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Development and simulation of a magnetorheological damper for segment erector vibration control

Journal: Transactions of the Canadian Society for Mechanical Engineering

Manuscript ID TCSME-2018-0131.R1

Manuscript Type: Article

Date Submitted by the Author: 03-Sep-2018

Complete List of Authors: Yang, Bo; Northeastern Universityzhang, Ao ; Northeastern UniversityBai, Yan; Northeastern UniversityZhang, Kuo; Northeastern UniversityLi, He; Northeastern University, School of Mechanical Engineering & Automation

Keywords: Takagi-Sugeno fuzzy control, Magnetorheological damper, Segment erector, disturbance observer

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Transactions of the Canadian Society for Mechanical Engineering

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Development and simulation of a magnetorheological damper

for segment erector vibration controlBo Yang, Ao zhang, Yan Bai, Kuo Zhang and He Li *

School of Mechanical Engineering & Automation, Northeastern University, Shenyang 110819, China; [email protected] (B.Y.) [email protected] (A.Z.); [email protected] (Y.B.); [email protected](K.Z.)* Correspondence: [email protected]; Tel.: +86-24-8637-4338

Abstract: In this paper, magnetorheological dampers (MR) applied to the segment erector and used to replace passive vibration dampers. Because MR damper dynamics is highly nonlinear, the design of direct control system is impossible. In order to the linear control theory to be applied to design the MR damper controller directly, the Takagi-Sugeno fuzzy (TS) model to be represented by analytical the segment erector models. In addition, a disturbance observer based on TS fuzzy controller is proposed for this system. Both simulation and experiments validate the performance enhancement and stability of the controller. The results show that the acceleration of the segment erector was reduced by 59.6% and 32.1% in oblique wave excitation and random excitation, respectively, compared with a conventional passive damper. The proposed fuzzy controller and magnetorheological dampers have great potential and are very practical for application because it can significantly improve the performance of segment erector.

Keywords: TS fuzzy control, Magnetorheological damper, Segment erector, disturbance observer.

1.IntroductionSegment erector plays the role of segment assembly in tunnel constructions for decades. The segment

erector, installed on tunnel boring machine (TBM), often faces various failure and unable to work properly when it runs in mixed rock formations (e.g., Liu et al. 2016; Zhao et al. 2007). Because the boring process is accompanied with huge torque, thrust, and large impact load from blocky rock mass, sticky soil, and high groundwater pressure, such unbalance loads lead to serious vibration of TBM cutter head, and the same to segment erector (e.g., Ling et al. 2015; Huo et al. 2017).

In order to reduce the amplitude of vibrations, using isolation system is a kind of effective method. There are commonly three different kinds of methods often used: passive, active and semi-active vibration control technology. Passive isolation system generally includes springs, rubbers or air mounts which are used on a large amount of equipment. However, these system delivers limited performance in vibration control applications due to uncontrollable shock absorber parts (Luo 2013). By contrast, active isolation system is mostly effective at low frequencies and large amplitudes of vibrations (Bazinenkov 2015), which need closed-loop control system with feedback, such as displacement or vibration sensor, and positioning mechanisms. Whereas High cost and complex technology obviously put forward a major hindrance for the commercial use of active isolation system (Han 2017). Semi-active isolation system combines the advantages of active and passive isolation system. For one thing, semi-active isolation system can provide desirable performance with Open-loop control or Fuzzy control. For another thing, such system only need less power and few components so as to improve efficiency and reliability as well as to control cost.

Magnetorheological (MR) damper has already some applications in the field of transportation industry and construction industry. One of the recent and promising technologies in the design of semi-active isolation systems, which have been proposed by many investigators, is utilization of magneto-rheological (MR)

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materials (Carlson 2000). Ning et al. (2017) have designed a kind of MR damper and applied to a semi-active quarter-car suspension, compared the performance of two different dampers including passive damper and MR damper. Experimental results show MR damper can reduce point-to-point values of acceleration about 45.5% compared with the conventional passive suspension. Prabakar et al. (2009) demonstrated a method of designing an optimal H∞ controller with preview using a Non-dominated Sorting Genetic Algorithm II (NSGA II) for a half car vehicle model with MR damper. Zareh et al. (2012) proposed a novel method of neuro-fuzzy (NF) control strategy based on an eleven degrees of freedom passenger car’s suspension system using a MR damper. In addition, MR damper has been employed in some civil structures to control the seismic response or braking-induced vibrations such as Dongting Lake Bridge (Chen et al. 2004) and Wuhan Tianxingzhou Yangtzi River Bridge (Qu et al. 2009).

In practical applications, MR damper as a nonlinear component owing to its hysteresis nonlinear behavior, is hard to predict the dynamical performance (Nguyen and Choi 2009). In order to control such system, a wide range of nonlinear control strategies is engaged in controller design such as fuzzy logic, neural networks, H∞ control and sliding mode control. Pour and Behbahani (2016) designed a fuzzy controller to compute the best supply voltage to MR damper based on the measured vibration signals and used it as a semi-active device in a machine tool. Phu et al. (2005) present a new adaptive fuzzy controller which integrated H∞ control technique and sliding mode control methodology, Experiment shown that the proposed controller can provide more better performance.

This paper is organized as follows. In Section 2, an optimized structure of MR damper is developed in terms to the model of the segment erector. The hysteretic behavior of the MR damper is modelled by Bouc-Wen models and the parameters can be obtained from tension and compression experiment. The TS fuzzy modeling of the segment erector and the controller are described in section 3. In section 4, numerical simulation is carried out by means of Runge-Kutta, and the results show the validity of the designed system. Section 5, presents the experiment preparation and results including the ramp exciting test and random exciting test. Finally, the conclusion is provided in Section 6.

2 Design and Parameter Identification of MR Damper

2. 1 MR Damper Structure

A MR damper is developed according to the model of the segment erector. The design takes into account the space size and drive capability of system. Figure 1 shows the structure of the MR damper designed for this experimental platform. It includes a spindle, a spring, a cylinder cap and base, a piston, a solenoid coil, a balance tab, and two O-rings. Note that unlike many other type of MR dampers, a balance tab which is a specially design can avoid radial shaking of the spindle effectively, reducing liquid leaks.

Figure 1. Structure of the MR damperThe barrel is filled with MR fluid. In order to achieve the best performance of the system, a type of MR

fluid is prepared, and the parameters are shown in Table 1.

Table 1. The parameters of MR fluid

2.2 MR damper model and simplify

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The modeling of hysteretic behavior is becoming increasingly important for analysis of dynamic response and design of MR devices. In practical applications, involving Bingham, Ramberg-Osgood (RO), Bouc-Wen and modified Bouc-Wen model can be used to define the nonlinear hysteresis model (Spencer et al. 1997). Bingham model could satisfactorily predict the operational force range of the MR damper. this model, however, is unable to capture the hysteretic behavior perfectly for controller design (Cesmeci and Engin 2010). Ramberg-Osgood model can be employed to fit the Bouc-Wen model to same hysteresis behavior, but the mathematical expression of RO model is not very suitable for engineering calculation (Sireteanu et al. 2014). Bouc-Wen model is not accurate enough to fit hysteretic loop in low speed and lack hysteretic behavior in high speed. Therefore, the modified Bouc-Wen model which contains acceleration parameter was established to extend the range of Bouc-Wen model. Considering the speed of the segment erector is 0.25 m/s to 0.35 m/s, this paper employed the Bouc-Wen model to portray the behavior of a prototype MR damper (Dominguez et al. 2004). The phenomenological model (Ikhouane and Rodellar 2007) is governed by the following equations:

𝑓𝑚 = 𝑐0𝑥 + 𝑘0𝑥 + 𝛼𝑧

𝑧 = ―𝛾|𝑥|𝑧|𝑧|𝑛 ― 1 ― 𝛽𝑥|𝑧|𝑛 + 𝐴𝑥

α = 𝑎𝑎 + 𝑎𝑏𝑣

𝑐0 = 𝑐0𝑎 + 𝑐0𝑏𝑣

𝑣 = ―𝜂(𝑣 ― 𝑢)

|𝑧| = 𝑧 ∙ 𝑠𝑖𝑔𝑛(𝑧)

(1)

Where is the restoring force; is the viscous damping observed at higher velocities; is the 𝑓𝑚 𝑐0 𝑘0 stiffness at large velocities; Parameters , and control the size and shape of the hysteretic loop. is γ β 𝐴 αthe evolutionary coefficient. is the output of a first-order filter; and is the input voltage. 𝑣 𝑢

Considering the particular case and the linearization technique of the Bouc-Wen (Bajrić and Høgsberg 2018), we set . Reducing the nonlinear restoring force in (1) ton = 1

𝑧 = 𝐴𝑥 ― 𝛽|𝑥|𝑧 ― 𝛾𝑥|𝑧| (2)

2.3 parameter identification

The MR damper was tested by using a 5KN material fatigue testing machine, see Figure 2. The force and displacement data were measured and send to computer by RS-232 serials communication. Using the MATLAB parameter estimating function, parameters were determined to fit this model of the MR damper, shown in Table1. The loaded state is 1Hz and 10mm under three different input voltages, see Figure 3.

Figure 2. MR damper testing

Figure 3. Experimental dynamic behavior of MR damper

The estimated model parameters of the MR damper are ; ; ; 𝐴 = 91.3 𝛾 = 319.7m ―1 𝛽 = 3.9m ―1 𝑐0𝑎

; ; ; .= 215 Ns/m 𝑐0𝑏 = 215 Ns/m 𝑎𝑎 = 215 N/m 𝑎𝑏 = 215 N/m

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3 Fuzzy modelling of segment erector with MR damper

3.1 Segment erector model with MR damper

Segment erector plays the role of segment assembly in shield tunneling. When the shield machine is advancing a ring, the segment erector then segments the precast concrete segment with a segment assembly machine into a tunnel lining to protect the inner surface of the tunnel just excavated with the cutterhead. The segment erector system is mainly composed of walking beam, fixed ring, rotary ring, lifting oil cylinder, lifting crossbeam and sucker. The mounting position of MRs is shown in Figure 4.

Figure 4. The structure of segment erectorAlthough the realistic structure and running state are very complicated, the model of vibration system

can be established by simplifying some factors. The following assumptions can be applied: (1) The segment erector and tunnel boring machine are simplified to rigid body respectively. (2) Only vertical vibration is considered in this model. (3) Joint surfaces of mechanical structure should be ignored. The segment erector model can be simplified to a two Degree of Freedom (DOF) model with nonlinear terms. The schematic of the model is shown in Figure 5.

Figure 5. Nonlinear model of segment erector systemThe differential equations give the nonlinear vibration model with MR damper as follows.

𝑚𝑒𝑧𝑒 + 𝑘𝑒(𝑧𝑒 ― 𝑧𝑏) + 𝑓𝑚 = 0 (3)

𝑚𝑏𝑧𝑏 + 𝑘𝑏𝑧𝑏 ― 𝑘𝑒(𝑧𝑒 ― 𝑧𝑏) + 𝑐𝑏𝑧𝑏 ― 𝑓𝑚 = 𝐹𝑧(𝑡) (4)

Where, is the mass of segment erector; is the mass of main beam of tunnel boring; and 𝑚𝑒 𝑚𝑏 𝑧𝑒

are the displacements of the segment erector and the main beam, respectively; and is the 𝑧𝑏 𝑘𝑒 𝑘𝑏 structural stiffness of the segment erector and the main beam, respectively. In order to simplify the representation of the hysteresis nonlinear system, this paper take advantage of the Bouc-Wen model to predict the mechanical behavior of MR dampers like the phenomenological model (1). The equations for the system are:

𝑚𝑒𝑧𝑒 + 𝑘𝑒(𝑧𝑒 ― 𝑧𝑏) + 𝑘0(𝑧𝑒 ― 𝑧𝑏) + 𝑐0(𝑧𝑒 ― 𝑧𝑏) + 𝛼𝑧 = 0

m𝑏𝑧𝑏 + 𝑘𝑏𝑧𝑏 ― 𝑘𝑒(𝑧𝑒 ― 𝑧𝑏) + 𝑐𝑏𝑧𝑏 ― 𝑘0(𝑧𝑒 ― 𝑧𝑏) ― 𝑐0(𝑧𝑒 ― 𝑧𝑏) ― 𝛼𝑧 = 𝐹𝑧(𝑡)

α = 𝛼𝑎 + 𝑎𝑏𝑣

𝑐0 = 𝑐0𝑎 + 𝑐0𝑏𝑣

𝑣 = ―𝜂(𝑣 ― 𝑢)

(5)

Define state variables of the segment erector system by:

𝑥1 = 𝑧𝑏, 𝑥2 = 𝑧𝑒 ― 𝑧𝑏, 𝑥3 = 𝑧𝑒, 𝑥4 = 𝑧𝑏, 𝑥5 = 𝑧, 𝑥6 = 𝑣 (6)

The control voltage is the input variable and disturbance variable by:𝑢 𝑤

w = 𝑧𝑏 (7)

Then, the state-space equation of the segment erector system can be represented as

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𝑥 = 𝐴𝑥 + 𝐵1𝑤 + 𝐵2𝑢 (8)

The system motion equations can be rewritten as follows:

𝑚𝑒𝑥3 = ― (𝑘𝑒 + 𝑘0)𝑥2 ― 𝑐𝑎𝑥3 + 𝑐𝑎𝑥4 ― 𝛼𝑎𝑥5 ― 𝑔1𝑥6

m𝑏𝑥4 = 𝐹𝑧(𝑡) ― 𝑘𝑒𝑥1 + (𝑘𝑒 + 𝑘0)𝑥2 + 𝑐0𝑎𝑥2 ― (𝑐0𝑎 + 𝑐𝑏)𝑥4 + 𝛼𝑎𝑥5 + 𝑔1𝑥6

𝑥5 = 𝐴𝑥3 ― 𝐴𝑥4 + 𝑔2𝑥5

𝑥6 = ―𝜂(𝑥6 ― 𝑢)

(9)

Where

𝑔1 = 𝑐0𝑏(𝑥2 ― 𝑥3) + 𝛼𝑏𝑥5

𝑔2 = ―𝛽|𝑥3 ― 𝑥4| ― 𝛾(𝑥3 ― 𝑥4)sign(𝑥5)(10)

In practice, the parameters and are bounded and continuous, therefore they can be replaced 𝑔1 𝑔2

by linear subsystems through the Takagi-Sugeno fuzzy modeling (Kawamoto et al. 1992) and segment erector represented as follows:

𝑔1 = 𝑀1𝑔1𝑚𝑎𝑥 + 𝑀2𝑔1𝑚𝑖𝑛

𝑔2 = 𝑁1𝑔2𝑚𝑎𝑥 + 𝑁2𝑔2𝑚𝑖𝑛

𝑀1 + 𝑀2 = 𝑁1 + 𝑁2 = 1

(11)

Where , , and are fuzzy membership functions. and represent the upper bound 𝑀1 𝑀2 𝑁1 𝑁2 𝑔𝑚𝑎𝑥 𝑔𝑚𝑖𝑛

and the lower bound of the nonlinearity , respectively. The membership functions are defined as𝑔

𝑀1 = [𝑐0𝑏(𝑥2 ― 𝑥3) + 𝛼𝑏𝑥5 ― 𝑔1𝑚𝑖𝑛]/(𝑔1𝑚𝑎𝑥 ― 𝑔1𝑚𝑖𝑛)

𝑀2 = 1 ― 𝑀1

𝑁1 = {[ ―𝛽|𝑥3 ― 𝑥4| ― 𝛾(𝑥3 ― 𝑥4)sign(𝑥5)] ― 𝑔2𝑚𝑖𝑛}/(𝑔2𝑚𝑎𝑥 ― 𝑔2𝑚𝑖𝑛)

𝑁2 = 1 ― 𝑁1

(12)

The nonlinear system model can be represented by the following models:IF is and is , THEN 𝑔1 𝑀1 𝑔2 𝑁1 𝑥 = 𝐴(1)𝑥 + 𝐵1𝜛 + 𝐵2𝑢IF is and is , THEN 𝑔1 𝑀1 𝑔2 𝑁2 𝑥 = 𝐴(2)𝑥 + 𝐵1𝜛 + 𝐵2𝑢IF is and is , THEN 𝑔1 𝑀2 𝑔2 𝑁1 𝑥 = 𝐴(3)𝑥 + 𝐵1𝜛 + 𝐵2𝑢IF is and is , THEN 𝑔1 𝑀2 𝑔2 𝑁2 𝑥 = 𝐴(4)𝑥 + 𝐵1𝜛 + 𝐵2𝑢The system can be put together as

𝑥 =4

∑𝑖 = 1

ℎ𝑖[𝐴(𝑖)𝑥 + 𝐵1𝜛 + 𝐵2𝑢]

= 𝐴ℎ𝑥 + 𝐵1𝜛 + 𝐵2𝑢

(13)

Where are defined as , , , ℎ𝑖 ℎ1 = 𝑀1𝑁1 ℎ2 = 𝑀1𝑁2 ℎ3 = 𝑀2𝑁1 ℎ4 = 𝑀2𝑁2

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A𝑖 = [0 0 0 1 0 00 0 1 ―1 0 0

0― (𝑘𝑒 + 𝑘0)

𝑚𝑒

― 𝑐𝑎

𝑚𝑒

𝑐𝑎

𝑚𝑒

―𝑎𝑎

𝑚𝑒

―𝑔1

𝑚𝑒― 𝑘𝑒

𝑚𝑏

𝑘𝑒 + 𝑘0 + 𝑐0𝑎

𝑚𝑏0

― (𝑐0𝑎 + 𝑐𝑏)𝑚𝑏

𝑎𝑎

𝑚𝑏

𝑔1

𝑚𝑏0 0 𝐴 ―𝐴 𝑔2 00 0 0 0 0 ―𝜂

]𝐵1 = [0 0 0 1 0 0]

𝐵2 = [0 0 0 0 0 𝜂]

3.2 State Observer Design

Vibration isolation design for the segment erector has two main points. For one thing, equipment stability, which can be measured by accelerometer. For another thing, the separation between the segment erector and tunnel boring machine react whether the equipment collide. In terms of the segment erector, and 𝑧𝑒 𝑧𝑒

can be measured by accelerometer and laser displacement sensors. Therefore, the controlled output is ― 𝑧𝑏

defined as

y = [𝜆1𝑧𝑒 𝜆2(𝑧𝑒 ― 𝑧𝑏)]𝑇

=4

∑𝑖 = 1

ℎ𝑖𝐶1(𝑖)𝑥

= C1ℎ𝑥

(14)

Where

C1(𝑖) = [0― (𝑘𝑒 + 𝑘0)

𝑚𝑒

― 𝑐𝑎

𝑚𝑒

𝑐𝑎

𝑚𝑒

―𝛼𝑚𝑒

― 𝑔1

𝑚𝑒0 0 1 ―1 0 0 ] (15)

The estimation error can be defined as

𝑒 = 𝑥 ― 𝑥 (16)

Then, the state observer can be designed as

𝑥 = 𝑓(𝑥,𝑢) + 𝐻( ∙ )[𝑦 ― ℎ(𝑥)]

= Aℎ𝑥 + 𝐵2𝑢 + 𝐻( ∙ )[𝑦 ― 𝐶1ℎ𝑥](17)

Where is the observer gain matrix.𝐻( ⋅ ) Differentiating (14) and substituting into (17), we obtain the dynamic equation of the state estimation error

𝑒 = 𝑥 ― 𝑥

= (𝐴ℎ ― 𝐻( ∙ )𝐶1ℎ)𝑒 + 𝐵1𝜛(18)

3.3 Takagi-Sugeno fuzzy controller design

The Parallel distributed compensation (PDC) offers a procedure to design a fuzzy controller from a

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given T-S fuzzy model (Wang et al. 1995). The observer-based controller can be represented as

u =4

∑𝑖 = 1

ℎ𝑖𝐹𝑖𝑥 (19)

Where are the state feedback gain matrices to be designed.𝐹𝐼

Combining the controller (19) and the fuzzy observers (15) – (18), we obtain the following system representations:

𝑥 =4

∑𝑖 = 1

ℎ1{(𝐴ℎ ― 𝐵2𝐹ℎ)𝑥 + 𝐵2𝐹ℎ𝑒} (20)

𝑒 =4

∑𝑖 = 1

ℎ1{𝐴ℎ ― 𝐻( ∙ )𝐶1ℎ}𝑒 (21)

In order to design an optimized control system to perform adequately in a wide range of shock and vibration environments, the norm is chosen as the performance measure. The gain of the system 𝐻∞ 𝐿2

(12) with (14), which is defined as

‖𝑇𝑆𝑊‖∞ = sup‖𝑦‖2

‖𝜔‖2, 𝜔2 ≠ 0

(22)

Where

‖𝑧‖22 = ∫

∞

0𝑦𝑇(𝑡)𝑦(𝑡)𝑑𝑡

‖𝑤‖22 = ∫

∞

0𝑤𝑇(𝑡)𝜔(𝑡)𝑑𝑡

The aim is to design a fuzzy controller such that the fuzzy system with controller is quadratically stable and the is minimized. Defined a Lyapunov function for the system (20) as𝐿2

𝑉(𝑥) = 𝑥𝑇𝑃𝑥 (23)

Where is a positive define matrix. By differentiating, we obtain𝑃

𝑉(𝑥) = 𝑥𝑇𝑃𝑥 + 𝑥𝑇𝑃𝑥 (24)

Adding to the two sides of (25) yields𝑠𝑇𝑠 ― 𝜆2𝜔𝑇𝜔

𝑉(𝑥) + y𝑇𝑦 ― 𝜆2𝜔𝑇𝜔 = 𝑥𝑇𝑃𝑥 + 𝑥𝑇𝑃𝑥𝑦𝑇𝑦 ― 𝜆2𝜔𝑇𝜔 (25)

Substituting (20) and (21) into (26), we obtain

𝑉(𝑥) + 𝑦𝑇𝑦 ― 𝜆2𝜔𝑇𝜔 = (𝐺ℎ𝑥 + 𝐵1𝜔)𝑇𝑃𝑥 + 𝑥𝑇(𝐺ℎ𝑥 + 𝐵1𝜔) + (𝐶1ℎ𝑥)𝑇(𝐶1ℎ𝑥) ― 𝑦2𝜔𝑇𝜔 (26)

Rearrangement of (27) gives

𝑉(𝑥) + 𝑦𝑇𝑦 ― 𝜆2𝜔𝑇𝜔 = [𝑥𝑇

𝜔𝑇]𝑇[𝐺ℎ𝑃 + 𝑃𝐺ℎ + 𝐶𝑇1ℎ𝐶1ℎ 𝐵1𝑃

∗ ―𝜆2][𝑥𝜔] (27)

Considering

Ω = [𝐺ℎ𝑃 + 𝑃𝐺ℎ + 𝐶𝑇1ℎ𝐶1ℎ 𝐵1𝑃

∗ ―𝜆2] (28)

When the disturbance is zero, then , it can conclude that if , then , and the close-loop 𝜔 = 0 Ω < 0 𝑉(𝑥) < 0

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control system (20) is quadratically stable. By considering Schur complement equivalence (Zhang 2005), (28) can be further arranged to linear matix inequalities (LMIs), as follows:

Ω = [𝐺ℎ𝑃 + 𝑃𝐺ℎ 𝐶𝑇1ℎ 𝐵1𝑃

∗ ―𝐼 0∗ ∗ ― 𝜆2] < 0 (29)

The state feedback gains and the observer gains of the state-observer-based TS fuzzy 𝐹(𝑖) 𝐻( ∙ )controller are determined by solving the LMIs using MATLAB software.

4. Numerical simulation and discussion

Based on the equations of dynamic in Equation (3) and (4), a Matlab/Simulink model of segment erector system can be established. A bump profile is employed to reveal the transient response characteristic shown in Figure 6. The response of the passive and MR isolators are shown in Figure 7 and Figure 8, respectively. Comparing the peak values of responses, the controlled model can reduce about 47% of response compared with passive isolators. It can be assured that MR isolator can reduce the response of the segment erector.

In addition, the resonance frequency of system is calculated by FFT transform, as shown in Figure 9. The first natural frequency is 7.3Hz, and the second natural frequency is 57Hz. Therefore, the range of isolation is defined in 1 – 20Hz, and the sampling rate of sensors is 100Hz.

Figure 6. The input signal of the system

Figure 7. The displacement of segment erector with passive isolators

Figure 8. The displacement of segment erector with MR isolators

Figure 9. Resonance frequency of system

5. Description of the test

5.1 Experimental setup

The experimental platform consists of two parts, excitation device and tested model in the figure 10. For one thing, the excitation source comes from an eccentric rotor, with two eccentric masses. Its exciting frequency can be changed by adjusting a frequency converter (model: VFD015M43B, Delta Group) which is controlled by a PLC ((Model: CP1L-L20DT-D, OMRON Corp). For another, the control system of MR damper is a multiple input and single output system. Four input signals, and , are measured by the 𝑧𝑏 𝑧𝑒

two accelerometers (Model: 4508-B, B&K Corp). and , are measured by two laser displacement 𝑧𝑏 𝑧𝑒

sensors ((Model: ZX-LD40, OMRON Corp). An output signal that is pulse-width modulation type can be outputted by 9403 model (NI, Corp). By running the control algorithm based on input parameters, the real-time controller (model: 9014, 9103, NI Crop.) calculates the duty circle and sends to driver board. The primary parameters of structure are , , , , 𝑚𝑏 = 120kg 𝑚𝑒 = 40kg 𝑘𝑏 = 53 726N/m 𝑘𝑒 = 11 691N/m 𝑐𝑏

. = 217N ∙ s/mIn addition, to validate the performance of the controller, a passive isolator with a damping coefficient

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of and a fuzzy control without observer is adopted for a comparison. 𝑐𝑠 = 500Nsm ―1

Figure 10. Experimental setup

5.2 Dynamic property test

In order to obtain the natural frequency of the text object and the performance of the MR dampers, it is necessary to carry out the following experiments. First of all, excitation frequency can be adjusted by frequency transformer from 4.5Hz to 21.5Hz, which contains sprung mass resonant frequency, with 0.5Hz step is applied. Secondly, Controlling the voltage input of the MR dampers from 0V to 2V with 0.25V step is applied. The acceleration response at different excitation frequencies is shown in Figure 11. The resonant frequency of the system is 7.5Hz. In the low frequency phase, 4.5 Hz– 12 Hz, higher input voltage can achieve better damping effect. In contrast, In the high frequency phase, 12.5 Hz - 21.5 Hz, lower input voltage can achieve better damping effect. In addition, Figure 12 shows the optimal control surface in different voltage and excitation frequencies.

Figure 11. Acceleration response in different voltage and excitation frequencies

Figure 12. Optimal control surface

5.3 Oblique wave excitation test

Taking into account the process from the start to the normal operation of the segment erector, it is necessary to use the oblique wave function to stimulate the equipment. Therefore, setting inverter output from 0 to 20Hz in 20s.

Figure 13 presents the performances of the passive and TS fuzzy under the oblique excitation. In order to more clearly present the effect of different dampers, the envelope of the vibration response curve is shown in figure 14. It can be shown that the transmissibility measured on the passive isolator shows a peak at 7.5Hz. The passive suspension acts as a low-pass filter which means that reducing the transmissibility around the system resonance frequency is difficult. However, TS fuzzy controller with observer and TS fuzzy without observer reduces the maximum peak-to-peak 59.6% and 23.5% respectively, compared with the conventional passive suspension. In the non-resonant area, The semi-active damper exhibits more than 20% lower vibration transmissibility than passive dampers.

The input current value of MRs between observers and non-observers at oblique wave excitation are presented in Figure 15. There is more fluctuation when control system is not use the observer. Meanwhile, it can cause the consumption of more energy.

In addition, damping force is measured by the transducer of the resistance strain gauge. As shown in Figure 16, It is observed that the variation trend of damping force agrees well with the change in acceleration. Note that the maximum value of the force is near the resonance region.

Figure 13. Acceleration responses at oblique wave excitation

Figure 14. Envelope curve of acceleration response

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Figure 15. Input current plots of MR isolator at oblique wave excitation

Figure 16. Damping force plots of MR isolator at oblique wave excitation

5.4 Random excitation test

Owing to complex geological conditions and mechanical structures, vibration excitation of the segment erector often cannot be calculated and measured accurately. We can only analyze the frequency range of excitation by calculated multi-degree of freedom dynamic model (Huo et al. 2015). Therefore, a random excitation method is used to better characterize the effects of different dampers and control strategies.

Figure 17 presents the performances of the passive and TS fuzzy under the random excitation. The envelope of the vibration response curve is shown in figure 18. It can be shown the TS fuzzy controller with observer and without observer reduces the maximum peak-to-peak 32.1% and 18.5% respectively, compared with the conventional passive suspension. By comparing the results, it is further conformed that good random response is achieved by the TS fuzzy control method.

As shown in figure 19, the values of input current are slightly different at random excitation. In addition, the variation trend of damping force agrees well with the change in acceleration in Figure 20.

Figure 17. Acceleration responses at random excitation

Figure 18. Envelope curve of acceleration response

Figure 19. Input current plots of MR isolator at random excitation

Figure 20. Damping force plots of MR isolator at random excitation

6. Conclusions

In this paper, a MR damper for the segment erector platform is designed and produced. a state-observer-based TS fuzzy controller is designed and applied to this nonlinear system. By comparing the performance of the MR damper and conventional passive damper in practice, the main conclusions of this study can be summarized as follows:1. Applying MR damper can achieve great vibration reduction effect, it can reduce the maximum peak-

to-peak of Acceleration about 59.6% in oblique wave excitation experiment and reduce the maximum peak-to-peak of Acceleration about 32.1% in random excitation experiment.

2. When the segment erector is running near the resonant frequency, the effect of vibration reduction is most obvious.

3. Disturbance observer based on TS fuzzy controller can improve control effect, compared with traditional methods.

Acknowledgments: This work was supported by the National Natural Science Foundation of China (Grant No. 51675091).

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Parameters ValueIron powder size ≤10um

Iron content 25%Dynamic viscosity ≤1.5Pa·s

Density 2.96g/cm3

1. Figure 1. Structure of the MR damper

2. Figure 2. MR damper testing3. Figure 3. Experimental dynamic behavior of MR damper

4. Figure 4. The structure of segment erector5. Figure 5. Nonlinear model of segment erector system6. Figure 6. The input signal of the system7. Figure 7. The displacement of segment erector with passive isolators8. Figure 8. The displacement of segment erector with MR isolators9. Figure 9. Resonance frequency of system10. Figure 10. Experimental setup

11. Figure 11. Acceleration response in different voltage and excitation frequencies12. Figure 12. Optimal control surface13. Figure 13. Acceleration responses at oblique wave excitation14. Figure 14. Envelope curve of acceleration response 15. Figure 15. Input current plots of MR isolator at oblique wave excitation16. Figure 16. Damping force plots of MR isolator at oblique wave excitation17. Figure 17. Acceleration responses at random excitation18. Figure 18. Envelope curve of acceleration response19. Figure 19. Input current plots of MR isolator at random excitation

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20. Figure 20. Damping force plots of MR isolator at random excitation

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Parameters ValueIron powder size ≤10um

Iron content 25%Dynamic viscosity ≤1.5Pa·s

Density 2.96g/cm3

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Structure of the MR damper

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MR damper testing

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Experimental dynamic behavior of MR damper

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The structure of segment erector

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Nonlinear model of segment erector system

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The input signal of the system

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The displacement of segment erector with passive isolators

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The displacement of segment erector with MR isolators

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Resonance frequency of system

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Experimental setup

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Acceleration response in different voltage and excitation frequencies

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Optimal control surface

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Acceleration responses at oblique wave excitation

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Envelope curve of acceleration response

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Input current plots of MR isolator at oblique wave excitation

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Damping force plots of MR isolator at oblique wave excitation

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Acceleration responses at random excitation

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Envelope curve of acceleration response

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Input current plots of MR isolator at random excitation

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Damping force plots of MR isolator at random excitation

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