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′yfighly=

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′withnn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ea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

=χ

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

1. Suhir, E., “Response of a Flexible Printed Circuit Board to Periodic Shock Loads

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

of PCBs in Portable Electronic Products during Drop Impact”. Proceedings 4th

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2002, 120-125

5. Wang, Y.Q., “Modeling and Simulation of PCB Drop Test”, Proc. 5-th EPTC,

Singapore, 2003, 263-268

6. Luan J.E., Tee, T.Y., Pek E., Lim C.T., Zhong, Z.W., “Modal Analysis and Dynamic

Responses of Board Level Drop Test”, 5-th EPTC Conference4 Proc., Singapore,

2003, 233-243

7. Tee, T.Y., Luan, J.E., Pek, E., Lim, C.T., Zhong, Z.W., “Novel Numerical and

Experimental Analysis of Dynamic Responses under Board Level Drop Test”,

EuroSime Conference Proc., Berlin, Germany, 2004, 671-677

8. Marjamaki, P., Mattila, T., Kivilahti, J., “FEA of Lead-Free Drop Test Boards”,

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10. Timoshenko, S.P., and J.M.Gere, “Theory of Elastic Stability”, 2-nd ed.,

McGraw-Hill, 1988.

11. Suhir, E., “Structural Analysis in Microelectronics and Fiber Optics”, Van-Nostrand,

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12. Pars, L.A., “A Treatise of Analytical Dynamics”, Heinemann, London, 1965.

Journal of Solid Mechanics

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1397

13. Kauderer, H., “Nichtlineare Mechanik”, Springer, Berlin, 1958 (in German)

14. Sneddon, I.N., Special Functions of Mathematical Physics and Chemistry, 3-rd ed.,

Longman, New York, 1980.

15. Appell P., Lacour E., “Principes de la theorie des functions elliptiques et

applications”, 2-nd ed., Gauthier-Villars, Paris, 1922 (in French)

16. Oberhettinger, F., und Magnus, W., “Anwendung der elliptischen Funktionen in

Physik und Technik”, Springer Verlag, 1949 (in German).

17. Spanier, J., and Oldham, K.B., “An Atlas of Functions”, Hemisphere Publ. Corp.,

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Academic Press, 1980.

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