Intelligent Reduction of Tire Noise

Transcription

1 Intelligent Reduction of Tire Noise Matthias Becker and Helena Szczerbicka University Hannover Welfengarten 3067 Hannover, Germany Abstract. In this paper we report about deployment of intelligent optimisation algorithms for noice reduction in tire manufacturing. Since the complexity of the problem grows exponentially (the search space is typically of the order of a 65-dimensional vector space), a complete search for the optimal tread profile is not possible even with today s computers. Thus heuristic optimization algorithms such as Genetic Algorithms and Simulated Annealing are an appropriate means to find (near) optimal tread profiles. We discuss approaches of speeding up the generation and analysis of tread profiles, and results using various optimization algorithms. Introduction Modern tires have to fulfill a lot of basic requirements concerning adhesion, abrasion, stability, etc. If the basic design of a tire is finished, there are still some fine points to be taken into account. One area of research is the noise characteristics of a tire. This paper is concerned with application of intelligent optimization algorithms in order to find designs of tread profiles for tires, that will produce a low, unobtrusive noise inside a car. Though the noise design comes after determination of the basic functional properties, the noise characteristics are an important marketing issue. The subjective impression of the quality of a car is highly influenced by a nice low sound inside the car. This lets the car appear comfortable and of high quality, thus car manufacturers are interested to choose a tire for their cars that makes them appear more valuable. It is therefor important for tire manufacturers to design low noise tires, in order to get a share of the large market of supplying car manufacturers. Problem Statement A tread profile usually consists out of a number of given basic building blocks (called pitches, normally an elevated part followed by a hole ). There is a set of different pitch types (two to approx. ten different types). Pitches of different types normally have the same structure, but vary in their length. The length is B. Gabrys, R.J. Howlett, and L.C. Jain (Eds.): KES 006, Part I, LNAI 45, pp , 006. c Springer-Verlag Berlin Heidelberg 006

2 Intelligent Reduction of Tire Noise 707 usually given as a length ratio, the first pitch type is assigned length, the second e.g. length. etc. Typically a tread profile consists out of 50 to 70 pitches. This is called a pitch sequence (of length n). A tread profile (or the corresponding (,,,,,... ) Fig.. Mathematical representation of a tread profile... pitch sequence of length n) can be represented by an n-dimensional vector (see fig. ), thus the problem can be mathematically formulated as a search in an n-dimensional vector space. Let k be the number of different pitch types, then the number of different possible pitch sequences is equal to k n. Such search space is too large for an exhaustive search in a reasonable amount of time, even with todays computers. The pure generation (without evaluation) of sequences reaches rates of millions of pitch sequences per second, however even at this rate the complete search would last thousands of years. The rest of the paper is organized as follows. In the next section we will show how to make the generation and analysis of pitch sequences more efficient. This is to be used for enumerative search. In section 4, a more directed search using heuristic optimization algorithms is presented. rotational symmetry mirror symmetry Fig.. Symmetry of pitch sequences 3 Efficient Analysis and Generation of Pitch Sequences The prediction of the noise characteristics of a pitch sequence is done by spectral analysis using the Fourier transform (cf. e.g. [3] or any mathematical standard

3 708 M. Becker and H. Szczerbicka literature). Roughly the noise model assumes that each pitch issues a noise impulse when hitting the ground. The details of the noise model vary among the tire manufacturers and are confidential or covered by patents (e.g. tire designs where the individual pitch lengths do not have a common divisor, or are prime numbers, EP 08059/3.0.84, EP / ). Some works about details of noise models have been published at international conferences such as the regular Internoise, or on national workshops (cf. e.g. [] for a study of physical mechanisms of noise generation by moving tires on the road). 3. Efficient Generation Since the noise model results in a periodic function whose spectrum is then analyzed, two rotational symmetric pitch sequences result in the same spectrum. Furthermore the noise spectrum remains the same, if the tire is moved forward or backward, or in other words, two pitch sequences that are mirror symmetric result in the same noise spectrum (see fig. ). Thus for efficiency, it is enough to generate only one representative pitch sequence for each equivalence class of rotational and mirror symmetry. Algorithms for that can be found in mathematical books of permutation theory, under the keyword bracelet and necklace, e.g. in []. Other keywords are Lyndon word (= an aperiodic necklace representative). It is important to use an algorithm that produces exactly one (and only one) representative of each equivalence class, instead of e.g. generating all sequences and checking for symmetry afterwards, and throwing equivalent sequences away then. Furthermore there are a number of constraints to be taken into account. Common constraints on pitch sequences are a maximal number of pitches of the same time in series, or how many pitches of each type should occur in a pitch sequence maximally/minimally, etc. 3. Efficient Analysis The assessment of the quality of a given pitch sequence (analysis) is based on Fourier analysis, i.e. the noise spectrum of the tire is estimated. The noise model for a pitch sequence is an irregular rectangular function, that results in an impulse function, if discretized via sampling. Fourier analysis of that function yields the frequency spectrum, i.e. the value of the Fourier coefficients that indicates how much the corresponding frequency contributes to the overall sound. Since the noise of pitch sequences are roughly rectangular functions (See upper part of fig. 3 for an example of a rectangular function with amplitude A over time t.), the noise spectrum of a sequence has the peak around the Fourier coefficient equal to the number of pitches, and sub-peaks at multiples of it (harmonics) (See lower part of fig. 3 where the amplitude A for the corresponding fourier coefficients FC is shown.). Ideally there should not be one peak frequency but the energy of the noise should be distributed rather equally over all frequencies.

4 Intelligent Reduction of Tire Noise 709 A t A FC Fig. 3. Square function and Fourier spectrum Evaluation Order of Fourier Coefficients. Since we are interested in a pitch sequence that has the lowest maximal Fourier coefficient (that mostly is located around the basic period equal to the pitch sequence length) one obvious improvement of efficiency is not to calculate the Fourier coefficients from to lets say 00, but to start with the nth Fourier coefficient, then evaluate n + / etc. (given a pitch sequence length of n). Sine Table. The calculation of each Fourier coefficient is a sum of sines. We achieve a significant speed up when using a table with pre-calculated values for sines, instead of calculating each value again and again with much too high precision. Re-use of Partial Sums. If pitch sequences are enumerated, so that the ith pitch sequence differs only in one or two digits from the i th, then we can re-use most of the calculations/of the sum evaluated for the ith sequence. This measure also increase efficiency a lot. Note that this is not possible if pitch sequences are generated randomly. 4 Heuristic Optimization of Pitch Sequences Often, there exists a pitch sequence that is in use already, and tire manufacturers would like to improve the existing pitch sequence. This can be done by local heuristic optimization algorithms, such as hill-climbing. Global heuristic optimization algorithms can also be used as alternative to random search or enumeration of pitch sequences. For more information on heuristic optimization algorithms see standard books on optimization algorithms, e.g. [4]. 4. Local Optimization In order to increase the quality of existing pitch sequences, we implemented local optimization algorithms, namely hill-climbing variants. The basic mechanism of hill-climbing is to check all neighbors of the momentarily best solution, and advance to the next better neighbor. If no better neighbor is found, then

5 70 M. Becker and H. Szczerbicka the algorithm ends. The crucial step in this case is the efficient generation of neighbors. When checking the neighborhood of one pitch sequence for better solutions, it is again important not to generate lots of invalid pitch sequences and test constraints after generation, but to generate only valid pitch sequences. We evaluated three variants of hill-climbing: SAHC:Steepest ascent hill-climbing proceeds from one vector to the best of all of its neighbors. NAHC:Next ascent hill-climbing proceeds from one vector to the next best found neighbor. RMHC:Random mutation hill-climbing modifies the current vector randomly and proceeds to the next best found vector which, different to NAHC, does not have to be a neighbor of the current vector. RMHC is not a strictly local search anymore, since by the random mutations also not direct neighbors might be chosen as next vector. 4. Global and Combined Optimization Local heuristic algorithms get stuck easily in a local optimum. Thus global heuristic algorithms are an alternative, if not a local optimization is wanted, but a global search (more intelligent than random search) should be accomplished. We implemented Simulated Annealing (SA), Genetic Algorithms (GA), and also combined algorithms, that first do a global search via random search/sa/ga, and afterwards do a local optimization using hill-climbing. These combined algorithms are known to achieve good results in terms of quality of solution, and execution speed quite often (see. e.g. [5]). We studied hill-climbing, SA, GA and two hybrid variants of both GA and RMHC. In the first variant a hill-climber is used for fine-tuning after the global search stops, and in the second a hill-climber is used to generate fine-tune new points during one step of the global search. All parameters used in the algorithms cannot be given. However we point out some import ones for GA: gene size of three, gene drift and crossover has been used, population size was 500. In detail, the algorithms used in the following test cases were: RMHC SAHC 3 NAHC 4SA 5GA 6 RMHC with SAHC for optimizing each new neighbor 7 RMHC with SAHC after RMHC has stopped 8 GA with SAHC for optimizing new members of the population 9 GA with SAHC after GA has stopped 4.3 Test Cases We tested the optimization algorithms on two problems, one smaller problem (pitch vector size of ) where the optimal solutions is known, and on a large problem (pitch vector size of 66).

6 Intelligent Reduction of Tire Noise 7 Small Vector Size. This is a small test problem which is not very realistic, but which is small enough to search the complete search space for the global optimum, giving a known reference point for the performance of the individual algorithms. The vector size is, and there are three different types of pitches. Constraints were that each element should occur between one and seven times, each pitch type may occur maximal six times in a row. The global optimum is with a value of Each heuristic optimization algorithm has been run 50 times, with different random seeds. Results for Small Problem. Fig. 4 shows the results. It shows that often HC based algorithms have difficulties to find valid neighbors with regard to the constraints, therefore they get stuck and stop with a bad result (this explains the large standard deviation in the right graph for some of the HC based results.). Best results showed the GA based algorithms, in 8% the global optimum has been reached, the median is only.% away from the optimum. Pure GA was slightly better than hybrid GA. SA performed well, the found vectors were about 7.7% away from the optimum. NAHC performed worst, being around 6.6% away from the optimum, and getting stuck very often because of the constraints. # evaluations goal function value Fig. 4. Performance of algorithms. Left graph shows the number of evaluations of the goal function (minimum, median, maximum). Right graph shows the found optima (cross), median of found solution (circle), and standard deviation.,,3,6,7 are algorithms based on hill-climbing, 4 is SA, 5 is pure GA, 8 and 9 are hybrid algorithms based on GA. Large Vector Size. Here the vector size is 66 with three different types of pitches. There are several constraints. Each vector should contain exactly 6 elements of type one, of type two and 8 of type three. For each pitch type, there should not be more than six in a row. Pitches of type one and three may not be neighbors.

7 7 M. Becker and H. Szczerbicka Results for Large Problem. The global optimum is not known, we tried to find the best vector by a very long search prior to the 5 (shorter) test for each algorithm. The best found vector is with a value of 4.3. The outcome is shown in fig. 5. Best results showed again GA, finding a vector only 0.9% away from the optimum (mean distance to optimum is 4.%). Second was RMHC, the best solution was 5.9% away (mean distance to optimum 5.9%). Both algorithms yielded good results with a small standard deviation in short time. Hybrid algorithms of GA and RMHC let the standard deviation become larger, but also increase the quality of the best found solution. SA got stuck in the same local optimum, despite including randomness in the search. Obviously this optimization problem was relatively hard, regarding the structure of the search space. Some very attractive suboptimal solutions exist and make it harder to find the best solution # evaluations goal function value Fig. 5. Performance of algorithms. Left graph shows the number of evaluations of the goal function (min, med, max). Left graph shows the found optima (cross), the median of the found solutions (circle), and the standard deviation. Again,,,3,6,7 are algorithms based on hill-climbing, 4 is SA, 5 is GA, 8 and 9 are hybrids based on GA. 5 Conclusion In this work, we studied the applicability of heuristic optimization algorithms in order to design low noise tire profiles. It showed that this particular problem should be best approached with genetic algorithms. We furthermore give hints on how to improve efficiency of generation and analysis of pitch sequences. We implemented a prototype software from scratch with regard to efficiency (see fig. 6), and we achieved a speed that is several magnitudes faster (in terms of generated and analyzed pitch sequences per second) than a grown bundle of existing software.

8 Intelligent Reduction of Tire Noise 73 Fig. 6. Screen shot of prototype In the future, we will try to exploit the mathematical properties of the Fourier analysis to possibly find good pitch sequences by construction. References. Combinatorical object server. cos/inf/neck/necklaceinfo.html. Beckenbauer, T.: Reifen-Fahrbahn-Geräusche Minderungspotentiale der Strassenoberfläche. In: Proceedings of the Deutsche Arbeitsgemeinschaft für Akustik (DAGA 03), Aachen, Germany, Briggs, W., Henson, V.E.: The DFT. SIAM, Philadelphia, Michalewicz, Z., Fogel, D.: How to solve it: Modern Heuristics. Springer, London., Szczerbicka, H., Syrjakow, M., Becker, M.: Genetic algorithms, a tool for modelling, simulation and optimization of complex systems. Cybernetics and Systems: An International Journal, Special Issue: Intelligent modelling and simulation for complex systems, II(7): , 998