Modeling of Nonlinear Isolation System Based on Bouc-Wen Differential Model
A feedforword neural network of multi-layer topologies for systems with hysteretic nonlinearity was constructed based on the Bouc-Wen differential model. The proposed model not only reflects the hysteresis force characteristics of the Bouc-Wen model, but can also determine the corresponding paramete...
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Інститут проблем міцності ім. Г.С. Писаренко НАН України
2017
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| Cite this: | Modeling of Nonlinear Isolation System Based on Bouc-Wen Differential Model / Z. Peng, C.G. Zhou // Проблемы прочности. — 2017. — № 1. — С. 228-233. — Бібліогр.: 10 назв. — англ. |
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| citation_txt | Modeling of Nonlinear Isolation System Based on Bouc-Wen Differential Model / Z. Peng, C.G. Zhou // Проблемы прочности. — 2017. — № 1. — С. 228-233. — Бібліогр.: 10 назв. — англ. |
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| description | A feedforword neural network of multi-layer topologies for systems with hysteretic nonlinearity was constructed based on the Bouc-Wen differential model. The proposed model not only reflects the hysteresis force characteristics of the Bouc-Wen model, but can also determine the corresponding parameters. The simulation results demonstrate that the restoring force-displacement curve hysteresis loop closely represents real curves. The trained model can accurately predict the time response of the system. By comparing results obtained by the proposed model with real responses, the model was validated in the presence of noise and exhibits increased modeling precision, good generalizability and anti-interference capability.
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UDC 539.4
Modeling of Nonlinear Isolation System Based on Bouc-W en Differential
Model
Z. P eng an d C . G. Z hou
College of Mechatronic Engineering, North University of China, Shanxi Taiyuan, China
A feedforword neural network o f multi-layer topologies fo r systems with hysteretic nonlinearity was
constructed based on the Bouc-Wen differential model. The proposed model not only reflects the
hysteresis force characteristics o f the Bouc-Wen model, but can also determine the corresponding
parameters. The simulation results demonstrate that the restoring force-displacement curve hysteresis
loop closely represents real curves. The trained model can accurately predict the time response o f the
system. By comparing results obtained by the proposed model with real responses, the model was
validated in the presence o f noise and exhibits increased modeling precision, good generalizability
and anti-interference capability.
Keywords'. B ouc-W en model, nonlinear. isolation system, modeling.
In tro d u c tio n . Piezoelectric ceram ic actuators such as m agneto rheological damper, as
w ell as dry friction dam ping steel w ire rope and nonlinear delay systems exist in
mechanical isolation systems, earthquake engineering, civil engineering, aerospace structural
dam ping systems, etc. [1, 2]. A ccurate m odeling is im portant to the analysis and response
prediction o f a dynam ic system, and has attracted w ide research attention. The B ouc-W en
m odel is a w idely used non-linear phenom enological m odel w hich describes the smooth
hysteresis behavior o f the lag elem ent according to a nonlinear differential equation [3, 4].
The nonlinear restoring force is divided into two com ponents' the nonlinear an hysteretic
restoring force related only to the instantaneous displacem ent and speed o f the structure,
and the pure lag restoring force related to the structure o f the displacem ent tim e history
w hich can be described by a first-order nonlinear differential equation [5-7].
In the present study, the use o f B ouc-W en m odel is used for the topological design
o f the neural netw ork layer. The corresponding relationship betw een netw ork w eights and
the m odel param eters was established. A neural netw ork m odel is obtained by network
training, w hich reflects not only the hysteresis force characteristics o f the B ouc-W en
m odel, but also the corresponding m odel parameters.
1. M a th em atic a l M odel o f H ysteresis N o n lin ear System s. In practical engineering
applications, it is necessary to establish the m athem atical description o f the hysteretic
nonlinear force in order to analyze the hysteresis nonlinear dynam ics o f the system. The
B ouc-W en differential m odel can describe the various forms o f smooth hysteresis
nonlinearity [8-10]. As long as it is appropriate to change its param eters, the proposed
m odel can describe the various types o f hysteresis loops, described as follows'
R (t ) = b x ( t)+ z ( t ), (1)
Z = T]X(t)— P\X( t )| z| z |n 1 — yX( t )| z |n . (2)
Equation (2) can be rew ritten as follows'
Z ( t ) = 1]X( t ) —f t X ( t ) | |Z ( t ) |n sgn[Z (t) ]—y X (t) |Z (t) |n . (3)
© Z. PENG, C. G. ZHOU, 2017
228 ISSN 0556-171X. npo6n.eubi 2017, N2 1
Modeling o f Nonlinear Isolation System
In Eqs. (1)-(3), R ( t ) is the system lag restoring force, bx(t ) is the non-lag
component, Z (t ) is the lag component, b, 3 , y , and n are the parameters to be identified.
A m ong the identifiable param eters, b, ^ , 3 , and y control the shape o f the hysteresis
curve, while n controls smoothness o f the transition zone in the hysteresis curve.
2. M odeling P rinc ip les B ased on th e B ouc-W en M odel. The B ouc-W en differential
m odel reflects the relationship betw een the lag force and the deform ation displacement.
The relationship betw een the restoring force and deform ation is determ ined by the five
unknow n parameters.
A ccording to the relationship betw een the restoring force and deform ation, by
constructing a series o f activation function, describing the differential equation by specific
neural network topology structure, correspond to the network weights and model parameters.
The neural netw ork m odel o f the system is able to obtain the lag resilience by the training
o f the custom network. The m odeling principle is shown in Fig. 1.
Fig. 1. Principle of hysteresis nonlinear system modeling.
3. N eu ra l N etw o rk Topology B ased on th e B ouc-W en M odel. In order to construct
a neural netw ork topology, Eq. (1) m ust be discretized to obtain the following:
R (t ) = b x ( t)+ z ( t ). (4)
A fter the first order differential forward on Eqs. (2) and (3) can be w ritten as follows:
r(k ) - r(k - 1) b[x (k ) - x (k - T ) ] , . ^ _
T = t + z (t ̂ (5)
z ( t ) = -?№ y « - 1)] - № ( t )| z ( t ) |z (t ) r - i - rl-z ( t >r № ) - x <k - 1 ) ] , (6)
where T is the sam pling interval, and k and k - 1 define the sam pling time. Equations (5)
and (6) are then com bined, and the difference equation indicates the relationship between
the restoring force, displacem ent and speed as follows:
R (k ) = R (k - 1)+ b[x( k ) - x( k - 1)] + TqX (k - 1 ) - Tfi\X (k - 1)||[ R (k - 1 ) - bx(k - 1)]|r X
X sgn[ R ( к — 1)— bx( к — 1)]— Tyx ( к — 1)|[R (k — 1)— bx(k — 1)]|"
ISSN 0556-I7IX. Проблемы прочности, 2017, N2 1
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229
Z. Peng and C. G. Zhou
4. C o n s tru c tio n o f N eu ra l N etw o rk Topology. A ccording to Eq. (7), the neural
netw ork topology shown in Fig. 2 can be constructed to achieve hysteresis nonlinearity
m ultilayer feedforw ard neural network m odeling betw een the restoring force and
displacement.
A s shown in Fig. 2, the m odel param eter inform ation and structural inform ation is
em bedded in the m ultilayer feedforw ard neural network w hich is integrated into the
structure, and m ust be identified by previous knowledge of the model. In the M ATLAB
environm ent, a custom neural netw ork is generated by the com m and net = network, known
as the init function, w hich initializes the network w ith a w eight-defined initialization
function to create a hybrid network, training and learning, until the requirem ents of training
perform ance indicators are met.
230
Fig. 2. Multilayer feedforward neural network modeling.
ISSN 0556-171X. Проблемы прочности, 2017, N2 1
Modeling o f Nonlinear Isolation System
5. C o n s tru c tio n o f N eu ra l N etw o rk Topology. U sing three groups o f experimental
response data, custom neural network training is achieved as shown in Fig. 2. Because the
created network is static, training is achieved through the im proved BP algorithm. The
training result parameter values are presented in Table 1. With the exception o f parameter y,
all param eters are nearly identical to their nom inal values.
T a b l e 1
Neural Network Modeling Results Based on the Bouc-W en Model
Parameter Parameter values training of differential model
Nominal value No noise £ = 5% £ =10% £=15%
b 0.1 0.1089 0.1317 0.1613 0.1668
1.0 0.9894 1.0385 1.0710 0.9468
ß 0.8 0.9954 1.1689 1.7103 2.2500
n 1.5 1.4980 1.6111 1.6860 1.3010
V 0.2 0.0060 0.1746 0.4144 0.3021
By com paring Fig. 3, results indicate that the restoring force-displacem ent hysteresis
loop curve and the real hysteresis loop curve are nearly identical.
A s show n in Fig. 4, the contrast betw een the predicted steady-state response and the
real system response under the three levels o f m otivation indicates that the training model
can accurately predict the tim e response o f the system.
Real curve Predicted curve
Fig. 3. Three types of real and predicted horizontal excitation resilience-displacement hysteresis
curves.
30 35 //s 40 45 50
Fig. 4. Comparison of steady-state responses under the three levels of motivation (solid lines
correspond to real curve and dashed lines - predicted curve).
ISSN 0556-171X. Проблемы прочности, 2017, № 1 231
Z. Peng and C. G. Zhou
2.0
1.5
1.0
^0 .5
-0.5
- 1.0
-1.5
- 2.0
' 8 ' 4 °x(t)4 8 30 35 40 t 45 50
Real curve--* -Prediction curve— Predicted curve
a b
Fig. 5. Hybrid models to predict compared with real response for Xg4 = 6.0: (a) restoring force;
(b) acceleration.
In order to test the training ability o f the hybrid netw ork m odel, the predicted response
is calculated and com pared to the real response, as show n in Fig. 5. The hybride network
m odel is still able to accurately predict the system response and the hysteresis curve, and
exhibits good generalizability.
6. M odel P erfo rm a n ce in th e P resence of Noise. It is assum ed that the restoring
force data representing the three levels o f m otivation is polluted by noise, expressed as
follows:
R j (t ) = R j (t )+£rj R j o , (8)
where rj is the norm al distribution w ith zero m ean unit variance random signal sequence,
R j o is the m agnitude o f the restoring force j , and £ is the noise level.
Fig. 6. Hybrid models to predict compared with real response for £ = 5%: (a) restoring force; (b)
acceleration.
The hybrid netw ork is separately trained in the use o f data w here £ is equal to 5,
10, and 15%. The training param eters are shown in Table 1. W hen the noise level is equal
to 5%, the training param eters exhibit some deviation. However, the sim ulation m odel is
still able to accurately predict the response o f the system w ith hysteresis characteristics as
shown in Fig. 6. Additionally, the results depicted in Table 1 also indicate that the value of
the error param eter during training gradually increases w ith noise level. Thus, the influence
o f noise can be reduced by increasing the training sample data.
232 ISSN 0556-171X. npoÖÄeubi 2017, N2 1
Modeling o f Nonlinear Isolation System
C o n c l u s i o n s
1. In the present study, a B ouc-W en differential m odel o f delayed nonlinear systems
was presented, a m ultilayer feedforward neural network m odel o f neural network topology
was constructed and the proposed model w as trained w ith experim ental response data.
2. Sim ulation results indicate that the obtained restoring force-displacem ent hysteresis
loop curves and the real hysteresis loop curves were nearly identical, dem onstrating that the
trained model can accurately predict the tim e response o f the system.
3. Results indicate that the m odel exhibits good generalizability based on com parison
with real response data.
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8. Z. Z. Wang, “The study o f com plex system m odeling and sim ulation on evolution
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Received 30. 08. 2016
ISSN 0556-171X. npoôëeMbi 2017, N2 1 233
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| id | nasplib_isofts_kiev_ua-123456789-173601 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0556-171X |
| language | English |
| last_indexed | 2025-12-07T18:16:33Z |
| publishDate | 2017 |
| publisher | Інститут проблем міцності ім. Г.С. Писаренко НАН України |
| record_format | dspace |
| spelling | Peng, Z. Zhou, C.G. 2020-12-12T15:21:09Z 2020-12-12T15:21:09Z 2017 Modeling of Nonlinear Isolation System Based on Bouc-Wen Differential Model / Z. Peng, C.G. Zhou // Проблемы прочности. — 2017. — № 1. — С. 228-233. — Бібліогр.: 10 назв. — англ. 0556-171X https://nasplib.isofts.kiev.ua/handle/123456789/173601 539.4 A feedforword neural network of multi-layer topologies for systems with hysteretic nonlinearity was constructed based on the Bouc-Wen differential model. The proposed model not only reflects the hysteresis force characteristics of the Bouc-Wen model, but can also determine the corresponding parameters. The simulation results demonstrate that the restoring force-displacement curve hysteresis loop closely represents real curves. The trained model can accurately predict the time response of the system. By comparing results obtained by the proposed model with real responses, the model was validated in the presence of noise and exhibits increased modeling precision, good generalizability and anti-interference capability. en Інститут проблем міцності ім. Г.С. Писаренко НАН України Проблемы прочности Научно-технический раздел Modeling of Nonlinear Isolation System Based on Bouc-Wen Differential Model Использование дифференциальной модели Бука-Вена для моделирования петли гистерезиса кривой усилие-смещение для нелинейной механической системы сейсмозащиты Article published earlier |
| spellingShingle | Modeling of Nonlinear Isolation System Based on Bouc-Wen Differential Model Peng, Z. Zhou, C.G. Научно-технический раздел |
| title | Modeling of Nonlinear Isolation System Based on Bouc-Wen Differential Model |
| title_alt | Использование дифференциальной модели Бука-Вена для моделирования петли гистерезиса кривой усилие-смещение для нелинейной механической системы сейсмозащиты |
| title_full | Modeling of Nonlinear Isolation System Based on Bouc-Wen Differential Model |
| title_fullStr | Modeling of Nonlinear Isolation System Based on Bouc-Wen Differential Model |
| title_full_unstemmed | Modeling of Nonlinear Isolation System Based on Bouc-Wen Differential Model |
| title_short | Modeling of Nonlinear Isolation System Based on Bouc-Wen Differential Model |
| title_sort | modeling of nonlinear isolation system based on bouc-wen differential model |
| topic | Научно-технический раздел |
| topic_facet | Научно-технический раздел |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/173601 |
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