Structural optimization of an automobile transmission case to minimize radiation noise using the model reduction technique

Transcription

1 Journal o Mechanical Science and Technology 25 (5) (2011) 1247~ DOI /s Structural optiization o an autoobile transission case to iniize radiation noise using the odel reduction technique Jung-Sun Choi 1, Hyun-Ah Lee 1, Ji-Yeong Lee 2,*, Gyung-Jin Park 2, Junhong Park 1, Chae-Hong Li 3 and Ki-Jong Park 3 1 Departent o Mechanical Engineering, Hanyang University, 17 Haengdang-dong, Seongdong-gu, Seoul , Korea 2 Departent o Mechanical Engineering, Hanyang University, 1271 Sa 3-dong, Sangnok-gu, Ansan City, Gyeonggi-do , Korea 3 Power Train R&D Center, Hyundai Motors, 772-1, Jangduk-Dong, Hwaseong-Si, Gyeonggi-do , Korea (Manuscript Received June 29, 2010; Revised Noveber 15, 2010; Accepted January 17, 2011) Abstract Vehicles should provide a coortable environent or passengers. The noise ro the transission case is one o the causes o an uncoortable environent. The transission is coposed o gears, shats, bearing and cases. When transission activity occurs, noise is transerred to the passengers through the transission case. Design o the transission case is perored in order to reduce the transission noise. Acoustic analysis is carried out and structural optiization is utilized or the design to reduce the noise. Generally, the boundary eleent ethod (BEM) has been utilized or acoustic analysis. However, it is diicult to utilize the boundary eleent ethod in structural optiization because the cost to calculate the sensitivity inoration is airly expensive. Instead, the inite eleent ethod (FEM) is eployed or calculating the radiation noise o the transission. Radiation noise is considered as the total noise ro the transission. Radiation noise is calculated at the outside o the transission case and it can be expressed indirectly by ultiplication o the velocity in the noral direction o the inite eleents, the radiation eiciency and the characteristic acoustic ipedance. The high requencies are doinant or the transission noise and the radiation eiciency is 1 at the high requency range. Since the characteristic acoustic ipedance has a constant value, the noise is the sae as the nor o the velocity. The velocity o each inite eleent is calculated ro odal analysis and the noise is expressed based on the average velocity o the vibrating structure. However, a long coputation tie is required to calculate the noise in a large scale structure such as a transission. Thus, the entire odel o transission is condensed into the reduced odel by the odel reduction technique. The coponent ode synthesis (CMS) ethod is eployed or the odel reduction technique. The CMS ethod is an eective ethod or dynaic analysis o large and/or coplex structures. The reduced odel keeps the dynaic characteristics o the entire structure and it is used or structural optiization. In structural optiization, the design variables are the thicknesses o the groups o the transission cases, the obective unction is the ass o the structure and a constraint is iposed on the noise. An alternative orulation is ade by exchanging the obective and constraint unctions. The optiization results are discussed in ters o practical application. Keywords: Structural optiization; Transission case; Model reduction; Acoustic power Introduction This paper was recoended or publication in revised or by Associate Editor Yeon June Kang * Corresponding author. Tel.: , Fax.: E-ail address: iyeongl@hanyang.ac.kr KSME & Springer 2011 Autoobiles should provide a coortable environent to drivers. Transission noise is one o any probles that ake an uncoortable environent [1]. Transission noise is a aor concern o autootive anuacturers [2], and contains gear whine noise, gear rattle noise, chain noise and others [3-5]. When a transission operates, transission noise is generated ro the inside coponents and transerred to the passenger through the transission case [6]. Thereore, optiization o the transission case to reduce the noise is needed. The acoustic power represents the noise o the structure along with the transission [7]. The acoustic power is iniized in structural optiization to reduce noise. The boundary eleent ethod (BEM) has been utilized or calculation o the acoustic power. However, it is diicult to eploy the BEM in structural optiization because the cost to calculate the sensitivity inoration is airly expensive. Because, the BEM took 10 hours when the transission case analyzed per 1 requency, while the FEM took hours when the nuber o requency is 2,500. The analysis tie o FEM is uch aster than BEM [8]. Also, i we want to use a coercial syste or optiization, the acoustic power cannot be directly selected as the obective unction due to the liitation o the

2 1248 J.-S. Choi et al. / Journal o Mechanical Science and Technology 25 (5) (2011) 1247~1255 current coercial solvers [9]. Masaki et al. proposed a new ethod or indirectly calculating the acoustic power using the noral velocity o the surace, and copared the acoustic power o the new ethod with the acoustic power o the BEM [8]. Hiroasa et al. calculated the gear noise o a transission by using noral velocities o the vibrated surace [10]. Makoto et al. have identiied parts o the transission case, which are related to noise, by sensitivity analysis and iproved the structure o the transission case [11]. Dai et al. perored topographical structural optiization to reduce the sound power by reducing the velocity o the surace in transaxles [9]. Thus, the inite eleent ethod (FEM) is utilized or optiization o the transission case in this research, because the calculation tie o the acoustic power using the FEM is shorter than the BEM. When the acoustic power is calculated or a large scale structure such as a transission using the FEM, we need a long coputation tie. In order to reduce the tie, any researchers had studied the reduced ethod o the FE odel using supereleents [12, 13]. In general, the construction ethods o the supereleents use the Guyan reduction, the generalized dynaic reduction (GDR) and the coponent ode synthesis (CMS) ethod [14, 15]. In this study, the coponent ode synthesis (CMS) ethod, which was proposed by Hurty in 1960, is used [16-18]. The CMS ethod is an eective ethod or dynaic analysis with very large and/or coplex structures. Coparisons o the natural requency and odes between an initial odel and a reduced odel using the CMS ethod have been carried out and the CMS ethod was ound to be valid [14, 15]. The transission noise is generated ro the inside coponents o the transission and transerred to the outside o the transission through the transission case. A design change o the inside coponents can reduce the acoustic power. However, it is quite diicult because the inside coponents should be very copact and there is not uch roo or a design change. Thereore, transission noise is reduced by the design change o the transission case. In order to iprove the design o the transission case, engineers have ocused on the indirect calculation o the acoustic power using FEM. However, the FE odel o the transission has a large nuber o degrees o reedo (DOFs) and needs a long CPU tie. Thereore, it is airly diicult to apply the conventional design process to the design o a transission case. Moreover, aster design processes are required in the autoobile industries. In this research, a new design process is proposed or size optiization o the transission case. The proposed design process is deined or size optiization using the supereleent and indirect calculation o the noise. The supereleent is useul to reduce the analysis tie or the structure o a large nuber o DOFs. In order to peror size optiization, the thickness o the shell eleent is generally deined or the design variable. The exterior suraces o the solid eleents are deined by the shell eleents and the rest o the transission case is odeled by the solid eleents. The solid eleents are assigned to supereleents. The atrices and boundary condition inoration o the solid eleents are stored in the database o supereleents and only the shell eleents reain. In optiization or dynaic analysis, the atrices o solid eleents are assebled into the atrices o the shell eleent by the boundary conditions. Thereore, the characteristics o the entire structure are represented by the shell eleents. The indirect noise is calculated using the surace velocity o the shell eleents. A coercial sotware syste enables the indirect calculation o the noise. In the indirect calculation ethod, the surace velocity is calculated by steady-state vibration analysis and the acoustic power is indirectly calculated by the surace velocity. Size optiization o the transission case is perored to reduce the noise by using the indirect calculation ethod. Two exaples are solved to validate the design process. First, the design process is veriied by a sipliied exaple. Second, the design process is applied to a real transission odel and size optiization is perored to iniize the acoustic power or ass. MD. NASTRAN [19] is used to reduce the DOFs o the FE odel using the CMS ethod. Steady-state vibration analysis and size optiization are perored by using MD. NASTRAN [19]. A C progra is developed to calculate the acoustic power and the progra is interaced to MD. NASTRAN [20]. 2. Model order reduction by using supereleents The inite eleent analysis o a structure with a large nuber o DOFs needs a long coputation tie and the coputation tie increases even ore when optiization is utilized. The coputation tie can be reduced by using supereleents. The coponent ode synthesis (CMS) ethod is one o the ethods aong the construction ethods o supereleents. This considers the dynaic behavior o the structure. As entioned earlier, the CMS ethod is used. 2.1 Model order reduction technique The supereleents are used to reduce the coputation tie. The construction ethods or supereleents are classiied into two ethods such as static reduction and dynaic reduction. The CMS ethod is one o the dynaic reduction ethods. The CMS ethod is used because the dynaic behavior o a transission ust be considered when calculating the acoustic power. The basic process o the CMS ethod is briely described [12]. A structure consists o ultiple coponents as illustrated in Fig. 1. The behavior o each coponent o the structure is approxiated by a set o basis vectors (Ritz vectors) and then coponent approxiations are assebled to obtain a global approxiation or a large coplex structure. Hence, the CMS ethod reduces the coputational size and cost o the proble.

3 J.-S. Choi et al. / Journal o Mechanical Science and Technology 25 (5) (2011) 1247~ x n n = F, x R, q R, TF R, < n x (4) i {} x = [ T ]{} q Fig. 1. Division o the entire structure into ultiple coponents. Fig. 2. Degrees o reedo o the rth coponent. The governing equation o an undaped structural coponent can be written as ollows: [ ]{ x } + []{ k x} = { } (1) where [] and [k] are n n ass and stiness atrices, respectively, {x} is the displaceent vector o diension n, { } is a tie dependent orce vector o diension n, and n is the nuber o equations. The reduction procedure divides the original syste DOF into two sets as ollows: ii i i x k i ii + x k i ki xi i = k x where i eans the interace degrees o reedo and eans the internal degrees o reedo as illustrated in Fig. 2, i + is equal to the total nuber o DOF n. Each coponent is associated with the ollowing eigenvalue proble: k k ii i (2) ki Φ 2 ii i 0 ω = (3) k i Φ 0 i where ( ω 2, Φ) indicates the eigenvalue-eigenvector pair. Eigenvector Φ indicates the set o retained coponent noral odes ro (3). The ai o the CMS ethod is to reduce the proble size; that is, < n should be kept. Thereore, a suitable set o basis coordinates are needed to replace the large set o physical coordinates. The coordinate transoration between the coponent physical coordinates {x} and reduced coponent basis coordinates {q} is deined by where the diension o x is n and the diension o q is, is less than n, and [ T F ] is the coordinate transoration atrix. The coordinate transoration atrix [ T F ] usually consists o Φ and ψ as ollows: F r [ Φ ] [ T ] = ψ (5) where Φ is a type o pre-selected coponent noral odes. The other coponent Ritz basis vectors are ψ which is classically constituted by a cobination o static response (constraint odes, attachent odes, rigid-body odes, inertia-relie odes and quasi-static odes) [14]. Substituting Eq. (4) into Eq. (1) and preultiplying the transpose o T ], the ollowing equation is obtained: [ F [ ]{ q } + [ k ]{ q} = { }, R, R, k R. (6) It is noted that the diension o Eq. (6) is saller than that o Eq. (1). Each sybol in Eq. (6) is = F i F (7) k kii k T i = TF TF k i k (8) T i = TF (9) T ii i [ ] [ T ] [ T ] [ ] [ ] [ ] { } [ ]. Eqs. (7), (8) and (9) represent the reduced ass, the reduced stiness atrix and the reduced orce vector, respectively. The governing equation, Eq. (6), o each coponent is assebled into the governing equation or the entire structure by the boundary conditions. Thereore, the diension o the assebled governing equation is less than that o the original governing equation o the entire structure. In this way, the CMS ethod reduces the coputation tie and the cost. 2.2 Application o supereleents to optiization The FE odel o a structure is divided into the design region and the non-design region or optiization. The nondesign region is assigned to the supereleents to condense the DOFs and the reaining design region is utilized in optiization. In this anner, analysis o the design region could include the odes o the entire structure. In this research, the entire structure is divided into the coponents o the shell eleents and the coponents o the solid eleents as illustrated in Fig. 3. The region o the solid ele-

4 1250 J.-S. Choi et al. / Journal o Mechanical Science and Technology 25 (5) (2011) 1247~1255 Fig. 3. Model reduction or optiization. ents is deined as the non-design region and the area o the shell eleent is deined as the design region. As entioned earlier, the non-design region is assigned to the supereleents. The non-design region is assebled into the design region and only the shell eleents reain. Thereore, the optiization process only considers the shell eleents. A odule in MD. NASTRAN is utilized or the construction o the supereleents [19]. 3. Calculation o the acoustic power using noral velocity o the surace The acoustic power is indirectly calculated by the average velocity which is calculated by steady-state vibration analysis [8]. 3.1 Steady-state vibration analysis The governing equation or steady-state vibration analysis using FEM is M x + Kx = F(ω)e ω i t (10) where M is the ass atrix, K is the stiness atrix, x is the dynaic displaceent vector. The ass and stiness atrix is deterined by the type o the FE eleent and the boundary condition. The external orce vector is F (ω) and ω is the excitation requency. The excitation requency range is selected according to the characteristic o the proble and Eq. (10) is solved. Eq. (10) also eans haronic analysis. Consequently, the velocities o the all the nodes are obtained at each ode. The velocities are used or the indirect calculation o the acoustic power. 3.2 Indirect calculation o the acoustic power Fig. 4 illustrates the nodal velocities, the noral vector and coordinates o the nodes o an eleent. In Fig. 4, v i (=1,,NEL; NEL=the nuber o eleents) is a velocity vector o the ith node in the th eleent. The coordinate vector o the ith node in the th eleent is x i. The unit noral vector o the th eleent is n. The ollowing equation is or calculation o the average velocity vector: ( v + v + v v ) 1 v, average = , (=1,., NEL). (11) 4 The average velocity vector is obtained ro the velocities o Fig. 4. The node velocity o an eleent. the edge nodes. The unit noral vector is calculated ro the ollowing equations: A = ( x 2 x 4) ( x 3 x 1) (12) n = A. (13) A The noral velocity and noral velocity vector o the th eleent is calculated as ollows: v = v, n (14) average v n = v n. (15) The noral averaged velocity v av, o the speciic requency is calculated as ollows: NEL 1 T v = Α av, ( v n v n), ( = p,, q Hz) (16) A total = 1 where Atotal is the total surace area. In the ollowing equation, V is the acoustic power o the speciic requency: V = A v ρ c, ( = p,, q Hz) (17) total av, 0 where ρ 0 is 1.21 kg/ 2 which is the density o the air and c is 340 /s which is the wave speed in the air. Thereore, ρ 0 c is a constant value called the characteristic acoustic ipedance [8]. The unit o the acoustic power is watt (W). The radiation eiciency is assued to be 1.0 at a high requency range because the critical requency o an autoobile transission case is in the range between 1000 and 3000 Hz. Thus, the radiation eiciency is not ultiplied in the equation. The calculation tie o Eq. (17) is uch shorter than the BEM and the calculation results o the acoustic power are siilar [8]. Thereore, an indirect calculation ethod is suitable or size optiization. Size optiization is perored using this acoustic power.

5 J.-S. Choi et al. / Journal o Mechanical Science and Technology 25 (5) (2011) 1247~ Fig. 6. The plate exaple or veriication o the design process. The FE odel reduction o Step 2 is only perored at the irst iteration o the optiization process and not perored ro the second iteration. The non-design region disappears and only the design region reains in Step 2. The process o size optiization is applied to a plate exaple to veriy the proposed process. 4.2 Exaple or veriication o the design process Fig. 5. The low o size optiization. 4. Process o design or reducing acoustic power with the reduced odel Size optiization is perored to reduce the noise in the FE odel with a large nuber o DOFs. The supereleents using the CMS ethod and the indirect calculation ethod o the acoustic power are utilized and the entire design process is deined. In order to veriy the process, a plate exaple is solved. 4.1 Process o size optiization Fig. 5 shows the process o size optiization and the steps are as ollows: Step 1. Deine the FE odel, boundary conditions and optiization orulation. Step 2. Reduce the FE odel size by the supereleent using the CMS ethod. The design doain o the FE odel uses the original inite eleents and the non-design doain uses supereleents. Step 3. Peror steady-state vibration analysis. Hence, the velocities o each node are obtained. Step 4. Calculate the acoustic power indirectly using the velocities o each node. Step 5. Solve the size optiization proble to reduce the indirect acoustic power. Step 6. I the convergence criteria are satisied then terinate the process. Otherwise, go to Step 3. When the design variables do not change and all the constraints are satisied, the convergence criteria are satisied. Fig. 6 shows an exaple or veriication o the proposed design process. The odel is a plate, which has a length o 0.7, width o 0.35 and unior thickness o The odel is coposed o 7,350 solid eleents which are consisted o three layers o solid eleents. The thickness o one layer is In order to peror size optiization, the upper surace o the solid eleents is odeled by the shell eleents. As entioned earlier, the atrix and boundary condition inoration o solid eleents are cobined into the shell eleents by the CMS ethod. Thereore, the characteristics o the entire structure are represented by the atrix o the shell eleents. The thickness o the shell eleent is. and 2,450 shell eleents. The boundary condition is iposed on the edge nodes o the botto where all the degrees o reedo in the six directions are ixed. The Young s odulus o the plate odel is 7.2E+10 Pa, the Poisson s ratio is 0.34 and the aterial density is 2770 kg/ 3. In size optiization, the design variables are deined by the thicknesses ro the edge to the center o the shell eleents. The dv in Fig. 6 eans the design variable. A unit orce in the z-direction is iposed between the 4th and 5th design variables. The requency range o the orce is ro 1000 Hz to 3000 Hz and the requency interval is 1 Hz. An optiization proble is orulated as ollows: Find to iniize subect to t ; i = 1,..., 7 i V = where V 0.0 t 3000Hz 2 ( V ) = 1000Hz = A ass ass 2 i total v av, ρ 0c (18) = 1000,, 3000 Hz o allthickness where ( i = 1,..., 7) is the ith design variable or the thickness t i

6 1252 J.-S. Choi et al. / Journal o Mechanical Science and Technology 25 (5) (2011) 1247~1255 Table 1. Coparison o the results o optiization between the initial odel. Design Variables [] dv1 dv2 dv3 dv4 dv5 dv6 dv7 Constraint violation Obective unction [W] o the ith layer, V is the acoustic power at a speciic requency o the orce, V is the obective unction, and 0.0 and are the lower and upper bounds o the design variables, respectively. The initial value o the design variables is. A constraint condition is deined so that the ass o the structure should be saller than 50% o the ass where all the design variables are That is, the ass o the structure in the optiization results have the sae ass which have the initial value o design variables. The obective unction is the total sound power, which is calculated by suation o the acoustic power ro 1000 Hz to 3000 Hz. When the acoustic power is calculated, the radiation eiciency is regarded as 1 because only the high requency range is considered. Two kinds o size optiization are perored using the orulation in Eq. (18). One is size optiization o the initial odel that has 7,350 solid eleents and 2,450 shell eleents. The other is size optiization o the reduced odel that has 2,450 shell eleents. 4.3 Results and discussion Iteration nuber Initial value Initial odel Reduced odel % -0.26% % The optiization results o the two odels are copared in Table 1. In the initial odel, the obective unction decreases ro W to W, and the obective unction decreases ro W to W in the reduced odel. The changing tendencies o the initial odel are siilar to those o the reduced odel in the design variables. Thereore, the reduced odel using supereleents can be used or size optiization. 5. Size optiization o the transission case The process o Fig. 5 is applied to the transission. The FE odel o the transission only has solid eleents. Shell eleents are deined at the surace o the transission case. The solid eleents o the FE odel are assigned to the supereleents. The FE odel o the transission is reduced by the supereleent using the CMS ethod. The FE odel o the Table 2. The CPU tie or analysis o the initial and reduced odels. Steady-state vibration analysis o the initial odel transission case is changed to the reduced odel which only has shell eleents. Steady-state vibration analysis is perored with the reduced odel. The acoustic power is indirectly calculated by the velocity o nodes. Size optiization o a transission case to reduce the acoustic power is perored. 5.1 The FE odel o the transission Supereleent Reduction o the initial odel by using supereleents Steady-state vibration analysis using reduced odel Tie 721 inutes 483 inutes 2 inutes (a) Fig. 7. The FE odel o the transission (a) The FE odel o the transission including the inside coponent; (b) The reduced FE odel o the transission. The odel is a anual transission o a Front Engine Front Drive vehicle. Fig. 7 illustrates the FE odel o the transission. The FE odel in Fig. 7(a) is coposed o solid and rigid eleents. Rigid eleents connect the inside coponents o the transission. The FE odel in Fig. 7(b) is ade or size optiization and is ade o shell eleents. In general, the noise easureent experient o the transission is perored by ixing the botto. In Fig. 7, the botto is part A in the elliptical region. The boundary condition is iposed on the botto o the transission, which is attached to the engine as illustrated in Fig. 7. All the degrees o reedo in the six directions are ixed. A ew orces are applied on the bearings aong the inside coponents o the transission. In order to indirectly calculate the acoustic power, the velocities o the surace are obtained by steady-state vibration analysis. Table 2 shows the CPU tie or analysis between the initial and reduced odel. Steady-state vibration analysis with the initial FE odel in Fig. 7(a) takes 721 inutes. In order to reduce the analysis tie, the size o the FE odel is reduced by the supereleent using the CMS ethod. It takes 483 inutes to construct the reduced odel. Construction o the reduced odel is Step 2 o Section 4.1. Steady-state vibration analysis takes 2 inutes or the reduced odel in Fig. 7(b). The utilized coputer is a cluster o Windows server 2008 based coputer with the HP Blade Syste c-class BL460c. The reduced odel is used in optiization. (b)

7 1253 J.-S. Choi et al. / Journal o Mechanical Science and Technology 25 (5) (2011) 1247~1255 Fig. 8. The natural requency between the initial odel and the reduced odel. Fig. 9. The node velocity between the initial and the reduced odel. 5.2 Coparison o the results o the odal analysis o the transission As entioned earlier, the velocities o the surace are obtained by vibration analysis and the acoustic power is indirectly calculated by using the velocities o the surace. The optiization process considers the dynaic behavior o the structure. Modal analysis is perored or the initial and the reduced odels in order to copare the dynaic behavior. Fig. 8 shows the natural requencies o the initial odel and the reduced odel. They are dierent by 7.52% at 3,500 Hz. However, optiization o the transission case does not usually consider the requencies above 3,500 Hz. The velocity o the surace is iportant or calculation o the acoustic power. Thereore, the velocity o an arbitrary node is copared between the initial odel and the reduced odel. Fig. 9 illustrates the nodal velocities o the initial odel and the reduced odel, and they are siilar. 5.3 Design orulation Size optiization to reduce the acoustic power is orulated as ollows: Find ti ; i = 1,..., 20 to iniize subect to volue V = A total vav, ρ 0 c x, = 400,..., 3000 Hz ti (19) Fig. 10. The graph o the constraint condition. where ti (i = 1,..., 20) is the ith design variable which is the thickness o the ith group o the surace, the volue is the obective unction, V is the acoustic power at a speciic requency, and and are the lower and upper bounds o the design variables, respectively. In Eq. (19), x eans the value o the sound power as shown the Fig. 10. The value depending on a requency is the constraint condition o size optiization and eans the excitation requency. When the acoustic power is calculated, the radiation eiciency is regarded as 1, and ρ 0c is the characteristic acoustic ipedance in the air, which is constant [8]. Then, the unit o the acoustic power is watt (W). Thereore, the unit is changed ro watt (W) to decibel (db) or constraints in optiization. The unit o the acoustic power can be changed as ollows: V [db] = 10log V [W] Reerence acoustic power. (20) In Eq. (20), the reerence acoustic power is W. Fig. 10 shows the graph o the constraint condition. The graph shows the desired acoustic power ro the low requency range to the high requency range. As illustrated in Fig. 10, the graph is linear. The vertical axis o the graph is the noralized acoustic power. 5.4 Results and discussion The history o the obective unction is illustrated in Fig. 11. The nuber o iterations is 6 and size optiization o the transission case takes 29 in. 32 sec. by using the reduced odel o Fig. 7(b). As a result, the noralized volue o the transission case decreases ro to The changes o the constraint violation are illustrated in Fig. 12. The initial constraint violation is % and the inal constraint violation is %. It is noted that the constraints are satisied when the calculation o the acoustic power is perored with the optiization solution. Fig. 13 shows the results o the design variables. Parts which are close to the dierential gear have the upper bound values. The optiization result veriies that reduction o the transission noise can be obtained by changing the thicknesses o the transission case.

8 1254 J.-S. Choi et al. / Journal o Mechanical Science and Technology 25 (5) (2011) 1247~1255 Fig. 11. The history o the obective unction. Fig. 12. The constraint violation. 6. Conclusions Fig. 13. The results o the design variables One o the iportant goals in the current autootive industry is to iprove the noise level o a vehicle. Transission noise is one o the noises that need to be iproved. In order to produce a low noise ro the transission, siulation techniques and an optiization ethod have been ade and applied. However, conventional ethods need a long coputation tie and are expensive. Moreover, the current coercial solvers are not appropriate or such optiization. Thereore, the conventional ethods using structural optiization are not suitable to reduce noise. In this research, a new design process is proposed to overcoe the diiculties. In order to reduce the coputation tie and high cost, the size o the FE odel is reduced by the supereleent using the CMS ethod. The noise is replaced by the acoustic power because the noise cannot be directly used in the current coercial solvers. In order to calculate the acoustic power, the surace velocity is calculated by steadystate vibration analysis and the acoustic power is indirectly calculated by the surace velocity. The acoustic power is used as a constraint or the obective unction in the optiization process. Size optiization is perored to reduce the acoustic power. First, the new design process is veriied by a sipliied exaple, and then the design process is applied to the entire transission odel. Size optiization o the transission case is perored. As a result, the constraint violation is decreased ro % to %. The noralized volue is decreased by 3.2% ro to Consequently, the ollowing conclusions are ade: (1) The proposed optiization process reduces the acoustic power, volue and coputation tie. It is noted that the obtained design is better than the existing one. (2) A new design process can be applied to the structure which has a large nuber o DOFs. (3) The proposed design process can be easily carried out by using the coercial solvers or optiization. Although the ethod is an approxiation ethod based on the acoustic power and supereleents, the results o optiization indicate the design direction o the transission case. In the uture, it will be necessary to peror various structural optiizations (e.g. shape optiization and topology optiization) considering the acoustic power o the transission using indirect noise calculation. Acknowledgeent This research was supported by the WCU (World Class University) progra through the Korea Science and Engineering Foundation unded by the Ministry o Education, Science and Technology (No. R ). The authors are thankul to Mrs. MiSun Park or her English correction o the anuscript. Noenclature A total : Total surace area c : Wave speed : Excitation requency { } : Tie dependent orce vector. K : Stiness atrix [k] : Stiness atrix M : Mass atrix [] : Mass atrix n : Unit noral vector o the th eleent

9 J.-S. Choi et al. / Journal o Mechanical Science and Technology 25 (5) (2011) 1247~ {q} : Reduced coponent basis coordinate t i : Thickness o the ith group o the surace [ T F ] : Coordinate transoration atrix v av, : Noral averaged velocity V : Acoustic power v i : Velocity vector o the ith node in the th eleent ω : Eigenvalue x : Displaceent vector x i : Coordinate vector o the ith node in the th eleent {x} : Displaceent vector Φ : Eigenvector ψ : Ritz basis vector ρ : Density o the air 0 Reerences [1] P. Belloo, N. D. Vito, C. H. Lang and L. Scaardi, In depth study o vehicle powertrains to identiy causes o loose coponents rattle in transissions, SAE TECHNICAL PAPER SERIES, No (2002). [2] A. Forcelli, C. Grasso and T. Pappalardo, The transission gear rattle noise: paraetric sensitivity study, SAE TECHNICAL PAPER SERIES, No (2004). [3] E. I. Rivin, Analysis and reduction o rattling in power transission systes, SAE TECHNICAL PAPER SERIES, No (2000). [4] K. Steinel and G. Tebbe, New torsional daper concept to reduce idle rattle in truck transissions, SAE TECHNICAL PAPER SERIES, No (2004). [5] M. R. Beacha, D. J. Bell and N. N. Powell, Developent o transission whine prediction tools, SAE TECHNICAL PAPER SERIES, No (1999). [6] S. C. Kostic and M. Ognanovic, The noise structure o gear transission units and the role o gear walls, Faculty o Mechanical Engineering Transactions, 35 (2) (2007) [7] J. W. Lee, Y. J. Ki and B. H. Ahn, A study on sound power easureent o copressor based on the sound intensity, The Korean Society or Power Syste Engineering (1999) (in Korean). [8] K. Masaki and H. Tooaki, Useul ethod to calculate the radiation noise o transission, The Society o Autootive Engineers o Japan Annual Congress, 66 (3) (2003) (in Japanese). [9] Y. Dai and D. M. Ranath, A topographically structural optiization ethodology or iproving noise radiation in transaxles, SAE TECHNICAL PAPER SERIES, No (2007). [10] K. Hiroasa, M. Kisao, K. Atsushi, H. Yoshinori and K. Naohito, Radiation noise o transission gear analysis technique, The Society o Autootive Engineers o Japan Annual Congress, 112 (4) (2004) 5-8 (in Japanese). [11] H. Makoto and K. Nobuyuki, Iproveent o the transission radiation noise using FEM, The Society o Autootive Engineers o Japan Annual Congress, 66 (3) (2003) (in Japanese). [12] Y. Cunedioglu, A. Mugan and H. Akcay, Frequency doain analysis o odel order reduction techniques, Finite Eleent in Analysis and Design, 42 (5) (2006) [13] Z. Q. Qu and R. P. Selva, Dynaic supereleent odeling ethod or copound dynaic systes, AIAA Journal, 38 (6) (2000) [14] G. Masson, B. A. Brik, S. Cogan and N. Bouhaddi, Coponent Mode Synthesis (CMS) based on an enriched ritz approach or eicient structural optiization, Journal o Sound and Vibration, 296 (4-5) (2006) [15] E. S. Cho, S. Baik, H. J. Yi and H. S. Ki, A study on odal analysis o a large structural syste with contribution actor analysis o substructures, The Korean Society o Mechanical Engineers, 1 (2) (2001) (in Korean). [16] W. C. Hurty, Vibration o structural syste by coponent ode synthesis, Journal o the Engineering Mechanics Division ASCE, 85 (1960) [17] W. C. Hurty, Dynaic analysis o structural syste using coponent odes, AIAA Journal, 3 (4) (1965) [18] W. C. Hurty, J. D. Collins and G. C. Hart, Dynaic analysis o large structures by odal synthesis techniques, Coputers and Structures, 1 (4) (1971) [19] MD R3 Nastran, User s Guide, MSC. Sotware corporation (2008). [20] B. Stroustrup, The C++ prograing language, Addison Wesley, U.S.A. (2000). Jung-Sun Choi received the B.S. degree in echanical engineering ro Hanyang University, Korea, in He is currently pursuing the Ph.D. degree at Hanyang University. His research interests include design optiization, design ethodology and luid structure interaction analysis or enhancing the perorance o the lapping wing MAV. Ji-Yeong Lee received the B.S. and M.S. degrees in echanical design and production engineering ro Seoul National University, Korea, in 1991 and 1993, respectively, and the Ph.D degree in echanical engineering ro Carnegie Mellon University, Pittsburgh, PA in He is currently an assistant proessor at Hanyang University, ERICA in Korea. His current research interests include sensor-based otion planning, path planning or redundant systes and ulti-robot systes, anipulation planning, and navigation or nonholonoic systes.