YANG Zhi-rong,LU Kun,RAO Zhu-shi,Yü Hong-liang

(1.Provincial Key Laboratory of Naval Architecture＆Ocean Engineering,Marine Engineering Institute, Jimei University,Xiamen 361021,China;2.State Key Laboratory of Mechanical System and Vibration, Shanghai Jiaotong University,Shanghai 200240,China)

Experiment Research of Axial Dynamic Vibration Absorbers Based on Magneto-Rheological Elastomers Using for Ship Shafting

YANG Zhi-rong1,2,LU Kun2,RAO Zhu-shi2,Yü Hong-liang1

(1.Provincial Key Laboratory of Naval Architecture＆Ocean Engineering,Marine Engineering Institute, Jimei University,Xiamen 361021,China;2.State Key Laboratory of Mechanical System and Vibration, Shanghai Jiaotong University,Shanghai 200240,China)

Magneto-Rheological Elastomers(MREs)are a new kind of Magneto-Rheological materials mainly composed of polymer rubber and micro-sized magnetizable iron particles.Dynamic vibration absorbers(DVAs)based on MREs are widely used in vibration systems with small amplitude since they have the advantages of no sealing equipments,good stability,and rapid response.This paper aims to research on the application of MREs in SDVAs for reducing axial vibration of ship shafting by experimental methods.The dynamic design principle of an axial SDVA attached to ship shafting, which is modeled as a continuous beam system,is studied.A DVA with adjustable stiffness based on MREs is proposed and its natural frequency-shift performance is tested.And,an experimental research on a model ship with scaled ratio 1:4 is carried out to verify the vibration attenuation capacity of SDVAs while propulsion shaft working on different rotation speeds.The results show that the MRE-based DVA has frequency-shift of 12.3 Hz and performs better than classic passive DVA on all rotation speeds.

magneto-rheological elastomers(MREs);intelligent materials; semi-active dynamic vibration absorber(SDVA);axial vibration; ship shafting

0 Introduction

The propeller is the main propulsion of ships and underwater vehicles,which works in spatially non-uniform wake field wherein axial pulse forces are produced and transferred to the hull through the propulsion shaft,thrust bearing,and bases.Consequently,axial vibration of shafting occurs and propagates through the supporting structure to the hull,where the vibration is radiated as structure-borne noise[1].In a series of means to reduce the axial vibrationof shafting,the installation of a dynamic vibration absorber(DVA)is found to be an effective and feasible method.Many types of vibration absorbers exist,including classical passive,active,and semi-active absorbers.Since classical passive absorber operates at a single excitation frequency,resonances in the system appear just above and/or below the excitation frequency,making the effective bandwidth of passive absorber very narrow and becoming inefficient as the excitation frequency shift[2].To overcome these disadvantages and extend the effective bandwidth of passive absorbers,active and semi-active absorbers are developed and introduced.Active DVAs[3]require a physically large envelope,power requirement,and cost. Additionally active systems have an inherent bandwidth limitation due to mechanical inertia (i.e.mass of actuator rod)and power electronic constraints.This bandwidth problem and weak control algorithms lead to active systems becoming unstable in the sense that under wide operation conditions a passive system will often significantly outperform the active system.Often the instability of an active system can lead to catastrophic results[4].Semi-active DVA(SDVA) controls the structural vibration by changing its dynamic parameters,such as the stiffness or damping.Some advantages of semi-active control are that it requires less energy,which lowers the costs,and its complexity is reduced in comparison with active systems,while being nearly as effective.There exist two major categories of variable stiffness elements,namely,mechanically variable spring and intelligent materials.Walsh and Lamancusa[5]reduced transient vibrations using a mechanically variable spring but found disadvantage of long response time of the mechanical structure and difficulty in meeting the design requirements of the system, which rapidly changes the vibration characteristics.Intelligent materials can also be used as variable stiffness elements such as piezoelectric ceramics,magneto-rheological elastomers (MREs),etc.Piezoelectric elements have relatively high stiffness that can lead to relatively large absorber masses.Also,piezoelectric materials should not be put into tension,as they are fragile and have displacement amplitude limitations.To make the device more robust and reduce the mass of absorber,magneto-rheological elastomers(MREs)can be used as variable stiffness elements[2,6].

MREs are composites whose highly elastic polymer matrices are filled with magnetic particles.Typically,magnetic fields are applied to the polymer composite during cross-linking so that chainlike structures can be formed and fixed in the matrix after curing.The unique characteristic of MRE is that its shear storage modulus can be controlled by the external magnetic field rapidly,continuously,and reversibly[7-8].Such properties make MREs promising in many applications,such as SDVAs,stiffness tunable mounts and suspensions,and variable impedance surfaces.Watson[9]applied a patent using MREs for a suspension bushing.Ginder et al[10]developed an adaptive tunable vibration absorber(TVA)using MREs.Li et al[11-12]designed a soft MR elastomer,fabricated and incorporated in the laminated structure of the new MRE base isolator,which aims to obtain a highly adjustable shear modulus under a medium level of magnetic field,with the ability for real-time adaptive control of base isolated structures against various types of earthquakes including near-or far-fault earthquakes.Kim et al[13]proposed a real time control system for the MRE based tuned vibration absorber(TVA)to suppress the cryogenic cooler vibrations.Research has also been conducted on optimizing the tuning parameters of vibration absorbers applied to continuous systems.Jacquot[14]modeled a continuous beam as an one-degree of freedom system and determined the optimum absorber parameters for this equivalent lumped mass system.To minimize the response for the first two modes,Kitis et al[15]employed a gradient-based optimization to tune two TVAs attached to a cantilever beam.Rice[16]used a simplex algorithm to optimize the position,stiffness,and damping of absorbers attached to a beam.Esmailzadeh and Jalili[17]fixed the absorbers’position along the beam and optimized their stiffness and damping by means of a direct update method. A genetic algorithm was used by Hadi and Arfiadi[18]to optimize the parameters of a TVAs applied to multi-degree of freedom structures.

This work aims to research on the application of MREs in SDVAs for reducing axial vibration of ship shafting by experimental methods.The dynamic design principle of an axial SDVA attached to ship shafting,which is modeled as a continuous beam system,is studied.A SDVA with adjustable stiffness based on MREs is proposed and its natural frequency-shift performance is tested.Furthermore,an experimental research on a model ship with scaled ratio 1:4 is carried out to verify the vibration attenuation capacity of SDVAs while propulsion shaft working on different rotation speeds.

1 Design principle of vibration absorption of SDVA attached to propulsion shaft

A typical ship-shafting system with SDVA is illustrated in Fig.1.As shown in the figure, the propulsion shaft mainly consists of a propeller,propeller shaft,intermediate shaft,flange,thrust shaft,thrust bearing and its bases.The propeller is assumed as a lumped mass M while the propulsion shaft is modeled as a distributed mass and flexibility homogeneous beam.The thrust bearing and the bases are modeled by linear spring with a stiffness K taken in the longitudinal direction.The SDVA is simplified as mass-spring-damper system,in which the ma,kaand cadenote the mass,stiffness,and damping coefficients,respectively.The simplified model of ship shafting-SDVA system is shown in Fig.2.

Fig.1 The installation of SDVA on propulsion shaft

Fig.2 The simplified model of ship shafting-SDVA system

An assumed mode method is employed to derive the discretized equations of motion.The axial displacement field of the shaft is expanded in the following form:

where N is the maximum number of modes taken for shaft.φiis the ith mode function and q denotes the corresponding generalized displacement.

The total kinetic energy of the system,including the energy contributions of the shipshafting and the attached absorbers,is given as:

The potential energy of the shaft and springs due to deformation can be expressed as:

where E is the modulus of elasticity,K is the longitudinal stiffness of thrust bearing,kais the absorber’s stiffness,u（ l,t）and u（ a,t）denote the axial displacements of the shaft at the location of x=l and x=a respectively,and uais the axial displacement of the absorber.

Generally,the propeller forces mainly excite the first-order vibration mode of the shipshafting system with the normal working speed.Consequently,only the first-order vibration mode of the shaft is employed in this work.In doing so,the Eq.(1)can be rewritten as:

Substituting Eq.(4)into Eqs.(2-3),the equations of motion can be obtained by using the Lagrange method.Considering the harmonic concentrate excitation force F0eiωtapplied at location x=0 on the shaft,the equations of motion of shafting-absorber system can be written in following matrix form:

For simplicity in the mathematical formulation,the following dimensionless parameters are introduced:

where K is the longitudinal stiffness of thrust bearing and δstis the static deformation.

Substituting Eq.(7)into Eq.(6),one obtains:

2 Structure of a single SDVA

Fig.3 The structure of a single SDVA

The schematic and fabricated diagram of the MRE-based SDVA is shown in Fig.3 and Fig.4 respectively.As shown in the figure,the MRE-based SDVA consists of three main components:the oscillator or dynamic mass,smart spring elements with MREs,and the base attached to the shafting indirectly.The smart spring elements(MREs)connect the dynamic mass and the base.The dynamic mass and the base are made of low-carbon steel.It can be seen thatthe dynamic mass,MREs,and the base form the closed circuit.The current coils are strapped on the base to emit the magnetic field acting on MREs and some grooves are slotted in the dynamic mass to install the current coils to increase the magnetic field intensity. This development makes SDVA more compact and more efficient for shafting axial vibration control as no additional oscillators and complex mechanical devices are required.The mass of the oscillator is 7.7 kg and the total current coils of the SDVA are 460 rounds.Furthermore,the outer and inner diameters of MREs are 50 mm and 40 mm,respectively.The length and out diameters of SDVA are 70 mm and 80 mm,respectively.

Fig.4 The fabricated MRE-based dynamic absorber

3 Frequency-shift property test of a single SDVA

To investigate the frequency-shift property of SDVA,the experimental setup is shown in Figs.(5～6)and the experimental procedure is as below.The developed MRE-based SDVA is placed at the centre of the vibration test rig.The sweep-frequency sine excitation applied to the vibration test rig is generated by an exciter via the power amplifier as the vibration test rig produces the horizontal direction movement.Two accelerometers are placed on the oscillator and the base to measure their responses,respectively.The measured signals are sent to the dynamic signal analysis system.The transfer function can be achieved by using FFT analysis and the natural frequency of SDVA is subsequently obtained.The control currents applied to the MRE-based SDVA are supplied from the adjustable direct current power source.The whole system can measure the natural frequencies of SDVA under different control currents.

Fig.5 The principle diagram of frequency-shift property test of dynamic absorber

Fig.6 The frequency-shift property test of dynamic absorber

The results shown in Fig.7 show that the natural frequency of SDVA has increasing trend with control currents.In the initial stage,natural frequency increases and then the rate of slope tends toward stabilization.Moreover,the natural frequencies of SDVA increase from 85 Hz at 0A to 98 Hz at 5A,which cover the normal working frequencies of ship shafting.

4 The test of vibration attenuation of ship shafting with SDVAs

To experimentally evaluate the vibration attenuation of propulsion shaft with SDVAs,a model shaft with scaled ratio 1:4 is employed and the schematic diagram is shown in Fig.8.

Fig.7 Shift-frequency of SDVA

Fig.8 The principle diagram of the vibration attenuation capacity test

The total mass of the model shaft is about 386 kg and the first step resonance frequency of axial vibration of ship shafting is about 94 Hz.Since the mass of ship shafting is large,several SDVAs are linked in parallel and connected to ship shafting to increase the mass ratio however the stiffness of SDVAs remains the same as a single SDVA.Fig.9(a)shows that three SDVAs are linked in parallel to assemble the ship shafting-SDVAs system.The control currents are applied by the conductive ring from the power source as shown in Fig.9(b)so that the SDVAs can rotate with the propulsion shaft.

Fig.9 The combination SDVAs and the control current supply

The excitation system of ship shafting is shown in Fig.10.The propeller is loaded by the exciter through the plunger and the air spring.The separated type thrust bearing is installed between the propeller and the exciter,so the propeller can be loaded a static thrust about 0.8 t in rotating condition.A force sensor is fixed on the plunger to measure the exciter force amplitude,while the accelerometer is placed on the flange of the thrust bearing of ship shafting.

Fig.10 Excitation system of propulsion shaft

The measured signals are sent to the wireless dynamic signal analysis system.The transfer function can be achieved by using FFT analysis to acquire the vibration attenuation of the ship shafting as follows.

where TAis the absolute transmissibility,a is the acceleration of the axial vibration of the shaft,and F is the force of the exciter.Generally,the vibration acceleration level(dB)is used to measure the vibration of the system.The Eq.(10)can be expressed as follows:

where a0is the reference acceleration,and its value is a0=10-6m·s-2.

To experimentally evaluate the vibration attenuation of the ship shafting with different DVAs,the passive DVAs are realized by fixing the applied currents of SDVAs with 3.5A as the natural frequencies of SDVAs are about 94 Hz(seen from Fig.7)and are equal to the first resonance frequency of axial vibration of ship shafting,while the SDVAs are achieved by changing the currents of SDVAs whose frequencies are tuned to trace the excitation frequency.The vibration absorption capability of SDVAs is compared with the passive DVAs and ship shafting without DVAs under different rotational speeds of shaft(0 rpm,100 rpm,200 rpm)is shown in Fig.11.

Fig.11 Comparison of vibration attenuation of SDVAs,passive DVAs and without DVAs

It can be seen that the ship shafting without DVAs has the primary resonance peak of the shafting at 94 Hz(0 rpm)and at 92.2 Hz(100 rpm,200 rpm)because of oil film damp due to shaft rotating.The ship shafting without DVAs has the relatively large displacement responses in the whole excitation frequency range.For the ship shafting with classic passive DVAs,the best vibration attenuation efficiency occurs near the natural frequency of the primary shafting system,but the effect becomes worse sharply while the excitation frequency is apart from the natural frequency specially in the low frequency range where the acceleration level increases about 3.5 dB.For the ship shafting with SDVAs,the SDVAs have better vibration absorption capability than the classic passive DVAs in the whole excitation frequency bandwidth and the vibration attenuation of resonance peaks is about 10 dB.

5 Conclusions

In this work,a SDVA with adjustable stiffness based on MREs is proposed and its natural frequency-shift performance is tested.It is found that the natural frequency of SDVA has increasing trend with control current and its natural frequency bandwidth is about 85～98 Hz.Furthermore,an experimental research on a model ship with scaled ratio 1:4 is carried out to verify the vibration attenuation capacity of SDVAs while propulsion shaft working on different rotation speeds.It shows that the SDVAs have better vibration absorption capability than the classic passive DVAs in the whole excitation frequency bandwidth and the vibration attenuation of resonance peaks is about 10 dB.

Acknowledgement

This work is supported by the Natural Science Foundation of Fujian Province,China (Grant No.2016J05130)and Scientific Research Fund of Fujian Provincial Education Department(Grant No.JA15272).

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基于磁流變彈性體的船舶軸系縱振動(dòng)力吸振器的實(shí)驗(yàn)研究

楊志榮1，2，盧坤2，饒柱石2，于洪亮1
（1.集美大學(xué)輪機(jī)工程學(xué)院福建省船船與海洋工程重點(diǎn)實(shí)驗(yàn)室，福建廈門(mén)361021；2.上海交通大學(xué)機(jī)械系統(tǒng)與振動(dòng)國(guó)家重點(diǎn)實(shí)驗(yàn)室，上海200240）

磁流變彈性體是一種新型的智能材料，由微米級(jí)的羰基鐵粉和聚合物合成。由于它具有彈性模量可調(diào)、無(wú)需封裝、穩(wěn)定性好、響應(yīng)快等優(yōu)點(diǎn)，已被廣泛應(yīng)用于機(jī)械振動(dòng)控制等領(lǐng)域。文中針對(duì)船舶推進(jìn)軸系縱振動(dòng)力學(xué)特性，提出一種固有頻率可調(diào)的磁流變彈性體動(dòng)力吸振器用于軸系縱振控制。首先理論分析了安裝動(dòng)力吸振器的推進(jìn)軸系的動(dòng)力學(xué)模型，其次對(duì)動(dòng)力吸振器進(jìn)行了移頻特性試驗(yàn)，并在1:4的船舶軸系縮比模型上進(jìn)行了變轉(zhuǎn)速工況下吸振效果試驗(yàn)。實(shí)驗(yàn)結(jié)果表明，該吸振器具有12.3 Hz的移頻范圍，并且在不同軸系工作轉(zhuǎn)速下相比于被動(dòng)式吸振器均有更好的吸振效果。

:磁流變彈性體；智能材料；半主動(dòng)式動(dòng)力吸振器；縱向振動(dòng)；船舶軸系

U664.21

:A

楊志榮（1981-），男，博士，集美大學(xué)輪機(jī)工程學(xué)院講師；

U664.21

:A

10.3969/j.issn.1007-7294.2017.06.009

1007-7294（2017）06-0750-11

盧坤（1991-），男，上海交通大學(xué)機(jī)械與動(dòng)力工程學(xué)院碩士研究生；

date：2017-01-20

Supported by the National Natural Science Foundation of Fujian Province,China (Grant No.2016J05130);Scientific Research Fund of Fujian Provincial Education Department(Grant No.JA15272)

Biography：YANG Zhi-rong(1981-),male,lecturer of Marine Engineering Institute of Jimei University, E-mail:yzhirong2000@126.com;RAO Zhu-shi(1962-),male,professor/tutor.

饒柱石（1962-），男，上海交通大學(xué)機(jī)械與動(dòng)力工程學(xué)院教授，博士生導(dǎo)師；

于洪亮（1963-），男，集美大學(xué)輪機(jī)工程學(xué)院教授，博士生導(dǎo)師。