: GAs The term, GAs, was first used by J.D. Bagley in 1967 in the field of computer science. Holland and his colleagues have further studied and extended it [13]—[15]. In recent years the engineering application has been studied [16]. A simple GAs is composed of three main operators as shown in Figure 4: (1) reproduction(selection), (2) crossover, and (3) mutation [14], [15]. The GAs begins by generating a population of N strings of length 1, where each string is composed of random numbers. Then each string is decoded yielding the actual design problem, the objective function and the constraints are evaluated, and the string is assigned a fitness value. These fitness values are then used to determine a new population of strings. New strings produced in subsequent generations are again decoded, evaluated, and transformed using the same operators. The process continues until convergence or suitable design is gained.
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initial population reproduction (selection) crossover (have children)
I mutation Fig. 4. Flowchart of simple genetic algorithms.
4.1. String representation
The coding method of the design variables is important in a GAs and affects the efficiency and reliability of a GAs. Binary representation is popular in a GAs. To obtain a bit string representation for a set of design variables, each pitch can be expressed as a fixed length string of zeros and ones which comprise components of a binary coded number. When the number of pitch sizes is five, three digits are necessary to express a pitch size. Binary numbers 000 and 100, for example, correspond to 1 and 5 of pitch sizes, but binary numbers 101 and 111 do not correspond to any pitch sizes. Hence the decimal representation is used in this paper. The decimal numbers of the strings in the first generation are determined by random number. 4.2. Reproduction Once a population of N strings is presented and the fitness value of each string is assigned, the genetic search can proceed to reproduction. Reproduction is the process where those strings with a higher level of fitness receive correspondingly large possibility of remaining in a mating pool where they await the action of the other two operators. In this process, the roulette wheel selection is used. The area occupied by an ith string on the wheel is assigned in proportion to the ratio, f^ , where f2 is the fitness value of ith string. The roulette wheel, which can be regarded as a random number generator, is invoked to determine the string to be reproduced as shown in Figure 5. To assign a fitness value to each string, a suitable function must be devised. This will be explained in the section of the fitness function.
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Fig. 5. Reproduction operator.
4.3. Crossover The second operator is crossover, and corresponds to allowing survival strings in a population to exchange their characteristics. Crossover proceeds in three steps as shown in Figure 6. In the first step, two newly reproduced strings are selected at random out of the mating pool. In the second step, selecting these two strings from the mating pool, arbitrary positions on the two strings are selected at random. The
Fig. 6. Crossover operator.
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third step is the exchange of all characters following the crossing site. Let us assume that sequence (a) and sequence (b) in (27) become the couple and the crossing sites are 2 and 4. Then the process of two point crossover is illustrated as follows:
parent(a) = 12 I 54 1 2343521 • parent(b) = 42 113 1 5223413.
(27)
where I indicates the crossing site. After the crossover two children are generated:
child(a') = 12 I 13 1 2343521 •
child(b') = 42 1 54 1 5223413....
(28)
String (a') is made up of the first part of string (a), the middle of string (b), and the tail of string (a). Similarly, string (b') is made up of the first part of string (b), the middle of string (a), and the tail of string (b). As shown in (27) and (28), the crossover can create a new pitch sequence design by exchanging the part of the strings. Though other crossover operations such as one point crossover and uniform crossover are proposed, two point crossover is applied in this paper. 4.4. Mutation Mutation is an insurance policy against the premature loss of the valuable genetic material during the reproduction and crossover. It corresponds to selecting quite a few strings in the population, determining at random a location on the strings, and changing the kind of pitch size at that location by the random number as shown in Figure 7. Although mutation is an indispensable process in a GAs, it occurs with a very small probability, merely 4% in this paper, and is secondary to the reproduction and crossover.
Fig. 7. Mutation operator.
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4.5. Fitness function
In the beginning of the optimization process, the condition of the strings is so random that the fitness values present a large diversification. Then a few strings having high fitness value have early domination and that causes premature convergence or premature loss of valuable genetic material. Thus, during the early stage of optimization process, the fitness values must be made to have less diversification by some transformation process. On the contrary, in the end of optimization process, the fitness values converge to the vicinity of the mean values, then there is high possibility that the bad strings will survive. During the end stage in the optimization process, the fitness values, hence, must have more diversification by some transformation process. The optimization problem defined by the equations (11) — (15) is a constrained problem. In a GAs of this paper, the design variables yielded from the strings are sent to the constrained optimization problem, evaluated with the objective function and the constraints, and assigned the fitness values. Those fitness values can be acquired from the fitness function, and an unconstrained maximization function is created by transforming a constrained optimization problem. The exterior penalty function is used for the transformation in this paper. The penalty function of the ith string in the tth generation is defined by the following equation: (i= 1 ^'N)
=f(Xfl+max[gj(XZ),0]
(29)
j=1
where 4i is the penalty function corresponding to the ith string in the tth generation, Xi is the vector of the design variables of the ith string in the tth generation, y is the penalty parameter, and N is the population size. In the general exterior penalty function method, the value of the penalty parameter is changed in proportion to the iteration. However, in this paper, the parameter value is fixed to be 9 x f through all generations, where 7 is an average objective function of the whole population in the first generation. The penalty function is transformed to the fitness function value by using the following fitness function and is shown in Figure 8: -
*z = —a(P? + b ^
avg (C
(i = 1 N N)
(30)
— 1)
(31)
a ^avg — min
b
—
4 avg ( C ^ avg — q5 min ^avg — Amin
)
(32)
where is the fitness function corresponding to the ith string in the tth generation. a, vg and $,,,,i n are the average and minimum values of the penalty functions
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respectively, and they can be expressed as follows: N
(33)
Oav g = N ^ i=l
min = min [-Pz]
(i = 1 — N).
(34)
Parameter C in the equations (31) and (32) is a coefficient for controlling the transformation, and 1.5 is adopted in this paper. Fitness Function 4)
C
avg
avg
L0 min
(P avg
(P max
Penalty Function Fig. 8. Fitness function.
5. Genetic Algorithms With Growth 5.1. Idea of growth A simple GAs worked properly in the optimization of the pitch sequence as shown in Figure 9. This is the history of convergence, when the objective function is the spectrum deviation. The population size, crossover and mutation probability are 500, 0.8 and 0.04 respectively. The number of pitch sizes, the ratio of the maximum and minimum pitch sizes, and the number of pitches are 5,11/7, and 66 respectively. Moreover, the ratio of the adjacent pitch sizes is constrained to be less than 9/7 in order to prohibit the big change in the adjacent pitches. The minimum value of the objective function decreases with generations. In the optimization process, the diversity of genetic materials is maintained, judging from the fact that there is the difference between the maximum and minimum value of the objective function in
every generation.
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Fig. 9. History of convergence. But a simple GAs required the large population size to get a good solution. In order to solve this problem, we introduced the growth operator, which was firstly proposed by Sugimoto [17], as shown in Figure 10. In the flowchart the initial population is generated by the random number and each string satisfies the constraints, and the convergence criteria are one of the following conditions. (a) Number of the generations reaches to the predetermined value. (b) Number of the strings having the highest fitness reaches more than ten percent of the population size. (c) The best value of the fitness is not updated in the consecutive 20 generations. In this paper the growth has the function that the selected pitch sequences are improved through the hill-climbing technique to reach the local optimum. The growth operator is controlled by the probability of the growth. For example, when the probability of each process is 0.5, half population are selected to improve through the hill climbing technique.
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I population initialization at random
improve some strings
I
I improve no string
assign fitness values reproduction selection of strings yes
no
crossover?
characters are not exchanged
exchange all characters following crossing site
mutation. yes
no
mutate at random no
check all strin s yes converge. es
no
optimal solution
Fig. 10. Flowchart of GA with growth operator. 5.2. Hill-climbing technique for growth operator : Flip-flop algorithm
The flip-flop algorithm used in the growth operator is a kind of hill-climbing techniques that begin at randomly generated point and climb the function in the steepest permissible direction until they cannot climb any more. The flip-flop algorithm is based on the following procedure and is shown in Figure 11. (a) Settle the range in which an exchange operation is performed. (b) Select a pair of pitches in which an exchange operation is performed. (c) Generate a new sequence by exchanging pitches. (d) When the new sequence satisfies the constraints, go to the next step. Otherwise, go to step (b) to select a different pair of pitches. (e) Evaluate the objective function of the new sequence.
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If the exchange range is not exceeded the given value, increase the range and go to the step (a). Otherwise, go to the next step. (g) If the objective function of the new sequence is better than the current best sequence, the new sequence becomes the best sequence in the iteration. (h) Otherwise, go to step (a) to select a different pair of pitches. (i) If the objective function is not improved by the new sequence, store the final results. Otherwise, go to step (a). (f)
Fig. 11. Flip-flop operation.
6. Experiment and Discussion 6.1. Parameter study on GAs Several parameters are required to supply for a GAs such as population size, crossover probability, mutation probability, and growth probability. Since these parameters are dependent on the application, we need to do the parameter study in the beginning. Hence we did the parameter study for a simple GAs. Figure 12 shows the effect of population size on the value of the objective function. The objective function is to minimize the spectrum peak, and the crossover and mutation probability are 0.8 and 0.04 respectively. The larger number of population is used, the larger improvement of the objective function can be obtained. Figure 13 shows the effect of population size on the convergence history of the objective function. When the population size is less than 500, the second convergence criterion is applied such that the number of the strings having the highest fitness reaches more than ten percent of the population size. On the other hand, when the population size is 1,000, the third convergence criterion is applied such that the best
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A. ABE
Fig. 12. Effect of population size on the value of the objective function.
Fig. 13. Effect of population size on history of convergence in a simple GA.
value of the fitness is not updated in the consecutive 20 generations. Hence the premature convergence is occurred in the small population size. When the population size becomes larger, the larger generations are necessary to get a convergence and the larger improvement is obtained. Since the CPU time is increased with the population size, the appropriate population size must be selected to obtain an optimum solution within the reasonable CPU time.
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Fig. 14. Effect of crossover probability on the value of the objective function.
Fig. 15. Effect of mutation probability on the value of the objective function. Figure 14 shows the effect of the crossover probability on the value of the objective function. The larger crossover probability is used, the larger improvement is obtained up to 0.8. If the crossover probability is too big, the optimum sequence in the previous generation is disappeared due to the crossover. The very large crossover probability, hence, does not effectively improve the value of the objective function. Figure 15 shows the effect of the mutation probability on the value of the objective function. The small mutation probability is better than zero mutation probability and the optimum of mutation probability is around 0.04. The small mutation probability seems to increase the search range and keep the solution from the premature convergence.
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6.2. Improvement of simple GA by growth operator Figure 16 shows the effect of growth operator on the value of the objective function. The population size, crossover, mutation, and growth probability are 100, 0.8, 0.04 and 0.25 respectively. We intentionally chose the small population to see the effect of the growth operator. When the growth operator is not applied, the second convergence criterion is applied around 200th generation. Since the second convergence criterion is that the number of the strings having the highest fitness reaches more than ten percent of the population size, the premature convergence is occurred in this case. On the other hand, when the growth is applied, the first convergence criterion is applied at the 300th generation. Since the first convergence criterion is that the number of the generations reaches to the predetermined value, the premature convergence is not occurred. Because the diversity can be maintained in all generations due to the better sequence generated by the growth operator. The value of the objective function is improved even in the younger generation and is gradually improved in the optimization process due to the growth operator. We found that the GA with growth can be practically applied in a small or medium population size. The CPU time and the optimum value of the objective function are compared in the different population size without growth and with growth as shown in Figure 17. In a simple GAs, the CPU time is increased with the population size and the population size must be large in a simple GAs, in order to obtain the better solution. However, the GA with growth can obtain the better solution in a small population size within the relatively small CPU time. Furthermore, we made a comparison between the GA with growth and the
Fig. 16. Effect of growth operator on the value of the objective function.
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crossover O.8 mutation=0.04 growth=0.25
Fig. 17. Comparison of CPU time and optimum value of objective function.
Fig. 18. Comparison of optimums by GA with growth and random number generation.
random number generation currently used in the tire manufacturers. In Figure 18 the horizontal axis is the value of the objective function and the vertical axis is the frequency which corresponds to the value of the objective function. The objective function is to minimize the spectrum peak in this example. In the random number generation, 5 x 10 5 pitch sequences are evaluated. On the other hand, in the GA with growth 3.7 x 10 5 pitch sequences are evaluated to get the optimum. Further-
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more the best sequence generated by the GA with growth possesses lower value of the objective function than the best sequence generated by the random number generation. Hence the GA with growth is better and more effective search method than the current method.
6.3. Effect of pitch sizes on the value of the objective function The ratio of a maximum and a minimum pitch size affects on the pitch noise. But we could not know whether the optimum ratio exists or not. Hence the effect of the pitch sizes on the value of the objective function was investigated by comparing the optimum sequences with different ratios. The objective function is to minimize the spectrum peak. When the ratio of a maximum and a minimum pitch size becomes larger, the value of the objective function becomes smaller. But there seems to have the optimum around 1.6 as shown in Figure 19.
Fig. 19. Effect of pitch size (max/min) on the optimum value.
This general tendency is explained by the theory of the frequency modulation [3]. The carrying wave is expressed by the following equation. e(t) = A sin ¢(t)
(35)
0(t)
(36)
=J
W(t)dt
Since the amplitude is constant and the frequency is changed in time, the angular frequency is defined as follows: w(t) = W e + Wd COs W m t
(37)
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where w, is the frequency of the carrying wave with evenly spaced tread design in the circumferential direction. wd is the parameter of frequency modulation range and is related to the frequency deviation from the frequency of carrying wave through the pitch variation. w,,,, is the modulation frequency. Substituting the equation (36) and (37) to the equation (35), the next equation is obtained: e(t) = A sin I w c t
\
+ wd sin w,,,.t I . wm
(38)
Defining mf by -, mf is called the frequency modulation index. The equation (38) is transformed to the following equation: e(t)
= A>2 J(m1) sin(w, + nw m )t
(39)
n=—oo
where Jn (mf) is the first kind Bessel function and is the spectrum amplitude at the frequency w, + nwm. Figure 20 shows the effect of mf on the spectrum dispersion. Since Wd is related to the difference of the maximum pitch size: pmay and the minimum pitch size: p„, mf is approximately expressed by the following equation: mf — Pmax — Pmin•
(40)
When m1 becomes large, the spectrum becomes uniform and the spectrum peak is reduced.
Fig. 20. Effect of m1 on the spectrum dispersion.
6.4. Effect of optimum sequence on noise reduction We applied two kinds of optimum sequences to the molded tires in which the spectrum deviation and the spectrum peak are selected as the objective functions respectively. Then we made a comparison with the current sequence obtained by
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Fig. 21. Noise improvement by the optimum sequence.
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the random number generation. The number of pitch sizes is five, the ratio of the maximum and the minimum pitch sizes is 11/7, and number of pitches is 66. The constraints of the ratio of the adjacent pitches is 9/7. Figure 21 shows that the optimum sequences are better than the current sequence in the coasting down pass by the noise measurement and the indoor noise measurement. We also found that other tire performances are not affected by changing the pitch sequence. Though tire noise has been considered to be difficult improve without sacrificing other performances, tire noise can be improved by applying the proposed sequence without sacrificing any other performances. Comparing the sequences with two different objective functions, the spectrum deviation is better objective function than the spectrum peak to improve the noise performance. Figure 22 shows the noise pressure contour predicted by the noise simulation system in which the FEM is combined with the BEM [1]. The optimum sequence posses the smaller noise contour surface than the current sequence. Since the noise contours have the same noise pressure, the effect of the optimum contour is also verified by the simulation.
Fig. 22. BEM simulation of noise pressure contour. 7. Summary A simple genetic algorithms (GAs) has been applied to generate the optimum pitch sequence. Since the parameters of a simple GAs are dependent on the application, the parameter study was done. In the optimization of the tire pitch sequence, the crossover and the mutation probability should be around 0.8 and 0.04 respectively. Though a simple GAs worked properly, there was the problem of the premature
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convergence. To solve the problem, we introduced the new operator named the growth. The growth operator is developed by the flip-flop algorithm which is a kind of the hill-climbing technique. The GA with growth generates better sequence than a simple GAs. Furthermore, the GA with growth does not have the premature convergence even in the smaller population size. By using the GA with growth, the way to design the pitch sequence is clarified. For example, the larger ratio of a maximum and a minimum pitch size is used, the larger improvement of the value of the objective function is obtained. Furthermore we found that the optimum pitch sequence improves the noise performance such as pass-by noise compared with the current pitch sequence. The spectrum deviation is better objective function than the spectrum peak to improve a tire noise performance. References [ 1 ] Y. Nakajima, Y. Inoue and H. Ogawa, Application of the boundary element method and modal analysis to tire acoustics problems. Tire Science and Technology, 21 (1993), 66. [2] JATMA, On Noise due to Tire and Road (4th version). (in Japanese). [3] J.H. Varterasian, Quieting noise mathematically - Its application to snow tires. SAE paper, No.690520, 1969. [4] Japanese Patent No.3-23366. [5] Japanese Patent No.4-232105. [6] Japanese Patent No.4-363234. [7] European Patent No.0 543 493 Al. [8] Y. Nakajima, T. Kamegawa and A. Abe, New tire design procedure based on optimization technique. SAE Technical Paper Series, 960997, 1996. [9] USA Patent No.F01613US. [10] A. Abe, T. Kamegawa and Y. Nakajima, Optimum Young's modulus distribution in tire design. Tire Science and Technology, 24 (1996), 204. [11] Y. Nakajima, T. Kamegawa and A. Abe, Theory of optimum tire contour and its application. Tire Science and Technology, 24 (1996), 184. [12] G.N. Vanderplaats, Numerical Optimization Techniques for Engineering Design with Applications. McGraw-Hill, 1984. [13] L. Davis (ed.), Handbook of Genetic Algorithm. Van Nostrand Reinhold, 1991. [14] D.E. Goldberg, Genetic Algorithm in Search, Optimization & Machine Learning. AddisonWesley, 1989. [15] J.H. Holland, Adaptation in Natural and Artificial Systems. The University of Michigan Press, Ann Arbor, 1975. [16] E. Sandgren and E. Jensen, Automotive structural design employing a genetic optimization algorithm. SAE Technical Paper, No.920772, 1992. [17] H. Sugimoto, Discrete optimization of truss structures and genetic algorithms. Proceedings of the Korea-Japan Joint Seminar on Structural Optimization, 1992.