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Shock amplification, curve veering and the role of damping

Authors:
Shock amplification, curve veering and the role of damping
K. Vijayan
n
, J. Woodhouse
Cambridge University Engineering Department, Trumpington Street, Cambridge CB2 1PZ, UK
article info
Article history:
Received 29 March 2013
Received in revised form
29 September 2013
Accepted 30 October 2013
Handling Editor: W. Lacarbonara
Available online 18 November 2013
abstract
The circumstances are investigated under which high peak acceleration can occur in the
internal parts of a system when subjected to impulsive driving on the outside. Previous
work using a coupled beam model has highlighted the importance of veering pairs of
modes. Such a veering pair can be approximated by a lumped system with two degrees of
freedom. The worst case of acceleration amplification is shown to occur when the two
oscillators are tuned to the same frequency, and for this case closed-form expressions are
derived to show the parameter dependence of the acceleration ratio on the mass ratio
and coupling strength. Sensitivity analysis of the eigenvalues and eigenvectors indicates
that mass ratio is the most sensitive parameter for altering the veering behaviour in an
undamped system. Non-proportional damping is also shown to have a strong influence on
the veering behaviour. The study gives design guidelines to allow permissible acceleration
levels to be achieved by the choice of the effective mass and damping of the indirectly
driven subsystem relative to the directly driven subsystem.
&2013 Elsevier Ltd. All rights reserved.
1. Introduction
There are many situations in which an object with fragile internal components may be subjected to external impacts. It is
important to be able to predict the shock level on these internal components, and to design the whole system to limit this to
an acceptable level. A typical example would be the design of packaging systems for the transportation of fragile art objects
to cope with, for example, airport freight handling facilities. Any such system can be broadly divided into two coupled
subsystems: the external case which is directly subjected to impact, and the internal system which is indirectly excited
through the coupling. Their response could explain the performance of many physical systems. An important engineering
application is in the transmission of vibration between coupled structures. An issue of general interest is whether under
some unfavourable circumstances the peak internal acceleration may be as great as or even greater than that directly
imposed on the external case. This is the phenomenon of shock amplification.
This problem represents a particular small corner of the study of vibration in coupled systems, which of course has a
huge literature. Methods like Statistical Energy Analysis (see for example [1]) explicitly involve vibration analysis in terms of
coupled subsystems. Numerical approaches such as the Finite Element method may use substructuring as a way to improve
efficiency. However, the particular problem treated here does not seem to have received a definitive treatment in the
literature, despite its relevance in a number of industrial settings.
Contents lists available at 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.2013.10.037
n
Corresponding author. Present address: College of Engineering, Swansea University, Singleton Park, Swansea SA2 8PP, UK. Tel.: þ44 1792 602969;
fax: þ44 1792 295676.
E-mail addresses: kiran.vijayan@gmail.com,kv247@cam.ac.uk (K. Vijayan),jw12@cam.ac.uk (J. Woodhouse).
Journal of Sound and Vibration 333 (2014) 13791389
In an earlier study [2], an idealised system consisting of two coupled beams was analysed both experimentally and
theoretically to explore the phenomenon of shock amplification. It was shown that amplification can indeed occur, even
within the limits of linear vibration theory and when the coupling between the two subsystems is weak. The analysis
carried out in that paper [2] indicated that the worst amplification was generallyassociated with one or more pairs of modes
exhibiting veeringbehaviour: when a mode of one beam falls close in frequency to a mode of the other beam, and these
couple to produce a pair of modes each showing significant modal amplitudes on both beams.
The phenomenon of veering has also generated its own significant literature. Although related phenomena were already
well known in atomic physics, where the effect is usually called level repulsion, curve veering was first identified within
the structural dynamics context in the 1970s, for a rectangular membrane problem [3]. Initially there were concerns about
whether it was an artefact of mathematical approximations used to estimate frequencies. However studies of the vibration
of orthotropic rectangular plates [4] and sagged cables [5] confirmed that veering was a physical phenomenon. Triantafyllou
noted that when the ends of a sagged cable are at the same level, the eigenvalues exhibit mode cross-over but when the
cable is inclined (ends at different levels), mode crossing never occurs. The existence of curve veering in continuous models
was verified by Perkins and Mote [6].
The pattern by which mode shapes change during a veering event relates to a wider phenomenon of mode localisation
[7], in which the modal or forced response within a coupled system is sometimes found to be restricted to one region rather
than distributed throughout the system. The localisation phenomenon for vibration in mechanical systems has received
considerable attention in recent years, especially in relation to periodic structures [811] made up of assemblies of
nominally identical substructures: see for example the review by Bendiksen [12].
It is now clearly understood that veering is the norm, not the exception: equal natural frequencies, implying the crossing
of eigenvalue locus curves, only occurs under special circumstances. Balmes [13] stated that three types of conservative
structures allow modal crossing:
(1) Symmetric structures have multiple modes that can be characterised using the algebraic properties of the group of
symmetryin particular, cyclically symmetric structures.
(2) Multidimensional substructures for which motion in different orientations uncouple, such as a beam having a bending
and a torsional mode at the same frequency.
(3) Structures with fully uncoupled substructures.
In the context of the coupled-beam problem studied earlier [2], a veering pattern was found to be a prerequisite for good
excitation on the directly driven beam simultaneously with high response on the indirectly driven beam. The specific
aspects of veering relevant to the shock amplification problem do not appear to have been studied in the existing literature,
and these aspects are investigated here using the simplest possible theoretical model, given by a two-degree-of-freedom
system. This system allows closed-form solutions that reveal important aspects of the parameter dependence on the shock
amplification effect.
The simplified springmass system shown in Fig. 1 will be used. In the first instance the system will be undamped, but
the effect of damping will be investigated later. This system is an idealisation for the coupling of any pair of modes from two
coupled subsystems as described above. The mass and stiffness matrices of the system are given by
M¼m
1
0
0m
2
"#
;K¼k
1
þk
c
k
c
k
c
k
2
þk
c
"#
:(1)
The case of most interest will involve weak coupling, represented in the model by small values of k
c
relative to k
1
and k
2
.
When the two subsystems, represented by the two masses here, are strongly coupled it would not be surprising if external
impacts resulted in high internal acceleration. The design of any protection and cushioning system tends to incorporate
weak coupling, precisely with the intention of isolating the internal system from external excitation. The aim here is to
explore when this isolation fails. The simplified system is familiar from analysis of tuned-mass dampers (see e.g. [14]),
but the question of interest is different here: not to achieve low motion on the directly driven mass, but how to get high
acceleration on the indirectly driven one. The present problem also differs from the tuned-mass damper in terms of the
relative masses of the two systems: a damper will normally have a much lower mass than the main structure, but there is no
reason to expect this to be the case for a fragile object being transported in, and protected by, an outer case. A wide range of
mass ratios could be of interest, and this study will not emphasise the low-mass case particularly.
The veering phenomenon is readily illustrated with this system. For simplicity the case m
1
¼m
2
is chosen for this initial
plot. Fig. 2 shows how the two natural frequencies vary when k
2
is varied over a range encompassing the value of k
1
, all
other parameters being kept fixed. In the absence of coupling, i.e. with k
c
¼0, the result is simply two lines that cross, but
k1c
k2
k
m1m2
Fig. 1. The two degree of freedom system.
K. Vijayan, J. Woodhouse / Journal of Sound and Vibration 333 (2014) 137913891380
with non-zero coupling veeringor avoided crossingoccurs. At both edges of the diagram the mode shapes are more or
less the same as in the uncoupled case, but when k
1
¼k
2
the character of the modes changes. Considerations of symmetry
in this case dictate that the modes must take the form ½11
T
and ½11
T
. Instead of localised modes, where the energy is
largely confined to one oscillator or the other, the modes become global, with equal amplitudes on each mass. To satisfy
orthogonality there must be a sign reversal between the two modes.
2. Undamped analysis
One might guess that the worst case for internal acceleration will occur when the two separate oscillators are tuned to
the same frequency in the absence of coupling: the resonant interaction between the two tuned oscillators leads to the
highest acceleration on the indirectly driven mass. This can be readily verified by computational studies. A typical example
is shown in Fig. 3. The directly driven oscillator is kept unchanging, a given value is chosen for the coupling spring, then the
mass and stiffness of the indirectly driven oscillator are varied over a large range. A unit impulse is applied to the directly
driven oscillator, and for each case of mass and stiffness ratio the peak acceleration on the indirectly driven oscillator is
calculated (by the method to be described explicitly shortly). The variation of this acceleration over the explored parameter
space is then plotted. It is immediately obvious that the largest values occur on or near the main diagonal, where the mass
and stiffness of the varied oscillator have the same ratio as for the fixed one.
This suggests carrying out a more detailed analysis of this tuned case, achieved bychoosing k
2
¼λk
1
and m
2
¼λm
1
so that
k
1
=m
1
¼k
2
=m
2
¼ω
2
n
say. The coupling strength can be expressed using a non-dimensional parameter α¼k
c
=k. This equal-
frequency case can be analysed in closed form. The mass and stiffness matrices reduce to
M¼m0
0λm

;K¼kð1þαÞαk
αkkðλþαÞ
"#
:(2)
The modes and natural frequencies are then determined by solving
Ku
1
u
2
"#
¼ω
2
Mu
1
u
2
"# (3)
as usual. This gives a quadratic equation in ω
2
whose roots are
ω
2
1
¼ω
2
n
;ω
2
2
¼ω
2
n
1þαþα
λ
 (4)
The corresponding eigenvectors can be written as u
1
¼½11
T
and u
2
¼½λ1
T
. The displacement response of the jth
mass to a unit impulse on mass 1 is given by
y
j
¼
2
n¼1
u
n;1
u
n;j
sin ω
n
t
ω
n
;tZ0 (5)
where u
n;j
is the jth component of the nth eigenvector u
n
, normalised with respect to the mass matrix so that
u
T
n
Mu
n
¼1:(6)
0.5 1 1.5
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
k2
Natural frequency
[1 0]T
[0 1]T
[0 1]T
[1 0]T
[1 −1]T
[1 1]T
Fig. 2. Illustration of veering behaviour. Natural frequencies of the system of Fig. 1 with m
1
¼m
2
¼1, k
1
¼1 and k
c
¼0:05 (solid lines), and the same system
with k
c
¼0 (dashed lines), over a range of values of k
2
.
K. Vijayan, J. Woodhouse / Journal of Sound and Vibration 333 (2014) 13791389 1381
The corresponding velocity response is
_
y
j
¼
2
n¼1
u
n;1
u
n;j
cos ω
n
t;tZ0:(7)
Each modal contribution to this velocity undergoes a step change at the instant of impact. Differentiation of the step input
gives a Dirac delta function contribution to the modal acceleration so that the acceleration takes the form
y
j
¼
2
n¼1
u
n;1
u
n;j
ðδðtÞω
n
sin ω
n
tÞ;tZ0:(8)
Mass-normalising the mode vectors found above gives
u
1
¼1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
mð1þλÞ
p
1
1

;u
2
¼1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
mλð1þλÞ
pλ
1

:(9)
The response of the directly driven mass is thus given by
y
1
¼1
mδtðÞω
1
sin ω
1
t
mð1þλÞλω
2
sin ω
2
t
mð1þλÞ

:(10)
For this ideal impulsive excitation the peak acceleration will occur at the instant of impact since the delta function
dominates the response. The acceleration response of the indirectly driven mass is given by
y
2
¼ω
n
mðλþ1Þsin ω
1
t1þαþα
λ

1=2
sin ω
2
t
 (11)
because in this case the delta-function contributions cancel out, from the form of the normalised eigenvectors in Eq. (9). This
cancellation must happen for any discrete system at a point remote from the applied impulse, as is clear on physical grounds
and was proved in [2].
If the two frequencies are close, the pair of sinusoids will combine to give a beating pattern: they have approximately
equal amplitudes. The peak of the envelope of acceleration on the indirectly driven mass then occurs when the two terms
add in phase, giving a simple estimate of the peak value:
y
ðmaxÞ
2
ω
n
mðλþ1Þ1þ1þαþα
λ

1=2

:(12)
The acceleration of the directly driven mass may also display a beating pattern, but from Eq. (10) the amplitudes of the two
sinusoids are only similar if the mass ratio λis close to unity. Comparison of the acceleration response on the directly driven
mass in Eq. (10) and the indirectly driven mass given in Eq. (11) indicates that the beating envelope between the responses
has a different phase in the two cases. Typical examples to illustrate these features of the response pattern are shown in
Fig. 4. The first uses values λ¼0:7 and α¼0:05, with m¼1 and k¼1, while the second has λ¼0:2 and α¼0:01. Beating is
much more apparent on the directly driven mass in the first case, but both show the phase difference of the envelope
between the responses of the two masses.
For an ideal impulse the amplitude will be infinite at t¼0, while the response of the indirectly driven beam remains
finite. Under those circumstances the ratio of the two peak accelerations is not a useful measure of the effectiveness of shock
isolation. However it is impossible in practice to excite the system with a delta function; real signals are of finite duration
and amplitude. The response of the model system to an input pulse of finite width is obtained by convolving the impulse
response with the input pulse. Convolution smooths the response signal: the input signal acts as a low pass filter. This limits
Stiffness ratio
Mass ratio
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
−35
−30
−25
−20
−15
−10
−5
0
5
10
15
Fig. 3. Peak acceleration on the indirectly driven mass, for a range of mass and stiffness ratios. The system is undamped, m
1
¼k
1
¼1, and the coupling
stiffness is held constant at k
c
¼0:01. Amplitude ratio is shown as logarithmically spaced contours at 3 dB intervals.
K. Vijayan, J. Woodhouse / Journal of Sound and Vibration 333 (2014) 137913891382
the amplitude of the initial spike associated with an ideal impulse and only the modes within the bandwidth of the input
pulse are excited. This suggests that a more useful measure of the shock performance of the coupled system would be the
peak absolute acceleration level on the indirectly excited mass, as given by Eq. (12), for an applied impulse of given area.
This result is independent of the input pulse shape provided only that it is narrow enough to excite both modes. This
definition would lend itself to experimental testing by, for example, a drop test.
An alternative measure, which gives a similar answer but lends itself better to computational examples, is the ratio of
peak accelerations disregarding the delta function contribution: in other words the ratio of the peaks of the oscillatory
contributions from the two sinusoids. Using the analogous argument to Eq. (12), this ratio is well approximated by
R¼1þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þαþα=λ
p
1þλffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þαþα=λ
p:(13)
The case of most interest has weak coupling, α1, and also has low mass on the indirectly driven oscillator so that λ1. If in
addition αλ, the ratio Rstays close to the constant value 2, describing modest acceleration amplification. However, in the
opposite extreme when λαthe ratio can become large, roughly ffiffiffiffiffiffiffi
α=λ
p. The pattern is illustrated in Fig. 5, showing Ras a
function of αand λover large ranges of both. The case to be avoided in the design of shock protection systems occurs
towards the bottom of this plot, when a resonance of an internal component with low effective mass falls close in frequency
to a resonance of the outer case. In that case, even if the coupling between the two is fairly weak, significant amplification of
internal acceleration can occur.
The results for this simplified system can be compared with the behaviour found in the earlier study of coupled beams
[2], in which the worst-case acceleration on the indirectly excited beam always occurred when at least one veering pair of
modes was present: the characteristic beating pattern of response, similar to Fig. 4, was shown in Fig. 11 of that reference.
The tightnessof the veering is determined by the coupling strength. In the simplified model this is represented by
the coupling stiffness parameter α, while in the beam model it was influenced by several factors: the thickness ratio, the
stiffness of the point couplings, and also by symmetry-breaking due to added masses or attached sensors. For coupled
beams of the same thickness, each mode of the directly excited beam will naturally coincide in frequency with a mode of the
0 50 100 150 200 250 300
−5
−4
−3
−2
−1
0
1
t (s)
Acceleration
0 50 100 150 200 250 300
−5
−4
−3
−2
−1
0
1
t (s)
Acceleration
Fig. 4. Example impulse responses for undamped systems with m¼1, k¼1: upper curve: acceleration of directly driven mass; lower curve: acceleration of
indirectly driven mass.
K. Vijayan, J. Woodhouse / Journal of Sound and Vibration 333 (2014) 13791389 1383
indirectly excited beam, so one would expect a high degree of modal veering. However, a higher number of veering pairs do
not necessarily lead to higher peak acceleration, since the mass ratio λis equal to unity in this case. A larger thickness ratio
between the beams was in fact found to lead to higher maximum acceleration, because this made λsmaller.
3. Sensitivity analysis
Since the peak acceleration response is dependent on the eigenvalues and eigenvectors of the systems their sensitivity to
changes in system parameters could give useful insight into the response. Various authors have derived expressions for
the sensitivity of eigenvalues and eigenvectors in, for example, aerodynamic flutter and damage analysis [1518]. Since the
intention of the present work is to give design guidance for practical applications based on the simplest reasonable model,
attention will be restricted to first-order perturbation theory. Higher-order theory could be developed, but it is argued that
in a real application this would provide rather little additional information, and that it is more likely that numerical studies
would be used to optimise a design.
The general equation for an eigenvalue problem is
Ku
j
¼ω
2
j
Mu
j
:
If the system is now perturbed by changing Kto KþδKand Mto MþδM, the equation governing the corresponding
perturbations to the mode vector and the squared frequency is
ðKþδKÞðu
j
þδu
j
Þ¼ðω
2
j
þδω
2
j
ÞðMþδMÞðu
j
þδu
j
Þ(14)
Rayleigh's principle states that given an approximation to a mode shape we get a rather good approximation to the natural
frequency by evaluating the Rayleigh quotient:
ω
2
j
þδω
2
j
u
T
j
ðKþδKÞu
j
u
T
j
ðMþδMÞu
j
:(15)
Removing higher order terms and simplifying Eq. (15) give
ω
2
j
u
T
j
δMu
j
þu
T
j
Mu
j
δω
2
j
u
T
j
δKu
j
:
Since the normalised mode u
j
satisfies Eq. (6), the final result is
δω
2
j
u
T
j
ðδKω
2
j
δMÞu
j
:(16)
If the perturbation is the result of a small change to a design parameter γ, the sensitivity to that change is thus
d
dγω
2
j
¼u
T
j
d
dγKω
2
j
d
dγM

u
j
:(17)
α
λ
0.01 0.03 0.10 0.32 1.00
0.01
0.03
0.10
0.32
1.00
0
2
4
6
8
10
12
14
16
18
Fig. 5. Ratio Rfrom Eq. (13) of peak acceleration on the indirectly driven mass to that of the driven mass, ignoring the delta-function contribution, for a
range of mass and coupling ratios. The system is undamped. Amplitude ratio is shown as logarithmically spaced contours at 3 dB intervals. The scale bar
shows 20 log
10
Rso that the value 0 corresponds to R¼1.
c
1
k1kck2
m1m2
c2
Fig. 6. The system of Fig. 1 with the addition of local damping on the two masses.
K. Vijayan, J. Woodhouse / Journal of Sound and Vibration 333 (2014) 137913891384
The next step is to determine the eigenvector sensitivity. From Eq. (14) removing higher order terms and simplifying give
ðKω
2
j
MÞδu
j
¼δω
2
j
Mu
j
ðδKω
2
j
δMÞu
j
:(18)
Since the eigenvectors span the domain the differential of the eigenvector can be written as a linear combination of
eigenvectors:
δu
j
¼
n
k¼1
δa
jk
u
k
:(19)
Substituting in Eq. (18) gives
ðKω
2
j
MÞ
n
k¼1
δa
jk
u
k
¼δω
2
j
Mu
j
ðδKω
2
j
δMÞu
j
:
Pre-multiplying by u
T
j
gives δa
jj
¼0, while pre-multiplying by u
T
k
for kajgives δa
jk
¼
jak
u
T
k
ðδKω
2
j
δMÞu
j
=ðω
2
j
ω
2
k
Þ. Thus
δu
j
¼
jak
u
T
k
ðδKω
2
j
δMÞu
j
ω
2
j
ω
2
k
u
k
:
Thus the sensitivity of the eigenvector to changing the design parameter γis
d
dγu
j
¼
jak
u
T
k
d
dγ
Kω
2
jd
dγ
M

u
j
ω
2
j
ω
2
k
u
k
:(20)
Applying these results to the original problem, the coupling strength αcan first be taken as the design parameter γwhile the
mass ratio λis kept constant. The stiffness matrix from Eq. (1) on being differentiated with respect to αgives
d
dαK¼kk
kk

¼k11
11

:(21)
The mass matrix from Eq. (1) is independent of αso
d
dαM¼0:(22)
Substituting these expressions into Eqs. (17) and (20) gives the eigenvalue and eigenvector sensitivities corresponding to the
first natural frequency ω
1
¼ω
n
as
d
dαω
2
1
¼d
dαu
1
¼0:(23)
These eigenvalue and eigenvector sensitivities are compatible with the modelled system since for the first natural frequency
the coupling does not influence the response. However for the natural frequency ω
2
¼ω
n
ð1þαþα=λÞthe coupling spring
influences the response since the two masses are in antiphase for this mode. The eigenvector sensitivity is still zero, but the
eigenvalue sensitivity is
d
dαω
2
2
¼ω
2
n
1þ1
λ

:(24)
The results of the sensitivity analysis indicate that the eigenvectors are not sensitive to the coupling strength. A more
interesting design parameter is the mass ratio λ. Both the stiffness and mass matrices depend on λ. The derivatives are now
d
dλK¼00
0k
 (25)
and
d
dλM¼00
0m

:(26)
The eigenvalue sensitivity for ω
1
¼ω
n
from Eq. (17) is
d
dλω
2
1
¼0:(27)
The corresponding eigenvector sensitivity from Eq. (20) is
d
dλu
1
¼1
λ
1=2
ð1þ1=λÞu
2
:(28)
K. Vijayan, J. Woodhouse / Journal of Sound and Vibration 333 (2014) 13791389 1385
The expression for the eigenvalue sensitivity for the second mode ω
2
¼ω
n
ð1þαþα=λÞis
d
dλω
2
2
¼αω
2
n
λ
2
:(29)
The corresponding expression for the eigenvector sensitivity is
d
dλu
2
¼0:(30)
The results from the sensitivity analysis indicate that with changes in λthere is a variation in both the eigenvalue and
eigenvector. The former alters the gap between the interacting modes and the latter causes changes in mode shape. Both
these factors influence the peak acceleration.
4. Damping inuence on mode veering
In the analysis considered so far, the two-degree-of-freedom system was assumed undamped. It is well known that
damping can have a positive influence on suppressing vibration in any system, but for this particular problem there might
be a less obvious effect of damping. If the damping on the two separate subsystems, and thus on the two separate oscillators
in the simplified model, is very different, it has been shown [19] that under some circumstances this can suppress the
phenomenon of veering. It is not easy to guess how strongly this will influence the acceleration ratio.
The simplest model to investigate this possibility is to assume linear viscous damping applied to the two masses, so that
the equation of motion can be written in the form
m0
0λm

u
1
u
2
"#
þc
1
0
0λc
2
"#
_
u
1
_
u
2
"#
þkð1þαÞαk
αkkðλþαÞ
"#
u
1
u
2
"#
¼0 (31)
where c
1
and c
2
are the damping coefficients.
Assuming time dependence of the form e
iωt
as usual, the complex natural frequencies are given by the determinant
condition jω
2
MþiωCþK0 where Cis the damping matrix from the middle term in Eq. (31). The result becomes a
quadratic equation in ω
2
if the damping terms are approximated, in a form appropriate to low damping, by
iωc
1
miω
2
n
Q
1
and iωλc
2
miω
2
n
Q
2
;(32)
0 10 20
0.99
1
1.01
1.02
1.03
1.04
1.05
1.06
1.07
1.08
1.09
Q1/Q2Q1/Q2
Re(ω)
Im(ω)
010 20
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
Fig. 7. Variation of the real and imaginary parts of the complex eigenvalues for the damped system of Fig. 6 with α¼0:05, m¼1, k¼1 and a range of values
of damping ratio Q
1
=Q
2
. The mass ratio λhas value 1 for the dashed curves, then values 0.9, 0.8, 0.7, 0.6 and 0.5 for the remaining curves.
K. Vijayan, J. Woodhouse / Journal of Sound and Vibration 333 (2014) 137913891386
where Q
1
and Q
2
are the modal Qfactors of the two decoupled oscillators. The roots of this quadratic equation are
ω
2
¼ω
2
n
22þαþα
λþi
Q
1
þi
Q
2
7ffiffiffi
D
p
 (33)
where
D¼α
2
1þ1
λ

2
1
Q
1
1
Q
2

2
þ2iα11
λ

1
Q
1
1
Q
2

:(34)
Note that setting the two Qfactors to infinity recovers the undamped frequencies given earlier.
For the equal-mass case λ¼1, Dis a real quantity. If it is positive, the two values of ω
2
have the same imaginary part but
different real parts: the modes veer as before, and both have the same decay constant given by the imaginary part. However,
if Dis negative the two roots have the same real part but different imaginary parts: the frequencies are equal but the two
damping factors are different, and veering has been suppressed. The condition determining the sign of Dis very similar to
the criterion given in earlier work [19]: suppression of veering requires
4α
2
o1
Q
1
1
Q
2

2
:(35)
For the case of proportional damping, Q
1
¼Q
2
, this condition can never be satisfied so veering always occurs. However,
when the two masses are unequal ðλa1Þthe situation changes. The discriminant Dis then a complex number, and no such
simple interpretation of the results is possible. The two roots are separated in both real and imaginary parts.
The behaviour is illustrated in Fig. 7. The equal-mass case, shown by the dashed lines, reveals a bifurcation at
Q
1
=Q
2
11, which agrees with the prediction of Eq. (35) for the parameter values of this particular example. However, the
other lines plotted in this figure show that λdoes not have to get very far from unity before the bifurcation behaviour
becomes scarcely visible. Both the real and imaginary parts of the two eigenvalues diverge across the entire range of the plot
by the time λ¼0:5.
0 50 100 150 200 250 300
−2.5
−2
−1.5
−1
−0.5
0
0.5
1
t (s)
Acceleration
0 50 100 150 200 250 300
−2
−1.5
−1
−0.5
0
0.5
1
t (s)
Acceleration
Fig. 8. Example impulse responses for damped systems with α¼0:05, λ¼0:7, m¼1, k¼1, Q
1
¼100: Upper curve: acceleration of directly driven mass
(without delta function contribution); lower curve: acceleration of indirectly driven mass. The response of the same system without damping was shown
in Fig. 4a.
K. Vijayan, J. Woodhouse / Journal of Sound and Vibration 333 (2014) 13791389 1387
A numerical study on the tuned two degree-of-freedom system was carried out to explore the consequence of this
complicated behaviour for peak acceleration ratio. The impulse response of the damped system can be calculated
straightforwardly by converting the equations into first-order form and following the procedure described, for example,
by Newland [20]. The acceleration response may produce either a decaying or a beating envelope, depending on detailed
parameter values. Examples are shown in Fig. 8. Both plots relate to the same system illustrated in Fig. 4a, but now with
damping. If the ratio Q
1
=Q
2
¼1 the system is proportionally damped, the eigenvectors remain exactly as in the undamped
case, and beating is still evident provided the damping is low enough to allow enough cycles of oscillation to be seen. As the
ratio departs from unity, the beating behaviour gradually vanishes: for the case illustrated it is residually present in Fig. 8a
with Q
1
=Q
2
¼5, and gone in Fig. 8b with Q
1
=Q
2
¼15.
Fig. 9 shows the acceleration ratio, as in Fig. 5, for a fixed coupling strength α¼0:05 and a wide range of values of mass
ratio λand damping ratio Q
1
=Q
2
. Every horizontal row in these results shows a monotonic decrease of the acceleration ratio
from left to right as the damping on the indirectly driven mass increases. However, the pattern across the whole parameter
plane shows some structure that would not be easy to guess. On the left-hand side the result is very sensitive to the value of
λ, whereas on the right-hand side it is almost independent of λ. The most pronounced acceleration amplification occurs
where λis small and when Q
2
Q
1
. This makes intuitive sense: the indirectly driven system has low effective mass and low
local damping. On the other hand, when the indirectly driven system has high damping relative to the directly driven mass,
its mass makes very little difference to the acceleration level.
5. Conclusion
A study of the influence of veering pairs of modes on the peak acceleration level in a coupled system has been carried out
using a simple two-degree-of-freedom model. The worst case arises when the two separate oscillators are tuned to the same
natural frequency ω
n
. The simple model gives a closed-form expression for the acceleration amplification on the indirectly
driven system, for the case without damping. This expression, given in Eq. (13), indicates a dependence on the eigenvalues
and eigenvectors of the system via two crucial parameters: the mass ratio ðλÞand normalised coupling strength ðαÞ.
A sensitivity analysis of the eigenvalues and eigenvectors with respect to λand αwas carried out. The results indicate that
the eigenvectors are not sensitive to the coupling strength, but the results from the sensitivity analysis with λindicated that
the sensitivity varies in a quadratic manner for the eigenvalue and inverse quadratic manner for the eigenvector. The former
alters the gap between the interacting modes and the latter causes changes in mode shape. Both these factors influence the
peak acceleration.
The study suggests possible design guidelines for suppressing high acceleration on the indirectly driven system.
Acceleration response was found to be dependent on the mass ratio: the highest peak levels can arise when the indirectly
driven system has low effective mass relative to the directly driven system. For systems in which the scope for altering
masses is limited, the study also revealed an alternative approach, by altering the damping strategically. In addition to the
familiar effect of damping on any vibration response, non-proportional damping within the coupled system can act to
suppress eigenvalue veering. The combination of these two factors gives a somewhat complicated pattern in the parameter
space, but the most beneficial effect is gained by adding damping to the indirectly driven subsystem, especially in cases
when that subsystem has low effective mass relative to the directly driven subsystem.
Q1/Q2
λ
0.05 0.22 1.00 4.47 20.00
0.01
0.03
0.10
0.32
1.00
−10
−5
0
5
Fig. 9. Ratio of peak acceleration on the indirectly driven mass to that of the driven mass, ignoring the delta-function contribution, for coupling strength
α¼0:05 for a range of mass and damping ratios. The directly driven mass has Q
1
¼100 throughout. Amplitude ratio is shown as logarithmically spaced
contours at 3 dB intervals. The 0 dB level is marked with the heavy line.
K. Vijayan, J. Woodhouse / Journal of Sound and Vibration 333 (2014) 137913891388
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K. Vijayan, J. Woodhouse / Journal of Sound and Vibration 333 (2014) 13791389 1389
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