A hybrid method for mechanical testing of electronic packaging.

Tudose, Virgil ; Gheorghiu, Horia ; Visan, Dana 等

1. INTRODUCTION

The most effective way to determine the increase of life of the EP structures in exploitation is the experimental one, which consists in determinations of the physical-mechanical characteristics of the materials and testing in situ, with load tests (Sommer et al., 2005). The study of the static and dynamic behavior in the EP exploitation, by numerical modeling and simulation, using computer codes, requires experimental verification of the results (Lau et al., 2003).

A method based on modal analysis and algorithms for identification is proposed in order to study the mechanical behavior of the plates with integrated circuits. The method was successfully tested in the Laboratory of Mechanical Vibration from the Department of Strength of Materials at the University "Politehnica" of Bucharest. Further reference is made to this method.

2. NUMERICAL METHOD

The scheme of the principle of this method (Parausanu, 1991) is shown in Figure 1. Starting from the response function in frequency, obtained by direct measurements carried out on the real structure, the modal model can be built. Then, on this basis, the material model, characterized by the inertia matrix [M], damping matrix [C] and the stiffness matrix [K] is obtained.

With these matrices, one can regenerate or synthesize responses of the structure to improve the model, by comparing the curves of frequency response and by applying some specific methods. The cycle can be repeated until one obtains a good model for study. Finally, one can predict the dynamic response of the structure by making changes in the three matrices of the physical model.

[FIGURE 1 OMITTED]

The dynamic response of a structure, measured at the j coordinate, which is produced by the excitation force applied at coordinate l, can be expressed through the mobility:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where [[PSI].sup.(r).sub.j] is the j-th element of the eigenvector {[[PSI].sup.(r)]}, [[omega].sup.r] is the eigenfrequency of the order r, [m.sub.r] = [{[[PSI].sup.(r)]}.sup.T] [M]{[[PSI].sup.(r)]} is the modal mass and [A.sub.jl.sup.(r)] is a modal constant defined as:

[A.sup.(r).sub.jl] = [[PSI].sup.(r).sub.j] [[PSI].sup.(r).sub.l]/[m.sub.r] (2)

In the relation (1), [zeta] = c/[c.sub.cr] represents the fraction of the critical damping, whose values, compared to unity, give important indications about the damping of structure. To start the calculation of [zeta], it can be initially estimated in the range of values 0.008 - 0.9. Its real value should be calculated experimentally and subsequently introduced in the existing mathematical model, contributing to the improvement of the dynamic response. The experimental calculation of the fraction from the critical damping is relatively easy if one manages to trace the movement vibrodiagram of the structure. Measuring effectively on the vibrodiagram two successive values regarding the amplitude of the movement, x1 and x2, one can calculate the logarithmical decrement, using the following formula: [DELTA] = ln([x.sub.1]/[x.sub.2]), to deduct then the value of the fraction from the critical depreciation so: [zeta] = [DELTA]2[pi], (Parausanu, 1995). If using an electromagnetic shaker (Fig. 2) one can measure the mobility for P values of separate frequency excitation [[OMEGA].sub.1], [[OMEGA].sub.2], ... [[OMEGA].sub.P] and the following matrix equation can be written:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

where {[M.sub.jl]} is a column vector of order p; [[OMEGA]] is a rectangular matrix of order pxN, {[A.sub.jl]} is a column vector of order N containing the obscures from the matrix equation (4). Values [[omega].sub.1], [[omega].sub.2], ... [[omega].sub.N] are the corresponding frequencies for the most responding curves in frequencies set on the experimentally measured data.

[FIGURE 2 OMITTED]

Modal constants can be easily determined according to the measured values as follows:

{[A.sub.jl]} = [[[OMEGA]].sup.+] x {[M.sub.jl]} (5)

where [[[OMEGA]].sup.+] is the pseudo inverse of the matrix [[OMEGA]]. By extending equation (2), the matrix of the modal constants can be built for each mode:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

The matrix group [??][A.sup.(1)][??], [??][A.sup.(2)] [??], ... [??][A.sup.(N)][??], where N is the number of degrees of freedom of a system, completely characterizing the modal model and represents the basic modal matrix [PSI] = [??]{[[PSI].sup.(1)][??]} {[[PSI].sup.(2)]} ... {[[PSI].sup.(N)]}[??], where {[[PSI].sup.(r)]} represents the modal vector orthonormalized according to the r module. Using the relations (7) and (8) one can determine the modal masses as follows:

[m.sub.r] = {[[PSI].sup.(r).sub.j]} {[[PSI].sup.(r).sub.l]}/{[A.sup.(r).sub.jl]} (7)

as well as the elastic modal constants:

[k.sub.r] = [m.sub.r] x [[omega].sup.2.sub.r], (8)

then the diagonal matrices [[m.sub.r]] = diag ([m.sub.r]) and [[k.sub.r]] = diag ([k.sub.r]) can be built. Based on the modal model, one can obtain physical model characterized by the matrices [M] and [K], which can be obtained using the modal diagonal matrices and the modal matrix [[PSI]], as follows:

[M]=[[[PSI]].sup.-T] [[m.sub.r]][[PSI].sup.]-1] [K]= [[PSI].sup.]-T][[k.sub.r]][[PSI].sup.]-1] (9)

For testing purposes: using the [M] and [K] matrices previously written in the eigenvalues issue:

([K]-[[omega].sup.2.sub.r][M]){[PSI].sup.(r)]} = 0 (10)

leads to the identified, eigenfrequency [[omega].sub.r] and the eigenvector {[PSI].sup.(r)]}. The effects of the structural changes in the real system, as such, can be predicted through the proper changes made to the elements from the main diagonal of the matrices [M] and [K] of the physical model.

3. CONCLUSIONS

An important advantage of the proposed hybrid method is that the experimental stage as well as the numerical computing can be started simultaneously, the purpose being to create the physical model of the structure, whose dynamic answer would be as close as possible to the reality. Thus, the experimental stage includes measuring the vibrations of the EP in order to trace the response curves in the frequency domain, and to determine the resonance frequency, further used in the numerical computing simulation (Sorohan et al., 2005).

The proposed hybrid method combines the experimental computing with the theoretical one, thus succeeding, in the end, in obtaining the physical model of the structure as close as possible to the real one. At the core of this method, is the analytic and experimental analysis of the structure, which has two branches that can be started simultaneously: while a team executes static and dynamic measurements in situ, another team, in the laboratory, starting from the execution drawing of the current structure, create a mathematical model using the finite element method (Parausanu et al., 2005). Then, the static and dynamic response of both analyses are compared, and decisions regarding the modifications of the mathematical model are taken (as well as the module from which these modifications are being made) only if the results are not close to the real structure. In the next stage, the basic calculations regarding the carrying capacity and the lifespan are made.

Moreover, the proposed hybrid method, whose computing algorithm was presented above, has been extended and generalized for the study of any structure, by obtaining the physical model of the respective structure. This model allows the structural hanges, at the numerical simulation level, which are put in use when the optimal configuration has been attained.

4. REFERENCES

Buzdugan, Gh.; Mihailescu, E.; Rades, M. (1986). Vibration measurement, Martinus Nijhoff Publishers, ISBN 90-2473111-9

Lau, J.H.; Wong, C.P.; Prince, J.I. & Nakayama, W. (2003). Electronic Packaging: Design, Materials, Process and Reliability, McGraw-Hill, ISBN 0-07-037135-0, New York

Parausanu, I.; Sorohan, S.; Gheorghiu, H.; Hadar, A. & Caruntu, D. (2005). Improvement of the Dynamic Behaviour of a Fan Using Finite Element Method, ASME International Mechanical Engineering Congress and RD&D Expo, November 05-11, Orlando, Florida, USA, IMECE2005-82254 (Technical Publication) ISBN: 0-7918-3769-6

Sommer, J.; Manessi, O. & Michel, B. (2005). Thermo mechanical analysis of advanced electronic packages in early system design, TECHNICAL PAPER, Microsyst Techn 2005, DOI10.1007/s00542-005-0024-8

Sorohan, S.; Parausanu, I.; Motomancea, A. & Caruntu, D. (2005). Dynamic Analysis of Composite Plates, ASME International Mechanical Engineering Congress and RD&D Expo, November 05-11, Orlando, Florida, USA, IMECE2005-82252 (Technical Publication) ISBN: 0-7918-3769-6