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The Optimal Design for Low Noise Intake System Using Kriging Method with Robust Design∗

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873 The Optimal Design for Low Noise Intake System Using Kriging Method with Robust Design∗ Kyung-Joon CHA∗∗, Chung-Un CHIN∗∗∗,Je-SeonRYU∗∗ and Jae-Eung OH∗∗∗∗ This paper proposes an optimal design scheme to improve an intake’s capacity of noise reduction of the exhaust system by combining the Taguchi and Kriging method. As a measur- ing tool for the performance of the intake system, the performance prediction software which is developed by Oh, Lee and Lee (1996) is used. In the first stage, the length and radius of each component of the current intake system are selected as control factors. Then, the L18 table of orthogonal arrays is adapted to extract the effective main factors. In the second stage, we use the Kriging method with the robust design to solve the non-linear problem and find the optimal levels of the significant factors in intake system. The L18 table of orthogonal arrays with main effects is proposed and the Kriging method is adapted for more efficient results. We notice that the Kriging method gives noticeable results and another way to analyze the intake system. Therefore, an optimal design of the intake system by reducing the noise of its system is proposed. Key Words: Intake System, Sound Reduction, Kriging Model, Robust Design to reduce the intake noise has been applied by the method 1. Introduction of trial and error after the design of the engine room is The noise from the exhaust system and the intake sys- finished. In addition methods of the excessive noise re- tem of an automotive vehicle has a uncomfortable effect duction cause a bad effect rather than reduce the intake on the riding quality of the passenger well as causes the noise. environmental noise. In addition, as the number of an au- It is difficult to design an optimal automotive intake tomotive vehicle is increasing, the quietness in a vehicle’s system because the design of the intake system affects the passenger compartment becomes one of the most impor- engine performance and the space of the engine room is tant performances of high quality vehicle, and recently the limited. Recently, various analysis methods (the trans- intake noise is being considered the important object of fer matrix method, the acoustic finite element method and the research. etc.) and the experimental method using the simulator are In general, the intake noise is the low frequency noise proposed(9), (25) – (28), (33). below 500 Hz. The booming noise generated by the intake However, it requires a lot of time and cost so noise transferring to the interior of the vehicle has a un- far to design the intake system optimally because vari- comfortable effect on the riding quality. However, it is a ous analysis methods and the experimental method de- difficult to reduce the time and the cost for the develop- pend on the method of trial and error. Therefore, the ment of the low noise intake system because the method Taguchi method which can improve the performance of the system with a low cost and time is frequently ∗ Received 21st November, 2003 (No. 03-5145) applied(16), (18), (23), (37) – (39), (43). ∗∗ Department of Mathematics, Hanyang University College In this study, the characteristics of the noise reduction of Natural Sciences, 17 Haengdang-Dong, Seongdong- are evaluated using the intake system (1 500 cc, DOHC en- Gu, Seoul 133–791, Korea. gine) as shown Fig. 1. Also, in order to solve the non- E-mail: [email protected], [email protected] linear optimization problem which is inefficient in the ro- ∗∗∗ BK21 Division in Mechanical Engineering, Hanyang Uni- bust design for the intake system, the Kriging method is versity, 17 Haengdang-Dong, Seongdong-Gu, Seoul 133– ff 791, Korea. E-mail: lada @hanmail.net applied. That is, after the most e ective factor is selected ∗∗∗∗ Division of Mechanical Engineering, Hanyang Univer- using the robust design, the level of the significant param- sity, 17 Haengdang-Dong, Seongdong-Gu, Seoul 133– eter is subdivided and the non-linear optimal condition of 791, Korea. E-mail: [email protected] parameters is obtained in the limit condition by applying JSME International Journal Series C, Vol. 47, No. 3, 2004 874 one confidence that the design is “infilterating” the design space well and important in the robust design. Nomenclature β : the unknown vector and should be estimated from n observed values βˆ : the usual generalized least squared estimate of β C12 : a real part of the cross spectrum for inlet f (x) : a known linear function of input x k : wave number T {pr vr} : the state vector at the upstream point r T {pr−1 vr−1} : the state vector at the downstream point r−1 Fig. 1 Overview of an intake system + p1 : an inlet sound pressure − p2 : an outlet sound pressure the Kriging method, which can estimate the mutually cor- Q12 : an imaginary part of cross spectrum for inlet related data. r(x) : the correlation vector between the response val- Kriging is based on the field of geostatistics, in ex- ues at the observed points x1,···, xn and the re- amples, hybrid discipline of mining engineering, geol- sponse at a given location x ogy, mathematics and statistics. The approach to pre- R : correlation matrix of input x diction advocated in this paper is known as Kriging af- R(xi, x j) : correlation function between any two points xi ter Dr. D.G. Krige’s work(17) on the Rand gold deposit, and x j of n sampled points in southern Africa. He developed an empirical method S aa : an incident spectrum for inlet for determining a true ore grade distribution from distri- S bb : a reflected spectrum for inlet butions based on sampled ore grade in the 1950’s. Since S cc : an incident spectrum for outlet (22) 1963, Matheron have developed this Kriging technique S dd : a reflected spectrum for outlet in France. This performs well in predicting the value of wi : the energy of inlet a possible but actually not taken observation of a spatially wt : the energy of outlet (17) distributed variable such as a mine grade , a soil char- yij : the transmission loss acteristic(41), rain fall(1), gene frequency(31),orimagese- y(x) : the unknown function of interest quence coding(11). Z(x) : the realization with mean 0, variance σ2, nonzero Recently, Kriging goes by a variety of names in- covariance cluding DACE (Design and Anlaysis of Computer Ex- periments) model, which is the title of the inaugural pa- 2. Backgrounds of Simulator and Experiments (34) per by Sacks, et al. This is derived from geostatistics 2. 1 Analysis of simulator and used for fitting the model of the deterministic output As a method of modeling the transfer characteristic of ffi as the realization of random process for e cient predict- acoustics, the transfer matrix method which introduces the ing. There have been some studies for DACE model at concept of impedance is used. It is widely used for acous- AIAA (American Institute of Aeronautics and Astronau- tic systems for its computational simplicity. This method tics, inc.), in particular. Sacks, et al., first introduced Krig- makes a design easy since modeling by each factor makes ing as a tool of interpolation of deterministic computer up the whole system. experiments in 1989. They proposed to use the Kriging Adopting acoustic pressure p and mass velocity v method with space filling design such as minimizing inte- as the two state variables in the transfer matrix method, grated mean squared error (IMSE), minimizing maximum we could find the four-pole parameters from the condi- mean squared error (MMSE) and maximizing entropy. tions of both sides which can be written as Eq. (1), where Giunta(14) and Giunta, et al.(15), performed a preliminary T {pr vr} is called the state vector at the upstream point r investigation into the use of Kriging for the multidisci- T and {pr−1 vr−1} is called the state vector at the downstream plinary design optimization of a high speed civil transport point r −1. aircraft. Booker(3) and Booker, et al.(5), use the Kriging method to study the aeroelastic and dynamic response of pr = Transfer matrix pr−1 v × v (1) a helicopter rotor with orthogonal arrays. Simpson(36) and r 2 2 r−1 Ryu, et al.(32), perform the Kriging method with evenly The transmission loss is an energy loss of acoustic spaced samples. elements, so the ratio of sound pressure between the in- The experimental design we used in this paper is let and outlet of acoustic elements can be expressed in dB orthogonal arrays and the Taguchi method. These give scale. Equation (2) shows a ratio between incident and re- Series C, Vol. 47, No. 3, 2004 JSME International Journal 875 Fig. 2 Schematics of transmission loss measurement flective pressure through acoustic elements. Also, a two- microphone method is used at the end of the acoustic ele- ment to remove the influence of reflected waves. + w p = i = 1 TL[dB] 10log10 w 20log10 − (2) t p2 where wi is the energy of inlet, wt is the energy of outlet, (a) Block diagram of the experimental set-up + − p1 is an inlet sound pressure, and p2 is an outlet sound + − pressure. Here, p1 and p2 are derived from Eq. (1). Fig- ure 2 shows schematics of transmission loss measurement. Transmission loss obtained from Eq. (2) is used to inter- pret a intake system. 2. 2 Analysis of experimental results The two-microphone method separates the incident wave and the reflected wave in the pipe.
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