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Shock transmission in a coupled beam system

Authors:
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,YÞX
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
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
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[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.
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K. Vijayan, J. Woodhouse / Journal of Sound and Vibration 332 (2013) 3681–3695 3695
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The dependence of eigenvalues on a system parameter is frequently illustrated by a family of loci. When two loci approach each other, they often cross or abruptly diverge. The latter case, called “curve veering”, has been observed in approximate solutions associated with discretized models. The influence of discretization in producing curve veering has raised doubt on the validity of many approximate solutions. The existence of curve veering in continuous models is illustrated by presenting the exact solution of an elementary eigenvalue problem. Veering is then examined in a general eigenvalue problem. Criteria are established to distinguish veerings from crossings in both continuous and discretized models. The application of the criteria is illustrated by examples.
Mechanical Vibration Analysis and Computation, Longman Scientific and Technical
  • D E Newland
  • K Vijayan
  • J Woodhouse
D.E. Newland, Mechanical Vibration Analysis and Computation, Longman Scientific and Technical, Harlow, 1994. K. Vijayan, J. Woodhouse / Journal of Sound and Vibration 332 (2013) 3681–3695