Microelectronics Reliability 50 (2010) 86–97

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Microelectronics Reliability

journal homepage: www.elsevier .com/locate /microrel

Accuracy of simplified printed circuit board finite element models

Robin Alastair Amy a,*, Guglielmo S. Aglietti a, Guy Richardson b

a Astronautical Research Group, School of Engineering Sciences, University of Southampton, UKb Surrey Satellite Technology Limited, Guildford, Surrey, UK

a r t i c l e i n f o a b s t r a c t

Article history:Received 14 October 2008Received in revised form 23 February 2009Available online 17 October 2009

0026-2714/$ - see front matter � 2009 Elsevier Ltd. Adoi:10.1016/j.microrel.2009.09.001

* Corresponding author.E-mail address: robinamy@soton.ac.uk (R.A. Amy)

Electronic components are prone to failure due to shock or vibration loads. To predict when this failuremay occur it is necessary to calculate the vibration response of the printed circuit board (PCB); this ismost usually achieved through use of simplified finite element (FE) models. The accuracy of these FEmodels will be mainly dependant on various sources of error, including: manufacturing variability, whichwill cause supposedly identical printed circuit boards to behave differently (including variability in mate-rials and assembly, as well as dimensional tolerances); inaccuracy in the model input parameters, whichis caused by either the modelling assumptions used or poor measurement technique; and errors in thesolution process (e.g. linear solutions in non-linear situations). This paper investigates experimentallythe contribution of these effects, this is achieved by first looking at measurement of input parametersand to what accuracy a PCB can reasonably be modelled, and then secondly measuring the extent of man-ufacturing and assembly induced variability. When these contributions have been defined, it will be pos-sible to assess the confidence in any FE PCB model.

� 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Shock and vibration loads imposed on a printed circuit boardmay, if severe enough, cause the attached electronic componentsto fail. The probability of this failure occurring is strongly influ-enced by the local PCB vibration response; therefore, if equipmentvibration reliability is to be predicted then so too must the PCB lo-cal vibration response. The most convenient and commonly usedmethod of predicting the local vibration response is through thecreation of finite element models.

Many good examples of detailed finite element models alreadyexist in the current literature [1–8], where the complexity of thesemodels is justified as the component internal stresses are required;however, the long time required to build and solve the models can-not be justified when only the PCB response is required. It is pos-sible to greatly simplify the creation of these models by usinglead spring constants in place of detailed lead models [9–11]. Fur-ther simplification can be achieved by removing the detailed 3Dmeshes of the component from the FE model altogether: insteadtheir effects can be included by artificially increasing the stiffnessand density of the underlying PCB properties in the FE model[12,13]. These artificial properties can either be included at the ex-act component location or averaged (sometimes called ‘‘smearing”)over the entire area of the board. As well as the detailed andsmeared models there are also models that use simple block 3Delements to recreate component effects [4,14–16], these models

ll rights reserved.

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are especially useful when relatively tall or big components arepresent. In addition to all the work on accurately incorporatingcomponent effects into the model, other work has looked theimportance of accurately incorporating boundary conditions intothe model [17,18], or specifically considering the effect of bound-ary condition modification with respect to reducing the vibrationresponse [18–21]. Finally, some previous research has noted theimportance of considering non-linear effects during modelling[22,23].

It is shown in this work that the first major difficulty in creatinggood PCB FE models is in specifying the input parameters, namely:stiffness, density, damping and boundary conditions. The accuracywith which each of these parameters is specified will determinethe accuracy of the response predicted by the final model. Anyinaccuracy in the input parameters will occur due to either poorinitial measurements or, when measurements are not possible,poor estimation of these factors.

A secondary difficulty in creating accurate PCB FE models is dueto variation that may occur in each of the input parameters, wherethe variation will cause supposedly identically manufactured andassembled PCBs to have different responses.

Finally, the model accuracy will be limited by parameters andsources of variation that cannot be easily specified or modelled.These may include non-linear effects, limitations of the FE meshand other effects too complicated to be easily included in the mod-el (i.e. air or acoustic influences). Most often these are accountedfor by including appropriate safety factors.

Using an experimental approach and keeping in mind the threepoints above, this paper will assess how accurately simplified mod-

R.A. Amy et al. / Microelectronics Reliability 50 (2010) 86–97 87

els can predict a given PCB’s response. The approach will involvetwo stages: the first stage will look at the creation of idealised the-oretical FE models of PCBs and how accurately these models pre-dict the actual response. The second stage will quantify thevariations that occur due to small differences in manufacturingand assembly, allowing quantification of the expected variabilityand its effect on response. There are three main contributions ofthis work: (1) future researchers in this area can follow a similarprocess to the one shown here and obtain expected accuracy val-ues specific to their own equipment and manufacturing tech-niques, (2) the observations presented here allows the modellingeffort spent on future PCB models to be used much more effec-tively, and (3) the expected accuracies published here can be usedas ball-park starting values for future work.

2. Current practice

The whole FEA process can be broken into three main parts: (1)modelling (including estimating input parameters, creating a meshand incorporating measured parameters in the model), (2) analysis(type of solution, linear or non-linear, etc), and (3) post-processing(looking at appropriate response parameters, such as: acceleration,curvature or deflection). Each of these three parts will be consid-ered in turn.

2.1. Creating FE models of PCBs

In creating any FE model of a PCB it is convenient to break themodel down into five separate areas: PCB properties, Componenteffects, chassis, damping and boundary conditions.

2.1.1. PCB propertiesThe main difficulty in creating a model of a PCB is not in creat-

ing the mesh but in specifying the properties: stiffness moduli,poisson ratio, density and thickness. These properties may be pro-vided by the PCB manufacturer but those that are not should bemeasured.

The Young’s modulus is most simply calculated using a staticbend test [13]. It is important to note that as most PCBs are lami-nates their properties may vary depending on the direction of load-ing, this necessitates that the Young’s modulus be measured inboth the x and y axis of the PCB (where axis are defined as in Fig. 1).

The shear modulus of a PCB may most conveniently be deter-mined through use of a static four point bend test [13] as shownin Fig. 1, allowing the shear modulus to be calculated using:

Gxy ¼3Ktab

4t3 ð1Þ

where Kt is the slope of the load displacement curve, a and b are thespecimen edge lengths and t is the specimen thickness.

Fig. 1. Diagram of experimental set-up of torsion test. Loads are placed on the fourcorners of the specimen and the corner deflection is measured. Total load = 2F.

The density can easily be found if the PCB mass, width, lengthand thickness are known, where there is the implicit assumptionthat the thickness is constant over the PCB.

Additionally, removal of the PCB’s copper surface during manu-facturing will reduce both the mass and stiffness. Because of thisfact it is important that any bend testing should be performed onthe etched PCB, as this reduces possible errors.

2.1.2. Component effectsThe following brief discussion on component modelling is only

included to provide the reader with the minimum necessary com-prehension of the subject. This subject is too broad to satisfactorilycover in this work; therefore, whilst reading this work, it should beremembered that component modelling is not within its scope.

When components are soldered to a PCB they will locally in-crease the mass and stiffness of the PCB. For good accuracy, espe-cially when large numbers of components are present, the FEmodel should include this added mass and stiffness effect. Theoret-ically, this could be achieved by including each individual compo-nent in the model. Several good examples of such detailed modelsexist [1–4], the high level of detail in these models is justified asthese works look at stresses within the component. However, if onlythe PCB response is required then detailed models require too muchtime and effort then can normally be justified. One solution thatavoids excessively complicated models is to assume that the addi-tional mass and stiffness of the component can be included by arti-ficially increasing the PCB Young’s modulus and density [12,13].This increase may be simplified by averaging (or ‘‘smearing”) theadditional mass and stiffness over the entire area of the board.The accuracy of the method depends on the level of simplificationused and the mass and stiffness of the components present [24].

As a general rule, small components (discrete resistors andcapacitors) may usually be ignored whilst larger and heavier com-ponents (heat-sinks, power transformers, large capacitors, etc)should always be modelled. The method of including the largercomponents depends on the information required, i.e. if only thePCB response is required then usually either smeared or simple3D models are sufficient, whilst if component lead stresses are re-quired then detailed 3D models are necessary. However, these areonly very general rules, and ideally the strengths and weaknessesof each method should be understood and considered each timethey are utilised.

2.1.3. ChassisA general rule of thumb for the FE modelling of any piece of

equipment is to always model the next level up from the objectof interest [25]. In the case of electronic equipment that meansthe chassis (sometimes referred to as enclosure) should also bemodelled. Failure to include the chassis response in the modelmay severely affect the accuracy, unless the chassis is extremely ri-gid relative to the PCB. Generally, modelling a chassis is fairly sim-ple as they are relatively straightforward structures, although it isalways highly recommended to validate the FE model predicted re-sponse of the bare chassis with experimentally measured values.

2.1.4. PCB boundary conditionsOnce the FE models of the PCB and chassis have been created,

the next step is to combine the two together so that the overall re-sponse can be calculated. To achieve this, rigid and/or spring FEelements are used to connect the two models, where these ele-ments are intended to represent the effect of the PCB-chassis fixingmethod (e.g. bolts or card-lok systems).

In terms of translational displacement, most fixings will be stiffenough that they will effectively rigidly constrain the PCB transla-tional displacement to that of the chassis (this is proved in Section4 of this work); this means that simple rigid FE elements are suffi-

88 R.A. Amy et al. / Microelectronics Reliability 50 (2010) 86–97

cient to constrain the x, y and z displacement of the PCB to thechassis. However, in terms of rotational displacement all fixingmethods will display some flexibility, therefore rotational springelements are required to tie the two models together. The main dif-ficulty in incorporating these rotational spring elements into the FEmodel is in specifying the value of the spring stiffness. There aretwo possible approaches to this based on whether a prototype ofthe PCB exists or not: first, if a prototype of the PCB and chassisdoes not exist then the users options are very limited and the onlyway to obtain spring constants is to either estimate the value basedon subjective experience of previous fixings or to create a detailedFE model of the joint (which would probably be too time consum-ing to be practical). Secondly, if a real example of the PCB and chas-sis does exist then the accuracy may be improved, as experimentsmay be performed on this combined structure to calculate therotational stiffness of that fixing [17].

This second approach requires an experimental set-up incorpo-rating the fixing method of interest attached to a PCB to be created.It is then possible to use a trial and error (‘‘tuning”) approach on anFE model of this structure, where the spring rotational stiffnesstuned until the predicted frequencies match those measuredexperimentally. When the two frequencies match, assuming every-thing else in the model is correct, the stiffness used in the model isassumed the same as the real stiffness. This method has been illus-trated in works by various authors [17,18], and also extended to al-low calculation of a non-dimensional parameter for the rotationalstiffness [17]. It is also pertinent to mention that previous studieshave attempted to calculate the boundary rotational stiffnessbased on the static deflection of an experimental set-up, but theprocess was deemed impracticable [17].

The method described in this previous work [17] was originallyintended for use with card-lok style fixing mechanisms, which pro-vide clamping force along the entire edge of a PCB, not just in dis-crete location as happens with bolted PCBs. Thus, the methodshould be used with caution if it is applied to bolted-down PCBs,as it may not be a correct application of the method. If a large num-ber of bolts are present on the edge of the PCB then the situationmay be considered similar to that of a card-lok fastened PCB,whereas if the PCB is fixed in only a few locations such an assump-tion could prove unrealistic.

Other works relevant to boundary conditions also exist. The ef-fects of very small variations in screw tightness on shock responsehave been examined [26], showing that even half an M3 screw pitchof variation can alter the PCB response. The same work also showedthat the number and location of fixings dramatically alters the re-sponse, similar results have also been shown in other work [27].

2.1.5. DampingAlthough there are several techniques to experimentally mea-

sure the damping, in this paper we shall focus on the three mostconvenient methods: the logarithmic decrement method, the mag-nification-factor method and the bandwidth method [28]. The log-arithmic decrement method is a time domain approach thatcalculates damping based on the free decay of oscillation in a struc-ture. The following Eqs. (2) and (3) are used to calculate the damp-ing (fld logarithmic decrement).

r ¼ 1r

lnxn

xnþr

� �¼ 2pfldffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1� f2ld

q ð2Þ

fld ¼1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ ð2p=rÞ2q ð3Þ

where r is the logarithmic decrement per unit cycle, x is the ampli-tude of the structures response and r is the number of cycles apartin the time history.

The bandwidth method is a frequency domain method that cal-culates damping from the frequency response of the PCB. Once thefrequency response has been obtained, the response at the halfpower points is measured and the damping (fb bandwidth) calcu-lated using Eq. (4).

fb ¼12

Dxxr

ð4Þ

where Dx is the width of the frequency response curve at the halfpower level, and xr is the resonant frequency.

The Magnification-factor method is also a frequency domainmethod where the damping (fmf magnification factor) is calculatedfrom the peak transmissibility at resonance.

Q ¼ 1

2fmf

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� f2

mf

q ð5Þ

or for low damping ðf < 0:1Þ this simplifies to

Q ¼ 12fmf

ð6Þ

which is simply re-arranged to

fmf ¼1

2Qð7Þ

When choosing which of the three methods to use the followingpoints should be considered [28]. The frequency domain methodsare poor for low damping (<1%) as the curves will be difficult tomeasure accurately due to the high rate of change of the frequencyresponse curve. When such low damping is present then the timedomain methods are preferred. Another point to consider is thatthe damping may increase with increasing deflection in the struc-ture, necessitating the damping to be measured at more than onelevel of input vibration. Finally, all the methods here assume thatonly one mode is being excited, if other modes exist close to theone of interest then more detailed analysis is required outsidethe scope of this paper (see [28] for more information).

Analytical methods of finding the damping value of a structurealso exist. Steinberg [29] states that the transmissibility at reso-nance of an electronic sub-assembly is equal to two times thesquare root of the resonant frequency:

Qpeak ¼ affiffiffiffiffiffixrp

ð8Þ

where a is a fitting factor based on xr and Qpeak is the transmissibil-ity at resonance. The factor a equals 0.5 if ðxr 6 100Þ, 0.75 ifð100 < xr 6 200Þ, 1 if ð200 < xr 6 400Þ and 2 if ð400 < xrÞ. Sub-sequently the damping can be calculated from the transmissibilityby Eq. (7), provided that the level of damping is low ðf < 0:1Þ.Unfortunately the data on which Steinbergs method is based isunavailable and therefore the method is unverifiable.

2.2. Analysis stage

Typically, the dynamic response of the board will be calculatedby first performing a modal analysis and then a frequency responseanalysis using a mode superposition method. To avoid large errorswhen solving for the dynamic response of a PCB, two points shouldbe considered [23]: first, the modal solution of the model shouldconsider enough modes so that a significant fraction (roughly atleast 90%) of the total mass of the structure is excited. In addition,when the board deflections are comparable to that of the boardthickness a non-linear analysis is preferred [23]; however, no suchanalysis can be found in the current literature, most likely due tothe difficulty in specifying the non-linear input parameters (e.g.non-linear edge rotational stiffness, frequency).

Fig. 2. Set-up A, a non-populated PCB integrated into an enclosure attached to theshaker head expander.

R.A. Amy et al. / Microelectronics Reliability 50 (2010) 86–97 89

In this work, model optimisation in terms of computer re-sources is not considered as the solutions times are very short,even on relatively modest hardware (<30 s). It is assumed, there-fore, that any optimisation efforts would consume more time thanis saved. However, more computer intensive modelling strategies(detailed 3D modelling of components for example) might benefitfrom such optimisation, in this case sub-structuring methods aremost commonly used [12,13].

2.3. Post-processing stage

The final stage of the FE process is to convert the FEA output toobtain suitable output parameters; this depends on the purpose ofthe analysis. Most users will wish to compare the results withsome pre-defined failure criteria; for example, relating the acceler-ation experienced by a component to probable time for it to fail.These failure criteria may take the form of local acceleration expe-rienced by a component [30], local bending moments [31], localboard surface strain [5,6] or board deflection [29].

The choice of which failure criteria to use depends on many fac-tors, including: component size, component mass, mounting tech-nology, manufacturing quality and board thickness. The abundanceof these influencing factors makes it very difficult to recommendusing one failure criteria over another. As general rule, however,the failure of lightweight components (e.g. QFP or BGA) correlatesmost strongly with either board curvature, bending moments orboard deflection (the choice between which of these criteria touse is mainly based on ease of measurement). This is in contrastto heavy components, these are increasingly likely to fail from highaccelerations as their mass increases, and this can be attributed tothe difference in inertia between light and heavy components. It isimportant to remember that the choice of failure criteria requires athorough understanding of the previous research and should notbe taken lightly, for the strengths and weaknesses of each criteriashould be fully considered, whilst a criteria that is correct for onesituation may be incorrect for another. The difficulty in recom-mending specific failure criteria in no way discredits the workhere, this is because the models created in this work are able tocreate any of these metrics with ease.

2.4. Additional software methods

It is relevant to note that a commercial software package – Calc-ePWA – also exists for the calculation of the PCB response [2,32].This software automates the creation of PCB FE models through aGUI interface. The modelling approach used in such software isnot known as the software is proprietary, although accurateboundary condition modelling is assumed to be incorporated.

2.5. Current practice: shortcomings

The primary shortcoming of the current practice is that it is dif-ficult to find appropriate experimental data in the literature anddetailed calculations of the PCB response. With few exceptions,most past analyses have been limited to finding the natural fre-quencies and very few determine the actual frequency response[20,21]; therefore, it is difficult to estimate the accuracy of the re-sults. The work presented here attempts to overcome this issue byusing the aforementioned techniques to model two typical PCBs,leading to an analysis between the predicted response curvesand those measured experimentally.

The second issue is that the influence of manufacturing andassembly variability has not been assessed. This variability, if se-vere enough, could have a significant impact on the results, so aseries of experiments measuring variability has been carried out

and it is reported here to illustrate and discuss some typical PCBsresponse.

3. Properties determination

In the first part of this section, the material and damping prop-erties of two typical PCB set-ups will be measured, and then the re-sults from these tests are to be used to create FE models of thePCBs. An analysis of manufacturing and assembly variability willbe made after this comparison.

Throughout all the tests in this work, the frequency responsewas measured with a dynamic signal Acquisition Board (NI PCI-4472) and several small accelerometers attached to the boards(Piezotronics 0.6 g or 0.2 g). Where multiple tests were performedon identical boards the accelerometers exact position was ensuredto be identical between each test.

The two PCBs to be modelled will be hereafter referred to as Set-up A and Set-up B, they consist of the following:

Set-up A consists of an unpopulated PCB attached to an alumin-ium enclosure as shown in Fig. 2. The PCB is made of FR4 lam-inate and is attached to the enclosure with M2.5 bolts, itmeasures 289 mm by 316 mm and weighs 359 g. The enclosurehas cross-members as shown in Fig. 10 (also known as an anti-vibration frame) to provide extra support for the centre of thePCB, it was attached to the shaker head expander with 8 M5bolts and using 30 mm stand-offs to minimise air-pumpingeffects. In later tests the PCB and enclosures free–free responsesare measured by hanging each from elastic bands.Set-up B consists of an unpopulated PCB directly attached toshaker head expander with 28 M3 bolts evenly spaced aroundthe perimeter as shown in Fig. 3. The bolts gave the PCB a20 mm stand-off to minimise air-pumping effects as shown inFig. 4. Seven supposedly identical PCBs existed to allow vari-ability tests later, all measured 250 mm by 250 mm andweighed within ±1 g of 190.5 g.

3.1. Determination of PCB properties test

The aforementioned PCB modelling approach was first appliedto Set-up A (PCB and enclosure as in Fig. 2). The first step was toexperimentally determine the material properties and comparethese experimental results with the manufacturer’s provided val-

Fig. 3. Set-up B, a non-populated PCB attached to the shaker head expander, fixingmethod is shown in greater detail in Fig. 4.

Fig. 4. Detail of fixing method for Set-up B.

Fig. 5. Damping of Set-up A measured using different measurement techniques,plotted against maximum acceleration of centre of PCB.

90 R.A. Amy et al. / Microelectronics Reliability 50 (2010) 86–97

ues (see Table 1 for a comparison of manufacturers and actualmaterial properties). The manufacturer’s provided values werenot justified by giving any expected percentage variation, norwas it stated whether these values were maximum or minimumlimits. During the experimental tests the following points weremade.

The material exhibited 15% stiffness variation between the xand y axis, highlighting the significant amount of anisotropy pres-ent in the material. It was observed that during both the Youngsmodulus and shear tests that the FR4 had relatively high level ofhysteresis, and that it was necessary to consider both the loadingand unloading cycles.

Table 1Material properties of PCB in Set-up A, experimentally measured values andmanufacturers given values.

Property Measured Manufacturers values Units

Young’s modulus (x axis) 4:02� 1010 3:50� 1010 N=m2

Young’s modulus (y axis) 4:56� 1010 3:50� 1010 N=m2

Density 2481 2400 kg=m3

Shear modulus 1:21� 1010 Not given N=m2

Poisson ratio 0.3 0.3Thickness 1.54–1.66 1.65 mm

The thickness of the PCB was measured at multiple points overthe specimen; surprisingly it was found that the thickness variedover the material by 0.12 mm, if this amount of variation wasnot considered throughout the test it would drastically affect theaccuracy of the results.

The density of this particular sample measured to be2480 kg=m3 which was 3% higher than the value of 2400 kg=m3 gi-ven by the manufacturers.

Overall, it was noted that the experimentally measured proper-ties of the PCB in Set-up A were appreciably different from themanufacturers published values (see Table 1).

Using the damping measurement techniques introduced earlier(formulas (2)–(8)), the damping was measured for the combinedstructure. To investigate whether the damping varies with PCB re-sponse the measurements were performed over a range of differ-ent input accelerations and using different vibration inputs (seeFig. 5). When using the logarithmic decrement method it wasfound that decay over several cycles should be used and multiplemeasurements taken and averaged if good results are to beobtained.

The whole process of properties measurement was repeated forSet-up B (set-up shown in Fig. 3, as seven identical PCBs existed itwas also possible to measure the variation in the properties. Mea-sured values are given in Table 2 and Fig. 6). The manufacturer couldnot give the material properties of its boards, although it did statethe boards were built according to standard IPC-4101B/21. Whenmeasuring the damping properties on this second set-up it wasfound that a much higher responses could be obtained, this alloweddamping values to be given up to much higher board accelerations.

4. Finite element modelling

4.1. Set-up B

The first PCB to be modelled was Set-up B (see Fig. 3), with theintention of obtaining a plot of the predicted frequency responsethat could be compared against the real response. The model was

Table 2Average material properties of seven PCBs in Set-up B.

Property Measured Units rð%Þ

Young’s modulus (x axis) 2:4� 1010 N=m2 6.4

Young’s modulus (y axis) 2:0� 1010 N=m2 4.0

Density 1820 kg=m3 Neg.Shear modulus 4:4� 910 N=m2 4.35

Poisson ratio 0.3 –Thickness 1.69 mm 0.26

Fig. 6. Damping of PCB in Set-up B measured using different measurementtechniques, plotted against maximum acceleration of centre of PCB (using samelegend as in Fig. 5).

Fig. 7. Comparison of response predicted by FE model (dotted line) and realresponse of Set-up B. Low base acceleration input was used.

R.A. Amy et al. / Microelectronics Reliability 50 (2010) 86–97 91

built in the PATRAN modelling environment and solved using NAS-TRAN. The model of the PCB was simply created using QUAD4 shellelements. Material properties were created and assigned to theseelements using the experimentally derived material properties, a2D Isotropic material was used.

The next step after creating the PCB model was to apply theboundary conditions. The PCB translational boundary conditionswere assumed to be rigidly grounded because the PCB in the exper-iment was directly attached to a 30 mm thick Aluminium plate,thus it was assumed that the response of the plate would be neg-ligible compared to that of the PCB. The PCB rotational boundaryconditions were modelled by spring elements (CBUSH) attachedto the mesh at each fixing location, and the rotational stiffness ofthe elements increased until the model frequencies matched theexperimental frequencies (see Table 3). The tuned model had arotational spring constant value of 45 Nm=rad.

The final step was to apply damping to the model based on theexperimentally derived values (see Fig. 6); unfortunately however,several different damping measurement techniques were used andeach gave a different value. To overcome this problem all the dif-ferent predicted values were tried in the model and the correlationcompared to see which gave the best results. The logarithmic dec-rement method gave a damping of 0.5% (derived using Eqs. (2) and(3)) and was found to give the best results, which agrees with con-ventional theory that this damping measurement method is themost appropriate for such low values of damping [28]. The finalpredicted response correlated well with the actual response (seeFig. 7 and Table 3).

The previous comparison was based on a very low base excita-tion (0:01 g2=Hz flat input spectrum), permitting the assumption ofnegligible non-linear effects due to the very small board deflec-tions. What would happen if the small deflection assumption couldnot be made? To answer this question the test was repeated at amuch higher vibration level so that the deflections were signifi-cantly greater than the board thickness, ensuring that there shouldbe some non-linear effects [23]. The FE analysis was also repeatedwith updated damping values based on the new acceleration input

Table 3Comparison of experimental and predicted response of Set-up B. Figures inparentheses are percentage error values.

Mode fexp (Hz) fmodel Qexp Qmodel

1 146.3 146.8 (�0.31%) 164.3 149.8 (8.83%)2 547.1 534.3 (2.34%) 75.4 60.7 (19.49%)3 564.8 572.2 (�1.31%) 47.2 64.09 (�35.8%)

and damping values from Fig. 6. A comparison of the results (asshown in Fig. 8) shows that the non-linear effects present at thehigher excitation levels significantly alter the response nearresonance.

The final test on this setup examines the effect of componentson model accuracy. The board was populated with 74 g of variousdifferent small components, these components were a mixture ofdifferent SMT and PTH components, none with an edge lengthgreater than 10 mm. This effect was then incorporated into theFE model by artificially raising the density to simulate the effectof the components. Any additional stiffening effects of the compo-nents were ignored. The effectiveness of this method can be seen inFig. 9 and Table 4.

4.2. Set-up A

Set-up A was modelled next (see Fig. 2), again with the inten-tion of examining the difference between the predicted and exper-imentally measured responses. The PCB FE model was built in asimilar way to the previous model (see Fig. 10), using QUAD4 ele-ments and 2D Isotropic material. The free–free response of this PCBwas calculated and compared to the actual free–free response (seeTable 5 Column A). Additionally, two more predictions of the free–free response were made, one prediction was made using thematerial properties provided by the manufacturer (Column C)and another made using the material properties measured here

Fig. 8. Comparison of response predicted by FE model (dotted line) and realresponse of Set-up B. High base acceleration input was used to investigate non-linear effects.

Fig. 9. Comparison of response predicted by FE model (dotted line) and realresponse of Set-up B when populated with components. Low base accelerationinput was used.

Table 4Comparison of experimental and predicted response of Set-up B when populated withcomponents. Figures in parentheses are percentage error values.

Mode fexp (Hz) fmodel Qexp Qmodel

1 129.2 129.5 (0.21%) 170.4 171.1 (0.36%)2 476.2 467.0 (�1.97%) 73.9 60.9 (�21.2%)3 492.9 500.1 (1.42%) 62.5 65.0 (3.82%)

Fig. 10. First torsional mode of finite element model of free–free PCB from Set-up A.

Table 5Table showing difference in natural frequencies between experimental test on free–free unpopulated PCB and various FE models. Model A uses experimentally derivedmaterial properties and PCB thickness, Model B assumes constant PCB thickness andmodel C uses the material properties provided by the PCB manufacturer.

Mode Experimental frequency (Hz) Percentage discrepancy

A B C

1 41.3 1.19 6.94 �15.662 66.6 0.15 �6.39 �31.623 93.3 �0.75 �13.64 �37.414 110 0.20 1.11 �22.365 119 0.79 �2.18 �26.466 204 �3.13 �9.05 �33.077 210 �0.82 �0.17 �23.678 243 1.12 �10.65 �15.179 280 �0.14 �8.48 �2.56

Fig. 11. Finite element model of Set-up A enclosure without PCB attached, notecentral cross-members to provide additional PCB support (cross-beams and internalstiffening ribs were modelled as 1D beam elements but for clarity are displayedhere as representative solid elements).

Fig. 12. Finite element model of Set-up A enclosure with PCB attached.

92 R.A. Amy et al. / Microelectronics Reliability 50 (2010) 86–97

and the material thickness provided by the manufacturer (ColumnB). The relatively poor accuracy of these last two models highlightsthe risk of using the material property values provided by themanufacturer.

The chassis was modelled using a combination of QUAD4 shellelements for the chassis walls and Bar2 beam elements (seeFig. 11). The only mode shape of any importance to the modelling

of the PCB response was that of the central cross-beams, thesewere experimentally measured to have a natural frequency of210.4 Hz whilst the FE model of the chassis gave frequencies of209.8 Hz, a difference of less than 0.5%.

To predict the response of the PCB in the enclosure the twomodels must be combined (see Fig. 12). As before, this wasachieved using rigid translational elements and tuned rotationalspring elements at the location of each fixing. Unlike the previousmodel, it was not initially possible to easily tune the spring ele-ments to correlate the frequencies. The initial attempts to modelgave significant errors in the frequency of second and third modes,whilst also significantly underestimating the response of the latermodes.

With some effort it was found that the PCB could be modelled toa good accuracy in the following way: first, in addition to tuningthe rotational stiffness it was found that modelling the PCB fixingswith a translational spring element allowed good frequency corre-lation, these additional spring elements also required tuning. Thissuggests that the initial assumption of the PCB fixings being effec-tively rigid in translation was incorrect for this set-up. Secondly, itwas found that the amplitude response prediction could be im-proved by specifying lower damping for higher frequency modes.This suggests that damping drops off with frequency, as the higher

Table 6Comparison of experimental results and FE models using different boundaryconditions. Subscripts ss, ff and tuned refer to simply supported, fully fixed andtuned edge conditions respectively. Italic values in parentheses are the peaktransmissibility for that mode, where these were known.

Mode Frequency (Hz)

Experiment Tuned SS FF

1 196 195.6 190 207(31.49) (31.17) (31.17) (31.17)

2 322 314 301 3423 357 365 349 3814 478 478 460 515

(35.86) (34.15) (11.71) (11.27)5 637 635 618 6786 1010 884 942 1066

Fig. 13. Comparison of response predicted by FE model (dotted line) and realresponse of Set-up A.

Fig. 14. Comparison of experimental results of Set-up A with two different FEmodels, the ‘‘Simply supported” model assumes the PCB to chassis connection haszero rotational stiffness, whilst the ‘‘Fully fixed” model assumes a rigid rotationalconnection.

Fig. 15. Set-up C attached to the shaker head expander.

R.A. Amy et al. / Microelectronics Reliability 50 (2010) 86–97 93

modes have lower displacement this would agree with the resultsof the earlier damping tests. The correlation of the final tuned mod-el can be seen in Table 6 and Fig. 13.

Finally, two additional models were made to investigate twocommonly made assumptions, namely, that the PCB to chassis con-nection is either rotationally fully rigid or flexible. Fig. 14 showsthe comparison of two models, one using rotationally rigid ele-ments and one using elements that do not constrain rotation. A fre-quency variation of up to 5% was seen for the first mode and 10%

for the second mode. Variability in the response magnitude wasas before (see Table 6).

5. Variability experiments

The next set of experiments measure the variation present insome typical PCBs. In addition to the Set-ups A and B used in theprevious experiments two more set-ups will also be introduced:

Set-up C consists of an MS-6323 Micro-ATX motherboardattached to shaker head expander by six M4 machine screwsas shown in Fig. 15. In this test the board only had a 6 mmstand-off as this is how the board is usually mounted. Sevenof these boards existed to allow the differenced between eachto be measured, each measures 244 mm by 205 mm and weighswithin ±1 g of 465 g. In some tests the motherboards free–freeresponses were measured, this was achieved by hanging eachboard from elastic bands.Set-up D consists of a graphics card (Vanta TNT2M64) sus-pended by elastic bands to measure free–free response (shownin Fig. 16), ten cards were tested this way, each weighs within±0.5 g of 69 g and measures 150 mm by 83 mm and are1.6 mm thick.

5.1. Boundary condition variability test

This first test illustrates the sensitivity of a PCBs response tosmall changes in boundary conditions. This involved removingand re-installing the same Set-up C (as shown in Fig. 15) severaltimes and testing the vibration response after each installation.Any slight variations in the boundary conditions between testswould be apparent as the vibration response would also change be-tween tests (see Fig. 17 and Table 7). The boards were subjected toa 0.5 g vibration input swept from 20 to 400 Hz at two Octaves perminute; this frequency was chosen as it contained the most signif-icant modes, whilst the value of 0.5 g vibration input was a com-promise between a good signal to noise ratio and possibledamage to the boards. The first test involved removing and re-installing Set-up C three times, the bolts were tightened to thesame torque each time. The second of tests was the same exceptthat not only was the torque identical but also the tightening pat-tern. Consequently it was found that only by tightening the screwsin exactly the same order and to exactly the same torque (1.5 N/m)could any repeatability be achieved (see Table 7), thus all experi-ments in this work use exactly the same tightening pattern andtorque.

Fig. 16. Set-up D for measuring free–free response of graphic cards.

Fig. 17. Frequency response of Set-up C over successive removal and re-installationattempts, to show sensitivity of board to small variations in boundary conditions(see Table 7).

Table 7Frequencies and peak transmissibilities of first two modes for Set-up C, between eachattempt the PCB was removed and then re-installed with bolts re-tightened to thesame torque. The first three attempts used a random bolt tightening pattern; the lastthree attempts used exactly the same bolt tightening pattern.

Attempt Mode 1 Mode 2

f (Hz) Q f (Hz) Q

1 92.6 9.20 128.8 15.252 93.4 7.92 131.6 18.843 97.4 8.4 130.0 10.204 97.6 8.40 130.0 16.05 98.0 8.72 131.0 16.256 97.8 9.60 131.0 18.0

Table 8Statistical parameters of the vibration response of different set-ups after threesuccessive re-installation attempts.

Mode �f (Hz) rf rf ð%Þ Q rQ rQ ð%Þ

Set-up C 1 94.5 2.57 2.72 8.50 0.64 7.90(random pattern) 2 130.3 1.40 1.08 14.76 4.34 29.40

Set-up C 1 97.8 0.20 0.20 8.91 0.62 6.982 130.7 0.58 0.44 16.75 1.09 6.51

Set-up A 1 194.4 1.35 0.70 21.14 0.70 3.302 333.0 1.17 0.35 4.80 0.15 3.163 361.9 0.65 0.18 3.67 0.26 7.064 483.5 1.28 0.26 48.14 2.25 4.68

Set-up B 1 145.3 1.72 1.19 147.3 6.67 4.532 543.5 6.20 1.14 69.78 9.19 13.18

Fig. 18. Frequency response of seven supposedly identical motherboards (Set-upC), each board was installed in exactly the same manner, with identical torques andbolt tightening pattern (see Table 9 table for specific information).

94 R.A. Amy et al. / Microelectronics Reliability 50 (2010) 86–97

It was suspected that different test set-ups might be more sen-sitive to slight changes in boundary conditions than others. Toinvestigate this assumption the tests were repeated for two Set-ups A and B (the statistics of variation are shown in Table 8). Pre-liminary testing showed the sensitivity to bolt tightening patternwas found to be insignificant in both these tests, so this part ofthe original test was not repeated.

5.2. Manufacturing variability test

The second set of tests involved testing and comparing the re-sponse of seven supposedly identical motherboards as in Set-upC, these were tested in a mounted condition similar to the previoustests. It was immediately apparent that identical manufacture didnot mean identical response (see Fig. 18 and statistics of variation

Table 9Statistics of variation for seven identical Micro-ATX motherboards in Set-up C, eachwith identical installation procedures.

Mode �f (Hz) rf rf ð%Þ Q rQ rQ ð%Þ

1 94.74 2.94 3.10 11.34 2.54 22.422 129.31 1.45 1.12 14.80 0.82 5.56

Fig. 19. Thickness against first natural frequency for identical Micro-ATX PCBs as inSet-up C.

Fig. 20. Graph to show decrease in first resonant frequency of a Set-up B due todamage from intense vibrations. Arrow points in the direction of increasing levels ofvibration.

R.A. Amy et al. / Microelectronics Reliability 50 (2010) 86–97 95

in Table 9). Additionally, some correlation was noted between themotherboard thickness and first natural frequency (see Fig. 19),leading to the conclusion that small differences in motherboardthickness were partly responsible for the variation in the response.

The tests were repeated on Set-ups B and D to investigatewhether this level of variation was typical (set-ups shown in Figs.15 and 16 but free–free). To remove any possible boundary condi-tion effects this second set of tests was performed using free–freeconditions (results are shown in Table 10).

5.3. Variations in response after high acceleration

A test was carried out to show how the response of a PCB mightbe irreversibly altered due to damage from high vibration loading.Set-up B was attached to a shaker head expander and excited at

Table 10Statistics of modes of several identical PCBs for free–free conditions. Seven boardswere tested in Set-up B and C, whilst ten were tested in Set-up D.

Set-up Mode �f (Hz) rf rf ð%Þ

Set-up C 1 53.51 1.50 2.802 76.00 0.86 1.133 103.10 1.57 1.524 134.03 1.68 1.255 157.1 4.06 2.58

Set-up B 1 23.5 0.54 2.312 48.0 1.41 2.943 62.1 0.62 1.004 72.0 1.33 1.845 75.2 0.75 1.006 133.0 0.79 0.597 149.2 0.82 0.558 157.5 0.70 0.499 179.1 0.89 0.50

10 230 1.57 0.68

Set-up D 1 202.4 3.22 0.792 262.1 3.74 0.71

several levels of vibration from 0.2 g to 40 g sinusoidal input, thevibration input was a sinusoidal input at just below the first reso-nant frequency of the board, each level lasted ten minutes. A visualinspection of the PCB after each test did not see signs of obviousdamage. The response of the board was then measured betweeneach stress test using a low-level sine sweep at 0.2 g. It was appar-ent that the response of the PCB was affected when more intensevibrations were applied (see Fig. 20), as the first resonant fre-quency of the PCB fell 3.2% and the acceleration response fell11%. This first test used a set-up that incorporated nylon washersthat were suspected of being susceptible to yielding, so the testwas repeated with another set-up that had no washers. Over thissecond test the frequency dropped by only 1.5% while the responsedropped by a significant 18.6%. It was observed that the majority ofthe reduction for the second test occurred when the board centreresponse reached over 160 g peak acceleration, whilst for the firsttest the reduction – and therefore damage – occurred evenlythroughout the test.

In summary, the response was irreversibly altered due to highaccelerations even though no visible damage occurred. This dam-age could take many possible forms: delamination of the glass fibrelayers within the PCB, failure of the epoxy to fibre bond, local yield-ing of the PCB around the fixing that resulted in lower bolt tension.Although the exact cause of failure can not be determined it is en-ough to know that at high responses such variations can occur andshould be measured if possible, if measurement is not possiblethen appropriate safety factors should be included. Also, it shouldbe noted that the damage did not affect any of the electrical con-nections on the PCB in this case study, but for multi-layer PCBsthe possibility of damage to electrical connections could be anissue.

6. Discussion

Say the response of a piece of equipment is known, to what ex-tent should responses of other identical pieces of equipment ex-pected to be the same?

First, let’s consider a worst-case scenario based on the resultsreported here. For example, if the assembly of a piece of equipmentis not tightly controlled, resulting in different bolt torques andtightening patterns, the response magnitude may have a standarddeviation of up to 30% (based on the highest variation seen in Table8). Low manufacturing tolerances may contribute up to another22% standard deviation (based on the highest variation seen in Ta-ble 9). These percentages are then corrected for the small samplesize by using a Student’s t-distribution, at a 99% confidence level

96 R.A. Amy et al. / Microelectronics Reliability 50 (2010) 86–97

this distribution requires that the percentages be multiplied by3.14 to safely account for any possible variability. Based on thesevalues, for a worst-case scenario, the actual response may be184% higher or lower than the measured value. Furthermore, thisdoes not consider that the frequencies may vary by up to 18% high-er or lower than the measured value, possibly placing the resonantfrequencies of the structure into a region of higher input accelera-tion. Finally, if large accelerations are present damage may occur tothe structure that could alter the response further.

Now let’s consider a best-case scenario. For equipment that isassembled in a very controlled environment, the response magni-tude may vary by as little as 3% (based on the lowest variationseen in Table 8), whilst the lowest deviation observed due to man-ufacturing variations was 3% (based on the lowest first mode var-iation seen in Table 10). At three sigma this amounts to theresponse being possibly 18% higher or lower than the measuredresponse.

Thus, to answer the original question of how similar responsesmay be, is not so straightforward, as it can be seen that the vari-ability depends on the specific case at hand. For example, themotherboards in Set-up C seemed to be more sensitive to smallvariations in boundary conditions, possibly due to their relativelysmall number of fixing points. Therefore, any estimate of variabil-ity should be based on set-ups and assembly processes as similaras possible to the equipment to which they will be applied.

Now let’s consider the question that was posed earlier: ‘‘Withwhat accuracy can the vibration response reasonably be predictedby an idealised theoretical PCB FE model?” First, if a specimen ofthe PCB material is available for bend testing, then the PCB mate-rial properties (stiffness, density, and thickness) can be measuredand specified accurately. With this information it is then possibleto get very good correlation between the predicted and actualPCB free–free response; thus removing one significant source of er-ror. Next, to overcome the problem of measuring damping, this canbe measured from an already existing piece of equipment. Thistechnique is only valid if the existing equipment is sufficiently sim-ilar to the proposed design, i.e. similar board size, fixtures andchassis. Fortunately, most companies only use a few different formfactors, so it is very likely that similar pieces of equipment alreadyexist.

With these two pieces of information (PCB properties anddamping) a full FE model of the combined PCB/enclosure systemcan be created; with this model it is now possible to attempt topredict the edge rotational stiffness. This can be achieved usingthe same technique as was just described for damping: using esti-mates based upon existing equipment with similar fixings. Allthese measured values can be saved in a database and then bere-used for any future modelling situations, without the need tobe measured every time. As it is assumed that most manufacturersuse only a few different types of PCBs, form factors, fitting mecha-nisms and chassis types, it is possible that with a relatively modestamount of effort, a database of all the different input parameterscould be created, thus allowing future models to be created quicklyand effectively.

In this paper, the FE modelling has been demonstrated for twodifferent examples of electronic equipment. In one example it waspossible to predict the edge rotational stiffness and the resultantresponse prediction was very good. In the second example theboundary conditions were more complicated and required a morecomplicated modelling approach, this approach had to includeboth translational spring constants at the PCB boundary conditionsand also frequency dependant damping. Finally, at high levels ofexcitation, non-linear effects may become significant and furtherdecrease accuracy. Fortunately, the non-linear effects seem to onlyreduce the response, so ignoring them is a conservativeassumption.

Again, the answer to the posed question is not straightforward.If a structure is submitted to extensive preliminary testing and hasrelatively simple PCB boundary conditions that permit trial and er-ror calculation, then it is possible to create a relatively accuratemodel. However, in a real situation, it is unlikely that such exten-sive testing will be practical and even if such testing is undertaken,there remains a possibility that the boundary conditions anddamping are relatively complicated to specify.

In addition, the following points have been noted during themeasurement process: accurate damping measurement is vital ifthe FE model is to accurately predict response magnitude. Mea-surements should be made at different power input levels to reflecthow damping varies with increased deflections. The Logarithmicmethod is the most suitable method for doing this, especially whenthe damping is low, although it is recommended to average multi-ple readings when using this method.

In terms of determining the PCB material properties the follow-ing points have been noted. The laminate material is noted to havestrongly anisotropic properties and an accurate model cannot becreated without using the stiffness moduli in both the x and y axis.The importance of measuring and including shear stiffness in themodel should not be overlooked. Thickness variations can signifi-cantly affect the accuracy of results if not considered.

7. Conclusion

The aim of this work was to assess the accuracy of typical FEmodels used to predict the response of PCBs to harsh randomvibration environment. It has been shown that if good data onthe board properties exists, then even simple PCB FE models candeliver very accurate response prediction. In practice, however,various input parameters are difficult to obtain: namely, reliabledata concerning boundary conditions, stiffness or damping. It hasbeen shown that with enough preliminary testing it is possible todetermine the stiffness and damping values, whilst the boundaryconditions may be calculated by a ‘‘tuning” approach. Unfortu-nately, this does create the requirement that an experimentalset-up similar the one being modelled already exists. Ultimately,with the approach given here, an estimate of the expected accuracyof a PCB FE model can be made, from this estimate the confidencein future models can be known. In addition, tests were performedto show that manufacturing and assembly variability can producesignificant differences in the response of supposedly identical PCBassemblies, highlighting the importance in measuring thisvariability.

Acknowledgements

The author would like to warmly thank Surrey Satellite Tech-nology Limited (SSTL) for their vital contribution to this research.Special thanks also go to Ian Black for kindly providing free equip-ment and technical support.

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