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10 Energy Finite Element Method

Robert Bernhard1 and Shuo Wang2

1 2

Purdue University, Hovde Hall, 610 Purdue Mall, West Lafayette, IN 47907, USA bernhard@purdue.edu Science Applications International Corporation (SAIC), 2450 NASA Road One, Houston, TX 77058, USA shuowang usa@yahoo.com

Summary. The energy ?nite element method was developed to predict the average response of built-up structural acoustic systems consisting of subsystems such as rods, beams, plates, and acoustical enclosures. The methodology for predicting the behavior in the subsystems is based on a diffuse energy ?eld approximation that is most appropriate for high frequency analysis where traditional ?nite element approaches become expensive. Subsystems are coupled together using net energy ?ow and energy superposition principles [3].

10.1 Background and Motivation

At high frequency, acoustical and structural wavelengths are short relative to the dimensions of the systems of interest. The model size and cost of traditional ?nite element models become prohibitively large. In addition, system response is highly sensitive to environmental and dimensional variation and deterministic methods have less value than statistical methods for predicting performance. Also, many noise applications are described in terms of one–third octave band or octave band metrics. Such analyses are most ef?ciently done using methods that are able to accurately predict average results. To obtain an accurate and simple mathematical model representing the average energy propagation in systems, signi?cant effort has been made to develop equations that govern the average energy ?ow in continuous structures. Belov and Rybak developed transport equations utilizing the Green’s function for energy ?ow in in?nite vibrating plates [1], and formulated conduction–like equations for the energy ?ow in ribbed plates [2]. From these equations Nefske and Sung [10] developed a ?nite element method to predict the energy ?ow in homogeneous ?nite beams in terms of energy variables. Wohlever and Bernhard [23] derived locally averaged energy governing equations using a method that is consistent with classical mechanics and obtained a second order differential equation, which governs the smoothed energy distribution in rods and beams. Coupling of the energy densities for rods and beams was resolved by Cho and Bernhard [6]. By this approach, energy conduction is simple to predict and

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can be implemented using standard ?nite element methods with relatively few elements. The technique is capable of predicting the spatial distribution of the energetics of built–up structures. The energy ?nite element method (EFEM) was developed to utilize available ?nite element geometric models for high frequency structural– acoustic analysis, based on the energy governing equations and the ?nite element formulation given by Wohlever and Bernhard and the coupling methods developed by Cho and Bernhard. The EFEM techniques have been successfully applied to various engineering problems. Wang and Bernhard [16, 17] applied the EFEM techniques to the realistic case study of a heavy equipment cab. Vlahopoulos et al. [14] applied the EFEM to a complex double–hull ship structure. Wang et al. [20] used the EFEM formulation to compute the structural response of an aluminum cylinder shell structure with periodic stiffeners. Wang and Bernhard [15, 18] also reformulated the numerical implementation of the equations of Wohlever and Cho using a simpli?ed energy ?nite element method, referred to as EFEM0 (the superscript ”0” denotes that a zero–order ?nite element interpolation or the ?nite volume method is used). The resulting formulation results in less degree of freedom for a model and can be implemented using Statistical Energy Analysis (SEA) software. Wang et al. combined the EFEM0 technique with SEA for sound package optimization of a trimmed van [22], and also for exterior acoustic modeling of a truck [19]. Klos applied the EFEM0 technique for point–excited shells [8]. The EFEM is complementary to low frequency FEM models since it can use existing FEM databases. A ?nite element geometric model can be applied to both low and high frequency analyses. The prediction of a spatially varying energy level within a subsystem is predicted with the EFEM computation. The post–processing of EFEM results also provides straightforward visualization of the energy ?ow in a system, which is convenient for diagnosis and control of noise propagation. EFEM can be used to model relatively highly damped or non–uniformly damped materials, and to model distributed masses as well as multi–point power input. Due to the utilization of the ?nite element technique, EFEM also has the other advantages of traditional FEM. It can be easily applied to irregular domains and geometries that are composed of different materials or mixed boundary conditions. The structural energy density and the power radiation obtained from an EFEM model can be used as boundary conditions of energy boundary element method (EBEM) to compute the radiated far-?eld sound pressure. An energy boundary element method was developed by Wang et al. [21] using numerical boundary element principles and is a useful complement to energy ?nite element methods for high frequency acoustical analysis in large or open spaces. Franzoni et al. have also formulated an acoustic boundary element method based on time–averaged energy and intensity variables [7]. Statistical energy analysis (SEA) is the most widely used energy-based methods for predicting average behavior at high frequencies [9]. SEA is based on the physical concept of power balance. However, the assumptions used to develop SEA limit application to lightly coupled, low damping applications. When a more detailed

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model is needed to include behavior such as the excitation location effects, spatial distribution of response for high–damping system, etc., SEA is not applicable. The assumptions of SEA limit the applicable range of the method.

10.2 Governing Equations

For steady state vibrational energy propagation within a control volume V , the principle of conservation of energy requires that the total power Πin entering the control volume must be balanced by the summation of the power dissipated within the volume, Πdiss , and the energy ?ow through the boundary S Πdiss +

S

I · ndS = Πin

(10.1)

where I is the intensity of the ?eld. For a general case, assuming the intensity function has continuous ?rst partial derivatives, the divergence theorem can be applied such that I · ndS = ? · IdV . (10.2)

S V

Writing each term in (10.1) in the format of volume integrals, πdiss dV +

V V

? · IdV =

V

πin dV ,

(10.3)

then, the energy balance equation is obtained for steady state vibrational energy propagation such that (10.4) πdiss + ? · I = πin , where πin is the input power density (power input per unit volume) and πdiss is the dissipated power density (power dissipated per unit volume). For EFA implementations, a simple loss factor model of damping is used for power dissipation (10.5) πdiss = ηωe , where η is the damping loss factor, ω is the angular frequency, and e is the time– averaged and locally space–averaged energy density. To develop an EFA equation and express (10.4) in terms of energy density, a relationship is required between the divergence of intensity and the energy density. For an omni–directional source in an in?nite medium, these relationships are known. In a two–dimensional medium this relationship is 1 d (rcg ed ) , (10.6) ?·I = r dr where cg is the group speed. In a three-dimensional medium, this relationship is ?·I = 1 d 2 r cg e d , r2 dr (10.7)

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where ed is the time–averaged energy density in the direct ?eld, and r is the radial distance from the excitation. Thus, the governing differential equation for the energy density distribution in the direct ?eld of a point–excited in?nite plate is 1 d (rcg ed ) + ηωe = πin . r dr The solution for the direct-?eld energy density due to a point source is ed = and the radial intensity is Irad = cg ed = Πin ? ηωr e cg 2πhr (10.10) Πin ? ηωr e cg 2πhrcg (10.9) (10.8)

Equations (10.6) through (10.10) potentially could be basis of the simplest versions of the Energy Boundary Element Method. For certain cases, such as in a reverberant ?eld, the response can be assumed to be the superposition of moderately damped plane waves. Bouthier and Bernhard [4] showed that the smoothed intensity I is related to the energy density by I = ? c2 g ηω ?e (10.11)

where e is the time–averaged and locally space–averaged energy density, ?e is the gradient of the smoothed energy density, and cg is the group speed. Using equations (10.4), (10.5), and (10.11), the general form of the differential equation governing the energy ?ow through an isotropic, homogeneous system can be written as c2 g ?e ? ηωe + πin = 0 . ?· (10.12) ηω Equation (10.12) is one of the governing equations of so–called energy ?ow analysis (EFA) and is applicable where the wave ?eld can be modeled as a superposition of moderately damped plane waves.

10.3 Energy Finite Element Method (EFEM) Formulations

The EFA governing differential equation (10.12) for damped plane wave behavior can be rewritten as ? ? (10.13) ? · D?e ? Ge + πin = 0 ? ? where the coef?cients are de?ned as D = c2 /(ηω) and G = ηω. Finite element g approximations of the EFA governing differential equation (10.13) can be developed by using the Galerkin’s method or the ?nite volume method.

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10.3.1 EFEM for Continuous Systems For a continuous system, using a conventional ?nite element formulation, the residual integral is expressed on an element (e) as [11] R(e) = ?

V

? ? N T ? · D?e ? Ge + πin dV

(10.14)

where N is the row vector of the shape functions (interpolation and test functions), and e = N en (10.15) where en denotes the nodal value vector of the energy density. Substituting (10.15) into (10.14) and forcing the residual integral to vanish results in the scalar equation as (e) (e) (10.16) R(e) = q (e) + kD + kG e(e) ? f (e) = 0 . π n ? ? For element (e), q (e) represents the inter–element energy ?ow along the element boundary ? (10.17) N T dq = N T ?Dn · ?e dS q (e) =

S S

The remaining element matrices are kD = ? and kG = ? The element input power vector is f (e) = π

V (e) (e)

? B T DBdV

V

(10.18)

? N T GN dV .

V

(10.19)

N T πin dV .

(10.20)

? In equation (10.18), B is a derivative vector and D is a diagonal matrix as for 1-D systems ?N ? ? B = and D = Dx , (10.21) ?x for 2-D systems ? ?N ? ?x ? ? B = ? ? ?N ? ?y and for 3-D systems ? ? Dx 0 ? D= ? 0 Dy

and

,

(10.22)

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Fig. 10.1 Two collinear beam elements coupled by a joint.

? ?N ? ?x ? ? ? B = ? ?N ? ?y ? ? ?N ?z

? ? ? ? ? ? ? ? ? ?? Dx 0 ? = ? 0 Dy ? D 0 0 ? 0 0 ? . ?z D

and

(10.23)

For clamped, free, and simply supported boundaries, the energy ?ow is zero and ? DAde/dn = 0. 10.3.2 Coupling Relations for Joined Systems at Discontinuities As described by Cho [5], the energy density discontinuities that occur when there is a discontinuity of material or geometric properties can be modeled using approximations that are consistent with the EFA equations. The joint relationships will be illustrated for a simple case of two coupled collinear beams as shown in Figure 10.1. The energy density and energy ?ow are expressed in terms of components associated with positive and negative traveling waves ei = e+ + e? i i

+ ? qi = qi ? qi = cgi e+ Ai ? cgi e? Ai i i

(10.24) with i = 1, 2 . (10.25)

At the joint position, the ei are the nodal values of energy density on either side of the joint and qi are the net energy ?ows out of the joint from beam i. The net energy ?ow away from the joint in each beam can be expressed as

? + + q2 = τ12 q1 + r22 q2 ? + + q1 = τ21 q2 + r11 q1

(10.26) (10.27)

where τij is the power transmission coef?cient from beam i to beam j with i, j = 1, 2, and rii is the power re?ection coef?cient in beam i. Ai represents the cross section area of beam i. Substituting equation (10.25) into (10.26) and (10.27) gives cg2 e? A2 = τ12 cg1 e+ A1 + r22 cg2 e+ A2 2 1 2 cg1 e? A1 = τ21 cg2 e+ A2 + r11 cg1 e+ A1 1 2 1 (10.28) (10.29)

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The values of e± are obtained by solving equations (10.24), (10.28) and (10.29). Subi stituting the values of e± into (10.25) yields the expression of energy ?ow through i the joint from beam 1, which is also the energy ?ow from beam 1 to 2 q1→2 = q1 = 1 τ12 cg1 A1 , ? τ21 cg2 A2 r11 + r22 e1 e2 (10.30)

For steady–state energy conduction, it is required that the energy ?ow is continuous at the joint. The energy ?ow from beam 2 to beam 1 is q2→1 = q2 = ?q1. The derivation of q1 and q2 does not require the joint to be conservative. Thus, it is possible to model a dissipative joint using EFA methods. However, for conservative coupling of rods and beams τ12 = τ21 , r11 = r22 and τ + r = 1. For this discussion the joint is assumed to be conservative. Using these conservative coupling relationships, the relationship between the energy and energy ?ow at a joint can be reduced to q1 q2 = 1 2 ? τ12 ? τ21 τ12 cg1 A1 ?τ12 cg1 A1 ?τ21 cg2 A2 τ21 cg2 A2 e1 e2 (10.31)

This approach has been generalized to cases of multiple structural members connected to the joint and multiple wave types in each structural member as well as line joints and area connections between plates and acoustical spaces. Cho [5] shows the extension of the approach for the joint matrix K J for these more complex and general joints. When all of the effects of energy dissipation and inter–element energy ?ow are accounted for, the EFEM formulation takes the form Ken = K D + K G + K J en = F . ? ? (10.32)

The EFEM approach is essentially a simple conduction ?nite element program with a unique joint to account for the discontinuity of energy density at discontinuous joints, which are accounted for using the joint matrix K J . 10.3.3 Simpli?ed Energy Finite Element Method (EFEM0 ) Model A simpler version of the EFEM formulation was developed by Wang [15] using the ?nite volume formulation of equation (10.12). To illustrate this, a one–dimensional form of the EFA equation (10.12) will be utilized d dx ? de D dx ? ηωe + πin = 0 . (10.33)

Using the ?nite volume method [13], the problem domain is discretized into a number of control volumes. The center of each control volume is treated as a node. A capital letter (P , W , or E) is used to represent both a volume and its center (node), as shown in Figure 10.2. The lower case letters w and e denote the west and east

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Fig. 10.2 One–dimensional ?nite volume grids.

boundaries of volume (or element) P . The spacing between the nodes is ΔxW P and ΔxP E . The length of element I (I is any of P , W or E) is identi?ed by LI . Integration of the 1–D EFA governing equation (10.33) over the control volume P yields d ? de D dV ? dx dx ηωedV +

V

V

? de πin dV = DA dx V

x=e

? ηωeV + πin V = 0

x=w

(10.34) where dV = Adx and A is the cross–sectional area of the one–dimensional system. The volume of P is V = ALP . Equation (10.34) is an energy balance equation, which requires that the sum of the energy leaving the control volume and the energy dissipated in the domain per unit time is equal to the power input. The derivatives in the energy ?ow terms in (10.34) can be rewritten as de dx =

e

eE ? eP ΔxP E

and

de dx

=

w

eP ? eW . ΔxWP

(10.35)

Substitution of (10.35) into (10.34) yields ? eE ? eP ? D eP ? eW ? ηωeP LP + πin LP = 0 ? D ΔxP E ΔxWP which can be rearranged as ηωLP + ? ? D D + ΔxP E ΔxWP eP ? ? ? D D eW ? eE = πin LP . ΔxWP ΔxP E (10.37) (10.36)

In order to solve the energy distribution within the system, an equation of this form can be developed for each nodal point. As demonstrated here, one of the major advantages of the ?nite volume method is that the numerical algorithm is closely related to the underlying physical conservation principle. This feature makes the method easy to apply and adapt to novel problems. To illustrate the comparison of the EFEM0 to SEA, the energy ?ow terms c in (10.36) can be restated in terms of two types of coupling coef?cients ηe and ?nite volume total energy as ω

c c c c ηeP E EP ? ηeEP EE + ηeP W EP ? ηeW P EW

+ωηEP = ΠPin . (10.38)

where EI = ALI eI , and

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Fig. 10.3 Coupled three–element beam model with continuous and discontinuous joints.

c ηeIJ =

? ? 2c2 DA D g = = ωVI ΔxIJ ωLI ΔxIJ ηω 2 LI (LI + LJ )

dc ηeij =

(10.39)

and

τij τij cgi cg ≈ i ωLi rii + rjj ωLi 2 ? τij ? τji

(10.40)

where I and J denote the central node numbers of adjacent control volumes, i.e. E, W , P . Equation (10.38) is similar to an SEA equation, which has the general form

m

ω (ηij Ei ? ηji Ej ) + ωηi Ei = Πiin .

j=1,=i

(10.41)

The EFEM0 equations for the three–element system shown in Figure 10.3 can be written in matrix form as ? ?? ? ? ? c c η1 + ηe12 ω ?ηe21 ω 0 E1 Πin1 ? ?? ? ? ? c c dc dc ?ηe32 ω ? ? E2 ? = ? Πin2 ? . η2 + ηe21 + ηe23 ω ? ?ηe12 ω dc dc E3 Πin3 η3 + ηe32 ω 0 ?ηe23 ω (10.42) Equation (10.42), which has been derived using a ?nite volume implementation of an EFA equation, has the same structure as the SEA matrix equation, which has been derived for lightly damped, weakly coupled modal systems with high modal overlap. Many of the terms are the same, particularly those associated with power input and dissipation of energy within the element. In this paper, we derived generic equations for 1–D, 2–D and 3–D problems based on structural–acoustic theories. The 3–D theory and equations included the ?uid or acoustic problems, although some derivations and examples may start with 1–D problem. In EFEM0 formulation one can change the coupling factors to account for acoustics based on radiation ef?ciency. In EFEM one can change the coupling terms in the joint matrix. 10.3.4 EFEM0 Models for One–dimensional Systems In order to illustrate the performance of the EFEM0 , a convergence study for a single uniform damped beam with simply supported boundary conditions is done using EFEM0 theory. An analytical EFA solution is also obtained for validation. The beam

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Fig. 10.4 Predictions of energy density at x = 3m using EFEM0 when dividing one uniform damped beam into n subsystems. (E = 7.1 · 1010 Pa, L = 4m, η = 0.24, A = 4 · 10?4 m2 ).

Fig. 10.5 Two coupled beams with simply supported boundary conditions.

is divided into an increasing number of elements. As shown in Figure 10.4, the accuracy of EFEM0 prediction increases as more elements are used, and converges to the EFA analytical solution. The coupled beams shown in Figure 10.5 with simply supported boundary conditions were also studied by Wang and Bernhard [18]. The physical properties are listed in Table 10.1. Analytical solutions were obtained using a wave solution for Euler–Bernoulli beam theory. As shown in Figure 10.6, the EFEM0 predictions were compared to analytical solutions for three cases: strong coupling (τ12 = 0.999), medium coupling (τ12 = 0.629) and weak coupling (τ12 = 0.236). The predictions using in equation (10.40) for discontinuous joint match the analytical solutions well for all three cases. In the study each beam was discretized into 100 elements to obtain a converged solution. A simpli?ed EFEM (EFEM0 ) has been developed from the energy ?ow analysis equations using the ?nite volume method. The resulting systems of equations have a similar form to SEA equations. The formulation results in apparent coupling loss factors for both continuous and discontinuous systems for relatively heavily damped, strongly coupled systems where conventional SEA coupling loss factors are not suitable. For a one–dimensional system, the prediction by EFEM0 agrees very well with the analytical solution and is numerically shown to converge to the exact EFA solution as the number of elements is increased.

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Fig. 10.6 Comparison of EFEM0 predictions with analytical solutions, dashed line: EFEM0 , solid line: analytical wave equation solution. Table 10.1 Physical parameters of the two beams, results for three cases: Case I for τ12 = 0.999, Case II for τ12 = 0.629, and Case III for τ12 = 0.236. Beam 1 Case I cross–sectional area [m ] A1 = 4 · 10 area moment of inertia [m4 ] I1 = 1.33 · 10?8 Young’s modulus [Pa] E1 = 7.1 · 1010 3 density [kg/m ] ρ1 = 2.7 · 103 loss factor η1 = 0.25 length [m] L1 = 2

2 ?4

Beam 2 Case II A2 I2 E2 ρ2 η2 L2 = = = = = = 6A1 36I1 E1 ρ1 η1 L1 Case III A2 I2 E2 ρ2 η2 L2 = = = = = = 16A1 256I1 E1 ρ1 η1 L1

A2 I2 E2 ρ2 η2 L2

= = = = = =

1.2A1 1.44I1 E1 ρ1 η1 L1

Generally, EFEM0 includes the advantages of both EFEM and SEA. Its global matrix has a simple band matrix form similar to SEA. Thus, computational cost is low. The method can be used to predict the local response in each subsystem and to deal with the local effects such as patch damping treatments and partial joints. The coupling relations are also generalized to heavily coupled systems. The EFEM0 potentially can be integrated into existing ?nite element model preparation software and will be able to utilize models that have been created for other purposes, such as

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Fig. 10.7 Hybrid–EFEM0 model for single plate.

low frequency structural acoustic models. EFEM0 coupling coef?cients can also be used in existing SEA programs to increase the capability of the programs. Therefore the EFEM0 is a good combination of the two methods, and potentially will be quite useful to model built–up, complex systems. 10.3.5 Hybrid EFEM0 Model for Single Plate The EFA plane wave equation was derived using a damped plane wave assumption, and thus, represents well the energy distribution in a damped reverberant ?eld. When the direct ?eld of a source is dominant, the EFA equation based on superimposed damped plane waves is not a good model of the response of such systems. A hybrid energy modeling method was developed using the approached described by Smith [12] to address this problem. The approach is to superimpose the contributions of the direct ?eld and the reverberant ?eld. The direct ?eld is calculated assuming the plate is in?nitely large. The reverberant ?eld is modeled assuming damped plane wave behavior and the input power from the direct ?eld re?ected at the boundary of the ?nite system. The overall response is the superposition of the two ?elds. For numerical implementation, the two–dimensional problem domain is discretized into ?nite volumes. The power input to the EFEM0 is modeled as the intensity at the boundary due to the direct ?eld, which is assumed to be re?ected into the reverberant ?eld at the boundary. The direct ?eld energy density for each volume is calculated from the energy density at the location of the center node of the ?nite volume relative to the source. The general strategy of this hybrid–EFEM0 technique is demonstrated in Figure 10.7. As a simple example of this implementation, a plate, is considered as shown in Figure 10.8. For S on the boundary surface of a region T in space, n is the outer unit normal vector of S. The direct ?eld power ?ow from a point source to a small

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Fig. 10.8 One element on the plate boundary.

surface area ΔA is equal to I · nΔA, where I · n is the normal component of I in the direction of n. For the EFEM0 numerical implementation, the magnitude of the intensity at a point (j) due to the direct ?eld is Id

(j)

= cg e j = d

(j) Πin e?ηωr /cg 2πhr(j)

(10.43)

where r(j) is the distance from the excitation point (x0 , y0 ) to the center of the element (x(j) , y (j) ), and is calculated by r(j) = x(j) ? x0

2

+ y (j) ? y0

2

,

(10.44)

r(j) can be expressed as a function of angle θ as shown in Figure 10.8 r(j) = d , cos θ (10.45)

where d is the normal distance from the excitation point to the boundary. The power re?ected into the element (j) of the reverberant ?eld along the boundary is (j) (j) Πν = r11 (θ)Id cos θΔA(j) (10.46)

Γ

where ΔA is the boundary elemental area, ΔA(j) = hL(j) , and θ is the angle between a radial vector from the source and the normal vector of the boundary. Substitution of equations (10.43)– (10.45) into (10.46) gives

(j) Πν (θ) Γ

(j)

= r11 (θ)D(θ)β (j) Πin

(10.47)

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R Bernhard, S Wang Table 10.2 Physical properties of the square plate properties. properties Young’s modulus [Pa] Poisson’s Ratio loss factor density [kg/m3 ] thickness [m] edge length [m] E μ η ρ h a=b case = = = = = = 2 · 108 0.5 0.02 1100 0.01 1.5

where D(θ) is a directivity function expressed as D(θ) = cos2 θ e and is de?ned as

ηωd ? cg cos θ

(10.48)

L(j) . (10.49) 2πd The elemental reverberant power is a function of angle θ and can be incorporated into the input power vector of the global matrix of the EFEM0 model. In [15], Wang applied the hybrid–EFEM0 models to the predictions for 8 square plates with different properties. One of the plates with moderate damping has the properties listed in Table 10.2. The predictions of the plate with simply supported boundaries are shown in Figure 10.9, compared with the frequency–averaged modal solutions for the plate. The results represent well the frequency– and space–averaged value of the modal solution, and are accurate in both the near ?eld of the excitation point and at the far ?eld of the boundary region. β (j) =

10.4 Engineering Application

10.4.1 Applications to Cab Rear Window The hybrid–EFEM0 model was used to model two coupled plates at the rear window of a heavy equipment cab. The top panel is tempered glass and the bottom panel is steel. For the EFEM0 model each panel is discretized into a 27 by 27 array of elements. This level of discretization is not necessary for typical systems but is used here to ensure that convergence occurred. The hybrid method is used for the prediction. The re?ected and transmitted power at the joints is computed from the direct ?eld using diffuse ?eld transmission and re?ection coef?cients. The experimental setup is shown in Figure 10.10. The rear window is divided into 12 subsystems. On each subsystem 5 points are picked at random. Acceleration responses to force excitation at the center of the glass plate are measured and converted to energy density. Both the average response and the response variation are computed.

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Fig. 10.9 Energy density predictions for square plate.

Fig. 10.10 Test setup to measure the responses of cab rear window.

The predicted values are averaged for each of the 12 experimental subsystems and compared to the test data. The comparisons for steel panel subsystem are shown in Figure 10.11. Generally the predicted energy density distributions lie in the region bounded by the experimental variation over the entire frequency domain. The spatial distributions of the direct ?eld, reverberant ?eld and total ?eld of the predicted responses of the glass panel are shown in Figure 10.12 for 630 Hz and 2500 Hz. Since the damping of the steel panel is light, its response is nearly uniform for most of the frequency range. It is not necessary to do a dense discretization of the steel plate. However, it is important to discretize the glass panel and use the hybrid method in order to account for the effects of damping and the direct ?eld of

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Fig. 10.11 Comparison of the EFEM0 prediction versus experimental data for cab rear window subsystem 11.

the source as demonstrated in Figure 10.12. The reverberant ?eld is more important at low frequency than at high frequency. However, in both cases, both ?elds are important. 10.4.2 EFEM Models for Cab Front Window The cab front and rear windows are typical structures and contain representative joints. In [16], the power transmission and re?ection coef?cients for joints were calculated and incorporated into both EFEM and SEA models to obtain results for comparison with each other and with measured data. When the transmission and re?ection coef?cients are obtained, the key step for EFEM is to calculate the general joint matrix (GJM). The key step for SEA is to calculate the coupling loss factor (CLF). The calculated power transmission and re?ection coef?cients are substituted into the joint matrix of EFEM. An existing cab NASTRAN model was imported into the EFEM preprocessor to generate a model with 20,225 elements and 19,405 nodes. The front window was divided into four subsystems, which are coupled with each other through ?ve joints. The joint between the middle and side glass panels is a rubber–like sealant which is modeled as a spring joint between plates. Due to the unique materials of the glass plates and the sealant between them, it is only possible to estimate the stiffness and loss factors by measurement of typical samples. The boundaries of the entire front window system were modeled as simply supported boundaries. Thus, the energy ?ow boundary condition of the front window was zero. The power input was computed using the input force and the real part of the impedance of an in?nite plate.

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Fig. 10.12 The energy density ?elds predicted by EFEM0 for the glass panel of the cab rear window: (a) the direct ?eld at 630 Hz; (b) the direct ?eld at 2500 Hz; (c) the reverberant ?eld at 630 Hz; (d) the reverberant ?eld at 2500 Hz; (e) the overall ?eld at 630 Hz; (f) the overall ?eld at 2500 Hz. The unit is in dB, re 10?12 J/m3 .

A test was conducted to validate the EFEM model of the cab front window. A mini–shaker was used to generate power input at the excitation point. Pretests were made to monitor the background noise and choose the excitation level to ensure good coherence. Coherence data was carefully monitored for each measurement. The acceleration responses of two points on each side of the spring joint were measured. The EFEM results after post–processing are shown in Figure 10.13 for the front window excited at the center. The spatial variation of energy density within each subsystem is clearly demonstrated in the EFEM results [16].

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Fig. 10.13 The EFEM prediction results of cab front window (800Hz).

10.5 Conclusions

The EFEM methods were developed to complement traditional ?nite element methods for high frequency prediction of average responses of structural acoustic systems. There are several distinct energy element methods. The Energy Boundary Element Methods superimpose energy ?elds from a combination of known and ?ctitious sources using the known fundamental solutions for energy ?ow from a point source in a free ?eld. The ?ctitious source strengths are solved to satisfy the boundary conditions, and when known, fully specify the solution. The EFEM methods are based on governing equations that were derived assuming the wave ?eld can be modeled as a superposition of moderately damped plane waves. The governing equations under these conditions are conduction–like equations. Traditional ?nite element methods can be used to develop numerical approximations of these equations. Due to the simplicity of the behavior of the energy for such conditions, a ?nite volume approximation of the vibrational conductivity equations also gives a satisfactory approximation of the behavior. The ?nite volume approximation results in an SEA–like set of linear equations that have the advantage of relative simplicity. For certain problems where a direct ?eld is important, hybrid EFEM methods have been developed. For these implementations it is assumed that the energy ?eld can be expressed as the superposition of a direct ?eld, which is predicted using the fundamental solution for energy propagation from a point source, and a reverberant ?eld, which is assumed to be modeled by an EFEM solution where the input is assumed to be the energy ?ow re?ected at the boundary of the subsystem that is driven.

10 Energy ?nite element method

305

The entire family of EFEM and EBEM methods makes up a simple, but complete complementary set of methods to traditional acoustical FEM and BEM methods. With this family of methods, the analyst is able to make dual use of models and predict structural acoustic response across the spectrum for low frequency to high frequency at a level of detail that is appropriate and useful to engineering applications.

References

1. Belov VD, Rybak SA (1975) Applicability of the transport equation in the one– dimensional wave–propagation problem. Journal of Soviet Physics–Acoustics 21:110– 114 2. Belov VD, Rybak SA, Tartakovskii BD (1977) Propagation of vibrational energy in absorbing structures. Journal of Soviet Physics–Acoustics 23:115–119 3. Bitsie F, Bernhard RJ (1998) Sensitivity calculations for structural–acoustic EFEM predictions. Noise Control Engineering Journal 46:91–96 4. Bouthier OM, Bernhard RJ (1995) Simple models of the energetics of transversely vibrating plates. Journal of Sound and Vibration 182:149–164 5. Cho PE (1993) Energy ?ow analysis of coupled structures. PhD Thesis, Purdue University, Herrick Laboratories, West Lafayette 6. Cho PE, Bernhard RJ (1998) Energy ?ow analysis of coupled beams. Journal of Sound and Vibration 211:593–605 7. Franzoni LP, Bliss DB, Rouse JW (2001) An acoustic boundary element method based on energy and intensity variables for predicting broadband sound ?elds. Journal of the Acoustical Society of America 110:3071–3080 8. Klos J (2001) Modeling cylindrical shell dynamics using energy ?ow methods: An analytical and experimental validation. MSME Thesis, School of Mechanical Engineering, Purdue University 9. Lyon RH, Dejong RG (1995) Statistical energy analysis. Butterworth–Heinemann, Boston, 2nd edition 10. Nefske DJ, Sung S (1989) Power ?ow ?nite element analysis of dynamic systems: Basic theory and application to beams. Journal of Vibration, Acoustics, Stress, and Reliability in Design 111:94–100 11. Segerlind, LJ (1984) Applied ?nite element analysis. John Wiley & Sons Inc., New York 12. Smith MJ (1997) A hybrid energy method for predicting high frequency vibrational response of point–loaded plates. Journal of Sound and Vibration 202:375–394 13. Versteeg HK, Malalasekera W (1995) An introduction to computational ?uid dynamics: the ?nite volume method. John Wiley & Sons Inc., New York 14. Vlahopoulos N, Wang A, Wu K (2005) An EFEA formulation for computing the structural response of complex structures. In: Proceedings of 2005 ASME International Mechanical Engineering Congress and Exposition, Orlando 15. Wang S (2000) High frequency energy ?ow analysis methods: Numerical implementation, applications, and veri?cation. PhD Thesis, Purdue University, Herrick Laboratories, West Lafayette 16. Wang S, Bernhard RJ (1999) Energy ?nite element method (EFEM) and statistical energy analysis (SEA) of a heavy equipment cab. In: Proceedings of SAE 99 Noise & Vibration Conference, Traverse City 443–450

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17. Wang S, Bernhard RJ (1998) The energy ?nite element method applied to a heavy equipment cab. In: Proceedings of Internoise 98, Christchurch, Paper No. 229 18. Wang S, Bernhard RJ (2002) Theory and applications of a simpli?ed energy ?nite element method and its similarity to SEA. Noise Control Engineering Journal 50:63–72 19. Wang S, Ebbitt G, Powell RE (2002) Development of a generic truck SEA model with EFEA–SEA hybrid exterior. In: Proceedings of the Internoise 2002, Dearborn 20. Wang A, Vlahopoulos N, Buehrle RD, Klos J (2005) Energy ?nite element analysis of the NASA aluminum test–bed cylinder. In: Proceedings of 2005 SAE Noise and Vibration Conference, Traverse City, SAE Paper No. 2005-01-2372 21. Wang A, Vlahopoulos N, Wu K (2004) Development of an energy boundary element formulation for computing high–frequency sound radiation from incoherent intensity boundary conditions. Journal of Sound and Vibration 278:413–436 22. Wang S, Xu H, Ebbitt G (2001) Sound package optimization using an EFEM0 –SEA model for a trimmed van. In: Proceedings of SAE 01 Noise & Vibration Conference, Traverse City 23. Wohlever JC, Bernhard RJ (1992) Mechanical energy ?ow models of rods and beams. Journal of Sound and Vibration 153:1–19

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