حوراء وعدنان Al-Khwarizmi Engineering Journal Al-Khwarizmi Engineering Journal, Vol. 7, No. 4, PP41-53 (2011) The Estimation of Vibrational Energy of Two Coupled (Welded) Plates Using Statistical Energy Analysis Adnan D. Mohammed Hawraa S. Ibraham Department of control and system engineering/University of Technology (Received 24 March 2011; Accepted 24 August 2011) Abstract This paper deals with a method called Statistical Energy Analysis that can be applied to the mechanical and acoustical systems like buildings, bridges and aircrafts …etc. S.E.A as a tool can be applied to the resonant systems in the circumstances of high frequency or/and complex structure». The parameters of S.E.A such as coupling loss factor, internal loss factor, modal density and input power are clarified in this work ; coupled plate sub-systems and explanations are presented for these parameters. The developed system is assumed to be resonant, conservative, linear and there is an equipartition of energy between all the resonant modes within a given frequency band in a given sub- system. The aim of this work is to find the energy stored in the sub-systems for two coupled (welded) plates in rectangular angle systems and study the effect of changing sub-systems dimensions, the results shows that as surface area of directly driven plates A1 increases energy level of plate 1 increases while a reduction in the energy level of indirectly driven plate (plate 2) is noticed. This is because of the strength of coupling decreases towards the weak coupling condition and this leads to a reduction in the power transferred from plate 1 to plate 2 and consequently a lower energy level for plate 2. In addition the effect of changing the internal loss factor for a range of (0.00001-0.1) causes a reduction of the values of energy level in these sub-systems. because the increasing of internal loss factor values led to the increasing of the material resistance and that will dissipate the energy flow across those sub-systems. A comparison is made between S.E.A models built by FORTRAN program and Finite Element model solved by ANSYS package. Keywords: - Type I diabetes, Backstepping, Bergman’s model, oral glucose tolerance test. 1. Introduction Of the techniques used to predict vibration levels in mechanical structures, are those based on the assumption of idealized mathematical in which the properties of the structures and the excitation are known exactly, and proceed effectively by numerical solutions of the equations of motion. For many structures, however the very detailed physical properties required such a model may not be available, or relevant, for the particular problem at hand. The vibrational behavior of complex structures involves modes high up in the modal series at frequencies of practical interest, and is very sensitive to structural detail. Variations of detail between a structure and its mathematical model, or between different realizations of nominally the same structures, may then account for significant quantitative differences in response. The computational effort involved in direct numerical solution may also be prohibitively large for these structures. Under these circumstances, a more appropriate technique is often one which provides an understanding of broad features of the vibration levels and transmission, given only a relative coarse and uncertain description of the structure and its excitation. Such technique is Statistical Energy Analysis (S.E.A) Ref.[1] and Ref.[4]. The method Statistical Energy Analysis (S.E.A) involves the division of a complex system into a number of inter-connected sub-systems with similar characteristics, for example plates, beams, acoustic spaces etc. It is based on power flow between sub-systems, the power dissipative by damping and the excitation power input within the system of interest Ref.[9]. Adnan D. Mohammed Al-Khwarizmi Engineering Journal, Vol. 7, No. 4, PP 41 - 53 (2011) 42 In this paper, the energy stored in sub-systems is found by using Statistical Energy Analysis, and a comparison is made between analytical solution by using Statistical energy Analysis (S.E.A) and numerical analysis by using Finite Element Method (F.E.M) built by (ANSYS 5.4) . For the case of the two coupled (welded) plates joined along a straight edge, one of the plates is excited and that excitation is harmonic at radian frequency ω and of constant amplitude Fo. The Statistical Energy Analysis gives us the energy in average manner over all the points of the plate and does not give any information about concentration of energy in particular parts of the sub-systems. The Finite Element Method gives the response in any particular point. Therefore, The ANSYS model is built with an application of point harmonic force at different locations on the driven sub-system. Results of the response (energy levels), for a given force location, are predicted at several locations on coupled sub-systems. An average of response at different locations that ensemble averages over the different force locations is obtained. 2. S.E.A Theory In an energy distribution model the sub- system energies and input powers are related by the following matrix equation, { } [ ]{ }EAP in ω= …(1) Equation (1) gives the relationship between the power injected matrix { }inP , and the total energy matrix at center frequency of the band { }E , noticing that, [ ]A is a matrix of energy influence coefficients in the relevant frequency band and the symbol (─) indicates time average quantity. In S.E.A a power balance equation is written for each sub-system, so that, for sub-system i, ijtransidissiin PPP ,,, += … (2) The power balance equation means that the input power into sub-system i equals to the summation of power dissipated due to damping in sub-system i, idissP , and the power transmitted from sub-systems i to sub-system j, ijtransP , , due to coupling loss factor between sub-systems. The equation for the dissipation of power in a subsystem can be written as Ref. [5] ׃­ iiidiss EP ωη=, … (3) Where, iη is the damping loss factor of sub- system i, iE represents the total stored energy in sub-system i and ω is the center frequency of the band of interest. The equation for the net power transmitted from sub-systems i to j is׃­ jjiiijijtrans EEP ωηωη −=, … (4) Where, ijη , jiη is the coupling loss factor from sub-systems i to j and from sub-systems j to i respectively and it depends on the type of connection between sub-systems, the material properties, the dimensions of the system and the center frequency of the band, ω (rad/sec) Ref. [5]. The power flow relationship of a structure consisting of two sub-systems is shown in fig.(1). Fig. 1.Basic S.E.A model Ref.[ 3]. Where 1E and 2E are the total stored energy of sub-systems (1) and (2) respectively, inP is the power injected into the first sub-system, dissP ,1 and dissP ,2 are the power dissipated in sub- systems due to internal damping and 12P and 21P are the power transmitted between sub-systems. The equation for the power flows between the sub-systems 1 and 2 under typical conditions is expressed as follows Ref. [3] ׃ ­ ( ) ( )221112111 EEEPin ηηωωη −+= ( ) ( )112221222 EEEPin ηηωωη −+= …(5) So the coupling loss factors must be related by the consistency relation Ref. (5), jijiji nn ηη = …(6) 1E 2E diss,1P diss,2P 12P 21P ( )2inP ( )1inP Adnan D. Mohammed Al-Khwarizmi Engineering Journal, Vol. 7, No. 4, PP 41 - 53 (2011) 43 Where in and jn are the modal densities of sub- systems i and j respectively. So, the equation of input power injected into sub-systems can be written as Ref. [10]:- ( )         −+= 2 2 1 1 112111 n E n E nEinP ωηωη ( )       −+= 1 1 2 2 221222 n E n E nEP in ωηωη …(7) A frequently encountered situation is when one sub-system is directly driven by an external force and the other sub-system is driven only through the coupling. The equation then reduced to Ref. (10), ( )       −+= 2 2 1 1 112111 n E n E nEP in ωηωη       −+= 1 1 2 2 221220 n E n E nE ωηωη …(8) Equation (7) and (8) for the two sub-system results cannot be extended to more than two sub- systems, although the power flow energy difference relationship is true even when the two sub-systems are connected through a third un damped sub-system Ref. [11]. By expanding our way of thinking in the previous section, it is possible to formalize, in the same way, the power flow relationships of a structure consisting of multiple sub-systems as shown in fig. (2). Fig. 2. Three coupled sub-systems Ref[11]. The power flow equation for a structure made up of N sub-systems is expressed with the following equation in matrix form,               =                 ×                             +− −      +− −−        + ∑ ∑ ∑ − ≠ ≠ ≠ iN i i N N N N Ni NiNnN N N i N N i i P P P n E n E n E nn nnn nnn MM LL MOMM L L 2 1 2 2 1 1 1 1 222 2 212221 111121 1 11 ηηη ηηηη ηηηη ω …(9) Where i i n E is the energy density of sub-system i. Equation (9) represents a set of N-simultaneous linear equations with the independence variable of interest i i n E . This set of equations (matrix equation) can be solved to obtain the stored energies in the coupled sub-systems Ref. [3]. E1 E2 E3 13P 12P 23P ( )2inP ( )3inP diss,3P ( )1inP diss,1P diss,2P 21P 32P 31P Adnan D. Mohammed Al-Khwarizmi Engineering Journal, Vol. 7, No. 4, PP 41 - 53 (2011) 44 3. System Modeling and S.E.A Parameters SEA models are all based on the balance of energy and power among groups of natural modes in a system. Complex systems can be modeled as coupled mode groups. There are three main steps in defining an SEA model. The first is to divide the system into groups of components with similar characteristics, for example plates, beams, acoustic spaces and etc. Similar subsystems are used because they have similar modes, similar structures also carry similar wave types like compression, shear and bending for example, a beam has the main mode of bending, as do all other beams. The second step is to define the physical connection between the subsystems for example line connection, bolt connection or weld connection. The last step is to determine or define the external excitation of the system like the noise or vibration generated by wind, engine, transmission, tire ...etc. The parameters that define the sub-systems are the average frequency spacing between modes (center frequency of third octave band), the average modal damping factor can be taken from table (1) that shows some typical values of structural loss factor for some common materials presented by Ref. [8], and Ref. [6]. The modal density of flat plate in flexural vibration is given by Ref.(3):- ( ) tC A n Lπ ω 2 12 = …(10) Where; )1( 2υρ − Υ =LC …(11) Where, LC is the longitudinal wave speed, Y is Young's modulus, υ is Poisson's ratio, A is the surface area of plate, and, t is the plate thickness. The time average input power can be written as, { }12 ~Re~ 2 1 −= min ZFP …(12) The real part of drive–point mechanical impedance of an infinite plate of thickness t and mass per unit area ρa in flexural vibration Ref. [6], is:- { } ( )2 3 112 8 ~ Re υ ρ − Υ = am t Z …(13) And the coupling loss factor between mode groups for line junction is evaluated by Lyons and Cremer et al., and it is conveniently given in terms of the wave transmission coefficient, 12τ (the ratio of transmitted power to the incident power) for a l i n e j u n c t i o n , L R e f . [7]. I t i s : - 1 12 12 2 A LC B πω τ η = …(14) Equation (14) reveals that the coupling loss factor depends on the surface area, 1A and the bending wave speed, BC of the first plate for two connected plates with line junction as a function of center frequency, f Ref. [6]. Where; tfCC LB 8.1= … (15) Table 1, Structural loss factor for some common materials Ref. [6]. Material Structural Loss Factor Aluminum 1.0×10-4 Brick, Concrete 1.5×10-2 Cast Iron 1.0×10-3 Copper 2.0×10-3 Glass 1.0×10-3 Plaster 5.0×10-3 Plywood 1.5×10-2 PVC 0.3×100 Sand (dry) 0.02-0.2 Steel 1-6×10-4 Tin 2.0×10-3 And the normal incidence transmission coefficient for two coupled flat plates at right angles to each other is given in Ref. [6]:- ( ) { } 2212112 20 −−+= ψψτ …(16) Where; 2 5 2 2 3 22 2 5 1 2 3 11 tC tC L L ρ ρ ψ = …(17) Adnan D. Mohammed Al-Khwarizmi Engineering Journal, Vol. 7, No. 4, PP 41 - 53 (2011) 45 The random incidence transmission coefficient 12τ is approximated by Ref. [6]:- ( ) X X 24.31 754.2 01212 + = ττ …(18) Where; 2 1 t tX = …(19) These parameters must be modeled correctly because they are the coupling of the sub-systems that determines how much energy is transmitted from one subsystem to another. When all these steps are defined correctly and submitted into the major equation of S.E.A eq(9), the conclusion will be easy to estimate when we find the energy stored in each sub-system. It should be noticed that the results of vibrational energies are obtained in decibels (dB) rather than any other physical units because it can show any small variation in the energy level and also it can mark the sensitivity of the model quite easily. This is done by using the formula Ref. [2],       = refE mNELogdBE ).(10)( 10 …(20) Where refE =10E-7 N.m The above equations give the most desired results of Statistical Energy Analysis (S.E.A) model. 4. System Details The system considered here is shown in fig. (3). It comprises two thin, flat, uniform, plates joined along a straight edge, each plate is forming one subsystem. The "Thin" assumption implies that the effects of shear deformation and rotational inertia are both negligible. The length of the coupling is L and the areas of the plates are A1 and A2. It is assumed that only plate 1 is excited and that the excitation is harmonic at frequency ω and of constant amplitude Fo. And the material of the system is steel. The developed system is assumed to be resonant, conservative, linear and there is an equpartition of energy between all the resonant modes within a given frequency band in a given sub-system. Fig. 3. Two Coupled Plates System. A1 A2 b1 b2 a t2 t1 Line junction (welded) a = 0.3 m b1= 0.4 m b2= 0.2 m t1= 0.003m t2= 0.002m A1= 0.12m2 A2= 0.06m2 ρ = 7800 kg/m3 υ = 0.3 Y=2×1011N/m2 η1= η2= 0.0001 Fo=1N Y Z X Foejωt Driven plat Adnan D. Mohammed Al-Khwarizmi Engineering Journal, Vol. 7, No. 4, PP 41 - 53 (2011) 46 5. Finite Element Modeling In this paper, the energy stored in sub-systems is found by using Statistical Energy Analysis, and a comparison is made between analytical solution (S.E.A) and numerical analysis (F.E.M) ,built by ANSYS 5.4 for the case of the two coupled (welded) plates. Because, the Statistical Energy Analysis gives us the energy in average manner over all the points of the plate and does not give any information about concentration of energy in particular parts of the sub-systems and that the Finite Element Method gives the response in any particular point. Therefore, the loading of harmonic forces will apply at different points on the first plate (five different points) and the average responses over different locations (ten different points for each plate) are obtained. A job name and analysis title is specified from the file option on the Utility menu and clarifies the type of analysis (structural analysis). Then, PREP7 preprocess is used to define the element types, element real constant, material properties for steel after consider it as isotropic material, and the model geometry as shown in figure (4). their properties will be: Element type = Shell Elastic. Young modulus of elasticity, Y=2×1011 (N/m2). Poisson's ratio, υ=0.3. Mass density, ρ=7800 (Kg/m3). Fig. .4.A Tested model of Two Coupled Plates Built by ANSYS Package. In applying the Finite Element method, it is necessary to divide the structure into a set of interconnected elements sufficiently small to adequately model, the structural geometry and to adequate represent loads and structural deformations. So, the whole system was divided and meshed (free mesh) as follows: Time of the run of program= 105 minutes. Number of elements for the first sub-system (plate 1)=192. Number of elements for the second sub-system (plate 2)=96. Figure (5) shows the final model after it is meshed and the load is applied to it, and by that the model building is completed. Fig. .5. The Final Model Built by ANSYS 5.4. 6. S.E.A Results And Discussion 6.1. The General Behavior Of Energy Flow The trend of all energy versus frequency curves is similar for all cases studied in this work. As the center frequency of third octave band increases the energy content of all sub-systems decreases. An obvious explanation for this behavior is due to the fact that as frequency increases the amplitude of vibration of any structural members decreases and consequently a reduction in the level is observed. 6.2. Effect of Variation of Surface Area on Energy Level Figures (6) and (7) demonstrate the effect of variations of the surface area of the directly driven plate A1 on the energy levels of the coupled plate- plate system. The figures display the variation as a function of center frequency of 1/3 octave bands. It shows that as A1 increases energy level of plate 1 increases while a reduction in the energy level of the indirectly driven plate is noticed. This is because of the fact that as A1 increases the Adnan D. Mohammed Al-Khwarizmi Engineering Journal, Vol. 7, No. 4, PP 41 - 53 (2011) 47 strength of coupling decreases towards the weak coupling condition. This leads to a reduction in the power transferred from plate 1 to plate 2 and consequently a lower energy level for plate 2. Figures (8) and (9) show that as the surface area of the indirectly driven plate A2 increases its energy level increases and a reduction in energy level of the directly driven plate is noticed. This is due to the same reason described above (i.e.:- coupling strength). Table (2) displays some of the predicted results at 500 Hz center of frequency of 1/3 octave band. 101 102 103 104 105 Ceter freqencyof 1/3 octave band(Hz) 0 5 10 15 20 25 30 35 40 45 50 55 60 E ne rg y le ve l, E 1 (d B ) A1=0.06 m2 A1=0.12 m2 A1=0.18 m2 A1=0.24 m2 A1=0.30 m2 Case 1 Plate-Plate system Fig. .6. The Effect of Variation Of Surface Area of Directly Driven Plate on First Plate Energy Levels. 101 102 103 104 105 Ceter freqency of 1/3 octave band (Hz) 0 5 10 15 20 25 30 35 40 45 50 55 60 E ne rg y le ve l, E 2 (d B ) A1=0.06 m2 A1=0.12 m2 A1=0.18 m2 A1=0.24 m2 A1=0.30 m2 Case 1 Plate-Plate system Fig. 7. The Effect of Variation of Surface Area of Directly Driven Plate on Second Plate Energy Levels. 101 102 103 104 105 Ceterfreqencyof1/3 octave band(Hz) 0 5 10 15 20 25 30 35 40 45 50 55 60 E ne rg y le ve l, E 1 (d B ) A2=0.06 m2 A2=0.12 m2 A2=0.18 m2 A2=0.24 m2 A2=0.30 m2 Case 1 Plate-Platesystem Fig. 8.The Effect of Variation of Surface Area of Indirectly Driven Plate on First Plate Energy Levels. 101 102 103 104 105 Ceter freqencyof 1/3 octave band(Hz) 0 5 10 15 20 25 30 35 40 45 50 55 60 E ne rg y le ve l, E 2 (d B ) A2=0.06 m2 A2=0.12 m2 A2=0.18 m2 A2=0.24 m2 A2=0.30 m2 Case 1 Plate-Plate system Fig. 9 The Effect of Variation of Surface Area of Indirectly Driven Plate on Second Plate Energy Levels. Plate-Plate system Plate-Plate system Plate-Plate system Plate-Plate system Center frequency of 1/3octave band (Hz) Center frequency of 1/3octave band (Hz) Center frequency of 1/3octave band (Hz) Center frequency of 1/3octave band (Hz) Adnan D. Mohammed Al-Khwarizmi Engineering Journal, Vol. 7, No. 4, PP 41 - 53 (2011) 48 Table 2, Effect of Surface Area Variation on the Energy Level at 500 Hz. Values of Area M2 0.06 0.09 0.12 0.15 0.18 ΔA1 E1 28.701640 30.247600 30.915370 31.292170 31.535020 E2 30.444710 28.980370 27.887220 27.014640 26.288390 ΔA2 E1 30.247600 28.712280 27.588370 26.702860 25.973140 E2 28.980370 30.437570 31.056880 31.403130 31.624950 6.3. Effect of Variation of Internal (Damping) Loss Factor on Energy Level The second part of the parametric study is concerned with the effect of variation of material dissipation loss factor on the predicted results of total vibrational energy for case studied in this work. This is done by increasing (alternatively) the dissipation loss factor (ηi) of one sub-system in each case while other parameters kept fixed. Figures (10)-(13) present the effect of variation of internal loss factor for a range of (0.00001-0.1). This variation causes a reduction of the values of the energy levels for the two plates and that is because the increasing of internal loss factor values led to increasing of the material resistance and that will dissipate the energy flow across that sub-system. Of course the amount of the energy dissipated depends on the sub-system that the variation lies on. Table (3) displays some of the predicted results at 500 Hz center of frequency of 1/3 octave band. 101 102 103 104 105 Ceter freqencyof 1/3 octave band(Hz) -10 -5 0 5 10 15 20 25 30 35 40 45 50 55 60 E ne rg y le ve l, E 1 (d B ) ILF1=0.00001 ILF1=0.0001 ILF1=0.001 ILF1=0.01 ILF1=0.1 Case1 Plate-Platesystem Fig. 10. The Effect of Variation of Directly Driven Plate Internal Loss Factor on the Plate 1 Energy Levels. 101 102 103 104 105 Ceter freqencyof 1/3 octave band(Hz) -20 -15 -10 -5 0 5 10 15 20 25 30 35 40 45 50 55 60 E ne rg y le ve l, E 2 (d B ) ILF1=0.00001 ILF1=0.0001 ILF1=0.001 ILF1=0.01 ILF1=0.1 Case1 Plate-Plate system Fig. 11. The Effect of Variation of Directly Driven Plate Internal Loss Factor on the Plate 2 Energy Levels. Plate-Plate system Center frequency of 1/3octave band (Hz) Plate-Plate system Center frequency of 1/3octave band (Hz) Adnan D. Mohammed Al-Khwarizmi Engineering Journal, Vol. 7, No. 4, PP 41 - 53 (2011) 49 101 102 103 104 105 Ceter freqencyof 1/3 octave band(Hz) 0 5 10 15 20 25 30 35 40 45 50 55 60 E ne rg y le ve l, E 1 (d B ) ILF1=0.00001 ILF1=0.0001 ILF1=0.001 ILF1=0.01 ILF1=0.1 Case1 Plate-Plate system Fig. 12. The effect of variation of indirectly driven plate internal loss factor on the plate 1 Energy levels. 101 102 103 104 105 Ceterfreqencyof 1/3 octave band(Hz) -15 -10 -5 0 5 10 15 20 25 30 35 40 45 50 55 60 E ne rg y le ve l, E 2 (d B ) ILF1=0.00001 ILF1=0.0001 ILF1=0.001 ILF1=0.01 ILF1=0.1 Case1 Plate-Platesystem Fig. 13. The Effect of Variation of Indirectly Driven Plate Internal Loss Factor on The Plate 2 Energy Levels. Table 3, Effect of Dissipated Loss Factor Variation On The Energy Level at 500 Hz. Values of Internal Loss Factor 0.00001 0.0001 0.001 0.01 0.1 Δη1 E1 15.409160 15.336340 14.668310 10.819890 2.445565 E2 12.662200 12.589380 11.921350 8.072927 -0.301393 Δη2 E1 12.667110 12.638050 12.368280 10.819890 8.751205 E2 11.415940 11.370820 10.943660 8.072927 0.411249 6.4. Comparison Between S.E.A and F.E.M Results Figures (14)-(18) present the average of energy levels predicted by Statistical Energy Analysis and Finite Element averages over different response locations due to an applied load at a specific location (node). Figure (19) shows the comparison between the average energy levels predicted using S.E.A and the ensemble averages of energy levels over the different force and response locations using F.E.M. The noticed differences between results of the two methods are due to the fact that S.E.A obtains the response for the whole sub-system without going into the details of the modes of vibration involved while F.E.M results involve mode by mode calculations. The comparison shows that at high frequency domain the results of the two methods become more comparable because of the fact that at high frequency regimes the number of modes becomes higher than that at low frequency regimes. The higher the number of modes means that the coupled sub-system in that region of frequency has high modal overlap factor which is the condition that should be available when using S.E.A. This comparison proved that Statistical Energy Analysis is more acceptable tool of analysis at high frequencies than that at low frequencies. ILF2= 0.00001 ILF2= 0.0001 ILF2= 0.001 ILF2= 0.01 ILF2= 0.1 Center frequency of 1/3octave band (Hz) Plate-Plate system Center frequency of 1/3octave band (Hz) ILF2= 0.00001 ILF2= 0.0001 ILF2= 0.001 ILF2= 0.01 ILF2= 0.1 Plate-Plate system Adnan D. Mohammed Al-Khwarizmi Engineering Journal, Vol. 7, No. 4, PP 41 - 53 (2011) 50 101 102 103 104 105 Center freqencyof 1/3 octave band(Hz) -20 -15 -10 -5 0 5 10 15 20 25 30 35 40 45 50 55 60 E ne rg y le ve l, E 1, E 2 (d B ) E1, SEA E2, SEA E1, FEM E2, FEM Fig.14. The comparison between S.E.A results and F.E.M average results due to a load applied in the center point of the first sub-system. 101 102 103 104 105 Centerfreqencyof 1/3 octave band(Hz) -20 -15 -10 -5 0 5 10 15 20 25 30 35 40 45 50 55 60 E ne rg y le ve l, E 1, E 2 (d B ) E1, SEA E2, SEA E1, FEM E2, FEM Fig. 15. The Comparison Between S.E.A Results and F.E.M Average Results due to a Load Applied in the Upper Right Corner of the First Sub-System. 101 102 103 104 105 Centerfreqencyof1/3 octave band(Hz) -15 -10 -5 0 5 10 15 20 25 30 35 40 45 50 55 60 E ne rg y le ve l, E 1, E 2 (d B ) E1, SEA E2, SEA E1, FEM E2, FEM Fig.16. The comparison between S.E.A results and F.E.M average results due to a load applied in the upper left corner of the first sub-system. 101 102 103 104 105 Centerfreqencyof1/3octaveband(Hz) -20 -15 -10 -5 0 5 10 15 20 25 30 35 40 45 50 55 60 E ne rg y le ve l, E 1, E 2 (d B ) E1, SEA E2, SEA E1, FEM E2, FEM Fig. 17. The Comparison Between S.E.A Results and F.E.M Average Results due to a Load Applied in the Lower Right Corner of the First Sub-System. Center frequency of 1/3octave band (Hz) Center frequency of 1/3octave band (Hz) Center frequency of 1/3octave band (Hz) Center frequency of 1/3octave band (Hz) Adnan D. Mohammed Al-Khwarizmi Engineering Journal, Vol. 7, No. 4, PP 41 - 53 (2011) 51 101 102 103 104 105 Center freqencyof 1/3 octave band (Hz) -20 -15 -10 -5 0 5 10 15 20 25 30 35 40 45 50 55 60 E ne rg y le ve l, E 1, E 2 (d B ) E1, SEA E2, SEA E1, FEM E2, FEM Fig.18. The Comparison Between S.E.A Results and F.E.M Average Results due to a Load Applied in the Lower Left Corner in the First Sub-System. 101 102 103 104 105 Center freqencyof 1/3 octave band(Hz) -15 -10 -5 0 5 10 15 20 25 30 35 40 45 50 55 60 E ne rg y le ve l, E 1, E 2 (d B ) E1, SEA E2, SEA E1, FEM E2, FEM Fig. 19. The Comparison between S.E.A Results and F.E.M Ensemble Average. Table 4, The Values of Energy at Frequencies of Octave Band for S.E.A and F.E. FREQUENCY HZ 1000 2000 4000 8000 16000 S.E.A E1 27.240470 24.234590 21.230570 18.229050 15.231060 E2 25.965870 22.949580 19.930900 16.908710 13.881680 F.E.M E1 28.000000 23.200000 26.100000 9.9900000 2.6500000 E2 16.800000 10.600000 12.400000 7.4100000 1.7300000 7. Conclusions The main conclusions drawn from this work based on the results obtained in the present study are summarized as follows:- 1. Statistical Energy Analysis is a quite powerful tool in analyzing the dynamic behavior of coupled systems without going into great technical details as other approaches. 2. The main thing to remember is that Statistical Energy Analysis uses power flow from higher sub-system energy to lower sub-system energy for the basis of its calculations. It is also important to remember that all of the parameters are based on averages and that; this method cannot calculate an exact response at an exact location. 3. The general trends of all energy reverse frequencies decrease with increasing frequency. This behavior is due to the fact that by increasing frequency, the amplitude decreases and consequently causes a reduction in energy levels. 4. The energy flow between sub-systems is increased by increasing the coupling loss factor (strong coupling). While decreasing the coupling loss factor decreases the energy flow between sub-systems (weak coupling). 5. Changing of sub-system dimensions affects the coupling loss factor and modal density and Adnan D. Mohammed Al-Khwarizmi Engineering Journal, Vol. 7, No. 4, PP 41 - 53 (2011) 52 consequently affects the energy stored in that sub-system. 6. The energy stored in particular sub-system decreases with increasing the internal loss factor. 7. The suggested analytical solution gives good agreement with the numerical solution at the high frequency domain. 8. References [1] Cotonis v., Langley R.S. and Kinder M. R. F., (2005), "Numerical and Experimental Validation of Variance Prediction in the S.E.A of Built-Up System", journal of sound and vibration, vol. 288, pp. 701-728. [2] Fahys F.J., and Mohammed A.D., (1996), "A Study of Uncertainty in Application of S.E.A to Coupled Beam and Plate System, Part I: - Computational Example", journal of sound and vibration, vol. 158, pp. 45-67. [3] Koizumis T., Tsujiuchi N., Tanaka H., Okubo M. and Shinomiya M., (1992), "Prediction of the Vibration in Buildings using S.E.A", journal of sound and vibration. [4] Lyons R. H., (1975), "SEA of Dynamical Systems: Theory and Application", MIT Press, Cambridge MA. [5] Maces B., (2003), "S.E.A, Energy Distribution Models and System Modes", journal of sound and vibration, vol. 264, pp. 391-409. [6] Nortons M. P., (1989), "Fundamental of Noise and Vibration Analysis for Engineers", Cambridge University Press, New York, Port Chester Melbourne Sydney. [7] Parks W. S., Thompson D. J. and Ferguson N. S., (2005), "Variability of the Coupling Loss Factor between Coupled Plates", journal of sound and vibration, vol. 279, pp. 557-579. [8] Rankys M. F. and Clarkson B. L., (1983), "Frequency Average Loss Factor of Plates and Shells", journal of sound and vibration, vol. 89, no. 3, pp. 309-323. [9] Renjis K., (2004), "Estimation of Spectral Density using S.E.A", journal of sound and vibration, 9vol. 275, pp. 447-451. [10] Westers E. C. N. and Mace B., (2005), "S.E.A of Two Edge – Coupled Rectangular Plates:- Ensemble Average", journal of sound and vibration, vol. 193, no. 4, pp. 793- 822. [11] Whites R. G. and Walker J. G., (1982), "Noise and Vibration", Chichester Harwood Ltd. )2011( 41 – 53 ، صفحة4، العدد 7مجلة الخوارزمي الھندسیة المجلد عدنان داوود محمد 53 باستعمال طریقة تحلیل الطاقة اإلحصائي) ملحومة(تقدیر طاقة االھتزاز لصفیحتین متصلتین حوراء صفاء إبراھیم حمد عدنان داوود م الجامعة التكنولوجیة /قسم ھندسة السیطرة والنظم الخالصة , صائي والتي یمكن تطبیقھا على المنظومات المھتزة المیكانیكیة والصوتیة مث ل البنای ات ھذا البحث یتعامل مع طریقة تسمى طریقة تحلیل الطاقة اإلح العوام ل الت ي تعتم د ). او ھیك ل معق د / عالي أو اتردد(ھذه الطریقة كأداة یمكن تطبیقھا على المنظومات الخاضعة للظروف . الخ.....والطائرات, الجسور الكثافة الطوریة والقدرة الداخلة إلى النظام المھتز تم توضیحھا لصفیحتین متصلتین , معامل الخسارة الداخلي, طمعامل خسارة االرتبا(علیھا ھذه الطریقة ق دیمھا ف ي ھ ذا وتم تقدیم شرح لھذه العوامل في ھذا البحث و باإلضافة إلى ذلك الشروط والخطوات التي یجب أتباعھا للنمذجة باستخدام ھذه الطریق ة ت م ت و األھتزاز یتم بصورة خطیة و ھناك توزیع متساوي للطاقة بین كل , و یتم حفظ الطاقة فیھا, مة المھتزة تم أفتراض أنھا تھتز بصورة رنانةالمنظو. البحث قائم ة بزاوی ة ) ملحوم ة (الھدف من ھذا البحث ھ و أیج اد الطاق ة المخزون ة ف ي األج زاء المھت زة الص فیحتیین م رتبطتین . األجزاء المھتزة عند تردد معین ) الص فیحة األول ى (النتائج تبین أن عند زیادة المساحة السطحیة للصفیحة المثارة بطریقة مباش رة , ھذه األجزاء) طول وعرض(ودراسة تأثیر تغییر أبعاد ھ ذا بس بب أن ق وة األتص ال ب ین و). الص فیحة الثانی ة (یزید مستوى الطاقة للصفیحة األولى بینما یقل مستوى الطاقة للصفیحة التي لم تثار بطریقة مباشرة اقة قلی ل للص فیحة الصفیحتین یتجھ نحو األرتباط الضعیف و ھذا یقود الى قلة الطاقة المنتقلة من الصفیحة األولى الى الصفیحة الثانیة و بالتالي مستوى ط التغییر یسبب في قلة مستوى الطاق ة للص فیحتیین و ذل ك ھذا ) ٠,١-٠,٠٠٠٠١(باإلضافة إلى ذلك دراسة تغییر معامل الخسارة الداخلي لمدى من . الثانیة وتم عمل مقارنة بین طریقة . بسبب أن زیادة معامل الخسارة الداخلي یؤدي الى زیادة المقاومة الداخلیة للمادة و ھذا سیضیع الطاقة المنتقلة بین الصفیحتیین ANSYSفورتران وطریقة العناصر المحددة التي تم عمل نموذج لھل باستخدام برنامج تحلیل الطاقة اإلحصائي التي تم بناء نموذج لھا باستعمال برنامج .5.4 )2011( 41 – 53 ، صفحة4، العدد 7مجلة الخوارزمي الھندسیة المجلد عدنان داوود محمد