Journal of Solid Mechanics and Materials

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J
ournal of Soli
d
Mechanics
an
d
Materials

Engineering

Vol. 4, No. 9, 2010
1381
Duffing Oscillator: Could an Impact Load of
Finite Duration be Substituted with an
Instantaneous Impulse?*
E. SUHIR** and L. ARRUDA***
** Bell Laboratories, Murray Hill, NJ (ret), University of California, Santa Cruz, CA,
University of Maryland, College Park, MA, and ERS Co., Los Altos, CA, USA
727 Alvina Ct., Los Altos, CA 94024, E-mail:suhire@aol.com
*** Nokia Corp., Instituto Nokia de Tecnologia, Manaus, Brazil


Abstract
We show that a single degree of freedom system with a rigid cubic characteristic of
the restoring force (known in the physical literature on nonlinear vibrations as
“Duffing oscillator”) is a suitable analytical (“mathematical”) model that can be
used, when appreciable reactive in-plane (“membrane”) forces occur, to evaluate the
dynamic response of a printed circuit board (PCB) subjected to a drop or a shock
impact. When modeling such a response either of the PCB itself, or of a surface
mounted device (SMD) package, including ball-grid-array (BGA) or pad-grid-array
(PGA) structure, on a board level, there is an obvious incentive in trying to simplify
the modeling by substituting an impact load of finite duration with an instantaneous
impulse. On the other hand, when there is an intent to replace drop tests with shock
tests, one has to properly “tune” the shock tester, so that to adequately mimic the
drop test conditions. In this analysis we obtain exact solutions to the Duffing
equation for the cases of an instantaneous impulse and for a suddenly applied and
suddenly removed constant loading. We use these solutions to determine the error (in
terms of the predicted amplitudes and accelerations) from substituting an impact
load of finite duration with an instantaneous impulse. We consider an elongated
PCB, which is currently employed in the Nokia accelerated test vehicle
(experimental setup). The PCB’s short edges are simply supported, while its long
edges are support-free. The in-plane reactive forces arise because the PCB’s short
edges (supports) cannot get closer during its impact induced vibrations. We have
determined, based on the obtained model, that the nonlinear system in question is, in
general, less sensitive to the duration of the applied load than a linear system, and
that this sensitivity decreases with an increase in the degree of the nonlinearity. Since
the nonlinear frequency is strongly dependent on the magnitude of the applied load,
we suggest that the nonlinear analysis be carried out prior to the assessment of the
possible error in a modeling or a testing effort.
Key words: Duffing Oscillator, Impact Loading, Shock Tests, Drop Tests

1. Introduction
The dynamic response of printed circuit boards (PCBs) to shocks and vibrations
has been addressed in numerous investigations
91−
with different objectives in mind.
In the analysis that follows we consider the dynamic response of an elongated
flexible PCB to a drop impact load applied to its supports. The board is being used in
a Nokia accelerated test vehicle (experimental setup). Our objective is to evaluate
*Received 5 Feb., 2009 (No. 09-0056)
[DOI: 10.1299/jmmp.4.1381]
Copyright © 2010 by JSME



Journal of Solid Mechanics
and Materials Engineering

Vol. 4, No. 9, 2010
1382
the response of a PCB to a constant load of finite duration suddenly applied to, and
suddenly removed from, its supports. Particularly we intend to assess the error from
substituting such a load with an instantaneous impulse. We consider a simplified
situation, when the board is simply supported at its short edges and is support-free at
its long edges. When modeling the dynamic response of a PCB or a particular
surface mounted device (SMD) system on a board level (such as, say, a
ball-grid-array or a pad-grid-array structure) to a shock load applied to the PCB
supports during drop or shock tests, there is an obvious incentive in trying to
simplify the analysis by substituting the impact load of finite duration with an
instantaneous impulse. On the other hand, when there is an intent to substitute drop
tests with shock tests, one has to “tune” the shock tester, in terms of the magnitude
and duration of the applied impact load, so that physically meaningful results are
obtained. In either situation, it is important to find out the error (in terms of the
predicted maximum displacements, accelerations, stresses and strains, etc.) from
substituting the load of finite duration with an instantaneous impulse. When the
impact load is significant and the PCB supports cannot get closer during the induced
vibrations, appreciable in-plane reactive forces (stresses) arise, and the PCB
vibrations become essentially nonlinear. Intuitively it is felt that a nonlinear system
with a rigid characteristic of nonlinearity (such as Duffing oscillator) should be less
sensitive to the duration of the applied load than a linear system. Our analysis is, in
effect, an attempt to quantify such an intuitive feeling.
2. Analysis
2.1 Non-linear Response
The impact induced vibrations of an elongated printed circuit board (PCB) after
the load is removed are characterized by the following equation of motion
(equilibrium):

.0),(),()(),( =+
′′
− txwmtxwtTtxDw
IV

(1)
Here
),( txw
is the deflection function,
)(tT
is the reactive tensile force,
)1(12
2
3
ν−
=
Eh
D
is the flexural rigidity of the PCB,
E
and
ν
=慲攠瑨攠敬慳瑩′=
捯湳瑡湴猠潦⁴桥a瑥物慬Ⱐ
h
is the PCB thickness,
t
is time, and
m
is the PCB mass
per unite area. The origin of the coordinate
x
is in the PCB’s mid-cross-section.
Assuming that the vibrations are caused by the initial velocities
)0,(
xw

applied
instantaneously to the PCB short edges,
a
x
±
=
, we seek the solution to the
equation (1) in the form:

a
x
tftxw
2
cos)(),(
π
=
. (2)
Here
)(
tf
is the principal coordinate,
a
x
xX
2
cos)(
π
=
is the coordinate function,
and
a
is half the PCB length. The solution (2) satisfies the boundary conditions
,0),( =±
taw

0),(
=
±


taw
. From (2) we have:



Journal of Solid Mechanics
and Materials Engineering

Vol. 4, No. 9, 2010
1383

a
x
tftxw
2
cos)(),(
π


=
. (3)
If the shock load is due to a drop impact, the vertical velocities
)0,(xw

of all the PCB
points are the same at the initial moment of time,
0
=
t
:

gH
a
x
f
a
x
fxw 2
2
cos
2
cos)0()0,(
0
===
π
π


, (4)
where
H
is the drop height, and
g
is the acceleration due to gravity. Using Fourier
transform, we obtain:
gHf
2
4
0
π
=

,
where the factor
π
4
reflects the role of the coordinate function. Introducing the
sought solution (2) into the equation (1) we obtain the following equation for the
principal coordinate
)(
tf
:

0)(
)(
2
)()(
2
2
=






++ tf
m
tT
a
tftf
π
ω

, (5)
where
m
D
a
2
2






=
π
ω

is the linear frequency. Linear vibrations take place when the reactive tensile force
)(tT

is zero or small and could be neglected. This force can be found based on the
following simple reasoning. The length
s
of the deflected elastic curve can be
determined as
[ ]
[ ]

∫∫
+=






+=
=

+≈

+=
a
aa
tf
a
adx
a
x
tf
a
a
dxtxwadxtxws
0
2
2
22
2
0
2
0
2
)(
8
2
2
sin)(
2
2
),(2),(12
πππ

The force
)(tT
that results in an elongation
)(
8
2
2
2
tf
a
ass
π
=−=∆
can be found,
in accordance with the Hooke’s law, as

)(
42
)(
2
2
tEhf
a
s
a
h
EtT






=∆=
π
. (6)
Then the equation (5) results in the following (Duffing-type) nonlinear equation for
the principal coordinate
:)(tf




Journal of Solid Mechanics
and Materials Engineering

Vol. 4, No. 9, 2010
1384

0)()()(
32
=++ tftftf µω

, (7)
where
m
Eh
a 42
4






=
π
µ
is the parameter of nonlinearity. In the case of forced
vibrations, the equation (1) should be replaced by the following inhomogeneous
equation:

),(),(),()(),( tqtxwmtxwtTtxDw
IV
=+
′′


(8)
where the load
)(tq
is equal to zero at
.
a
x
±

In this case the function
)(
tf
could
be found from the equation

m
tQ
tftftf
)(
)()()(
32
=++
µω

, (9)
where
)(
4
)( tqtQ
π
=
is the generalized force.
2.2 Vibrations Caused by an Instantaneous Impulse: Vibration Amplitudes
In the case of an instantaneous impulse, the induced vibrations are free, and the
equation (7) should be used to evaluate the response. This equation can be written as

0)(
2
1
)()(
4222
=






++ tftftf
dt
d
µω

,
so that

Ctftftf =++ )(
2
1
)()(
4222
µω

, (10)
where
.ConstC =
With the zero displacement
)0(
0
ff =
and non-zero velocity
gHff 2
4
)0(
0
π
==

,
we have
.
32
2
gHC
π
=

On the other hand, when the displacement
)(tf
reaches its maximum value,
S
A
, it is
the velocity
)(tf


that is zero. Then the equation (10) results in the following
bi-quadratic equation for the amplitude
S
A
:

0
8
2
2
2
2
4
=






−+
µπµ
ω
gH
AA
SS
. (11)
The solution to this equation can be written as
0
SAS
AA
η
=
, where
ω
π
gH
A
S
2
4
0
=




Journal of Solid Mechanics
and Materials Engineering

Vol. 4, No. 9, 2010
1385
is the linear amplitude
)0(
=
µ
Ⱐ慮搠瑨攠晡捴潲†
µ
µ
η
121 −+
=
A

considers the effect of the nonlinearity on the amplitude of vibrations. In the last
formula,
2
0








=
ω
µµ
S
A

is the dimensionless parameter of nonlinearity. Thus, the vibration amplitudes could
be obtained without obtaining the complete solution to the equation (7), i.e., without
being able to establish the function
)(
tf
.

2.3 Vibrations Caused by a Suddenly Applied Constant Load:
Maximum Displacement
In the case of a suddenly applied load, the equation (9) should be used to
determine the principal coordinate
)(
tf
. This equation can be written as

0)(2)(
2
1
)()(
4222
=






−++
tf
m
Q
tftftf
dt
d
µω

.
so that

Ctf
m
Q
tftftf
=−++ )(2)(
2
1
)()(
4222
µω

. (12)
If the initial conditions
)0(f
and
)0(
f

are zero, the constant
C
is zero as well, and
the velocity
)(
tf

can be expressed through the displacement
)(
tf

as

)(
2
1
)()(2)(
422
tftftf
m
Q
tf µω −−=

.
(13)

The obtained equation determines the phase diagram of the nonlinear system in
question. The displacement
)(tf
reaches its maximum value
max
f

at the end of the
first quarter-period of vibrations, when the velocity
)(
tf

is zero. This results in the
equation:

0
4
2
max
2
3
max
=−+
m
Q
ff
µµ
ω
(14)
The maximum linear (
µ
㴰⤠摩獰污捥den琠楳†
2
0
浡m
2
ω
m
Q
f
=
,
and the solution to the cubic equation (14) can be written as

0
maxmax
ff
Q
η=
,
where the factor



Journal of Solid Mechanics
and Materials Engineering

Vol. 4, No. 9, 2010
1386

33
27
8
11
27
8
11
f
f
f
f
Q
µ
µ
µ
µ
η
+−
+
++
=
(15)
considers the effect of the nonlinearity on the maximum displacement. This factor
changes from one to zero, when the dimensionless parameter of nonlinearity
2
0
max








=
ω
µµ
f
f

changes from zero to infinity. Thus, we were able to obtain the expression for the
maximum displacement without obtaining the solution to the basic equation (9).

2.4 Dynamic Factor
If one puts the acceleration
)(
tf

in the equation (9) equal to zero and considers
that the force
QtQ
=
)(
is time-independent, then the corresponding static
displacement
st
f
can be found from the equation

0
2
3
=−+
m
Q
ff
stst
µµ
ω
. (16)
The maximum static linear (
)0
=
µ
摩獰污捥d敮e⁩猠
2
0
ω
m
Q
f
st
=
and turns out to be
half of the maximum dynamic displacement that occurs when the load
Q
is
suddenly applied to, and remains on, the system. The solution to the equation (16)
can be written as
0
ststst
ff η=
, where the factor
33
27
4
11
27
4
11
st
st
st
st
st
µ
µ
µ
µ
η
+−
+
++
=
,
2
0








=
ω
µµ
st
st
f
(17)
considers the effect of the nonlinearity on the maximum static displacement. The
factor
st
η
changes from one to zero, when the parameter
st
µ
changes from zero to
infinity. Calculations indicate that the factor
z
η
decreases more rapidly with an
increase in the parameter
µ
=ofon汩l敡物瑹⁴=慮⁴=攠晡捴e±=
st
η
, and therefore the
dynamic factor
st
z
st
d
f
f
K
η
η
2
max
==

(which is equal to 2 for a linear system) trends to 1 for a highly nonlinear system.
Hence, the effect of the dynamic application of the loading
)(tQ
is less significant in
a nonlinear system than in a linear one.




Journal of Solid Mechanics
and Materials Engineering

Vol. 4, No. 9, 2010
1387
2.5 Vibrations Caused by an Instantaneous Impulse:
Solution to the Basic Equation
The equation (7) lends itself to the following exact solution:

cnuAtcnAtf
SSS
== ),()(
εσ
, (18)
where
tu
s
σ
=
,
cnu
is an elliptic cosine
1714

,
S
σ
is the parameter of the nonlinear
frequency and
ε
⁩猠瑨攠楮楴楡i⁰ha獥⁡湧l攮eUs楮i⁴=攠景±mul慳†
( )
,snudnucnu −=


( )
,cnudnusnu =


( )
,
2
snucnukdnu
S
−=


of differentiating the elliptic functions, and the formulas
,1
22
=+ ucnusn

1
222
=+ usnkudn
S

of the “elliptic geometry” we obtain:

,)(
snudnuAtf
SS
σ
−=


(
)
usnkcnuAtf
SSS
222
21)(
−−=
σ

. (19)
Here
s
nu
is the elliptic sine,
dnu
is the function of delta-amplitude and
S
k
is the
modulus of the elliptic function. Introducing (18) and the second formula in (19) into
the equation (7), we conclude that this equation is fulfilled, if the following
relationships take place:

2
22
21
S
SS
k
A

=+=
ω
µωσ
,








−==
2
2
1
2
1
2
S
S
S
S
A
k
σ
ωµ
σ
. (20)
The period of the function
),(
ε
tcn
is
)(4
S
kK
, where


=
2
0
22
sin1
)(
π
ξ
ξ
S
S
k
d
kK


is the complete elliptic integral of the first kind. Hence, the period of the function
),(
εσ
tcn
S
, i.e., the period of the vibrations, is
)(
4
kK
S
σ
, and the frequency is
)(2
S
S
S
kK
p
πσ
=
.
The maximum acceleration (deceleration) can be found directly from the equation
(7):

SSSSS
AAAf
2
3
2
σµω −=−−=


.

Thus, we were able to obtain the exact solution, given by the formula (18), to the
basic equation (7).


2.6 Vibrations Caused by a Suddenly Applied Constant Load: Solution to the
Basic Equation
Using the formula (13), one can represent the solution to the equation (9) in the
form:



Journal of Solid Mechanics
and Materials Engineering

Vol. 4, No. 9, 2010
1388

Q
f
u
tftftf
m
Q
df
t
σ
µω
=
−−
=

0
422
)(
2
1
)()(2
, (21)
where


===
θ
θ
θ
θεσ
0
22
sin1
),(),(
Q
QQ
k
d
kFtuu

is the elliptic integral of the first kind
1714

,
Q
k
is the modulus of the elliptic function,
amu=
θ
is the amplitude of this function, and
Q
σ
is the parameter of the nonlinear
frequency. In order to express the modulus
Q
k
and the parameter
Q
σ
through the
characteristics of the system (9) we seek the inversion of the integral (21) as

cnu
cnu
ftf
)1(1
1
)(
max
δδ −−+

=
, (22)
where
0
maxmax
ff
Q
η=
is the maximum displacement, and
δ
=楳⁴i攠ea牡±e瑥爠tf⁴=攠
m潤畬畳u
Q
k
. From (22) we obtain:

[ ]
2
max
)1(1
2)(
cnu
snudnu
ftf
Q
−−+
=
δδ
δσ

, (23)
(
)
[ ]
3
2222
max
2
)1(1
])21(1)[1()21(1
2)(
cnu
usnkcnuusnk
ftf
QQ
Q
−−+
−+−+−+
=
δδ
δδ
δσ

. (24)
Introducing the formulas (22) and (24) into the equation (9) we conclude that this
equation is fulfilled if the relationships
,33
2
1
max
2
3
max Q
f
Q
m
f
Q
m
η
ωµ
δ −=−=+=
( )
,
3
1
8
2
0
max
3
2
2










=
f
ω
δ
δ
µ

,
8
)3)(1(
δ
δδ
−−
=
Q
k

2
3
2
δ
δ
ωσ

=
Q
(25)
take place. From the first formula in (25) we have:

( )
3
2
2
2
0
max
3
1
8
δ
δ
ω
µµ


=








=
f
f

When this parameter changes from zero (linear system) to infinity (highly nonlinear
system), the parameter
δ
=捨anges⁦牯m‱⁴=
㜳㈱.13 =
, the modulus
Q
k
changes
from zero to
,2588.0
2
32
=

=k




Journal of Solid Mechanics
and Materials Engineering

Vol. 4, No. 9, 2010
1389
the maximum dynamic displacement
max
f
changes from
2
2
ωf
Q
to zero, the angle
Q
karcsin=α
changes from zero to 15˚, and the parameter
Q
σ
of the nonlinear
frequency changes from
ω
=⡬楮敡爠晲敱uen′y)⁴=⁩=晩fi瑹
晲敱uen′yf⁡⁨ighly=
湯湬楮敡爠獹獴sm).†
周攠景牭ul愠景±⁴= 攠emp汩瑵de=
amu
=
θ
can be obtained from (22) by putting
.cos
θ
=cnu
This yields:








−= 1
1
2
max
f
f
arcctg
δ
θ
.
The amplitude
θ
=牥慣桥猠楴猠ia硩mum⁶a汵攠
,
2
浡m
π
θ
=
when the displacement
)(tf

reaches
.
max
f
In this case the integral


=
θ
θ
θ
θ
0
22
sin1
),(
Q
Q
k
d
kF

becomes the complete elliptic integral of the first kind:







==
QQ
kFkKu
,
2
)(
π
.
Since the time required for the angle
θ
=瑯⁣=慮ge⁦牯m=≠e牯⁴==
2
π
=楳⁥iua氠瑯⁴=攠
煵慲t敲=⁴h攠灥ei潤o⁶i扲慴i潮猬⁷e⁣潮捬畤攠uh慴⁴his⁰敲i潤⁩s=
,
)(4
Q
kK
σ
so that
the frequency of the nonlinear vibrations is
.
)(2
Q
Q
Q
kK
p
σ
π
=

At the moment
0
tt =

of time

the solution (21) yields:



==
==
−−
=
0
0
0
22
0
0
0
422
0
sin1
1
),(
),(
)(
2
1
)()(2
θ
θ
θ
σσ
θ
σ
εσ
µω
Q
k
d
kF
tu
tftftf
m
Q
df
t
QQ
Q
Q
Q
f
, (26)
where








−= 1
1
2
0
max
0
f
f
arcctg
δ
θ
,



Journal of Solid Mechanics
and Materials Engineering

Vol. 4, No. 9, 2010
1390
so that
2
1
0
2
max
0
θ
δctg
f
f
+
=

After the displacement
0
f
is found, the velocity
0
f

can be determined by the
formula (13).

2.7 Free Vibrations with Non-Zero Initial Conditions
Let us revisit the equation (7). With the initial conditions
0
)0( ff
=
and
0
)0( ff

==
the formula (10) yields:
4
0
2
0
22
0
2
1
fffC µω ++=

.
If the initial conditions
0
f
and
0
f

are due to the removal of the previously applied
constant impact loading, then we have:

0
2 f
m
Q
C =
, and the equation (13) yields:

0
4222
2)(
2
1
)()( f
m
Q
tftftf =++ µω

. (27)
The maximum displacement
Q
Af =
max
takes place at the moment of time when the
velocity
)(tf

is zero, and the relationship (27) yields:

042
0
2
2
4
=−+ f
m
Q
AA
QQ
µµ
ω
. (28)
This equation has the following solution:

,141
0
4
0
2
QQQ
A
m
Qf
A η
ω
µ
µ
ω
=








−+=
(29)
where
0
2
0
2 f
m
Q
A
Q
ω
=


is the linear amplitude, and the factor
Q
Q
Q
µ
µ
η
121 −+
=

considers the effect of the nonlinearity. This factor changes from 1 to zero, when the
dimensionless parameter
4
0
2
ω
µµ
m
Qf
Q
=




Journal of Solid Mechanics
and Materials Engineering

Vol. 4, No. 9, 2010
1391
of nonlinearity changes from zero to infinity. The maximum acceleration
(deceleration) can be found as
QQQQQ
AAAf
2
3
2
σµω −=−−=

,
where
22
QQ
A
µωσ
+=

is the parameter of the nonlinear frequency.

2.8 Error from Substituting an Impact Load of Finite Duration with an
Instantaneous Impulse
The equation (11) can be written as

,0
2
2
2
0
2
2
4
=−+
VAA
SS
µµ
ω
(30)
where
gHV 2
4
0
π
=
is the initial velocity. This velocity can be determined as
m
S
V =
0
when the applied load is caused by an instantaneous impulse,
S
. In a
situation, when the actual impulse
S
is due to a suddenly applied and suddenly
removed constant force
Q
acting during the time
0
t

we have:
0
QtS
=
, so that the
equation (30) can be written as

.0
2
2
2
2
0
2
2
2
4
=−+
m
tQ
AA
SS
µµ
ω

(31)
This bi-quadratic equation has the following solution:









−+= 121
42
2
0
2
2
ω
µ
µ
ω
m
tQ
A
S
. (32)
Comparing the solutions (29) and (32) we conclude that the ratio

=
−+
−+
==
121
141
42
2
0
2
4
0
ω
µ
ω
µ
χ
m
tQ
m
Qf
A
A
S
Q
A
( )
( )
1),(
3
1
21
1
2
1
1
3
1
81
0
2
2
2
2
0
2
2
2
2



+

+


+
Q
kF
ctg
θ
δδ
δ
θ
δ
δ
δ
(33)
characterizes the error, as far as the impact induced amplitudes are concerned, from
substituting a constant impact load
Q
applied for a short period
0
t

of time (and
removed when the displacement of the system is equal to
0
f
) with an instantaneous
impulse
0
QtS
=
. The ratio of the corresponding maximum accelerations
(decelerations) can be found as



Journal of Solid Mechanics
and Materials Engineering

Vol. 4, No. 9, 2010
1392

2
2
2
2
2
1
1
S
SA
A
S
Q
f
A
A
f
f
ω
µ
ω
µ
χ
χχ
+






+
==



(34)
The function
),(
0
θδχχ
AA
=
is tabulated in Table 1. This table enables one to
determine, for the given degree of nonlinearity (characterized by the parameter
δ
潦o
th攠e潤畬畳u
0
Q
k
of the elliptic function during the time of forced vibrations, i.e.,
prior to the removal of the constant load Q) and the given duration of loading
(characterized by the phase angle
0
θ
⤮†䉯)h⁴h攠湵me牡±o爠慮搠±h攠摥湯mi湡no±=
畮摥爠uh攠獱畡牥⁲潯u⁩n⁴he=景牭畬愠⠳㐩⁲敤u捥′with⁡n⁩n捲敡獥⁩n⁴h攠摥g±敥映=h攠
湯湬楮敡物瑹Ⱐ,h楣栠楳⁣潮獩d敲敤⁢e⁴桥⁡=gl攠
.
α
⁔=楳⁳楴i慴楯n,⁡猠敶楤敮琠晲tm⁴=e=
θabl攠e⁤a瑡Ⱐ牥獵汴猠楮⁴=攠晡捴⁴e慴Ⱐ景爠獨o±琠汯慤楮gs
=ma汬l
0
θ
va汵敳⤬⁴桥⁦慣瑯l=
A
χ
has a minimum at certain
α
⁶=lues.=χo±=
δ
=va汵敳⁣汯獥⁴漠ㄠ⡬楮敡爠獹獴sm⤬⁴桥)
景牭畬愠⠳㌩ayi敬摳d†
†††††=††††
,
2
2
獩s
0
0
0
θ
θ
χχ ==
AA
.
00
t
ωθ =
.
(35)
The factor
A
χ
changes from 1 to 0.9004, when the phase angle
0
θ
changes from
zero (instantaneous impulse) to
0
90
(the duration of loading is equal to a quarter of
the period of vibrations). For
δ
va汵敳⁣汯獥⁴l=
,㜳㈱.13=
the formula (34)
yields:

( )
( )
2
5774.02588.0,
4142.1
2
1
,
2
0
2
0
0
2
0
θ
θ
θ
δ
θ
χ
ctgF
ctgkF
Q
A
+
=
+
=
. (36)
The factor
A
χ
changes from 1 to 0.9972, when the phase angle
0
θ
changes from
zero (instantaneous impulse) to
0
90
(the duration of loading is equal to a quarter of
the period of vibrations).

3. Numerical Example
Input data

PCB length
,1012 mma
=
thickness
,00.1 mmh =
Young’s
modulus
2
/2020
mmkgE
=
, Poisson’s ratio
,3.0=
ν
mass per unit area
,sec1095.2
10210
−−
=
xmmkgxxm
phase angle when the constant load is removed



Journal of Solid Mechanics
and Materials Engineering

Vol. 4, No. 9, 2010
1393
.
2
0
π
θ =

Table 1.
Error in the predicted amplitudes from substituting an impact load of finite duration
with an instantaneous impulse
0
arcsin
Q
k=
α
0 5 10 14 14.5 15
0
Q
k

0

0.0872 0.1736 0.2419 0.2504 0.2588
δ
=
ㄠㄮ〳ㄹ ㄮㄴ㤸 ㄮ㐲ㄸ1ㄮ㔰㐶
3
㴱⸷㌲=
,
0
θ

A
χ

0 1 1 1 1 1 1
5 0.9998 0.9992 0.9994 0.9996 0.9996 1
10 0.9988 0.9989 0.9988 0.9995 0.9996 1
20 0.9950 0.9949 0.9962 0.9983 0.9988 0.9999
30 0.9887 0.9888 0.9918 0.9961 0.9973 0.9998
40 0.9799 0.9804 0.9855 0.9934 0.9955 0.9997
50 0.9686 0.9696 0.9775 0.9900 0.9932 0.9995
60 0.9550 0.9567 0.9680 0.9860 0.9905 0.9993
70 0.9390 0.9416 0.9570 0.9815 0.9874 0.9989
80 0.9208 0.9245 0.9445 0.9762 0.9837 0.9983
90 0.9004 0.9055 0.9305 0.9700 0.9792 0.9972

Computed data

Flexural rigidity
kgxmm
xEh
D 982.184
)3.01(12
00.12020
)1(12
2
3
2
3
=

=

=
ν

Linear frequency

1
10
22
sec143.766
1095.2
982.184
1012


=






=






=
x
m
D
a
ππ
ω

Parameter of nonlinearity
22
10
44
sec418.1602440
1095.24
00.12020
10142
−−

=






=






= mm
xx
x
m
Eh
a
ππ
µ


The computed data in Table 2 is obtained for different drop heights.
The duration
0
t
of loading in the line 14 is accepted in such a way that the phase
angle
00
t
Q
σθ
=
is equal to
.90
2
0
=
π
The Table 3 illustrates how this duration was
obtained for the drop height
.10mmH
=

The parameters of the nonlinear frequencies and the frequencies themselves, as well
as the quarters of the corresponding times (periods of vibrations) in milliseconds, are
shown, for the assumed drop heights, in Table 4. As evident from this table the
frequencies of the vibrations caused by constant loads are approximately twice as
high as the frequencies of vibrations due to instantaneous impulses.





Journal of Solid Mechanics
and Materials Engineering

Vol. 4, No. 9, 2010
1394
Table 2.
Response characteristics for different drop heights
1
mmH,

10 50 500 900
2
sec/
,2
4
0
mm
gHV
π
=


563.688

1260.444

3985.874

5347.611
3
2
7
0
sec
10,

=
xmmkgx
xmVS

1.6629

3.7183

11.7583

15.7754

4
mm
V
A
S
,
0
0
ω
=


0.7358

1.6452

5.2026

6.9800
5
2
0








=
ω
µµ
S
A


1.4780

7.3892

73.8930

133.0066
6
µ
µ
η
121 −+
=
A


0.8180

0.6342

0.3893

0.3396
7
mmAA
SAS
,
0
η
=

0.6019 1.0434 2.0253 2.3705
8
1
2
sec,1







+=
ω
µωσ
S
S
A

1080.0022

1526.9334

2675.8044

3097.0181
9
22
sec/,mmAf
SSS
σ
−=


-71.5673g -248.2361g -1479.6943g -2320.0715g
10








−=
2
2
1
2
1
S
S
k
σ
ω


0.4984

0.6117

0.6775

0.6851
11
0
,arcsin
SS
k=
α

29.893 37.709 42.649 43.246
12
)(
S
kK

1.6858 1.7637 1.8262 1.8351
13
1
sec,
)(2

=
S
S
S
kK
p
π
σ


1006.325

1359.925

2301.577

2650.962
14
mst
,
0

0.7183 0.5010 0.2890 0.2492
15
24
0
/,10 mmkg
t
S
Q =

2.3151

7.4218

40.8083

63.3142

16
mm
m
Q
f,
2
2
0
max
ω
=


2.6740

8.5722

47.1343

73.1289

17
2
0
max








=
ω
µµ
f
f


19.5203

200.6100

6065.0821

14599.6083
18
Q
η


0.3956

0.1999

0.0675

0.0507
19
mmff
Q
,
0
maxmax
η
=

1.0578 1.7133 3.1816 3.7076
20
Q
ηδ
−= 3

1.6138 1.6733 1.7125 1.7173
21
),(
0
θδχχ
AA
=

0.9878 0.9925 0.9956 0.9960
22
mmAA
SAQ
,
χ
=

0.5946 1.0356 2.0165 2.3611

23
S
Q
f
f
f






0.9759

0.9814

0.9876

0.9885






Journal of Solid Mechanics
and Materials Engineering

Vol. 4, No. 9, 2010
1395
Comparing the corresponding times (vibration periods) with the loading durations,
0
t
, that result in phase angles of
0
0
90=θ
, we conclude that an impact load of finite
duration can be substituted by an instantaneous impulse, if the duration does not
exceed the time
S
t
σ
π
2
0
=
, i.e., a quarter of the “period” that corresponds to the
parameter
S
σ
of the nonlinear frequency of vibrations caused by an instantaneous
impulse.

Table 3.
Example of how the duration of the impact was computed for the given phase angle
and the given degree of non-linearity
mst
,
0

3636.0
8
=
S
σ
π
0.7183
7272.0
4
=
S
σ
π
4544.1
2
=
S
σ
π
24
0
/,10 mmkgx
t
S
Q =

4.5734


2.3150


2.2867


1.1434

mm
m
Q
f,
2
2
0
max
ω
=

5.2824 2.6739 2.6412 1.3206
2
0
max








=
ω
µµ
f
f

76.1772 19.5191 19.0443 4.7611
Q
η

0.2679 0.3956 0.3983 0.5667
mmff
Q
,
0
maxmax
η
=
1.4150 1.0578 1.0520 0.7484
Q
ηδ
−= 3

1.6529 1.6138 1.6130 1.5599
2
3
2
δ
δ
ωσ

=
Q


2691.2

2188.2

2180.6

1797.6
00
t
Q
σθ
=

0.9785
5708.12/
=
π

1.5857 2.6144

Table 4.
Parameter of the non-linear frequency and the non-linear frequency itself for an
instantaneous impulse and an (suddenly applied and suddenly removed constant) impact load
of finite duration
Drop height, H, mm 10 50 500 900
Parameter of
the nonlinear
frequency,
1
,
sec

S
σ

1080
(1.4544ms)
1527
(1.0287ms)
2676
(0.5870ms)
3097
(0.5072ms)
Instantaneous
impulse
Nonlinear
frequency,
1
,
sec

S
p
1006
(1.5614ms)
1360
(1.1550ms)
2302
(0.6824ms)
2651
(0.5925ms)
Parameter of
the nonlinear
frequency,
1
,
sec

Q
σ
2188
(0.7179ms)
3133
(0.5014ms)
5464
(0.2875ms)
6295
(0.2495ms)
Constant load
Nonlinear
frequency,
1
,
sec

Q
p

2079
(0.7556ms)
2977
(0.5276ms)
5193
(0.3025ms)
5982
(0.2626ms)




Journal of Solid Mechanics
and Materials Engineering

Vol. 4, No. 9, 2010
1396
4. Conclusions

The Duffing oscillator is a suitable analytical (“mathematical”) model that can
be used to analyze the nonlinear dynamic response of a printed circuit board (PCB)
subjected to a drop or a shock impact. Simple and physically meaningful solutions
are obtained for this oscillator when the excitation force can be idealized as an
instantaneous impulse or a constant suddenly applied load of finite duration. The
obtained solutions and the calculation procedures can be used to model the dynamic
response of a PCB or a particular surface-mounted-device (SMD) package,
including ball-grid-array (BGA) and pad-grid-array (PGA) systems, on a board level
to an impact load applied to the PCB supports during drop or shock tests. Although
the analysis is carried out for to a simplified case of a simply supported elongated
PCB employed in a specific accelerated test vehicle, whose ultimate goal is to
predict the physical behavior of a SMD package and especially the performance of
the BGA solder joint interconnections, it can be easily generalized for a PCB with
other boundary conditions and/or with a finite aspect ratio and/or with different
boundary conditions at the support contour.
References
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Applied to Its Support Contour”, ASME Journal of Applied Mechanics, vol. 59, No.
2, 1992.
2. Suhir, E., “Nonlinear Dynamic Response of a Flexible Thin Plate to a Constant
Acceleration Applied to Its Support Contour, with Application to Printed Circuit
Boards Used in Avionic Packaging”, Int. Journal of Solids and Structures, vol. 29,
No. 1, 1992.
3. Suhir, E., “Predicted Fundamental Vibration Frequency of a Heavy Electronic
Component Mounted on a Printed Circuit Board”, ASME Journal of Electronic
Packaging, vol.122, No.1, 2000, 1-3
4. Seah S.K.W. Lim C.T. Wong E.H. Tan V.B.C. Shim V.P.W. “Mechanical Response
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Journal of Solid Mechanics
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Vol. 4, No. 9, 2010
1397
13. Kauderer, H., “Nichtlineare Mechanik”, Springer, Berlin, 1958 (in German)
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