SEA Subsystem Attribute Correction Based on Stiffness Multipliers

Transcription

1 Internationa Conference on Eectrica, Mechanica and Industria Engineering (ICEMIE 016) SEA Subsystem Attribute Correction Based on Stiffness Mutipiers Jie Gao1,* and Yong Yang 1 Coege of hysics and Information Technoogy, Shaanxi Norma University, Shaanxi, Shanxi,Xian,710119,.R. China T&S Technoogies Co. Ltd, Beijing, ,.R. China * Corresponding author whereas the energy transfer is determined by oca attributes near the structura connection (such as oca impedance at the structura connection position, transfer oss, and radiation efficiency to the she structure and acoustic couping). Therefore, with SEA structures having the same energy storage capacity and transmission properties, accurate SEA subsystems can be estabished to represent compex structures. This approach estabishes a simpified SEA mode for compex structures and guarantees accuracy and computationa efficiency (main physica parameters that affect structura response). Abstract This study introduces the principes and appication procedures of the stiffness mutipier method used to correct the statistica energy anaysis (SEA) subsystem attributes. Using the cockpit foor of a certain type of fighter aircraft as an exampe, this study combines the finite eement method and stiffness mutipier correction technique to modify the moda density of the SEA mode through theoretica derivation and numerica simuation anaysis. Resuts of the numerica simuation show that the SEA subsystem modified with the stiffness mutipier method can perfecty refect the dynamic characteristics of compex stiffened pates. Moreover, the resuts of the cacuation are identica to those obtained with the finite eement method. The stiffness mutipier method is found to be appicabe to the construction of the compex structures of the SEA subsystem mode. Over the past decades, severa studies have focused on estimating the parameters of SEA subsystems. V. Cotoni and R. Langey estimated the SEA parameters of a periodic structure and presented an anaysis theory for periodic structure vibration [6]. Taner Onsay and Anab Akanda et a. deveoped an estimation method for SEA parameters based on equivaent stiffness [7] and adopted fine finite eements to estimate moda density so as to achieve equivaence in stiffness with a stiffness mutipier. B. R. Mace and. J. Shorter adopted the method of energy finite eements to study the couping reationship between subsystems[8]. On the basis of the work of Taner Onsay and Anab Akanda et a., the present study anayzes the correction of SEA attributes with the stiffness mutipier method and conducts numerica verification using practica structures. Keywords-SEA system correction; stiffness mutipier; moda number; radiation efficiency. I. INTRODUCTION As an effective prediction technique for medium-high frequency responses, statistica energy anaysis (SEA) is widey used in the aerospace and auto industry[1-3]. SEA divides compex systems into different moda groups. In cases of statistica significance, SEA disassembes arge systems into severa independent subsystems that are easy to anayze instead of precisey determining each moda response. In the initia engineering design stage of SEA appications, the subsystems constituted by moda groups must be defined. The SEA mode estabished in this way can ceary express the characteristics of input, storage, oss, and transmission of vibration energy. The attribute parameters of subsystems can aso be cacuated accuratey to achieve precise dynamic prediction resuts. However, substructures in practica appications often have compex geometrica and physica properties, and no reated attribute parameter is avaiabe for reference[4]. Moreover, experiments for testing the attribute parameters of substructures are often time consuming and are thus not widey used in engineering practice. This imitation emphasizes the need to deveop a fast and effective method that coud predict the attribute parameters of subsystems with compex structures. II. SEA CORRECTION BASED ON STIFFNESS MULTILIER The main theory of the SEA method is based on the power fow baance equation, with which the energy fow reationship among subsystems can be soved[9]. The power baance reationship among N subsystems is η1n η1 + η1i N1 i 1 η1 N1 η + ηi N ω i η1k N1 In predicting the attribute parameters of subsystems with compex structures, dynamic response (frequency and space average) is usuay controed by a series of tota attributes even when structures are extremey compex from the physica aspect[5]. The energy storage capacity of structures is controed by the frequency band within the ocaized mode, 016. The authors - ubished by Atantis ress E1 N 1 in,1 = Ek in,k ηk + ηki Nk Nk i k ηk1nk (1) where ω is the mid-band frequency, η k is the damping oss factor (DLF) of the subsystem, η ki is the couping oss factor (CLF) among subsystems, n is the moda density, and 11

2 is the input power of the subsystem oad. For increasingy compex structures, their DLF vaues can be tested with the method of steady energy fow[10]. The key to this method is to measure the input power in of the excitation source in the system and the system energy E. The input power of the subsystem is = R ( τ = 0) = S ( ω) dω in = G ( ω) dω = f() t v() t The energy of the subsystem can be expressed as () E = M v () t (3) where S ( ω ) is the cross-spectrum density of the biatera force f () t and speed vt () at the excitation point and ( ω ) is the uniatera cross-spectrum density. According to G the definition of the DLF, we can obtain the foowing from Formuas () and (3): η = in (4) ωe As for a three-dimensiona space with voume V, superficia area A, and tota side ength, the formua of its moda density is 4π f V π fa n( f) = + + (5) 3 c c 8c For the uniform rectange pate, the moda density formua of the fexura vibration is A n( f) = (6) Rc both the radiated sound power and the mean square veocity can be cacuated with the finite eement mode and experimenta measurement. As for pates with specia structures, such as stiffened pates, their moda densities cannot be cacuated with the formua for uniform fat structures. Such cacuation woud resut in arge errors and necessitate extensive corrections. If a compex pate structure is substituted with a simpe two-dimensiona pate structure, then the cacuation is simpified. The equivaent bending stiffness of a stiffened pate is defined as B eq, and that of an ordinary pate is B. The ratio of the equivaent bending stiffness of the stiffened pate to that of an ordinary two-dimensiona fat pate is defined as the stiffness mutipier α, which is written as foows: B eq n( f) α = = B neq ( f) B = Eh 3 1(1 υ ) where n( ω ) is the moda density of the structure, B is the bending stiffness of the pate, υ is the oisson s ratio of materia, and h is the thickness of the pate. The easticity moduus of a stiffened pate structure for SEA can be determined by mutipying the easticity moduus of two-dimensiona fat materias with a stiffness mutipier. The equivaent stiffness can then be corrected. III. EXAMLE OF NUMERICAL SIMULATION ANALYSIS A. Mode of Cockpit foor of Fighter Aircraft We estabish a cockpit foor of a fighter aircraft as Figure I shows. The spatia distribution of the stiffeners on the base pate is extremey compicated, their thickness is not uniform, and a corresponding subsystem based on standard SEA subsystems is acking. In this exampe, we use the stiffness mutipier method to determine the equivaent SEA subsystem for the base pate. (8) (9) where of the section, A is the area of the pate, R is the turning radius c is the ongitudina wave veocity in the c = E ρ, where E denotes the easticity materia, and / moduus and ρ denotes the mass density. The radiation efficiency of the two-dimensiona pate is σ rad = (7) ρ rad oca o p v n In Formua (7), rad is the radiated sound power, and v n is the mean square veocity on the pate surface. Here, FIGURE I. COCKIT FLOOR OF FIGHTER AIRCRAFT 1

3 We combine the finite eement method with the stiffness mutipier to modify the moda density. First, a detaied finite eement mode of the stiffened pate substructure must be estabished. Then, the cacuation is conducted, incuding the cacuation of the moda number within the frequency band, inherent frequency, and mode shape of the finite eement subsystem. Subsequenty, the equivaent SEA subsystem is estabished. At this point, a standard SEA subsystem is defined, and the moda number within the frequency band of the defined standard SEA subsystem is obtained with the VA One software. Then, the ratio of the moda number within the frequency band of the standard SEA subsystem and that of the finite eement subsystem are defined as the stiffness mutipiers. The materia attributes of the SEA subsystem are modified, and the equivaent SEA subsystem is finay estabished. For acoustic probems, the radiation efficiency and transfer oss of the SEA subsystem can be used to (a) cacuate the radiation efficiency of the SEA subsystem after correcting materia attributes and (b) determine the radiation efficiency from the cacuation of the finite eement subsystem. Consequenty, the CLF of the equivaent SEA subsystem can be modified. The finite eement subsystem mode of the foor can be created with the VA One software, as shown in Figure II. The finite eement subsystem mode comprises 591 nodes and 584 units (incuding she and beam eements). The sound fieds on both sides of the subsystem are created with the SIF function of the VA One SEA software. The attributes of the structure are simpified to uniform materia, thickness, and stiffened section characteristics (which are simiar to those of the base pate materias on the foor). The thickness of the pate is m, the materia density is 757 kg/m3, the easticity moduus 70Ga, and the oisson s ratio is 0.3. The moment of inertia of an area of the beam sections are Ixx 8.004e-8 m4, Iyy 4.169e-008 m4, Jzz 1.73e-007 m4, and Qzz 3.76e-010 m4, the area is m, and the perimeter is 0.56 m. FIGURE II. MODEL OF THE VAONE SUBSYSTEM. The materia and thickness of the non-stiffened SEA pate subsystem (uniformed SEA) created with the VA One software are identica to those of the finite eement subsystem. The sound fieds on both sides of the pate subsystem are created with the SIF function as we. The anaysis frequency domain is set at 1/3 OCT and 15 1,50 Hz. The VA One buit-in finite eement sover is adopted to cacuate the moda number of the frequency bands of the finite eement, which shoud then be compared with that of the uniformed SEA subsystem, as shown in Figure Ш (A). Subsequenty, the stiffness mutipiers reated to the frequencies can be obtained with Formua (1), as shown in Figure Ш. Finay, the mean of the stiffness mutipiers reated to the frequencies is used to modify the stiffness mutipier of the SEA subsystem. (A) FIGURE III. (A) MODAL NUMBER OF FREQUENCY BANDS AND COMARISON BETWEEN FINITE ELEMENTS AND SEA; MEANS OF STIFFNESS MULTILIERS AND FREQUENCY CHANGES AND DOMAINS The average stiffness mutipier is appied to modify the uniformed SEA subsystem and finay obtain the updated SEA. The subsystem mode is then created with VA One (Figure II). B. Comparative anaysis of resuts The equations are an exception to the prescribed specifications of this tempate. You wi need to determine whether or not your equation shoud be typed using either the Times New Roman or the Symbo font (pease no other font). To create mutieveed equations, it may be necessary to treat the equation as a graphic and insert it into the text after your paper is styed. Figure IV (A) shows 0 decouping concentrated force excitations imposed on the finite eement system and 4 corner points imposed with simpe support restraints. Eighteen sensors are imposed uniformy on the finite eement subsystem to obtain the dynamic response at the response position. The dynamic response, radiation efficiency, and radiation sound pressure eve of the structure can be cacuated with the hybrid finite eement/sea in the VA One software (Figure V). Equivaent excitations are imposed on two SEA subsystems (uniformed and updated SEA). The spatia average dynamic response, radiation efficiency, and radiation sound pressure eve of the structure can then be obtained (Figure V). 13

4 (A) (D) FIGURE IV. (A) MODAL LING CONCENTRATED FORCE; VIRTUAL SENSOR AT A RANDOM LOCATION (A) (E) FIGURE V. (A) SENSOR RESONSE OF THE FINITE ELEMENT SUBSYSTEM; COMARISON OF THE MODAL NUMBERS OF THE FREQUENCY BANDS OF THE FINITE ELEMENT AND UDATED SEA; (C) COMARISON OF THE RADIATION EFFICIENCIES OF THE FINITE ELEMENT AND UDATED SEA; (D) COMARISON OF THE RADIATION SOUND RESSURE LEVELS OF THE FINITE ELEMENT AND UDATED SEA; AND (E) COMARISON OF THE SATIAL AVERAGE DYNAMIC RESONSES OF THE FINITE ELEMENT AND UDATED SEA (THE SATIAL AVERAGE DYNAMIC RESONSE OF THE FINITE ELEMENT CAN BE OBTAINED WITH THE DYNAMIC RESONSES OF EACH SENSOR) Figure V indicates that the moda number of the updated SEA pate subsystem fits perfecty with that of the finite eement subsystem. This resut is a good refection of the dynamic characteristics of the stiffened pate structure. The radiation efficiencies of these two subsystems are consistent. The ow radiation efficiency at 1,000 Hz is caused by the compression of the pate structure. The difference in the radiation efficiencies at 150 Hz refects the compex radiation characteristics at a coincidence frequency. Under this coincidence frequency, the SEA is mainy corrected with the stiffeners, irreguar regions, or boundary radiations, which cannot fuy refect the characteristics of stiffeners. (C) FIGURE VI. RADIATION FREQUENCIES (EQUAL BANDWIDTH, 5 HZ) 14

5 Figure VI shows the comparison of the equa bandwidth radiation efficiencies (bandwidth at 5 Hz) and indicates changes in the ow frequency radiation efficiencies. The differences in the radiation efficiencies are refected in the radiated sound pressures. The prediction of the radiation sound pressure eve of the updated SEA pate subsystem is consistent with that of the finite eement subsystem. The differences in the radiation frequencies at 150 Hz reach neary 6 db. The dynamic responses of the structure in Figure Ⅴ (e) show a good fit and an obvious correation. The sma difference can be eiminated through the we-distributed arrangement of sensors. [7] T. Onsay, A. Akanda, G. Goetchius, Vibro-Acoustic behavior of bead-stiffened fat panes: FEA, SEA and experimenta anaysis, roc. SAE Noise and Vibration conference, [8] B. R. Mace,. J. Shorter, Energy Fow Modes From Finite Eement Anaysis. J. Sound Vib, vo. 33, pp , 000. [9] R.H. Lyon and R.G.DeJong, Theory and Appication of Statistica Energy Anaysis, Butterworth-Heinemann, Newton, 1995, pp [10] B. Mace, Statistica energy anaysis: couping oss factors, indirect couping and system modes, J. Sound Vib, vo. 79, pp , 005. IV. CONCLUSIONS This study introduces the principe and appication of correcting the attributes of the SEA subsystem with the stiffness mutipier method. Taking the cockpit foor of a certain type of fighter aircraft as an exampe, the SEA subsystem of the foor (updated SEA subsystem) is created with the stiffness mutipier method. Numerica verification is conducted on the stiffness mutipier with the hybrid finite eement/sea method of the VA One software. Through the comparative anaysis of the cacuation resuts, the foowing concusions can be drawn. First, the SEA subsystem corrected with the stiffness mutipier method can effectivey refect the dynamic characteristics of the compex stiffened pate. Second, the radiation noise and vibration response of the structure anayzed with the updated SEA subsystem fit we with those anayzed with the finite eement method. Third, the stiffness mutipier method can be used to create a SEA subsystem, which is simiar to a compex stiffened structure. In this study, the transmission oss of the updated SEA subsystem is not expored but wi be studied in future research. ACKNOWLEDGMENT This work was financiay supported by the Nationa Natura Science Foundation of China (Grant.No ) and the Fundamenta Research Funds for the Centra Universities of Ministry of Education of China (Grant No. GK014001). REFERENCES [1] X. Chen, D. Wang, Z. Ma, Simuation on a car interior aerodynamic noise contro based on statistica energy anaysis, Chin. J. Mech. Eng. vo.5, pp , 01. [] Cristina Díaz-Cereceda, Jordi obet-uig, Antonio Rodríguez-Ferran Automatic subsystem identification in statistica energy anaysis, Mechanica Systems and Signa rocessing. vo , pp , 015. [3] H. Yan, A. arrett, W. Nack. Statistica energy anaysis oss by finite eements for midde frequency vibration. Finite Eements in Anaysis and Design. vo. 35, pp , 000. [4] M. Kassem, C. Soize, L. Gagiardini, Structura partitioning of compex structures in the medium-frequency range An appication to an automotive vehice, J. Sound Vib, vo. 330, pp , 011. [5] R. Lyon, Statistica Energy Anaysis of Dynamica Systems, MIT ress, Cambridge, Massachusetts, [6] V. Cotoni, R.S. Langey,.J. Shorter, A statistica energy anaysis subsystem formuation using finite eement and periodic structure theory, J. Sound Vib, vo. 318, pp ,