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1、Finite Elements in Analysis and Design 45 (2009) 456-462 Contents lists available at ScienceDirect FiniteElementsinAnalysisandDesign journal homepage: Simplified modelling of joints and beam-like structures for BIW optimization in a conceptphaseofthevehicledesignprocess D. Mundoa, R. Hadjitb, S. Don

2、dersb, M. Brughmansb, P. Masb, W. Desmetc aDepartment of Mechanical Engineering, University of Calabria, 87036 Arcavacata di Rende, Italy bLMS International, Interleuvenlaan 68, B-3001 Leuven, Belgium cDepartment of Mechanical Engineering, Katholieke Universiteit Leuven, Division PMA, B-3001, Leuven

3、, Belgium A R T I C L EI N F OA B S T R A C T Article history: Received 3 March 2008 Received in revised form 3 December 2008 Accepted 10 December 2008 Available online 7 February 2009 Keywords: Beam Joint Conceptual design NVH Vehicle body The paper proposes an engineering approach for the replacem

4、ent of beam-like structures and joints in a vehicle model. The final goal is to provide the designer with an effective methodology for creating a concept model of such automotive components, so that an NVH optimization of the body in white (BIW) can be performed at the earliest phases of the vehicle

5、 design process. The proposed replacement method- ology is based on the reduced beam and joint modelling approach, which involves a geometric analysis of beam-member cross-sections and a static analysis of joints. The first analysis aims at identifying the beam center nodes and computing the equival

6、ent beam properties. The second analysis produces a simplified model of a joint that connects three or more beam-members through a static reduction of the detailed joint FE model. In order to validate the proposed approach, an industrial case-study is presented, where beams and joints of the upper r

7、egion of a vehicles BIW are replaced by simplified models. Two static load-cases are defined to compare the original and the simplified model by evaluating the stiffness of the full vehicle under torsion and bending in accordance with the standards used by automotive original equipment manufac- ture

8、r (OEM) companies. A dynamic comparison between the two models, based on global frequencies and modal shapes of the full vehicle, is presented as well. 2009 Elsevier B.V. All rights reserved. 1. Introduction In highly competitive markets, design engineers face the chal- lenging problem of developing

9、 products, which must fulfil complex and even conflicting design criteria. In the field of automotive indus- try, the task of improving various functional performance attributes, such as safety, noise and vibrations, ecological impact etc., is made more and more difficult by the necessity of launchi

10、ng new products or renewing existing models in an increasingly short time frame. In order to make the complexity of the design criteria compatible with the necessity of reducing the time-to-market, predictive computer- aided engineering (CAE) methods must be already available in the early phases of

11、the design process. Traditional computer-aided design (CAD) software packages have a very limited applicability in early design stages, since they require detailed data of the vehicle. Besides, they are based on the traditional definition of geometry via points, lines and surfaces, thus making Corre

12、sponding author. Fax: +39984494673. E-mail address: d.mundounical.it (D. Mundo). 0168-874X/$-see front matter2009 Elsevier B.V. All rights reserved. doi:10.1016/j.finel.2008.12.003 the parameterization of models difficult and time-consuming 1. As a result, the experience of engineers is a key factor

13、 for the selection of proper structural concepts at the beginning of the design process. Recently, research efforts have been spent to enable designers to use CAE as a support in the conceptual phase of the design process, when functional performance targets are defined, while detailed ge- ometrical

14、 data are still unavailable. The objective is to improve the initial CAD design, hence shortening the design cycle 25. In the field of NVH and crashworthiness prediction, several con- cept modelling approaches have been proposed by researchers. They can be classified into three categories: methods b

15、ased on predecessor FE models, methods from scratch, and methods concurrent with CAD. Methods belonging to the first category, which includes mesh morphing and concept modification approaches 68, are used to de- sign a variant or incremental improvement of an existing vehicle model. By using a prede

16、cessor FE model, early CAE predictions can be performed to identify issues and to include possible countermea- sures already in the initial CAD design. If a new car concept is to be designed and a predecessor FE model is not available, methods “from scratch” can be used to support the design process

17、 during the early design phases. Two classes of meth- ods are distinguished. The first class is topology design optimization, D. Mundo et al. / Finite Elements in Analysis and Design 45 (2009) 456-462457 where material is eliminated from an initial admissible design domain in order to make the struc

18、ture lighter without violating functional requirements 911. Performance based on the opti- mized topological information is usually improved by optimizing shape and size. The second class of methods “from scratch”, known as functional layout design, aims at building a simplified concept model, consi

19、sting of beams, joints and panels, which represents the functional layout and which is used to predict the performance of the model 12. Methods concurrent with CAD are CAE tools available in an early phase of the design process. These methods provide simulation re- sults as soon as component-level C

20、AD models are available, while vehicle-level models are still unavailable 13. Among the methods based on predecessor FE models, the “reduced beam and joint modelling” approach has been recently proposed by Donders et al. 14 to improve the fundamental NVH behavior of a vehicle BIW. The proposed appro

21、ach creates a reduced modal model at the beam center nodes, to which beam elements and joint superelements can be added, thus enabling a concept modification of the body and an accurate prediction of dynamic NVH performance. The commercial software program LMS Virtual Lab. 15 includes a user-friendl

22、y implementation of the reduced beam and joint modelling approach. Design engineers can define a beam and joint layout, calculate the body reduced modal model and perform efficient design modification and optimization of the body beam-like sections and joint connections. In this paper, the reduced b

23、eam and joint modelling approach is employed to replace beams and joints of the predecessor FE model with concept models. After identifying the beam center nodes as the geometric center of the cross-sections, the equivalent beam properties are calculated through a geometric analysis and applied to s

24、implified beam elements that connect the beam center nodes. The stiffness parameters of thin-walled beams, as computed by means of a geometric approach, need a correction that takes into account section variations and discontinuities (holes, spot-welds, stiffeners) 16,17. For this purpose, proper co

25、rrection factors are defined and estimated for each beam-member by means of an iterative model updating procedure. In a next step, a simplified model of joints, connecting two or more beam-members, is then obtained through a static reduction of the detailed FE model of the joint. In order to validat

26、e the proposed approach, a case-study is pre- sented, in which beams and joints of the upper region of a vehicle BIW are replaced by simplified models. A static comparison between the original and the simplified model is performed by evaluating the static stiffness of the full FE vehicle BIW under t

27、orsion and bend- ing. A dynamic comparison between the two models, based on the global frequencies and mode shapes of the full vehicle, is performed as well. 2. The reduced beam and joint modelling approach The reduced beam and joint modelling approach is proposed by Donders et al. 14 for efficient

28、modification of beams and joints of a vehicle, based on the reduced modal model of the nominal vehicle. The basic idea is to identify the so-called beam center nodes, and to create a reduced modal model at these beam center nodes. Subsequently, the mass and stiffness properties of the structure are

29、modified by connecting the beam center nodes through simple beam elements and joint superelements. In this paper, simplified beam and joint models are created to completely replace the original FE model (without the necessity of the reduced modal model), so that an optimization of the vehicle can be

30、 performed in the early phase of conceptual design, when a detailed model of the structure is not yet available. yi zi xi x B.C.N. z y Fig. 1. Schematic representation of a beam end-section. In this section, an overview of the procedure that is used to es- timate the mass and stiffness properties of

31、 the simplified beam and joint models is provided. 2.1. Beam property estimation Beam-like members, i.e. structures for which the dimension in the longitudinal direction is much larger than the characteristic di- mension of the cross-sectional area, are the primary structural ele- ments in a BIW. Th

32、ey strongly influence the natural frequencies of the vehicle body. In the FE model of a vehicle, beam-like members are typically thin-walled structures, formed by shell elements. In order to replace the detailed mesh of such components by sim- plified beam elements, a number of beam cross-sections a

33、re consid- ered and the equivalent beam properties are computed for each of them. For this purpose the following procedure is implemented: (1) a cut node is selected in the region of the beam-member where an intersection plane is to be applied, (2) an axis system that defines the approximate beam di

34、rection and intersection plane is defined, (3) the primary members shell elements along the intersection plane are cut and analyzed to locate the beam center node in the geometric center of the original cross-section, (4) the following equivalent beam properties w.r.t. the beam center node are compu

35、ted: A: cross-section area; Ixx: torsional moment of inertia; Iyy, Izz: moments of inertia of area; and Iyz: product of inertia of area. Here, x denotes the beam direction, and the yz plane is the intersection plane, as shown in Fig. 1. For an arbitrary cross- section, the calculation of the propert

36、ies can be implemented by computing the equivalent beam properties for each shell element that belongs to the cross-section, according to the local principal axes (xi, yi, zi). Then, a transformation from the local axis system to that of the intersection plane (x, y, z) is performed. Finally, a summ

37、ation over all shell elements is performed to find the global properties for that cross-section. (5) the beam center node is connected to the surrounding mesh by means of interpolation relations (Nastran superelements RBE3). These relations are defined between each beam center node and a particular

38、node group, formed by all nodes of the shell ele- ments that are defined at the intersection plane at the consid- ered cross-section. Typically, along each primary beam-member a number of intersec- tion planes are defined, for which equivalent beam properties are computed. The entire beam member can

39、 then be represented as a series of linear beam elements taken from a standard FE library. An example is shown in Fig. 2, where both the original detailed and the simplified FE model of B-pillars of a vehicle BIW are represented. 458D. Mundo et al. / Finite Elements in Analysis and Design 45 (2009)

40、456-462 Fig. 2. (a) Original and (b) conceptual models of a BIW B-pillars. Fig. 3. Original FE model of a joint group, extracted from the vehicle model for static reduction. 2.2. Joint property estimation Complementary to the simplified beam modelling approach de- scribed in Section 2.1, a procedure

41、 for simplifying joints connecting beam-like structural members in a vehicle body is proposed. After evaluating the equivalent beam properties of all beam-members con- nected by the joint, a joint group is created that includes the inter- polation elements to the beam center nodes at the joint ends

42、15. In Fig. 3 an example is shown, in which the mesh of the joint that con- nects the left B-pillar of the vehicle to the roof-rails is extracted from the rest of the vehicle body. For this isolated joint model, Guyan re- duction is used to calculate a small-sized representation of the joint. Guyan

43、reduction 18, also known as static condensation, is a method to reduce the finite element stiffness and mass matrices of structures. For an arbitrary structure, the basic static FE matrix equation is given by K x = F(1) where K is the stiffness matrix, F and x are the force and the dis- placement ve

44、ctors, respectively. By identifying ntboundary degrees of freedom (DOFs), which must be retained in the solution, and no internal DOFs, which are to be removed by static condensation, the system of Eq. (1) can be partitioned as follows: ?K oo Kot KtoKtt ? ?x o xt ? = ?F o Ft ? (2) where subscripts t

45、 and o are used to designate the boundary and the internal DOFs, respectively. From the first line of Eq. (2), the internal displacement vector can be determined as xo= K1 oo(Fo Kot xt) (3) By introducing the static reduction matrix Got=K1 ooKotand substi- tuting Eq. (3) into the second line of Eq.

46、(2), the following equation is obtained: Ktt,red xt= Ft,red(4) where Ft,red=Ft+GT otFo is the reduced loading vector, while Ktt,red= KtoGot+ Kttis the ntx ntreduced stiffness matrix. Physically this matrix represents the stiffness values between each pair of boundary DOFs. This way, the stiffness of

47、 the structure has been condensed to the boundary DOFs. The same transformation can be used to condense the mass ma- trix on the boundary DOFs, to obtain a reduced system also for dy- namic analyses. However, while exact for the stiffness matrix, the Guyan reduction is an approximation for the mass

48、matrix. By re- ducing the mass matrix, it is assumed for the considered structure that inertia forces on internal DOFs are less important than elastic forces transmitted by the boundary DOFs. This is true for very stiff components or in cases where local dynamic effects can be ignored. Therefore, th

49、e accuracy of the result is case dependent. For each isolated joint model, a Guyan reduction is performed, with the DOF of the joints end nodes (i.e. beam center nodes) as the boundary DOFs to be retained in the solution. The FE model of the joint is thus reduced to a small superelement, consisting

50、of a reduced stiffness and mass matrix. For typical automotive joints, the stiffness relations between the end points of the joint have a much stronger influence on the global body behavior than the exact distribution of mass on the joint. For this reason, Guyan reduction of the joint structure to i

51、ts joint end-nodes (i.e. beam center nodes) seems an appropriate choice to create a small-sized representation of the actual joint 14. D. Mundo et al. / Finite Elements in Analysis and Design 45 (2009) 456-462459 3. Case-study 3.1. Model description Fig. 4 shows an industrial BIW model, consisting o

52、f 123 panels that are modelled with linear shell elements. The constituent panels are assembled by means of about 3000 spot-weld connections 19, which are represented in the FE model by means of Hexa solid ele- ments15.Inordertovalidatethereducedbeamandjointmodelling approach, as described in the pr

53、evious section, a group of beam-like structures, labelled in Fig. 4 as B1.B5, are selected and replaced by equivalent simple beams. In total, 10 beams are selected for the replacement, namely the A and B-pillars and the longitudinal and transversal roof-rails. Four joints, symmetrically arranged w.r

54、.t. the longitudinal plane of the vehicle, connecting these beams are labelled in Fig. 4 as J1, J2, J3 and J4,are statically reduced. Fig. 5 shows the simplified BIW model, where the detailed shell models of the beam-like structures have been replaced by simple two-node beam elements. The number and

55、 length have been selected based on the geometric characteris- tics (i.e. length and cross-section variations) of the original mesh. The original FE joint models have been removed from the BIW FE model, and the joints have been represented by static superelements (i.e., the equivalent mass and stiff

56、ness matrices of each joint). 3.2. Static comparison To validate the proposed approach, static and dynamic indica- tors of the full vehicle performance are considered. These indicators are evaluated for both the original BIW model and the simplified (or conceptual) model. To assess the static behavi

57、or, the torsional and bending stiffness of the BIW are calculated. The body is clamped at the rear suspensions, while static vertical forces are applied at the Fig. 4. Original FE model of the BIW. Fig. 5. Conceptual FE model of the BIW. The original meshes of 10 beam-members and four joints are rep

58、laced by simplified beam elements and joint superelements. Fig. 6. Static load-cases defined to estimate the BIW stiffness under (a) torsion and (b) bending. front suspensions (points A and B in Fig. 6). Based on the estimation of the vertical displacements vAand vBat the excitation points, the bend

59、ing and torsion deflection angles?band?tare determined as ?b= arctan ?v A+ vB 2L ? (5) ?t= arctan ?v A vB W ? (6) where L and W denote the wheelbase and the width of the car, respectively, measured at the front suspension points. Based on the torsional deflection?t, the torsional stiffness Ktis determined as Kt= M ?t (7) where M = F W is the moment applied at the front suspension, resulting from two oppositely oriented forces F. Similarly, the bending stiffness Kbis determined from?bas Kb= 2FL ?b (8) where F is the vertical force applied at the frontal suspension loca- tion.