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Shock transmission in a coupled beam system

K. Vijayan

n

, J. Woodhouse

Cambridge University Engineering Department, Trumpington Street, Cambridge CB2 1PZ, UK

article info

Article history:

Received 21 May 2012

Received in revised form

15 February 2013

Accepted 18 February 2013

Handling Editor: D.J. Wagg

Available online 21 March 2013

abstract

This paper investigates the circumstances under which high peak acceleration can occur

in the internal parts of a system when subjected to impulsive driving on the outside.

Motivating examples include the design of packaging for transportation of fragile items.

The system is modelled in an idealised form using two beams coupled with point

connections. A Rayleigh–Ritz model of such coupled beams was validated against

measurements on a particular beam system, then the model was used to explore the

acceleration response to impulsive driving in the time, frequency and spatial domains.

This study is restricted to linear vibration response and additional mechanisms for high

internal acceleration due to nonlinear effects such as internal impacts are not considered.

Using Monte Carlo simulation in which the indirectly driven beam was perturbed by

randomly placed point masses a wide range of system behaviour was explored. This

facilitates identiﬁcation of vulnerable conﬁgurations that can lead to high internal accel-

eration. The results from the study indicate the possibility of curve veering inﬂuencing the

peak acceleration ampliﬁcation. The possibility of veering within an ensemble was found to

be dependent on the relative coupling strength of the modes. Understanding of the

mechanism may help to avoid vulnerable cases, either by design or by preparatory vibration

testing.

&2013 Elsevier Ltd. All rights reserved.

1. Introduction

There are many contexts in which it is important to understand, predict and control the peak acceleration within a

system in response to external excitation. Examples include the design of protective packaging for transportation of fragile

items (e.g. [1]), and the fatigue life estimation of structures exposed to extreme environments, such as vibro-impact

equipment and spacecraft electronic circuits [2].

A common testing procedure for such systems is a drop test of some kind. An external shock is administered, with

perhaps some monitoring and control of the peak acceleration level thus imposed. Internal accelerometers record the

response, and the peak level recorded. A qualiﬁcation standard may require that the ratio of peak internal acceleration to

the peak external acceleration does not exceed a speciﬁed level.

High internal accelerations can arise within such systems by two routes: via linear vibration mechanisms and via

nonlinear effects such as internal impacts. This paper concerns the former case. The task is to study a simpliﬁed model

system to understand the conditions that lead to worst-case peak accelerations at points remote from the direct excitation.

The intention is to stand back from detailed modelling of realistic systems with all their essential but confusing detail, and

Contents lists available at SciVerse ScienceDirect

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

Journal of Sound and Vibration

0022-460X/$ - see front matter &2013 Elsevier Ltd. All rights reserved.

http://dx.doi.org/10.1016/j.jsv.2013.02.024

n

Corresponding author. Tel.: þ44 1223 332642; fax: þ44 1223 332662.

E-mail addresses: kiran.vijayan@gmail.com, kv247@cam.ac.uk (K. Vijayan), jw12@cam.ac.uk (J. Woodhouse).

Journal of Sound and Vibration 332 (2013) 3681–3695

to highlight the role of general factors such as coupling strength, relations between natural frequencies, and mass ratio

between the subsystems.

The chosen system consists of two beams coupled by bolts or other local connections near the two ends. Impacts are

applied directly to one of the beams and the resulting spatial and temporal pattern of acceleration on both beams is

examined. This system seems to be about the simplest that captures the essence of the applications mentioned above:

for example the directly excited beam could represent a transit case, while the other beam represents the internal fragile

object. The couplings, which could be stiff or soft, can represent the attachment or cushioning between the two

components. The coupled-beam system readily lends itself both to theoretical modelling and to experiment and both

approaches will be used here.

It should be emphasised that the dynamic response of coupled beams with bolted joints is not as such the main

interest: the aim is to use this system to deduce parametric trends and general design guidelines for protection of the

whole class of structures against high internal accelerations. With this in mind, the beam system will be randomised in a

Monte Carlo study using added masses. The process generates an ensemble of related systems, and serves to ﬂush out

particular conditions that lead to the worst cases of high acceleration. Study of those worst cases reveals a link to the

phenomenon of ‘‘curve veering’’ [3–5].

2. Modelling and validation

2.1. Theoretical model

The ﬁrst stage of the study was to establish a simple theoretical model for the linear vibration behaviour of the coupled

beam system. The details of the model were chosen with an eye to ease of making a physical system against which to test

it. The system that will be used for initial validation tests consists of two steel beams of different thicknesses, coupled

using nylon bolts and spacer washers as shown in Figs. 1 and 2. The main parameter values of this test rig are listed, and

their symbols deﬁned, in Table 1. For clarity in later discussion, the two beams are designated ‘‘directly driven’’ and

‘‘indirectly driven’’. The ﬁrst, the thick beam, will be directly excited with an impulsive force, while the second is only

excited via the couplings.

A theoretical model of these coupled beams was developed using the Rayleigh–Ritz method. This approach was chosen

in preference to, for example, ﬁnite-element modelling because of the ease and efﬁciency of later Monte Carlo studies with

randomly placed added masses. The motion of the beams is constrained by the coupling bolts. This constraint imposes a

restriction on the relative translation and rotation of the coupled beams, modelled using point-connected linear and

torsional spring elements. Linear springs are needed to represent stiffness in the normal direction (stiffness k

1

) and

torsional springs (stiffness k

2

) to represent rotational stiffness. To obtain good ﬁts to experimental results it turned out

that additional linear springs (stiffness k

3

) needed to be included to represent the shear stiffness across the bolted

connections: the differential horizontal motion on the surface of the beams from the tension and compression on either

side of the neutral axis as they bend produces relative shearing motion.

Vibration of the two beams can be described by lateral displacements u(x) and v(x), deﬁned to be positive in the upward

direction in Fig. 1, where xis the axial position measured from one end of the beams. These displacements can be

approximated in terms of a set of basis functions consisting of Nterms of a truncated Fourier sine series, plus linear terms

needed to represent rigid body translation and rotation:

uðxÞ¼ X

N

n¼1

a

n

sin n

p

x

L

þb

0

þb

1

x,vðxÞ¼ X

N

n¼1

c

n

sin n

p

x

L

þd

0

þd

1

x(1)

L1

L2

L

Ls

Fig. 1. Sketch of coupled beams.

K. Vijayan, J. Woodhouse / Journal of Sound and Vibration 332 (2013) 3681 –36953682

where a

n

,b

0

,b

1

and c

n

,d

0

,d

1

represent modal coefﬁcients of the basis function of the directly and indirectly driven beam

respectively. The inclusion of the ‘‘constraint modes’’ via the linear terms in Eq. (1) allows the linear combination of basis

function to satisfy the boundary conditions at the free ends. The model uses Bernoulli–Euler beam theory, in which

transverse plane sections of the beam remain plane and normal to the longitudinal ﬁbres of the beam after bending. Thus

the horizontal surface displacement on the upper surface of the directly driven beam of thickness t

1

is given by

qu

qx

t

1

2(2)

There is a corresponding expression for the horizontal displacement on the lower surface of the indirectly driven beam.

The linear shear spring k

3

acts between these two at the coupling point.

Using these basis functions, the mass and stiffness matrices can be readily calculated via the potential (V) and kinetic

(T) energies:

V¼1

2E

1

I

1

Z

L

0

q

2

u

qx

2

!

2

dxþ1

2E

2

I

2

Z

L

0

q

2

v

qx

2

!

2

dxþ1

2k

1

ðuvÞ

2

x¼x

1

þ1

2k

1

ðuvÞ

2

x¼x

2

þ1

2k

2

qu

qxqv

qx

2

x¼x

1

þ1

2k

2

qu

qxqv

qx

2

x¼x

2

þ1

2k

3

qu

qx

t

1

2þqv

qx

t

2

2

2

x¼x

1

þ1

2k

3

qu

qx

t

1

2þqv

qx

t

2

2

2

x¼x

2

(3)

where E

1

,I

1

and E

2

,I

2

are Young’s modulus and second moment of area of the directly and indirectly driven beam

respectively; k

1

,k

2

and k

3

are the longitudinal, rotational and shear stiffness of the coupling springs respectively; and

T¼T

beam

þT

mass

where

T

beam

¼1

2m

1

Z

L

0

_

u

2

dxþ1

2m

2

Z

L

0

_

v

2

dx(4)

Fig. 2. Test rig with steel beams coupled by nylon bolts, showing impulse hammer used in testing.

Table 1

Parameters of the test rig.

Parameter Symbol Value

Thickness of directly driven beam t

1

6mm

Thickness of indirectly driven beam t

2

3mm

Width of beams w25 mm

Length of beams L0.5 m

Bolt positions from end L

1

,LL

2

15 mm

Young’s modulus E210 GN m

2

Density

r

7800 kg m

3

K. Vijayan, J. Woodhouse / Journal of Sound and Vibration 332 (2013) 3681–3695 3683

and

T

mass

¼

1

2

m

s

_

v

2

9

x¼L

s

(5)

where m

1

is the mass per unit length of the directly driven beam and m

2

is the mass per unit length of the indirectly

driven beam.

The last kinetic energy term (5) represents a point attachment of mass m

s

at position L

s

on one beam. It could represent

the accelerometer used in the measurements, but also in a later stage of the study randomised point masses are used to

perturb the system and these are modelled by adding additional kinetic energy terms of the same form as Eq. (5) with the

appropriate masses and positions.

From the mass and stiffness matrices a standard eigenvalue/eigenvector routine can be used to extract the natural

frequencies and mode shapes (in terms of the coefﬁcients a

n

,b

n

,c

n

,d

n

). For this study the Matlab function EIG was used.

For comparison with the measurements it is sufﬁcient to represent modes accurately up to 2 kHz, which translates into the

ﬁrst 15 or so of the modal series. A convergence study showed that it was sufﬁcient to include terms up to N¼30 in the set

of basis functions, giving a total of 64 coefﬁcients.

2.2. Experimental comparisons

Measurements were made on the coupled beams by excitation with an impulse hammer as shown in Fig. 2. Response

was measured using an accelerometer; quite a heavy one, but the mass perturbation thus introduced was regarded as part

of the process of exploring variations in the system dynamics, and it was included in the theoretical modelling. Tapping

was done at an array of points on the directly driven beam, and response measured at the same points and on a similar

array on the indirectly driven beam. Time histories and transfer functions were recorded for each combination of tapping

and observing point: transfer functions were obtained as an average of three taps.

For model validation purposes the next step was to establish suitable values for the coupling stiffnesses in order to give

a reasonable match to the observed natural frequencies of the coupled beams, deduced from the transfer function

measurements. These speciﬁc stiffness values will not be important when it comes to using the model to explore general

issues of shock transmission, but it is reassuring to have validation for a particular case. The nylon bolts have a high

stiffness in the longitudinal direction, enough that the exact value of k

1

does not matter very much in the frequency range

studied: a somewhat arbitrary high value, 100 MN/m, was assigned. An iterative search was carried out to determine the

best stiffness of the rotational springs k

2

and shear springs k

3

. The optimal coefﬁcients were chosen to minimise the mean

square deviation of the predicted natural frequencies of the ﬁrst three modes, resulting in values k

2

¼500 Nm/rad and

k

3

¼30 MN/m.

Using these values, the ﬁrst few vibration modes of the coupled beams are shown in Fig. 3. For the ﬁrst three modes,

experimentally determined mode shapes are also shown, extracted from the measured transfer functions using circle

ﬁtting [6]. The results show a reasonable match in frequency and mode shape. Experimental versions of modes 4 and 5 are

not plotted here because the mode shapes could not be separated reliably due to the degree of modal overlap between this

closely spaced pair, but the observed shapes were consistent with a linear combination of the two theoretical mode

shapes. More will be said about the sensitivity of these mode shapes in Section 4, in the discussion of Fig. 14.

The transfer function of accelerance at point x¼Yin response to excitation at point x¼Xat frequency

o

is given by the

standard modal summation formula

HðX,Y,

o

Þ¼

o

2

X

n

j

n

ðXÞ

j

n

ðYÞ

o

2

n

þi

oo

n

=Q

n

o

2

(6)

where

j

n

ðxÞis the mass normalised nth mode of the coupled beams, with natural frequency

o

n

and modal Qfactor Q

n

.

Damping will be assumed small throughout this study (i.e. Q

n

b1 for all modes), for algebraic simplicity. The Qfactors of

the ﬁrst few modes of the experimental rig were determined as part of the modal analysis shown in Fig. 3, giving values of

the order of 200.

A particular combination of tapping point and sensing position will be used to illustrate the response of the coupled

beams: excitation on the directly driven beam at X¼60 mm and measurement on the indirectly driven beam at

Y¼260 mm. A comparison of the measured and theoretical transfer functions is shown in Fig. 4. For this comparison

the measured Qfactors have been used for the ﬁrst six modes, and higher modes were all given the ﬁxed value Q

n

¼100.

Generally satisfactory agreement is seen. A conspicuous peak around 1100 Hz in the measurement is missing from the

theory, but this is attributable to a torsional mode not included in this Bernoulli–Euler beam model but seen in the

measurement as a consequence of not driving and measuring precisely on the centre line of the beams.

The corresponding impulse response of the system obtained by tapping at Xand observing displacement at Ycan also

be obtained using modal summation. The result, using the small damping approximation and assuming an ideal impulse, is

gðX,YÞX

n

j

n

ðXÞ

j

n

ðYÞsin

o

n

t

o

n

e

o

n

t=2Q

n

,tZ0 (7)

K. Vijayan, J. Woodhouse / Journal of Sound and Vibration 332 (2013) 3681 –36953684

The velocity response of the system is thus

_

gX

n

j

n

ðXÞ

j

n

ðYÞcos

o

n

te

o

n

t=2Q

n

,tZ0 (8)

The acceleration response is obtained by differentiating again. Each modal velocity of the system undergoes a step change

at the instant of impact, and differentiation of the step input gives a Dirac delta function contribution:

€

gX

n

j

n

ðXÞ

j

n

ðYÞð

d

ðtÞ

o

n

sin

o

n

te

o

n

t=2Q

n

Þ,tZ0 (9)

0 0.1 0.2 0.3 0.4 0.5

Expt 73 Hz, theory 71.5 Hz

0 0.1 0.2 0.3 0.4 0.5

Expt 150 Hz, theory 150.8 Hz

0 0.1 0.2 0.3 0.4 0.5

Expt 190 Hz, theory 193.9 Hz

0 0.1 0.2 0.3 0.4 0.5

Theory 376.4 Hz

0 0.1 0.2 0.3 0.4 0.5

Theory 376.6 Hz

x (m)

0 0.1 0.2 0.3 0.4 0.5

Theory 617.7 Hz

x (m)

Fig. 3. Comparison of experimental and theoretical mode shapes. Upper curves: directly excited beam; lower curves: indirectly excited beam; dashed

lines indicate undeformed positions. Discrete points show measured mode shapes for the ﬁrst three modes.

0 200 400 600 800 1000 1200 1400 1600 1800 2000

30

20

10

0

10

20

30

40

50

Fre

q

uenc

y

(Hz)

Accelerance (dB)

Fig. 4. Comparison of experimental and theoretically predicted frequency response for the beams coupled using nylon bolts.

K. Vijayan, J. Woodhouse / Journal of Sound and Vibration 332 (2013) 3681–3695 3685

At the driving point all the delta function contributions have positive coefﬁcients and so the net response starts with a

delta function. However at locations other than the driving point the situation is different. At a remote point the delta

function contributions cancel out

X

n

j

n

ðXÞ

j

n

ðYÞ¼0 for XaY(10)

and so

€

g

X

n

o

n

j

n

ðXÞ

j

n

ðYÞsin

o

n

te

o

n

t=2Q

n

,XaY,tZ0 (11)

It follows from Eq. (11) that a necessary condition to obtain high acceleration response is that at least one modal amplitude

must be signiﬁcant both at the location of tapping and of observation. This will prove important shortly.

It is clear on physical grounds that Eq. (10) must hold: for any system with a ﬁnite group velocity, there cannot be an

instantaneous response at a remote position. The proof of Eq. (10) is simplest in terms of a general discrete system with

mass matrix M. The statement of orthogonality in terms of the matrix Uwhose columns are the mass-normalised mode

vectors is

U

T

MU ¼I

where Iis the unit matrix. Any matrix commutes with its inverse, so

UU

T

M¼I

So UU

T

is the inverse of the mass matrix, and for any problem with a diagonal mass matrix the result must be diagonal.

This is the result (10) expressed in a discrete form. It is clear from the form of the kinetic energy integral (4), involving only

the square of velocity at each point with no cross terms, that the mass matrix of any discrete version of this model will

automatically be diagonal.

For theoretical studies it is convenient to use the time-domain summation (11) directly, truncating the mode sum at a

chosen cutoff frequency. However, to obtain a direct comparison with measured results to check the accuracy of modelling

it is necessary to take account of the shape and ﬁnite duration of the actual hammer pulse. This could be done by

performing a convolution of the hammer pulse with the ideal impulse response, but it is simpler to work in the frequency

domain. The magnitude of a typical force spectrum for a hammer pulse from the experiment is plotted in Fig. 5, showing

input force tailing off around 2 kHz. The accelerance calculated according to Eq. (6) can be multiplied by this force

spectrum (including the phase information, of course), then an inverse FFT performed to give a simulated acceleration time

history. A comparison is shown in Fig. 6, for the same excitation and observation points as in Fig. 4. The agreement is

excellent.

2.3. Spatial response prediction

Having validated the theoretical model, it can be used to give a spatial map of peak acceleration in response to

any impulsive excitation. Fifty equally spaced positions covering the entire length of each beam were selected. For every

excitation point on the directly driven beam and for every observation point on both beams the impulse response is

calculated. The peak value is found, and these peak values plotted. For this calculation it is no longer necessary to follow

0 500 1000 1500 2000 2500 3000 3500 4000

20

15

10

5

0

5

10

Frequency (Hz)

Hammer spectrum (dB)

Fig. 5. Typical measured hammer spectrum.

K. Vijayan, J. Woodhouse / Journal of Sound and Vibration 332 (2013) 3681 –36953686

the exact details of the experimental rig since the aim is to understand generic features. Rather than using the

experimental hammer impulse and inverse FFT, it is computationally more efﬁcient to use the ideal impulse from

Eq. (9), limiting the modal sum to natural frequencies below 2 kHz.

Contour maps of the peak acceleration variation are shown in Fig. 7, for the directly and indirectly excited beams. The

values cover quite a wide range so the contour spacing is logarithmic. Absolute levels are not important here, but the same

amplitude scale is used for both plots. The positions of the coupling points are indicated by white lines. Parameter values

for the beams and coupling positions match the experimental beams, but no added sensor mass is included.

Fig. 7(a), for the directly driven beam, shows a symmetric pattern with respect to the main diagonal resulting from the

standard reciprocal theorem: driving and observing points are on the same beam and the response is unchanged if the two

are interchanged. It is also symmetric with respect to the other main diagonal, as a result of the bilateral symmetry of the

beam system. The highest levels occur when driving at the ends of the beam, as is to be expected with free boundary

conditions. The pattern is diagonally dominant in each row. This is not surprising because in the case of driving-point

responses the theoretical acceleration should be inﬁnite at t¼0 whereas all off-diagonal responses are ﬁnite. However,

inﬁnities do not occur in the computed driving-point responses because the calculation is being carried out in discrete

time with a particular sampling frequency, so the delta function contributions turn into one-sample pulses with height

proportional to the sampling frequency. By contrast, the response at any remote position, and indeed at later times at a

driving point, does not depend signiﬁcantly on the sampling rate provided it is fast enough that all included modes fall

well below the Nyquist frequency. This means that the apparent degree of diagonal dominance depends on the sampling

rate, and the particular diagonal values in the plot are of no physical signiﬁcance. They were computed with a sampling

rate of 10 kHz, so that the highest frequency included is 2/5 of the Nyquist frequency. The factor

o

n

in Eq. (9) means that in

discrete time, the magnitude of the Dirac delta function in the ﬁrst sample becomes equal to the peak amplitude of the

sinusoidal term when the natural frequency reaches 1=

p

times the Nyquist frequency.

Fig. 7(b) shows the corresponding map for the indirectly driven beam. It exhibits 1801rotational symmetry, which

follows from the bilateral symmetry of the beam system: the response at Yto driving at Xis the same as the response at

LYto driving at LX. The spatial response is relatively featureless, except for higher levels near the ends of the beam. This

enhancement near the edges of the plot conceals two different effects, as is made clear by shifting the coupling points

further away from the ends of the beams. Fig. 8 shows results in the same format as Fig. 7 for such a case. The change has

not made very much difference to the response of the directly excited beam, but the pattern on the indirectly excited beam

is now more complicated. The highest levels occur near the main diagonal at the coupling positions, corresponding to more

or less direct driving of the ‘‘indirect’’ beam through the rather stiff coupling springs k

1

.Fig. 8(b) also shows enhanced

response around the edges, associated with the free boundary condition of the indirectly driven beam.

Since the main purpose of this study is to understand when high accelerations can occur on the indirectly driven beam

relative to the directly driven one it is of interest to show this ratio from the simulation results. Fig. 9 shows the

‘‘acceleration ampliﬁcation factor’’: the results from Fig. 8(b) have been normalised by dividing each row by the

corresponding peak acceleration at the driving point. The results are now dimensionless, and the highest value shows as

0 dB at the ‘‘hot spots’’ near the coupling positions. This is as would be expected from the description just given: the peak

acceleration near the coupling point is approximately the same on both beams. These particular coupled beams never

show signiﬁcant acceleration ampliﬁcation, in this sense. This raises the question of whether that is always the case, or

whether parametric changes of some kind might induce high ampliﬁcation. To explore that question, the next stage of the

study involved a Monte Carlo investigation with random perturbations.

0 5 15

2

1.5

1

0.5

0

0.5

1

1.5

2

time (ms)

Acceleration (m/s2)

10

Fig. 6. Comparison of experimental and theoretically predicted temporal response for the beams coupled using nylon bolts.

K. Vijayan, J. Woodhouse / Journal of Sound and Vibration 332 (2013) 3681–3695 3687

3. Monte Carlo studies

The coupled beam system was perturbed in a manner motivated by the example problems mentioned in the Introduction.

One would expect that a transit case might have vibration behaviour that is reasonably consistent and predictable, but that the

particular fragile object contained within it might vary on different occasions. This variability of the indirectly driven subsystem

is crudely simulated by positioning ﬁve equal masses on the indirectly driven beam at random locations drawn from a uniform

distribution. A wide range of parametric combinations was explored; ratio of beam thicknesses, position of coupling points and

total added mass.

It was discovered that the behaviour was generically similar across this wide range, and the worst case from each

Monte Carlo ensemble showed a consistent pattern. Some typical examples will be chosen to illustrate this. Fig. 10 shows

histograms of the maximum acceleration ampliﬁcation factor, deﬁned in the previous section, for two cases. The difference

in the two cases lies in the thickness of the directly excited beam, as detailed in the ﬁgure caption. For both, 10,000

different randomised systems were simulated. Both histograms show a range of ampliﬁcation factors from around unity,

as seen in Fig. 9 for the unperturbed beams, up to values in excess of 3. The shapes are different in the two cases: Fig. 10(a)

shows a mean value of ampliﬁcation factor around 1.7 with a relatively symmetrical distribution about that value, whereas

Fig. 10(b) shows a much lower mean value but a distribution with a long tail. In both cases it is clear that there are

occasional distributions of the added masses which produce anomalously high acceleration ampliﬁcation. Understanding

those exceptional cases is the next task, because they point to the possibility of internal systems that are not well

protected against damage by the enclosing structure.

Sensing position (m)

Tapping position (m)

0 0.1 0.2 0.3 0.4 0.5

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0

-5

5

10

15

20

25

Sensin

g

position (m)

Tapping position (m)

0 0.1 0.2 0.3 0.4 0.5

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0

-5

5

10

15

20

25

Fig. 7. Spatial map of peak acceleration for observing points on (a) the directly driven beam and (b) the indirectly driven beam. Contours are

logarithmically spaced. The scale bar indicates the level expressed in dB, with the same datum for both plots. White lines indicate the positions of the

coupling points between the beams.

K. Vijayan, J. Woodhouse / Journal of Sound and Vibration 332 (2013) 3681 –36953688

Sensing position (m)

Tapping position (m)

0 0.1 0.2 0.3 0.4 0.5

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0

-5

5

10

15

20

25

30

Sensin

g

p

osition (m)

Tapping position (m)

0 0.1 0.2 0.3 0.4 0.5

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0

-5

5

10

15

20

Fig. 8. Spatial map of peak acceleration in the same format as Fig. 7, with the coupling points moved to positions 0.1 m from the ends of the beams.

Sensin

g

p

osition (m)

Tapping position (m)

0 0.1 0.2 0.3 0.4 0.5

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

25

20

15

10

5

0

Fig. 9. The peak acceleration data of Fig. 8, normalised on each row by the peak acceleration at the driving point to give a dimensionless ‘‘acceleration

ampliﬁcation factor’’. Scale bar gives values expressed in dB.

K. Vijayan, J. Woodhouse / Journal of Sound and Vibration 332 (2013) 3681–3695 3689

The worst case from Fig. 10(a) shows the typical features. Fig. 11 shows a few examples of the cumulative sum of

Eq. (11) as successive modal contributions are added in, with the particular combination of driving point and observing

point that gave the highest acceleration ampliﬁcation. The traces have been separated vertically for clarity. The lowest

trace shows the sum including all modes up to the 12th (ignoring the two rigid-body modes at zero frequency). It can be

seen that these lower modes contribute rather little to the acceleration waveform. Adding in mode 13, in the next trace,

gives a much larger result. Since the waveform is dominated by a single mode a clear exponential decay is seen. Adding in

mode 14, in the next trace, makes the amplitude signiﬁcantly bigger, and produces a characteristic beating envelope.

The top trace shows the effect of adding mode 15: very little changes.

This pattern has been seen again and again in different Monte Carlo studies. For the worst case, the largest contribution

to the acceleration waveform comes from an adjacent pair of modes, excited to comparable amplitudes and fairly close

in frequency so that a beating envelope is produced. The mode numbers are not always the same as in this particular

example, and in some cases more than one pair showing similar behaviour contributes to the total peak acceleration.

The nature of the mode shapes underlying this pattern is illustrated in Fig. 12. The majority of modes show strong

motion on one beam or the other, but not both. Fig. 12(a) shows the ﬁrst mode, looking more or less like a clamped–

clamped mode of the indirectly driven beam with the directly driven beam moving essentially as a rigid body to balance

the linear momentum of the indirectly driven beam motion. Many higher modes are similar in character to this, showing

successively shorter wavelengths on the indirectly driven beam with little motion in the directly driven beam. Occasional

modes show the opposite pattern, and one is illustrated in Fig. 12(b). This is approximately the second mode of the free–

free directly driven beam, carrying the indirectly driven beam along with it and only producing signiﬁcant motion there

near the ends, where there are free ‘‘cantilevers’’ that are vibrated by the rotation at the coupling points.

0.5 1 1.5 2 2.5 3 3.5

0

20

40

60

80

100

120

140

Amplification factor

Number

1 1.5 2 2.5 3 3.5 4

0

100

200

300

400

500

600

700

800

900

Amplification factor

Number

Fig. 10. Histograms of peak acceleration ampliﬁcation factor for 10,000 cases of Monte Carlo simulations: parameters as in Table 1 except t

1

¼5 mm for

case (a) and t

1

¼9 mm for case (b), with t

2

¼1 mm for both and connection positions L

1

¼50 mm. Five equal masses were added, each of mass 42 g.

K. Vijayan, J. Woodhouse / Journal of Sound and Vibration 332 (2013) 3681 –36953690

The picture changes for modes 13 and 14 shown in Fig. 12(c) and (d): these are the two that contributed the strongest

signals in Fig. 11. Both modes show signiﬁcant motion on both beams, in a characteristic pattern. The shapes on the two

separate beams are similar in the two modes, but in mode 14 the phase relation is reversed relative to mode 13. This is

familiar behaviour from many contexts involving coupled oscillators, such as tuned-mass dampers (see e.g. [7,8]), as will

be discussed further in the next section.

Notice how the added masses are acting on the different modes in Fig. 12. For the lower modes the effect of the masses

is relatively slight, reducing the frequency but not changing the mode shape very much from that of unperturbed beams.

As frequency rises the effectiveness of the masses increases, and eventually each mass tends to create a constraint, pinning

the indirectly driven beam. The different random positions of the masses within a Monte Carlo ensemble will naturally

change the mode shapes accordingly. The frequencies will also change, and the special feature which makes this into the

0 0.1 0.2 0.3 0.4

−10

0

10

18.8 Hz

0 0.1 0.2 0.3 0.4

0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4

−10

0

10

272.1 Hz

−10

0

10

902.8 Hz

−10

0

10

959.2 Hz

Fig. 12. Selected modes for the case shown in Fig. 11: (a) mode 1 (19 Hz); (b) mode 6 (272 Hz); (c) mode 13 (903 Hz); and (d) mode 14 (959 Hz). Upper

trace shows the directly driven beam and lower trace the indirectly driven beam. Stars mark the positions of the added masses.

0 0.02 0.04 0.06 0.08

170

180

190

200

210

220

230

240

Time (s)

Acceleration

0.1

Fig. 11. Acceleration waveforms after adding the ﬁrst 12 (lowest trace), 13, 14 and 15 modes (highest trace) for the mass positions corresponding to the

worst-case ampliﬁcation in Fig. 10(a).

K. Vijayan, J. Woodhouse / Journal of Sound and Vibration 332 (2013) 3681–3695 3691

worst case for acceleration ampliﬁcation is that this particular set of mass positions has brought the pair of modes 13,14

into sufﬁciently close proximity to create the ‘‘sum and difference’’ coupled pair seen in Fig. 12(c) and (d). This is the

mechanism of high acceleration ampliﬁcation: both modes of the pair show signiﬁcant motion on both beams, whereas

this is not in general the case for the other modes. This mechanism for producing large motion on both beams

automatically involves two (or possibly more) modes, so the beating envelope seen in Fig. 11 ceases to be surprising.

Fig. 13 shows spatial maps of peak acceleration for the same case as Figs. 11 and 12. The pattern on the directly excited

beam, shown in Fig. 13(a), is hardly changed from the earlier plots of unperturbed beams. However, the pattern on the

indirectly excited beam, plotted in Fig. 13(b) as the acceleration ampliﬁcation factor, is much more complicated than in

earlier cases. The scale bar shows that the ampliﬁcation factor reaches approximately 10 dB, corresponding to a linear

factor 3.26, the rightmost point in the histogram of Fig. 10(a). The dark vertical lines in Fig. 13(b) indicate the positions of

the masses, as can be seen by comparing with Fig. 12.

4. Coupling, veering and symmetry breaking

The effect shown in Figs. 11–13 is often associated with the phenomenon of ‘‘curve veering’’ (see e.g. [5]) when the

behaviour of natural frequencies is examined as a function of some varying system parameter. This can be illustrated

clearly by considering the coupled beams without added masses, and varying the ratio of thicknesses of the two beams.

Fig. 14(a) shows results for the coupled beams detailed in Table 1, except that the thickness of the indirectly driven beam

is varied from 6 mm (the same thickness as the directly driven beam) down to 1 mm. A qualitative description of this plot

is that there are two families of approximately straight lines. One set is approximately horizontal, corresponding to natural

frequencies of the unchanging directly driven beam. The second set of lines rises in a fan radiating from the origin of

coordinates, showing natural frequencies of the indirectly driven beam.

Sensing position (m)

Tapping position (m)

0 0.1 0.2 0.3 0.4 0.5

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

−5

0

5

10

15

20

25

30

Sensin

g

p

osition (m)

Tapping position (m)

0 0.1 0.2 0.3 0.4 0.5

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

−30

−25

−20

−15

−10

−5

0

5

10

Fig. 13. Spatial patterns of peak acceleration for the case shown in Figs. 11 and 12; (a) acceleration of directly driven beam and (b) acceleration

ampliﬁcation factor for indirectly driven beam in the format of Fig. 9. Contour spacing is logarithmic, scale bar gives values expressed in dB.

K. Vijayan, J. Woodhouse / Journal of Sound and Vibration 332 (2013) 3681 –36953692

In some places the lines of the two families cross, but at other positions ‘‘veering’’ occurs: the two lines avoid the

crossing point. Crossing or veering is determined by considerations of symmetry. The system is mirror-symmetric about

the centre of the beams, and every mode is either symmetric or antisymmetric with respect to reﬂection in this ‘‘mirror’’.

The plot symbols indicate the symmetry of each mode in the ﬁgure. It is immediately clear that lines corresponding

to different symmetries can cross, but that when lines with the same symmetry approach one another, they veer. The

features of Fig. 14(a) are all qualitatively explained by this one fact, bearing in mind that at the right-hand edge the two

beams have equal thickness, so each pair of lines with matching symmetry at this edge is ‘‘in the middle of veering’’ as the

lines reach the edge of the plot.

If the exact symmetry of the system is disturbed the pattern changes. Fig. 14(b) shows the result of positioning a 20 g

point mass at position x¼0 on the directly driven beam. There are now no crossings: every pair of curves veers, although in

some cases it is not easy to see at the scale of this plot. Again, different plot symbols are used to give an indication of the

character of mode shapes. Since modes are no longer exactly symmetric or antisymmetric a somewhat ad hoc criterion has

been used to show ‘‘approximate symmetry’’ of each mode. The Fourier functions used in the Rayleigh–Ritz calculation,

from Eq. (1), can be divided into symmetric terms (odd values of n) and antisymmetric terms (even values of n). For each

calculated eigenvector, the sums of absolute values of the coefﬁcients of the symmetric and antisymmetric terms are

computed. If the ratio of these two sums exceeds a chosen threshold value, either way round, the mode can be labelled as

having a ‘‘predominant symmetry’’ corresponding to the larger of the two terms. If the ratio is closer to unity than the

threshold, the mode is labelled as having no predominant symmetry. The chosen threshold value was 3, but the results are

not very sensitive to this choice.

1 2 3 4 5 6

100

200

300

400

500

600

700

800

900

1000

Thickness (mm)

Frequency (Hz)

1 2 3 4 5 6

100

200

300

400

500

600

700

800

900

1000

Thickness (mm)

Frequency (Hz)

Fig. 14. Natural frequencies as a function of thickness of the indirectly driven beam, for (a) symmetric coupled beams and (b) the system as in (a)

perturbed by an added mass of 20 g at one extreme end of the directly excited beam. All other parameter values are as in Table 1. Black circles indicate

modes that are symmetric, or predominantly so; red stars denote modes that are antisymmetric or predominantly so; blue dots denote modes without

clear symmetry. (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.)

K. Vijayan, J. Woodhouse / Journal of Sound and Vibration 332 (2013) 3681–3695 3693

The associated plot symbols reveal the expected pattern in Fig. 14(b). Much of the diagram, including the regions where

veering was seen in Fig. 14(a), shows the same symbol as the corresponding place in Fig. 14(a). However, everywhere that

crossing was previously seen a small region of ‘‘no symmetry’’ is seen where a new veering event has appeared. In each of

these regions there is now a pair of modes close in frequency, with no predominant symmetry because both modes involve

(approximately) a mixture of symmetric motion on one beam and antisymmetric motion on the other. Note that the

original beam system had a thickness of 3 mm for the indirectly driven beam, and this plot shows why it was not easy to

get clear experimental shapes for modes 4 and 5 of these beams (see Fig. 3). This particular thickness happens to place

these two modes very close in frequency so that they are prompted into veering by a relatively small added mass. Since

the accelerometer mass was moved during the measurements, the veering details were not constant throughout the

measurement.

Qualitatively similar plots to Fig. 14 can be created by varying other system parameters, such as the stiffness or position

of the coupling springs, or the position of an added mass. If the system retains exact symmetry, curves can cross when they

correspond to modes with different symmetry. But in all other cases veering is the rule, and exact crossing never occurs.

Any parametric combination that leads to at least one veering event will contribute a pair of modes to the impulse

response that will behave qualitatively like Fig. 11, and will tend to lead to high acceleration ampliﬁcation on the indirectly

excited beam.

The two shapes of histogram seen in Fig. 10 can tentatively be explained in these terms. First, the mean values of

ampliﬁcation factor were very different. This suggests that for the case of Fig. 10(a) the unperturbed beam system had at

least one pair of modes quite close to veering, so that most cases in the ensemble shows signiﬁcant ampliﬁcation. Certain

extreme cases involve ﬁne tuning of the veering event to improve the frequency match and hence the ampliﬁcation.

The case of Fig. 10(b), on the other hand, shows most ensemble members having virtually no ampliﬁcation because the

unperturbed system had no ready-made veering pairs. But occasional combinations of the positions of the added masses

were still able to create a pair of modes with close frequency that showed the characteristic coupled mode shapes

associated with veering.

The rarity of these exceptional cases, resulting in the long tail of the distribution in Fig. 10(b), suggests that the veering

event thus created was in some sense very tight in the relevant parametric space, so that it required very precise values of

the mass positions to create the ampliﬁcation effect. Fig. 14 gives examples: the tightness of the veering events seen in this

plot varies quite a bit. In broad terms, tight veering is associated with weak coupling, whereas stronger coupling causes a

wider veering region. In the case of the coupled beams several different routes to ‘‘weak coupling’’ can be identiﬁed. Most

obvious is the stiffness of the coupling springs: soft springs, for example representing cushioning within a transit case,

would allow modes of the two separate beams, corresponding to modes of the internal object and modes of the case, to

remain largely independent. Alternatively, a large ratio of beam thicknesses tends to keep the modes of the two beams

distinct as seen in Fig. 12(a), (b). This leads to a kind of weak coupling characterised by a large impedance jump between

the subsystems even if the coupling springs are stiff. Finally, and least obviously, the transition between Fig. 14(a) and (b)

points to a kind of weak coupling arising in almost-symmetrical systems. Modes that are uncoupled for symmetry reasons

in a perfectly symmetric system can interact when the symmetry is broken, and if the symmetry-breaking is weak then the

coupling will be correspondingly weak.

5. Conclusions

A study has been carried out to investigate the circumstances under which high peak acceleration can occur in the

internal parts of a system in response to impulsive driving on the outside. Motivating examples include the design of

packaging for transportation of fragile items. Such systems have been modelled in an idealised form using two beams

coupled with point connections. A Rayleigh–Ritz model of such coupled beams was validated against measurements on a

particular beam system, then the model was used to explore the acceleration response to impulsive driving in the time,

frequency and spatial domains. This study was restricted to linear vibration response: additional mechanisms for high

internal acceleration are of course created if nonlinear effects such as internal impacts can occur.

In order to explore a wide range of system behaviour and reveal any particular conﬁgurations that lead to high internal

acceleration, Monte Carlo simulations were carried out in which the indirectly driven beam was perturbed by randomly

placed point masses. The results showed generally similar patterns in many different cases. The distribution of peak

acceleration ampliﬁcation factors, typiﬁed by Fig. 10, generally shows high ampliﬁcation in certain cases. When examined

in detail, the acceleration waveform of these extreme cases turned out to be dominated by one or more pairs of modes that

were close in frequency and excited to comparable amplitudes, leading to a beating envelope in the response.

This can be broadly interpreted as a resonance phenomenon: the indirectly driven beam has a resonance close to one of

the directly excited beam’s leading to signiﬁcantly coupled response. This effect is associated with the phenomenon of

curve veering in some relevant parameter space. If coupling is weak, by one of several possible mechanisms, the veering

event is conﬁned to a small region of the parameter space and this may make its occurrence rare within a given Monte

Carlo ensemble. These are the rare but potentially important systems for which high internal acceleration can be generated

by external excitation. Not only is the internal subsystem not being protected by its outer packaging, but it may even

experience higher acceleration than it would have if directly excited. Understanding of the mechanism may help to avoid

these special cases, either by design or by preparatory vibration testing.

K. Vijayan, J. Woodhouse / Journal of Sound and Vibration 332 (2013) 3681 –36953694

Acknowledgments

The authors thank Dr. Tore Butlin and Prof. Robin Langley for stimulating discussion on the work, and two anonymous

reviewers for constructive comments.

References

[1] M.F. Mecklenburg, M. Richards, R.M. Merrill (Eds.), Art in Transit: Handbook for Packing and Transporting Paintings, National Gallery of Art, Washington,

DC, 1991.

[2] D.S. Steinberg, Vibration Analysis for Electronic Equipment, Wiley-Interscience, Chichester, Sussex, UK, 1973.

[3] J.R. Kuttler, V.G. Sigillito, On curve veering, Journal of Sound Vibration 75 (1981) 585–588.

[4] A.W. Leissa, On a curve veering abberation, Journal of Applied Mathematics and Physics (ZAMP) 25 (1974) 99.

[5] N.C. Perkins, C.D. Mote, Comments on curve veering in eigenvalue problems, Journal of Sound and Vibration 106 (1986) 451–463.

[6] D.J. Ewins, Modal Testing:Theory and Practice, Research Studies Press, Letchworth, 1986.

[7] L. Meirovitch, Element of Vibration Analysis, McGraw-Hill, New York, NY, USA, 1986.

[8] D.E. Newland, Mechanical Vibration Analysis and Computation, Longman Scientiﬁc and Technical, Harlow, 1994.

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