Dynamic FE-based method for concept modelling of vehicle ...past.isma-isaac.be/downloads/isma2012/papers/isma2012_0352.pdf · Dynamic FE-based method for concept modelling of vehicle

Dynamic FE-based method for concept modelling of vehicle beam-like structures

G. De Gaetano1, F.I. Cosco2, C. Maletta1, D. Mundo1, S. Donders3 1 University of Calabria, Department of Mechanical Engineering Ponte Pietro Bucci,46/C, 87036 Rende, Italy e-mail: [email protected], [email protected], [email protected] 2 G&G Design and Engineering, via G. Barrio, 87100 Cosenza, Italy e-mail: [email protected] 3 Simulation Division - LMS International Interleuvenlaan 68, 3001 Leuven, Belgium e-mail: [email protected]

Abstract Vehicle body structures are characterized by load-carrying thin-walled beam-members, typically formed by spot-welded panels, with sections variations and discontinuities that highly influence both the static and the dynamic behaviour of the entire structure. Vehicle body concept modelling is an active field of research. Different methodologies are available, either based on the geometric analysis of the spatial mass distribution at each relevant cross-section or on static finite element (FE) based characterization of the beam properties. This paper presents a new methodology for estimating the cross-section properties of automotive structures with a beam-like behaviour. It is based on a dynamic FE approach, which allows estimating the stiffness characteristics (e.g. quadratic moments of inertia, torsional modulus, etc.) of equivalent 1D beam elements, using the natural frequencies estimated by means of a modal analysis on the original 3D FE model of the structure. The proposed method has been validated through an application case, comprising the analysis of a thin-walled beam with double-symmetric cross-section, formed by two spot-welded panels.

1 Introduction

In the literature of the last decade the term concept modelling has referred to a growing variety of modelling techniques, which are exploitable as predictive tools during the earliest phases of product design, when geometric data are limited or even unavailable. In the automotive industry, such modelling techniques are crucial for reducing the time-to-market, since they enable designers to anticipate the optimization of several vehicle performance attributes (NVH, dynamics, safety, energy efficiency, etc.) already at the beginning of design process. In the field of vehicle body engineering, concept modelling techniques commonly aim at obtaining a simplified model of the vehicle Body-in-White (BIW). The static and dynamic behaviour of the BIW are dominated by the stiffness properties of primary structural elements, such as beam-like members. In a typical vehicle body, these load-carrying members are thin-walled structures, formed by two or more curved panels connected to each other by means of spot welds. Therefore, the vehicle body concept modelling consists of replacing the detailed 3D FE models of beam members with simplified models, also referred to as equivalent 1D concept beams. The stiffness properties of these beams are computed by applying proper analytical or numerical procedures to a set of cross-sections extracted from the original mesh. Methods reported in the literature can be grouped in two main classes: the static FE-based approaches and the geometric methods.

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Static FE-based concept modelling methods rely on running static FE analyses of the original structure [1-2]. A set of static load cases is generated by applying bending, torsion and axial loads at the end sections of each beam segment. Here, a segment refers to any portion of the beam with constant geometric and structural properties. At each cross section, a central node is created, which is connected to the other nodes by means of multi-point constraint methods. In this way, external and reaction loads are applied directly to the central node and transferred to the rest of the beam structure through the connection elements. Finally, the stiffness properties of the concept beam are estimated by applying the linear elastic load-deformation relationship of the equivalent beam model, starting from the static deformations predicted for the original structure. The advantage of this approach is that it can be easily applied to any beam, whatever the shape of its cross-section, always providing results that can be sufficiently accurate to steer the initial design choices. However, the addition of auxiliary connection elements within the structure prevents the end sections from deforming. Such a modification of the original mesh may result in a significant overestimation of the beam stiffness properties, especially for automotive structures, which are usually formed by spot welded thin-walled panels. Geometric concept modelling methods are based on a geometric analysis of the beam cross-sections [3-5]. The mass and stiffness properties of the equivalent 1D beam are calculated by analysing the mass distribution along the section and considering whether the section has a single or a multi-connected closed shape. These methods are characterized by their simple implementation and application, but in the specific case of vehicle beam-like structures, typically constituted by spot welded profiles, the assumption of closed cross-section rarely corresponds to real cases and produces an overestimation of the equivalent beam stiffness properties. To remedy this overestimation, correction parameters have been proposed in the literature, in order to improve the accuracy of the equivalent models [6-7]. In this paper, a dynamic FE-based method for the estimation of cross-section properties in structures with a beam-like behaviour is proposed, overcoming the limitations of both the approaches mentioned above. The method consists of two steps:

1. Firstly, the natural frequencies of the given beam-structure are estimated by means of a modal analysis of the detailed 3D FE model in free-free conditions.

2. Secondly, the flexural and torsional frequencies are derived as a function of the cross-sectional stiffness properties, using the differential equations of beam vibrations [8-9]. In particular, the modal model of thin-walled beams is used to derive the properties of the equivalent 1D concept beam from the natural frequencies previously estimated.

The main advantage of the method is that it takes into account all possible discontinuities and variations that may occur along a beam and that affect its stiffness, especially under torsional loads. In facts, in both static and dynamic comparisons between the original and the simplified models, the dynamic method results in an increased accuracy as compared to the both the geometric and the static FE-based methods. The rest of the paper is organized as follows: section 2 describes the main steps of the dynamic FE-based approach and its mathematical background; section 3 deals with the definition of the spot welded beam model that is used for method validation purposes. The validation results are reported in section 4, demonstrating the improvements that can be obtained with the proposed method. Section 5 concludes the paper, by reviewing the obtained results and providing an outlook on the foreseen next steps.

2 The dynamic FE-based method

The focus of this paper is to develop a dynamic FE-based method for concept modelling of vehicle beam-like structures. This is motivated by the need to develop vehicle concept models that can accurately predict the dynamic behaviour of the vehicle body in a concept design phase. The proposed method relies on the analytical Timoshenko model to describe the flexural and torsional vibrations of prismatic beams. For sake of simplicity, in this work we limited the methodology to consider only beams with double-symmetric sections, where the cross-section centre of gravity is coincident with the shear centre. This assumption leads to a set of uncoupled differential equations for flexural and torsional vibrations in free-free conditions. We implemented a numerical procedure that estimates the stiffness properties of the

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equivalent 1D concept model by using the modal equations of the beam and the targeted values of natural frequencies, which are previously calculated through an FE modal analysis of the original 3D model.

2.1 Analytical Model

According to the Timoshenko beam model, let’s define a Cartesian reference frame (x, y, z), with the origin at the centre of gravity of the left end section of the beam, where x is the neutral axis, y and z are the principal and the secondary bending directions respectively. By setting the external excitations to zero in the flexural and torsional equations of motion, we get the following uncoupled ordinary differential equations, governing the modal behaviour of the beam [8]:

022

22

4

4

bfb

s

bf

bb mw

dx

wd

GA

EIm

dx

wdEI (1)

0022

2

4

4

tttw Jdx

dGI

dx

dEI (2)

where equations (1) and (2) refer to the flexural and the torsional vibrations, respectively. The physical meaning of each parameter is given in Table 1.

As is the shear area Iw is the warping modulus

E is the young’s modulus 0tJ is the polar moment of inertia about the centre of gravity

G is the shear modulus ψ is the twist angle

m is the distributed mass w is the beam deflection

Ib is the cross-section moment of inertia ωf is a generic flexural natural frequency

It is the torsional modulus ωt is a generic torsional natural frequency

Table 1: Nomenclature

By solving the characteristic equation for flexural vibrations, the following non-linear frequency equations are obtained [9]:

0tantanh ll (3)

0tanhtan ll (4)

where equations (3) and (4) refers to symmetric and anti-symmetric modes respectively. In both equations l is the half length of the beam, whereas the functions and are defined as follows:

bba 24 1 (5)

bba 24 1 (6)

Here, the coefficients a and b are given by:

b

f

EI

ma

2 (7)

VEHICLE CONCEPT MODELLING 3727

s

bf

GA

mEIb

2

(8)

By solving the characteristic equation for torsional vibrations, the following equation is obtained to calculate the eigenfrequencies:

t

w

t

tnt

GI

EI

L

n

J

GI

L

n2

0, 1

,...2,1,0n (9)

where frequencies for both symmetric and anti-symmetric modes are considered, depending on whether the value of n is even or odd, respectively.

2.2 Numerical method

By assuming that the material properties of beam are known, equations (3) to (9) are adopted as the mathematical foundation, which is used as a basis to develop two different numerical methods for the flexural and the torsional modes, respectively. The method developed for the flexural vibrations is based on an unconstrained nonlinear minimization algorithm where Ib and As are the design parameters that will be optimized. The objective function is defined as the modulus of a vector, for which the components are computed by evaluating the left side of equations (3) or (4) for the first three flexural frequencies, as obtained from the 3D FE model of the beam structure. The optimal values of Ib and As are computed after setting initial guess values and adopting the Nelder-Mead simplex algorithm as described in [10]. The same algorithm is applied twice, once for each bending direction. Once the optimal values of the shear areas Asy and Asz are computed, the shear factors of the conceptual model in the two bending directions are calculated as:

t

sy

yA

AK (10)

t

szz

A

AK (11)

where At is the total cross-sectional area. For the torsional vibrations, a similar approach is used. It and Iw are considered as the design parameters to be optimized, and the objective function is defined by equation (9). The goal is to minimize the squared sum of the differences between the frequencies reference vector, obtained again from the initial dynamic simulation on the 3D FE model, and the frequencies vector iteratively computed by applying equation (9). In particular, the value of the polar moment of inertia, 0

tJ , is assumed to be given by the sum of the two moments of inertia that were previously estimated in the flexural analysis cases, multiplied by the density of material. Note that for the estimation of the beam section properties only the first two torsional frequencies and the first three flexural frequencies in each bending plane have been used. The motivation for this is the need to limit the effect of the local modes on the accuracy of the model, which tend to yield an overestimate in the eigenfrequency prediction.

3 Application Case

This section describes an application model, on which the proposed methodology has been applied and validated. The geometry of the reference beam structure is described in section 3.1. Thereafter, in section


3.2 a sensitivity analysis is performed, aimed at understanding the influence of the spot weld layout on the dynamic behaviour of the beam.

3.1 Model Description

According to the double symmetry assumption stated in section 2, a thin-walled beam with rectangular cross-section has been considered to define a validation case. The model consists of two thin-walled steel beams, with identical C-shaped section, whose geometry and dimensions are shown in figure 1a). Each beam has been meshed by 4-node shell elements, resulting in a model with 48000 degrees of freedom (DOFs). The two parts have then been connected by a set of equally spaced welding points along each of the longitudinal walls, resulting into a two-meter long girder. For the purpose of model preparation, the Structures environment of LMS Virtual.Lab software [11] has been used in order to create each welding point as a Hexa solid element, connected to both flanges by interpolation elements. Figure 1b) shows a part of the final FE model, and the simulations have been carried out under the assumption of homogeneous and isotropic material by using the material properties of steel:

Elasticity modulus: E = 210000 MPa ; Poisson’s ratio: ν = 0.3 ; Mass density: ρ = 7.9 x 10-9 ton/mm3 .

a) b)

Figure 1: Application model: cross-section geometry (a) and FE model (b)

3.2 Dynamic sensitivity analysis with respect to the welding layout

In order to quantify to what extent the spot welds distribution may influence the dynamic behaviour of a beam-like structure, in particular the first flexural and torsional natural frequencies, a sensitivity analysis has been carried out, relying on the simulation model described in the previous sub-section. Four different cases were analysed, with a distance between two consecutive spot welds of 10, 50, 100 and 200 mm, and compared to the simulation results obtained on a beam with closed rectangular section, which has been considered as the reference case. The computed natural frequencies are given in Table 2, whereas figure 2 shows the relative variation of each natural frequency of the four models with respect to the values in the reference case.


These results show that the distance between two spot welded sections significantly affects the dynamic behaviour of the beam, since natural frequencies increase with the increasing spot weld density. For both flexural and torsional modes, the variation of frequencies values increases moving towards higher order modes. In particular, for the bending case this variation is larger on the x-y plane than on the x-z plane, because the spot welds have a larger stiffness in their axial direction, parallel to the x-z plane, than in their tangential direction. On the other side, the torsion modes show even a higher sensitivity, with the frequency variations for the first torsion mode ranging from 2% for the model with higher density of welding points to more than 60% for the model with 200 mm distance between two spot welds, due to the decreased torsion stiffness caused by the transition from a closed to a partially open section. The overall results of this sensitivity analysis suggest that the natural frequencies of spot welded beam-like structures may differ significantly from those of thin-walled beam with a closed rectangular section, especially at high frequencies and for torsional modes.

Natural frequencies (Hz)

Mode Reference model

(closed section)

Distance between spot welds

10 mm 50 mm 100 mm 200 mm

x-z

flexural

plane

1 76.63 76.63 76.54 76.51 76.47 2 208.60 208.17 207.65 207.23 206.38 3 399.49 396.04 393.24 390.18 383.16

x-y

flexural

plane

1 90.54 90.37 90.29 89.69 87.99 2 246.54 246.06 243.80 238.01 223.77 3 473.55 472.54 462.58 440.81 396.51

torsional 1 637.34 621.41 530.07 388 251.64 2 1392.10 1388.60 1139.30 848.46 575.81

Table 2: Natural frequencies estimated for the reference model (beam with closed rectangular section) and for the different spot welded models

Figure 2: Frequency variations between each spot welded model and the reference model


4 Method validation

As already stated in the previous section, the dynamic behaviour of spot welded beam-like structures is highly affected by the distance between spot welds, especially for the torsional modes. This result suggests that the geometrical method could be inaccurate for this kind of applications. For a geometric method, it is assumed that the beam has a closed cross-section, such that the spot weld layout is not taken into account. In contrast, both the static and the dynamic FE-based modelling approaches take into account the spot weld layout by including the impact of the spot welds on the static and dynamic stiffness, respectively. This section presents a comparative analysis, aimed at assessing the accuracy of the proposed dynamic method as compared to both the geometrical and the FE-static methods. For this purpose, the third model presented in the previous section has been analysed, which is characterized by a distribution of welded sections equally spaced by 100 mm, a typical layout in automotive beams. As shown in figure 2, this spot welded model differs significantly from the beam with a closed cross-section. When comparing the two models, differences of about 7% and 40% are obtained for the bending and torsion modes, respectively. Figures 3, 4 and 5 show the modal shapes corresponding to the eight natural frequencies, the first three in each flexural plane and the first two for torsional vibrations, which were used to estimate all the stiffness properties of the equivalent concept model trough the dynamic FE-based method illustrated in section 2. In particular, both torsion modes show significant local deformations, in the regions of the model around the spot welds, in opposite to the more regular shape of the flexural modes. This increased local flexibility of the model is expected to be responsible for the high differences between the spot welded model and the

a) b) c)

Figure 3: First (a), second (b) and third (c) vibration modes in x-z flexural plane

a) b) c)

Figure 4: First (a), second (b) and third (c) vibration modes in x-y flexural plane

a) b)

Figure 5: First (a) and second (b) torsion modes


closed-section beam in terms of torsion frequencies. The natural frequencies of the full FE model have been determined by modal analysis and recognized through graphical representation. Then the two shear factors, Ky and Kz, as well as the stiffness properties of the equivalent concept model were computed by using the numerical procedures described in section 2.2. The values estimated by the dynamic FE-based method for each parameter are listed in table 3. For comparison, the results obtained with the geometric method and the static FE-based method are shown as well. Here, it is noted that the geometric method has been executed considering a thin-walled beam with rectangular cross-section with the same dimensions of the spot welded cross-section. The static FE-based method has been implemented by defining a proper set of static load cases for the 3D beam model. Following the approach suggested by Corn et al. [1], the structure has been clamped at one end section and loaded at the other end, where a central node was created at the section centre of gravity and connected to the other nodes of the same section through rigid spiders. Finally, the equivalent concept beam parameters were estimated by first measuring the displacements and the rotations resulting after applying a set of bending and torsional loads on the centre node, followed by an inversion of the linear static relations based on the Timoshenko model [12]. In particular, for bending properties the following equations have been used:

bEI

FLL

2

2

(12)

sb GA

FL

EI

FLLv

3

3

(13)

where the beam rotation (L) and deflection v(L) at the free end section are expressed as a function of the unknown parameters, Ib and As, the applied load F, the beam length L and the material elastic moduli E and G. Similarly, the torsional modulus of the equivalent beam has been estimated by applying a torque T on the central node along the longitudinal direction, and then inverting the following equation:

tGI

TLL

(14)

where ϑ(L) is the angle of twist of the free end section of the beam. Table 3 summarizes the parameters of the concept beam as estimated by the three different methods. Note that the larger differences among FE-based methods, both the static and the dynamic one, and the geometric method regard the shear factors and the torsion parameters.

Parameters

Concept modelling methods Geometric FE-based static FE-based dynamic

Iy (mm4) 50695 50745.9 51177

Iz (mm4) 70865 70827.7 70920

Ky 0.5444 0.1258 0.1260 Kz 0.4333 0.4233 0.1028

It (mm4) 88888.9 35115.3 26891

Iw (mm6) 0.0 0.0 2.921e+08

Table 3: Equivalent beam properties estimated by different methods

In particular, the dynamic FE-based method provided an estimate of the shear factors that was about four times lower than the estimate provided by the geometric method. The same trend has been observed for the torsion modulus, with an estimated value more than 3 times lower than the value calculated for the


closed rectangular section. In particular, the main difference between the dynamic method and the other two approaches concerns the warping modulus, for which the estimated value is not equal to zero, as assumed by the other two methods. This proves that the dynamic method estimates the equivalent beam torsion-properties without assuming a constant closed cross-section along the entire beam [13]. All the parameter values listed in Table 3 have been used to define three different concept models of the beam, formed by 20 one-dimensional, 100 mm long beam elements (1D model). Each beam element has 2 nodes, for a total of 126 DOFs for each concept model. This number is almost three orders of magnitude lower than in the original 3D model, which results in a significant reduction of the computational time need for static or dynamic FE simulations.

4.1 Dynamic Validation

A modal analysis has been run for each of the three different concept FE models. The obtained results have been used to assess the accuracy of the models for predicting the modal behaviour of the original 3D

Mode

Natural frequencies (Hz)

3D Shell

Model

Geometric

1D Beam

Model

FE Static 1D

Beam Model

FE Dynamic

1D Beam

Model

x-z

flexural

plane

1 76.51 76.83 76.86 76.70 2 207.23 210.31 210.37 206.53 3 390.18 407.97 407.98 391.26

x-y

flexural

plane

1 89.69 90.81 89.93 89.74 2 238.01 248.39 240.07 237.96 3 440.81 481.29 449.57 441.74

torsional 1 388 684.26 430.05 384.08 2 848.46 1372.80 862.76 820.53

Table 4: Natural frequencies of simplified beam models

Figure 6: Accuracy of the three concept models in terms of frequency predictions


beam structure. The predicted flexural and torsional natural frequencies are listed in Table 4, together with the frequencies of the original shell model. Figure 6 shows the predictive accuracy of each concept model, estimated as the percentage difference between the eigenfrequencies of each 1D beam model and the values predicted by the 3D model. It is clear that the first bending frequency in each flexural plane have been accurately predicted by all concept models. On the contrary, the geometric and the static FE method yield an increasing prediction error for the higher order modes, whereas the dynamic FE model results in accurate predictions (as compared to the 3D shell model) for all of the modes in the evaluated range. The most significant improvement obtained by the dynamic FE method is clearly related to the prediction of torsion frequencies, for which a maximum difference of 3.2% is noticed. This result is 3 times and 25 times more accurate than the static FE-based and the geometric method, respectively.

4.2 Static Validation

A further validation of the proposed dynamic FE-based method has been achieved by comparing the static deformation provided by both the concept and the 3D model under static bending loading conditions. For this purpose, both models have been clamped at one end section and loaded by a resultant vertical force of 220N at the other end section, as shown in figure 7. In order to estimate the deformation of the beam

Figure 7: Bending load case used for the static validation of the model

Figure 8: Static deflection of the beam centre line as predicted by

the 3D model (solid blue line) and by the 1D concept model (red markers)


centre line in the 3D model, 10 equally-spaced central nodes have been created along the beam longitudinal direction and connected to the nodes of each cross section by using interpolation spiders [14]. The results of the static simulations are shown in figure 8, in which the vertical displacements of the centre nodes in the two models are shown. It is noted that the concept model provides an excellent estimate of the structure deformation under static bending, resulting in an approximation error between the two predicted displacements at the loaded section with a value lower than 0.1%.

5 Conclusions

In this paper, a new methodology for the estimation of the stiffness properties of automotive beam-like structures has been presented and validated. The presented methodology allows defining concept beam models that can reproduce the static and the dynamic behaviour of spot welded thin-walled automotive beam models, with an increased accuracy as compared to concept modelling methods reported in the state of the art, based on either geometric or static FE-based approaches. A sensitivity analysis has been performed, which demonstrated that the dynamic behaviour of spot welded beam-like structures differs significantly from the behaviour of the corresponding closed section. This analysis provided high errors for the values of the natural frequencies, especially for the torsional frequencies. The proposed method was validated by analysing an application case, consisting of a spot welded beam composed by two C-shaped panels, with a spot weld layout that is representative for automotive applications. By comparing the natural frequencies of the 3D model with the values predicted by the three concept models, the dynamic FE-based method proved to be much more accurate than both the geometric and the static FE-based methods. A static validation of the method has been provided as well, which has shown that the static flexural behaviour of the concept beam is very close to the behaviour of the 3D reference model. The next steps of this research will comprise the extension of the proposed methodology to beams with general cross-section: by removing the double-symmetry assumption, a coupling of flexural and torsional vibrations, and therefore of the frequency equations, can be obtained. Another important step will be the application of this methodology to beams with variable section properties in the longitudinal direction.

Acknowledgements

We gratefully acknowledge the European Commission for their support of the Marie Curie IAPP project “INTERACTIVE” (Innovative Concept Modelling Techniques for Multi-Attribute Optimization of Active Vehicles), with contract number 285808; see http://www.fp7interactive.eu/.

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