45

CHAPTER 3

MODAL ANALYSIS OF A PRINTED CIRCUIT BOARD

3.1 INTRODUCTION

This chapter describes the methodology for performing the modal

analysis of a printed circuit board used in a hand held electronic components

by analytical and experimental methods.

3.2 NEED FOR PERFORMING MODAL ANALYSIS

In the past two decades, modal analysis has become a major tool in

the quest for determining, improving and optimizing dynamic characteristics

of engineering structures. Not only has it been recognized in mechanical and

aeronautical engineering, but also modal analysis has discovered profound

applications for civil and building structures, biomechanical problems, space

structures, acoustical instruments, electronic components, and nuclear plants.

According to Ewins (2001), modal analysis is the process of

determining the inherent dynamic characteristics of a system in the form of

natural frequencies, damping factors and mode shapes, and using them to

formulate a mathematical model for its dynamic behavior. The formulated

mathematical model is referred to as the modal model of the system and the

information for the characteristics is known as its modal data. Modal analysis

is based upon the fact that the vibration response of a linear time-invariant

dynamic system can be expressed as the linear combination of a set of simple

harmonic motions called the natural modes of vibration. The natural modes of

46

vibration are inherent to a dynamic system and are determined completely by

its physical properties (mass, stiffness, damping) and their spatial

distributions. Each mode is described in terms of its modal parameters:

natural frequency, the modal damping factor and characteristic displacement

pattern, namely mode shape. The mode shape may be real or complex. Each

corresponds to a natural frequency. The degree of participation of each natural

mode in the overall vibration is determined both by properties of the

excitation source(s) and by the mode shapes of the system.

Modal analysis embraces both theoretical and experimental

techniques. The theoretical modal analysis anchors on a physical model of a

dynamic system comprising its mass, stiffness and damping properties. These

properties may be given in forms of partial differential equations. A more

realistic physical model will usually comprise the mass, stiffness and damping

properties in terms of their spatial distributions, namely the mass, stiffness

and damping matrices. These matrices are incorporated into a set of normal

differential equations of motion. The superposition principle of a linear

dynamic system is used to transform these equations into a typical eigen value

problem. Its solution provides the modal data of the system.

Modern finite element analysis empowers the discretization of

almost any linear dynamic structure and hence has greatly enhanced the

capacity and scope of theoretical modal analysis. On the other hand, the rapid

development over the last two decades of data acquisition and processing

capabilities has given rise to major advances in the experimental realm of the

analysis, which has become known as modal testing.

In case of hand held electronic components, the fine solder joints of

IC packages used in these components are susceptible to failure when the

components are subjected to vibration loads or when the products are

suddenly dropped. So the reliability of these packages and their board level

interconnections when subjected to vibration loads or drop impacts has

47

become a vital issue. The failure of the solder joints under vibration or drop

impact is mainly due to the bending of the printed circuit boards (PCBs) on

which the packages are mounted. The bending of the PCBs is in turn depends

on the modal shapes of the PCBs. Studies have shown that the effect of the

strain rate on the material properties of solder joint is significant. The strain

rate of solder joints is dependent on its strain amplitude and the natural

frequency of PCB vibration (Steinberg 2000). Hence the mode shapes of the

PCB are the base for understanding the dynamic response characteristics of

the PCB and the solder joints. For a particular IC package, PCB and its

mounting conditions, the natural frequencies and the mode shapes are its

inherent characteristics which can be obtained by modal analysis. These

dynamic characteristics in turn decide the response of the electronic

equipments when subjected to vibration loads or when they are subjected to

drop impact conditions. The response of the electronic equipment in turn

determines the vulnerable or critical areas in the equipment that are subjected

to high stresses and may fail prematurely during its service life. Designers can

then concentrate on these areas to reduce the stresses developed in the design

stage itself which ensures that the equipment performs its intended function

for the specified time rather than failing prematurely.

In this research, a PCB custom made as per JEDEC standard

(JESD22-B111, 2003) with 5 electronic packages mounted on it is subjected

to modal analysis. The modal analysis is carried out by (i) FE method and (ii)

experimental method. While performing the modal analysis using FE method,

the complete PCB is modeled in FEM software ANSYS 10.0. The natural

frequencies and mode shapes are extracted from the FE model. Experimental

modal analysis is done by exciting the actual PCB using an impulse hammer.

The response of the PCB for the given impulse is captured using an

accelerometer. The natural frequencies, mode shapes and damping ratios of

the PCB are then extracted from the Frequency Response Function (FRF)

obtained from the impulse used to excite the PCB and the response measured

48

from the accelerometer. The methodology used in the modal analysis of the

PCB is shown in Figure 3.1.

Figure 3.1 Methodology adopted in the modal analysis of the PCB

Modal analysis of a PCB

Finite Element Method

Develop a FE model of the PCB

Apply material properties and boundary conditions

Solve the FE model for modal analysis

Extract the natural frequencies and mode shapes

Experimental method

Fabricate a PCB with packages mounted

Mount the PCB on a rigid fixture

Apply an impulse using an impact hammer

Acquire the PCB response using an accelerometer

Calculate the FRF from the signal acquired from the

impact hammer and accelerometer

Extract the mode shapes, natural frequencies and

damping ratios from the FRF

Compare results from FE and experimental methods

49

- Location of the packages mounted on PCB

3.3 MODAL ANALYSIS BY FINITE ELEMENT METHOD

In this method, a FE model of a PCB was developed using ANSYS

10.0. The PCB considered in this research is custom made as per JEDEC

standard (JESD22-B111, 2003). The size of the PCB is 132 x 77 x 1.6 mm.

Five electronic packages of Ball Grid Array (BGA) type were mounted on the

PCB at specific locations as per JEDEC standard. The layout of the PCB, the

location where the packages are mounted and the locations where the PCB is

supported is shown in Figure 3.2 and the actual PCB custom made for this

research is shown in Figure 3.3. The BGA package mounted on the PCB has a

size of 2.5 x 2.5 x 0.74 mm and is mounted on to the board by means of 15

solder balls. The details of the BGA package are shown in Figure 3.4.

Figure 3.2 Detailed layout of PCB with location of packages mounted

50

Figure 3.3 Printed Circuit Board with BGA packages

Figure 3.4 Detailed layout of BGA Package

Figure 3.5 shows the FE model of the PCB together with the

packages. The boundary conditions applied to the FE model are the same as

that of the actual PCB.

PCB

BGA Package

51

Figure 3.5 Finite Element (FE) Model of the PCB with BGA Package

The material properties of the various components in the PCB are

listed in Table 3.1. One of the important assumptions in the modal analysis by

FE method is that the materials are assumed to be linear isotropic in nature

even though the PCB itself is a composite material made of FR4/Epoxy. This

assumption is important because the system as a whole is considered as a

linear system which is the precondition for performing modal analysis.

BGA Package

PCB

Solder ball

52

Table 3.1 Material Properties of PCB and packages used in FEA

(Pecht et al 1999)

S.No Component Young’s

modulus, E (Pa)

Poisson’s ratio,

Density, (kg/m3)

1. PCB 1.7 x 1010 0.35 2200

2. Package 2.2 x 1010 0.29 2100

4. Solder ball 3.2 x 1010 0.38 8440

The maximum frequency of vibration to which hand held electronic

devices are exposed is found to be 1000 Hz (JESD22-B103B, 2006). Hence, it

is decided to extract the natural frequencies with in 1000 Hz from FEM. The

first, second, third and fourth mode shapes and the corresponding natural

frequencies obtained from the FE model are shown in Figures 3.6, 3.7, 3.8

and 3.9 respectively and the same is tabulated in Table 3.2.

Figure 3.6 First natural frequency (341.08 Hz) and mode shape of the

PCB

53

Figure 3.7 Second natural frequency (573.483 Hz) and mode shape of

the PCB

Figure 3.8 Third natural frequency (821. 479 Hz) and mode shape of

the PCB

54

Figure 3.9 Fourth natural frequency (876.465 Hz) and mode shape of

the PCB

Table 3.2 Natural frequencies from FE Model of the PCB

Mode Natural frequency (Hz)

1 341.08

2 573.48

3 821.48

4 876.47

3.4 MODAL ANALYSIS BY EXPERIMENT METHOD

Experimental modal analysis can be carried out either in the

frequency domain or in the time domain (He J. et al, 2001). When modal

analysis is carried out in frequency domain it depends on the Frequency

Response Function (FRF) of the system. From the measured FRF’s it is

possible to extract the system dynamic characteristics such as the natural

frequencies, mode shapes and damping ratios by using the mathematical

55

models available. When the same modal analysis is carried out in time

domain, the extraction of system dynamic characteristics is difficult. Also,

data inaccuracies and presence of noise in the acquired data may lead to

computational errors in the extraction of system dynamic characteristics.

Hence, in this research modal analysis was carried out in the frequency

domain.

Experimental modal analysis in the frequency domain can be

classified into three different test methods based on the number of FRFs

which are to be included in the analysis (Ewins 2001). The simplest of the

three methods is referred to as SISO (Single Input, Single Output) which

involves measuring a single FRF for a single input given. A SISO data set is

made of a set of FRFs which are measured individually but sequentially. The

second test method is referred to as SIMO (Single Input, Multiple Output).

This refers to a set of FRFs measured simultaneously at different locations for

a single input given at a specific location. The third method is referred to as

MIMO (Multiple Input, Multiple Output) in which the FRFs at various points

are measured simultaneously while the structure is excited at several points

simultaneously. Since the PCB is a light weight structure, mounting several

accelerometers to acquire FRFs simultaneously will result in erroneous

natural frequencies as the mass of the accelerometers will affect the natural

frequencies of the PCB. Hence in this research, SISO method was adopted to

perform experimental modal analysis.

3.4.1 Experimental setup

The experimental setup for performing the modal analysis consists

of the following equipments:

(a) Impulse hammer

56

(b) Accelerometer

(c) Signal conditioner

(d) Data Acquisition System

(e) Base plate fixture

The schematic representation of the experimental setup is shown in

Figure 3.10 and the actual test setup is shown in Figure 3.11.

The impulse/impact hammer (PCB Piezotronics Model 086C03) is

used in the modal analysis for giving the necessary excitation to the PCB. The

impact hammer is provided with several types of heads typically made up of

plastic, steel or rubber to give different excitation levels depending upon the

structure being tested. The accelerometer is used to pick up the vibration

signals from various locations on the PCB. To minimize the effect of the mass

of the accelerometer on the natural frequencies of the PCB a miniature

accelerometer (B&K Model 4517) weighing 0.6 gm was used in the test. The

PCB is clamped to an aluminium base plate fixture at its four support

locations and the base plate is in turn clamped rigidly to a table to isolate the

PCB from external excitations.

The signal acquired from the impact hammer is fed into a signal

conditioner and after conditioning the data is fed into NI PXI 4472 Data

Acquisition System (DAQ) which is mounted on NI PXI-1042Q Chassis. The

NI PXI 4472 DAQ is an 8-channel dynamic signal acquisition module

suitable for acquiring high frequency signals at higher rates.

57

Figure 3.10 Schematic of the experimental set up for modal analysis

Figure 3.11 Experimental set up for modal analysis

NI PXI 1042Q Chassis

Base plate

Electronic package

PCB Clamping Screws

Impact Hammer

Signal Conditioner

Data Acquisition System DAQ NI

PXI 4472

Accelerometer

58

It acts as a bridge between the sensors and the computer. It is

capable of acquiring eight simultaneously sampled analog inputs at a rate of

102.4 kS/s. NI PXI-1042Q is a stand alone, portable computer operating on

Windows XP. The chassis supports the LabVIEW 8.6 software which is used

for analyzing the input signals from the accelerometer and the impact hammer

and to calculate the FRFs based on the input signals. It is also capable of

displaying acquired signals and the calculated FRFs either in time domain or

in the frequency domain.

3.4.2 Experimental procedure

The modal testing based on SISO method can be performed in two

ways. One is called the roving hammer technique and the other is roving

accelerometer technique (Bruel & Kjaer, 2003). Both the techniques rely on

the principle of Maxwell’s reciprocity theorem (Rao S.S., 2004). According to

this principle, for a linear system, the FRF measured at a point ‘j’ for an

excitation given at a point ‘i’ (Hij) will be equal to the FRF measured at point

‘i’ for an excitation given at point ‘j’ (Hji).

i.e. Hij = Hji

In the roving hammer technique, the accelerometer is fixed at

specific location and the impact hammer is used to excite the structure at

various predetermined locations sequentially. In the roving accelerometer

technique, impact hammer will be exciting the structure at one specific

location whereas the accelerometer will be moved to various predetermined

locations to measure the response sequentially. Since it is difficult task to

move the accelerometer from one place to another as it has to be mounted on

to the PCB using bees wax, roving hammer technique was adopted in this

research to perform modal test.

59

Before starting the test, two important steps need to be completed. One is to identify a suitable location for mounting the accelerometer and the other is to determine various locations on the PCB to excite it using the impact hammer. Choosing a proper location for the accelerometer is a very important step because if the accelerometer is mounted at a location least disturbed by the excitation, then it will become difficult to measure FRFs properly. For this reason, the accelerometer should not be mounted on a point in the nodal line of the PCB mode shapes. To determine the locations on the PCB for giving excitation, the PCB was divided into small segments. The layout of the PCB after it was divided into smaller segments; the locations for exciting the PCB using impact hammer and the location for mounting the accelerometer were shown in Figure 3.12. The location of the accelerometer was decided by analyzing the mode shapes of the PCB obtained from FEM. Altogether, the PCB was excited at 48 different points sequentially in the order given in Figure 3.12 and the response was measured by the accelerometer for each excitation.

Figure 3.12 Layout of the PCB showing the excitation and response points

1 2 3 5 4 6 7 8

9 10 11 12 13 14 15 16

17 18 19 20 21 22 23 24

25 26 27 28 29

39

30 31 32

33 34 35 36 40 37 38

47 41 42 43 44 48 45 46

- Location of the packages mounted on PCB - Location of excitation by impact hammer

- Location of accelerometer

60

The next step in the modal testing is to determine the head to be

used in the impact hammer for exciting the PCB. The impact hammer was

provided with various types of heads made of different materials and was

detachable for using in different types of structures. The head to be used for

exciting the PCB should excite as many natural frequencies as possible. This

depends on the time duration of the exciting impulse. If the time duration of

the exciting impulse is too large, then the frequency spectrum curve which

shows the variation of the exciting impulse with respect to the frequency will

decrease rapidly thereby making the measured FRF unreliable.

Figure 3.13 Frequency spectrum curves obtained from various hammer

heads (a) rubber-I (b) plastic (c) steel (d) rubber-II

61

The frequency-spectrum curve of exciting impulse should not decay more than 30 dB within the frequency range of interest. From the FE method it is clear that the range of frequency to be considered is between 340 Hz and 877 Hz. Hence the signal from the impulse hammer should not decay more than 30 dB within the frequency range (Zhang et al 2008). Figure 3.13 shows the frequency spectrum curve obtained for various types of heads when used to excite the PCB.

The decay in the spectrum curve for the frequency range of interest for the plastic head was around 15 dB. For the other heads the decay is around 20 dB. Hence, for performing the modal test on the PCB, the plastic head was used in this research.

3.4.3 Experimental results

The modal test on the PCB was carried out by hitting the PCB at the specified points sequentially from point 1 to point 48 as shown in Fig 3.9 and the response is measured for each of the given impulse. At each point three impacts were given and the results were averaged to reduce the effect of noise in the acquired signal. In experiment modal testing, one of the tools used to ensure the quality of the acquired signal is coherence (Ewins 2001). Coherence indicates how much the measured response is correlated to the input excitation. If there is another signal present in the response, either from noise or from other signal, the quality of the response measured will be poor. Figure 3.14 shows the measured excitation and response.

Figure 3.14 Schematic representation of measurement of input excitation (hammer) and the response (accelerometer)

Applied excitation Structure under modal test

Measured response B

Measured excitation A

62

Frequency response function (H(f)) can then be defined as:

H(f) = SAB(f) / SAA(f) (3.1)

where SAB(f) = cross power spectrum of A and B

SAA(f) = power spectrum of A

Coherence function(2) can then be defined as:

2 = ( SAB(f))2 / SAA(f)* SBB(f) (3.2)

where SBB(f) = power spectrum of B

Coherence can have a maximum value of 1 and a minimum value of 0.

Coherence value of 1 indicates that the response measured is entirely due to

the given input excitation and value of 0 indicates that the measured response

is entirely due to some other excitation than the given excitation. Coherence

function was measured several times during the modal impact test and some

of the measured coherence is shown in Figure 3.15.

Figure 3.15 Measured coherence during modal testing

63

From Figure 3.15 it can be inferred that for the frequency range of

interest from 340 Hz to 877 Hz, the coherence function was near to one for

most of the trials conducted. Hence the measure response was due to the

given input excitation.

Figure 3.16 shows one of the measured the FRF from the modal

test. The FRF consists of two parts; one real part and the other imaginary part.

[

Figure 3.16 FRF measured at one particular location on the PCB

showing the real and imaginary curves

According to the theory of modal analysis (Ewins 2001), the natural

frequencies will occur at the peak of the imaginary curve and at the middle of

adjacent positive and negative peaks of the real curve. From Figure 3.16 the

first three natural frequencies measure by the FRF were found to be 340 Hz,

Rea

l (G

/N)

Imag

inar

y (G

/N)

Frequency (Hz)

64

615 Hz and 840 Hz respectively. The fourth natural frequency was not excited

by the given impulse and hence was not shown in the FRF (Figure 3.16).

Similar trend was reported in various literature were experimental modal

testing was not able to excite all natural frequencies within a specified range

(Li 1999), (Chen 2006).

3.4.4 Extraction of modal parameters

Once the modal test was completed, the next step is to extract the

modal parameters namely the natural frequencies, the mode shapes and the

damping ratios. These parameters are to be extracted by post processing the

FRFs captured in the modal test. For post processing, DIAMOND software

(Diamond 1997) was used. DIAMOND is a graphical user interface toolkit

developed using Matlab by Los Alamos National Laboratory, Los Alamos,

USA. DIAMOND stands for (Damage Identification And MOdal aNalyis for

Dummies) and it has the capability of analyzing modal data from single or

multiple reference experiments to determine natural frequencies, damping

ratios and mode shapes by using a variety of modal curve fitting algorithms.

The first step in extracting the modal parameters is to define the

geometry of the PCB in DIAMOND toolkit. The layout of the PCB used in

modal testing as shown in Figure 3.12 was reproduced in DIAMOND. The

various points on which excitation were given and the point from which

response was measured are created in DIAMOND. The lines joining these

points and the areas created by the joining these lines represent the geometry

of the PCB and it is shown in Figure 3.17. Once the geometry is created, the

measured FRFs were given as input at the respective points from where they

are measured. The extraction of modal parameters from the measured FRFs is

a curve fitting problem (Ewins 2001). There are several curve fitting methods

available for extracting the modal parameters. Rational Fractional Polynomial

(RFP) is one such method for extracting modal parameters from Multi Degree

65

of Freedom (MDOF) systems (Richardson 1982) (Richardson 1985). RFP

method of curve fitting was adopted to extract the mode shapes and natural

frequencies from the geometry and it was implemented in DIAMOND. The

extracted mode shapes are shown in Figure 3.18, 3.19 and 3.20 and the

extracted natural frequencies were tabulated in Table 3.3.

Figure 3.17 Geometry of the PCB created in DIAMOND software

Figure 3.18 First natural frequency (341 Hz) and the extracted mode

shape

1 2

5

7 8

3

6

4

9 10

13

15 16

11

14

12

17 18

21

23 24

19

22

20

25 26

29

31 32

27

30

28

33 34

37

39 40

35

38

36

41 42

45

47 48

43

46

44

66

Figure 3.19 Second natural frequency (615 Hz) and the extracted mode

shape

Figure 3.20 Third natural frequency (830 Hz) and the extracted mode

shape

67

Table 3.3 Natural frequencies from experimental modal analysis of

the PCB

Mode Natural frequency (Hz)

1 341

2 615

3 830

3.5 COMPARISON OF RESULTS

The extracted mode shapes were compared with mode shapes

obtained from FEM as shown in Table 3.4 and were found to be in good

agreement. Similarly, the extracted natural frequencies from experiments

were then compared with that obtained from FE method and is shown in

Table 3.5. The next step in the experimental modal analysis is the extraction

of damping ratios. For extracting the damping ratios, half power band width

technique was used.

According to this technique (Rao 2004), the damping ratio is

given by the expression

1 2

r2

(3.3)

where 1, 2 are frequencies at which the magnitude of the frequency

response curve is (1/2) times the peak response obtained at the natural

frequency r. The half power bandwidth for the first natural frequency is

shown in Figure 3.21. The damping ratios calculated for first, second and

third natural frequencies by the above method are shown in Table 3.6.

68

Table 3.4 Comparison of mode shapes obtained from FEM and experiments

Mode Number

Mode shape from FEA Mode shape from

experiment 1

2

3

Table 3.5 Comparison of natural frequencies from FE method and experimental method

Mode Natural

frequency from FEA (Hz)

Natural frequency from

experiments (Hz)

Deviation %

1 341.08 341 0.023 2 573.48 615 6.7 3 821.48 830 1.03 4 876.47 - -

69

Table 3.6 Damping ratios for the extracted modes

Mode Natural frequency (Hz) Damping ratio ()

1 341 0.012

2 615 0.0072

3 830 0.0032

3.6 CONCLUDING REMARKS

In this chapter, a comprehensive method for performing modal

analysis on a PCB by both finite element and experimental methods was

described. The natural frequencies and mode shapes were extracted from the

FE model of the PCB with packages mounted on it. The first four natural

frequencies obtained from the FE model were found to be 341.08 Hz, 573.48

Hz, 821.48 Hz and 876.47 Hz respectively.

Experimental model testing was conducted on a custom made PCB

using SISO technique. The PCB was excited by an impulse hammer and the

response of the PCB was measured using an accelerometer mounted on the

PCB. The measured FRFs were then used to extract the natural frequencies,

mode shapes and damping ratios of the PCB. The extraction of these dynamic

characteristics of the PCB was done using DIAMOND software. The first,

second and the third natural frequencies obtained from the experimental

method were found to be 341 Hz, 615 Hz and 830 Hz respectively. The

corresponding damping ratios were found to be 0.012, 0.0072 and 0.0032

respectively.

The natural frequencies and mode shapes extracted from FE

method were found to be in good agreement with those extracted from the

70

experimental method. The maximum percentage of variation between the two

methods was found to be 6.7% occurring at the second natural frequency.

Figure 3.21 Half power bandwidth for calculating the damping ratio at

the first natural frequency

1 r 2

71

As there was good correlation between the FE and experimental

results, the FE model can be considered to be a validated model and hence

can be used for further dynamic simulation studies. The dynamic responses of

hand held electronic components when subjected to random vibration loads

and drop impact loads were then analyzed using the validated FE model

developed in this chapter. The details of these dynamic simulation studies

were explained in chapters 4 and 5 respectively.