Nonlinear vibration and dynamic performance analysis of the inerter-based multi-directional vibration isolator

Abstract

Motivated by the demand of improving the multi-directional vibration dynamic performance, an inerter-based multi-directional (IMD) vibration isolator is proposed in this paper, which is composed of the inerter, damper and spring structures in multiple directions. The dynamic equation of the IMD vibration isolator is established using the Lagrange theory, its dynamic response under base harmonic excitation is obtained using the harmonic balance method (HBM) and pseudo arc length (PAL) method, the stability of the dynamic response is considered. The dynamic performance of the IMD vibration isolator under harmonic and shock excitations are studied and compared with those of the conventional multi-directional (MD) vibration isolator consist of the damper and spring structure, the effect of structural parameters on its dynamic performance is investigated in detail. The results show that the IMD vibration isolator has nonlinear inertial, damping and stiffness characteristics, it further reduces the dynamic displacement and absolute displacement transmissibility peaks, widens the isolation frequency band than the MD vibration isolator, also has better shock performance in the middle severity parameter range. In order to obtain better isolation and shock performance, the vertical and horizontal inertance-to-mass ratios are chosen as larger values, the stiffness ratio and the horizontal spring compression ratio are chosen as smaller values. Therefore, the design of the proposed IMD vibration isolator exhibits the advantages of applying the inerter and provides excellent isolation and shock performance in multiple directions.

Full text

Archive of Applied Mechanics
https://doi.org/10.1007/s00419-022-02252-9
ORIGINAL
Yong Wang ·Peili Wang ·Haodong Meng ·Li-Qun Chen
Nonlinear vibration and dynamic performance analysis
of the inerter-based multi-directional vibration isolator
Received: 31 March 2022 / Accepted: 29 August 2022
© The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2022
Abstract Motivated by the demand of improving the multi-directional vibration dynamic performance, an
inerter-based multi-directional (IMD) vibration isolator is proposed in this paper, which is composed of the
inerter, damper and spring structures in multiple directions. The dynamic equation of the IMD vibration isola-
tor is established using the Lagrange theory, its dynamic response under base harmonic excitation is obtained
using the harmonic balance method and pseudo-arc-length method, and the stability of the dynamic response
is considered. The dynamic performance of the IMD vibration isolator under harmonic and shock excitations
is studied and compared with those of the conventional multi-directional (MD) vibration isolator consist of the
damper and spring structure, and the effect of structural parameters on its dynamic performance is investigated
in detail. The results show that the IMD vibration isolator has nonlinear inertial, damping and stiffness char-
acteristics, and it further reduces the dynamic displacement and absolute displacement transmissibility peaks,
widens the isolation frequency band than the MD vibration isolator and also has better shock performance in
the middle severity parameter range. In order to obtain better isolation and shock performance, the vertical
and horizontal inertance-to-mass ratios are chosen as larger values, and the stiffness ratio and the horizontal
spring compression ratio are chosen as smaller values. Therefore, the design of the proposed IMD vibration
isolator exhibits the advantages of applying the inerter and provides excellent isolation and shock performance
in multiple directions.
Keywords Inerter ·Multi-directional vibration isolator ·Dynamic response ·Isolation performance ·Shock
performance
1 Introduction
Vibration exists widely in practical engineering, and it could lead to fatigue damage of the engineering struc-
tures, deteriorate the dynamic performance and shorten their service time. Therefore, the vibration mitigation
and isolation devices are used to reduce the vibration amplitude, which increases the reliability and durability of
the engineering structures [1]. Traditional vibration mitigation and isolation devices focus on one-directional
vibration, while in some practical engineering, the vibration is multi-directional, for instance, seismic and
Y. Wa ng (B
)·P. Wa ng
Automotive Engineering Research Institute, Jiangsu University, Zhenjiang 212013, China
e-mail: wangy1921@126.com
H. Meng
School of Automotive Engineering, Changzhou Institute of Technology, Changzhou 213032, China
L.-Q. Chen (B
)
School of Mechanics and Engineering Science, Shanghai University, Shanghai 200444, China
e-mail: lqchen@shu.edu.cn

Y. Wang et al.
wind vibrations in civil engineering [2]. For multi-directional vibration, the vibration mitigation and isolation
devices should have effectiveness in all directions.
Here, the vibration isolator is considered and some researchers have conducted studies in multi-directional
vibration isolation area. Xu et al. [3] presented a comprehensive assessment of recent developments of multi-
directional (MD) vibration isolator and the material used in these devices. Sun and Jing [4] designed a three-
directional quasi-zero-stiffness (QZS) vibration isolator, and it is composed of two symmetrically scissor-like
structures in the horizontal direction and a spring–mass–damper system in the vertical direction, which reduces
the natural frequencies and resonant peak in both directions. Furthermore, Xu and Sun [5] exploit the potential
benefits of time-delayed active control for the three-directional QZS vibration isolator. Wu et al. [6] constructed
a 6 degree-of-freedom (DOF) vibration isolator which combines the X-shaped structure and Stewart platform,
and it has good isolation performance in 6 directions. Zhou et al. [7] devised a 6 DOF vibration isolator using
a cam-roller-beam mechanism, which broadens the vibration isolation bandwidth and has higher effectiveness
in the lower frequency band. Dong et al. [8] constructed a MD vibration isolator with QZS structure and spatial
pendulum, which achieves better low-frequency vibration isolation performance in multiple directions. Lu et al.
[9] proposed an electromagnetic Stewart platform to reduce the vibration in 6 directions and simultaneously
harvest energy. Chai et al. [10] presented a 3DOF X-shaped structure-based vibration isolator, which achieves
low-frequency vibration isolation in 3 directions. Yang and Cao [11] designed a 6DOF micro-vibration isolator
based on the hexagon structure and obtained a broader isolation bandwidth. In these researches, the MD
vibration isolator is based on the spring and damper structure and has nonlinear stiffness or nonlinear damping
characteristic, and the further improvement of the isolation performance is restricted by its inherent structure.
As a two-terminal inertial structure, the inerter possesses the characteristic that the generated force between
its two ends is proportional to its relative acceleration [12]. This proportionality is expressed as inertance and
has the unit of kilogram. The inerter retains the mass amplification affect and supplies a larger inertance
compared with its own mass, and the total inertial of the system would be increased without the need of adding
more mass and thus satisfies the lightweight requirement. The inerter has been used in different areas and
illustrated favorable effects owing to its mechanical property. Wagg [13] presented a review of the mechanical
inerter and analyzed its physical realizations and nonlinear applications. Hu et al. [14], Qin et al. [15]and
Wang et al. [16] studied the dynamic response of the inerter-based suspension, and a better vehicle ride comfort
can be realized than the traditional suspension. Li et al. [17] introduced an inerter-based mechanical passive
suppression device to the design of the landing gear, and better shimmy performance can be obtained as the
aircraft is landing. Zhang et al. [18], Wang and Giaralis [19] and Zhao et al. [20] utilized the inerter in the
building isolation device, which reduces the seismic and wind vibrations. Hu and Chen [21], Barredo et al.
[22], Shi et al. [23] proposed the inerter-based dynamic vibration absorber, which expands the bandwidth
and reduces the vibration amplitude than the classic dynamic vibration absorber. Lewis et al. [24]usedthe
inerter-based suspension to enhance both track wear and passenger comfort in the high-speed train. Dai et al.
[25] designed a nonlinear tuned mass damper inerter to suppress the longitudinal vibration transmission in
propulsion shaft system. Liu et al. [26] constructed linear and geometrically nonlinear inerter-based resonator
in metamaterials and obtained a lower-frequency bandgap.
Some scholars have designed different kinds of inerter-based vibration isolators and acquired some fruitful
results. Hu et al. [27] and Wang et al. [28] constructed different types of inerter-based linear vibration isolators,
studied their dynamic characteristics and found that a better dynamic performance is obtained compared with
the linear ones. Furthermore, Wang et al. [29] put forward a semi-active inerter-based linear vibration isolator
based on the acceleration–velocity switch control strategy to improve the isolation performance. ˇ
Cakmak et al.
[30,31] conducted a dynamic performance and optimization of vibration-induced fatigue in a 2DOF inerter-
based vibration isolator. Dai et al. [32] proposed two kinds of inerter-based piecewise vibration isolators,
which has both bilinear and mass magnification characteristics, and found that the parallel-connected one
has smaller transmissibility peak than the traditional linear and piecewise vibration isolators. Moraes et al.
[33], Wang et al. [34] and Yang et al. [35] designed a nonlinear vibration isolator with lateral inerters, which
has nonlinear inertial characteristic, a wider isolation bandwidth and a smaller transmissibility which can be
acquired. In addition, Wang et al. [36,37] devised three types of inerter-based QZS vibration isolator, which
further reduces the transmissibility and widens the isolation bandwidth than the QZS one. Dong et al. [38]
proposed an inerter-based nonlinear passive joint device and applied it in the coupled systems to suppress the
vibration transmission. Shi et al. [39] arranged an inerter in a diamond-shaped linkage mechanism to construct
an inerter-based nonlinear vibration isolator, which can shift and bend transmissibility peaks and power flow
to the lower frequency band and achieve a wider isolation frequency band.

Nonlinear vibration and dynamic performance analysis
x
y
z
Base
Isolation Object
Support platform
Spring
Damper
Inerter
Fig. 1 Structural diagram of the IMD vibration isolator
Until now, the inerter has been applied in the linear and nonlinear vibration isolation areas, based on the
layout of inerter, damper and spring structure, different kinds of inerter-based linear and nonlinear vibration
isolators have been constructed, and the results indicate that the dynamic performance can be improved as the
inerter is added. In the above researches, the vertical-directional vibration isolation is mainly considered, while
the MD vibration isolation has been rarely studied. Thus, the authors introduce the inerter in the MD vibration
isolation area and propose a novel inerter-based multi-directional (IMD) vibration isolator in this paper. In the
three directions, the IMD vibration isolator is composed of the inerter, damper and spring structure, respectively,
and the three elements are in the parallel-connected. The dynamic performance of the IMD vibration isolator
under base harmonic and shock excitations is studied in detail, and the influence law of the structural parameters
on its isolation performance is elucidated, especially the inerter. The purpose of this paper is to study how
the inerter affects the isolation performance in the multiple directions, and whether the IMD vibration isolator
could have beneficial dynamic performance than the conventional MD vibration isolator consisting of the
damper and spring.
The paper is arranged as follows. In Sect. 2, the IMD vibration isolator is presented and the Lagrange
theory is used to establish its dynamic equation. In Sect. 3, combining the harmonic balance method (HBM)
and the pseudo-arc-length (PAL) method, the dynamic response of the IMD vibration isolator subjected to
base harmonic excitation is acquired and the stability of the dynamic response is investigated; the isolation
performance is analyzed and compared with the MD vibration isolator. The dynamic performance of the IMD
vibration isolator under base shock excitation is investigated in Sect. 4. Section 5summarizes the conclusions.
2 Structure and modeling
2.1 Structural diagram
Figure 1shows the structural diagram of the IMD vibration isolator, which is designed for three-directional
vibration isolation. The isolation object is loaded in the support platform, and the support platform is connected
with the base through the inerter-based isolation structure, which is composed of the inerter, damper and
spring. The structural parameters of the IMD vibration isolator are displayed in Table 1. The displacement of
the isolation object in the three directions is xa,yaand za, respectively, and the displacement of the base in the
three directions is xb,yband zb, respectively. Due to the same structural parameters of the inerter, damper and
spring in the xand zdirections, the dynamic response of the isolation object in the xand zdirections is the
same, so the three-directional vibration isolation can be simplified as the xand ydirections vibration isolation
for brevity. Figure 2shows the plane diagram of the IMD vibration isolator in the xand ydirections. Aware
that the pre-deformation of the horizontal spring is λh, it is pre-extended as λh>0 or pre-compressed as λh
<0.

Y. Wang et al.
Table 1 Structural parameters of the IMD vibration isolator
Symbol Structural parameters
mMass of isolation object
kvStiffness of the vertical spring
cvDamping coefficient of the vertical damper
bvInertance of the vertical inerter
lv0Original length of the vertical spring
khStiffness of the horizontal spring
chDamping coefficient of the horizontal damper
bhInertance of the horizontal inerter
lh0Original length of the horizontal spring
λhPre-deformation of the horizontal spring
a
x
v
c
h
k
h
b
h
c
0h
l
h
λ
m
v
k
0v
l
v
b
a
y
b
y
b
x
Support
Platform
Fig. 2 Plane diagram of the IMD vibration isolator in the xand yplane
r
x
r
y
Fig. 3 Deformation of the vertical and horizontal springs for the IMD vibration isolator
2.2 Dynamic modeling
The dynamic equation of the IMD vibration isolator is established using the Lagrange theory, and the kinetic
energy of the IMD vibration isolator in the xand ydirections is
T1
2m˙x2
a+1
2m˙y2
a.(1)
Denote the relative displacement of the isolation object and base in the xand ydirections as
xrxa−xbyrya−yb,(2)
The deformation of the vertical and horizontal springs for the IMD vibration isolator is shown in Fig. 3,
the length of the left and right horizontal spring changes from lh0 to lhl and lhr, respectively, and the length of
the vertical spring changes from lv0to lv. The lengths lhl,lhr and lvare expressed as
lhl (lh0 +λh+xr)2+y2
rlhr (lh0+λh−xr)2+y2
rlv(lv0 +yr)2+x2
r.(3)

Nonlinear vibration and dynamic performance analysis
Thus, the potential energy of the IMD vibration isolator in the xand ydirections is
V1
2kh(lhl −lh0)2+1
2kh(lhr −lh0)2+1
2kv(lv−lv0)2
1
2kh(lh0 +λh+xr)2+y2
r−lh02
+1
2kh(lh0 +λh−xr)2+y2
r−lh02
+1
2kv(lv0+yr)2+x2
r−lv02
,(4)
which includes the elastic potential energy of the horizontal and vertical springs.
The absolute displacements of the isolation object in the xand ydirections are chosen as the generalized
coordinates, based on the Lagrange theory, and the Lagrange equations of the IMD vibration isolator in the
two directions are given as
d
dt ∂T
∂˙xa−∂T
∂xa
+∂V
∂xa−cv
dlv
dt ·dlv
dxa−chdlhl
dt ·dlhl
dxa−chdlhr
dt ·dlhr
dxa
−bv
d2lv
dt2·dlv
dxa−bh
d2lhl
dt2·dlhl
dxa−bhd2lhr
dt2·dlhr
dxa
,(5)
d
dt ∂T
∂˙ya−∂T
∂ya
+∂V
∂ya−cv
dlv
dt ·dlv
dya−ch
dlhl
dt ·dlhl
dya−chdlhr
dt ·dlhr
dya
−bv
d2lv
dt2·dlv
dya−bhd2lhl
dt2·dlhl
dya−bhd2lhr
dt2·dlhr
dya
.(6)
A detailed derivation process of the dynamic equation for the IMD vibration isolator in the xdirection is
provided, the partial derivative of kinetic energy Twith respect to the displacement xaand velocity ˙xais given
as
∂T
∂xa0∂T
∂˙xam˙xam(˙xr+˙xb),(7)
which yields
d
dt ∂T
∂˙xam(¨xr+¨xb),(8)
and the partial derivative of potential energy Vwith respect to the displacement xais given by
∂V
∂xa∂V
∂xr·∂x
∂xa∂V
∂xrkvxr⎡
⎣1−lv0
(lv0+yr)2+x2
r
⎤
⎦
+kh(lh0+λh+xr)⎡
⎣1−lh0
(lh0 +λh+xr)2+y2
r
⎤
⎦
−kh(lh0 +λh−xr)⎡
⎣1−lh0
(lh0+λh−xr)2+y2
r
⎤
⎦,(9)
Equation (9) can be approximated by using the Taylor series expansion, which yields
∂V
∂xrkvxryr
lv0+kvx3
r
2l2
v0−kvxry3
r
l2
v0
+2khxr−2khlh0 xry2
r
(lh0 +λh)3,(10)
using Eq. (3), the generalized force provided by the damper is obtained as
cv
dlv
dt ·dlv
dxa
+chdlhl
dt ·dlhl
dxa
+chdlhr
dt ·dlhr
dxacv
(lv0+yr)xr˙yr+x2
r˙xr
(lv0+yr)2+x2
r

Y. Wang et al.
+ch(lh0 +λh+xr)2˙xr+(lh0 +λh+xr)yr˙yr
(lh0 +λh+xr)2+y2
r
+ch(lh0 +λh−xr)2˙xr−(lh0 +λh−xr)yr˙yr
(lh0 +λh−xr)2+y2
r
,(11)
and using the Taylor series expansion, Eq. (11) can be approximated as
cv
dlv
dt ·dlv
dxa
+chdlhl
dt ·dlhl
dxa
+chdlhr
dt ·dlhr
dxacvx2
r˙xr
l2
v0
+xr˙yr
lv0−xryr˙yr
l2
v0
+2ch1−y2
r
(lh0 +λh)2˙xr−2chxryr˙yr
(lh0 +λh)2,(12)
The generalized force provided by the inerter is given as
bv
d2lv
dt2·dlv
dxa
+bhd2lhl
dt2·dlhl
dxa
+bhd2lhr
dt2·dlhr
dxabv
(lv0+yr)2xr¨yr+x2
r¨xr
(lv0+yr)2+x2
r
+bv
xr[(lv0+yr)˙xr−xr˙yr]2
(lv0+yr)2+x2
r2
+bh(lh0 +λh+xr)2¨xr+(lh0 +λh+xr)yr¨yr
(lh0 +λh+xr)2+y2
r
+bh(lh0 +λh+xr)[(lh0 +λh+xr)˙yr−˙xryr]2
(lh0 +λh+xr)2+y2
r2
+bh(lh0 +λh−xr)2¨xr−(lh0 +λh−xr)yr¨yr
(lh0 +λh−xr)2+y2
r
−bh
(lh0 +λh−xr)[(lh0+λh−xr)˙yr+˙xryr]2
(lh0 +λh−xr)2+y2
r2,(13)
Equation (13) can be approximated by using the Taylor series expansion, which leads to
bv
d2lv
dt2·dlv
dxa
+bhd2ll
dt2·dll
dxa
+bhd2lr
dt2·dlr
dxabvxr
lv0−xryr
l2
v0¨yr+bv
x2
r
l2
v0¨xr
+bv
xr˙x2
r
l2
v0
+2bh1−y2
r
(lh0 +λh)2¨xr
−2bhxryr¨yr
(lh0 +λh)2−2bhxr˙y2
r
(lh0 +λh)2−4bhyr˙xr˙yr
(lh0 +λh)2,(14)
Combining Eqs. (7), (8), (10), (12)and(14) leads to the dynamic equation of the IMD vibration isolator
in the xdirection, which is given by
m¨xr+bvxr
lv0−xryr
l2
v0¨yr+bv
x2
r
l2
v0¨xr+bv
xr˙x2
r
l2
v0
+2bh1−y2
r
(lh0 +λh)2...
xr−2bhxryr¨yr
(lh0 +λh)2−2bhxr˙y2
r
(lh0 +λh)2−4bhyr˙xr˙yr
(lh0 +λh)2
+cvx2
r˙xr
l2
v0
+xr˙yr
lv0−xryr˙yr
l2
v0+2ch1−y2
r
(lh0 +λh)2˙xr−2chxryr˙yr
(lh0 +λh)2
+kvxryr
lv0+kvx3
r
2l2
v0−kvxry2
r
l2
v0
+2khxr−2khlh0 xry2
r
(lh0 +λh)3−m¨xb.
(15)

Nonlinear vibration and dynamic performance analysis
The dynamic equation of the IMD vibration isolator in the ydirection can be obtained following the same
derivation process, which yields
m¨yr+bv1−x2
r
l2
v0¨yr+bvxr
lv0−xryr
l2
v0¨xr+bv1
lv0−yr
l2
v0˙x2
r
−2bvxr˙xr˙yr
l2
v0−2bhxryr¨xr
(lh0 +λh)2+2bhy2
r¨yr
(lh0 +λh)2+2bhyr˙y2
r
(lh0 +λh)2
+cv˙yr−x2
r˙yr
l2
v0
+xr˙xr
lv0−xryr˙xr
l2
v0+2chy2
r˙yr
(lh0 +λh)2−2chxryr˙xr
(lh0 +λh)2
+kv+2kh−2khlh0
lh0 +λhyr+kvx2
r
2lv0
−kv
l2
v0
+2khlh0
(lh0 +λh)3x2
ryr+khlh0 y3
r
(lh0 +λh)3−m¨yb.
(16)
3 Base harmonic excitation
3.1 Dynamic equation
Firstly, the base harmonic excitation is considered, the excitation in the xand ydirections is expressed as
xbmcos(ωt)andybmcos(ωt), respectively, which have the same excitation frequency ω,and the base amplitudes
are xbm and ybm, respectively. Substituting the base harmonic excitations into Eqs. (15)and(16), also using
the following non-dimensional parameters
ωnkv
mζvcv
2mωn
ζvcv
2mωn
δvbv
mδhbh
mγkh
kv
ω
ωn
Tωntηlh0
lv0μλh
lh0 Xrxr
lv0Yryr
lv0Xbm xbm
lv0Ybm ybm
lv0,
(17)
Equations (15)and(16) can be written in a non-dimensional form
1+δvX2
r+2δh1−Y2
r
η2(1+μ)2X
r+δv(Xr−XrYr)−2δhXrYr
η2(1+μ)2Y
r+δvXrX2
r−2δhXrY2
r
η2(1+μ)2
−4δhYrXrYr
η2(1+μ)2+2ζvX2
rXr+XrY
r−XrYrY
r+4ζh1−Y2
r
η2(1+μ)2X
r−4ζhXrYrYr
η2(1+μ)2
+XrYr+X3
r
2−XrY2
r+2γXr−2γXrY2
r
η2(1+μ)32Xbm cos(T),
(18)
δv(Xr−XrYr)−2δhXrYr
η2(1+μ)2X
r+1+δv1−X2
r+2δhY2
r
η2(1+μ)2Y
r+δv(1−Yr)X2
r−2δvXrXrYr
+2δhYrY2
r
η2(1+μ)2+2ζvY
r−X2
rYr+XrXr−XrYrXr+4ζhY2
rYr
η2(1+μ)2−4ζhXrYrXr
η2(1+μ)2
+1+2γ−2γ
1+μYr+X2
r
2−1+ 2γ
η2(1+μ)3X2
rYr+γY3
r
η2(1+μ)32Ybm cos(T),
(19)
where (·)  d2(·)/dT2and (·)d(·)/dT. As shown in Eqs. (18)and(19), for the IMD vibration isolator,
adding the inerter generates additional nonlinear acceleration and velocity terms compared with those of the
MD one, which yields nonlinear inertial and damping characteristics, respectively.

Y. Wang et al.
Fig. 4 Horizontal natural frequency with different δhand γ:athree-dimensional diagram, bplane diagram
3.2 Natural frequency
For the IMD vibration isolator, the dynamic equation in the xand ydirections is two nonlinear coupled
equations, and if the excitation base amplitude and dynamic response are smaller; compared with the linear
terms, the nonlinear terms in the dynamic equation are smaller, then the nonlinear and higher-order terms can
be neglected. Thus in this case, the nonlinear dynamic equations can be simplified into the linear one, and the
corresponding linear equations of Eqs. (18)and(19)are
(1 + 2δh)X
r+4ζhXr+2γXr2Xbm cos(T),(20)
(1 + δv)Yr+2ζvYr+1+2γ−2γ
1+μYr2Ybm cos(T),(21)
Equations (20)and(21) are two uncoupled linear equations. For a linear vibration isolation system, its
isolation performance depends on its natural frequency and yields a beneficial isolation affect when the exci-
tation frequency is larger than √2 times its natural frequency. The natural frequencies of the simplified linear
vibration isolator in the xand ydirections are given as
x2γ
1+2δhy1+(1+2γ)μ
(1 + δv)(1 + μ),(22)
which relies on the stiffness ratio γ, horizontal spring compression ratio μ, inertance-to-mass ratio (δv,δh).
The horizontal natural frequency of the simplified linear vibration isolator with a different horizontal
inertance-to-mass ratio δhand stiffness ratio γis shown in Fig. 4. The horizontal natural frequency becomes
smaller as increasing the horizontal inertance-to-mass ratio δhor reducing the stiffness ratio γ. The vertical
natural frequency also decreases as increasing the vertical inertance-to-mass ratio δv, and its changing tendency
with other two structural parameters is shown in Fig. 5. So as to maintain the vertical natural frequency positive,
the horizontal spring compression ratio μshould be equal to or larger than -1/(1 + 2γ). The vertical natural
frequency becomes larger as increasing the horizontal spring compression ratio μ, and if the horizontal spring
is pre-compressed, it increases as the stiffness ratio γdecreases, while if the horizontal spring is pre-extended,
it increases as the stiffness ratio γincreases.
3.3 Approximate solution
For the IMD vibration isolator, Eqs. (18)and(19) are strongly coupled nonlinear equations, the HBM is
acquired to obtain its dynamic response [40,41], and taking the first-order and third-order harmonics, the
approximate solutions are expressed as
Xra1cos(T)+a2sin(T)+a3cos(3T)+a4sin(3T)
Yrb1cos(T)+b2sin(T)+b3cos(3T)+b4sin(3T),(23)

Nonlinear vibration and dynamic performance analysis
Fig. 5 Vertical natural frequency with different μand γ:athree-dimensional diagram, bplane diagram (δv1)
Substituting Eq. (23) into Eqs. (18)and(19), balancing the same harmonic terms in the two equations
based on cos(ΩT), sin(ΩT), cos(3ΩT) and sin(3ΩT) yields the following equations
F1(a1,···a4,b1,···b4,
)cos(T)+F2(a1,···a4,b1,···b4,
)sin(T)
+F3(a1,···a4,b1,···b4,
)cos(3T)+F4(a1,···a4,b1,···b4,
)sin(3T)0
F5(a1,···a4,b1,···b4,
)cos(T)+F6(a1,···a4,b1,···b4,
)sin(T)
+F7(a1,···a4,b1,···b4,
)cos(3T)+F8(a1,···a4,b1,···b4,
)sin(3T)0,
(24)
Letting the coefficients of the first-order and third-order harmonic terms equal to zero derives the following
eight nonlinear equations
F1(a1,···a4,b1,···b4,
)0,···Fi(a1,···a4,b1,···b4,
)0,···F8(a1,···a4,b1,···b4,
)0,
(25)
The expressions of the eight nonlinear equations are shown in Appendix. Equation (25) can also be expressed
as
F(A,
)0,(26)
where F[F1,···F8]Tand A[a1,···a4,b1,···b4], Eq. (26) can be rewritten as
F(w)0,(27)
where F:R9→R8,w[w1,···w9]andw(A,
).Inthe R9space, the solution of Eq. (27)isa
one-dimensional manifold composed of the intersection of eight hypersurfaces, which derives
DF(w)⎡
⎢
⎣
∂F1
∂w1
∂F1
∂w2··· ∂F1
∂w9
.
.
..
.
.....
.
.
∂F8
∂w1
∂F8
∂w2··· ∂F8
∂w9
⎤
⎥
⎦8×9
,(28)
Defining the following nine-dimensional vector
H(w)[H1,···H9]THi(−1)i+1 det∂F
∂w1,··· ∂F
∂wi−1,∂ˆ
F
∂wi
,∂F
∂wi+1 ,···,∂F
∂w9,(29)
where ∂ˆ
F/∂ w iindicates that this column vector is omit. The relationship between the matrix DF(w) and vector
H(w)is
∂Fi
∂w1
∂Fi
∂w2··· ∂Fi
∂w9·H0i1···8,(30)

Y. Wang et al.
which denotes that the vector H(w) is the tangential vector of the solution curve for Eq. (27)inthe R9space,
and the corresponding unit tangent vector is
ε(w)H
H,(31)
where ·indicates the modulus.
The PAL method [42] is utilized to solve Eq. (25) which includes eight nonlinear equations, especially
there exist folding points in the frequency response curve (FRC), and the arc length of the one-dimensional
curve in the R9space is defined as
ds2
9

i1
dw2
i,(32)
Equation (27) contains nine variables (the eight amplitudes and the excitation frequency Ω), and an
additional constraint equation should be added to make Eq. (27) solvable, which is given as
ε·w−s0,(33)
where (·) denotes smaller change of the variable; then, solving Eq. (27) can be converted to
F(w)0ε·w−s0,(34)
Equation (34) can also be transformed into solving the Cauchy problem
dw
ds ε(w0)w(0)w0,(35)
Using the modified Euler method to solve Eq. (35), the predicted solution is given as
wiwi−1+ε(wi−1)(si−si−1)+(si−si−1)2
2·ε(wi−1)−ε(wi−2)
si−1−si−2i1,2,..., (36)
Using the Newton-type iterative correction to dominate the precision of the solution
w0
iwiwj
iwj−1
i−DF(wi)
H(wi)−1Fwj−1
i
0j1,2,···,(37)
after a finite number of iteration steps, the solution converges to a point w∗which satisfies F(w∗)0, it
indicates that the point w∗is the solution of Eq. (35) in a certain precision; then, the solution of Eq. (27)is
given, and the steady-state amplitudes of the IMD vibration isolator under base harmonic excitation can be
obtained as
Xrm a2
1+a2
2+a2
3+a2
4,Yrm b2
1+b2
2+b2
3+b2
4.(38)
3.4 Stability analysis
In order to analyze the stability of the steady-state amplitudes, the formal approximate solutions (Eq. 23)are
expressed as the time-varying ones, which are given as
Xr(T)a1(T)cos(T)+a2(T)sin(T)+a3(T)cos(3T)+a4(T)sin(3T)
Yr(T)b1(T)cos(T)+b2(T)sin(T)+b3(T)cos(3T)+b4(T)sin(3T).(39)
Substituting Eq. (39) into Eqs. (18)and(19), balancing the same harmonic terms in the two equations
based on cos(ΩT), sin(ΩT), cos(3ΩT) and sin(3ΩT), letting the coefficients of the first-order and third-order
harmonic terms equal to zero yields
F∗A(t),A(t),A(t),
0,(40)

Nonlinear vibration and dynamic performance analysis
The second derivative of the time-varying amplitude is equal to zero (A(t)0)in spite of the stability
of the first derivative, which gives
F∗A(t),A(t),
0,(41)
Equation (41) can be transformed into an explicit form which is composed of eight first-order differential
equations
A(t)F∗∗(A(t),
),(42)
where F∗∗ denotes a group of algebraic equations including the time-varying amplitudes A(t)and excitation
frequency Ω.
Therefore, the stability of the steady-state amplitudes transforms into the stability of the first derivative of
the time-varying amplitude, which is defined by Eq. (42). Then, the first method of Lyapunov is adopted to
judge the stability of Eq. (42), and the eigenvalues of the Jacobian determinant for Eq. (42) are given as
J(A(t),)−λjI0j1···8,(43)
J⎡
⎢
⎢
⎣
∂F∗∗
1
∂A1(t)
∂F∗∗
1
∂A2(t)··· ∂F∗∗
1
∂A8(t)
.
.
..
.
.....
.
.
∂F∗∗
8
∂A1(t)
∂F∗∗
8
∂A2(t)··· ∂F∗∗
8
∂A8(t)
⎤
⎥
⎥
⎦8×8
,(44)
If all the eigenvalues of Eq. (43) are negative, the steady-state amplitude is stable, and if there exists at
least one positive eigenvalue, the steady-state amplitude is unstable.
3.5 Dynamic response
Figure 6shows the FRC of the IMD vibration isolator under different base amplitudes, which shows the
representative changing trends. The horizontal and vertical inertance-to-mass ratios (δh,δv) are chosen as 0.5,
which are relatively smaller values. The horizontal and vertical damping ratios (ζh,ζv) are chosen as 0.05,
the stiffness ratio γis chosen as 2, the length ratio ηis chosen as 1, the horizontal spring compression ratio
μis chosen as 0 which indicates that it is in the original length state, and the vertical and horizontal base
amplitudes are chosen as the same values. As can be seen from Fig. 6, the resonance frequency of the IMD
vibration isolator in the ydirection is smaller than that of the xdirection, and while the resonance peak shows
the reverse tendency. When the base amplitude is relatively smaller, the FRCs of the IMD vibration isolator
in the xand ydirections are single-valued, display linear characteristics and seem linear vibration system, and
the steady-state amplitudes are stable, which is shown in Fig. 6a.
When the base amplitude increases, the FRC of the IMD vibration isolator in the ydirection turns to the
right, displays hardening characteristic and seems hardening Duffing vibration system; while the FRC in the
xdirection displays linear characteristic, except in a frequency band that is consistent with the resonance one
of the ydirection, there exists a folding point in the FRC and there are unstable steady-state amplitudes in this
frequency band, and this tendency is exhibited in Fig. 6b. The eigenvalues in the resonance frequency band of
the ydirectionareshowninTable2, and the stability of the steady-state amplitudes can be determined based
on the eigenvalues.
When the base amplitude is a larger value, the FRC of the IMD vibration isolator in the ydirection turns to
the left, displays softening characteristic and seems softening Duffing vibration system; while the FRC in the
xdirection has two resonance frequencies, the FRC around the smaller resonance frequency turns to the left
and shows softening characteristic, this resonance frequency band corresponds to the resonance one of the y
direction, the FRC around the larger resonance frequency is linear, and this tendency is exhibited in Fig. 6c.
The eigenvalues in the resonance frequency band can be also determined and are not shown for simplicity.
Aware that when the excitation frequency falls into the range [0.95, 1], there exist five steady-state amplitudes,
among the five steady-state amplitudes, three ones are stable and two ones are unstable, and the stability can
be determined by calculating the corresponding eigenvalues. When the excitation frequency Ω0.98, there
exist three stable steady-state amplitudes and the Fourier spectra for the three stable steady-state amplitudes
are shown in Fig. 7.

Y. Wang et al.
Fig. 6 FRC of the IMD vibration isolator in the xand ydirections for smaller inertance-to-mass ratio (δhδv0.5, γ2, η
1, μ0, ζhζv0.05)
As the horizontal and vertical inertance-to-mass ratios (δh,δv) are chosen as 10, which are relatively larger
values, the corresponding FRC of the IMD vibration isolator under different base amplitudes is shown in
Fig. 8, and the other structural parameters are chosen as the same values with those of Fig. 6. When the base
amplitudes (Xbm,Ybm) are chosen as 0.1, the FRCs of the IMD vibration isolator in the xand ydirections are
single-valued and show linear characteristics, which is shown in Fig. 8a, and this changing trend is similar
to Fig. 6a. As shown in Fig. 8b, when the base amplitudes (Xbm,Ybm) increase to 0.4, a different changing
trendoftheFRCinthexdirection is observed. The FRC in the xdirection turns to the left, displays softening
characteristic and seems softening Duffing vibration system; the FRC in the ydirection turns to the right,
displays hardening characteristic and seems hardening Duffing vibration system, another resonance frequency

Nonlinear vibration and dynamic performance analysis
Table 2 The eigenvalues of the IMD vibration isolator with Xbm 0.07 and Ybm 0.07 (δhδv0.5, ζhζv0.05, γ
2, η1, μ0)
Excitation frequency Amplitude Eigenvalues Stability analysis
0.92 X0.031, Y0.535 −0.121 + 0.615i, −0.121 −0.615i, −0.047 + 0.046i−
0.047−0.046i, −0.025 + 0.965i, −0.025 −0.965i −
0.0193 + 1.067i, −0.0193 −1.067i
Stable
0.92 X0.029, Y0.36 0.149 −2.877i, −0.142 + 0.706i, −0.142 −0.706i −
0.084, −0.027 + 0.949i, −0.027−0.949i −0.017 +
1.089i, −0.017 −1.089i,
Unstable
0.93 X0.032, Y0.552 −0.113 + 0.579i, −0.113 −0.579i, −0.047 + 0.036i −
0.047 −0.036i, −0.025 + 0.983i, −0.025 −0.983i −
0.019 + 1.079i, −0.019–1.079i,
Stable
0.93 X0.031, Y0.442 0.159 −2.931i, −0.126 + 0.644i, −0.126 −0.644i −
0.099, −0.026 + 0.972i, −0.026 −0.972i −0.018 +
1.094i, −0.018 −1.094i,
Unstable
0.94 X0.032, Y0.566 −0.106 + 0.548i, −0.106 −0.548i, −0.047 + 0.014i −
0.047 −0.014i, −0.027 + 1i, −0.027 −1i −0.019 +
1.092i, −0.019 −1.092i
Stable
0.94 X0.033, Y0.499 0.167 −2.986i, −0.114 + 0.589i, −0.114–0.589i −
0.102, −0.025 + 0.994i, −0.025 −0.994i −0.019 +
1.101i, −0.019 −1.101i
Unstable
is found and its resonance peak is relatively smaller, and the stability of the FRC can be also determined by
calculating the corresponding eigenvalues, which is not shown for brevity.
The fourth-order Runge–Kutta method is used to solve Eqs. (18)and(19) to acquire the numerical results,
which are shown as circles in Figs. 6and 8. The analytical results exhibit good consistent with the numerical
results, and it denotes that adopting the HBM and PAL method to acquire the analytical results can represent the
real dynamic responses, which is an effective method to solve this type of strongly coupled nonlinear dynamic
system. The asterisks shown in Figs. 6and 8denote the unstable analytical results, which is determined by
the eigenvalues of the Jacobian determinant for Eq. (42). It should be noted that the non-dimensional dynamic
displacement is normalized by the original length of the vertical spring, and if the base amplitude is larger, the
dynamic displacement in the xand ydirections can be larger, especially for the resonance peaks.
3.6 Isolation performance
For the IMD vibration isolator, its isolation performance in this paper is evaluated by three performance criteria:
(1) dynamic displacement peak, (2) displacement transmissibility peak and (3) isolation frequency band, and
the absolute displacements in the xand ydirections are given as
Xam Xrm +Xbm cos(T)(a1+Xbm)cos(T)+a2sin(T)+a3cos(3T)+a4sin(3T)
Yam Yrm +Ybm cos(T)(b1+Ybm)cos(T)+b2sin(T)+b3cos(3T)+b4sin(3T),(45)
Then, the corresponding absolute displacement transmissibilities are obtained as
Tax 
xa
xb|Xam|
Xbm (a1+Xbm)2+a2
2+a2
3+a2
4
Xbm
Tay 
ya
yb|Yam|
Ybm (b1+Ybm)2+b2
2+b2
3+b2
4
Ybm ,
(46)
The dynamic displacement and absolute displacement transmissibility peaks should maintain smaller for
the IMD vibration isolator, which determines the maximum dynamic displacement and absolute displacement
transmissibility, respectively. The isolation frequency band determines the bandwidth where the IMD vibration
isolator provides a advantageous isolation performance, which in this frequency band the absolute displacement
transmissibility is smaller than 1.
For the IMD vibration isolator, the horizontal and vertical inertance-to-mass ratios (δh,δv) determine
its inertial characteristic and the effect of inerter in the xand ydirections, respectively; the stiffness ratio γ

Y. Wang et al.
Fig. 7 Fourier spectra of the IMD vibration isolator for the three stable steady-state amplitudes in the xand ydirections with Ω
0.98 (δhδv0.5, γ2, η1, μ0, ζhζv0.05, Xbm Ybm 0.01)
Fig. 8 FRC of the IMD vibration isolator in the xand ydirections for larger inertance-to-mass ratio (δhδv10, γ2, η
1, μ0, ζhζv0.05)

Nonlinear vibration and dynamic performance analysis
Fig. 9 a Dynamic displacement in xdirection, bdynamic displacement in ydirection, cabsolute displacement transmissibility
in xdirection and dabsolute displacement transmissibility in ydirection of the IMD vibration isolator with different δh(δv
0.5, γ2, η1, μ0, ζhζv0.05, Xbm Ybm 0.01)
determines its stiffness characteristic; the horizontal spring compression ratio μand the length ratio ηdetermine
the initial state of the horizontal spring and the acceleration, velocity and displacement terms in the dynamic
equation (see Eqs. (18)and(19)), which has effect on the inertial, damping and stiffness characteristics.
The isolation performance of the IMD vibration isolator is compared with the conventional MD vibration
isolator composed of the damper and spring, and denoting the horizontal and vertical inertance-to-mass ratios
(δh,δv)equalto0inEqs.(18)and(19) yields the corresponding non-dimensional dynamic equation subjected
to base harmonic excitation
X
r+2ζvX2
rX
r+XrY
r−XrYrY
r+4ζh1−Y2
r
η2(1+μ)2X
r−4ζhXrYrY
r
η2(1+μ)2
+XrYr+X3
r
2−XrY2
r+2γXr−2γXrY2
r
η2(1+μ)32Xbm cos(T),
(47)
Y
r+2ζvY
r−X2
rY
r+XrX
r−XrYrX
r+4ζhY2
rY
r
η2(1+μ)2−4ζhXrYrX
r
η2(1+μ)2
+1+2γ−2γ
1+μYr+X2
r
2−1+ 2γ
η2(1+μ)3X2
rYr+γY3
r
η2(1+μ)32Ybm cos(T),
(48)
Following the same solving procedure shown in Sect. 3.3, the HBM and PAL methods are used to acquire
the dynamic response, and then, the absolute displacement transmissibility can be obtained.
The dynamic displacement and absolute displacement transmissibility of the IMD vibration isolator in
the xand ydirections with a different inertance-to-mass ratio (δh,δv), stiffness ratio γ, horizontal spring
compression ratio μand length ratio ηare shown in Figs. 9,10,11,12 and 13, respectively. In Figs. 9,10,11,
12 and 13, the horizontal and vertical base amplitudes (Xbm,Ybm) are chosen as 0.01, which are smaller base
amplitudes, and the FRC of the IMD vibration isolator displays linear characteristic.

Y. Wang et al.
Fig. 10 a Dynamic displacement in xdirection, bdynamic displacement in ydirection, cabsolute displacement transmissibility
in xdirection and dabsolute displacement transmissibility in ydirection of the IMD vibration isolator with different δv(δh
0.5, γ2, η1, μ0, ζhζv0.05, Xbm Ybm 0.01)
As shown in Fig. 9, for the IMD vibration isolator, when the horizontal inertance-to-mass ratio increases,
the dynamic displacement and absolute displacement transmissibility peaks in the xdirection decrease, the
resonance frequency in the xdirection becomes smaller, and the isolation frequency band in the xdirection
becomes wider, while the high-frequency absolute displacement transmissibility in the xdirection increases.
It should be noted that for the chosen horizontal inertance-to-mass ratio range (δh[0, 10]), the isolation
performance criteria in the ydirection remain almost the same, which indicates that the horizontal inertance-
to-mass ratio has less effect on the ydirection isolation performance than that of the xdirection.
As shown in Fig. 10, for the IMD vibration isolator, as the vertical inertance-to-mass ratio increases,
the dynamic displacement and absolute displacement transmissibility peaks in the ydirection decrease, the
resonance frequency in the ydirection decreases, and the isolation frequency band in the ydirection widens,
while the high-frequency absolute displacement transmissibility in the ydirection increases. For the chosen
vertical inertance-to-mass ratio range (δv[0, 10]), the isolation performance criteria in the xdirection remain
almost the same, which suggests that the vertical inertance-to-mass ratio has less effect on the xdirection
isolation performance than that of the ydirection.
Compared with the MD vibration isolator (δhδv0), the IMD vibration isolator further reduces the
dynamic displacement and absolute displacement transmissibility peaks and also widens the isolation frequency
band; however, only the high-frequency absolute displacement transmissibility is larger. Therefore, adding the
vertical and horizontal inerters on the basis of the MD vibration isolator to constitute the IMD one could further
improve the isolation performance.
For the IMD vibration isolator, by increasing the stiffness ratio, the dynamic displacement and absolute
displacement transmissibility peaks in the xdirection increase, the resonance frequency in the xdirection
becomes larger and the isolation frequency band in the xdirection becomes narrower, while the dynamic dis-
placement and absolute displacement transmissibility peaks in the ydirection increase a bit, and the resonance

Nonlinear vibration and dynamic performance analysis
Fig. 11 a Dynamic displacement in xdirection, bdynamic displacement in ydirection, cabsolute displacement transmissibility
in xdirection and dabsolute displacement transmissibility in ydirection of the IMD vibration isolator with different γ(δhδv
0.5, η1, μ0, ζhζv0.05, Xbm Ybm 0.01)
frequency and isolation frequency band in the ydirection remains almost the same. This trend is shown in
Fig. 11.
For the IMD vibration isolator, when the horizontal spring compression ratio increases, the isolation
performance criteria in the xdirection maintain the same which indicates that the horizontal spring compression
ratio has little effect on the xdirection isolation performance, while the dynamic displacement and absolute
displacement transmissibility peaks in the ydirection increase, the resonance frequency in the ydirection
becomes larger and the isolation frequency band in the ydirection becomes narrower. It should be noted that if
the horizontal spring compression ratio is equal to −1/(1 + 2γ) for a fixed stiffness ratio, the natural frequency
of the corresponding linear vibration isolator in the ydirection is equal to 0 (see Eq. (22) and Fig. 5), and
the IMD vibration isolator can achieve the full frequency band vibration isolation in the ydirection, which is
clearly shown in Fig. 12d. The overall trend is exhibited in Fig. 12.
For the IMD vibration isolator, the length ratio has little effect on the isolation performance in the xand
ydirections. The isolation performance criteria in the xdirection retain the same with different length ratios.
When the length ratio increases, the dynamic displacement and absolute displacement transmissibility peaks
in the ydirection decrease a bit, the resonance frequency in the ydirection becomes a little smaller, and the
isolation frequency band in the ydirection becomes a little wider. This trend is illustrated in Fig. 13.
As the inertance-to-mass ratio is chosen as smaller and larger values, the dynamic displacement and
absolute displacement transmissibility of the IMD vibration isolator in the xand ydirections with different base
amplitudes (Xbm,Ybm ) is shown in Figs. 14 and 15, respectively. When the inertance-to-mass ratio is chosen as
smaller value, increasing the base amplitude results in larger dynamic displacement and absolute displacement
transmissibility peaks in the xand ydirections, while it has bit effect on the isolation frequency band. It should
be noted that for larger horizontal base amplitude, there exists an additional smaller resonance frequency in the
FRC for the xdirection and the corresponding resonance peak is larger, which further deteriorates the isolation
performance. When the inertance-to-mass ratio is chosen as larger value, increasing the base amplitude results
in larger dynamic displacement peak in the xand ydirections, while leads to smaller absolute displacement

Y. Wang et al.
Fig. 12 a Dynamic displacement in xdirection, bdynamic displacement in ydirection, cabsolute displacement transmissibility
in xdirection and dabsolute displacement transmissibility in ydirection of the IMD vibration isolator with different μ(δhδv
0.5, γ2, η1, ζhζv0.05, Xbm Ybm 0.01)
transmissibility peak in the xand ydirections, widens the isolation frequency band in the xdirection, while
narrows the isolation frequency band in the ydirection.
Overall, the IMD vibration isolator can enhance the isolation performance of the MD vibration isolator
significantly. The horizontal and vertical inertance-to-mass ratios are selected as larger values for better xand
ydirections isolation performance, the stiffness ratio is selected as smaller value that is especially better for
xdirection isolation performance, the horizontal spring compression ratio is selected as smaller value that is
especially better for ydirection isolation performance, and the length ratio is selected as larger value which is
a bit better for ydirection isolation performance.
4 Base shock excitation
Then, the base shock excitation is considered and the rounded displacement pulse [43] is used here, which can
denote a bump or discrete irregularity of the road in practical engineering. The shock excitation in the xand y
directions is expressed as
xbxbme24(νωnt)2e−νωntyb(t)ybme24(νωnt)2e−νωnt,(49)
where ωn√kv/mand νis the severity parameter; if the shock excitation is more severer, the parameter
νis larger. For the IMD vibration isolator, substituting Eq. (49) into Eqs. (15)and(16), and combined with
Eq. (17) yields its non-dimensional dynamic equation subjected to shock excitation

Nonlinear vibration and dynamic performance analysis
Fig. 13 a Dynamic displacement in xdirection, bdynamic displacement in ydirection, cabsolute displacement transmissibility
in xdirection and dabsolute displacement transmissibility in ydirection of the IMD vibration isolator with different η(δhδv
0.5, γ2, μ0, ζhζv0.05, Xbm Ybm 0.01)
1+δvX2
r+2δh1−Y2
r
η2(1+μ)2Xr+δv(Xr−XrYr)−2δhXrYr
η2(1+μ)2Yr+δvXrX2
r−2δhXrY2
r
η2(1+μ)2
−4δhYrXrYr
η2(1+μ)2+2ζvX2
rXr+XrYr−XrYrYr+4ζh1−Y2
r
η2(1+μ)2Xr−4ζhXrYrY
r
η2(1+μ)2
+XrYr+X3
r
2−XrY2
r+2γXr−2γXrY2
r
η2(1+μ)3−e2ν2
42−4νT+ν2T2Xbme−νT,
(50)
δv(Xr−XrYr)−2δhXrYr
η2(1+μ)2X
r+1+δv1−X2
r+2δhY2
r
η2(1+μ)2Y
r+δv(1−Yr)X2
r−2δvXrX
rY
r
+2δhYrY2
r
η2(1+μ)2+2ζvY
r−X2
rY
r+XrX
r−XrYrX
r+4ζhY2
rY
r
η2(1+μ)2−4ζhXrYrX
r
η2(1+μ)2
+1+2γ−2γ
1+μYr+X2
r
2−1+ 2γ
η2(1+μ)3X2
rYr+γY3
r
η2(1+μ)3−e2ν2
42−4νT+ν2T2Ybme−νT,
(51)
Using the fourth-order Runge–Kutta method to acquire the corresponding dynamic response, then the
absolute displacements in the xand ydirections are given as
Xam Xrm +e2ν2T2
4Xbme−νTYam Yrm +e2ν2T2
4Ybme−νT.(52)

Y. Wang et al.
Fig. 14 a Dynamic displacement in xdirection, bdynamic displacement in ydirection, cabsolute displacement transmissibility
in xdirection and dabsolute displacement transmissibility in ydirection of the IMD vibration isolator with different Xbm and
Ybm for smaller inertance-to-mass ratio (δhδv0.5, γ2, η1, μ0, ζhζv0.05)
The time history of the absolute displacement in the xand ydirections with different inertance-to-mass
ratios is displayed in Fig. 16,whereν5 denotes less severe impact. The IMD vibration isolator can further
reduce the displacement peak than the MD one, and the displacement peak has decreasing trend by increasing
the inertance-to-mass ratio.
The shock performance of the IMD vibration isolator under shock excitation is assessed using the maximum
absolute displacement ratio (MADR), which is defined as
MADRxmax|Xam|
Xbm MADRymax|Yam |
Ybm .(53)
Similar to the harmonic excitation, the effect of the inertance-to-mass ratio (δh,δv), stiffness ratio γ,
horizontal spring compression ratio μandlengthratioηon the shock performance is mainly analyzed. The
MADR of the IMD vibration isolator in the xand ydirections is calculated in the νrange from 0.1 to 100, which
covers the slight and severe impact, and Figs. 17,18,19 and 20 show the changing tendency of the MADR
with different δ,γ,μand η, respectively. The horizontal and vertical inertance-to-mass ratios are selected
as the same values for brevity in this section. The horizontal and vertical damping ratios (ζh,ζv) are chosen
as 0.05, and the horizontal and vertical base amplitudes (Xbm,Ybm) are chosen as 0.01. When the severity
parameter increases, the MADR of the IMD vibration isolator first increases, then reaches a peak value and
finally decreases to a fixed value.
As shown in Fig. 17, for smaller severity parameter, the MADR decreases by increasing the inertance-to-
mass ratio, while for larger severity parameter, the MADR increases by increasing the inertance-to-mass ratio.
Compared with the MD vibration isolator (δhδv0), the IMD vibration isolator further reduces the MADR
in the middle severity parameter range, while it increases the MADR in the higher severity parameter range.
As displayed in Fig. 18, the stiffness ratio has little effect on the shock performance in the ydirection.
When the severity parameter is smaller, the MADR in the xdirection decreases by increasing the stiffness

Nonlinear vibration and dynamic performance analysis
Fig. 15 a Dynamic displacement in xdirection, bdynamic displacement in ydirection, cabsolute displacement transmissibility
in xdirection and dabsolute displacement transmissibility in ydirection of the IMD vibration isolator with different Xbm and
Ybm for larger inertance-to-mass ratio (δhδv10, γ2, η1, μ0, ζhζv0.05)
Fig. 16 Time history of the absolute displacement for the IMD vibration isolator under shock excitation with different δvand δh
(γ2, η1, μ0, ν5, ζhζv0.05, Xbm Ybm 0.01)
ratio; when the severity parameter increases, the MADR in the xdirection increases by increasing the stiffness
ratio; as the severity parameter increases to a larger value (e.g., ν>10), the MADR in the xdirection remains
almost the same with different stiffness ratios.
As exhibited in Fig. 19, the horizontal spring compression ratio has little effect on the shock performance
in the xdirection. When the severity parameter is smaller, the MADR in the ydirection decreases as the
horizontal spring compression ratio increases; when the severity parameter increases, the MADR in the y
direction increases as the horizontal spring compression ratio increases; as the severity parameter increases to
a larger value (e.g., ν>10), the MADR in the ydirection remains almost the same with different horizontal
spring compression ratios. It should be noted that if the horizontal spring compression ratio is equal to −1/(1

Y. Wang et al.
Fig. 17 MADR of the IMD vibration isolator with different δvand δh(γ2, η1, μ0, ζhζv0.05, Xbm Ybm 0.01)
Fig. 18 MADR of the IMD vibration isolator with different γ(δhδv0.5, η1, μ0, ζhζv0.05, Xbm Ybm 0.01)
Fig. 19 MADR of the IMD vibration isolator with different μ(δhδv0.5, γ2, η1, ζhζv0.05, Xbm Ybm 0.01)
+2γ) for a fixed stiffness ratio, the MADR in the ydirection is smaller than 1 in the full severity parameter
range, which achieves excellent shock performance and is the same with those of harmonic excitation shown
in Sect. 3.6.
As illustrated in Fig. 20, the length ratio has little on the shock performance in the xand ydirections, which
the MADR in the xand ydirections remains almost the same with different length ratios.
Overall, the IMD vibration isolator can improve the shock performance of the MD vibration isolator in the
middle severity parameter range; in order to acquire a better shock performance, the vertical and horizontal
inertance-to-mass ratios are chosen as larger values, and the stiffness ratio and thehorizontal spring compression
ratio are chosen as smaller values.

Nonlinear vibration and dynamic performance analysis
Fig. 20 MADR of the IMD vibration isolator with different η(δhδv0.5, γ2, μ0, ζhζv0.05, Xbm Ybm 0.01)
5Conclusion
This paper adds the vertical and horizontal inerters on the basis of the MD vibration isolator and proposes the
IMD vibration isolator consisting of the inerter, damper and spring structures. The Lagrange theory is used
to establish its dynamic equation, combining the HBM and PAL method, the dynamic response subjected to
base harmonic excitation is acquired and the stability of the dynamic response is investigated, the dynamic
performance under harmonic and shock excitations is analyzed, and compared with those of the MD vibration
isolator, the effect of structural parameters on its dynamic performance is studied in detail. This work yields
the following conclusions:
(1) The dynamic equation of the IMD vibration isolator in the multiple directions is strongly coupled nonlinear
dynamic equations, the HBM and PAL method could be used to conveniently obtain its dynamic response,
and the analytical results exhibit good accuracy with the numerical results, which confirms the validity
of the analytical method.
(2) The IMD vibration isolator has nonlinear inertial, damping and stiffness characteristics, and it further
reduces the dynamic displacement and absolute displacement transmissibility peaks, widens the isolation
frequency band than the MD vibration isolator and also has better shock performance in the middle
severity parameter range.
(3) In order to achieve better isolation and shock performance, the vertical and horizontal inertance-to-mass
ratios (δv,δh) are chosen as larger values, and the stiffness ratio γand the horizontal spring compression
ratio μare chosen as smaller values.
In summary, the proposed IMD vibration isolator is a original device and exhibits the advantages of applying
the inerter, which provides excellent isolation and shock performance in multiple directions.
Acknowledgements The research described in this paper is supported by the National Natural Science Foundation of China
(Grant No. 12172153, 51805216), Major Project of Basic Science (Natural Science) of the Jiangsu Higher Education Institutions
(22KJA410001) and the project funded by the Youth Talent Cultivation Program of Jiangsu University.
Declarations
Conflict of interest The authors declare that they have no known competing financial interests or personal relationships that
could have appeared to influence the work reported in this paper.
Appendix
The expressions of the eight nonlinear equations in Eq. (25) are given as
F1(a1,···a4,b1,···b4,
)− 1
8η2(μ+1)
3(F11 +F12 +F13),(A1)

Y. Wang et al.
F11 4a3
1δvη2μ3+12a2
1a3δvη2μ3+4a1a2
2δvη2μ3+24a1a2a4δvη2μ3+40a1a2
3δvη2μ3+40a1a2
4δvη2μ3
−6a1b2
1δvη2μ3−20a1b1b3δvη2μ3−2a1b2
2δvη2μ3−20a1b2b4δvη2μ3−36a1b2
3δvη2μ3−36a1b2
4δvη2μ3
−12a2
2a3δvη2μ3−4a2b1b2δvη2μ3−20a2b1b4δvη2μ3+20a2b2b3δvη2μ3−2a3b2
1δvη2μ3−40a3b1b3δvη2μ3
+2a3b2
2δvη2μ3−4a4b1b2δvη2μ3−40a4b1b4δvη2μ3+12a3
1δvη2μ2+36a2
1a3δvη2μ2+12a1a2
2δvη2μ2
+72a1a2a4δvη2μ2+ 120a1a2
3δvη2μ2+ 120a1a2
4δvη2μ2−18a1b2
1δvη2μ2−60a1b1b3δvη2μ2+8η2Xbm
−6a1b2
2δvη2μ2−60a1b2b4δvη2μ2−108a1b2
3δvη2μ2−108a1b2
4δvη2μ2−36a2
2a3δvη2μ2+8a1η2
−12a2b1b2δvη2μ2−60a2b1b4δvη2μ2+60a2b2b3δvη2μ2−6a3b2
1δvη2μ2−120a3b1b3δvη2μ2−20a3b2
1δh
+6a3b2
2δvη2μ2−12a4b1b2δvη2μ2−120a4b1b4δvη2μ2+12a3
1δvη2μ+36a2
1a3δvη2μ+12a1a2
2δvη2μ
+72a1a2a4δvη2μ+ 120a1a2
3δvη2μ+ 120a1a2
4δvη2μ−18a1b2
1δvη2μ−60a1b1b3δvη2μ−6a1b2
2δvη2μ
−60a1b2b4δvη2μ−108a1b2
3δvη2μ−108a1b2
4δvη2μ+16a1δhη2μ3−36a2
2a3δvη2μ−12a2b1b2δvη2μ
−60a2b1b4δvη2μ+60a2b2b3δvη2μ−6a3b2
1δvη2μ−120a3b1b3δvη2μ+6a3b2
2δvη2μ−12a4b1b2δvη2μ
−120a4b1b4δvη2μ+4a3
1δvη2+12a2
1a3δvη2+4a1a2
2δvη2+24a1a2a4δvη2+40a1a2
3δvη2+40a1a2
4δvη2
−6a1b2
1δvη2−20a1b1b3δvη2−2a1b2
2δvη2−20a1b2b4δvη2−36a1b2
3δvvη2−36a1b2
4δvη2+48a1δhη2μ2
+8a1η2μ3−12a2
2a3δvη2−4a2b1b2δvη2−20a2b1b4δvη2+20a2b2b3δvη2−2a3b2
1δvη2−40a3b1b3δvη2
+2a3b2
2δvη2−4a4b1b2δvη2−40a4b1b4δvη2+8η2μ3Xbm −12a1b2
1δhμ−8a1b1b3δhμ−4a1b2
2δhμ
−8a1b2b4δhμ−8a1b2
3δhμ−8a1b2
4δhμ+48a1δhη2μ+24a1η2μ2−8a2b1b2δhμ−8a2b1b4δhμ+20a3b2
2δh
+8a2b2b3δhμ−20a3b2
1δhμ−80a3b1b3δhμ+20a3b2
2δhμ−40a4b1b2δhμ−80a4b1b4δhμ+24η2μ2Xbm
−12a1b2
1δh−8a1b1b3δh−4a1b2
2δh−8a1b2b4δh−8a1b2
3δh−8a1b2
4δh+16a1δhη2+24a1η2μ−8a2b1b2δh
−8a2b1b4δh+8a2b2b3δh−80a3b1b3δh−40a4b1b2δh−80a4b1b4δh+24η2μXbm 2,(A2)
F12 −4a2
1a2ζvη2μ3−4a2
1a4ζvη2μ3+8a1a2a3ζvη2μ3+8a1b1b2ζvη2μ3+8a1b1b4ζvη2μ3
−8a1b2b3ζvη2μ3−4a3
2ζvη2μ3+4a2
2a4ζvη2μ3−8a2a2
3ζvη2μ3−8a2a2
4ζvη2μ3−4a2b2
1ζvη2μ3
−8a2b1b3ζvη2μ3+4a2b2
2ζvη2μ3−8a2b2b4ζvη2μ3+8a3b1b2ζvη2μ3+24a3b1b4ζvη2μ3−32a3b1b2ζh
+8a3b2b3ζvη2μ3−4a4b2
1ζvη2μ3−24a4b1b3ζvη2μ3+4a4b2
2ζvη2μ3+8a4b2b4ζvη2μ3−4a3
2ζvη2
−12a2
1a2ζvη2μ2−12a2
1a4ζvη2μ2+24a1a2a3ζvη2μ2+24a1b1b2ζvη2μ2+24a1b1b4ζvη2μ2−48a3b1b4ζh
−24a1b2b3ζvη2μ2−12a3
2ζvη2μ2+12a2
2a4ζvη2μ2−24a2a2
3ζvη2μ2−24a2a2
4ζvη2μ2−4a2
1a2ζvη2
−12a2b2
1ζvη2μ2−24a2b1b3ζvη2μ2+12a2b2
2ζvη2μ2−24a2b2b4ζvη2μ2+24a3b1b2ζvη2μ2−32a2ζhη2
+72a3b1b4ζvη2μ2+24a3b2b3ζvη2μ2−12a4b2
1ζvη2μ2−72a4b1b3ζvη2μ2+12a4b2
2ζvη2μ2−4a4b2
1ζvη2
+24a4b2b4ζvη2μ2−12a2
1a2ζvη2μ−12a2
1a4ζvη2μ+24a1a2a3ζvη2μ+24a1b1b2ζvη2μ+24a1b1b4ζvη2μ
−24a1b2b3ζvη2μ−12a3
2ζvη2μ+12a2
2a4ζvη2μ−24a2a2
3ζvη2μ−24a2a2
4ζvη2μ−12a2b2
1ζvη2μ
−24a2b1b3ζvη2μ+12a2b2
2ζvη2μ−24a2b2b4ζvη2μ−32a2ζhη2μ3+24a3b1b2ζvη2μ+72a3b1b4ζvη2μ
+24a3b2b3ζvη2μ−12a4b2
1ζvη2μ−72a4b1b3ζvη2μ+12a4b2
2ζvη2μ+24a4b2b4ζvη2μ−4a2
1a4ζvη2
+8a1a2a3ζvη2+8a1b1b2ζvη2+8a1b1b4ζvη2−8a1b2b3ζvη2+4a2
2a4ζvη2−8a2a2
3ζvη2−8a2a2
4ζvη2
−4a2b2
1ζvη2−8a2b1b3ζvη2+4a2b2
2ζvη2−8a2b2b4ζvη2−96a2ζhη2μ2+8a3b1b2ζvη2+24a3b1b4ζvη2
+8a3b2b3ζvη2−24a4b1b3ζvη2+4a4b2
2ζvη2+8a4b2b4ζvη2+16a2b2
1ζhμ+16a2b2
2ζhμ+16a2b2
3ζhμ
+16a2b2
4ζhμ−96a2ζhη2μ−32a3b1b2ζhμ−48a3b1b4ζhμ+16a3b2b3ζhμ+16a4b2
1ζhμ+48a4b1b3ζhμ
−16a4b2
2ζhμ+16a4b2b4ζhμ+16a2b2
1ζh+16a2b2
2ζh+16a2b2
3ζh+16a2b2
4ζh+16a3b2b3ζh+16a4b2
1ζh
+48a4b1b3ζh−16a4b2
2ζh+16a4b2b4ζh, (A3)
F13 −3a3
1η2μ3−3a2
1a3n2μ3−3a1a2
2η2μ3−6a1a2a4η2μ3−6a1a2
3η2μ3−6a1a2
4η2μ3+6a1b2
1η2μ3
+4a1b1b3η2μ3+2a1b2
2η2μ3+4a1b2b4η2μ3+4a1b2
3η2μ3+4a1b2
4η2μ3+3a2
2a3η2μ3+4a2b1b2η2μ3

Nonlinear vibration and dynamic performance analysis
+4a2b1b4η2μ3−4a2b2b3η2μ3+2a3b2
1η2μ3+8a3b1b3η2μ3−2a3b2
2η2μ3+4a4b1b2η2μ3+8a4b1b4η2μ3
−9a3
1η2μ2−9a2
1a3η2μ2−9a1a2
2η2μ2−18a1a2a4η2μ2−18a1a2
3η2μ2−18a1a2
4η2μ2+18a1b2
1η2μ2
+12a1b1b3η2μ2+6a1b2
2η2μ2+12a1b2b4η2μ2+12a1b2
3η2μ2+12a1b2
4η2μ2−16a1η2γμ
3+9a2
2a3η2μ2
+12a2b1b2η2μ2+12a2b1b4η2μ2−12a2b2b3η2μ2+6a3b2
1η2μ2+24a3b1b3η2μ2−6a3b2
2η2μ2+6a1b2
2η2μ
+12a4b1b2η2μ2+24a4b1b4η2μ2−9a3
1η2μ−9a2
1a3η2μ−9a1a2
2η2μ+8a1b2b4γ−18a1a2
3η2μ+4a1b2
2γ
−18a1a2
4η2μ+18a1b2
1η2μ+12a1b1b3η2μ+12a1b2b4η2μ+12a1b2
3η2μ+12a1b2
4η2μ+8a1b2
4γ+12a1b2
1γ
+9a2
2a3η2μ+12a2b1b2η2μ+12a2b1b4η2μ−12a2b2b3η2μ+6a3b2
1η2μ+24a3b1b3η2μ+8a1b2
3γ+8a1b1b3γ
+12a4b1b2η2μ+24a4b1b4η2μ−3a3
1η2−3a2
1a3η2−3a1a2
2η2−6a1a2a4η2−6a1a2
3η2−6a1a2
4η2+6a1b2
1η2
+4a1b1b3η2+2a1b2
2η2+4a1b2b4η2+4a1b2
3η2+4a1b2
4η2−48a1η2γμ+3a2
2a3η2+4a2b1b2η2+4a2b1b4η2
−4a2b2b3η2+2a3b2
1η2+8a3b1b3η2−2a3b2
2η2+4a4b1b2η2+8a4b1b4η2−18a1a2a4η2μ−48a1η2γμ
2
−6a3b2
2η2μ−16a1η2γ+8a2b1b2γ+8a2b1b4γ−8a2b2b3γ+4a3b2
1γ+16a3b1b3γ+8a4b1b2γ+16a4b1b4γ
(A4)
F2(a1,···a4,b1,···b4,
)− 1
8η2(μ+1)
3(F21 +F22 +F23),(A5)
F21 4a2
1a2δvη2μ3+12a2
1a4δvη2μ3−24a1a2a3δvη2μ3−4a1b1b2δvη2μ3−80a4b2b4δh+8η2a2
−20a1b1b4δvη2μ3+20a1b2b3δvη2μ3+4a3
2δvη2μ3−12a2
2a4δvη2μ3+40a2a2
3δvη2μ3+40a2a2
4δvη2μ3
−2a2b2
1δvη2μ3+20a2b1b3δvη2μ3−6a2b2
2δvη2μ3+20a2b2b4δvη2μ3−36a2b2
3δvη2μ3−36a2b2
4δvη2μ3
+4a3b1b2δvη2μ3−40a3b2b3δvη2μ3−2a4b2
1δvη2μ3+2a4b2
2δvη2μ3−40a4b2b4δvη2μ3+12a2
1a2δvη2μ2
+36a2
1a4δvη2μ2−72a1a2a3δvη2μ2−12a1b1b2δvη2μ2−60a1b1b4δvη2μ2+60a1b2b3δvη2μ2+12a3
2δvη2μ2
−36a2
2a4δvη2μ2+ 120a2a2
3δvη2μ2+ 120a2a2
4δvη2μ2−6a2b2
1δvη2μ2+60a2b1b3δvη2μ2−18a2b2
2δvη2μ2
+60a2b2b4δvη2μ2−108a2b2
3δvη2μ2−108a2b2
4δvη2μ2+12a3b1b2δvη2μ2−120a3b2b3δvη2μ2−8a2b2
4δh
−6a4b2
1δvη2μ2+6a4b2
2δvη2μ2−120a4b2b4δvη2μ2+12a2
1a2δvη2μ+36a2
1a4δvη2μ−72a1a2a3δvη2μ
−12a1b1b2δvη2μ−60a1b1b4δvη2μ+60a1b2b3δvη2μ+12a3
2δvη2μ−36a2
2a4δvη2μ+ 120a2a2
3δvη2μ
+ 120a2a2
4δvη2μ−6a2b2
1δvη2μ+60a2b1b3δvη2μ−18a2b2
2δvη2μ+60a2b2b4δvη2μ−108a2b2
3δvη2μ
−108a2b2
4δvη2μ+16a2δhη2μ3+12a3b1b2δvη2μ−120a3b2b3δvη2μ−6a4b2
1δvη2μ+6a4b2
2δvη2μ
−120a4b2b4δvη2μ+4a2
1a2δvη2+12a2
1a4δvη2−24a1a2a3δvη2−4a1b1b2δvη2−20a1b1b4δvη2−8a2b2
3δh
+20a1b2b3δvη2+4a3
2δvη2−12a2
2a4δvη2+40a2a2
3δvη2+40a2a2
4δvη2−2a2b2
1δvη2+20a2b1b3δvη2
−6a2b2
2δvη2+20a2b2b4δvη2−36a2b2
3δvη2−36a2b2
4δvη2+48a2δhη2μ2+8η2a2μ3+4a3b1b2δvη2
−40a3b2b3δvη2−2a4b2
1δvη2+2a4b2
2δvη2−40a4b2b4δvη2−8a1b1b2δhμ−8a1b1b4δhμ+8a1b2b3δhμ
−4a2b2
1δhμ+8a2b1b3δhμ−12a2b2
2δhμ+8a2b2b4δhμ−8a2b2
3δhμ−8a2b42δhμ+48a2δhη2μ
+24a2η2μ2+40a3b1b2δhμ−80a3b2b3δhμ−20a4b2
1δhμ+20a4b2
2δhμ−80a4b2b4δhμ−8a1b1b2δh
−8a1b1b4δh+8a1b2b3δh−4a2b2
1δh+8a2b1b3δh−12a2b2
2δh+8a2b2b4δh+16a2δhη2+24η2a2μ
+40a3b1b2δh−80a3b2b3δh−20a4b2
1δh+20a4b2
2δh2,(A6)
F22 4a3
1ζvη2μ3+4a2
1a3ζvη2μ3+4a1a2
2ζvη2μ3+8a1a2a4ζvη2μ3+8a1a2
3ζvη2μ3+8a1a2
4ζvη2μ3
−4a1b2
1ζvη2μ3−8a1b1b3ζvη2μ3+4a1b2
2ζvη2μ3−8a1b2b4ζvη2μ3−4a2
2a3ζvη2μ3−8a2b1b2ζvη2μ3
−8a2b1b4ζvη2μ3+8a2b2b3ζvη2μ3+4a3b2
1ζvη2μ3−8a3b1b3ζvη2μ3−4a3b2
2ζvη2μ3+24a3b2b4ζvη2μ3
+8a4b1b2ζvη2μ3−8a4b1b4ζvη2μ3−24a4b2b3ζvη2μ3+12a3
1ζvη2μ2+12a2
1a3ζvη2μ2+12a1a2
2ζvη2μ2
+24a1a2a4ζvη2μ2+24a1a2
3ζvη2μ2+24a1a2
4ζvη2μ2−12a1b2
1ζvη2μ2−24a1b1b3ζvη2μ2+12a1b2
2ζvη2μ2
−24a1b2b4ζvη2μ2−12a2
2a3ζvη2μ2−24a2b1b2ζvη2μ2−24a2b1b4ζvη2μ2+24a2b2b3ζvη2μ2−16a1b2
1ζh
+12a3b2
1ζvη2μ2−24a3b1b3ζvη2μ2−12a3b2
2ζvη2μ2+72a3b2b4ζvη2μ2+24a4b1b2ζvη2μ2+16a3b2
2ζh
−24a4b1b4ζvη2μ2−72a4b2b3ζvη2μ2+12a3
1ζvη2μ+12a2
1a3ζvη2μ+12a1a2
2ζvη2μ+24a1a2a4ζvη2μ

Y. Wang et al.
+24a1a2
3ζvη2μ+24a1a2
4ζvη2μ−12a1b2
1ζvη2μ−24a1b1b3ζvη2μ+12a1b2
2ζvη2μ−24a1b2b4ζvη2μ
+32a1ζhη2μ3−12a2
2a3ζvη2μ−24a2b1b2ζvη2μ−24a2b1b4ζvη2μ+24a2b2b3ζvη2μ+12a3b2
1ζvη2μ
−24a3b1b3ζvη2μ−12a3b2
2ζvη2μ+72a3b2b4ζvη2μ+24a4b1b2ζvη2μ−24a4b1b4ζvη2μ−72a4b2b3ζvη2μ
+4a3
1ζvη2+4a2
1a3ζvη2+4a1a2
2ζvη2+8a1a2a4ζvη2+8a1a2
3ζvη2+8a1a2
4ζvη2−4a1b2
1ζvη2
−8a1b1b3ζvη2+4a1b2
2ζvη2−8a1b2b4ζvη2+96a1ζhη2u2−4a2
2a3ζvη2−8a2b1b2ζvη2−8a2b1b4ζvη2
+8a2b2b3ζvη2+4a3b2
1ζvη2−8a3b1b3ζvη2−4a3b2
2ζvη2+24a3b2b4ζvη2+8a4b1b2ζvη2−8a4b1b4ζvη2
−24a4b2b3ζvη2−16a1b2
1ζhμ−16a1b2
2ζhμ−16a1b2
3ζhμ−16a1b2
4ζhμ+96a1ζhη2μ−16a3b2
1ζhμ
−16a3b1b3ζhμ+16a3b2
2ζhμ−48a3b2b4ζhμ−32a4b1b2ζhμ−16a4b1b4ζhμ+48a4b2b3ζhμ−16a1b2
2ζh
−16a1b2
3ζh−16a1b2
4ζh+32a1ζhη2−16a3b2
1ζh−16a3b1b3ζh−48a3b2b4ζh−32a4b1b2ζh−16a4b1b4ζh
+48a4b2b3ζh), (A7)
F23 −3a2
1a2η2μ3−3a2
1a4η2μ3+6a1a2a3η2μ3+4a1b1b2η2μ3−4a4b2
2γ+16a4b2b4−12a3b1b2η2μ2
+4a1b1b4η2μ3−4a1b2b3η2μ3−3a3
2η2μ3+3a2
2a4η2μ3−6a2a2
3η2μ3−6a2a2
4η2μ3+2a2b2
1η2μ3
−4a2b1b3η2μ3+6a2b2
2η2μ3−4a2b2b4η2μ3+4a2b2
3η2μ3+4a2b2
4η2μ3−4a3b1b2η2μ3+8a3b2b3η2μ3
+2a4b2
1η2μ3−2a4b2
2η2μ3+8a4b2b4η2μ3−9a2
1a2η2μ2−9a2
1a4η2μ2+18a1a2a3η2μ2+12a1b1b2η2μ2
+12a1b1b4η2μ2−12a1b2b3η2μ2−9a3
2η2μ2+9a2
2a4η2μ2−18a2a2
3η2μ2−18a2a2
4η2μ2+6a2b2
1η2μ2
−12a2b1b3η2μ2+18a2b2
2η2μ2−12a2b2b4η2μ2+12a2b2
3η2μ2+12a2b2
4η2μ2−16a2η2γμ
3−8a2b2b4γ
+24a3b2b3η2μ2+6a4b2
1η2μ2−6a4b2
2η2μ2+24a4b2b4η2μ2−9a2
1a2η2μ−9a2
1a4η2μ+18a1a2a3η2μ
+12a1b1b2η2μ+12a1b1b4η2μ−12a1b2b3η2μ−9a3
2η2μ+9a2
2a4η2μ−18a2a2
3η2μ−18a2a2
4η2μ
+6a2b2
1η2μ−12a2b1b3η2μ+18a2b2
2η2μ−12a2b2b4η2μ+12a2b2
3η2μ+12a2b2
4η2μ−48a2η2γμ
2
−12a3b1b2η2μ+24a3b2b3η2μ+6a4b2
1η2μ−6a4b2
2η2μ+24a4b2b4η2μ−3a2
1a2η2−3a2
1a4η2+12a2b2
2γ
+6a1a2a3η2+4a1b1b2η2+4a1b1b4η2−4a1b2b3η2−3η2a3
2+3a2
2a4η2−6a2
3η2a2−6a2a2
4η2+2b2
1η2a2
−4b1η2a2b3+6b2
2η2a2−4a2b2b4η2+4η2a2b2
3+4a2b2
4η2−48a2η2γμ−4b2a3b1η2+8b2a3η2b3+2a4b2
1η2
−2a4b2
2η2+8a4b2b4η2+8a1b1b2γ+8a1b1b4γ−8a1b2b3γ+4a2b2
1γ−8a2b1b3γ+8a2b2
3γ+8a2b2
4γ
−16a2η2γ−8a3b1b2γ+16a3b2b3γ+4a4b2
1γ, (A8)
F3(a1,···a4,b1,···b4,
)− 1
8η2(μ+1)
34a3
1δvη2μ3+40a2
1a3δvη2μ3−12a1a2
2δvη2μ3−2a1b2
1δvη2μ3
−40a1b1b3δvη2μ3+2a1b2
2δvη2μ3+40a2
2a3δvη2μ3+4a2b1b2δvη2μ3−40a2b2b3δvη2μ3+36a3
3δvη2μ3
+36a3a2
4δvη2μ3−4a3b2
1δvη2μ3−4a3b2
2δvη2μ3−54a3b2
3δvη2μ3−18a3b2
4δvη2μ3−36a4b3b4δvη2μ3
+12a3
1δvη2μ2+ 120a2
1a3δvη2μ2−36a1a2
2δvη2μ2−6a1b2
1δvη2μ2−120a1b1b3δvη2μ2+6a1b2
2δvη2μ2
+ 120a2
2a3δvη2μ2+12a2b1b2δvη2μ2−120a2b2b3δvη2μ2+ 108a3
3δvη2μ2+ 108a3a2
4δvη2μ2
−12a3b2
1δvη2μ2−12a3b2
2δvη2μ2−162a3b2
3δvη2μ2−54a3b2
4δvη2μ2−108a4b3b4δvη2μ2+12a3
1δvη2μ
+ 120a2
1a3δvη2μ−36a1a2
2δvη2μ−6a1b2
1δvη2μ−120a1b1b3δvη2μ+6a1b2
2δvη2μ+ 120a2
2a3δvη2μ
+12a2b1b2δvη2μ−120a2b2b3δvη2μ+ 108a3
3δvη2μ+ 108a3a2
4δvη2μ−12a3b2
1δvη2μ−12a3b2
2δvη2μ
−162a3b2
3δvη2μ−54a3b2
4δvη2μ+ 144a3δhη2μ3−108a4a3b4δvη2μ+4a3
1δvη2+40a2
1a3δvη2
−12a1a2
2δvη2−2a1b2
1δvη2−40a1b1b3δvη2+2a1b2
2δvη2+40a2
2a3δvη2+4a2b1b2δvη2−40a2b2b3δvη2
+36a3
3δvη2+36a3a2
4δvη2−4a3b2
1δvη2−4a3b2
2δvη2−54a3b2
3δvη2−18a3b2
4δvη2+ 432a3δhη2μ2
+72a3η2μ3−36a4b3b4δvη2−20a1b2
1δhμ−80a1b1b3δhμ+20a1b2
2δhμ+40a2b1b2δhμ−80a2b2b3δhμ
−72a3b2
1δhμ−72a3b2
2δhμ−108a3b2
3δhμ−36a3b2
4δhμ+ 432a3δhη2μ+ 216a3η2μ2−72a4b3b4δhμ
−20a1b2
1δh−80a1b1b3δh+20a1b2
2δh+40a2b1b2δh−80a2b2b3δh−72a3b2
1δh−72a3b2
2bh −108a3b2
3bh
−36a3b2
4δh+ 144a3δhη2+ 216a3η2μ−72a4b3b4δh+72a3η22+−12a2
1a2ζvη2μ3−24a2
1a4ζvη2μ3

Nonlinear vibration and dynamic performance analysis
+8a1b1b2ζvη2μ3+24a1b1b4ζvη2μ3+8a1b2b3ζvη2μ3+4a3
2ζvη2μ3−24a2
2a4ζvη2μ3+4a2b2
1ζvη2μ3
−8a2b1b3ζvη2μ3−4a2b2
2ζvη2μ3+24a2b2b4ζvη2μ3−12a2
3a4ζvη2μ3+24a3b3b4ζvη2μ3−12a3
4ζvη2μ3
−12a4b2
3ζvη2μ3+12a4b2
4ζvη2μ3−36a2
1a2ζvη2μ2−72a2
1a4ζvη2μ2+24a1b1b2ζvη2μ2+48a4b2
3ζh
+72a1b1b4ζvη2μ2+24a1b2b3ζvη2μ2+12a3
2ζvη2μ2−72a2
2a4ζvη2μ2+12a2b2
1ζvη2μ2+48a4b2
4ζh
−24a2b1b3ζvη2μ2−12a2b2
2ζvη2μ2+72a2b2b4ζvη2μ2−36a2
3a4ζvη2μ2+72a3b3b4ζvη2μ2
−36a3
4ζvη2μ2−36a4b2
3ζvη2μ2+36a4b2
4ζvη2μ2−36a2
1a2ζvη2μ−72a2
1a4ζvη2μ+24a1b1b2ζvη2μ
+72a1b1b4ζvη2μ+24a1b2b3ζvη2μ+12a3
2ζvη2μ−72a2
2a4ζvη2μ+12a2b2
1ζvη2μ−24a2b1b3ζvη2μ
−12a2b2
2ζvη2μ+72a2b2b4ζvη2μ−36a2
3a4ζvη2μ+72a3b3b4ζvη2μ−36a3
4ζvη2μ−36a4b2
3ζvη2μ
+36a4b2
4ζvη2μ−96a4ζhη2μ3−12a2
1a2ζvη2−24a2
1a4ζvη2+8a1b1b2ζvη2+24a1b1b4ζvη2+8a1b2b3ζvη2
+4a3
2ζvη2−24a2
2a4ζvη2+4a2b2
1ζvη2−8a2b1b3ζvη2−4a2b2
2ζvη2+24a2b2b4ζvη2−12a2
3a4ζvη2
+24a3b3b4ζvη2−12a3
4ζvη2−12a4b2
3ζvη2+12a4b2
4ζvη2−288a4ζhη2μ2+32a1b1b2ζhμ+48a1b1b4ζhμ
−16a1b2b3ζhμ+16a2b2
1ζhμ+16a2b1b3ζhμ−16a2b2
2ζhμ+48a2b2b4ζhμ+48a4b2
1ζhμ+48a4b2
2ζhμ
+48a4b2
3ζhμ+48a4b2
4ζhμ−288a4ζhη2μ+32a1b1b2ζh+48a1b1b4ζh−16a1b2b3ζh+16a2b2
1ζh
+16a2b1b3ζh−16a2b2
2ζh+48a2b2b4ζh+48a4b2
1ζh+48a4b2
2ζh−96a4ζhη2+−a3
1η2μ3−6a2
1a3η2μ3
+3a1a2
2η2μ3+2a1b2
1η2μ3+8a1b1b3η2μ3−2a1b2
2η2μ3−6a2
2a3η2μ3−4a2b1b2η2μ3+8a2b2b3η2μ3
−3a3
3η2μ3−3a3a2
4η2μ3+4a3b2
1η2μ3+4a3b2
2η2μ3+6a3b2
3η2μ3+2a3b2
4η2μ3+4a4b3b4η2μ3−3a3
1η2μ2
−18a2
1a3η2μ2+9a1a2
2η2μ2+6a1b2
1η2μ2+24a1b1b3η2μ2−6a1b2
2η2μ2−18a2
2a3η2μ2−12a2b1b2η2μ2
+24a2b2b3η2μ2−9a3
3η2μ2−9a3a2
4η2μ2+12a3b2
1η2μ2+12a3b2
2η2μ2+18a3b2
3η2μ2+6a3b2
4η2μ2
−16a3η2γμ
3+12a4b3b4η2μ2−3a3
1η2μ−18a2
1a3η2μ+9a1a2
2η2μ+6a1b2
1η2μ+24a1b1b3η2μ−6a1b2
2η2μ
−18a2
2a3η2μ−12a2b1b2η2μ+24a2b2b3η2μ−9a3
3η2μ−9a3a2
4η2μ+12a3b2
1η2μ+12a3b2
2η2μ
+18a3b2
3η2μ+6a3b2
4η2μ−48a3η2γμ
2+12a4b3b4η2μ−a3
1η2−6a2
1a3η2+3a1a2
2η2+2a1b2
1η2+8a1b1b3η2
−2a1b2
2η2−6a3η2a2
2−4b2b1η2a2+8b2η2a2b3−3a3
3η2−3a3a2
4η2+4a3b2
1η2+4b2
2a3η2+6a3η2b2
3
+2a3b2
4η2−48a3η2γμ+4a4b3b4η2+4a1b2
1γ+16a1b1b3γ−4a1b2
2γ−8a2b1b2γ+16a2b2b3γ+8a3b2
1γ
+8a3b2
2γ+12a3b2
3γ+4a3b2
4γ−16a3η2γ+8a4b3b4γ,(A9)
F4(a1,···a4,b1,···b4,
)− 1
8η2μ3+3μ2+3μ+1
12a2
1a2δvη2μ3+40a2
1a4δvη2μ3−4a1b1b2δvη2μ3
−40a1b1b4δvη2μ3−4a3
2δvη2μ3+40a2
2a4δvη2μ3−2a2b2
1δvη2μ3+2a2b2
2δvη2μ3
−40a2b2b4δvη2μ3+36a2
3a4δvη2μ3−36a3b3b4δvη2μ3+36a3
4δvη2μ3−4a4b2
1δvη2μ3
−4a4b2
2δvη2μ3−18a4b2
3δvη2μ3−54a4b2
4δvη2μ3+36a2
1a2δvη2μ2+ 120a2
1a4δvη2μ2
−12a1b1b2δvη2μ2−120a1b1b4δvη2μ2−12a3
2δvη2μ2+ 120a2
2a4δvη2μ2−6a2b2
1δvη2μ2
+6a2b2
2δvη2μ2−120a2b2b4δvη2μ2+ 108a2
3a4δvη2μ2−108a3b3b4δvη2μ2+ 108a3
4δvη2μ2
−12a4b2
1δvη2μ2−12a4b2
2δvη2μ2−54a4b2
3δvη2μ2−162a4b2
4δvη2μ2+36a2
1a2δvη2μ
+ 120a2
1a4δvη2μ−12a1b1b2δvη2μ−120a1b1b4δvη2μ−12a3
2δvη2μ+ 120a2
2a4δvη2μ
−6a2b2
1δvη2μ+6a2b2
2δvη2μ−120a2b2b4δvη2μ+ 108a2
3a4δvη2μ−108a3b3b4δvη2μ
+ 108a3
4δvη2μ−12a4b2
1δvη2μ−12a4b2
2δvη2μ−54a4b2
3δvη2μ−162a4b2
4δvη2μ
+ 144a4δhη2μ3+12a2
1a2δvη2+40a2
1a4δvη2−4a1b1b2δvη2−40a1b1b4δvη2−4a3
2δvη2
+40a2
2a4δvη2−2a2b2
1δvη2+2a2b2
2δvη2−40a2b2b4δvη2+36a2
3a4δvη2−36a3b3b4δvη2
+36a3
4δvη2−4a4b2
1δvη2−4a4b2
2δvη2−18a4b2
3δvη2−54a4b2
4δvη2+ 432a4δhη2μ2
+72η2μ3a4−40a1b1b2δhμ−80a1b1b4δhμ−20a2b2
1δhμ+20a2b2
2δhμ−80a2b2b4δhμ
−72a3b3b4δhμ−72a4b2
1δhμ−72a4b2
2δhμ−36a4b2
3δhμ−108a4b2
4δhμ+ 432a4δhn2μ

Y. Wang et al.
+ 216a4η2μ2−40a1b1b2δh−80a1b1b4δh−20a2b2
1δh+20a2b2
2δh−80a2b2b4δh−72a3b3b4δh
−72a4b2
1δh−72a4b2
2δh−36a4b2
3δh−108a4b2
4δh+ 144a4δhη2+ 216a4η2μ+72a4η22
+4a3
1ζvη2μ3+24a2
1a3ζvη2μ3−12a1a2
2ζvη2μ3−4a1b2
1ζvη2μ3−24a1b1b3ζvη2μ3+4a1b2
2ζvη2μ3
+8a1b2b4ζvη2μ3+24a2
2a3ζvη2μ3+8a2b1b2ζvη2μ3−8a2b1b4ζvη2μ3−24a2b2b3ζvη2μ3
+12a3
3ζvη2μ3+12a3a2
4ζvη2μ3−12a3b2
3ζvη2μ3+12a3b2
4ζvη2μ3−24a4b3b4ζvη2μ3
+12a3
1ζvη2μ2+72a2
1a3ζvη2μ2−36a1a2
2ζvη2μ2−12a1b2
1ζvη2μ2−72a1b1b3ζvη2μ2
+12a1b2
2ζvη2μ2+24a1b2b4ζvη2μ2+72a2
2a3ζvη2μ2+24a2b1b2ζvη2μ2
−24a2b1b4ζvη2μ2−72a2b2b3ζvη2μ2+36a3
3ζvη2μ2+36a3a2
4ζvη2μ2−36a3b2
3ζvη2μ2+36a3b2
4ζvη2μ2
−72a4b3b4ζvη2μ2+12a3
1ζvη2μ+72a2
1a3ζvη2μ−36a1a2
2ζvη2μ−12a1b2
1ζvη2μ−72a1b1b3ζvη2μ
+12a1b2
2ζvη2μ+24a1b2b4ζvη2μ+72a2
2a3ζvη2μ+24a2b1b2ζvη2μ−24a2b1b4ζvη2μ−72a2b2b3ζvη2μ
+36a3
3ζvη2μ+36a3a2
4ζvη2μ−36a3b2
3ζvη2μ+36a3b2
4ζvη2μ+96a3ζhη2μ3−72a4b3b4ζvη2μ
+4a3
1ζvη2+24a2
1a3ζvη2−12a1a2
2ζvη2−4a1b2
1ζvη2−24a1b1b3ζvη2+4a1b2
2ζvη2+8a1b2b4ζvη2
+24a2
2a3ζvη2+8a2b1b2ζvη2−8a2b1b4ζvη2−24a2b2b3ζvη2+12a3
3ζvη2+12a3a2
4ζvη2−12a3b2
3ζvη2
+12a3b2
4ζvη2+ 288a3ζhη2μ2−24a4b3b4ζvη2−16a1b2
1ζhμ−48a1b1b3ζhμ+16a1b2
2ζhμ−16a1b2b4ζhμ
+32a2b1b2ζhμ+16a2b1b4ζhμ−48a2b2b3ζhμ−48a3b2
1ζhμ−48a3b2
2ζhμ−48a3b2
3ζhμ−48a3b2
4ζhμ
+ 288a3ζhη2μ−16a1b2
1ζh−48a1b1b3ζh+16a1b2
2ζh−16a1b2b4ζh+32a2b1b2ζh+16a2b1b4ζh
−48a2b2b3ζh−48a3b2
1ζh−48a3b2
2ζh−48a3b2
3ζh−48a3b2
4ζh+96a3ζhη2+−3a2
1a2η2μ3
−6a2
1a4η2μ3+4a1b1b2η2μ3+8a1b1b4η2μ3+a3
4ημ3−6a2
2a4η2μ3+2a2b2
1η2μ3−2a2b2
2η2μ3
+8a2b2b4η2μ3−3a2
3a4η2μ3+4a3b3b4η2μ3−3a3
4η2μ3+4a4b2
1η2μ3+4a4b2
2η2μ3+2a4b2
3η2μ3
+6a4b2
4η2μ3−9a2
1a2η2μ2−18a2
1a4η2μ2+12a1b1b2η2μ2+24a1b1b4η2μ2+3a3
2η2μ2−18a2
2a4η2μ2
+6a2b2
1η2μ2−6a2b2
2η2μ2+24a2b2b4η2μ2−9a2
3a4η2μ2+12a3b3b4η2μ2−9a3
4η2μ2+12a4b2
1η2μ2
+12a4b2
2η2μ2+6a4b2
3η2μ2+18a4b2
4η2μ2−16a4η2γμ
3−9a2
1a2η2μ−18a2
1a4η2μ+12a1b1b2η2μ
+24a1b1b4η2μ+3a3
2η2μ−18a2
2a4η2μ+6a2b2
1η2μ−6a2b2
2η2μ+24a2b2b4η2μ−9a2
3a4η2μ
+12a3b3b4η2μ−9a3
4η2μ+12a4b2
1η2μ+12a4b2
2η2μ+6a4b2
3η2μ+18a4b2
4η2μ−48a4η2rμ2−3a2
1a2η2
−6a2
1a4η2+4a1b1b2η2+8a1b1b4η2+η2a3
2−6a2
2a4η2+2b2
1η2a2−2b2
2η2a2+8a2b2b4η2−3a2
3a4η2
+4a3b3b4η2−3a3
4η2+4a4b2
1η2+4a4b2
2η2+2a4b2
3η2+6a4b2
4η2−48a4η2γμ+8a1b1b2γ+16a1b1b4γ
+4a2b2
1γ−4a2b2
2γ+16a2b2b4γ+8a3b3b4γ+8a4b2
1γ+8a4b2
2γ+4a4b2
3γ+12a4b2
4γ−16a4η2γ,(A10)
F5(a1,···a4,b1,···b4,
)1
4η2(μ+1)
33a2
1b1δvη2μ3+5a2
1b3δvη2μ3+2a1a2b2δvη2μ3+10a1a2b4δvη2μ3
+2a1a3b1δvη2μ3+20a1a3b3δvη2μ3+2a1a4b2δvη2μ3+20a1a4b4δvη2μ3+a2
2b1δvη2μ3−5a2
2b3δvη2μ3
−2a2a3b2δvη2μ3+2a2a4b1δvη2μ3+2a2
3b1δvη2μ3+2a2
4b1δvη2μ3+9a2
1b1δvη2μ2+15a2
1b3δvη2μ2
+6a1a2b2δvη2μ2+30η2a1a2b4δvμ2+6a1a3b1δvη2μ2+60η2a1a3b3δvμ2+6η2a1a4b2δvμ2+60η2a1a4b4δvμ2
+3a2
2b1δvη2μ2−15a2
2b3δvη2μ2−6a2a3b2δvη2μ2+6η2a2a4b1δvμ2+6η2a2
3b1δvμ2+6η2a2
4b1δvμ2
+9a2
1b1δvη2μ+15a2
1b3δvη2μ+6a1a2b2δvη2μ+30η2a1a2b4δvμ+6a1a3b1δvη2μ+60η2a1a3b3δvμ
+6η2a1a4b2δvμ+60η2a1a4b4δvμ+3a2
2b1δvη2μ−15a2
2b3δvη2μ−6a2a3b2δvη2μ+6η2a2a4b1δvμ
+6η2a2
3b1δvμ+6η2a2
4b1δvμ−4η2b1δvμ3+3a2
1b1δvη2+5a2
1b3δvη2+2a1a2b2δvη2+10η2a1a2b4δv
+2a1a3b1δvη2+20η2a1a3b3δv+2η2a1a4b2δv+20η2a1a4b4δv+a2
2b1δvη2−5a2
2b3δvη2−2a2a3b2δvη2
+2η2b1δva2a4+2η2b1δva2
3+2η2b1δva2
4−12η2b1δvμ2−4b1η2μ3−4η2μ3Ybm +6a2
1b1δhμ+2a2
1b3δhμ
+4a1a2b2δhμ+4μδha1b4a2+20a1a3b1δhμ+40μδhb3a3a1+20μδhb2a1a4+40μδha1b4a4+2a2
2b1δhμ
−2a2
2b3δhμ−20a2a3b2δhμ+20μδha2b1a4+36μδha2
3b1+36μδhb1a2
4−4b3
1δhμ−12b2
1b3δhμ

Nonlinear vibration and dynamic performance analysis
−4b1b2
2δhμ−24b1b2b4δhμ−40b1b2
3δhμ−40b1b2
4δhμ−12b1δvη2μ−12b1η2μ2+12b2
2b3δhμ
−12η2μ2Ybm +6a2
1b1δh+2a2
1b3δh+4a1a2b2δh+4a1a2b4δh+20a1a3b1δh+40a1a3b3δh+20a1a4b2δh
+40a1a4b4δh+2a2
2b1δh−2a2
2b3δh−20a2a3b2δh+20a2a4b1δh+36a2
3b1δh+36a2
4b1δh−4b3
1δh
−12b2
1b3δh−4b1b2
2δh−12b2
1b3δh−4b1b2
2δh−24b1b2b4δh−40b1b2
3δh−40b1b2
4δh−4b1δvη2
−12b1η2μ+12b2
2b3δh−12η2μYbm −4b1n2−4η2Ybm2+−4a2
1b2ζvη2μ3−4a2
1b4ζvη2μ3
+8a1a2b3ζvη2μ3−12a1a3b4ζvη2μ3+12a1a4b3ζvη2μ3−4a2
2b2ζvη2μ3+4a2
2b4ζvη2μ3−4a2a3b3ζvη2μ3
−4a2a4b4ζvη2μ3−4a2
3b2ζvη2μ3−4a2
4b2ζvη2μ3−12a2
1b2ζvη2μ2−12a2
1b4ζvη2μ2+24a1a2b3ζvη2μ2
−36a1a3b4ζvη2μ2+36a1a4b3ζvη2μ2−12a2
2b2ζvη2μ2+12a2
2b4ζvη2μ2−12a2a3b3ζvη2μ2
−12a2a4b4ζvη2μ2−12a2
3b2ζvη2μ2−12a2
4b2ζvη2μ2−12a2
1b2ζvη2μ−12a2
1b4ζvη2μ+24a1a2b3ζvη2μ
−36a1a3b4ζvη2μ+36a1a4b3ζvη2μ−12a2
2b2ζvη2μ+12a2
2b4ζvη2μ−12a2a3b3ζvη2μ−12a2a4b4ζvη2μ
−12a2
3b2ζvη2μ−12a2
4b2ζvη2μ+8b2ζvη2μ3−4a2
1b2ζvη2−4a2
1b4ζvη2+8a1a2b3ζvη2−12a1a3b4ζvη2
+12a1a4b3ζvη2−4a2
2b2ζvη2+4a2
2b4ζvη2−4a2a3b3ζvη2−4a2a4b4ζvη2−4a2
3b2ζvη2−4a2
4b2ζvη2
+24b2ζvη2μ2+4a2
1b2ζhμ+4a2
1b4ζhμ−8a1a2b1ζhμ−8a1a2b3ζhμ+8a1a3b2ζhμ+24a1a3b4ζhμ
−8a1a4b1ζhμ−24a1a4b3ζhμ−4a2
2b2ζhμ−4a2
2b4ζhμ+8a2a3b1ζhμ−8a2a3b3ζhμ+8a2a4b2ζhμ
−8a2a4b4ζhμ+4b2
1b2ζhμ+4b2
1b4ζhμ−8b1b2b3ζhμ+4b3
2ζhμ−4b2
2b4ζhμ+8b2b2
3ζhμ+8b2b2
4ζhμ
+24b2ζvη2μ+4a2
1b2ζh+4a2
1b4ζh−8a1a2b1ζh−8a1a2b3eζh+8a1ab2ζh+24a1a3b4ζh−8a1a4b1ζh
−24a1a4b3ζh−4a2
2b2ζh−4a2
2b4ζh+8a2a3b1ζh−8a2a3b3ζh+8a2a4b2ζh−8a2a4b4ζh+4b2
1b2ζh
+4b2
1b4ζh−8b1b2b3ζh+4b3
2ζh−4b2
2b4ζh+8b2b2
3ζh+8b2b2
4ζh+8b2ζvη2+−3a2
1b1η2μ3
−a2
1b3η2μ3−2a1a2b2η2μ3−2a1a2b4η2μ3−2a1a3b1η2μ3−4a1a3b3η2μ3−2a1a4b2η2μ3−4a1a4b4η2μ3
−a2
2b1η2μ3+a2
2b3η2μ3+2a2a3b2η2μ3−2a2a4b1η2μ3−2a2
3b1η2μ3−2a2
4b1η2μ3−9a2
1b1η2μ2
−3a2
1b3η2μ2−6a1a2b2η2μ2−6a1a2b4η2μ2−6a1a3b1η2μ2−12a1a3b3η2μ2−6a1a4b2η2μ2
−12a1a4b4η2μ2−3a2
2b1η2μ2+3a2
2b3η2μ2+6a2a3b2η2μ2−6a2a4b1η2μ2−6a2
3b1η2μ2−6a2
4b1η2μ2
+8b1η2γμ
3−9a2
1b1η2μ−3a2
1b3η2μ−6a1a2b2η2μ−6a1a2b4η2μ−6a1a3b1η2μ−12a1a3b3η2μ
−6a1a4b2η2μ−12a1a4b4η2μ−3a2
2b1η2μ+3a2
2b3η2μ+6a2a3b2η2μ−6a2a4b1η2μ−6a2
3b1η2μ
−6a2
4b1η2μ+16b1η2γμ
2+4b1η2μ3−3a2
1b1η2−a2
1b3η2−2a1a2b2η2−2a1a2b4η2−2a1a3b1η2
−4a1a3b3η2−2a1a4b2η2−4a1a4b4η2−a2
2b1η2+a2
2b3η2+2a2a3b2η2−2a2a4b1η2−2a2
3b1η2
−2a2
4b1η2+8b1η2γμ +12b1η2μ2−6a2
1b1γ−2a2
1b3γ−4a1a2b2γ−4a1a2b4γ−4a1a3b1γ−8a1a3b3γ
−4a1a4b2γ−8a1a4b4γ−2a2
2b1γ+2a2
2b3γ+4a2a3b2γ−4a2a4b1γ−4a2
3b1γ−4a2
4b1γ+3b3
1γ+3b2
1b3r
+3b1b2
2γ+6b1b2b4γ+6b1b2
3γ+6b1b2
4γ+12b1η2μ−3b2
2b3γ+4b1η2,(A11)
F6(a1,···a4,b1,···b4,
)1
4η2(μ+1)
3a2
1b2δvη2μ3+5a2
1b4δvη2μ3+2a1a2b1δvη2μ3−10a1a2b3δvη2μ3
−2a1a3b2δvη2μ3+2a1a4b1δvη2μ3+3a2
2b2δvη2μ3−5a2
2b4δvη2μ3−2a2a3b1δvη2μ3+20a2a3b3δvη2μ3
−2a2a4b2δvη2μ3+20a2a4b4δvη2μ3+2a2
3b2δvη2μ3+2a2
4b2δvη2μ3+3a2
1b2δvη2μ2+15a2
1b4δvη2μ2
+6a1a2b1δvη2μ2−30a1a2b3δvη2μ2−6a1a3b2δvη2μ2+6a1a4b1δvη2μ2+9a2
2b2δvη2μ2−15a2
2b4δvη2μ2
−6a2a3b1δvη2μ2+60a2a3b3δvη2μ2−6a2a4b2δvη2μ2+60a2a4b4δvη2μ2+6a2
3b2δvη2μ2+6a2
4b2δvη2μ2
+3a2
1b2δvη2μ+15a2
1b4δvη2μ+6a1a2b1δvη2μ−30a1a2b3δvη2μ−6a1a3b2δvη2μ+6a1a4b1δvη2μ
+9a2
2b2δvη2μ−15a2
2b4δvη2μ−6a2a3b1δvη2μ+60a2a3b3δvη2μ−6a2a4b2δvη2μ+60a2a4b4δvη2μ
+6a2
3b2δvη2μ+6a2
4b2δvη2μ−4b2δvη2μ3+a2
1b2δvη2+5a2
1b4δvη2+2a1a2b1δvη2−10a1a2b3δvη2
−2a1a3b2δvη2+2a1a4b1δvη2+3a2
2b2δvη2−5a2
2b4δvη2−2a2a3b1δvη2+20a2a3b3δvη2−2a2a4b2δvη2
+20a2a4b4δvη2+2a2
3b2δvη2+2a2
4b2δvη2−12b2δvη2μ2−4b2η2μ3+2a2
1b2δhμ+2a2
1b4δhμ

Y. Wang et al.
+4a1a2b1δhμ−4a1a2b3δhμ−20a1a3b2δhμ+20a1a4b1δhμ+6a2
2b2δhμ−2a2
2b4δhμ−20a2a3b1δhμ
+40a2a3b3δhμ−20a2a4b2δhμ+40a2a4b4δhμ+36a2
3b2δhμ+36a2
4b2δhμ−4b2
1b2δhμ−12b2
1b4δhμ
+24b1b2b3δhμ−4b3
2δhμ+12b2
2b4δhμ−40b2b2
3δhμ−40b2b2
4δhμ−12b2δvη2μ−12b2η2μ2+2δhb2a2
1
+2δhb4a2
1+4δha1a2b1−4δha1a2b3−20δhb2a1a3+20δha1a4b1+6δhb2a2
2−2δha2
2b4−20δha2a3b1
+40δha2a3b3−20δha2a4b2+40δha2a4b4+36δhb2a2
3+36δhb2a2
4−4δhb2b2
1−12δhb4b2
1+24δhb2b3b1
−4δhb3
2+12δhb2
2b4−40δhb2b2
3−40b2b2
4δh−4b2δvη2−12b2η2μ−4b2η22+4a2
1b1ζvη2μ3
+4a2
1b3ζvη2μ3+8a1a2b4ζvη2μ3+4a1a3b3ζvη2μ3+4a1a4b4ζvη2μ3+4a2
2b1ζvη2μ3−4a2
2b3ζvη2μ3
−12a2a3b4ζvη2μ3+12a2a4b3ζvη2μ3+4a2
3b1ζvη2μ3+4a2
4b1ζvη2μ3+12a2
1b1ζvη2μ2+12a2
1b3ζvη2μ2
+24a1a2b4ζvη2μ2+12a1a3b3ζvη2μ2+12a1a4b4ζvη2μ2+12a2
2b1ζvη2μ2−12a2
2b3ζvη2μ2
−36a2a3b4ζ2
vμ2+36a2a4b3ζvη2μ2+12a2
3b1ζvη2μ2+12a42b1ζvη2μ2+12a2
1b1ζvη2μ
+12a2
1b3ζvη2μ+24a1a2b4ζvη2μ+12a1a3b3ζvη2μ+12a1a4b4ζvη2μ+12a22b1ζvη2μ
−12a2
2b3ζvη2μ−36a2a3b4ζvη2μ+36a2a4b3ζvη2μ+12a2
3b1ζvη2μ+12a2
4b1ζvη2μ−8b1ζvη2μ3
+4a2
1b1ζvη2+4a2
1b3ζvη2+8a1a2b4ζvη2+4a1a3b3ζvη2+4a1a4b4ζvη2+4a2
2b1ζvη2−4a2
2b3ζvη2
−12a2a3b4ζvη2+12a2a4b3ζvη2+4a2
3b1ζvη2+4a2
4b1ζvη2−24b1ζvη2μ2+4a2
1b1ζhμ−4a2
1b3ζhμ
+8a1a2b2ζhμ−8a1a2b4ζhμ+8a1a3b1ζhμ+8a1a3b3ζhμ+8a1a4b2ζhμ+8a1a4b4ζhμ−4a2
2b1ζhμ
+4a2
2b3ζhμ−8a2a3b2ζhμ+24a2a3b4ζhμ+8a2a4b1ζhμ−24a2a4b3ζhμ−4b3
1ζhμ−4b2
1b3ζhμ
−4b1b2
2ζhμ−8b1b2b4ζhμ−8b1b2
3ζhμ−8b1b2
4ζhμ−24b1ζvη2μ+4b2
2b3ζhμ+4a2
1b1ζh−4a2
1b3ζh
+8a1a2b2ζh−8a1a2b4ζh+8a1a3b1ζh+8a1a3b3ζh+8a1a4b2ζh+8a1a4b4ζh−4a2
2b1ζh+4a2
2b3ζh
−8a2a3b2ζh+24a2a3b4ζh+8a2a4b1ζh−24a2a4b3ζh−4b3
1ζh−4b2
1b3ζh−4b1b2
2ζh−8b1b2b4ζh
−8b1b2
3ζh−8b1b2
4ζh−8b1ζvη2+4b2
2b3ζh+−a2
1b2η2μ3−a2
1b4η2μ3−2a1a2b1η2μ3+2a1a2b3η2μ3
+2a1a3b2η2μ3−2a1a4b1η2μ3−3a2
2b2η2μ3+a2
2b4η2μ3+2a2a3b1η2μ3−4a2a3b3η2μ3+2a2a4b2η2μ3
−4a2a4b4η2μ3−2a2
3b2η2μ3−2a2
4b2η2μ3−3a2
1b2η2μ2−3a2
1b4η2μ2−6a1a2b1η2μ2+6a1a2b3η2μ2
+6a1a3b2η2μ2−6a1a4b1η2μ2−9a2
2b2η2μ2+3a2
2b4η2μ2+6a2a3b1η2μ2−12a2a3b3η2μ2+6a2a4b2η2μ2
−12a2a4b4η2μ2−6a2
3b2η2μ2−6a2
4b2η2μ2+8b2η2γμ
3−3a2
1b2η2μ−3a2
1b4η2μ−6a1a2b1η2μ
+6a1a2b3η2μ+6a1a3b2η2μ−6a1a4b1η2μ−9a2
2b2η2μ+3a2
2b4η2μ+6a2a3b1η2μ−12a2a3b3η2μ
+6a2a4b2η2μ−12a2a4b4η2μ−6a2
3b2η2μ−6a2
4b2η2μ+16b2η2γμ
2+4b2η2μ3−a2
1b2η2−a2
1b4η2
−2a1a2b1η2+2a1a2b3η2+2a1a3b2η2−2a1a4b1η2−3a2
2b2η2+a2
2b4η2+2a2a3b1η2−4a2a3b3η2
+2a2a4b2η2−4a2a4b4η2−2a2
3b2η2−2a2
4b2η2+8b2η2γμ +12b2η2μ2−2a2
1b2γ−2a2
1b4γ−4a1a2b1γ
+4a1a2b3γ+4a1a3b2γ−4a1a4b1γ−6a2
2b2γ+2a2
2b4r+4a2a3b1γ−8a2a3b3r+4a2a4b2γ−8a2a4b4γ−4a2
3b2γ
−4a2
4b2γ+3b2
1b2γ+3b2
1b4γ−6b1b2b3γ+3b3
2γ−3b2
2b4γ+6b2b2
3γ+6b2b2
4γ+12b2η2μ+4b2η2,(A12)
F7(a1,···a4,b1,···b4,
)1
4η2(μ+1)
35a2
1b1δvη2μ3+18a2
1b3δvη2μ3−10a1a2b2δvη2μ3+20a1a3b1δvη2μ3
−5a2
2b1δvη2μ3+18a2
2b3δvη2μ3+20a2a3b2δvη2μ3+27a2
3b3δvη2μ3+18a3a4b4δvη2μ3+9a2
4b3δvη2μ3
+15a2
1b1δvη2μ2+54a2
1b3δvη2μ2−30a1a2b2δvη2μ2+60a1a3b1δvη2μ2−15a2
2b1δvη2μ2+54a2
2b3δvη2μ2
+60a2a3b2δvη2μ2+ 81η2a2
3b3δvμ2+54η2a3a4b4δvμ2+ 27η2a2
4b3δvμ2+15η2a2
1b1δvμ+54η2a2
1b3δvμ
−30η2a1a2b2δvμ+60η2a1a3b1δvμ−15η2a2
2b1δvμ+54η2a2
2b3δvμ+60η2a2a3b2δvμ+ 81η2a2
3b3δvμ
+54η2a3a4b4δvμ+ 27η2a2
4b3δvμ−36η2b3δvμ3+5η2δvb1a2
1+18η2b3δva2
1−10η2δvb2a2a1+20η2δva3b1a1
−5η2δvb1a2
2+18η2b3δva2
2+20η2δvb2a3a2+ 27η2b3δva2
3+ 18η2δvb4a3a4+9η2b3δva2
4−108η2b3δvμ2
−36η2b3μ3+2μδhb1a2
1+4μδhb3a2
1−4μδhb2a2a1+40μδha3b1a1−2μδhb1a2
2+4μδha2
2b3+40μδhb2a3a2
+54μδha2
3b3+ 36μδhb4a3a4+18μδhb3a2
4−4μδhb3
1−40μδhb2
1b3+12μδhb2
2b1−40μδhb2
2b3−36μδhb3
3

Nonlinear vibration and dynamic performance analysis
−36b3b2
4δhμ−108b3δvη2μ−108b3η2μ2+2a2
1b1δh+4a2
1b3δh−4a1a2b2δh+40a1a3b1δh−2a2
2b1δh+4a2
2b3δh
+40a2a3b2δh+54a2
3b3δh+36a3a4b4δh+18a2
4b3δh−4b3
1δh−40b2
1b3δh+12b1b2
2δh−40b2
2b3δh−36b3
3δh
−36b3b2
4δh−36b3δvη2−108b3η2μ−36b3η22+−4a2
1b2ζvη2μ3−12a2
1b4ζvη2μ3−8a1a2b1ζvη2μ3
−4a1a3b2ζvη2μ3−12a1a4b1ζvη2μ3+4a2
2b2ζvη2μ3−12a2
2b4ζvη2μ3+4a2a3b1ζvη2μ3−12a2a4b2ζvη2μ3
−12a2
3b4ζvη2μ3−12a2
4b4ζvη2μ3−12a2
1b2ζvη2μ2−36a2
1b4ζvη2μ2−24a1a2b1ζvη2μ2−12a1a3b2ζvη2μ2
−36a1a4b1ζvη2μ2+12a2
2b2ζvη2μ2−36a2
2b4ζvη2μ2+12a2a3b1ζvη2μ2−36a2a4b2ζvη2μ2−36a2
3b4ζvη2μ2
−36a2
4b4ζvη2μ2−12a2
1b2ζvη2μ−36a2
1b4ζvη2μ−24a1a2b1ζvη2μ−12a1a3b2ζvη2μ−36a1a4b1ζvη2μ
+12a2
2b2ζvη2μ−36a2
2b4ζvη2μ+12a2a3b1ζvη2μ−36a2a4b2ζvη2μ−36a2
4b4ζvη2μ−36a2
4b4ζvη2μ
+24b4ζvη2μ3−4a2
1b2ζvη2−12a2
1b4ζvη2−8a1a2b1ζvη2−4a1a3b2ζvη2−12a1a4b1ζvη2+4a2
2b2ζvη2
−12a2
2b4ζvη2+4a2a3b1ζvη2−12a2a4b2ζvη2−12a2
3b4ζvη2−12a2
4b4ζvη2+72b4ζvμ2μ2−4a2
1b2ζhμ
−8a1a2b1ζhμ+8a1a3b2ζhμ−24a1a4b1ζhμ+4a2
2b2ζhμ−8a2a3b1ζhμ−24a2a4b2ζhμ+12a2
3b4ζhμ
−24a3a4b3ζhμ−12a2
4b4ζhμ+12b2
1b2ζhμ+24b2
1b4ζhμ−4b3
2ζhμ+24b2
2b4ζhμ+12b2
3b4ζhμ
+12b3
4ζhμ+72b4ζvη2μ−4a2
1b2ζh−8a1a2b1ζh+8a1a3b2ζh−24a1a4b1ζh+4a2
2b2ζh−8a2a3b1ζh
−24a2a4b2ζh+12a2
3b4ζh−24a3a4b3ζh−12a2
4b4ζh+12b2
1b2ζh+24b2
1b4ζh−4b3
2eh +24b2
2b4ζh
+12b2
3b4ζh+12b3
4ζh+24b4ζvη2+−a2
1b1η2μ3−2a2
1b3η2μ3+2a1a2b2η2μ3−4a1a3b1η2μ3+a2
2b1η2μ3
−2a2
2b3η2μ3−4a2a3b2η2μ3−3a2
3b3η2μ3−2a3a4b4η2μ3−a2
4b3η2μ3−3a2
1b1η2μ2
−6a2
1b3η2μ2+6a1a2b2η2μ2−12a1a3b1η2μ2+3a2
2b1η2μ2−6a2
2b3η2μ2−12a2a3b2η2μ2−9a2
3b3η2μ2
−6a3a4b4η2μ2−3a2
4b3η2μ2+8b3η2γμ
3−3a2
1b1η2μ−6a2
1b3η2μ+6a1a2b2η2μ−12a1a3b1η2μ+3a2
2b1η2μ
−6a2
2b3η2μ−12a2a3b2η2μ−9a2
3b2η2μ−6a3a4b4η2μ−3a2
4b3η2μ+16b3η2γμ
2+4η2b3μ3−a2
1b1η2−2a2
1b3η2
+2a1a2b2η2−4a1a3b1η2+b1η2a2
2−2a2
2η2b3−4b2a3η2a2−3a2
3η2b3−2a3a4b4η2−a2
4b3η2+8b3η2γμ +12η2b3μ2
−2a2
1b1γ−4a2
1b3γ+4a1a2b2γ−8a1a3b1γ+2a2
2b1γ−4a2
2b3γ−8a2a3b2γ−6a2
3b3γ−4a3a4b4γ−2a2
4b3γ+b3
1γ
+6b2
1b3γ−3b1b2
2γ+6b2
2b3γ+3b3
3γ+3b3b2
4γ+12b3η2μ+4b3η2,(A13)
F8(a1,···a4,b1,···b4,
)1
4η2(μ+1)
35a2
1b2δvη2μ3+18a2
1b4δvη2μ3+10a1a2b1δvη2μ3+20a1a4b1δvη2μ3
−5a2
2b2δvη2μ3+18a2
2b4δvη2μ3+20a2a4b2δvη2μ3+9a2
3b4δvη2μ3+18a3a4b3δvη2μ3+27a2
4b4δvη2μ3
+15a2
1b2δvη2μ2+54a2
1b4δvη2μ2+30a1a2b1δvη2μ2−15a2
2b2δvη2μ2+54a2
2b4δvη2μ2+60a2a4b2δvη2μ2
+54a3a4b3δvη2μ2+27a2
3b4δvη2μ2+81a2
4b4δvη2μ2+54a2
1b4δvη2μ+30a1a2b1δvη2μ+60a1a4b1δvη2μ2
+15a2
1b2δvη2μ+60a1a4b1δvη2μ−15a2
2b2δvη2μ+54a2
2b4δvη2μ+60a2a4b2δvη2μ+27a2
3b4δvη2μ
+54a3a4b3δvη2μ+81a2
4b4δvη2μ+5a2
1b2δvη2+18a2
1b4δvη2+10a1a2b1δvη2+20a1a4b1δvη2−5a2
2b2δvη2
+9a2
3b4δvη2−36b4δvη2μ3+18a2
2b4δvη2+20a2a4b2δvη2+2a2
1b2δhμ+54a2
4b4δhμ−12b2
1b2δhμ
−40b2
1b4δhμ+4b3
2δhμ−40b2
2b4δhμ−36b2
3b4δhμ+4a2
2b4δhμ+18a3a4b3δvη2−108b4δvη2μ2−36b4η2μ3
+4a2
1b4δhμ+4a1a2b1δhμ−2a2
2b2δhμ+40a2a4b2δhμ+18a2
3b4δhμ+36a3a4b3δhμ−36b3
4δhμ−108b4δvη2μ
−108b4η2μ2+2a2
1b2δh+4a2
1b4δh+4a1a2b1δh+40a1a4b1δh−2a2
2b2δh+4a2
2b4δh+40a2a4b2δh+18a2
3b4δh
+36a3a4b3δh+54a2
4b4δh−12b2
1b2δh−40b2
1b4δh+4b3
2δh−40b2
2b4δh−36b2
3b4δh−36b3
4δh−36b4δvη2
−108b4η2μ−36b4η2+40a1a4b1δhμ2+4a2
1b1ζvη2μ3+12a2
1b3ζvη2μ3−8a1a2b2ζvη2μ3+12a1a3b1ζvη2μ3
−4a1a4b2ζvη2μ3−4a2
2b1ζvη2μ3+12a2
2b3ζvη2μ3+12a2a3b2ζvη2μ3+4a2a4b1ζvη2μ3+12a2
3b3ζvη2μ3
+12a2
4b3ζvη2μ3+12a2
1b1ζvη2μ2+36a2
1b3ζvη2μ2−24a1a2b2ζvη2μ2+36a1a3b1ζvη2μ2−12a1a4b2ζvη2μ2
−12a2
2b1ζvη2μ2+36a2
2b3ζvη2μ2+36a2a3b2ζvη2μ2+12a2a4b1ζvη2μ2+36a2
3b3ζvη2μ2+36a2
4b3ζvη2μ2
+12a2
1b1ζvη2μ−24a1a2b2ζvη2μ+36a1a3b1ζvη2μ−12a1a1b2ζvη2u−12a2
2b1ζvη2μ+36a2
2b3ζvη2μ+4a2
1b1ζvη2
+36a2a3b2ζvη2μ+12a2a4b1ζvη2μ+36a2
3b3ζvη2μ+36a2
4b3ζvη2μ−24b3ζvη2μ3+12a2
1b3ζvη2−8a1a2b2ζvη2
+12a1a3b1ζvη2−4a1a4b2ζvη2−4a2
2b1ζvη2+12a2
2b3ζvη2+12a2a3b2ζvη2+4a2a4b1ζvη2+4a2
1b1ζh−8a1a2b2ζh

Y. Wang et al.
+24a1a3b1ζh+8a1a4b2ζh−4a2
2b1ζh+24a2a3b2ζh−8a2a4b1ζh+12a2
3b3ζh+24a3a4b4ζh−12a2
4b3ζh−4b3
1ζh
−24b2
1b3ζh+12b1b2
2ζh−24b2
2b3ζh−12b3
3ζh−12b3b2
4ζh−24b3ζvη2+12a2
3b3ζvη2+12a2
4b3ζvη2−72b3ζvη2μ2
+4a2
1b1ζhμ−8a1a2b2ζhμ+24a1a3b1ζhμ+8a1a4b2ζhμ−4a2
2b1ζhμ+24a2a3b2ζhμ−8a2a4b1ζhμ+12a2
3b3ζhμ
+24a3a4b4ζhμ−12a2
4b3ζhμ−4b3
1ζhμ−24b2
1b3ζhμ+12b1b2
2ζhμ−24b2
2b3ζhμ−12b3
3ζhμ−12b3b2
4ζhμ
−72b3ζvη2μ+36a2
1b3ζvη2μ+−a2
1b2η2μ3−2a2
1b4η2μ3−2a1a2b1η2μ3−4a1a4b1η2μ3+a2
2b2η2μ3
−2a2
2b4η2μ3−4a2a4b2η2μ3−a2
3b4η2μ3−2a3a4b3η2μ3+27a2
4b4δvη2w2−3a2
4b4η2μ3−3a2
1b2η2μ2−6a2
1b4η2μ2
−6a1a2b1η2μ2−12a1a4b1η2μ2+3a2
2b2η2μ2−6a2
2b4η2μ2−12a2a4b2η2μ2−3a2
3b4η2μ2−6a3a4b3η2μ2+8b4η2γμ
3
−3a2
1b2η2μ−6a2
1b4η2μ−6a1a2b1η2μ−12a1a4b1η2μ+3a2
2b2η2μ−6a2
2b4η2μ−12a2a4b2η2μ−3a2
3b4η2μ
−6a3a4b3η2μ−9a2
4b4η2μ+ 16b4η2γμ
2+4b4η2μ3−a2
1b2η2−2a2
1b4η2−2a1a2b1η2−4a1a4b1η2+a2
2b2η2−2a2
2b4η2
−4a2a4b2η2−a2
3b4η2−2a3a4b3η2−3a42b4η2+8b4η2γμ +12b4η2μ2−2a2
1b2γ−4a2
1b4γ−4a1a2b1γ−8a1a4b1γ
+2a2
2b2γ−4a2
2b4γ−8a2a4b2γ−2a2
3b4γ−4a3a4b3γ−6a2
4b4γ+3b2
1b2γ+6b2
1b4γ−b3
2γ+6b2
2b4γ+3b2
3b4γ+3b3
4γ
+12b4η2μ+4b4η2−9a2
4b4η2μ2,(A14)
References
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