Experimental Modal Analysis and Random Vibration
Simulation of Printed Circuit Board Assembly
Fang Liu1,∗ and Rongping Fan2
School of Mechanical Engineering and Automation, Wuhan Textile University,
Wuhan, 430073, P R China
liufang408@163.com
2
Department of Mechanical Engineering, Hong Kong Polytechnic University, Hung Hong
1
Abstract
In this study, the impact hammer testing method and non-contact TV Laser holography technology were
utilized to conduct experimental modal analysis and investigate the dynamic characteristics of Printed
Circuit Board (PCB) assembly, respectively. The experimental results show that the weight and location
of the accelerometer have great influence on test results in the former, and test results of the latter
would be more accurate if the excitation points were selected well. Based on the modal test results, an
effective Finite Element (FE) model was established and was used to carry out random vibration analysis.
The simulation results of random vibration analysis are in good agreement with the experimental data.
The benefit of experimental modal analysis and FE simulation is validated and the current approach saves
modeling validation time while increasing accuracy. The FE simulation based on experimental modal
analysis will provide a guide for investigating the dynamic characteristics and optimization design of
the PCB assembly.
Keywords: PCB assembly, Experimental modal analysis, Non-contact, Impact hammer testing method,
Random vibration
1
Introduction
Electronic equipment could be subjected to many different forms of mechanical shock and vibration
during transportation and handling, such as in military or automotive applications[1] . In many applications microelectronic devices are used in severe vibration environments, which would not only cause
mechanical failures in the housing of the device but also create electrical failures in the Printed Circuit
Board (PCB) assemblies mounted inside the housing due to transfer of energy through PCB supports.
These would have great influence on the electronic devices’ reliability. More researchers devoted
themselves to the vibration reliability of PCB assemblies[2–5] .
Modal analysis is frequently utilized to abstract the modal parameters of a system, including natural
frequencies, mode shapes, and modal damping ratio, etc. These parameters depend only on the system
itself but dominate the response of the system to excitations Modal analysis is the fundamental of response
analysis and has therefore gained increasing attentions. Experimental Modal Analysis (EMA) can estimate the natural frequencies, mode shapes, and damping values of the measured structure[6–8] . The Finite
Element (FE) model of the structure can be validated by comparing experimental natural frequencies
∗
Corresponding author.
ADVANCES IN VIBRATION ENGINEERING, 12(5) 2013
© KRISHTEL eMAGING SOLUTIONS PRIVATE LIMITED
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and the mode shapes with the results of FE model analysis. Because the forced dynamic response of the
populated PCB is a function of both the mode shapes and the frequencies (among other factors), good
agreement of both modes and frequencies of FEM with experimental modes and frequencies is vital to
the reliable prediction of the response[9, 10] . The equivalent FE model was established by modal parameters of EMA[11, 12] . The experimental results were used to verify computational results and to validate
the computational model[13, 14] , and Liu et al.[15] built a valid FE model, verified the FE analysis results,
and made the model update when some deficiencies were found in the FE model[6] . EMA is very useful
to build the finite element model of electronic products and key devices.
The relationships between the peak amplitude and natural frequencies had been demonstrated by
Sternberg[1] . Yang[16] adopted impact hammer testing method to carry out EMA. A series of modal
parameters were obtained during EMA, such as the mode shapes, the natural frequencies and the damping
ratios. These parameters were useful for studying the dynamic response of PCB[15] . Experimental modal
tests of PCB assembly and PCB were conducted respectively, and FE simulation was used to study the
dynamics responses of PCB assembly[14, 15] . Impact hammer testing method was employed to carry
out the experimental modal test of PCB on one aeronautical device under free condition[18] . Vibration
reliability of electronic products was closely related to the natural frequency of PCB, the installation
locations of components, the sizes and weight of components. Based on modal analysis results, the
PCB layout was optimized to enhance vibration performance[19] . The dynamic characteristics of PCB
assembly were investigated, which was a necessary part in order to enhance the design. Although there
are some literatures relevant to FE simulations, few studies addressed EMA of board-level package in
great detail and random vibration simulation based on EMA.
In this paper, impact hammer testing method and non-contact TV Laser holography technology
were both employed to conduct EMA in order to investigate the dynamic characteristics. Based on the
experimental modal analysis results, a FE model could be established to carry out random vibration
analysis, and numerical simulation results are compared with experimental data.
2
Modal Test
In this section, impact hammer testing method and non-contact TV Laser holography technology were
both used to complete experimental modal analysis.
2.1 Basic theory
Dynamic behavior of mechanical systems is described by vector differential equations and suitable
boundary conditions. When an experimental approach is utilized to determined physical properties of a
structure, a simplified space discrete model is utilized. Consider a linear dynamic system with structural
damping and N degree-of-freedom. The governing equations of motion are
Mẍ(t) + Cẋ(t) + Kx(t) = F(t)
(1)
where M, C and K are N × N the mass, damping and stiffness matrices respectively, and x(t) is N × 1
vector of time-varying displacements. F(t) is N × 1 vector of the external applied forces to the system.
The dynamic characteristics of the system are described completely by the frequency response matrix
H(ω) = [K + j ωC − ω2 M]−1
where ω is the frequency parameter.
(2)
EXPERIMENTAL MODAL ANALYSIS AND RANDOM VIBRATION SIMULATION OF PRINTED CIRCUIT BOARD ASSEMBLY
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Experimental modal analysis is a procedure for determining the dynamic characteristics of a system
whose physical parameters are not known. It is based upon the fact that the system behavior can be
described entirely in terms of its modal properties-natural frequencies, damping loss factors, and mode
shapes. For example, the frequency response function relating the output at degree-of freedom j to an
input at degree-of-freedom k is
Hj k (ω) =
N
r=1
ϕj r ϕkr
mr [λ2r − ω2 ]
(3)
where, ϕj r and ϕkr are the ith and kth components of the corresponding mass-normalized eigenvector
or mode shape[5] , λr and mr is the modal frequency and mass of the rth mode. The damped modal
frequency λr is related to the natural frequency ωr and damping loss factor ηr
λ2r = ωr2 (1 + iηr )
(4)
where, ωr and ηr are the respective natural frequency and damping loss factor for the rth mode.
Modal analysis software packages use various curve-fit routines to extract the modal parameters
from the Frequency Response Functions (FRFs). Frequency analyzers, or special purpose computer
programs, compute the FRFs by using digitized input and output signals and a Fast Fourier Transform
algorithm[14] .
2.2 Test vehicle
Except the shape of the test vehicle, the board design would strictly meet
the requirement of the JEDEC standard. The test printed board used an
eight-layer round-shape FR4 board, whose diameter and thickness are
160 mm and 1mm respectively. The component used was 0.8 mm pitch
BGA with a size of 11 mm by 13 mm. The test vehicle with eight BGAs
is shown in Fig. 1. The eight BGA packages of every test board were
also numbered U 1, U 2, . . . , U 8. There are two daisy-chain loops for
every BGA package with outer loop (DC2) and inner loop (DC1). The
boards have Organic Solderability Preservatives (OSP) surface finish on
Fig. 1 Test vehicle
Non-Solder Mask Defined (NSMD) pads, while the components have
elecro-plated nickel-gold surface finish on Solder Mask Defined (SMD)
pads.
During the modal test, PCB assembly was fixed by eight screws. It is assumed that the pre-tightening
forces of all the eight screws are always equal.
2.3 Impact hammer testing method
Excitation point distribution of the PCB assembly was shown in Fig. 2 The number of excitation
points was 57. Acceleration transducer was placed at No. 36 excitation point A mini-accelerometer
was used to reduce the influence of its weight on the experimental results. The measured acceleration
response was obtained by the accelerometer, while the impact hammer would be moved, going
through all the 57 points. For each excitation point, five hammer impacts were performed and five
excitation-output pairs signals were acquired in order to perform adequate averaging on the obtained
data, and the excitation signal from hammer impact and response signal from the accelerometer
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Fig. 2
FANG LIU AND RONGPING FAN / ADVANCES IN VIBRATION ENGINEERING, 12(5) 2013
Excitation point distribution of the PCB
assembly
Fig. 3
Block diagram for modal testing system using
impact hammer
are both measured by the SignalCalc Mobilyzer dynamic signal analyzer simultaneously. The two
signals are sampled at the sampling frequency of 625 Hz. Damping parameters, elastic modulus and
natural frequencies could be obtained from modal test Fig. 3 depicted a block diagram for the testing
system.
2.4 Non-contact TV laser holography technology
The block diagram for non-contact TV Laser
holography technology was shown in Fig. 4.
PCB
The frequency and amplitude of the impulse
assembly
VibroMap 1000
excitation was controlled by the workstation. The
B&K
exciter
frequency and amplitude were set and digital
Analog video
Laser
signals of impulse excitation were converted
filter
beam
2712
into analog signals by the Digital/Analog
power
Converter (ADC). Then the exciter was driven
Monitor
amplifier
Work station
by these signals which were amplified through
2712 power amplifier. At the same time,
Signal
VibroMap 1000 Laser vibration system was
generator
controlled by the workstation in order to carry out
image acquisition. Images collected were filtered
through the analog video filter and displayed on Fig. 4 Block diagram for non-contact modal testing
system
the monitor. The distance between the Laser lens
and the specimen was 1 m. The excitation points
were described in Fig. 5. The No. 1 excitation
point was far away from the fixed boundary. The No. 2 excitation point was near the fixed boundary
and was the relative rigid place of the test vehicle. The No. 3 excitation point was on the fixture. Since
the first two modes were very important, the first two modal frequencies were obtained through modal
tests.
EXPERIMENTAL MODAL ANALYSIS AND RANDOM VIBRATION SIMULATION OF PRINTED CIRCUIT BOARD ASSEMBLY
493
Fig. 5 Excitation points of test vehicle
Fig. 6
3
The first order mode
Fig. 7
The frequency response
Test Results
3.1 Test results from impact hammer testing method
Since the shape of PCB assembly is round and very special, the effect of the accelerator position should
be considered. The modal test of round PCB under eight-screw fixed condition was conducted, and the
first three order modes would be obtained. If the accelerator was placed at the center of round PCB, the
second and third modes were not obvious because the center point was located at the second and third
nodal lines. Therefore, FE simulation should be conducted before the modal test in order to understand
the modes of the test vehicle. When impact hammer testing method was utilized to carry out modal
tests, the first order mode and the frequency response function were got and shown in Fig. 6 and Fig. 7,
respectively. The first order natural frequency is 174.51 Hz.
3.2 Non-contact modal analysis results
During the non-contact modal tests, the effect of the excitation point position on test results was
investigated. The following test results were acquired.
(1) For the No. 1 excitation point, the first two order natural frequencies were 183.0 Hz and 278.0 Hz,
respectively. The first two order mode shapes were shown in Fig. 8(a). It can be seen that the modes
were asymmetrical.
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Fig. 8 First two order mode shapes under three different excitation points
(2) For the No. 2 excitation point, the first two order natural frequencies were 200.0 Hz and 439.0 Hz,
respectively. The first two order mode shapes were shown in Fig. 8(b). It can be seen that the modes
were Symmetrical.
(3) For the No. 3 excitation point, the first order natural frequencies were 198.0 Hz and 432.5 Hz,
respectively. The first two order mode shapes were shown in Fig. 8(c). It can be seen that the modes
were Symmetrical. The test results accord with the results of the No. 2 excitation point.
3.3 Discussion
The weight of test vehicle is 50 g, and the accelerometer’s weight is 3 g. The first order natural frequency
from impact hammer testing method is 174.51 Hz while the first order natural frequency from non-contact
test method is 200 Hz. The error between these two methods is very large (> 10%). This indicates that
the weight of the accelerometer has great influence on the test results. In addition, the accelerometer
must not be located at nodes and nodal lines for the impact hammer method. For the non-contact test
method, the choice of the proper excitation point position is also very important. The selected excitation
point should be located at a rigid place. For example, the mode shapes were not well and the first two
order frequencies were smaller when No. 1 excitation point was selected. If the rigid of the fixture is
good enough, the excitation point could be able to be located on the fixture. For instance, the first order
natural frequency for No. 3 excitation point was similar to that of No. 2 excitation point, and the mode
shapes of these two excitation points were the same.
4
Random Vibration Simulation
Commercial finite element software ABAQUS was adopted to carry out random vibration analysis.
Because the random vibration analysis is calculated based on modal responses, a modal analysis is a
prerequisite.
4.1 FE modeling and modal analysis
The PCB assembly was fixed on a base plate by eight screws under modal test and random vibration
experiment. Boundary conditions of FE simulation were identical to the experimental boundary
conditions of PCB assembly. The experimental results were applied to determine screwing tightness
of the PCB assembly corresponding to a fixed boundary condition with the eight screws of the PCB
constrained. Based on the non-contact modal test results (No. 2 excitation point was excited), a FE
model was established and the C3D8R solid element type was selected in this model. The Block Lanczos
method was employed to carry out modal analysis. A numerical convergence analysis was carried out
EXPERIMENTAL MODAL ANALYSIS AND RANDOM VIBRATION SIMULATION OF PRINTED CIRCUIT BOARD ASSEMBLY
495
Table 1 Convergence analysis with respect to natural frequencies (Unit: Hz)
Cell
Mode
1
2
A
B
C
Number of Elements
20675
245937
26349
205.34
460.71
200.29
444.52
199.85
442.11
B vs. C
Relative Error (%)
0.22
0.55
Fig. 9 The First six order modes
with respect to natural frequencies of the first modes of the PCB assembly, as shown in Table 1. Clearly
from the results, the FE model with 245937 elements has achieved sufficient accuracy with moderate
computational efficiency. The first six order modes were shown in Fig. 9.
4.2 Frequency domain random vibration responses
Random vibration loads can be conveniently characterized as random processes in terms of the Power
Spectral Density (PSD) functions. Typical dynamic loadings are modeled as a correlated PSD complex
matrix. The random vibration responses, such as stresses and displacements of PCB assembly, are
simulated efficiently by employing ABAQUS frequency response module. The natural frequencies and
mode shapes of the finite element model are extracted from modal analysis. The excitation at the support
(base excitation) of FE model and the acceleration PSD input are used, and the random vibration
responses could be obtained by using mode superposition method. The statistical characteristics of
random vibration response are found through the moments of its PSD function.
When the amplitude of acceleration Power Spectral Density (PSD) inputs is 2((m/s2 )2 /Hz) and the
frequency range is between 50 Hz and 500 Hz, PSDs of center displacement and Root Mean Square
(RMS) of center displacement responses were shown in Fig. 10 and Fig. 11, respectively. It can be shown
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FANG LIU AND RONGPING FAN / ADVANCES IN VIBRATION ENGINEERING, 12(5) 2013
Fig. 10
Fig. 12
Fig. 11
The center displacement PSD
First two order modes for
less pre-tightening force
Fig. 13
RMS of center displacement response
Center displacement PSDs of PCB assembly
from these two figures that the FE results are in accord with the experimental data, FE modeling based
on EMA is very effective. In addition, it can be seen from Fig. 10 that the first order natural frequency
of PCB assembly is 200 Hz. This also proved that both No. 2 excitation point and No. 3 excitation point
were the right choices and the test results from non-contact test method were more accurate than from
the impact hammer method.
When electronic products were always subjected to vibration and mechanical shock, the screws
which were used to fix the PCB assembly would inevitably become looser and the pre-tightening force
of PCB would become small. The pre-tightening force that was produced by screws would affect the
natural frequencies of PCB assembly. Non-contact TV Laser holography technology was adopted and
the No. 2 excitation point was excited, modal tests were conducted under two kinds of pre-tightening
force condition (complete and looser). Test results showed that the mode shapes for these two kinds of
condition were similar and the natural frequencies became smaller when the screws got looser and the
pre-tightening force became less. The mode shapes and frequencies were shown in Fig. 12. Random
vibration simulations were conducted (the amplitude of acceleration PSD inputs and the frequency
range are the same as above). The center displacement PSDs of PCB assembly could be obtained
EXPERIMENTAL MODAL ANALYSIS AND RANDOM VIBRATION SIMULATION OF PRINTED CIRCUIT BOARD ASSEMBLY
497
(shown in Fig. 13). It can be seen that the center displacement PSDs under less pre-tightening force
were bigger. This indicated that PCB assembly would be more likely to be damaged when the screws
became looser.
5
Conclusions
Two kinds of test methods were used to conduct modal tests, and they each had their advantages and
shortcomings. For impact hammer testing method, the accelerometer’s weight should be as light as
possible if the test vehicle is very light. Otherwise, test results would have a great error. In addition, the
accelerometer’s location should avoid the nodes and nodal lines. For Non-contact TV Laser holography
technology, the excitation point should be located at the rigid part of the structure. The results from EMA
could be used to establish an equivalent FE model. Natural frequencies and mode shapes obtained by
modal test and FE analysis were generally in very good agreement. Random vibration simulation results
of PCB assembly were in accord with test data, which also proved that the equivalent FE model was an
effective model. So EMA is very significant to establish FE model. In addition, if the screws became
looser, PCB assembly would be more likely to be damaged. These will provide a guide for investigating
the dynamic characteristics and optimization design of PCB assembly.
Acknowledgments
This work is jointly supported by NSFC (No. 50775138 and No. 11102141), the science and technology
project of Hubei Educational Committee, China (No. B20111602). This research is supported in part
by a Young Scholars Funding of Wuhan Textile University.
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