Amplification in the Response of LRB Isolated
Structures due to Bidirectional Excitations due
to Lead Core Heating
Gokhan Ozdemir1 & Ugurhan Akyuz2
ABSTRACT
This study investigates the response of lead rubber bearings (LRBs) under bidirectional earthquake excitations when
lead core heating effect is of concern. For this purpose, series of nonlinear response history analyses were conducted
with a bilinear force-deformation relation for LRBs. In the considered bilinear representation, the strength of LRBs
deteriorates due to lead core heating under cyclic motions. Response of LRBs was studied in terms of maximum isolator
displacements (MIDs) and maximum lead core temperature as a function of isolator characteristics (characteristic
strength to weight ratio, Q/W, and post-yield isolation period, T). Nonlinear response history analyses were performed
using well known near field ground motion records. To quantify the interacted effects of coupled analysis and lead core
heating on MID, unidirectional analyses were also performed. The results demonstrate that the amount of amplifications
in MIDs and lead core temperatures of LRBs in bidirectional analyses are approximately 30% and 50% higher than that
of unidirectional ones, respectively. Furthermore, amount of amplifications increase with increasing Q/W ratio.
1
2
Department of Civil Engineering, Kocaeli University, Izmit, Turkey
Department of Civil Engineering, Middle East Technical University, Ankara, Turkey
Amplification in the Response of LRB Isolated Structures under
Bidirectional Excitations due to Lead Core Heating
Gokhan Ozdemir1,a and Ugurhan Akyuz2,b
1
Department of Civil Engineering, Kocaeli University, Turkey
2
Department of Civil Engineering, Middle East Technical University, Turkey
a
gokas3050@yahoo.com, bhan@metu.edu.tr
Keywords: Lead rubber bearing, seismic isolation, lead core heating, bidirectional excitation.
Abstract. This paper investigates the response of lead rubber bearings (LRBs) under bidirectional
earthquake excitations when lead core heating effect is of concern. For this purpose, series of
nonlinear response history analyses were conducted with a bilinear force-deformation relation for
LRBs. In the considered bilinear representation, the strength of LRBs deteriorates due to lead core
heating under cyclic motions. Response of LRBs was studied in terms of maximum isolator
displacements (MIDs) and maximum lead core temperature. Nonlinear response history analyses
were performed using well known near field ground motion records. To quantify the interacted
effects of coupled analysis and lead core heating on MID, unidirectional analyses were also
performed. The results demonstrate that the amount of amplifications in MIDs and lead core
temperatures of LRBs in bidirectional analyses are approximately 20% and 50% higher than that of
unidirectional ones, respectively. Furthermore, amount of amplifications increase with increasing
Q/W ratio.
Introduction
The superior performance of seismic isolated structures against adverse effects of ground
motions are noted during severe earthquake excitations. Specifically, structures isolated with lead
rubber bearings (LRBs), the most commonly used isolators among the various types of isolators,
performed very well during the 1994 Northridge and 1995 Kobe earthquakes [1,2]. Analytical
representations of LRBs have also been used in several studies to understand the dynamic response
of isolated structures [3-6]. Although these earlier studies provided invaluable experience and
knowledge about the response of LRB isolated structures, none of these studies considered a
deteriorating force-deformation relation to idealize the bilinear hysteretic behavior of LRBs. Instead,
a generic steady-state, non-deteriorating force-deformation relation was used to model the hysteretic
behavior of LRBs. However, experiments conducted with LRBs showed that the strength of LRBs
reduces under cyclic motion resulting in a deteriorating force-deformation relation [7]. The variation
in strength of LRBs is tried to be considered by employing bounding analyses where upper and
lower bound properties are used to construct the corresponding non-deteriorating hysteretic
representations. Such modeling approach provides an envelope for the response quantities of
seismic isolated structures and fulfills its intention to estimate the response of isolated structures
with some overestimation. On the other hand, results obtained from bounding analyses may not be
realistic in most of the time. Because, employed non-deteriorating hysteretic behavior of LRBs is
not realistic. To overcome such modeling difficulty, Kalpakidis [8] focused on the reason of the
reduction in strength of an LRB under cyclic motion. As a result, it is claimed that the main reason
for deterioration in hysteretic force-deformation relation of an LRB is the rise in lead core
temperature due to cyclic motion [9]. Authors also proposed a mathematical model that is capable
of simulating the reduction in strength of LRB instantly as a function of instantaneous lead core
temperature [9]. In a complimentary study, the model proposed by Kalpakidis and Constantinou [9]
was tested and verified by comparing the test results with that of the analytical ones [10].
In this study, the material model proposed by Kalpakidis and Constantinou [9] is used to quantify
the amplification in maximum isolator displacements (MIDs) when the isolated structure is
subjected to bidirectional excitations rather than unidirectional ones. Hence, a series of nonlinear
response history analyses are performed under both bidirectional and unidirectional excitations.
Analyses are conducted by the structural analysis program OpenSees. Results obtained from
nonlinear analyses are discussed in terms of both lead core temperature and MIDs. A further set of
analysis is also conducted to observe the effect of characteristic strength to weight (Q/W) ratio on
the amplification of considered response quantities when lead core heating is of concern.
Modeling of the Superstructure
The investigated building is adopted from the isolated structure given in Chapter 11 of NEHRP
Recommended Provisions: Design Examples [11]. The original building has 3 stories with a
penthouse at the top and consists of steel frame with concentric braces. In order to focus solely on
the isolator response, the superstructure is considered to be as simple as possible. In this sense, the
penthouse and braces are removed from the original structure and the remaining 3-story steel frame
is used in the analyses. Fig. 1 depicts the 3-D model of the considered LRB isolated structure.
Figure 1 Idealized 3-D model of the isolated structure.
The analyzed structure is symmetric in plan with the dimensions of 36mx54m. All story heights
are identical and equal to 3m. Span lengths in the plan are the same and equal to 9m. Total weight
of the superstructure acting on the isolators is 73000 kN. It is assumed that the weights of the
isolation level, first story, and second story are equal while the weight at the roof level is 75% of the
others. The corresponding floor masses are equally distributed to joints with rigid diaphragm
assumption at each story level. The considered superstructure has 35 columns at each floor and
LRBs were implemented under each column at the isolation level.
Modeling of the Lead Core Heating Effects
The material model proposed by Kalpakidis and Constantinou [9] enables to represent the
reduction in strength of LRBs subjected to cyclic motion. The model considers the heating of lead
core as the main reason for reduction in strength. Thus, variation in strength of LRBs is modeled as
a function of the instantaneous lead core temperature. The instantaneous lead core temperature is
used to update the yield stress of lead that is used to calculate the strength of the bearing. According
to model proposed in Reference [9], calculation of the instant strength of an LRB is performed by
the following set of equations:
T&L =
σ YL ( TL ) ⋅ Z x2 + Z y2 U& x2 + U& y2
ρ L ⋅ c L ⋅ hL
k s ⋅ TL
r ⋅ ρ L ⋅ c L ⋅ hL
−
⎛1
⎛t ⎞
⋅ ⎜⎜ + 1.274 ⋅ ⎜ s ⎟ ⋅ t +
⎝r⎠
⎝F
1/ 2
2
3
⎧
⎛t+ ⎞
t + ⎡ ⎛ t + ⎞ ⎛ t + ⎞ 15 ⎛ t + ⎞ ⎤
⎪
⎢
⎜
⎟
⎜
⎟
⎜
⎟
⎜
⎟
2 ⋅ ⎜ ⎟ − ⋅ 2 − ⎜ ⎟ − ⎜ ⎟ − ⋅ ⎜ ⎟ ⎥,
⎪⎪
4 ⎝ 4⎠ ⎥
π ⎢ ⎝4⎠ ⎝4⎠
⎝π ⎠
⎣
⎦
F =⎨
⎡
⎪ 8
1
1
1
1
⎢1 −
−
⋅
+
−
⎪
1/ 2
2
+
+
+
⎢⎣ 3 ⋅ 4 ⋅ t
6⋅ 4⋅t
12 ⋅ 4 ⋅ t +
⎪⎩ 3 ⋅ π 2 ⋅ π ⋅ t
(
t+ =
(
)
)
(
)
αs ⋅t
−1 / 3 ⎞
( )
(
⎫
t < 0.6⎪
⎪⎪
⎬
⎤
⎪
+
⎥
,
t
≥
0
.
6
⎪
3
⎥⎦
⎪⎭
⎟⎟
⎠
(1)
+
(2)
)
(3)
r2
σ YL (TL ) = σ YL 0 ⋅ exp(− E 2 ⋅ TL )
(4)
In the above equations, hL, r, ρL (11200 kg/m3), cL (130 J/kgoC) and σYL0 are the height, radius,
density, specific heat and yield stress at the reference (initial) temperature of the lead core,
respectively. ts is the total shim plate thickness, αs (1.41x10-5 m2/s) is the thermal diffusivity of
steel, ks (50 W/moC) is the thermal conductivity of steel, t+ is the dimensionless time, t is the time
since the beginning of the motion, and E2 (0.0069/oC) is a constant that relates the temperature and
yield stress.
Coupled Plasticity Model
The bidirectional bilinear hysteretic model developed by Park et al. [12] was used to model the
LRBs. According to the model developed by Park et al. [12], when isolators behave nonlinearly in
both of the horizontal directions, forces assembled with due account for bidirectional interaction
effects are computed as follows:
⎧ Fx ⎫ = K ⋅ ⎧U x ⎫ + (σ (T )A ) ⋅ ⎧ Z x ⎫
⎨F ⎬
⎨U ⎬
⎨Z ⎬
YL L
L
⎩ y⎭
⎩ y⎭
⎩ y⎭
(5)
⎧ . ⎫
⎧ . ⎫
⎪Z x ⎪
⎪U ⎪
Y ⋅ ⎨ . ⎬ = ( A ⋅ [I ] − B ⋅ [Ω]) ⋅ ⎨ . x ⎬
⎪⎩Z y ⎪⎭
⎪⎩U y ⎪⎭
(6)
[
]
[
]
⎧ Z x2 ⋅ sgn(U& x Z x ) + 1
[Ω] = ⎪⎨
⎪Z x Z y sgn(U& x Z x ) + 1
⎩
[ (
) ]
Z x Z y sgn U& y Z y + 1 ⎫
⎪
⎬
Z y2 sgn U& y Z y + 1 ⎪⎭
[ (
) ]
(7)
where Fx and Fy are the isolator forces and Ux and Uy are the displacements of the isolators in x and
y directions, respectively. Y and K are the yield displacement and post-yield stiffness of the bilinear
force-deformation relation of isolators, respectively. AL is the cross-sectional area of the lead core.
In Eq. (5), σYL(TL) stands for the instantaneous yield stress of the lead based on the instantaneous
lead core temperature, TL, and it is calculated through Eqs. (1)-(4).
Solution of Eqs. (6) and (7) provides a circular interaction surface for the forces Fx and Fy. Here,
Zx and Zy are hysteretic dimensionless quantities that account for the direction and the interaction of
hysteretic forces and vary between +1 and -1. In Eq. (6), A and B values should satisfy the relation
of A = 2B [13]. This assumption is essential because it assures that the force and displacement
vectors are in the same direction. In the above equations, [I] is the unit matrix, sgn stands for the
signum function and overdot means differentiation with respect to time.
Selection of Ground Motions
Ten ground motion records were selected from well known and extensively studied seismic
events. Magnitude Mw of the records are in between 6.7 and 7.6 and closest distance R of the records
to fault rupture is less than 20 km. The average shear wave velocities of the ground motions at the
upper most 30 m soil deposit are in the range of 360 m/s and 750m/s. Properties of the records used
in this study are given in Table 1 including peak ground acceleration (PGA), peak ground velocity
(PGV), and peak ground displacement (PGD) values.
Table 1 Characteristics of the selected ground motions.
#
Earthquake
Station
Mw
R
(km)
1
Cape Mendocino
Fortuna Blvd.
7.0
20.0
2
Gazli
Karakyr
6.8
5.5
3
Kocaeli
Gebze
7.5
10.9
4
Kocaeli
Izmit
7.5
7.2
5
Kobe
KJM
6.9
1.0
6
Loma Prieta
Saratoga Aloha Ave
6.9
8.5
7
Northridge
Newhall W Pico
Canyon
6.7
5.5
8
ChiChi
TCU057
7.6
11.8
9
ChiChi
TCU087
7.6
7.0
10
Tabas
Tabas
7.4
2.1
Component
0
90
0
90
0
270
180
90
0
90
0
90
46
316
N
W
N
W
LN
TR
PGA
(g)
0.12
0.11
0.61
0.72
0.24
0.14
0.15
0.22
0.82
0.60
0.51
0.32
0.46
0.33
0.09
0.12
0.12
0.13
0.84
0.85
PGV
(cm/s)
30.0
21.7
65.4
71.6
50.3
29.7
22.6
29.8
81.3
74.4
41.2
42.6
92.8
67.4
58.8
42.6
37.1
40.8
97.8
121.4
PGD
(cm)
27.6
12.8
25.3
23.7
42.7
27.5
9.8
17.1
17.7
20
16.2
27.5
56.6
16.1
56.2
56.7
25.5
62.6
36.9
94.6
Scaling of Ground Motions
Scaling of the selected records were performed in two complimentary steps. The procedure
followed in the first step was also utilized in Ozdemir and Constantinou [14] and seeks to minimize
a sum (ε) of the weighted squared errors between the geometric mean of the two horizontal
components and the target spectral values at a set of periods. Error ε is defined as:
n
(8)
ε = ∑ bi a ⋅ yi − yTi 2
i =1
(
)
where bi is the weighting factor for the squared error at period Ti; a is the scaling factor for the pair
of ground motions of interest; yi is the geometric mean of the spectral ordinates for the pair at period
Ti; yTi is the target spectral ordinate at period Ti; and n is the number of target spectral values
considered. The scaling factor (a) that results in the minimum value of ε is calculated by setting the
derivative of Eq. (8) equal to zero as given in Eq. (9).
n
a=
∑ bi ⋅ yi ⋅ yTi
i =1
n
∑ bi ⋅
i =1
(9)
y i2
This scaling was based on five target periods (Ti): 1, 2, 3, 4, and 5 sec. The weightings of factors
were determined such that the scaled spectra have the most compatible shape with that of the target
spectrum under consideration. In the second step of scaling, records were further scaled so that the
average of square-root-of-sum-of-squares (SRSS) spectra from all ground motion pairs does not fall
below 1.3 times the corresponding ordinate of the target response spectrum by more than 10%
within the periods bounded by 0.5TD and 1.25TM in accordance with ASCE-7 [15]. Target response
spectrum considered in this study was taken from the Turkish Earthquake Code [16] for the
corresponding soil class and presented in Fig. 2 together with scaled average SRSS of the
considered records. Table 2 presents the applied scale factors for the selected ground motion
records.
Acceleration Spectra (g)
2.5
scaled mean SRSS
0.9x1.3xdesign spectrum
design spectrum
2.0
1.5
1.0
0.5
0.0
0.0
1.0
2.0
Period (sec)
3.0
4.0
Figure 2 The scaled average SRSS spectra of the selected ground motions and corresponding 5%
damped design spectrum.
Table 2 Scale factors applied to the selected ground motions.
Earthquake #
1
2
3
4
5
6
7
8
9
10
Scale
2.95 1.27 2.78 2.46 0.65 1.88 0.88 2.39 2.40 0.85
Factor
Comparison of Unidirectional and Bidirectional Responses of LRBs
In this section, the variation in the response of an LRB (represented by deteriorating forcedeformation relation) subjected to both unidirectional and bidirectional ground motion excitations is
illustrated based on solutions with uncoupled (square surface) and coupled (circular interacted
surface) plasticity models, respectively. For this purpose, an LRB with characteristic strength (force
intercept at zero displacement in a bilinear force-deformation relation) to weight ratio, Q/W, of
0.075 and post-yield period of 2.50s (height of the bearing, hL, is 230 mm, radius of the lead core, r,
is 77.5 mm, total shim plate thickness, ts, is 66 mm, diameter of the bearing is 750 mm) was
subjected to orthogonal horizontal components of ChiChi, TCU057 record with a scale factor of
2.39 (see Table 2). To identify the effect of coupling explicitly, three analyses were conducted: i) X
component only, ii) Y component only, iii) X and Y components simultaneously. The variation in
300
300
150
150
Dy (mm)
Dx (mm)
the response of systems under both coupled and uncoupled analyses was studied in the literature
extensively [17, 19]. Previous studies showed that the coupled analyses result in reduced effective
stiffness and reduced effective damping compared to uncoupled analyses. Such reduction in
effective stiffness and damping leads to amplification in isolator displacements. However, the
amount of amplification highly depends on the magnitude of the force acting on the isolator and
phasing of the orthogonal ground motion components. Since the phase of the orthogonal horizontal
components of ground motions is random in nature, the amount of amplification in displacements is
also random [17,19]. The corresponding discussion for the considered isolator characteristics is
presented in Fig. 3. The comparison of displacement histories in X and Y directions is depicted in
Figs. 3(a) and (b) for coupled and uncoupled cases. Besides, the circular displacement and force
surfaces obtained from coupled (bidirectional) analysis are given in Figs. 3(c) and (d), respectively
where square displacement and force surfaces obtained from the uncoupled (unidirectional) analyses
are also shown and represented by dashed lines.
0
-150
0
-150
uni-dir. (X)
bi-dir.
uni-dir. (Y)
bi-dir.
-300
-300
0
10
20
30
40
50
0
60
10
Time (s)
800
bi-dir.
uni-dir.
50
60
0
0
-200
-400
-400
-400
-800
-800
0
Dx (mm)
200
bi-dir.
uni-dir.
400
Fy (kN)
200
Dy (mm)
40
(b)
400
(c)
30
Time (s)
(a)
-200
20
400
-400
0
400
800
Fx (kN)
(d)
Figure 3 Comparison of unidirectional (X and Y only) and bidirectional (X Y simultaneously)
responses in terms of isolator displacements and isolator forces.
Amplification in Lead Core Temperatures and MIDs due to Bidirectional Excitations
In this section, the amounts of amplification in lead core temperature and MIDs are studied when
the excitations are applied bidirectionally rather than unidirectionally. The investigated isolation
system has a period of 2.5s and Q/W ratio of 0.09. Height of the bearing, hL, is 230 mm, radius of
the lead core, r, is 77.5 mm, total shim plate thickness, ts, is 66 mm, diameter of the bearing is 750
mm. The corresponding initial yield strength of the considered LRB is 288.6 kN and reduces in
accordance with Eqs. (1)-(4) when the motion in the bearing initiates.
The seismic isolated structure under investigation is first subjected to both horizontal orthogonal
components of the selected and scaled records, individually. Then, these records are applied
simultaneously for bidirectional analyses. To quantify the amplifications in lead core temperatures
150
150
120
120
Temp. Rise (oC)
Temp. Rise (oC)
and MIDs, response quantities obtained from bidirectional analyses are normalized with the ones
obtained from the maximum of the unidirectional analyses (Xbi/Xuni).
The first investigated response quantity is the lead core temperature and its variation with
analysis time is given in Fig. 4 for both unidirectional and bidirectional analyses. Fig. 4.a reveals
that there is a significant increase in the lead core temperatures compared to Figs. 4.b and 4.c.
90
60
30
90
60
30
0
0
0
10
20
30
40
50
0
10
20
Time (sec)
30
40
50
Time (sec)
(a) bidirectional condition
(b) first unidirectional condition
Temp. Rise (oC)
150
120
90
60
30
0
0
10
20
30
40
50
Time (sec)
(c) second unidirectional condition
Figure 4 Variation of lead core temperatures with analysis time for both unidirectional and
bidirectional cases.
The maximum amount of rise in the lead core temperatures are given in Table 3 for all of the
considered ground motions listed in Table 1. The averages of the maximum lead core temperatures
are 78.1 oC, 50.3 oC, and 40 oC for bidirectional (Temp-bi), first unidirectional (Temp-uni1) and
second unidirectional (Temp-uni2) analyses. When the observed average temperature rise in
bidirectional analyses is divided by the max(Temp-uni1,Temp-uni2), it is found that the
amplification in lead core temperature is more than 50% due to coupling in bidirectional analyses.
Similar comparison for MIDs is given in Fig. 5. For the considered ground motions and LRB, the
amplification in MIDs due to bidirectional analysis varies in between 8% and 47% with an average
value of 22%.
Table 3 Maximum lead core temperatures for both unidirectional and bidirectional analyses.
Earthquake #
1
2
3
4
5
6
7
8
9
10
Temp-bi 80.6 76.5 53.3 67.0 46.4 78.0 64.5 112.5 108.9 93.5
Temp-uni1 54.0 50.8 32.8 39.2 33.9 56.1 49.8 68.6 64.9 58.8
Temp-uni2 32.9 50.3 25.3 38.3 18.8 32.1 27.6 56.5 59.1 52.8
MIDbi / MIDuni
1.6
1.4
1.2
1.0
0.8
0
2
4
6
8
Earthquake #
10
12
Figure 5 Variation in MIDbi/MIDuni ratios for all of the considered ground motions.
Effect of Q/W Ratio on Amplifications
The lead core diameter of an LRB is directly related to characteristic strength, Q, of that LRB. In
the design of an LRB, the characteristic strength, Q, is normalized by the weight, W, acting on the
LRB in order to work with a unitless and bearing independent parameter. Physically, increase in
Q/W ratio represents the increase in the lead core diameter. In this section, the effect of Q/W ratio
on the amplification in lead core temperature and MIDs are studied. For this purpose, distinct Q/W
ratios are considered namely, 0.045, 0.060, 0.075 and 0.090. The isolation period is kept constant at
2.50s to identify the effect of Q/W ratio, explicitly. Thus, the only variation in LRBs is the change
in diameter of the lead core. The considered LRBs have lead core diameters of 110 mm, 126 mm,
142 mm and 155 mm for Q/W ratios of 0.045, 0.060, 0.075 and 0.090, respectively. The hL, and ts
are the same for all LRBs and equal to 230 mm and 66 mm, respectively. Fig. 6 presents the
obtained lead core temperatures from bidirectional analyses against the corresponding Q/W ratios. It
is clear that there is a tendency to decrease for lead core temperature with increasing Q/W ratio.
To identify the corresponding amplification in lead core temperature, due to bidirectional
analyses, as a function of Q/W ratio, Fig. 7 is depicted. In Fig. 7, maximum lead core temperature
obtained from bidirectional analysis (Tempbi) over maximum lead core temperature obtained from
unidirectional analyses (Tempuni) ratios are given. Tempbi/Tempuni ratios vary in a wide range for all
of the considered Q/W ratios. In Fig. 7, the black solid line stands for the averages of the maximum
lead core temperature at each Q/W ratio. Although there is a slight increase in the average line, 50%
amplification in temperature is almost valid for all of the conditions.
160
o
Temp. Rise ( C)
200
120
80
40
0
0.030
0.045
0.060
0.075
0.090
0.105
Q/W ratio
Figure 6 Lead core temperatures versus Q/W ratios for bidirectional analyses.
Temp bi / Temp uni
1.8
1.6
1.4
1.2
1.0
0.030
0.045
0.060
0.075
0.090
0.105
Q/W ratio
Figure 7 Variation in Tempbi/Tempuni ratios as a function of Q/W ratio.
Results regarding the amplification in MIDs due to bidirectional excitations, when lead core
heating effect is considered, are discussed by means of Fig. 8. MIDbi/MIDuni ratios are depicted in
Fig. 8 as a function of the Q/W ratios. The black solid line in Fig. 8 represents the averages of
MIDbi/MIDuni ratios for all of the Q/W ratios. It is evident that the MIDbi/MIDuni has a tendency to
increase with increasing Q/W ratio, in average. The average values for MIDbi/MIDuni ratios are 1.14,
1.17, 1.20, and 1.22 for Q/W ratios of 0.045, 0.060, 0.075, and 0.090, respectively. MIDbi/MIDuni
has
MIDbi / MIDuni
1.6
1.4
1.2
1.0
0.8
0.030
0.045
0.060
0.075
0.090
0.105
Q/W ratio
Figure 8 Variation in MIDbi/MIDuni ratios as a function of Q/W ratio.
Conclusions
In the present study, the amplification in response quantities of an LRB isolated structure is
studied in terms of lead core temperatures and maximum isolator displacements when the
earthquake excitations are applied bidirectionally rather than unidirectionally. The considered
representative LRB isolated structure is analyzed under both bidirectional and unidirectional
excitations of recorded earthquake motions. Considered ground motions are selected and scaled to
match a target spectrum. The hysteretic force-deformation relations of the LRBs are modeled by a
recently proposed material behavior that enables to represent the deterioration in strength of LRBs
gradually as a function of the lead core temperature. The effect of Q/W ratio on the amplification of
the considered response quantities is also studied.
Results of the nonlinear response history analyses revealed that lead core temperatures obtained
from bidirectional analyses are approximately 50% more than those of the unidirectional ones. The
corresponding amplification in isolator displacements is slightly higher than 20%, in average. It is
also found that the amplification in MIDs due to bidirectional excitations increases with increasing
Q/W ratio. On the other hand, there is only a slight amplification, which can be negligible, in lead
core temperatures with increasing Q/W ratio.
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