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This paper presents a comparison between the experimental investigation and the Finite Element (FE) modal analysis of an automotive rear subframe. A modal correlation between the experimental data and the forecasts is performed. The present numerical model constitutes a predictive methodology able to forecast the experimental dynamic behaviour of the structure. The actual structure is excited with impact hammers and the modal response of the subframe is collected and evaluated by the PolyMAX algorithm. Both the FE model and the structural performance of the subframe are defined according to the Ferrari S.p.A. internal regulations. In addition, a novel modelling technique for welded joints is proposed that represents an extension of ACM2 approach, formulated for spot weld joints in dynamic analysis. Therefore, the Modal Assurance Criterion (MAC) is considered the optimal comparison index for the numerical-experimental correlation. In conclusion, a good numerical-experimental agreement from 50 Hz up to 500 Hz has been achieved by monitoring various dynamic parameters such as the natural frequencies, the mode shapes, and frequency response functions (FRFs) of the structure that represent a validation of this FE model for structural dynamic applications.

The employment of Finite Element (FE) models to predict the dynamic properties of a vehicle has continuously become more important in modern automotive industries. Whenever there is a new design or modification of an existing one, the structural dynamic properties of the car should be examined to fulfil some criteria proposed by the industry itself before the product can be launched on the market.

The traditional methodology for evaluating the structural dynamic properties of a vehicle is to perform various dynamic tests on prototypes of the product and to demonstrate their capacity to withstand these tests. Until the experimental results show that the prototypes can comply with the relevant criteria, the component has to be redesigned and another design-test loop must be followed. In this design-test-redesign loop, the higher percentage of time and financial resources is spent in producing prototypes and performing tests.

With the growing capabilities of computing techniques, and the strength of the competition between companies, FE model predictions are used more and more frequently to substitute practical dynamic test data. Furthermore, the FE modelling technique may also be used to predict the dynamic response of structures when working beyond a limit situation that makes the simulations by experiment extremely difficult, if not impossible. All of these results depend on the accuracy of FE model predictions.The validation of FE models and their capability to predict the dynamic behaviour of the structures are crucial topics for industrial purposes and especially for aerospace and automotive applications.

In 1990s, Baker [

Brughmans et al. [

Schedlinski et al. [

In addition to the previous works, [

The analyzed structure is an aluminium rear subframe that can be disassembled from the chassis; it is made of eleven extruded beams, plates, gussets, and two casting components, as shown in Figure

Aluminium rear subframe.

As pointed out by Schwarz and Richardson [

Structure test to obtain FRF measurements

FRF curve fitting to extract experimental modal parameters

A fast and convenient way to find the modes of a structure is impact testing. It was developed in the 1970’s and it has become the most popular modal testing used for evaluating the modal properties of a structure. The equipment required to perform an impact test is

Impact hammer

Accelerometers

FFT analyzer

Postprocessing modal software

Figure

EMA through impact test.

The modal parameter estimation is obtained from a set of frequency response function (FRF) measurements. The FRF, as explained in [

The modal test was performed at the NVH experimental department of Ferrari S.p.A. The structure was suspended by four soft elastic bungees in order to simulate the free-free condition; this condition means that the structure is not connected to the ground at any of its coordinates and it is, in effect, freely suspended in space. In this condition, the structure will exhibit rigid body modes, which are determined merely by its mass and inertia properties. It had been verified in a pretest phase that the suspension system did not interfere with the modes of vibration of the structure. A single impact point on the lower side of the left casting in the

Test geometry of the rear subframe.

Parameter estimation techniques for modal analysis are based on the extraction of natural frequency, the damping, and the mode shapes from the experimental data, which is in a processed form such as frequency response functions (FRFs). Figure

Plots of FRFs of the 35 reference points.

The estimation method used in this phase was PolyMAX (Peeters et al. [

Once the coefficients

The mode shapes could be determined by considering pole-residue model:

[LR] and [UR] are the lower and upper residuals which model the influence of the out-of-band modes.

Estimated poles are calculated from (

Figure

Natural frequencies (scaled) from EMA.

Mode | Frequency (scaled) [Hz] | Damping ratio (%) |
---|---|---|

1 | 0.244 | 0.20 |

2 | 0.254 | 0.35 |

3 | 0.519 | 0.04 |

4 | 0.623 | 0.08 |

5 | 0.834 | 0.08 |

6 | 0.849 | 0.03 |

7 | 1.023 | 0.09 |

8 | 1.037 | 0.07 |

9 | 1.236 | 0.13 |

Stabilization diagram for PolyMAX method; the blue curve represents the sum of FRFs.

The FE modelling techniques used for this study followed the internal criteria of the CAE Department of Ferrari S.p.A.

The extruded beams were modelled according to shell formulation using CQUAD4 and CTRIA3 elements. The castings were modelled with tridimensional elements; second-order CTETRA were used in order to compensate the spatial discretization. The description of the Finite Element model of welded joints is widely discussed in the next paragraph. Testing devices (e.g., suspension “biscuits” and gearbox mounting “clocks”) were added to the FE model in order to accurately reproduce the testing setup. They were connected to the structure through CBAR elements. Nonstructural mass NSML1 was applied to align the numerical mass to the weight of the physically tested structure, taking into account the mass added from coating process.

Figure

FE model of rear subframe.

The FE approach for modelling the welded joints is a relevant research topic for the present study. There are two different techniques for welding models: for stress analysis and for stiffness-based analysis. Obviously, the choice of the modelling technique depends on the aim of FE analysis performed. In literature [

The novel approach used in this paper is to extend the ACM2, proposed for spot welds, to MIG welding. A row of CHEXA elements is created for the body of welded joints and numerous RBE2/RBE3 elements for the head, as shown in Figure

Details of ACM2 extended method: CHEXA for the body of the welded joint and RBE2/RBE3 for the head.

This approach allows the development of a FE model which is useful both for stress analysis and for structural dynamic analysis.

Figure

Details of MIG welding on the structure: (a) the actual welded joint, (b) the ACM2 approach.

The FE model consists of 453852 nodes and 379708 elements. The structure was analyzed in free-free conditions, so six clear rigid body modes have been expected in the results. The results of the FE modal analysis, scaled, from 50 Hz up to 500 Hz are listed in Table

Natural frequencies (scaled) from FEA.

Mode | Frequency (scaled) [Hz] |
---|---|

7 | 0.244 |

8 | 0.252 |

9 | 0.503 |

10 | 0.612 |

11 | 0.817 |

12 | 0.833 |

13 | 1.000 |

14 | 1.019 |

15 | 1.234 |

A comparison between the novel ACM2 approach and the standard rigid one-dimensional elements has been performed. Figure

Details of MIG welding models: (a) the ACM2 approach, (b) the rigid elements approach.

Table

Comparison between natural frequencies (scaled) from FEA using different methods.

Mode | ACM2 method | Rigid elements method | Difference (%) |
---|---|---|---|

Scaled freq. [Hz] | Scaled freq. [Hz] | ||

7 | 0.244 | 0.237 | 2.742 |

8 | 0.252 | 0.249 | 1.297 |

9 | 0.503 | 0.500 | 0.609 |

10 | 0.612 | 0.617 | −0.896 |

11 | 0.817 | 0.787 | 3.630 |

12 | 0.833 | 0.820 | 1.610 |

13 | 1.000 | 0.978 | 2.193 |

14 | 1.019 | 1.019 | 0.029 |

15 | 1.234 | 1.226 | 0.655 |

The FE model using ACM2 method for MIG welded joints will be used for experimental-numerical correlation activity and FEA results will refer to it.

The correlation phase is focused on comparing, understanding, and evaluating the correlation between test and FE data. A modal based index, which is used for comparing experimental and numerical modal shapes, is Modal Assurance Criterion (MAC). MAC was originally developed for orthogonality check and, in the late 1970s, was proposed by Allemang and Brown [

Generally, values of MAC should be above 0.7 for representing a good correlation. It is also worth noting that every degree of freedom (DOF) gives a contribution to the MAC index. So, relatively considerable in-phase displacements of a DOF give a positive contribution to the correlation; relatively considerable out-of-phase displacements give a negative contribution; and relatively small displacements give less important contributions.

The correlation results have been evaluated comparing the natural frequency values and the error percentages, the mode shapes, and the MAC matrix.

Plotting the FEA-EMA values of natural frequencies, the theoretical best fit is represented by a 45-degree line. The natural frequency plot in Figure

Natural frequency pairs plot.

As pointed out in Figure

Absolute percentage error histogram.

Figure

Mode shapes for the first pair of modes: it represents the first torsional mode: (a) experimental model on the left, (b) FE model.

In addition, to a simple visual analysis of each mode shape, a comparison based on MAC values has been performed in order to deeply investigate the correlation achieved. Figure

MAC matrix.

Table

Summary of correlation between EMA and FEA results.

Mode pair | EMA scaled freq. [Hz] | FEA scaled freq. [Hz] | Error (%) | MAC |
---|---|---|---|---|

1 | 0.244 | 0.244 | 0.060 | 0.90 |

2 | 0.254 | 0.252 | 1.055 | 0.89 |

3 | 0.519 | 0.503 | 3.076 | 0.84 |

4 | 0.623 | 0.612 | 1.677 | 0.96 |

5 | 0.834 | 0.817 | 2.016 | 0.55 |

6 | 0.849 | 0.833 | 1.934 | 0.76 |

7 | 1.023 | 1.000 | 2.221 | 0.73 |

8 | 1.037 | 1.019 | 1.709 | 0.73 |

9 | 1.236 | 1.234 | 0.134 | 0.82 |

In the present paper, a correlation activity has been performed between experimental and numerical modal analyses of an aluminium rear subframe in free-free condition. In particular, a novel modelling technique of welded joints has been applied; the ACM2 approach, formulated by [

Further improvements should be made on the experimental side in order to reduce the data noise, exciting in a proper way all the modes of the structure in the frequency range of interest. Although the correlation between EMA and FEA has been found to be good, additional developments could regard the improvement of the FE model through sensitivity and updating procedure, so as to define a more accurate numerical model for dynamic purposes. Also, better correlation could be achieved using different correlation indexes: an example is the Coordinate Modal Assurance Criterion (CoMAC) that attempts to identify which measurements contribute negatively to a low value of MAC.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

The authors would like to thank the Ferrari NVH Team Leader Mr. Tarabra Marco and NVH Senior Engineer Mr. Roncaglia Valerio for the technical support and the guidance throughout the course of this research. The authors wish to express their sincere gratitude to Cavazzoni Luca from MilleChili Lab for the invaluable assistance in the final review of the paper.