NOISE-CON 98
Ypsilanti, Michigan 1998 April 5-8 Laboratorium 'icor Scheepshydromechj &rchi3f Makelwog 2,2628 CD Deft iW (Ì1 -T5 Fu 015- 181AN APPROACH FOR MODELING SPOT-WELDED JOINTS IN AN
ENERGY FINITE ELEMENT FORMULATION
Nickolas Viahopoulos Department of NA&ME The University of Michigan Ann Arbor, MI
Thomas Allen
Ford Research Laboratory Ford Motor Co.
Dearborn, MT
INTRODUCTION
The vibro-acoustics attributes of a vehicle are important
in its perceived value and itscompetitiveness. Simulation technology is utilized for predicting and improving these characteristics[1-3]. An emerging simulation technology is the Energy Finite Element Analysis (EFEA).[4-7] It is based on formulating the governing differential equations with respect to an energy variable, and utilizing a finite element numerical approach for solving them. In order for this method to be applicable to automotive structures it is important to be suitable for modeling spot-welded joints. This work presents a development effort in this area.
TECHNICAL BACKGROUND
The scope of this work was to develop a numerical approach for computing the energy
transmitted through spot-welded joints, and utilize the information in an energy finite element analysis (EFEA). The capability of modeling spot-welded joints in the EFEA is essential in order
to apply the method in car body structures. The EFEA is based on formulating the governing
differential equations for a structural-acoustic system in terms of the frequency and space average energy density.[5-7] The primary variable (energy density) varies exponentially with space, therefore, it is feasible to employ a finite element approach at high frequencies for obtaininga numerical solution. The required number of elements in the discretization remains small due to the slow variation of the energy density with space. In the EFEA a specialized formulation is required to account for discontinuities in the model. At locations where there are geometric or material changes (joints) the energy density becomes a discontinuous variable. Joints are modeled at the boundaries of finite elements by assigning a different primary variable to the nodes across
the joint, and taking into account continuity of power flow.[6,8,9} Based on this principle, relationships can be developed for associating the energy density variable across each joint. They allow to couple the discontinuous energy densities across each joint in the model. In order to develop the joint equations, the members connected at the joint are considered semi-infinite.
Thus, an analytical solution can be derived for their displacements. It is comprised by terms which represent a wave traveling towards the joint, and a wave traveling away from the joint. By prescribing an impinging wave on a particular member the continuity conditions for displacement,
Xi Zhao
Department of NA&ME The University of Michigan Ann Arbor, Ml
VIBRATION AND SHOCK
slope, rmment, and shear force can be utilized for deriving the joint equation. A relationship between the energy transmitted to the other members, and the energy which reflects back from the joint can be derived. In this manner transmission and reflection coefficients can be computed. They associate the energy density variables across a joint, and they express mathematically the physical mechanism of transferring energy through the joint. Due to the analytical nature of the joint formulation it is necessary to have a continuous connection at the joint. This is not the case
for spot-welded connections. Therefore, an alternative approach was developed.
This development was based on principles utilized for deriving coupling loss factors for
statistical energy analysis (SEA) from conventional finite element models.[lO-13] There are two components to the new development:
Deriving numerically information about the energy ratio between spot-welded members. Incorporating information for the energy transferred between members in the EFEA analysis. Technical information associated with this development, its implementation, and comparison of numerical results to test data are presented.
IMPLEMENTATiON
Computation of energy transferred between spot-welded members. In the past, conventional
finite element models have been successfully utilized to determine the amount of energy
exchanged between members in
plate assemblies, and for calculating SEA coupling loss
factors.[1Q-13] Conventional FEA models can not compute the discrete displacement at a
specific point and for a particular frequency. However, if the calculatd displacements are integrated over an area and simultaneously over a frequency band, then the;spatial and frequency
averages are sufficiently accurate to provide an estimate for the vibrational energy
of a
structure.[ll,121 This approach was utilized to calculate the ratio of the flexural energy in the receiving versus the excited plate in a "L" and a "H" plate configuration. Numerical results obtained from conventional FEA models were compared successfully to test data. Once theenergy ratio information was available, the SEA coupling loss factors
for the plates were
calculated a.s:[l 1,12](Ei)
ni(Ei) -
±Ln +flk<Ei)
where - energy ration between plate "k" and "j" computed from the conventional FEA
(Ei)
model, n = SEA coupling loss factor, and n3, nk =
modal densities in plates 'j" and "k"
respectively. Since the derivation of transmission and reflection coefficients can not be achieved
through an analytical process for spot-welded joints, the numerical approach utilized for
computing the coupling loss factors in SEA was employed in this project. A conventional FEA model for the spot-welded structures was developed, and a direct frequency response analysis was performed at multiple frequencies in each 1/3 octave band. The analysis was performed at
frequency increments of 2.5 Hz for the bands with center frequencies at 800Hz, 1000Hz, and 1250Hz, and at increments of 5Hz for the bands with center frequencies at 1,600Hz, 2,000Hz,
and 2,500Hz. Software was developed for computing the flexural energy of a structure from (1)
vibration results. The Section of the structural FEA model over which the energy is to be
computed, and the NASTRAN '.pch" file containing the vibration results, comprise the input data. The software calculates the kinetic energy for each frequency of analysis.
It then
summarizes the energy results for evaluating the energy over a frequency band. The ratio of the energies stored in the members connected by a spot-welded joint were utilized for deriving the appropriate transmission and reflection coefficients which demonstrate the same energy ratio between connected members.Implementation in an EFEA formulation.
The next Step in the process was to utilize theinformation about the energy transferred between spot-welded members in the EFEAformulation. This was accomplished by modifying the values of the reflection and transmission coefficients associated with the flexural energy. The EFEA matrix for two plate elements connectedby a joint can be expressed as:
where L = length of edge along the joint, c = wave speed in member "i", K = entries in the EFEA matrix, subscripts indicate the degrees of freedom and superscripts indicate the element of the joint. The spot-welded joints are modeled as regular plate joints. The difference is introduced by defining appropriate transmission and reflection coefficients. They are specified by requiring the resulting energy ratio to demonstrate the same value as the one computed munerically for the
spot-welded joints.
APPLICATION
In order to validate this predictive methodology, three sets of plate assemblies were constructed. Each set was comprised respectively, by a pair of fully welded, spot-welded at 2" intervals, and spot-welded at 4" intervals plates. Two measurement areas were scanned with a laser vibrometer.
Measured data for 100 point were collected for each measurement patch. The test data were
processed through specialized software, developed for this project. It added the square of the velocities over each area, and each 1/3 octave band. Since the thickness of both plates was the same, the mass corresponding to each measurement section was determined from its associated area. Test data were collected at 2.5Hz increments
for the 1/3 octave bands with center
frequencies at 800Hz, 1000Hz, 1250Hz. and at 3.125Hz increments for the bands with centerK111 K1 K1 K1 K112 K
K1 K3
K2 K3 O O O O K113 K3r Lc
K Kr121
K4 K4r Lc
K2 K221 O O i2Lc1 O O O O2I
O O O O K213 K2 K3 K3 K214 K224 KK,
(-r
Lc - 12r11r Lc
12r11 ni2LcI 6r1'i2'1
-
2'
2Lc1-
6r1 r12Lc2 l2ti K2 r12Lc2 12ri1i21
'i21
-
6r1'i22
12 - l2r1 K2 r12Lc2 6r1, O O 12r11 O O l2ij K1 K1 6i K2 K2350 VIBRATiON AND SHOCK
frequencies at 1600Hz, 2000Hz, and 2500Hz. Results for the energy ratio between the receiving and excited plates are presented in Figures 1-3. They include test data and numerical results.
o C > o C LU 0.8 0.2 -o Test AnalysE
Figure 1. Test and Numerical Results for Fully Welded Plates
Figure 2. Test and Numerical Results for Spot-Welded Plates at 2" Increments
0.8 o 0.6 0.4
-
02-Wo
o Q O O O O O Q u) Q O Q O C'.l 0 O aL) c',a c Frequency Test AnasEFigure 3. Test and Numerical Results for Spot-Welded Plates at 4" Increments Discussion of the results:
In both test and analytical results the fully welded pair of plates demonstrates a more even energy transfer over the entire frequency range.
Both sets of spot welded plates demonstrate decaying energy transfer at higher frequencies. This behavior is captured correctly by the analysis.
Magnitudes of the amount of energy transfer correlate well between test and analysis.
The behavior of the fully welded plates demonstrates increasing energy transfer between 2000 and 2500 bands. This characteristic is captured correctly by the analysis.
o o Q C o o
o o u o o o
o c'J (D o
5. In both test and analysis the spot welded plates present higher energy transfer than the corresponding fully welded pair at some of the lower frequency bands.
Once the information about the energy transfer become available, the corresponding transmission and reflection coefficients were computed. They were introduced in the EFEA analysis. Then, although the joints were defined as fully connected in the data file of the EFEA code,[14] during the analysis they exhibited the characteristics of spot-welded joints. Figures 4 and 5 present
respectively, results for the 2" and 4" spot-welded plates. The test data, the original EFEA results, and the EFEA results with adapted coefficients are presented in each Figure.
0.8 o 0.6 ,.. 0.4 -Ui I o o o o o o o o u, Q O Q O C'4 (
-
Q C4 C'J Frequency Test EA giìaJ----EAMdtd
Figure 4. Results for Spot-Welded Plates at 2" Increments
The modified EFEA data match exactly the energy ratio computed by the averaging finite element process. In a similar manner the EFEA reflection and transmission coefficients could have been modified based on the information supplied by the test data. Then, the results from the modified
EFEA analysis would have matched exactly the test data. The results demonstrate that the
current EFEA method can be enhanced by adapting the joint coefficients and including
information about the energy transferred between spot-welded joints either from numerical results or from testing. 0.8 .2 L 0.4 0.2 -W o Q O O O Q O O Q Q Q Q o c Q u -. ,- c4 c,a Frequency Test EA OrIaI----8EAMddcd
Figure 5. Results for Spot-Welded Plates at 4" Increments
This development can also be utflized to modify the predictions of the EFEA for the fully connected members if information can be available from another numerical approach or from test data. Figure 6 presents the energy ratio between the receiving and the excited plate for the fully welded pair over ail extended frequency range (250Hz - 2,500 Hz). This demonstrates that the
reflection and transmission values can be adapted in order to reflect the energy transmission exhibited by structures at lower frequencies.
352 VBRAT1ON AND SH
Figure 6. Results for the Fully Welded Plates over an Extended Frequency Range
CONCLUSIONS
A nunrical methodology was developed for modeling spot-welded joints
in an EFEA
formulation. A numerical approach utilized in the past in the SEA for computing the energy transferred between members and the coupling loss factors, was employed in this project forcomputing the energy transferred between spot-welded members. This information was utilized for adapting the values of the transmission and reflection coefficients in an EFEA formulation.
Numerical results for three sets of plates, fully welded, welded at 2" intervals, and spot-welded at 4" intervals were utilized to validate the development.
AKNOWLEDGEMENTS
This work was partially sponsored by Automated Analysis Corporation and Ford Motor Company. The authors would like to
express their gratitude to Alexander Petniunas of Fortl Motor Co. for providing the test data utilized io the validation.
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