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(National Key Laboratory on Ship Vibration and Noise,China Ship Scientific Research Center,Wuxi 214082,China)

Abstract:The extensive application of pipeline systems in ship industry has attracted growing research attentions on the vibration and acoustic radiation of fluid-filled pipes.In this paper,the transverse vibrations of pipes conveying fluid,based on Timoshenko beam theory,are calculated by using the transfer matrix method.The periodic characteristics of wave propagation,i.e.wave stop and wave bands are investigated for fluid-filled piping systems with periodic supports.Through validations against FEM and experiments,the present model considering shear deformation proves to be more accurate than the traditional slender body theory,especially in higher frequency domains.Meanwhile,it is found that the wave stop and propagation bands are significantly affected by the stiffness and equidistance of periodic spring supports.This work will provide some guidance for the vibration reduction of pipeline structures with periodic supports.

Key words:periodic support;pipeline system;transverse vibration;band gap

0 Introduction

Pipeline systems carry fluid in marine crafts and airplanes.However,excessive vibrations due to fluid-structure interactions could cause damage to the devices connected with the pipeline systems.Hence much attention has been focused on prediction of the dynamic responses of fluid-filled pipelines in order to seek for effective vibration control measures.In practical engineering,typical piping systems such as steam generator heat exchange tubes,oil pipe lines and marine risers,are supported in various ways,depending on environmental conditions and individual requirements[1].In general,these supports are arranged periodically along the axial direction of pipelines.With special dynamic characteristics of free wave propagation,such spatially periodic structures have been studied by many researchers.Yin et al[2]investigated the free stress and pressure waves in fluidfilled pipe systems.The flexible hoses have showed an excellent performance in controlling both stress and pressure waves,and the spring supports are of significance in pressure wave reduction.Du et al[3]calculated the band gaps of periodic pipelines made of functionally-graded materials by using the finite element method.The results reveal that the inclusion of functionally-graded material can improve the band gap properties compared with classic periodic pipelines.Sun et al[4]used the wave approach to investigate the effects of support stiffness and online mass on the transverse vibration propagation characteristics.Shen et al[5]used the transfer matrix method to study the vibration transfer properties of fluid-filled L-shaped pipes in the form of periodic arrangement,but did not consider the effects of supports.

In this paper,the transverse vibration of a straight pipe conveying fluid is modeled by Timoshenko beam theory and calculated by transfer matrix method.With these formulations,the vibration wave stop and propagation bands of periodically-supported pipe systems are analyzed.The harmonic responses are compared with experiment data to verify the present method.Afterwards,the influences of periodic support parameters on wave propagation are discussed.Through this research,the design of periodic supports might be an effective way to reduce vibration in pipeline systems.

1 Formulation of an infinite periodic pipe

1.1 Single pipe conveying fluid

Pipes are slender structures with relatively small sections,therefore they are usually modeled as beams in engineering calculations[6].While describing the flexural vibrations of a straight uniform fluid-filled pipe element,the Timoshenko beam theory is superior to the Euler-Bernoulli beam theory since the shear deformation and moment of inertia have been included.The governing equations of transverse vibration for fluid-filled pipes based on the Timoshenko beam theory are given as[7-8]:

whereyandzare the axial and transverse coordinates,trepresents time,uzandφxrepresent the transverse vibration and rotation angle respectively,fzandmxare the corresponding shear force and bending moment on the cross section,Gis the shear modulus,Eis the Young’s modulus,IPandIFare the moment of inertia of pipe and fluid respectively,ρPandρFare density of pipe and fluid respectively,APis the cross section area of pipe,andkPis the shear coefficient.As for pipes with circular cross sections,

whereμis the Poisson’s ratio,randRare the inner and outer radius of pipe sections,respectively.The general solution to Eq.(1)is written as

where,j is the imaginary unit,i.e.j2=-1,andωis the circular frequency.

With Eq.(3)substituted into Eq.(1),and for the case of harmonic excitation,the following equation can be obtained in the frequency domain:

where the coefficients are

The solution of Eq.(4)can be expressed as

whereAnare constants,and the flexural wave numbersλnare the four complex roots of the equation:

Substituting Eq.(6)into Eq.(3),the functions can be written in the matrix form as

Eq.(8)can be expressed asYis the column vector on the cross section,Ais the column vector for constants,Qis the square matrix of dimension 4×4.Suppose that the length of the pipe element isL,and then substitutey=0 andy=Linto the equation,the following two equations can be derived:

where the specific forms of the matrices are

Eq.(9)is combined as

whereTis the transfer matrix of a periodic cell,

1.2 Pipe systems with supports

Fig.1(a)shows the periodical supported piping system formed by periodic cells in Fig.1(b).Lis the length of the shell andKis the spring stiffness of the supports.The governing equation for the periodic shell without springs has been given in Eq.(10).Since the boundary conditions should be applied on the periodic shell,the new governing equation should be reformed to

Fig.1 Periodically supported piping system formed by periodic cells

whererepresents the sum of shear force resulting from pipes and the elastic force resulting from springs,T*is the new transfer matrix for this periodic shell with springs,and it can be calculated as follows:

Now in the periodic cell,the right column vector is represented by the left column vector in the form of Eq.(11).And according to Floquet theorem[9],the state vectors can also be represented as

whereZis a constant related with frequency.

Associating Eq.(11)with Eq.(12),it is easily derived that

It is indicated thatZare the eigenvalues of the transfer matrixT*.The wave propagation constantsμare obtained by these eigenvalues:

whereμRandμIare real and imaginary parts ofμrespectively.

2 Vibration responses in frequency domain

The calculation model in this example is a clamped-free pipe.A unit harmonic force and moment are applied on the free end respectively.In order to distinguish the Timoshenko and Euler models,the displacement response curves are plotted in Fig.2.The pipe has an outer radius of 0.05 m,a length of 2 m and a thickness of 0.005 m.The densities for the pipe and fluid are 8 000 kg/m3and 1 000 kg/m3respectively.The Young’s modulus and Poisson’s ratio are 208 GPa and 0.3 respectively.It is shown in Fig.2 that the results derived by Timoshenko beam model are more accordant with the results obtained by FEM software.In low frequency rarge,i.e.below 200 Hz,there are no differences between the two beam models.However,as the frequency increases,the peaks of Euler results are right shifted.So in the low frequency range,the two models are applicable,but in a higher frequency range,the Timoshenko model which considers shear deformation is more appropriate and accurate.

Fig.2 Displacement and rotation angle response at driving point under unit excitation

In addition,an experiment was carried out to verify the calculation method.Frequency response curve calculated by the present theory is compared with those derived by test.The dimensions of the pipe and material properties are the same with the data in the above example.The straight steel pipe was excited by a horizontal electric shaker at the top end while the other end was fixed.An accelerometer was used to measure vibration responses.Fig.3 shows acceleration responses of pipes with and without internal fluid.It is easily found that the two results are in agreement.Due to the mass added by the internal fluid,the vibration peaks moved to lower regions.Since there were no bends or branched pipe,the interaction between the pipe wall and fluid was weak.

Fig.3 Frequency response of the pipe at driving points

3 Characteristics of wave propagation

3.1 Sensitivity of pipe parameters on wave propagation

Sensitivity analyses are carried out to investigate the characteristics of the wave propagation of a periodically supported piping system due to the internal fluid,support stiffness and equidistance.Discussion of these parameters is significant to determine the wave stop bands,in which the excitation is not well propagated.

Fig.4 shows frequency-dependent curves of the real and imaginary parts of propagation constants in order to clarify the effect of internal fluid.It is found that internal fluid changes the wave propagation bands greatly,especially in high frequency rarge.With the inclusion of internal fluid,the propagation bands become narrower,for example,the first three wave bands without water are 82 Hz,129 Hz and 162 Hz while the first three bands with water are 67 Hz,105 Hz and 132 Hz.It is noted that the amplitude of imaginary part of the wave vector is the same,which means the wave propagation speed will not be affected by internal fluid.

Fig.4 Effects of internal water on wave propagation(——without internal water;----with internal water

Fig.5 shows frequency-dependent curves of propagation constants under different support stiffness.As the support stiffness decreases,the wave propagation bands move towards the lower frequency region.There is no change when the support stiffness increases from 1010N/m to 1012N/m,which means the supports are hard enough to the pipe.When the stiffness is less than 106N/m,the periodic pipe supports no longer show wave stop bands.

Fig.5 Effects of pipe support stiffness(——k=106 N/m;----k=108 N/m;····k=1010 N/m;-·-·-·k=1012 N/m)

The equidistance effect is shown in Fig.6.It is seen from Fig.6 that as the equidistance increases,the wave propagation bands move towards the low frequency region,and the length of the wave propagation band becomes shorter.Therefore,equidistance can be one of the most important design bases in the design of periodically supported piping systems.

Fig.6 Effects of equidistance of periodic system on free wave propagation

3.2 Response of periodic supported pipe system

The pipeline calculated in this example is a 12-metre long uniform straight pipe with six spring supports.The equidistance of the supports is 2 m..A unit force is applied on the end of the pipe as shown in Fig.7.The wave propagation constants and the vibration responses at the positions of six supports are compared in Fig.8.

Fig.7 Layout of the calculation piping system

Fig.8 Frequency-dependent vibration responses and free wave propagation of periodically supported pipe system

It is clearly found that the frequencydependent vibration response curves are in agreement with wave propagation and attenuation bands.For example,when the imaginary part of the wave number is 0,there is only the attenuation wave,so the vibration responses fade with the increasing distance away from the excitation.On the contrary,when the real part of the wave number is 0,there is only the propagation wave.As the frequency increases,both the wave propagation and attenuation bands get wider.Such characteristics can be applied to the vibration control of fluid-filled pipes.

4 Concluding remarks

The transverse vibration wave propagation of periodically supported pipes is studied in this paper by using the transfer matrix method.The present calculating model is verified by FEM results and experiment data.The stiffness and equidistance of periodic supports greatly influence the free wave propagation characteristics in frequency bands.The results presented by this paper can serve as a valuable reference for the design of piping systems with periodic supports,which may benefit the vibration control and reduction of fluid-filled piping systems.