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Exact tuning of a vibration neutralizer for the reductionof flexural waves in beams
Jimmy S. Issaa)
Department of Industrial and Mechanical Engineering, Lebanese American University, P.O. Box 36, Byblos,Lebanon
(Received 14 January 2019; revised 6 June 2019; accepted 26 June 2019; published online 26 July2019)
In this work, designs of vibration neutralizers for the reduction of flexural waves in beams are pro-
posed. The system considered consists of an Euler-Bernoulli beam experiencing a harmonically
travelling wave. The neutralizer, which consists of a mass, spring, and damper (viscous or hyster-
etic), is attached at a point on the beam. Two designs are considered. In the first, the aim is the min-
imization of the transmitted energy, and in the second, the aim is the maximization of the energy
dissipated in the neutralizer. For minimum energy transmission, the undamped neutralizer can be
tuned to reflect all the energy back to its source at the tuning frequency. The tuning stiffness ratio
and the neutralizer bandwidth is obtained in an analytical closed form. For maximum energy dissi-
pation, the neutralizer can dissipate up to 50% of the incident energy. The maximum energy dissi-
pated, i.e., 50% of the incident energy, is constant and occurs at the tuning frequency. It is neither
affected by the neutralizer mass nor by the tuning frequency. The tuning parameters are obtained
analytically. Finally, in all the above cases, the increasing of the neutralizer mass enhances the neu-
tralizer performance. All analytical findings are validated numerically.VC 2019 Acoustical Society of America. https://doi.org/10.1121/1.5116690
[BA] Pages: 486–500
I. INTRODUCTION
Vibration suppression has always been a research topic
of paramount importance because of the damaging effect of
unwanted vibration on both structures and human beings.
For example, vibration of machines reduces their efficien-
cies, increases wear, and can lead to failure due to fatigue.
Once initiated from a source, the vibrational energy will not
necessarily stay confined to a part of a structure, since it can
easily get transmitted from one part to another. In the pro-
cess, the vibration of these parts will interact with the sur-
rounding air which absorbs part of this energy in the form of
noise resulting in annoyance in the surrounding area and in
the extreme case to hearing impairments. The vibration
absorber is one of the most commonly used passive techni-
ques for vibration reduction. Since its invention in 1909,1
this device has been properly designed for maximum perfor-
mance2–4 when coupled to discrete single degree of freedom
main systems. It reduces the vibration of the main system by
applying forces passively that counteract the forces that are
transmitted to the main system from the excitation source.
When the main system can no longer be modeled as a
point mass, e.g., slender parts in structures, a different
approach for vibration suppression should be adopted. A
common example of slender parts is beam elements in struc-
tures, which are responsible for channeling and spreading
the vibrational energy initially generated at one location on
the structure. Once the main guides of the energy transmitted
in a structure are identified, vibration neutralizers can be
attached to them and designed to reduce energy propagation
across these guides. When a wave propagates in a beam, it will
not get disturbed unless it encounters an impedance mismatch
at a point, e.g., a point mass or a spring.5,6 Part of the energy
will get transmitted through this point and the remaining part
will get reflected and directed back to its source. The first
meticulous study of a vibration neutralizer for the reduction of
flexural waves in beams was conducted by Brennan.7 The neu-
tralizer is assumed to be lightly damped and approximate ana-
lytical expressions for the tuning frequency; bandwidth and
attenuation ratio are derived in two different frequency regimes
where these expressions can be simplified. El Khatib et al.8
used a hysteretically damped vibration neutralizer to either
reflect the energy transmitted at a point or to absorb it. The
effect of both the near field and far field waves is considered.
Salleh and Brennan9 considered four different neutralizer con-
figurations, namely, the force type, the moment type, the cou-
pled, and the uncoupled force moment types. The transmission
ratio was derived for each case; however, the study focused on
the force and the uncoupled force moment types only because
they can be easily realized. It is shown that the uncoupled force
moment type can be more effective than the force type.
Thompson10 attached a continuous vibration neutralizer with-
out bending stiffness to a beam and showed that a blocked fre-
quency region exists where waves will no longer be able to
freely propagate across the beam at these frequencies.
Approximate formulas for the waves decay rate and neutralizer
bandwidth are derived. The use of multiple vibration neutral-
izers for the reduction of flexural waves in beams was also con-
sidered, e.g.,11–16 the key parameter in this case is the
separation between the different oscillators. The reader is
referred to the cited works therein for a more exhaustive cover-
age of this part of the research.a)Electronic mail: [email protected]
486 J. Acoust. Soc. Am. 146 (1), July 2019 VC 2019 Acoustical Society of America0001-4966/2019/146(1)/486/15/$30.00
In this work, a comprehensive study on the reduction of
flexural waves in a beam using a vibration neutralizer is pre-
sented. The neutralizer is assumed to be attached at one point
on the beam. A wave propagating from one side of the beam
will encounter an obstacle, i.e., the neutralizer. The aim is
vibration reduction, and it is achieved by either minimizing
the energy transmitted passed the attachment point, or by
maximizing the energy dissipated in the neutralizer. It is
shown how one can select the neutralizer damping and stiff-
ness properties to achieve optimal performance. In both
cases, and to reduce errors resulting from simplifications, the
optimal tuning parameters of the neutralizer are obtained in
an analytical closed form without making any kind of
assumptions on the system damping size or on the system
operating regime. The results are obtained for the two com-
mon damping models, namely, viscous and hysteretic. The
exactness of the solution is validated for both objectives con-
sidered by a direct numerical resolution of the coupled beam
neutralizer equation of motion. In Sec. II, the dimensionless
parameters are defined and the relationships between the differ-
ent wave components are derived. In Sec. III, exact tuning for
minimum energy transmission is performed. The transmission
and reflection ratios are defined, and the tuning neutralizer
parameters are determined analytically. In Sec. IV, design for
maximum energy dissipation is considered. The dissipation
ratio is defined, and the tuning neutralizer parameters are deter-
mined in closed form. Numerical verification of the analytical
findings is done in Sec. V. Concluding remarks and directions
for future works are given in Sec. VI.
II. TRANSMISSION AND REFLECTION RATIOS
The schematic of an infinite beam coupled with a vis-
cously damped neutralizer is shown in Fig. 1(a) and with a hys-
teretically damped neutralizer in Fig. 1(b). In these figures, mand k depict the neutralizer mass and stiffness constant, respec-
tively. w(x, t) is the transverse motion of the beam, and y(t) is
the neutralizer displacement. The viscous damping constant is
depicted by c in Fig. 1(a), and the loss factor of the hysteretic
damping by d in Fig. 1(b). It is assumed that the neutralizer is
attached at the reference origin x¼ 0 in the far field of the
energy source and, hence, the incident wave wi comprises of its
propagating component only. The transmitted wave wt has two
parts, namely, wtp and wte, which are the propagating and eva-
nescent components, respectively. Similarly, wrp and wre
denote the propagating and evanescent components of the
reflected wave wr, respectively. Assuming harmonic excitation,
the incident, transmitted and reflected wave components can be
written as follows:
wiðx;tÞ¼Wi ei xt�bxð Þ;
wtðx;tÞ¼wtpðx; tÞþwteðx; tÞ¼Wt ei xt�bxð Þ þWte e ixt�bxð Þ;
wrðx; tÞ¼wrpðx;tÞþwreðx;tÞ¼Wr ei xtþbxð Þ þWre e ixtþbxð Þ:
(1)
In Eq. (1), Wi, Wt, and Wr are the amplitudes of the propagat-
ing component of the incident, transmitted and reflected
waves, respectively. Wte and Wre are the amplitudes of the
evanescent component of the transmitted and reflected
waves at x¼ 0, respectively, and x is the excitation fre-
quency. Let q and E be the material density and Young’s
modulus, respectively, and A and I the cross-sectional area
and its second moment of area, respectively. Then, the flex-
ural wave number is defined as b ¼ ðx2qA=EIÞ1=4. In the
remainder of the paper, the time term in Eq. (1) will be
ignored to reduce verbosity. The next step is to solve for the
reflected and transmitted amplitudes in terms of the incident
amplitude and the system parameters. This is achieved by
ensuring continuity of the displacement and slope of the
beam at x¼ 0 and by applying Newton’s Law to an infinites-
imal beam element centered at x¼ 0. The continuity equa-
tions on the displacement and slope of the beam at x¼ 0 are
written as follows:
wið0; tÞ þ wrð0; tÞ ¼ wtð0; tÞ;@wi
@xð0; tÞ þ @wr
@xð0; tÞ ¼ @wt
@xð0; tÞ: (2)
An infinitesimal element of the beam at x¼ 0 is shown in Fig. 2.
V and M denote the shear force and bending moment, respec-
tively, and Fd the dynamic force applied by the neutralizer on
the beam. The minus and plus superscripts represent variables
evaluated right before and right after x¼ 0, respectively.
Applying Newton’s law on this element results in the
following two equations:
M�ð0; tÞ �Mþð0; tÞ ¼ 0;
FdðtÞ þ V�ð0; tÞ � Vþð0; tÞ ¼ 0: (3)
Let Kd be the dynamic stiffness of the neutralizer, hence
Eqs. (2) and (3) can now be simplified to
FIG. 1. Schematic of an infinite beam coupled with a vibration neutralizer (a) with viscous damping, (b) with hysteretic damping.
J. Acoust. Soc. Am. 146 (1), July 2019 Jimmy S. Issa 487
Wi þWr þWre �Wt �Wte ¼ 0;
Wi �Wr þ iWre �Wt þ iWte ¼ 0;
Wi þWr �Wre �Wt þWte ¼ 0;
KdðWi þWr þWreÞ
� EIb3ðiWi � iWr þWre � iWt þWteÞ ¼ 0: (4)
Solving Eq. (4) yields
Wt
Wi¼ ð1� iÞða� 1Þ
2a� ð1� iÞ ;Wr
Wi¼ � ð1þ iÞa
2a� ð1� iÞ ;
Wte
Wi¼ Wre
Wi¼ � ð1� iÞa
2a� ð1� iÞ : (5)
In Eq. (5), a¼Kd/4EIb3 is a complex dimensionless ratio that
encompasses all the system parameters. It is the ratio of the
neutralizer dynamic stiffness to that of the beam, and hence it
will be referred to as the dynamic stiffness ratio. In the remain-
der of the paper, parameters with subscripts v and h refer to the
viscous and hysteretic damping cases, respectively. For exam-
ple, Kd is defined as the neutralizer dynamic stiffness, then Kdv
is the dynamic stiffness of the viscously damped neutralizer
and Kdhis that of the hysteretically damped neutralizer. Kdv
and Kdh take the form (see Appendix A)
Kdv ¼1
1
k þ icx� 1
mx2
; Kdh¼ 1
1
kð1þ idÞ �1
mx2
:
(6)
Now, the following dimensionless parameters are defined
and used instead of the dimensional ones to generalize the
results:
l ¼ m
4qA3=2; f ¼ kA3=2
4EI; f ¼ c
4ffiffiffiffiffiffiffiffiEIqp ;
r ¼ x
� ffiffiffiffiffiffiffiffiEI
qA3
s0@
1A
1=2
: (7)
In Eq. (7), l is the mass ratio which is the ratio of the neutral-
izer mass to that of a portion of this beam of length 4A1=2. f is
the stiffness ratio; it can be viewed as the ratio of the neutral-
izer stiffness to the static stiffness at the mid-span of a simply
supported portion of this beam of length 121=3A1=2. f is the
damping ratio, which is the ratio of the neutralizer viscous
damping constant to the damping constant 2ffiffiffiffiffiffiffiffiEIqp
. Finally, r is
the frequency ratio and is defined as the square root of the ratio
of the excitation frequency to the first natural frequency of a
simply supported beam made of the same material with a
length of pA1=2. It is important to note that these parameters
are conveniently defined in a way to ensure that each one
depends on only one of the design parameters. For example,
the effect of the neutralizer mass is taken into account in the
mass ratio l only, and hence varying m will change the lparameter only without affecting the remaining ones.
Similarly, f, f, and r separately account for the effect of k, c,
and x, respectively. The dynamic stiffness ratio can now be
written in terms of the dimensionless parameters for the vis-
cously and hysteretically damped cases as follows:
av ¼1
1
f
r3þ i
fr
� 1
lr
or ah ¼1
1
f
r3ð1þ idÞ
� 1
lr
:(8)
To conclude this section, the aim of this work is to propose
two different vibration neutralizer designs, i.e., find the most
appropriate mass, stiffness, and damping constants of the
neutralizer that will fit a given objective. The first design
aims at reducing the energy transmitted passed the attach-
ment point at some given excitation frequency. And for the
second design, the aim is to maximize energy absorption by
the neutralizer at some given excitation frequency. In Sec.
III, the first design is treated.
III. DESIGN FOR MINIMUM ENERGY TRANSMISSION
A. Transmission and reflection ratios
A harmonic flexural wave w(x, t) carries energy as it
travels across a beam from one point to another. The instan-
taneous power flowing past a point in an Euler-Bernoulli
beam in the positive x axis direction is defined in Ref. 17.
The average energy incident at a point can be calculated
as Pi ¼ kWik2 EIxb3. Similarly, the average transmitted
energy passed a point and that reflected at the point are
Pt ¼ kWtk2 EIxb3 and Pr ¼ kWrk2 EIxb3, respectively.
The evanescent components of the waves do not show up in
these equations simply because they do not carry energy.
The fraction of the incident energy that is transmitted
passed the attachment point is defined as the transmission
ratio and calculated as T ¼ Pt=Pi ¼ kWtk2=kWik2.
Similarly, the reflection ratio, R ¼ Pr=Pi ¼ kWrk2=kWik2,
is defined as the faction of the incident energy that is
reflected at the attachment point. When there is no energy
dissipation at the attachment point (zero damping), the
energy incident will partly get transmitted and the remain-
ing part will get reflected, hence in this case TþR¼ 1.
When damping is present, part of the incident energy will
get dissipated in the damper. The transmission and reflec-
tion ratios are calculated in terms of the dimensionless
parameters for the viscously and hysteretically damped
cases and take the forms
FIG. 2. Schematic of an infinitesimal element of the beam centered around
x¼ 0.
488 J. Acoust. Soc. Am. 146 (1), July 2019 Jimmy S. Issa
Tv ¼ðf þ frl� r4lÞ2 þ r4f2ð1þ rlÞ2
ðf þ frl� r3fl� r4lÞ2 þ r2ðrfþ f lþ r2flÞ2;
Rv ¼r6f2l2 þ r2f 2l2
ðf þ frl� r3fl� r4lÞ2 þ r2ðrfþ f lþ r2flÞ2;
or
Th ¼ðf þ frl� r4lÞ2 þ f 2d2ð1þ rlÞ2
f 2ðdþ rlþ rdlÞ2 þ ðr4lþ frdl� frl� f Þ2;
Rh ¼f 2r2l2 þ f 2r2d2l2
f 2ðdþ rlþ rdlÞ2 þ ðr4lþ frdl� frl� f Þ2:
(9)
Since the target in this section is the minimization of the
transmitted energy, the neutralizer parameters should be
properly chosen to reduce the transmission ratio T. The
undamped neutralizer design is treated in Sec. III B.
B. Undamped neutralizer
The transmission ratio of an undamped neutralizer is
referred to as T0 and is obtained from Eq. (9) from Tv (or Th)
by setting the damping parameter to zero, as
T0 ¼ðf þ frl� r4lÞ2
ðf þ frl� r4lÞ2 þ ðrflÞ2: (10)
In Fig. 3, T0 is plotted versus the frequency ratio for a given
set of parameters. The plot is first thoroughly analyzed and
examined in order to facilitate proposing optimized design
guidelines. The figure clearly shows that for low and high fre-
quencies, the transmission ratio tends towards 1, i.e., at these
frequencies, waves are completely transmitted and hence neu-
tralizers are useless. At some frequency ratio denoted by r0,
the transmission ratio is zero and hence a wave travelling
with a frequency ratio r0 will get completely reflected back to
its source. This frequency ratio is referred to as the tuning fre-
quency ratio. Furthermore, it is observed that waves with fre-
quency ratios falling in the vicinity of r0 will have a large
chunk of their energy reflected back to their source. It is con-
cluded that, if properly designed, these neutralizers are very
useful in reducing flexural waves with a narrow frequency
band defined by a main major frequency and some fluctuations
around it. In this case, one should choose the neutralizer
parameters such that r0 coincides with the major excitation fre-
quency and should ensure that the remaining frequency band
receives adequate amount of energy reflection. This is referred
to as tuning the neutralizer. The neutralizer bandwidth is
defined as the frequency range within which the transmission
ratio is less than �3 dB, i.e., T0 � 1=2. The boundaries of the
neutralizer bandwidth are depicted in the figure by the two key
frequency ratios r1 and r2. Often, the major excitation fre-
quency is well defined; however, its fluctuations, i.e., frequency
range is not, and hence the best design is achieved by maximiz-
ing the neutralizer bandwidth so as to encompass the largest
amount of fluctuations possible around the main major fre-
quency. The frequency ratios r0, r1, and r2 are first obtained in
terms of the neutralizer parameters, then design guidelines for
the undamped neutralizer are proposed.
Using Eq. (10), the tuning frequency r0 is calculated
from the equation T0¼ 0. This results in the following fourth
order polynomial in r:
f þ fl r � l r4 ¼ 0: (11)
This equation can be solved using Ferrari’s method and
yields only one real solution, which takes the form
c0 ¼ �27f 2l3 þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið27f 2l3 þ 128f Þ2 � ð128f Þ2
q;
b0 ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi128f 3
c0
!1=3
þ c0
54l3
� �1=3
vuut;
r0 ¼1
2b0 þ
1
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2f
b0
� b20
s: (12)
The bandwidth boundaries r1 and r2 are calculated form the
equation T0 ¼ 1=2, which after simplification, reduces to
ðf � r4lÞðf þ 2frl� r4lÞ ¼ 0: (13)
Solving this equation yields eight roots out of which two are
real positive and define the bandwidth of the neutralizer as
r1 ¼f
l
� �1=4
;
c2 ¼ �54f 2l3 þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið54f 2l3 þ 16f Þ2 � ð16f Þ2
q;
b2 ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi64f 3
c2
!1=3
� c2
27l3
� �1=3
vuut;
r2 ¼1
2b2 þ
1
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4f
b2
� b22
s: (14)
FIG. 3. Plot of the transmission ratio showing the tuning frequency ratio r0
and the boundary of the neutralizer bandwidth r1 and r2, for l¼ 1.25 and
f¼ 0.25.
J. Acoust. Soc. Am. 146 (1), July 2019 Jimmy S. Issa 489
For a given neutralizer mass and stiffness, i.e., l and f, the
tuning frequency ratio r0 can be calculated from the exact
expression given in Eq. (12) and the normalized bandwidth
(r2� r1)/r0 can be obtained in an exact closed form using
Eqs. (12) and (14). Since in a real scenario the major excita-
tion frequency is known, the neutralizer is tuned to this fre-
quency and therefore, r0 is known. Tuning the neutralizer is
achieved by choosing its mass and stiffness such that Eq.
(11) is satisfied after substituting r0 for r. Hence, for a given
neutralizer mass ratio l, the tuned stiffness ratio is calculated
as follows:
ft ¼l r4
0
1þ l r0
: (15)
The effect of l on the neutralizer bandwidth is examined by
inspecting the sign of @r1/@l and @r2/@l after tuning the sys-
tem, i.e., after substituting ft for f in the expressions of r1 and
r2. This results in, after simplifications
@r1
@l¼ � r2
0
4ð1þ l r0Þ5=4;
@r2
@l¼ r2ð2r2 � r0Þ
2ð1þ l r0Þð2þ 3lr2Þ:
(16)
It is concluded from Eq. (16) that 8l; @r1=@l < 0, since
both l and r0 are positive. This means that for a fixed r0, r1
decreases with the increasing of l. Furthermore, it is con-
cluded from the same equation that 8l; @r2=@l > 0, since
both l and r0 are positive and since by definition r2 > r0.
This means that r2 increases with the increasing of l. Hence,
in summary, when the neutralizer is tuned to some fixed fre-
quency ratio r0 by setting f to ft, the neutralizer bandwidth
increases with the increasing of the mass ratio l. Hence, one
should use the largest neutralizer mass possible to widen its
bandwidth as much as possible.
The transmission ratio is plotted in Fig. 4 for different
mass ratios of a neutralizer tuned to the frequency ratio r0¼ 1.
The figure clearly shows that as l increases, the bandwidth
increases to reach its maximum when l tends towards infinity.
In this case, the neutralizer mass is infinitely large, and this
scenario reduces to the case of a spring attached directly to a
fixed ground. r1 tends towards zero and r2 tends towards
21=3r0. Furthermore, it is important to note that the tuning stiff-
ness ratio ft depends on l and increases with increasing l.
When l tends towards infinity, the tuning stiffness ratio tends
towards r30. This concludes the discussion of the study of the
undamped neutralizer design. In Sec. III C, the case of the
damped neutralizer will be considered.
C. Damped neutralizer
When damping is present in the neutralizer, the trans-
mission ratio will no longer exhibit zero transmission at
some frequency, instead it will pass through a minimum
value which is higher than zero. The transmission ratio is
plotted in Fig. 5 for different neutralizer parameters, namely,
Fig. 5(a) with viscous damping and Fig. 5(b) with hysteretic
damping. For both damping cases, the mass ratio is chosen
to be l¼ 3 and the stiffness and damping ratios are properly
chosen so that the system remains tuned to r0¼ 0.5. Tuning
when damping is zero yields a transmission ratio passing
though zero at r0¼ 0.5; however, when damping is present,
tuning is achieved when the minimum of the transmission
ratio occurs at r0¼ 0.5. Each figure shows two damped cases
tuned to r0¼ 0.5. In the first case, the minimum transmission
ratio, which is denoted by h0, is equal to 0.1, and in the sec-
ond it is equal to 0.2. For example, in Fig. 5(a), the minimum
transmission ratio will equal 0.1 at r0¼ 0.5 when (f¼ 0.086,
f¼ 0.0756) and 0.2 when (f¼ 0.156, f¼ 0.0748). This means
that in these cases, the best scenario occurs when r¼ r0, at
this point zero transmission is not achieved; instead, 10% or
20% of the energy will still get transmitted through the point
of attachment of the neutralizer. Therefore, one might think
that the addition of damping is detrimental on the neutral-
izer. This is not entirely true because even though the addi-
tion of damping deteriorates the neutralizer performance at
the tuning frequency, it increases its bandwidth specially on
the right side of r0. The effect of the increase of the neutral-
izer damping on increasing r2 is higher than that on increas-
ing r1. The same observations are made for the hysteretically
damped cases shown in Fig. 5(b). Hence, the addition of
damping yields some type of trade-off in the neutralizer
design.
These damped neutralizers are very useful for the reduc-
tion of travelling waves with relatively larger frequency
bands compared to those treated by undamped neutralizers
which are considered to have narrow frequency bandwidths.
In this case, and since there is a trade-off between the mini-
mum transmission ratio and the neutralizer bandwidth, in
addition to the major frequency r0, which is considered to be
known, the sought after minimum transmission ratio h0 is
assumed to be known. Then, for a given l, the neutralizer
can now be tuned to r0 by a proper choice of the stiffness
and damping ratios. The tuning neutralizer parameters sat-
isfy the below two equations,
FIG. 4. Plot of the transmission ratio when the neutralizer is tuned to r0¼ 1
showing the effect of the mass ratio l on the neutralizer bandwidth.
490 J. Acoust. Soc. Am. 146 (1), July 2019 Jimmy S. Issa
@T
@r
����r¼r0
¼ 0; T
����r¼r0
¼ h0: (17)
The above equations are solved analytically in Appendix B
and the tuning stiffness and damping ratios are obtained in
closed form in terms of r0, h0, and l. The effect of the mass
ratio on the transmission ratio is depicted in Fig. 6. In this
figure, the neutralizer is tuned to r0¼ 0.5 and h0¼ 0.1. The
tuning neutralizer parameters corresponding to each mass
ratio, are calculated using the analytical expressions derived
in Appendix B. It is clearly shown that, as expected, the neu-
tralizer bandwidth increases with the increasing of the mass
ratio. The limiting case when l tends towards 1, i.e., when
the spring and damper are directly connected to a fixed sup-
port is shown too. The latter case yields the utmost perfor-
mance; however, connecting a spring and damper to a fixed
support is most of the time not physically achievable.
IV. DESIGN FOR MAXIMUM ENERGY DISSIPATION
A. Dissipation ratio
In this section, the aim is to maximize energy dissipa-
tion in the neutralizer. When coupled to a beam, a damped
vibration neutralizer will dissipate part of the travelling
energy in the damping element. Hence, in this case, the addi-
tion of a damped neutralizer is always beneficial; however,
FIG. 5. Plots of the transmission ratio (a) viscously damped for l¼ 3.0 and three different sets of stiffness and damping ratios, namely, (f¼ 0.0, f¼ 0.075),
(f¼ 0.086, f¼ 0.0756), and (f¼ 0.156, f¼ 0.0748), (b) hysteretically damped for l¼ 3.0 and three different sets of stiffness and loss factors, namely, (d¼ 0.0,
f¼ 0.075), (d¼ 0.29, f¼ 0.073), and (d¼ 0.53, f¼ 0.068).
FIG. 6. Plots of the transmission ratio of a vibration neutralizer tuned to r0¼ 0.5 and h0¼ 0.1 showing the effect of the mass ratio l (a) viscously damped for
(l¼ 0.1, ft¼ 0.000524, ft¼ 0.00595), (l¼ 2, ft¼ 0.059, ft¼ 0.0628), (l¼ 5, ft¼ 0.124, ft¼ 0.0908), and (l¼1, ft¼ 0.258, ft¼ 0.133); (b) hysteretically
damped for (l¼ 0.1, dt¼ 0.022, ft¼ 0.00595), (l¼ 2, dt¼ 0.236, ft¼ 0.0613), (l¼ 5, dt¼ 0.345, ft¼ 0.0863), and (l¼1, dt¼ 0.505, ft¼ 0.118).
J. Acoust. Soc. Am. 146 (1), July 2019 Jimmy S. Issa 491
the neutralizer performance can be maximized if the latter is
properly designed. The average energy incident, transmitted
and reflected at the attachment point, were calculated in
Sec. III as Pi ¼ kWik2 EIxb3; Pt ¼ kWtk2 EIxb3 and
Pr ¼ kWrk2 EIxb3, respectively. The relative motion of
the neutralizer with respect to its attachment point on the
beam is defined as z(t)¼ y(t)�w(0, t). The average energy
dissipated over one cycle in the damper element of the
neutralizer is calculated as Pdv ¼ 1=2cx2kzðtÞk2for vis-
cous damping and Pdh¼ 1=2kdxkzðtÞk2
for hysteretic
damping. The dissipation ratio is defined as the ratio of the
dissipated energy to the incident energy as, D¼Pd/Pi. The
neutralizer parameters will be obtained analytically with
the aim of maximizing the dissipation ratio D. Let yðtÞ¼ Y eixt be the neutralizer response, where Y is the neutral-
izer complex amplitude. y(t) can be directly related to the
motion w(0, t) of the attachment point. Assuming harmonic
motion, the two displacements are related through the
equation of motion of the neutralizer, which can be written
for the viscous and hysteretic damping cases, respectively,
as follows:
m€yðtÞ þ c _yðtÞ þ kyðtÞ ¼ c _wð0; tÞ þ kwð0; tÞ
m€y þ k 1þ idð ÞyðtÞ ¼ k 1þ idð Þwð0; tÞ: (18)
Assuming harmonic motion, wð0; tÞ ¼ W0eixt, the steady
state neutralizer response Y is written in terms of W0 by solv-
ing Eq. (18) as
Yv ¼kþ icx
k�mx2ð Þ þ icxW0; Yh ¼
k 1þ idð Þk�mx2ð Þ þ ikd
W0:
(19)
Now, the beam displacement at the attachment point w(0, t) can
be written using Eq. (1) in terms of the incident and reflected
wave as wð0; tÞ ¼ wið0; tÞ þ wrð0; tÞ ¼ ðWi þWr þWreÞ eixt
or in terms of the transmitted wave as wð0; tÞ ¼ wtð0; tÞ¼ ðWt þWteÞ eixt. For convenience, the second expression is
used resulting in W0¼WtþWte. This simplifies Eq. (19) to
Yv ¼k þ icx
k � mx2ð Þ þ icxðWt þWteÞ;
Yh ¼k 1þ idð Þ
k � mx2ð Þ þ ikdðWt þWteÞ: (20)
The relative neutralizer motion takes the form zðtÞ ¼ yðtÞ�wð0; tÞ ¼ ðY � ðWt þWteÞÞ eixt. Let Z¼ Y� (WtþWte) be
the complex amplitude of z(t), which takes the form
Zv ¼mx2
k � mx2ð Þ þ icxðWt þWteÞ;
Zh ¼mx2
k � mx2ð Þ þ ikdðWt þWteÞ: (21)
Using the expressions for Wt and Wte from Eq. (5), the dissi-
pation ratio is written in terms of the dimensionless parame-
ters as follows:
Dv ¼1=2cx2kZvk2
EIxb3 kWik2
¼ 2r7fl2
r2 flþ fr þ r2flð Þ2 þ f þ frl� r3ðr þ fÞl� �2
;
Dh ¼1=2kdxkZhk2
EIxb3 kWik2
¼ 2fr5dl2
f 2 dþ rlð1þ dÞð Þ2 þ r4lþ frlðd� 1Þ � f� �2
:
(22)
Alternatively, the dissipation ratio can be obtained using a dif-
ferent approach. Instead of directly calculating the energy dis-
sipated in the neutralizer, one can calculate it from the incident
energy Pi, transmitted energy Pt, and reflected energy Pr. It is
a fact that, as mentioned before, for an undamped neutralizer,
the incident energy is equal to the summation of the reflected
and transmitted energies, i.e., Pi¼ (PrþPt), and hence no
energy is lost. However, when damping is present, the energy
dissipated in the neutralizer can be obtained as Pi� (PrþPt).
This means that the dissipation ratio can be written in terms of
the reflection and transmission ratios as D¼ 1� (TþR).
Substituting the expressions of T and R in terms of the dimen-
sionless parameters from Eq. (9) into D¼ 1� (TþR) yields
expressions of the dissipation ratios as Dv¼ 1� (TvþRv) and
Dh¼ 1� (ThþRh) identical to those given in Eq. (22). The
objective is to maximize the dissipation ratios Dv and Dh by a
proper choice of the neutralizer parameters. It is important to
note that the dissipation ratios depend on the frequency ratio rand hence the optimization problem is formulated as follows.
For a given mass ratio l, find the optimal stiffness and damp-
ing ratios such that the dissipation ratio is maximized at the
tuning frequency r0. Each damping case is treated in a separate
section. In Sec. IV B, viscous damping is considered.
B. Viscous damping
In order to better understand the behavior of the shape
of the dissipation ratio curve, the latter is first plotted versus
the frequency ratio for three different sets of parameters in
Fig. 7. All curves start from zero and tend towards zero on
the limit as r tends towards infinity. In between, the curves
exhibit a maximum value at some frequency ratio. At this
frequency, the neutralizer reaches its maximum performance
by absorbing the largest amount of energy possible for a
given set of parameters. For example, in case I, the neutral-
izer reaches its maximum value, i.e., 0.37, at a frequency
ratio of 0.61. In case II, the maximum dissipation ratio is
0.45 and occurs at r¼ 1.03. And in case III, the maximum
value is 0.49 and occurs at r¼ 1.76. The aim is to maximize
the dissipation ratio at some r¼ r0 by a proper choice of f, fand l. First, the optimal solution is obtained for a given l,
then the effect of the mass ratio is studied.
At a given mass ratio l and tuning frequency ratio r0,
the dissipation ratio becomes function of two variables,
namely, f and f, and hence the tuning parameters ft and ft are
obtained from the below equations,
492 J. Acoust. Soc. Am. 146 (1), July 2019 Jimmy S. Issa
@Dv
@f
����r¼r0
¼ 0;@Dv
@f
����r¼r0
¼ 0: (23)
The solution of Eq. (23) will yield either a saddle point or
a local or global optimum (maximum or minimum). The goal is
to find first the set of parameters that yields a global maximum.
The two equations in Eq. (23) reduce to after simplifications
r40lð1þ r0lÞ � f ð1þ 2r0lþ 2r2
0l2Þ ¼ 0;
r80l
2 � 2fr40lð1þ r0lÞ þ f 2ð1þ 2r0lþ 2r2
0l2Þ
� r40f
2ð1þ 2r0lþ 2r20l
2Þ ¼ 0: (24)
Solving these equations results in a unique set of positive
stiffness and damping ratios as
ft ¼r4
0lð1þ r0lÞ1þ 2r0lþ 2r2
0l2; ft ¼
r30l
2
1þ 2r0lþ 2r20l
2: (25)
To ensure that at this set of solution the dissipation ratio
passes through a maximum, the determinant of the Hessian
matrix should be positive at this solution and the second
derivative of the dissipation ratio with respect to one of the
parameters, i.e., f or f, should be negative. Let H be the
Hessian matrix of the dissipation ratio function, it is defined
and evaluated at the tuning parameters as
H ¼
@2Dv
@f 2
@2Dv
@f@f
@2Dv
@f@f
@2Dv
@f2
266664
377775
¼�ð1þ 2r0lþ 2r2
0l2Þ2
4r100 l4
0
0 �ð1þ 2r0lþ 2r20l
2Þ2
4r60l
4
266664
377775:
(26)
It is clear from Eq. (26) that, 8r0 and 8l, the determinant of His positive, and @2Dv/@f2 (or @2Dv/@f
2) is negative. This means
that the tuning parameters results in a maximum. This maxi-
mum value is global because the solution of Eq. (24) is unique,
and hence it is the sole optimum of the dissipation ratio. The
maximum value of Dv when the system is tuned is obtained by
substituting the tuning parameters into Eq. (22), which yields
Dvmax¼ 1
2: (27)
It is clear that the maximum value of Dv, when the system is
tuned to r¼ r0, is constant and depends on neither l nor r0.
Hence, once tuned, 8l and 8r0 the neutralizer will dissipate
exactly 50% of the incident energy at the tuning ratio r0.
Finally, it can be easily shown that
@Dv
@r
����r¼r0;f¼ft;f¼ft
¼ 0: (28)
This means that at the tuning parameters, if the dissipation ratio
function is plotted versus r, the peak which is shown in Fig. 7
will occur at r¼ r0. It is concluded that after tuning the neutral-
izer, it reaches its utmost performance at r ¼ r0; 8r > 0.
Hence, 8r > 0, the maximum energy that can be absorbed by
a tuned neutralizer is 50% of the incident energy and occurs at
r¼ r0. Furthermore, it can be shown that at the tuning fre-
quency, Tv¼Th¼ 0.25 and Rv¼Rh¼ 0.25. Hence, at r¼ r0,
50% of the energy will get dissipated, 25% will get transmit-
ted, and 25% will get reflected back to its source.
The dissipation ratio is shown in Fig. 8 tuned to r0¼ 0.5
and r0¼ 2 for three different values of l in each case. The
figure clearly shows that the maximum energy dissipation is
50% at r0¼ 0.5 or r0¼ 2 for all the mass ratios used.
Furthermore, the slope of the curves is zero at the tuning fre-
quencies, which yields a dome like shape for the dissipation
ratio, with the dome peak occurring at the tuning frequency.
Even though the mass ratio has no effect on the maximum
value of Dv, its effect is tremendous on the dome size, which
is clearly depicted in the figure. The dome size increases
with increasing l; this means that the neutralizer perfor-
mance improves in the vicinity of the tuning frequency r0
when the neutralizer mass is increased. Furthermore, the
neutralizer performance enhances with the increasing of the
tuning frequency for a fixed mass ratio. This is clearly illus-
trated in the figure, for example, for l¼ 0.5, the dome size is
larger when the neutralizer is tuned to r0¼ 2.0 compared to
that when it is tuned to r0¼ 0.5. In Sec. IV C, tuning of the
hysteretically damped neutralizer is considered.
C. Hysteretic damping
The same procedure used in the determination of the
tuning parameters of the viscously damped neutralizer will
be followed. First, at a given tuning frequency r0 and for a
given mass ratio l, the tuning parameters are calculated
from the below equations,
@Dh
@f
����r¼r0
¼ 0;@Dh
@d
����r¼r0
¼ 0; (29)
FIG. 7. Plot of the dissipation ratio versus the frequency ratio for three dif-
ferent parameter sets, namely, (I) l¼ 0.5, f¼ 0.05, f¼ 0.01, (II) l¼ 0.5,
f¼ 0.3, f¼ 0.2, and (III) l¼ 0.5, f¼ 2, f¼ 0.4.
J. Acoust. Soc. Am. 146 (1), July 2019 Jimmy S. Issa 493
which reduce after simplifications to the following forms:
f 2ðd2 þ 1Þð1þ 2r0lð1þ r0lÞÞ � r80l
2 ¼ 0;
2fr40lð1þ r0lÞ þ f 2ðd2 � 1Þð1þ 2r0lð1þ r0lÞÞ� r8
0l2 ¼ 0: (30)
The tuning parameters are obtained by solving the above
equations as
ft ¼r4
0lð1þ r0lÞ1þ 2r0lþ 2r2
0l2; dt ¼
r0l1þ r0l
: (31)
To ensure that the tuning parameters yield a global maxi-
mum of the dissipation ratio, the Hessian matrix is first cal-
culated as
H ¼
@2Dh
@f 2
@2Dh
@f@d
@2Dh
@d@f
@2Dh
@d2
266664
377775
¼�ð1þ 2r0lþ 2r2
0l2Þ3
4r100 l4ð1þ r0lÞ2
�ð1þ 2r0lþ 2r20l
2Þ4r5
0l2
�ð1þ 2r0lþ 2r20l
2Þ4r5
0l2
�ð1þ r0lÞ2
4r20l
2
2666664
3777775:
(32)
The determinant of the Hessian matrix is now calculated and
simplified to the below form:
ð1þ r0lÞ2ð1þ 2r0lþ 2r20l
2Þ2
16r120 l6
: (33)
It is clear that the Hessian determinant is positive, further-
more @2Dh/@f2 (or @2Dh/@d2) is negative; this means that the
tuning parameters derived in Eq. (31) correspond indeed to a
maximum of the dissipation ratio. This maximum is global
because the solution of Eq. (30) is unique. Similar to the
case of viscous damping, it can be shown that when the neu-
tralizer is tuned, the slope of the dissipation ratio at r¼ r0 is
zero, hence its peak occurs at the tuning frequency. Finally,
the maximum dissipation ratio of a tuned neutralizer is equal
to 1/2 and occurs at the tuning frequency r0. For the sake of
completeness, the same plot shown in Fig. 8 is generated
using hysteretic damping as shown in Fig. 9. Six cases are
shown, three tuned to r0¼ 0.5 and the remaining three to
r0¼ 2.0. At each tuning frequency, three mass ratios are con-
sidered, namely, l¼ 0.1, l¼ 0.5, and l¼ 2.0. The same dis-
cussions and observations made for the viscous damping
case hold for this case and hence they will not be repeated to
reduce verbosity.
V. NUMERICAL VERIFICATION
In this section, the previously derived analytical
expressions are verified numerically by a direct resolution
of the coupled beam neutralizer system. The partial differ-
ential equation that governs the motion in a Euler-
Bernoulli beam and the equation of motion of a viscously
damped neutralizer shown in Fig. 1(a) are written as
follows:
qA €wðx; tÞ þ EI w0000ðx; tÞ þ ddðxÞ k ðwðx; tÞ � yðtÞÞðþc ð _wðx; tÞ � _yðtÞÞÞ ¼ 0;
m €yðtÞ þ k ðyðtÞ � wð0; tÞÞ þ c ð _yðtÞ � _wð0; tÞÞ ¼ 0:
(34)
For hysteretic damping, the above equations become
qA €wðx; tÞ þ EI w0000ðx; tÞ þ ddðxÞ kð1þ idÞð� wðx; tÞ � yðtÞÞð Þ ¼ 0;
m €yðtÞ þ kð1þ idÞ ðyðtÞ � wð0; tÞÞÞ ¼ 0: (35)
In Eqs. (34) and (35), the over-dot depicts the derivative
with respect to time and (0) the derivative with respect to
space x. dd(x) is the Dirac delta function. At this stage, the
following dimensionless time and space parameters are
defined as: s¼xt and X¼bx. Using this change of variables
and the dimensionless parameters defined in Sec. II, Eqs.
(34) and (35) reduce to the following forms after some
simplifications:
FIG. 8. Plot of the dissipation ratio
versus the frequency ratio tuned to
r0¼ 0.5 and r0¼ 2. For each case,
three mass ratios, namely, l¼ 0.1,
l¼ 0.5, and l¼ 2.0 are used.
494 J. Acoust. Soc. Am. 146 (1), July 2019 Jimmy S. Issa
€wðX;sÞþw0000ðX;sÞþddðXÞ 4f=r3 ðwðX;sÞ�
�yðsÞÞþ4f=r ð _wðX;sÞ� _yðsÞÞÞ¼ 0;
l€yðsÞþ f=r4 ðyðsÞ�wð0;sÞÞþ f=r2 ð _yðsÞ� _wð0;sÞÞ¼ 0:
(36)
€wðX; sÞ þ w0000ðX; sÞ þ ddðXÞ 4f=r3ð1þ idÞ�
� wðX; sÞ � yðsÞÞð Þ ¼ 0;
l €yðsÞ þ f=r4ð1þ idÞ ðyðsÞ � wð0; sÞÞ ¼ 0: (37)
In these equations, the over-dot and (0) denote now the deriv-
atives with respect to the dimensionless time s and space X,
respectively. To prepare these equations for the numerical
resolution, first the space X is discretized using finite ele-
ments. Hermitian beam elements are used, which results in
two degrees of freedom at each node, namely, the displace-
ment w(X, s) and the slope @w(X, s)/@X. Let me and ke be the
mass and stiffness matrices of a Hermitian beam element,
respectively. The elementary matrices associated with the
partial differential equation given in Eqs. (36) or (37) are
given below, where le is the element length
me ¼le
420
156 �22le 54 13le
�22le 4l2e �13le �3l2e
54 �13le 156 22le
13le �3l2e 22le 4l2e
2666664
3777775;
ke ¼2
l3e
6 �3le �6 �3le
�3le 2l2e 3le l2e
�6 3le 6 3le
�3le l2e 3le 2l2
e
2666664
3777775: (38)
To ensure good representation of the solution using the
discretized space, and since using the dimensionless space
parameter fixes the dimensionless wave length to 2p, 60 ele-
ments per wave length are used. This ensures that for any
material or geometric properties, the wave response is well
represented in the space dimension. Hence, the length of the
beam element used is le¼ 2p/60. The beam is assumed to
have infinite length; however, for calculation purposes, some
length should be assumed. A beam length of 120 wave
length (i.e., 240p) is used. This ensures that the length of the
beam will accommodate up to 120 cycles along its span.
This yields a total number of 7200 beam elements and 7201
nodes, thus 14 402 degrees of freedom for the beam only,
since as mentioned before, there are 2 degrees of freedom
per node. The X¼ 0 node, which is the neutralizer attach-
ment point, is selected to be the beam mid-span. The neutral-
izer equation, i.e., the second equation in Eqs. (36) or (37),
is an algebraic equation and hence does not need any spatial
discretization. It is simply added to the set of equations
(here, 14 402 equations) that results from the finite element
discretization of the beam. This increases the number of
degrees of freedom by one to Nd¼ 14 403, i.e., the absorber
motion. After assembling all elementary matrices of the
beam and adding the neutralizer equation, the total discrete
set of ordinary equations of motion of the whole system will
take the form
M €X þ K X ¼ 0 : (39)
In Eq. (39), M and K are the mass and stiffness matrices of
the whole system with sizes of (14 403� 14 403). The inci-
dent wave is generated by setting a prescribed harmonic
motion to the left end of the beam. This means that one of
the degrees of freedom is known and will become the driving
force of this equation. Here, the displacement of the first
node is assumed to be harmonic for example to have a
motion of the form of sin(s). Now, the forced set of equa-
tions of motion of the system become
�M €X þ �K X ¼ F : (40)
In Eq. (40), �M is obtained from M by simply setting to zero
all values in its first line. �K is obtained from K by setting to
zero all values of its first line except for the first value, which
is replaced by 1. Now, the forcing vector F has zero entries
except for its first value, which is the prescribed motion of
the first node, i.e., sin(s). Finally, Eq. (40), which is a second
order ordinary equation in time, is reduced to first order by
the classical change of variables Z ¼ ½X _X�T . The reduced
FIG. 9. Plot of the dissipation ratio
versus the frequency ratio tuned to
r0¼ 0.5 and r0¼ 2. For each case,
three mass ratios, namely, l¼ 0.1,
l¼ 0.5, and l¼ 2.0.
J. Acoust. Soc. Am. 146 (1), July 2019 Jimmy S. Issa 495
equation takes the form _Z ¼ AZ þ b where the (2Nd� 2Nd)
matrix A and (2Nd) vector b are given below
A ¼0ðNd�NdÞ IðNd�NdÞ
�M�1K 0ðNd�NdÞ
" #;
b ¼ cos ðsÞ 0 � � � 0 �sin s 0 � � � 0½ �: (41)
A is a (2Nd� 2Nd) matrix and vector b is a zero vector with
2Nd dimension except for the first entry, which is cos s and the
Ndþ 1 entry, which is �sin s. The set of ordinary differential
equations that will result from the finite element discretization
are solved using the Crank–Nicolson technique, which is an
implicit finite difference method. This technique is used to solve
the reduced equation _Z ¼ AZ þ b. Based on Crank–Nicolson,
the state vector Ziþ1 at the next time step is calculated from the
state vector at the previous time step Zi as
Ziþ1¼ I� dt
2A
�1dt
2biþ1þAZiþbið ÞþZi
: (42)
Time stepping is essential here in order to avoid misrep-
resentation of the solution in the time domain. Here, the
dimensionless time s fixes the dimensionless period of
motion to 2p for any material or geometric properties. The
time step is chosen to equal dt¼ 2p/60, this ensures that
each steady state cycle at a given node is described with 60
points. Now, test cases of the transmissibility ratios are con-
sidered to validate the analytical findings. In Fig. 10(a), the
undamped neutralizer tuned to r0¼ 1 is considered with
l¼ 3 and its corresponding transmission ratio plotted using
the analytical expression in Eq. (10). The tuning stiffness
ratio is calculated from Eq. (15) and is equal to f¼ 0.75. The
figure shows, as expected, that the wave will be completely
reflected at the tuning frequency and the neutralizer perfor-
mance deteriorates as the frequency moves away from r0. For
the same case, the transmission ratio was calculated numeri-
cally by solving either Eqs. (36) or (37) and setting the damp-
ing parameter to zero. The numerical solution was obtained at
nine different frequency ratios including the tuning frequency
ratio r0¼ 1. Each data point is obtained as follows: the beam is
given a prescribed harmonic motion at its left end. This will
initiate a wave that will propagate through the beam and will
encounter an obstacle at the attachment point of the neutral-
izer. Part of the wave will get reflected and part will get trans-
mitted. After ensuring that the steady state regime is achieved,
the amplitudes of the incident and transmitted waves are mea-
sured and the transmission ratio is obtained. The numerically
calculated transmission ratios are illustrated by the solid points
in Fig. 10(a). The impeccable match between the numerical
and analytical solution is clearly shown in the figure and stems
from the fact that the analytical solution obtained is exact since
no assumptions were made during the derivation process.
Similar validations are shown in Figs. 10(b) and 10(c), where
the neutralizer is hysteretically damped in Fig. 10(b) and tuned
to r0¼ 1 with h0¼ 0.1 and viscously damped in Fig. 10(c) and
tuned to r0¼ 1 with h0¼ 0.2. The tuning stiffness and damping
ratios used to plot these curves are obtained from the analyti-
cally derived expression given in Appendix B. For each case,
nine data points are obtained numerically by solving either
Eqs. (36) or (37). Again, for these two cases, an almost perfect
match between the analytical solution and numerical results is
achieved.
FIG. 10. Plots of the transmission ratio of a vibration neutralizer tuned to r0¼ 1 with a mass ratio of l¼ 3.0. (a) undamped neutralizer with tuning stiffness
ratio of f¼ 0.75. (b) hysteretically damped neutralizer with h0¼ 0.1 with tuning parameters f¼ 0.72 and d¼ 0.36. (c) viscously damped neutralizer, with
h0¼ 0.2 with tuning parameters f¼ 0.76 and f¼ 0.51.
496 J. Acoust. Soc. Am. 146 (1), July 2019 Jimmy S. Issa
Finally, the analytical expressions of the tuning parame-
ters pertaining to the dissipation ratio part of the analysis are
validated numerically. Four cases are considered, two for
each damping type. The dissipation ratio with viscous damp-
ing Dv is plotted in Fig. 11(a) for two mass ratios, namely,
l¼ 0.2 and l¼ 1. In this plot, the neutralizer is tuned to
r0¼ 1 using the analytical expressions in Eq. (25), which
yields the following tuning parameters (f¼ 0.16, f¼ 0.027)
for l¼ 0.2, and (f¼ 0.4, f¼ 0.2) for l¼ 1. In the same plot,
nine data points for each dissipation ratio curve are calcu-
lated numerically by solving Eq. (36). For each data point,
the dissipation ratio is calculated as follows: first the beam is
given a prescribed harmonic motion at its left boundary to
initiate a travelling wave. After reaching steady state condi-
tions, the amplitude kZvk of the relative neutralizer motion is
obtained along with that of the incident wave, i.e., kWik. The
dissipation ratio can now be calculated as follows:
Dv ¼ ð2f=rÞðkZvk2=kWik2Þ, which is deduced from Eq. (22).
For all cases considered, almost a perfect match between the
analytical and numerical results is achieved. The same vali-
dation is done for the analytical results derived for the
hysteretic damping case. The dissipation ratio is first
plotted in Fig. 11(b) for the two mass ratios l¼ 0.2 and
l¼ 1 tuned to r0¼ 1. The tuning parameters are calcu-
lated using the analytical expression in Eq. (31), which
yields (f¼ 0.16, d¼ 0.167) for l¼ 0.2 and (f¼ 0.4,
f¼ 0.5) for l¼ 1. Nine data points for each curve are
obtained numerically for this case from the resolution of
Eq. (37), as depicted by the solid points in the figure. The
dissipation ratio is calculated numerically for each data
point from a modified version of Eq. (22) as follows:
Dh ¼ ð2f d=r3ÞðkZhk2=kWik2Þ. Again, here, the figure
shows a great match between the numerical and analyti-
cal findings.
VI. CONCLUSION
In this paper, the use of vibration neutralizers for the
reduction of flexural waves in beam elements is investigated.
The neutralizer is attached to the beam and its parameters are
properly chosen to maximize either the energy reflected back to
its source or the energy dissipated in the neutralizer. The
undamped neutralizer is considered first and for a given mass
ratio, the neutralizer was tuned to a frequency at which energy
was completely reflected back to its source. The tuning stiffness
ratio is obtained in an analytical closed form along with the
neutralizer bandwidth. Then, the damped neutralizer is consid-
ered and tuned to a frequency by forcing the minimum value of
the transmissibility ratio to occur at that frequency. The tuning
stiffness and damping parameters are obtained analytically for
both the viscous and hysteretic damping cases. For both the
undamped and damped neutralizer, the increasing of its mass is
shown to widen its bandwidth. As for energy dissipation, the
bare fact that a neutralizer is attached to a beam will lead to
some energy dissipation in the damping element. However, this
energy can be maximized and can reach at most 50% of the
energy propagating in the beam at a given tuning frequency.
The tuning stiffness and damping ratios are obtained in a closed
form in terms of the system mass and tuning frequency ratios.
It is shown that, even though the neutralizer mass has no effect
on the maximum energy dissipated at the tuning frequency, its
effect is tremendous on the neutralizer performance in the
vicinity of the tuning frequency. Finally, the analytically
derived expressions are validated numerically where almost a
perfect match is achieved between the two solutions.
APPENDIX A
In this appendix, the dynamic stiffness of a damped neu-
tralizer is derived. The schematic of a damped neutralizer is
FIG. 11. Plots of the dissipation ratio of a damped vibration neutralizer tuned to r0¼ 1 for two different mass ratios, namely, l¼ 0.2 and l¼ 1.0. (a) viscously
damped neutralizer with the following tuning parameters (l¼ 0.2, f¼ 0.16, f¼ 0.027) and (l¼ 1, f¼ 0.4, f¼ 0.2), (b) hysteretically damped neutralizer with
the following tuning parameters (l¼ 0.2, f¼ 0.16, d¼ 0.167), and (l¼ 1, f¼ 0.4, d¼ 0.5).
J. Acoust. Soc. Am. 146 (1), July 2019 Jimmy S. Issa 497
shown in Figs. 12(a) and 12(b), with viscous and hysteretic
damping, respectively. The free end of the neutralizer is
depicted with point D. Let u(t) be the motion of D, F(t) the
force applied at D, and y(t) the motion of the neutralizer
mass. The force F(t) is written in terms of u(t) and y(t) as
FðtÞ ¼ kðuðtÞ � yðtÞÞ þ cð _uðtÞ � _yðtÞÞ for the viscous damp-
ing case and FðtÞ ¼ kð1þ idÞðuðtÞ � yðtÞÞ for the hysteretic
case. The dynamic stiffness Kd at point D is defined as the
ratio between the force and displacement assuming harmonic
motion, i.e., uðtÞ ¼ Ueixt and yðtÞ ¼ Yeixt. Hence, Kd¼F(t)/u(t) and after substituting the expression of F(t) and u(t), the
dynamic stiffness of the viscous and hysteretic damping
cases reduces to the following forms, respectively:
Kdv ¼ðk þ icxÞðU � YÞ
U; Kdh
¼ kð1þ idÞðU � YÞU
:
(A1)
The motions u(t) and y(t) are related through the equation of
motion of the neutralizer. Assuming a prescribed harmonic
support motion uðtÞ ¼ Ueixt and solving for y(t) yields the
following solution for the viscous and hysteretic damping
cases, respectively:
Y ¼ k þ icxk þ icx� mx2
U; Y ¼ kð1þ idÞkð1þ idÞ � mx2
U:
(A2)
Finally, substituting the expressions of Y in Eq. (A2) into Eq.
(A1) yields, after simplifications, the below dynamic stiff-
ness of the viscously and hysteretically damped neutralizer,
respectively
Kdv ¼1
1
k þ icx� 1
mx2
; Kdh¼ 1
1
kð1þ idÞ �1
mx2
:
(A3)
APPENDIX B
In this appendix, the tuning neutralizer parameters are
obtained analytically for both viscous and hysteretic damp-
ing with the aim of maximizing energy reflection at some
tuning frequency. For a given mass ratio of the system l,
tuning the system to r0 means forcing the transmission ratio
curve to pass through a minimum value at r¼ r0 and to have
a minimum transmission ratio h0 at this point. This can be
achieved if the stiffness and damping ratios are chosen such
that the below two equations are satisfied
@T
@r
����r¼r0
¼ 0; T
����r¼r0
¼ h0: (B1)
In order to simplify these equations, the transmissibility ratio
T is first written as T¼N/D since it is the ratio of two poly-
nomial equations as given in Eq. (9). Now, the equations in
Eq. (B1) are rewritten as
N00D0 � N0D00D2
0
¼ 0;N0
D0
¼ h0: (B2)
Here, prime denotes the derivative of the function with
respect to r and the subscript zero means that the correspond-
ing function is evaluated at r¼ r0. The equations in Eq. (B2)
are further simplified to the following forms:
N00D00¼ h0;
N0
D0
¼ h0: (B3)
And finally they are written as follows:
N00 � h0D00 ¼ 0; N0 � h0D0 ¼ 0: (B4)
At this stage, two cases are considered, namely, viscous
and hysteretic damping. The tuning parameters, i.e., the
stiffness and damping parameters, are obtained in closed
form for each case by solving the system of equations in
Eq. (B4).
1. Viscous damping
The expressions for N0 and D0 are obtained from Eq. (9)
as follows:
N0 ¼ ðf þ fr0l� r40lÞ
2 þ r40f
2ð1þ r0lÞ2;D0 ¼ ðf þ fr0l� r3
0fl� r40lÞ
2 þ r20ðr0fþ flþ r2
0flÞ2:
(B5)
The two equations in Eq. (B4) are rewritten as
a0 þ a1 f þ a2 f 2 þ a3 fþ a4 f2 ¼ 0;
b0 þ b1 f þ b2 f 2 þ b3 fþ b4 f2 ¼ 0: (B6)
Where
a0 ¼ ð1� h0Þr80l
2 ;
a1 ¼ 2ðh0� 1Þr40lð1þ r0lÞ ;
a2 ¼ ð1þ r0lÞ2� h0ð1þ 2r0lð1þ r0lÞÞ ;a3 ¼�2h0r7
0l2 ;
a4 ¼ r40ðð1þ r0lÞ2� h0ð1þ 2r0lð1þ r0lÞÞÞ ;
b0 ¼ 8ð1� h0Þr70l
2 ;
b1 ¼ 2ðh0� 1Þr30lð4þ 5r0lÞ ;
b2 ¼ 2lð1� h0þ ð1� 2h0Þr0lÞ ;b3 ¼�14h0r6
0l2 ;
b4 ¼ 2r30ðð1þ r0lÞð2þ 3r0lÞ � h0ð2þ r0lð5þ 6r0lÞÞÞ:
(B7)
FIG. 12. Schematic of a vibration neutralizer (a) with viscous damping, (b)
with hysteretic damping.
498 J. Acoust. Soc. Am. 146 (1), July 2019 Jimmy S. Issa
Now, the stiffness ratio is written in terms of f using the
equations in Eq. (B6) as follows:
f ¼ a0b2 � a2b0 þ ða3b2 � a2b3Þfþ ða4b2 � a2b4Þf2
a2b1 � a1b2
:
(B8)
The stiffness ratio is eliminated from the first equation of
Eq. (B6) using Eq. (B8). This yields the below fourth order
polynomial in f
a0 þ a1 fþ a2 f2 þ a3 f3 þ a4 f4 ¼ 0: (B9)
Where
a0 ¼ a2ða22b2
0 � a1a2b0b1 þ b2ða21b0 � a0a1b1 þ a2
0b2Þþ a0a2ðb2
1 � 2b0b2ÞÞ ;a1 ¼ a2ða2a3ðb2
1 � 2b0b2Þ þ 2a22b0b3
� a2ða1b1 þ 2a0b2Þb3
þ b2ð�a1a3b1 þ 2a0a3b2 þ a21b3ÞÞ ;
a2 ¼ a2ðb2ð�a1a4b1 þ ða23 þ 2a0a4Þb2 þ a2
1b4Þþ a2
2ðb23 þ 2b0b4Þ þ a2ða4ðb2
1 � 2b0b2Þ;�2a3b2b3 � a1b1b4 � 2a0b2b4ÞÞ ;
a3 ¼ 2a2ða3b2 � a2b3Þða4b2 � a2b4Þ ;a4 ¼ a2ða4b2 � a2b4Þ2: (B10)
The tuned damping ratio is obtained analytically by solving
Eq. (B9). The roots of a fourth order polynomial can be
obtained using Ferrari’s method. The positive root of this
polynomial is the tuning damping ratio, which can be shown
to take the form
ft ¼ �a3
4a4
� Sþ 1
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiq
S� 4S2 � 2p
r: (B11)
Where
S ¼ 1
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1
3a4
Qþ D0
Q
� �� 2
3p
s;
Q ¼D1 þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiD2
1 � 4D30
q2
!1=3
;
q ¼ a33 � 4a4a3a2 þ 8a2
4a1
8a34
;
p ¼ 8a4a2 � 3a23
8a24
;
D1 ¼ 2a32 � 9a3a2a1 þ 27a2
3a0 þ 27a4a21 � 72a4a2a0 ;
D0 ¼ a22 � 3a3a1 þ 12a4a0: (B12)
This concludes this section of the appendix. Once the tuning
damping ratio is calculated from Eq. (B11), the tuning stiff-
ness ratio can be calculated from Eq. (B8).
2. Hysteretic damping
The expressions for N0 and D0 for the hysteretic damp-
ing case are obtained from Eq. (9) as follows:
N0 ¼ f 2d2ð1þ rlÞ2 þ ðr4l� rfl� f Þ2;D0 ¼ f 2ðdþ rlþ rdlÞ2 þ ðr4lþ frdl� frl� f Þ2:
(B13)
The two equations in Eq. (B4) are then rewritten as
a0 þ a1 þ a2dð Þ f þ a3 1þ d2ð Þ f 2 ¼ 0;
b0 þ b1 þ b2dð Þ f þ b3 1þ d2ð Þ f 2 ¼ 0; (B14)
where
a0 ¼ ð1� h0Þr80l
2;
a1 ¼ 2ðh0 � 1Þr40lð1þ r0lÞ;
a2 ¼ �2h0r50l
2;
a3 ¼ ð1þ r0lÞ2 � h0ð1þ 2r0lð1þ r0lÞÞ;b0 ¼ 8ð1� h0Þr7
0l2;
b1 ¼ 2ðh0 � 1Þr30lð4þ 5r0lÞ;
b2 ¼ �10h0r40l
2;
b3 ¼ 2lð1� h0 þ ð1� 2h0Þr0lÞ: (B15)
The stiffness ratio is now written in terms of d using the
equations in Eq. (B14) as follows:
f ¼ a0b3 � a3b0
a3b1 � a1b3 þ a3b2 � a2b3ð Þd : (B16)
Substituting the above expression of f in the first equation of
Eq. (B14) yields the below second order polynomial in d2
a0 þ a1 dþ a2 d2 ¼ 0; (B17)
where
a0 ¼ ða3b1� a1b3Þ2a0þ ða0b3� a3b0Þða3b1� a1b3Þa1
þ ða0b3� a3b0Þ2a3 ;
a1 ¼ 2ða3b1� a1b3Þða3b2� a2b3Þa0þ ða0b3� a3b0Þ� ða3b2� a2b3Þa1þ ða0b3� a3b0Þða3b1� a1b3Þa2 ;
a2 ¼ ða3b2� a2b3Þ2a0þ ða0b3� a3b0Þða3b2� a2b3Þa2
þ ða0b3� a3b0Þ2a3: (B18)
The tuning damping ratio is the positive root of the second
order polynomial of Eq. (B17), which takes the form
dt ¼�a1 þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2
1 � 4a2a0
p2a2
: (B19)
Finally, for a given frequency ratio r0, minimum transmis-
sion ratio h0 and mass ratio l, the tuning damping ratio is
first calculated form Eq. (B19), then the tuning stiffness ratio
is calculated from Eq. (B16). It is important to note that sim-
ilar constant names are used in both Appendixes. These
J. Acoust. Soc. Am. 146 (1), July 2019 Jimmy S. Issa 499
constants are defined locally in each section, and these defi-
nitions hold in the sections where they are defined only.
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