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A sensitivity study of printed wiring board vibrations using a statistical method
Journal of Sound and Vibration (1995) 181(4), 593–604
A SENSITIVITY STUDY OF PRINTED WIRING BOARD VIBRATIONS USING A STATISTICAL METHOD A. O. C, N. S C. A. N IBM T. J. Watson Research Center, P.O. Box 218, Yorktown Heights, New York 10598, U.S.A. (Received 22 July 1993, and in final form 18 January 1994) The dynamic behavior of a printed wiring (PW) board plays an important role in its reliability. One key parameter in determining such behavior is the fundamental frequency of the PW board. This paper shows how the sensitivity of the natural frequencies of the board with respect to some design variables can be studied using a statistical approach and how this can guide the experimental and modeling phases of the design.
1. INTRODUCTION
The basic building block of most structures found in the electronics industry is the PW board. The dynamic behavior of such boards is becoming an important issue in determining the reliability of these structures. A number of failures at the connectors level between the board and the components (heat sinks, capacitors, transformers, etc.) can be attributed to vibration or fatigue resulting from dynamic loads. Thus, the need to study the dynamic behavior of a PW board as a function of its design variables is apparent. Previously, many authors have addressed different aspects of this problem. Suhir [1] has studied the non-linear response of boards subjected to a sudden acceleration applied at the boundary. Pitarresi et al. [2–4] have studied some modelling issues regarding the structural representation of boards plus components, as well as the relationship between the stiffness of the board and the location of its supports. Cifuentes and Kalbag [5] developed a method to optimize the location of the supports to maximize the natural frequency of the PW board. Singal and Gorman [6] have performed experimental studies to quantify the influence of concentrated masses on the fundamental frequency of plates (a model frequently used to represent PW boards). Wong et al. [7] have dealt with the application of modal methods to estimate the dynamic response of boards [7]. Roberts and Stillo [8] have done work focusing on the application of random analysis techniques to compute the dynamic response of these structures. A good review of current design practices can be found in the books by Steinberg, Dally or Pecht [9–11]. A key parameter in determining the dynamic behavior of a board is its fundamental natural frequency. Hence, in most cases it is used as a figure of merit to assess the goodness of a possible design. For example, in general one wants the board to be as stiff as possible (within certain constraints) in order to minimize the amplitude of the dynamic displacements. This is equivalent to maximizing the fundamental frequency of the board. In addition, requirements in terms of fatigue life of components as well as maximum acceptable accelerations in certain regions of the board often determine minimum design values for the fundamental frequency. As a result, it is important for the engineer to know 593 0022–460X/95/140593 + 12 $08.00/0
7 1995 Academic Press Limited
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which are—on a given board—the critical design variables that affect the value of the natural frequency. This knowledge is useful in two situations. First, if a board does not meet the design requirements, it is necessary to know which parameters (design variables) should be changed. One would prefer to change those design variables that are the most influential in the value of the fundamental frequency. Second, since in general the reliability of a board is studied using laboratory tests combined with computer simulations (finite elements, for instance) it is extremely useful to know which are the most important parameters to guide the experiments. One wants to concentrate the effort on measuring the parameters that will be crucial, to make a good model. Some parameters are quite easy to determine experimentally (for example, the mass of an unpopulated board), while others are more cumbersome to estimate (the stiffening effect of a component on a certain region of a board). Thus, this information is very valuable if one is to avoid spending resources measuring irrelevant quantities. This paper is concerned with studying the sensitivity of the fundamental frequency of a PW board using a statistical approach. The following sections describe the modelling strategy employed to represent PW boards using finite elements, the statistical method and several case studies to demonstrate the advantages of this approach. The last sections deal with a discussion of the results and conclusions. 2. MODELLING OF PW BOARDS
A PW board is a flat surface, normally of rectangular shape, on which components are mounted. Mechanically speaking, the PW board behaves as a thin plate (shear deformations are negligible). The components, the total mass of which can be significant compared to that of the board, are mounted on top of the board surface and attached to it by a number of techniques, such as pins, J-lead connections, gullwing connections and elastomeric interposes. Thus, a finite element representation of a board reduces to an assembly of plate elements. As far as the components are concerned, there is a wide variety of possibilities. These range from rather simple models in which the presence of a component is ignored or represented as a concentrated mass if the mass of such component exceeds a certain critical value, to more elaborate models in which each component is described in detail using an assembly of thick plate, beam and possibly solid elements. Pitarresi [2, 3] has studied this issue in detail. In this study, the presence of a component is modelled by considering an equivalent mass density and stiffness for the plate elements located in the area where the component is situated. This approach has been validated experimentally, with excellent results [4, 12]. In short, the PW board model reduces to an assembly of plate elements in which those elements corresponding to areas in which there are no components have the properties (mass density and Young’s modulus) of the unpopulated board; and the plate elements corresponding to areas in which components are mounted have a higher mass density and stiffness to account for the presence of such components. (In any event, the results presented in this paper are not tied to a particular modelling strategy.) In addition, one has to consider the support conditions—the way in which the board is connected to its mother structure. It is assumed that the supports can be defined in terms of two linear springs; a translational spring acting in the direction normal to the board plane and a rotational spring in which the rotary axis is parallel to the board edge. In this study, a board is characterized in terms of six design variables: (1) Eb , the Young’s modulus of the unpopulated board, (2) rb , the mass density of the unpopulated board,
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(3) Ec , the Young’s modulus of the area in which the component is located (that is, this value represents the stiffness of the board-plus-component assembly in that region), (4) rc , the mass density in the region in which the component is located, (5) KZ , the stiffness of the individual springs placed at each support node, and (6) Ku , the rotational stiffness of the springs at the support nodes. This idealization is shown in Figure 1. For simplicity, it has also been assumed that all board components are defined by the same local properties (in case there are several components). This study considers the relationship between the fundamental frequency of the board ( f1 ) and the six above-mentioned parameters (or design variables). In a way, one might consider these variables as subject to error. Consequently, one studies how an error in one of these variables influences the error in determining the value of f1 . There are, obviously, a number of other variables that play a role in the value of the fundamental frequency. In this study, however, they are not considered as variables. For example, the Poisson ratio of the material is not considered as a variable, simply because in general the value of f1 is not very sensitive to variations in n. The board dimensions (length, width and thickness), on the other hand, although they influence the value of f1 greatly, are quite easy to determine. In other words, one would expect the designer to have access to an accurate estimate of these variables without too much trouble. In any event, the treatment of a problem choosing a different set of design variables (or a frequency other than the fundamental frequency) would be analogous to what is presented here. 3. STATISTICAL METHOD
The approach used in this study is based on the design-of-experiments theory proposed by Fisher in the 1920’s [13]. The original motivation for Fisher’s work was to analyze problems arising in agricultural sciences. Later, Box et al. extended these ideas to treat a number of industrial problems while Yates—who also did some pioneering work in the field—addressed some computational aspects of these methods [14–17]. Recently, Srinivasan et al. [18] employed this approach to analyze the sensitivity of a thin-film structure in the context of thermal stresses. For the sake of clarity, the so-called full factorial design of experiments is explained here in reference to the PW vibration problem considered.
Figure 1. A typical finite element model of a PW board having two components.
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Let f1 be the ‘‘response’’ of interest. It is assumed then that the relationship between f1 and the six ‘‘factors’’ (design variables) can be expressed in terms of a multi-linear function. For simplicity, the factors can be re-labelled as follows: r b = X2 ,
Eb = X1 ,
rc = X 4 ,
Ec = X3 ,
K2 = X5 ,
Ku = X6 .
(1–6)
Hence, one can write f1 (X1 , . . . , X6 ) = C000000 + C100000 X1 + C010000 X2 + · · · + C000001 X6 +C110000 X1 X2 + · · · + C000011 X5 X6 + · · · + C111111 X1 X2 X3 X4 X5 X6 ,
(7)
where the C’s are unknown (in principle) but are constants. A ‘‘1’’ in the ith position of the six-digit subindex indicates the ith factor is considered in that term. In this case, there are 26 = 64 unknown coefficients. Assume that each factor (X1 , . . . , X6 ) is subjected to some uncertainty or error expressed in terms of an interval that contains the true value of the factor: that is, for i = 1, . . . , 6 it is considered that Xi $ [Xi min , Xi max , where Xi min represents the lower end of the interval and Xi max denotes the upper bound. Performing a change of variable Xi =
Xi
min
+ Xi 2
max
+
Xi
max
− Xi 2
min
Yi ,
(8)
one can normalize the factors and rewrite equation (7) as f1 (Y1 , . . . , Y6 ) = D000000 + D100000 Y1 + D010000 Y2 + · · · + D000001 Y6 + D110000 Y1 Y2 + · · · + D000011 Y5 Y6 + · · · + D111111 Y1 Y2 Y3 Y4 Y5 Y6 .
(9)
There are still 64 unknown coefficients. However, the factors (Y1 , . . . , Y6 ) vary only between −1 and +1. D000000 is called the average or mean (that is, the value obtained for f1 , when all the normalized factors take the value 0—center of the interval). Twice the coefficient of each of the linear factors is called the main effect of the associated factor; and twice the coefficient of the product of each combination of two factors is known as the two-factor interaction. (Note that one of the advantages of this method, in contrast with less sophisticated approaches in which variables are varied one at a time, is that one can detect—through the two-factor interactions—the influence of coupling effects that otherwise would go unnoticed.) Finally, the so-called sensitivities, that is, the derivatives of f1 (in this case) with respect to the factors (design variables), can be written as 1f1 = 1Xi Xi
2 − Xi min
1f1 , 1Y max i
(10)
where 1f1 = D00i000 + D100i00 Y1 + · · ·. 1Yi
(11)
If one considers equation (9) (the normalized equation) and all the possible combinations of −1 and +1 for the values of the factors (Y1 , . . . , Y6 ), one can establish 26 = 64 equations. These equations determine a linear system of the form [A]{D} = { f },
(12)
where [A] represents a 64 × 64 matrix, the elements of which take the value of −1 or +1; {D} denotes the vector that contains the unknown coefficients; and { f } denotes the vector with the values of the ‘‘response function’’ (the natural frequency in this case) for different combinations of −1 and +1 of the normalized factors. It can be shown that the matrix
Figure 2. PW board configuration, Case 1. Finite element mesh, location of supports (springs) and area occupied by the components (heavy line). W, Locations of the supports, which consist of a vertical spring (KZ ) plus a rotational spring (Ku ).
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Figure 3. PW board configuration, Case 2. Finite element mesh, location of supports (springs) and area occupied by the components (heavy line). W, Locations of the supports, which consist of a vertical spring (KZ ) plus a rotational spring (Ku ).
[A] is non-singular and orthogonal. This system can be solved using any of the methods available for linear systems; however, a more efficient approach is to employ Yates’ algorithm [17, 19]. (Yates’ algorithm requires a number of operations proportional to n log n, whereas non-specialized L–U decomposition methods require a number of operations proportional to n 2.) Finally, after solving equation (12) one can determine, by inspecting the value of the coefficients, the relative influence of each factor (design variable) in the value of f1 . In essence, this has been achieved by studying how the relative errors in the value of the factors influence the error in the value of f1 . 4. EXAMPLE OF APPLICATION
Six PW boards, depicted in Figures 2–7, are considered in this study. The properties of these PW boards are listed in Table 1. These PW boards represent a variety of realistic configurations. Their plane dimensions vary from 0·08 m to 0·4 m, whereas the
Figure 4. PW board configuration, Case 3. Finite element mesh, location of supports (springs) and area occupied by the components (heavy line). W, Locations of the supports, which consist of a vertical spring (KZ ) plus a rotational spring (Ku ).
Figure 5. PW board configuration, Case 4. Finite element mesh, location of supports (springs) and area occupied by the components (heavy line). W, Locations of the supports, which consist of a vertical spring (KZ ) plus a rotational spring (Ku ).
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Figure 7. PW board configuration, Case 6. Finite element mesh, location of supports (springs) and area occupied by the components (heavy line). W, Locations of the supports, which consist of a vertical spring (KZ ) plus a rotational spring (Ku ). (The supports marked with a × only have a vertical spring.)
Figure 6. PW board configuration, Case 5. Finite element mesh, location of supports (springs) and area occupied by the components (heavy line). W, Locations of the supports, which consist of a vertical spring (KZ ) plus a rotational spring (Ku ).
fundamental frequency goes from 33·34 Hz (a fairly flexible board) to 163·77 Hz (a rather rigid board). In addition, consider the figures of merit a, b and g, defined as follows. a=
f1 , frigid
(13)
where frigid is the fundamental frequency of the board obtained by substituting the springs by rigid supports (that is, KZ and Ku become infinity). A value of a close to 1 indicates that the actual springs (supports) are very rigid. Decreasing values of a relative to 1 correspond to increasing levels of flexibility of the supports. b = area of components/total area,
(14)
thus b indicates the fraction of the PW board that is occupied by the components; and g=
mass of components (r − rb )Ac = c , mass of unpopulated board rb A
(15)
T 1 Properties of the PW boards considered in the example Case
1
2
3
4
5
6
Eb (N/m2) × 109 20·7 17·0 22·0 16·0 20·0 20·0 rb (kg/m3) 6 092 5 120 4 000 1 800 4 000 4 500 Ec (N/m2) × 109 25·9 18·7 24·0 24·0 28·0 40·0 rc (kg/m3) 18 276 10 240 6 000 5 400 11 000 9 500 KZ (N/m) 68 000 1 200 12 500 25 000 15 500 11 300 Ku (N-m/radian) 30 11 120 35 0·5 5 Length (m) 0·17 0·4 0·3 0·3 0·1 0·25 Width (m) 0·17 0·15 0·2 0·2 0·1 0·08 Thickness (m) 0·0023 0·00127 0·0015 0·0015 0·0012 0·0018 nb 0·18 0·3 0·3 0·3 0·2 0·25 nc 0·33 0·3 0·3 0·3 0·2 0·25 a 0·57 0·31 0·93 0·83 0·53 0·70 b 0·25 0·25 0·166 0·208 0·64 0·166 g 0·5 0·25 0·08 0·41 1·12 0·18 f1 (Hz) 114·00 33·34 39·59 82·93 163·77 81·37
599
where A refers to the total area of the board and Ac denotes the area occupied by the components. Thus, g indicates how heavy the components are with respect to the unpopulated board. Inspection of Table 1 indicates that the PW boards chosen vary from designs with very flexible supports (a = 0·31) to fairly stiff designs (a = 0·93). In addition, the relative areas occupied by the components (b) range from 0·166 to 0·64 and the mass of the components compared to that of the board (g) varies between 0·08 and 1·12. In short, the six examples selected correspond to a very diverse set of configurations, that can be considered a representative sample. The fundamental frequency of each board was computed using a finite element model with the mesh indicated in the figures and with the help of ABAQUS (a general-purpose finite element code [20]). This value is shown for each board at the bottom of Table 1. Finally, the length, width and thickness of the boards as well as the value of the Poisson ratio are not considered as variables in the statistical analysis. 5. RESULTS
It was assumed that each one of the factors (design variables) was subjected to a 15% error; that is, for i = 1, . . . , 6, Xi
min
= (Xi )ref − 0·15 × (Xi )ref
(16)
= (Xi )ref + 0·15 × (Xi )ref ,
(17)
and Xi
max
where (Xi )ref represents the nominal value of the ith factor as given in Table 1. The choice of an error of 15% in the determination of the design variables is arbitrary; what is important is that the error is the same in all variables—in this way it is easier to make a meaningful comparison in terms of which design variables are more influential in determining the value of f1 . Hence, by computing the value of f1 for all the possible combinations of −1 and +1 of the normalized variables (see equation (9)) one creates a linear system of equations (see equation (12)) which was solved using Yates’s algorithm. This process was automated using several C++ programs—and a UNIX shell script to drive them—in combination with ABAQUS, which runs the eigenvalue analysis. The results for the six cases described in Figures 2–7 are summarized in Tables 2–7. In the tables the factors have been ranked in decreasing order of importance. Only the first seven factors have been included (the remaining 57 make negligible contributions—they account for a small portion of the mean value). For convenience, the coefficients have been identified in terms of the variables they are related to, instead of using the six-subindex notation. T 2 Values of the coefficients for Case 1
T 3 Values of the coefficients for Case 2
Factor Coefficients (D’s) Percent of mean
Factor Coefficients (D’s) Percent of mean
Mean rc Ec KZ Ku rb Eb Eb rc
Mean KZ rb rc Eb Ec Eb rc Ku
113·96 −6·76 2·81 2·67 2·21 −1·86 0·94 0·34
100·00 −5·94 2·46 2·34 1·94 −1·62 0·82 0·29
33·34 1·98 −1·60 −0·93 0·27 0·14 0·14 0·12
100·00 5·95 −4·80 −2·77 0·81 0·44 0·42 0·36
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600 T 4
T 5
Values of the coefficients for Case 3
Values of the coefficients for Case 4
Factor Coefficients (D’s) Percent of mean
Factor Coefficients (D’s) Percent of mean
Mean Eb rb rc Ec KZ Eb rc Ku
Mean rc Eb rb Ec KZ Ku Eb rc
39·56 2·04 −1·71 −1·28 0·56 0·28 0·17 0·10
100·00 5·17 −4·32 −3·25 1·42 0·70 0·44 0·26
82·70 −4·07 3·08 −2·19 1·26 0·98 0·92 0·34
100·00 −4·92 3·73 −2·65 1·53 1·18 1·11 0·41
(In Appendix A it is shown in more detail, using Case 1 as an example, how the numerical entries that define equation (12) were generated.) Note that the value listed as ‘‘mean’’ (actually the D000000 coefficient of equation (9)) coincides—except for some roundoff—with the value previously computed for f1 and shown in Table 1. (It is really the constant term in a Taylor’s series expansion.) Also, in any of these tables, summation of the absolute value of all the entries listed in the third column (except, of course, for the value 100·00 which corresponds to the mean) yields roughly 15·0%. Simply put, this indicates that if one computes the value of f1 assuming for each factor the value given by the lower or upper end of the interval, the error in f1 is roughly 15%. This is consistent with the standard result from error theory—that the relative error in the function to be evaluated is approximately equal to the sum of the relative errors of each one of the variables involved in evaluating that function (because of the normalization). Finally, consider that ultimately the value of f1 is proportional to the square root of the ratio between the modal stiffness and the modal mass associated with the first mode of vibration: that is, f1 = v1 /2p = (1/2p)zKmodal /Mmodal ,
(18)
where Kmodal denotes the modal stiffness, Mmodal denotes the modal mass and v1 refers to the fundamental frequency expressed in radians per second. Noting that 1v1 1 = 1Kmodal 2Mmodal v1
and
1v1 v1 =− 1Mmodal 2Mmodal
(19, 20)
one can conclude, using a Taylor’s expansion with only the linear terms, that
b b b Df1 E f1
1 2
b b
DKmodal + Kmodal
1 2
b
DMmodal , Mmodal
(21)
T 6 Values of the coefficients for Case 5
T 7 Values of the coefficients for Case 6
Factor Coefficients (D’s) Percent of mean
Factor Coefficients (D’s) Percent of mean
Mean rc Ec KZ Ku Eb Ec rc rb
Mean rb KZ Eb rc Ec Eb rc Eb K Z
164·36 −12·09 6·54 3·09 1·98 0·79 −0·46 −0·31
100·00 −7·35 3·97 1·88 1·20 0·48 −0·28 −0·19
81·29 −4·47 3·34 2·29 −1·68 0·50 0·29 0·27
100·00 −5·49 4·11 2·82 −2·07 0·62 0·35 0·32
601
T 8 Values of the coefficients for Case 1 (second vibration mode) Factor Coefficients (D’s) Percent of mean Mean KZ rb rc Eb Ec Ku Eb rc
264·13 12·32 −11·91 −8·09 3·51 2·85 1·30 1·19
100·00 4·66 −4·51 −3·07 1·33 1·08 0·49 0·45
where Df1 , DKmodal and DMmodal denote the absolute error in f1 , Kmodal and Mmodal . In essence, equation (21) says that if the relative errors in the mass and in the stiffness are the same, then their contributions to the error in f1—or, more precisely, to an upper bound of the error—are the same. This is indeed the case in the examples. In any of the tables one can verify that adding the positive coefficients asscociated only with linear factors (that is, those associated with the stiffness) from the third column (except for the value 100·00) one obtains approximately 7·5% (half of 15%), consistent with the initial error assumption and equation (21). In the same way, adding the negative coefficients (which are associated with the mass) one obtains approximately 7·5%.
6. DISCUSSION
Inspecting the results presented in Tables 2–7, one can see that the method does indeed identify the few influential factors that affect the value of f1 the most. In general, two and higher factor interactions between the design variables play a minor role. The most important conclusion—somewhat to be expected from a structural mechanics standpoint—is that there is no consistency in the order in which the variables are ranked for all six cases. Thus, a variable that can play a major role in one case (KZ in Case 2, for example) can be almost irrelevant in other situation (see Case 3). Therefore, there is no such thing as the most important parameter in the dynamic behavior of a PW board—it all depends on the specifics of the design. The highest value of a (Case 3) corresponds to a major role for Eb , whereas in the case of the smallest value of a (Case 2), KZ takes that place. Again, this is in agreement with intuition—in a board with flexible supports, the stiffness of the supports is more important; in a board that is more rigidly supported, variations in the stiffness of the board are dominant. Also, one notices that for higher values of b the importance of the stiffness of the components (Ec ) grows—compare Cases 5 and 6. Furthermore, increasing values of g reflect an increasing importance of rc —compare Case 5 with Cases 3 or 6, for instance. Of course, the values of a, b and g only reflect general trends and cannot be used to predict the rank of the design variables. Only the statistical model can do this. A specific result that one notes in these tables is that rc is always more influential than Ec in the value of f1 . This is in agreement with analytical results presented by Cifuentes elsewhere [21]. This result is very important for the modeller; it means that one should concentrate in obtaining a better estimate of the mass of the components rather than the stiffness of the area in which the components are located. This is also fortunate from the
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602
experimental point of view for, in general, it is a lot easier to determine the mass of a component than its contribution to the stiffness of the board. In summary, these results show that there cannot be rigid guidelines in terms of determining a priori which are the key design variables that control the dynamic response of a PW board. However, the statistical approach described here constitutes a powerful technique to identify the key variables (if a certain design needs to be modified). Moreover, an approximate model of a PW board can also be used in combination with this technique to guide the experimental testing. For instance, in a case such as Case 2 there is no point in trying to determine Ku very accurately; the effort should be concentrated on measuring Kz and rb . Finally, the statistical technique presented here is not restricted to investigating the first natural frequency of the PW board. In Table 8 are shown the results obtained for the second natural frequency in Case 1 (that is, the PW board shown in Figure 2). Comparing these results against those from Table 2, one can see that the design variables are ranked in quite a different sequence. In short, this demonstrates that what might be an important factor in the determination of f1 might not be so important in the determination of f2 (the second natural frequency). It is crucial to keep this in mind if a certain design needs to be improved—for instance, to increase the overall stiffness of the board. In such a case, the statistical analysis should be undertaken considering the natural frequency of the mode that accounts for the largest fraction of the total mass (which, in some instances, might not be the first mode of vibration). Notice also that rc is still more important than Ec when it comes to computing f2 and that the positive and negative coefficients from the third column (Table 8) still add up to approximately 7·5%. These two results hold for any normal mode. (Obviously in this case, equation (9) still holds, except that f1 needs to be replaced by f2 .)
7. CONCLUDING REMARKS
A statistical method based on the theory of experiments has been employed to investigate the sensitivity of the natural frequency of six PW boards as a function of several design variables. This approach is expected to be useful for designers of PW boards. First, it can identify the critical variables, should modifications to a design become necessary. Second, it can guide the laboratory testing by identifying the variables that need to be measured with more accuracy. Extension of this approach to study PW boards using a different set of design variables is straightforward.
T 8 Values of the coefficients for Case 1 (second vibration mode) Factor Coefficients (D’s) Percent of mean Mean KZ rb rc Eb Ec Ku Eb rc
264·13 12·32 −11·91 −8·09 3·51 2·85 1·30 1·19
100·00 4·66 −4·51 −3·07 1·33 1·08 0·49 0·45
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REFERENCES 1. E. S 1992 International Journal of Solids and Structures 29, 41–55, Nonlinear dynamic response of a flexible thin plate to constant acceleration applied to its support contour with application to printed circuit boards in avionic packaging. 2. J. M. P, D. V. C, R. C and D. E. S 1991 Transactions of the American Society of Mechanical Engineers, Journal of Electronic Packaging 113, 250–257. The ‘‘smeared’’ property technique for the FE vibration analysis of printed circuit cards. 3. J. M. P and A. V. D E 1993 Transactions of the American Society of Mechanical Engineers, Journal of Electronic Packaging 115, 118–123. A design approach for the systematic improvement of support locations for vibrating circuit cards. 4. J. M. P and A. A. P 1991 Proceedings of the ASME Winter Annual Meeting, Atlanta, paper 91-WA-EEP-34. Comparison of modelling techniques for the vibration analysis of printed circuit cards. 5. A. O. C and A. K 1993 Proceedings of the 43th Electronic Components and Technology Conference, Lake Buena Vista, Florida, 270–275. Dynamic behavior of printed wiring boards: increasing board stiffness by optimizing support locations. 6. R. K. S and D. J. G 1992 Transactions of the American Society of Mechanical Engineers, Journal of Electronic Packaging 114, 239–245. A general analytical solution for free vibrations of rectangular plates resting on fixed supports and with attached masses. 7. T. W, K. S and G. W 1991 Transactions of the American Society of Mechanical Engineers, Journal of Electronic Packaging 113, 244–249. Experimental modal analysis and dynamic response prediction of PC boards with surface mount electronic components. 8. J. C. R and D. M. S 1991 Journal of the IES Society 34, 25–31. Random vibration analysis of a printec circuit board with electronic components. 9. D. S. S 1988 Vibration Analysis for Electronic Equipment. New York: John Wiley; second edition. 10. J. W. D 1990 Packaging of Electronic Systems. New York: McGraw-Hill. 11. M. P 1991 Handbook of Electronic Package Design. New York: Marcel Dekker. 12. A. A. P 1991 Master Thesis, State University of New York, Binghamton, Mechanical Engineering Department. Comparison of modelling techniques for the vibration analysis of circuit cards. 13. R. A. F 1971 The Design of Experiments. New York: Hafner; ninth edition. 14. G. E. P. B, W. G. H and J. S. H 1978 Statistics for Experimenters. New York: John Wiley. 15. G. E. P. B and J. S. H 1961 Technometrics 3, 331–351. The 2k − p fractional factorial design, part I. 16. G. E. P. B and J. S. H 1961 Technometrics 3, 449–458. The 2k − p fractional factorial design, part II. 17. F. Y 1970 Selected Papers. New York: Hafner (Macmillan). 18. V. S, J. H. K, C. A. N, A. K and J. T 1992 IBM Research Division Report RC 18223. Yorktown Heights, New York. Stud stress problem: a statistical study. 19. F. Y 1937 Imperial Bureau of Social Science, Bulletin 35, Harpenden, Herts, U.K.: Hafner (Macmillan). The design and analysis of factorial experiments. 20. ABAQUS U’ M 1992, Hibbit, Karlsson and Sorensen Inc., Pawtucket, Rhode Island, U.S.A.. 21. A. O. C 1994 IEEE CHMT Transactions (in press). Estimating the dynamic behavior of printed circuit boards.
APPENDIX A: NUMERICAL CALCULATIONS
In what follows, it is explained in more detail, using Case 1 as an example (see Table 2), how the numerical calculations were carried out. First, recall that the design variables (factors) were redefined, for convenience, according to equations (1)–(6). Then, using equations (16) and (17), the lower and upper bounds for each factor were computed. This results in the following intervals: [17·595 × 109, 23·805 × 109], [5178·2, 7005·8], [21·993 × 109, 29·756 × 109], [15534·6, 21017·4], [57800·0, 78200·0] and [25·5, 34·5]. The first interval contains X1 , the second interval corresponds to X2 , and so on.
604
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With this information one can determine the rows of the linear system of equations given by equation (12). For example, the first row is determined by assuming Y1 = Y2 = Y3 = Y4 = Y5 = Y6 = −1 (that is, all the design variables are assumed to take the value given by the lower end of each interval). The corresponding value of f1 , (which is determined by the finite element model and ABAQUS) is 114·0 Hz. Thus, the first row of equation (12) becomes D000000 + D100000 × (−1) + D010000 × (−1) + · · · + D000001 × (−1) + D110000 × (−1)(−1) + · · · + D000011 × (−1)(−1) + · · · + D111111 × (−1)(−1)(−1)(−1)(−1)(−1) = 114.
(A1)
Then, the second row of equation (12) is determined assuming Y1 = +1 and Y2 = Y3 = Y4 = Y5 = Y6 = −1; and the corresponding value of f1 is 116 Hz. This yields D000000 + D100000 × (+1) + D010000 × (−1) + · · · + D000001 × (−1) +D110000 × (+1)(−1) + · · · + D000011 × (−1)(−1) +· · · + D111111 × (+1)(−1)(−1)(−1)(−1)(−1) = 116.
(A2)
Then, simply by covering all possible combinations of (−1) and (+1) for the variables Y1 , . . . , Y6 , one obtains a linear system of size 64 × 64. Solving this linear system, one obtains the values of the components for the vector D. For instance, D000000 (113·96) corresponds to the mean. The remaining D’s have been ranked according to their absolute values (in decreasing order); only the first seven coefficients are shown in Table 2. The coefficients have been identified using, again, equations (1)–(6). Thus, rc is associated with D000100 , Ec with D001000 , Eb rc with D100100 , and so on.
A SENSITIVITY STUDY OF PRINTED WIRING BOARD VIBRATIONS USING A STATISTICAL METHOD A. O. C, N. S C. A. N IBM T. J. Watson Research Center, P.O. Box 218, Yorktown Heights, New York 10598, U.S.A. (Received 22 July 1993, and in final form 18 January 1994) The dynamic behavior of a printed wiring (PW) board plays an important role in its reliability. One key parameter in determining such behavior is the fundamental frequency of the PW board. This paper shows how the sensitivity of the natural frequencies of the board with respect to some design variables can be studied using a statistical approach and how this can guide the experimental and modeling phases of the design.
1. INTRODUCTION
The basic building block of most structures found in the electronics industry is the PW board. The dynamic behavior of such boards is becoming an important issue in determining the reliability of these structures. A number of failures at the connectors level between the board and the components (heat sinks, capacitors, transformers, etc.) can be attributed to vibration or fatigue resulting from dynamic loads. Thus, the need to study the dynamic behavior of a PW board as a function of its design variables is apparent. Previously, many authors have addressed different aspects of this problem. Suhir [1] has studied the non-linear response of boards subjected to a sudden acceleration applied at the boundary. Pitarresi et al. [2–4] have studied some modelling issues regarding the structural representation of boards plus components, as well as the relationship between the stiffness of the board and the location of its supports. Cifuentes and Kalbag [5] developed a method to optimize the location of the supports to maximize the natural frequency of the PW board. Singal and Gorman [6] have performed experimental studies to quantify the influence of concentrated masses on the fundamental frequency of plates (a model frequently used to represent PW boards). Wong et al. [7] have dealt with the application of modal methods to estimate the dynamic response of boards [7]. Roberts and Stillo [8] have done work focusing on the application of random analysis techniques to compute the dynamic response of these structures. A good review of current design practices can be found in the books by Steinberg, Dally or Pecht [9–11]. A key parameter in determining the dynamic behavior of a board is its fundamental natural frequency. Hence, in most cases it is used as a figure of merit to assess the goodness of a possible design. For example, in general one wants the board to be as stiff as possible (within certain constraints) in order to minimize the amplitude of the dynamic displacements. This is equivalent to maximizing the fundamental frequency of the board. In addition, requirements in terms of fatigue life of components as well as maximum acceptable accelerations in certain regions of the board often determine minimum design values for the fundamental frequency. As a result, it is important for the engineer to know 593 0022–460X/95/140593 + 12 $08.00/0
7 1995 Academic Press Limited
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which are—on a given board—the critical design variables that affect the value of the natural frequency. This knowledge is useful in two situations. First, if a board does not meet the design requirements, it is necessary to know which parameters (design variables) should be changed. One would prefer to change those design variables that are the most influential in the value of the fundamental frequency. Second, since in general the reliability of a board is studied using laboratory tests combined with computer simulations (finite elements, for instance) it is extremely useful to know which are the most important parameters to guide the experiments. One wants to concentrate the effort on measuring the parameters that will be crucial, to make a good model. Some parameters are quite easy to determine experimentally (for example, the mass of an unpopulated board), while others are more cumbersome to estimate (the stiffening effect of a component on a certain region of a board). Thus, this information is very valuable if one is to avoid spending resources measuring irrelevant quantities. This paper is concerned with studying the sensitivity of the fundamental frequency of a PW board using a statistical approach. The following sections describe the modelling strategy employed to represent PW boards using finite elements, the statistical method and several case studies to demonstrate the advantages of this approach. The last sections deal with a discussion of the results and conclusions. 2. MODELLING OF PW BOARDS
A PW board is a flat surface, normally of rectangular shape, on which components are mounted. Mechanically speaking, the PW board behaves as a thin plate (shear deformations are negligible). The components, the total mass of which can be significant compared to that of the board, are mounted on top of the board surface and attached to it by a number of techniques, such as pins, J-lead connections, gullwing connections and elastomeric interposes. Thus, a finite element representation of a board reduces to an assembly of plate elements. As far as the components are concerned, there is a wide variety of possibilities. These range from rather simple models in which the presence of a component is ignored or represented as a concentrated mass if the mass of such component exceeds a certain critical value, to more elaborate models in which each component is described in detail using an assembly of thick plate, beam and possibly solid elements. Pitarresi [2, 3] has studied this issue in detail. In this study, the presence of a component is modelled by considering an equivalent mass density and stiffness for the plate elements located in the area where the component is situated. This approach has been validated experimentally, with excellent results [4, 12]. In short, the PW board model reduces to an assembly of plate elements in which those elements corresponding to areas in which there are no components have the properties (mass density and Young’s modulus) of the unpopulated board; and the plate elements corresponding to areas in which components are mounted have a higher mass density and stiffness to account for the presence of such components. (In any event, the results presented in this paper are not tied to a particular modelling strategy.) In addition, one has to consider the support conditions—the way in which the board is connected to its mother structure. It is assumed that the supports can be defined in terms of two linear springs; a translational spring acting in the direction normal to the board plane and a rotational spring in which the rotary axis is parallel to the board edge. In this study, a board is characterized in terms of six design variables: (1) Eb , the Young’s modulus of the unpopulated board, (2) rb , the mass density of the unpopulated board,
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(3) Ec , the Young’s modulus of the area in which the component is located (that is, this value represents the stiffness of the board-plus-component assembly in that region), (4) rc , the mass density in the region in which the component is located, (5) KZ , the stiffness of the individual springs placed at each support node, and (6) Ku , the rotational stiffness of the springs at the support nodes. This idealization is shown in Figure 1. For simplicity, it has also been assumed that all board components are defined by the same local properties (in case there are several components). This study considers the relationship between the fundamental frequency of the board ( f1 ) and the six above-mentioned parameters (or design variables). In a way, one might consider these variables as subject to error. Consequently, one studies how an error in one of these variables influences the error in determining the value of f1 . There are, obviously, a number of other variables that play a role in the value of the fundamental frequency. In this study, however, they are not considered as variables. For example, the Poisson ratio of the material is not considered as a variable, simply because in general the value of f1 is not very sensitive to variations in n. The board dimensions (length, width and thickness), on the other hand, although they influence the value of f1 greatly, are quite easy to determine. In other words, one would expect the designer to have access to an accurate estimate of these variables without too much trouble. In any event, the treatment of a problem choosing a different set of design variables (or a frequency other than the fundamental frequency) would be analogous to what is presented here. 3. STATISTICAL METHOD
The approach used in this study is based on the design-of-experiments theory proposed by Fisher in the 1920’s [13]. The original motivation for Fisher’s work was to analyze problems arising in agricultural sciences. Later, Box et al. extended these ideas to treat a number of industrial problems while Yates—who also did some pioneering work in the field—addressed some computational aspects of these methods [14–17]. Recently, Srinivasan et al. [18] employed this approach to analyze the sensitivity of a thin-film structure in the context of thermal stresses. For the sake of clarity, the so-called full factorial design of experiments is explained here in reference to the PW vibration problem considered.
Figure 1. A typical finite element model of a PW board having two components.
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Let f1 be the ‘‘response’’ of interest. It is assumed then that the relationship between f1 and the six ‘‘factors’’ (design variables) can be expressed in terms of a multi-linear function. For simplicity, the factors can be re-labelled as follows: r b = X2 ,
Eb = X1 ,
rc = X 4 ,
Ec = X3 ,
K2 = X5 ,
Ku = X6 .
(1–6)
Hence, one can write f1 (X1 , . . . , X6 ) = C000000 + C100000 X1 + C010000 X2 + · · · + C000001 X6 +C110000 X1 X2 + · · · + C000011 X5 X6 + · · · + C111111 X1 X2 X3 X4 X5 X6 ,
(7)
where the C’s are unknown (in principle) but are constants. A ‘‘1’’ in the ith position of the six-digit subindex indicates the ith factor is considered in that term. In this case, there are 26 = 64 unknown coefficients. Assume that each factor (X1 , . . . , X6 ) is subjected to some uncertainty or error expressed in terms of an interval that contains the true value of the factor: that is, for i = 1, . . . , 6 it is considered that Xi $ [Xi min , Xi max , where Xi min represents the lower end of the interval and Xi max denotes the upper bound. Performing a change of variable Xi =
Xi
min
+ Xi 2
max
+
Xi
max
− Xi 2
min
Yi ,
(8)
one can normalize the factors and rewrite equation (7) as f1 (Y1 , . . . , Y6 ) = D000000 + D100000 Y1 + D010000 Y2 + · · · + D000001 Y6 + D110000 Y1 Y2 + · · · + D000011 Y5 Y6 + · · · + D111111 Y1 Y2 Y3 Y4 Y5 Y6 .
(9)
There are still 64 unknown coefficients. However, the factors (Y1 , . . . , Y6 ) vary only between −1 and +1. D000000 is called the average or mean (that is, the value obtained for f1 , when all the normalized factors take the value 0—center of the interval). Twice the coefficient of each of the linear factors is called the main effect of the associated factor; and twice the coefficient of the product of each combination of two factors is known as the two-factor interaction. (Note that one of the advantages of this method, in contrast with less sophisticated approaches in which variables are varied one at a time, is that one can detect—through the two-factor interactions—the influence of coupling effects that otherwise would go unnoticed.) Finally, the so-called sensitivities, that is, the derivatives of f1 (in this case) with respect to the factors (design variables), can be written as 1f1 = 1Xi Xi
2 − Xi min
1f1 , 1Y max i
(10)
where 1f1 = D00i000 + D100i00 Y1 + · · ·. 1Yi
(11)
If one considers equation (9) (the normalized equation) and all the possible combinations of −1 and +1 for the values of the factors (Y1 , . . . , Y6 ), one can establish 26 = 64 equations. These equations determine a linear system of the form [A]{D} = { f },
(12)
where [A] represents a 64 × 64 matrix, the elements of which take the value of −1 or +1; {D} denotes the vector that contains the unknown coefficients; and { f } denotes the vector with the values of the ‘‘response function’’ (the natural frequency in this case) for different combinations of −1 and +1 of the normalized factors. It can be shown that the matrix
Figure 2. PW board configuration, Case 1. Finite element mesh, location of supports (springs) and area occupied by the components (heavy line). W, Locations of the supports, which consist of a vertical spring (KZ ) plus a rotational spring (Ku ).
597
Figure 3. PW board configuration, Case 2. Finite element mesh, location of supports (springs) and area occupied by the components (heavy line). W, Locations of the supports, which consist of a vertical spring (KZ ) plus a rotational spring (Ku ).
[A] is non-singular and orthogonal. This system can be solved using any of the methods available for linear systems; however, a more efficient approach is to employ Yates’ algorithm [17, 19]. (Yates’ algorithm requires a number of operations proportional to n log n, whereas non-specialized L–U decomposition methods require a number of operations proportional to n 2.) Finally, after solving equation (12) one can determine, by inspecting the value of the coefficients, the relative influence of each factor (design variable) in the value of f1 . In essence, this has been achieved by studying how the relative errors in the value of the factors influence the error in the value of f1 . 4. EXAMPLE OF APPLICATION
Six PW boards, depicted in Figures 2–7, are considered in this study. The properties of these PW boards are listed in Table 1. These PW boards represent a variety of realistic configurations. Their plane dimensions vary from 0·08 m to 0·4 m, whereas the
Figure 4. PW board configuration, Case 3. Finite element mesh, location of supports (springs) and area occupied by the components (heavy line). W, Locations of the supports, which consist of a vertical spring (KZ ) plus a rotational spring (Ku ).
Figure 5. PW board configuration, Case 4. Finite element mesh, location of supports (springs) and area occupied by the components (heavy line). W, Locations of the supports, which consist of a vertical spring (KZ ) plus a rotational spring (Ku ).
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Figure 7. PW board configuration, Case 6. Finite element mesh, location of supports (springs) and area occupied by the components (heavy line). W, Locations of the supports, which consist of a vertical spring (KZ ) plus a rotational spring (Ku ). (The supports marked with a × only have a vertical spring.)
Figure 6. PW board configuration, Case 5. Finite element mesh, location of supports (springs) and area occupied by the components (heavy line). W, Locations of the supports, which consist of a vertical spring (KZ ) plus a rotational spring (Ku ).
fundamental frequency goes from 33·34 Hz (a fairly flexible board) to 163·77 Hz (a rather rigid board). In addition, consider the figures of merit a, b and g, defined as follows. a=
f1 , frigid
(13)
where frigid is the fundamental frequency of the board obtained by substituting the springs by rigid supports (that is, KZ and Ku become infinity). A value of a close to 1 indicates that the actual springs (supports) are very rigid. Decreasing values of a relative to 1 correspond to increasing levels of flexibility of the supports. b = area of components/total area,
(14)
thus b indicates the fraction of the PW board that is occupied by the components; and g=
mass of components (r − rb )Ac = c , mass of unpopulated board rb A
(15)
T 1 Properties of the PW boards considered in the example Case
1
2
3
4
5
6
Eb (N/m2) × 109 20·7 17·0 22·0 16·0 20·0 20·0 rb (kg/m3) 6 092 5 120 4 000 1 800 4 000 4 500 Ec (N/m2) × 109 25·9 18·7 24·0 24·0 28·0 40·0 rc (kg/m3) 18 276 10 240 6 000 5 400 11 000 9 500 KZ (N/m) 68 000 1 200 12 500 25 000 15 500 11 300 Ku (N-m/radian) 30 11 120 35 0·5 5 Length (m) 0·17 0·4 0·3 0·3 0·1 0·25 Width (m) 0·17 0·15 0·2 0·2 0·1 0·08 Thickness (m) 0·0023 0·00127 0·0015 0·0015 0·0012 0·0018 nb 0·18 0·3 0·3 0·3 0·2 0·25 nc 0·33 0·3 0·3 0·3 0·2 0·25 a 0·57 0·31 0·93 0·83 0·53 0·70 b 0·25 0·25 0·166 0·208 0·64 0·166 g 0·5 0·25 0·08 0·41 1·12 0·18 f1 (Hz) 114·00 33·34 39·59 82·93 163·77 81·37
599
where A refers to the total area of the board and Ac denotes the area occupied by the components. Thus, g indicates how heavy the components are with respect to the unpopulated board. Inspection of Table 1 indicates that the PW boards chosen vary from designs with very flexible supports (a = 0·31) to fairly stiff designs (a = 0·93). In addition, the relative areas occupied by the components (b) range from 0·166 to 0·64 and the mass of the components compared to that of the board (g) varies between 0·08 and 1·12. In short, the six examples selected correspond to a very diverse set of configurations, that can be considered a representative sample. The fundamental frequency of each board was computed using a finite element model with the mesh indicated in the figures and with the help of ABAQUS (a general-purpose finite element code [20]). This value is shown for each board at the bottom of Table 1. Finally, the length, width and thickness of the boards as well as the value of the Poisson ratio are not considered as variables in the statistical analysis. 5. RESULTS
It was assumed that each one of the factors (design variables) was subjected to a 15% error; that is, for i = 1, . . . , 6, Xi
min
= (Xi )ref − 0·15 × (Xi )ref
(16)
= (Xi )ref + 0·15 × (Xi )ref ,
(17)
and Xi
max
where (Xi )ref represents the nominal value of the ith factor as given in Table 1. The choice of an error of 15% in the determination of the design variables is arbitrary; what is important is that the error is the same in all variables—in this way it is easier to make a meaningful comparison in terms of which design variables are more influential in determining the value of f1 . Hence, by computing the value of f1 for all the possible combinations of −1 and +1 of the normalized variables (see equation (9)) one creates a linear system of equations (see equation (12)) which was solved using Yates’s algorithm. This process was automated using several C++ programs—and a UNIX shell script to drive them—in combination with ABAQUS, which runs the eigenvalue analysis. The results for the six cases described in Figures 2–7 are summarized in Tables 2–7. In the tables the factors have been ranked in decreasing order of importance. Only the first seven factors have been included (the remaining 57 make negligible contributions—they account for a small portion of the mean value). For convenience, the coefficients have been identified in terms of the variables they are related to, instead of using the six-subindex notation. T 2 Values of the coefficients for Case 1
T 3 Values of the coefficients for Case 2
Factor Coefficients (D’s) Percent of mean
Factor Coefficients (D’s) Percent of mean
Mean rc Ec KZ Ku rb Eb Eb rc
Mean KZ rb rc Eb Ec Eb rc Ku
113·96 −6·76 2·81 2·67 2·21 −1·86 0·94 0·34
100·00 −5·94 2·46 2·34 1·94 −1·62 0·82 0·29
33·34 1·98 −1·60 −0·93 0·27 0·14 0·14 0·12
100·00 5·95 −4·80 −2·77 0·81 0·44 0·42 0·36
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600 T 4
T 5
Values of the coefficients for Case 3
Values of the coefficients for Case 4
Factor Coefficients (D’s) Percent of mean
Factor Coefficients (D’s) Percent of mean
Mean Eb rb rc Ec KZ Eb rc Ku
Mean rc Eb rb Ec KZ Ku Eb rc
39·56 2·04 −1·71 −1·28 0·56 0·28 0·17 0·10
100·00 5·17 −4·32 −3·25 1·42 0·70 0·44 0·26
82·70 −4·07 3·08 −2·19 1·26 0·98 0·92 0·34
100·00 −4·92 3·73 −2·65 1·53 1·18 1·11 0·41
(In Appendix A it is shown in more detail, using Case 1 as an example, how the numerical entries that define equation (12) were generated.) Note that the value listed as ‘‘mean’’ (actually the D000000 coefficient of equation (9)) coincides—except for some roundoff—with the value previously computed for f1 and shown in Table 1. (It is really the constant term in a Taylor’s series expansion.) Also, in any of these tables, summation of the absolute value of all the entries listed in the third column (except, of course, for the value 100·00 which corresponds to the mean) yields roughly 15·0%. Simply put, this indicates that if one computes the value of f1 assuming for each factor the value given by the lower or upper end of the interval, the error in f1 is roughly 15%. This is consistent with the standard result from error theory—that the relative error in the function to be evaluated is approximately equal to the sum of the relative errors of each one of the variables involved in evaluating that function (because of the normalization). Finally, consider that ultimately the value of f1 is proportional to the square root of the ratio between the modal stiffness and the modal mass associated with the first mode of vibration: that is, f1 = v1 /2p = (1/2p)zKmodal /Mmodal ,
(18)
where Kmodal denotes the modal stiffness, Mmodal denotes the modal mass and v1 refers to the fundamental frequency expressed in radians per second. Noting that 1v1 1 = 1Kmodal 2Mmodal v1
and
1v1 v1 =− 1Mmodal 2Mmodal
(19, 20)
one can conclude, using a Taylor’s expansion with only the linear terms, that
b b b Df1 E f1
1 2
b b
DKmodal + Kmodal
1 2
b
DMmodal , Mmodal
(21)
T 6 Values of the coefficients for Case 5
T 7 Values of the coefficients for Case 6
Factor Coefficients (D’s) Percent of mean
Factor Coefficients (D’s) Percent of mean
Mean rc Ec KZ Ku Eb Ec rc rb
Mean rb KZ Eb rc Ec Eb rc Eb K Z
164·36 −12·09 6·54 3·09 1·98 0·79 −0·46 −0·31
100·00 −7·35 3·97 1·88 1·20 0·48 −0·28 −0·19
81·29 −4·47 3·34 2·29 −1·68 0·50 0·29 0·27
100·00 −5·49 4·11 2·82 −2·07 0·62 0·35 0·32
601
T 8 Values of the coefficients for Case 1 (second vibration mode) Factor Coefficients (D’s) Percent of mean Mean KZ rb rc Eb Ec Ku Eb rc
264·13 12·32 −11·91 −8·09 3·51 2·85 1·30 1·19
100·00 4·66 −4·51 −3·07 1·33 1·08 0·49 0·45
where Df1 , DKmodal and DMmodal denote the absolute error in f1 , Kmodal and Mmodal . In essence, equation (21) says that if the relative errors in the mass and in the stiffness are the same, then their contributions to the error in f1—or, more precisely, to an upper bound of the error—are the same. This is indeed the case in the examples. In any of the tables one can verify that adding the positive coefficients asscociated only with linear factors (that is, those associated with the stiffness) from the third column (except for the value 100·00) one obtains approximately 7·5% (half of 15%), consistent with the initial error assumption and equation (21). In the same way, adding the negative coefficients (which are associated with the mass) one obtains approximately 7·5%.
6. DISCUSSION
Inspecting the results presented in Tables 2–7, one can see that the method does indeed identify the few influential factors that affect the value of f1 the most. In general, two and higher factor interactions between the design variables play a minor role. The most important conclusion—somewhat to be expected from a structural mechanics standpoint—is that there is no consistency in the order in which the variables are ranked for all six cases. Thus, a variable that can play a major role in one case (KZ in Case 2, for example) can be almost irrelevant in other situation (see Case 3). Therefore, there is no such thing as the most important parameter in the dynamic behavior of a PW board—it all depends on the specifics of the design. The highest value of a (Case 3) corresponds to a major role for Eb , whereas in the case of the smallest value of a (Case 2), KZ takes that place. Again, this is in agreement with intuition—in a board with flexible supports, the stiffness of the supports is more important; in a board that is more rigidly supported, variations in the stiffness of the board are dominant. Also, one notices that for higher values of b the importance of the stiffness of the components (Ec ) grows—compare Cases 5 and 6. Furthermore, increasing values of g reflect an increasing importance of rc —compare Case 5 with Cases 3 or 6, for instance. Of course, the values of a, b and g only reflect general trends and cannot be used to predict the rank of the design variables. Only the statistical model can do this. A specific result that one notes in these tables is that rc is always more influential than Ec in the value of f1 . This is in agreement with analytical results presented by Cifuentes elsewhere [21]. This result is very important for the modeller; it means that one should concentrate in obtaining a better estimate of the mass of the components rather than the stiffness of the area in which the components are located. This is also fortunate from the
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602
experimental point of view for, in general, it is a lot easier to determine the mass of a component than its contribution to the stiffness of the board. In summary, these results show that there cannot be rigid guidelines in terms of determining a priori which are the key design variables that control the dynamic response of a PW board. However, the statistical approach described here constitutes a powerful technique to identify the key variables (if a certain design needs to be modified). Moreover, an approximate model of a PW board can also be used in combination with this technique to guide the experimental testing. For instance, in a case such as Case 2 there is no point in trying to determine Ku very accurately; the effort should be concentrated on measuring Kz and rb . Finally, the statistical technique presented here is not restricted to investigating the first natural frequency of the PW board. In Table 8 are shown the results obtained for the second natural frequency in Case 1 (that is, the PW board shown in Figure 2). Comparing these results against those from Table 2, one can see that the design variables are ranked in quite a different sequence. In short, this demonstrates that what might be an important factor in the determination of f1 might not be so important in the determination of f2 (the second natural frequency). It is crucial to keep this in mind if a certain design needs to be improved—for instance, to increase the overall stiffness of the board. In such a case, the statistical analysis should be undertaken considering the natural frequency of the mode that accounts for the largest fraction of the total mass (which, in some instances, might not be the first mode of vibration). Notice also that rc is still more important than Ec when it comes to computing f2 and that the positive and negative coefficients from the third column (Table 8) still add up to approximately 7·5%. These two results hold for any normal mode. (Obviously in this case, equation (9) still holds, except that f1 needs to be replaced by f2 .)
7. CONCLUDING REMARKS
A statistical method based on the theory of experiments has been employed to investigate the sensitivity of the natural frequency of six PW boards as a function of several design variables. This approach is expected to be useful for designers of PW boards. First, it can identify the critical variables, should modifications to a design become necessary. Second, it can guide the laboratory testing by identifying the variables that need to be measured with more accuracy. Extension of this approach to study PW boards using a different set of design variables is straightforward.
T 8 Values of the coefficients for Case 1 (second vibration mode) Factor Coefficients (D’s) Percent of mean Mean KZ rb rc Eb Ec Ku Eb rc
264·13 12·32 −11·91 −8·09 3·51 2·85 1·30 1·19
100·00 4·66 −4·51 −3·07 1·33 1·08 0·49 0·45
603
REFERENCES 1. E. S 1992 International Journal of Solids and Structures 29, 41–55, Nonlinear dynamic response of a flexible thin plate to constant acceleration applied to its support contour with application to printed circuit boards in avionic packaging. 2. J. M. P, D. V. C, R. C and D. E. S 1991 Transactions of the American Society of Mechanical Engineers, Journal of Electronic Packaging 113, 250–257. The ‘‘smeared’’ property technique for the FE vibration analysis of printed circuit cards. 3. J. M. P and A. V. D E 1993 Transactions of the American Society of Mechanical Engineers, Journal of Electronic Packaging 115, 118–123. A design approach for the systematic improvement of support locations for vibrating circuit cards. 4. J. M. P and A. A. P 1991 Proceedings of the ASME Winter Annual Meeting, Atlanta, paper 91-WA-EEP-34. Comparison of modelling techniques for the vibration analysis of printed circuit cards. 5. A. O. C and A. K 1993 Proceedings of the 43th Electronic Components and Technology Conference, Lake Buena Vista, Florida, 270–275. Dynamic behavior of printed wiring boards: increasing board stiffness by optimizing support locations. 6. R. K. S and D. J. G 1992 Transactions of the American Society of Mechanical Engineers, Journal of Electronic Packaging 114, 239–245. A general analytical solution for free vibrations of rectangular plates resting on fixed supports and with attached masses. 7. T. W, K. S and G. W 1991 Transactions of the American Society of Mechanical Engineers, Journal of Electronic Packaging 113, 244–249. Experimental modal analysis and dynamic response prediction of PC boards with surface mount electronic components. 8. J. C. R and D. M. S 1991 Journal of the IES Society 34, 25–31. Random vibration analysis of a printec circuit board with electronic components. 9. D. S. S 1988 Vibration Analysis for Electronic Equipment. New York: John Wiley; second edition. 10. J. W. D 1990 Packaging of Electronic Systems. New York: McGraw-Hill. 11. M. P 1991 Handbook of Electronic Package Design. New York: Marcel Dekker. 12. A. A. P 1991 Master Thesis, State University of New York, Binghamton, Mechanical Engineering Department. Comparison of modelling techniques for the vibration analysis of circuit cards. 13. R. A. F 1971 The Design of Experiments. New York: Hafner; ninth edition. 14. G. E. P. B, W. G. H and J. S. H 1978 Statistics for Experimenters. New York: John Wiley. 15. G. E. P. B and J. S. H 1961 Technometrics 3, 331–351. The 2k − p fractional factorial design, part I. 16. G. E. P. B and J. S. H 1961 Technometrics 3, 449–458. The 2k − p fractional factorial design, part II. 17. F. Y 1970 Selected Papers. New York: Hafner (Macmillan). 18. V. S, J. H. K, C. A. N, A. K and J. T 1992 IBM Research Division Report RC 18223. Yorktown Heights, New York. Stud stress problem: a statistical study. 19. F. Y 1937 Imperial Bureau of Social Science, Bulletin 35, Harpenden, Herts, U.K.: Hafner (Macmillan). The design and analysis of factorial experiments. 20. ABAQUS U’ M 1992, Hibbit, Karlsson and Sorensen Inc., Pawtucket, Rhode Island, U.S.A.. 21. A. O. C 1994 IEEE CHMT Transactions (in press). Estimating the dynamic behavior of printed circuit boards.
APPENDIX A: NUMERICAL CALCULATIONS
In what follows, it is explained in more detail, using Case 1 as an example (see Table 2), how the numerical calculations were carried out. First, recall that the design variables (factors) were redefined, for convenience, according to equations (1)–(6). Then, using equations (16) and (17), the lower and upper bounds for each factor were computed. This results in the following intervals: [17·595 × 109, 23·805 × 109], [5178·2, 7005·8], [21·993 × 109, 29·756 × 109], [15534·6, 21017·4], [57800·0, 78200·0] and [25·5, 34·5]. The first interval contains X1 , the second interval corresponds to X2 , and so on.
604
. . ET AL .
With this information one can determine the rows of the linear system of equations given by equation (12). For example, the first row is determined by assuming Y1 = Y2 = Y3 = Y4 = Y5 = Y6 = −1 (that is, all the design variables are assumed to take the value given by the lower end of each interval). The corresponding value of f1 , (which is determined by the finite element model and ABAQUS) is 114·0 Hz. Thus, the first row of equation (12) becomes D000000 + D100000 × (−1) + D010000 × (−1) + · · · + D000001 × (−1) + D110000 × (−1)(−1) + · · · + D000011 × (−1)(−1) + · · · + D111111 × (−1)(−1)(−1)(−1)(−1)(−1) = 114.
(A1)
Then, the second row of equation (12) is determined assuming Y1 = +1 and Y2 = Y3 = Y4 = Y5 = Y6 = −1; and the corresponding value of f1 is 116 Hz. This yields D000000 + D100000 × (+1) + D010000 × (−1) + · · · + D000001 × (−1) +D110000 × (+1)(−1) + · · · + D000011 × (−1)(−1) +· · · + D111111 × (+1)(−1)(−1)(−1)(−1)(−1) = 116.
(A2)
Then, simply by covering all possible combinations of (−1) and (+1) for the variables Y1 , . . . , Y6 , one obtains a linear system of size 64 × 64. Solving this linear system, one obtains the values of the components for the vector D. For instance, D000000 (113·96) corresponds to the mean. The remaining D’s have been ranked according to their absolute values (in decreasing order); only the first seven coefficients are shown in Table 2. The coefficients have been identified using, again, equations (1)–(6). Thus, rc is associated with D000100 , Ec with D001000 , Eb rc with D100100 , and so on.