Helical Wave Spectra of the Radial Displacement of an Infinite Laminated Composite Cylindrical Shell Reinforced by Doubly Periodic Rings

FENG Yuan,CUI Hong-fei,YIN Xue-wen,HUA Hong-xing

(1 Wuxi Institute of Technology,Wuxi 214121,China;2 Wuxi Branch,Jiangsu Institute of Special Equipment Supervision and Inspection,Wuxi 214174,China;3 State Key Laboratory of Mechanical System and Vibration,Shanghai Jiao Tong University,Shanghai 200240,China)

Biography：FENG Yuan(1963-),male,associate professor of Wuxi Institute of Technology.

1 Introduction

The problems of vibration and acoustic radiation from infinite cylindrical shells[1-5]are the extension of those from isotropic cylindrical shells,which were studied in numerous articles[6-10].According to the thin shell theory,it is known that the shells,whatever they are,composite or isotropic,may carry three types of waves[5],namely,shear,compressional and flexural waves.Though the former two can be strongly coupled to the acoustic field in some cases,the latter contributes mainly to the transverse vibrations of shells.And flexural waves may propagate in the shell in helical paths determined by the circumferential and axial wavenumbers,which are usually termed as helical waves.

Particularly,Choi et al[12]proposed a modal-based method to analyze the acoustic radiation of an axisymmetric submerged cylindrical shell of finite length containing internal substructures,where a helical wave spectra expression for the finite cylindrical shell with two bulkheads was given in the following form,

where d is the distance between the two bulkheads,and α=0.76e0.72iwhich can be determined by two factors:the flexural wave dispersion and the location of the bulkheads.However,their work on helical wave spectra resorts to approximate numerical methods,where modal truncation was employed for prediction of the surface velocity.Yin et al[5]extended the work for an infinite isotropic plate done by Mace[13],as well as the corresponding work for an infinite laminated composite plate done by Yin et al[14]to develop an analytical expression for an infinite laminated composite cylindrical shell with doubly reinforcing rings,where the analytical expression for the helical wave spectra of the laminated composite cylindrical shell with and without the reinforcing rings was proposed in a more elaborate way.

In Ref.[5],the effects of internal reinforced rings on the helical wave spectra of radial displacement of the shells are discussed,where the effects of the ply schemes,i.e.,ply angle,are not addressed.Actually,one of the most distinct features of the laminated composite shells with comparison to the isotropic shells is that the former are made of layers with varying material orientations.In the present paper,a complementary work is conducted by addressing the effects of ply angles on the radial displacement of the composite shell with and without the reinforcing rings,which would be helpful for better understanding the characteristics of acoustic radiation from a composite shell with doubly rings.

2 Description of mathematical model

An infinite laminated composite cylindrical shell in Fig.1 is considered,whose total thickness h is much less than its mean radius a.Two sets of periodically spaced rings are attached to the inside surface of the shell.The first set of rings are the smaller rings with a spacing of l,and the second set of rings are the larger ones,which replace every qth ring in the smaller set.Thus,the spacing of the larger set of rings is ql.The shell is submerged in an unbound acous tic medium of density ρ0and acoustic phase velocity c0.The interior of the shell is in vacuo.The shell’s coordinate system is also shown in Fig.1,and its r-φ plane shall lie in the plane where one arbitrary larger ring is laid.

As illustrated in Fig.1,the cylindrical shell is driven by a combination of the following loadings:the external point force,the pressure by the acoustic fluid,and the reactive forces by two sets of rings.All of the loadings have a time dependence e-iωt.As an algebraic convenience,e-iωtis henceforth suppressed.

In the acoustic medium outside the shell,the pressure pa（r,φ,x ） satisfies the Helmholtz equation

and k0=ω/c0is the acoustic wave number.At the interface between the shell and the fluid,the continuity of radial displacement leads to the following condition

Based on the classical laminated composite shell theory,the motions of the laminated composite shell reinforced by doubly periodic rings and immersed in an acoustic field can be written in the following concise matrix form,where u and v are the axial and circumferential displacements of a point in the middle surface,respectively,w is the radial displacement of the shell.pa（a,x,φ）is the fluid loading due to the acoustic field,and peis the point force which is applied at point（x0, φ0）and with a magni-tude of Q.pr（x,φ）and pb（x,φ）are the reactive forces due to the two sets of rings,which are derived by Burroughs[6]and will not be repeated in this paper.The 3×3 matrix differential operators Lij（i=1,3;j=1,3）for a thin laminated composite shell will be defined in Appendix A of Ref.[5].

将研究所得各项数据纳入SPSS 19.0统计学软件中进行对比分析，将计量资料进行t检验，计数资料进行x2检验，数据差异有意义以P＜0.05为标准。

Eqs.(3)-(6),together with radiation conditions at infinity,uniquely constitute the problem of radiation form a point-driven,fluid-loaded,laminated composite cylindrical shell with doubly periodic rings.

For sake of brevity,the development of the solutions for the velocity and the radiated pressure of the laminated composite cylindrical shell with doubly rings will not be repeated here.The readers would review Ref.[5]if necessary.

3 Numerical calculation and discussion

In Ref.[5],the effects of ply schemes,Poisson’s ratios,and ply angles on acoustic radiation are addressed,besides,the contributions of the internal reinforcing rings on the helical wave spectra of the radial displacement of the laminated composite shells are discussed in detail.In engineering applications,one of the most remarkable features of the laminated composite plates and shells is that they are formed by stacking layers of different composite materials and/or fiber orientation.Since composite materials have different mechanical features in different material axes,ply angle is one of the most significant parameters in altering the dynamic characteristics.However,the problem that how the ply angle embodies itself in the helical wave spectra has not yet been adequately addressed.

Unfortunately,the ply angle is implicit in the governing Eq.(6),which would appear in the extensional stiffnesses Aij,the bending stiffnesses Dij,and the extension-bending coupling stiffnesses Bij

[5],so the relationship between the helical wave spectra and the ply angle can not be explicitly expressed.

wave spectra is expressed as follows:

Tab.1 Parameter values for the acoustic fluid,the rings,etc(all in S.I.units)

In this section,the parameters of the acoustic fluid and rings are listed in Tab.1 and a single layered shell with three different ply angles is depicted in Tab.2.The observation point is chosen 50m away from the origin of the shell’s coordinates and with the same circumferential and azimuthal angles of 45deg,and the acoustic pressure levels are calculated with reference to 1.0μPa,and corrected to a distance of 1m.

Tab.2 Parameters of the cylindrical shell(all in S.I.units)

Tab.3 Parameters of an one-layered composite shell(all in S.I.units)

3.1 Helical wave spectra of the shell’s radial displacement

In engineering applications,cylindrical shells are often reinforced by periodically or randomly spaced rings,which would convert the vibration responses at a single wavenumber into those at multiple wavenumbers,consequently,the acoustic characteristics of the original empty shells[5,9]would be altered.Williams et al[11]and Choi et al[12]pointed out that the helical wave spectra would exhibit eight-pattern for empty shells,and Yin et al[5]indicated that the lower wavenumber bright points would appear within the eight-pattern contour in addressing the acoustic problem of the laminated composite shells with periodic rings by utilizing numerical analysis.Yet,they all did not discover distinctive periodically spaced bright points in the contour of the helical wave spectra for stiffened shells.

When we check Eq.(22)in Ref.[5],we will find that the responses of shells with periodic rings（k） would comes from the contribution of the periodic rings with periodic wavenumbers,i.e.,（k-mkd）.Figs.2-3 show the helical wave spectra of the radial displacement of the shell depicted in Tab.2.From the contour plot it is shown that the presence of the periodic rings would increase periodically spaced bright points in lower wavenumbers.

With comparison to those given by Choi et al[12]for an isotropic shell,the eight-pattern in Figs.4 and 5 is not upright but with a sharp angle to the axial wave number axis because the shell is constructed with orthotropic plies with various ply angles,and the material axes of the shell are not directed along the shell’s geometry axes(see Tab.3).Moreover,with the increasing of the ply angles,the configuration of the contour would rotate in counter-clockwise direction.This would help the engineers in tailoring the composite materials for the optimization of sound design,as well as for attaining prescribed dynamic characteristics of the shells.

4 Conclusions

This work is a valuably complementary part for our previous publications for laminate composite cylindrical shells[5],as well as for laminated composite plates[14].The theory development is suppressed,but this would not affect the discussion of the features of the periodic rings and the ply angles in addressing the dynamic characteristics of the shells,namely,helical wave spectra of the radial displacement of the shells.The periodically spaced bright points due to the periodic rings,which are attributed to the wellknown effects of wavenumer conversion discovered by Mace[13]and Burroughs[9],are clearly identified.In addition,the configuration of the helical wave spectra in the contour plot would rotate in counter-clockwise direction with the increase of the ply angle.

[1]Lee Y S,Kim Y W.Vibration analysis of rotating composite cylindrical shells with orthogonal stiffeners[J].Computers and Structures,1998,69:271-281.

[2]Lee S J,Reddy J N.Vibration suppression of laminated shell structures investigated using higher order shear deformation theory[J].Smart Materials and Structures,2004,13:1176-1194.

[3]Singh A V,Shen L B.Free vibration of open circular cylindrical composite shells with point supports[J].J Aerospace Engineering,2005:120-128.

[4]Harari A,Sandman B.Vibratory response of laminated cylindrical shells embedded in an acoustic fluid[J].J Acoust.Soc.Am.,1978,60(1):117-128.

[5]Yin X W,Liu L J,Hua H X,Shen R Y.Acoustic radiation from an infinite laminated composite cylindrical shell with doubly periodic rings[J].J Vib.Acoust.,2009,131:011005-1.

[6]Smith P W.Phase velocity and displacement characteristics of free waves in a thin cylindrical shell[J].J Acoust.Soc.Am.,1955,27(6):1065-1072.

[7]Guo Y P.Approximate solutions of the dispersion equation for fluid-loaded cylindrical shells[J].J Acoust.Soc.Am.,1994,95(3):1435-1440.

[8]Guo Y P.Acoustic scattering from cylindrical shells with internal elastic plates[J].J Acoust.Soc.Am.,1994,95(5):2550-2559.

[9]Burroughs C B.Acoustic radiation from fluid-loaded infinite circular cylinders with doubly periodic supports[J].J Acoust.Soc.Am.,1984,75(3):715-722.

[10]Bjarnason J,Achenbach J D,Igusa T.Acoustic radiation from a cylindrical shell with an internal plate[J].Wave Motion,1992,15:23-41.

[11]Williams E G,Houston B H,Bucaro J A.Experimental investigation of the wave propagation on a point-driven,submerged capped cylinder using K-Space analysis[J].J Acoust.Soc.Am.,1990,87(2):513-522.

[12]Choi S H,Igusa T,Achenbach J D.Nonaxisymmetric vibration and acoustic radiation of a submerged cylindrical shell of finite length containing internal substructures[J].J Acoust.Soc.Am.,1995,98(1):353-362.

[13]Mace B R.Sound radiation form a plate reinforced by two sets of parallel stiffeners[J].J Sound Vib.,1980,71(3):435-441.

[14]Yin X W,Gu X J,Cui H F,et al.Acoustic radiation from a laminated composite plate reinforced by doubly periodic parallel stiffeners[J].J Sound Vib.,2007,306:877-889.