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School of Mechanical and Electric Engineering，Soochow University，Suzhou 215131，P.R.China

Abstract: The doubly curved shell（DCS）is a common structure in the engineering field. In a thermal environment，the vibration characteristics of the DCS will be affected by the thermal effect. The research on the vibration characteristics of DCS in thermal environment is relatively limited. In this paper，the thermal strain and the change of Young’s modulus caused by the changing of temperature are studied，and the DCS energy equation is established systematically. The displacement tolerance function of the DCS is constructed by the spectral geometry method，and the natural frequencies and mode shapes of the DCS with different structural parameters，such as thicknesses，ratios of Ra/Rb and a/b，at different temperatures are solved by the Rayleigh-Ritz method. The results show that the natural frequency of the DCS decreases with the increasing temperature，Ra/Rb and a/b ratios，and increases with the increasing thickness.

Key words：doubly curved shell；thermal environment；Rayleigh-Ritz method；natural frequencies

0 Introduction

The doubly curved shell is widely used in aero?space，marine engineering，vehicle engineering，civil construction，machinery and other related engi?neering fields. Therefore，it is necessary to study the vibration characteristics of the doubly curved shell（DCS）in thermal environments. Huu et al.［1］analyzed the free vibration of simply supported the doubly curved shell on elastic foundations in thermal environments and deduced the motion control equa?tion based on first-order shear deformation theory and the Hamiltonian principle. Alijani et al.［2-3］inves?tigated nonlinear forced vibrations of functionally graded material （FGM） doubly curved shallow shells with rectangular and thermal effects. Yazdi［4］investigated the large amplitude vibration of moder?ately thick three-phase multiscale composite doubly curved shells. Viola et al.［5］provided a general framework for the formulation and dynamic analysis computations of moderately thick laminated doublycurved shells and shells. Sofiyev et al.［6］analyzed the large amplitude vibration behavior of functional?ly graded orthotropic double-curved shallow shells，and various examples revealed that the influence of heterogeneity was noticeable. Ghorbanpour［7］stud?ied the critical temperature rise of cylindrical shells based on higher-order stability equations. Li et al.［8］analyzed the free vibration of cylindrical shells with temperature-dependent material properties. Torna?bene et al.［5］studied the differences between the well-known first-order shear deformation theory（FSDT）and several higher-order shear deforma?tion theories（HSDTs）and established［9］the dis?placement field formulation of laminated composite shell structures with variable radii of curvature by HSDTs. Dastjerdi et al.［10-11］studied the non-linear dynamic analysis of torus-shaped and cylindrical shell-like structures and presented［12］a single gener?al formulation for the analysis of various shellshaped structures，including cylindrical，conical，spherical，elliptical，hyperbolic，parabolic. Karimi?asl et al.［13］investigated the post-buckling behaviors of doubly curved composite shells in hygro-thermal environment by employing multiple scales perturba?tion method.

Based on the Rayleigh-Ritz method，four sets of springs with adjustable stiffness are introduced at the boundary of the DCS to simulate various bound?ary conditions in this paper. The displacement toler?ance function based on the spectral geometry meth?od is used to ensure the continuity of the boundary of the shell，and the vibration characteristics of dou?bly curved shells are calculated by considering the thermal strain.

1 Theoretical Analysis

The geometry and coordinate system of the doubly curved shell are shown in Fig. 1. Four groups of virtual springs are introduced around the doubly curved shell to simulate the boundary condi?tions.

Fig.1 Geometry and coordinate system of doubly curved shell

In Fig.1，φ，θandzrepresent the two circum?ferential and normal coordinates of the doubly curved shell，respectively；andu，vandwrepre?sent the displacements of the middle surface of the shell in the directionsφ，θandz，respectively；aandbrepresent the arc lengths of the shell in the di?rectionsφandθ，respectively；RaandRbrepresent the curvature radius of the shell in the directionsφandθ，respectively；his the shell thickness in the di?rection of thez-axis direction. In Fig.1，three sets of linear displacement constraint springs（kuφa，kvφa，kwφa）and one set of rotary constraint springs（Kwφa）are set along the three coordinatesφ，θ，andzdirections lo?cated atφ=a，respectively（stiffness constants for other springs are defined similarly）.

The Rayleigh-Ritz method is employed to solve the free vibration problem of the DCS. The Lagrangian for the shell can be written as［14］

whereVpis the total potential energy associated with the strain energy of the DCS and the deforma?tions of the restraint springs andTpdenotes the total kinetic energy of the shell. Both can be explicitly ex?pressed as［15］

whereDis the bending rigidity，D=E（t）h3/12（1-ν2）；andGthe extensional rigidity，G=E（t）h/（1-ν2）.E（t），ν，ρa(bǔ)ndωare the temperature de?pendent Young’s modulus［16］，the Poisson’s ratio，the mass density and the natural frequency of the DCS，respectively.φaandθbrepresent the value ofφatφ=aandθatθ=b，respectively.

According to the Flugge thin-shell theory，the strain on the middle surface of the doubly curved shell can be expressed as［17］

Considering the thermal effect caused by tem?perature change，the thermal strain in the middle surface of the shell is［18］

whereαφ，αθ，αφθare the thermal expansion coeffi?cients inφ，θand tangential directions.andindicates the strains at temperatureTand the su?perscriptTrepresents the temperature. Then，the mechanical strains in the middle surface of the shell are

The Spectro-Geometric method is adopted to construct the displacement functions［19］

whereλm=mπ/φa，λn=nπ/θb，Wmn，Umn，andVmnrepresent unknown coefficient vectors of displace?ment function series expansion of the shell；mandnare the wavenumbers in theφ-axis andθ-axis direc?tions；andMandNare the numbers of truncation coefficients that are used.

Substituting Eqs.（12—14） of the DCS dis?placement into Eqs.（1—3）and minimizing the re?sulting equations against the unknown Fourier coef?ficients will lead to a set of couple systems

whereKandMrepresent the stiffness matrix and mass matrix of the shell，respectively，andErepre?sents the unknown coefficient vector of displace?ment function series expansion.

By solving Eq.（15），the natural frequency and the mode shapes of the doubly curved shell in ther?mal environment can be obtained.

2 Results and Discussion

With the theoretical model and formulations in the previous section，the natural frequencies and mode shapes of the doubly curved shell with a ther?mal environment under classical boundary condi?tions are analyzed in this section. The Young’s modulusE（t）of the material used in this paper vary?ing with temperature is shown in Table 1. The ther?mal modal analysis of the doubly curved shell can be carried out according to the above material proper?ties.

Table 1 Young’s modulus of material at different tem?peratures

2.1 Method validation

The formulations and the resulting model will be first validated against the results obtained by the finite element method（FEM）. In Table 2，the first six natural frequencies for the DCS with completely free（FFFF）and clamped boundary（CCCC）condi?tions at 200 ℃are examined. The material proper?ties and geometrical dimensions used for the DCS are given as follows：ρ=2 700 kg/m3，ν=0.3，Ra=Rb=5 m，h=0.002 m，a=0.8 m，b=0.4 m.The simply supported boundary condition（S）could be derived by setting the translational stiffness and the rotational stiffness to infinitely large and zero，respec?tively. The clamped boundary condition（C）is real?ized by setting the translational and rotational stiff?nesses to infinitely large.For the free boundary condi?tion（F），the translational and rotational stiffnesses are both set to zero.

Table 2 Comparison and convergence of the first six natural frequencies for DCS with completely free and clamped boundary conditions

Fig.2 Comparisons of mode shapes of curved panel with completely clamped edges between present method and FEM(Upper plot:the present method;Lower plot:FEM)

It is clearly seen that the results agree well with each other. The convergence study of the truncated number of the improved Fourier series is also exam?ined in Table 2. The results converge atM=N=14 for the given four-digital precision and numerical sta?bility of the solution is evident. The above results completely prove the correctness of the method in this paper. To further validate the correctness of the present method，the first to sixth mode shapes of the curved panel under completely clamped edges（CCCC）are shown in Fig.2. Again，good agree?ment is observed between the two predictions.

2.2 Comparison of thermal modal frequency under classical boundary conditions

Fig.3 shows the first six natural frequencies for the DCS with a completely clamped boundary and a simply supported boundary at different temperatures.The material properties and geometrical dimensions used for the DCS are given as follows：ρ=2 700 kg/m3，ν=0.3，Ra=Rb=4 m，h=0.002 m，a=0.5 m，b=0.3 m. Whether in Fig.3（a）or Fig.3（b），it is seen that when the modal number is determined，the modal frequency of the DCS will decrease with in?creasing temperature，but the temperature will not affect the change trend of the modal frequency.

Fig.3 The first six natural frequencies for doubly curved shell at different temperatures

Fig.4 shows a comparison of the first eight nat?ural frequencies for the doubly curved shell under different boundaries at 200 ℃. The material proper?ties and geometrical dimensions used for the doubly curved shell are given as follows：ρ=2 700 kg/m3，ν=0.3，Ra=Rb=3 m，h=0.002 m，a=0.4 m，b=0.4 m. The CSCS represents the clamped boundary atφ=0 andφ=a，and the simply support?ed boundary atθ=0 andθ=b. The SCSC repre?sents the simply supported boundary atφ=0 andφ=a，and the clamped boundary at θ=0 and θ=b.It is clearly seen that the two curves of CSCS and SCSC coincide when the temperature is constant.This is because the DCS has symmetrical character?istics whena=bandRa=Rb. The SSSS boundary condition has the lowest modal frequency and as the number of fixed edges increases，the modal frequen?cy will also increase.

Fig.4 Comparison of the first eight natural frequencies for DCS under different boundaries at 200 ℃

The mode shapes of the DCS with different boundaries at 200 ℃are plotted in Table 3，in which the first，third，fifth，eighth order of the mode shapes are randomly selected. The material proper?ties are the same as in Fig.4.Since the frequencies of the doubly curved shell under CSCS boundary condi?tion is the same as that under SCSC boundary condi?tion，only the mode shapes of CSCS are analyzed. It is clearly seen that the mode shapes under different boundary conditions have small change but percepti?ble.With the increase of the numbers of fixed edges，the area where the peripheral displacement of the doubly curved shell is zero gradually increases，and the maximum vibration mode points gradually ap?proach the center of the shell.

Table 3 Mode shapes of DCS with different boundaries at 200 ℃

2.3 Effects of structural parameters

Fig.5 shows the first natural frequencies for the doubly curved shell with different thicknesses under different temperatures. The material properties and geometrical dimensions used for the doubly curved shell are given as follows：ρ=2 700 kg/m3，ν=0.3，Ra=Rb=4 m，a=0.2 m，b=0.2 m. The boundary condition is a completely simply supported boundary. Whenhis constant，the modal frequency decreases with increasing temperature，which is the same as that in Fig.3. When the temperature is con?stant，the modal frequency increases with increas?ingh.

Fig.5 Comparison of the first natural frequencies for DCS with different thickness

Fig.6 Comparison of the first natural frequencies for DCS with different curvature radius

Fig.6 shows the first natural frequencies for the doubly curved shell with different curvature radii un?der different temperatures. The material properties and geometrical dimensions used for the doubly curved shell are given as follows：ρ=2 700 kg/m3，ν=0.3，h=0.002 m，a=0.2 m，b=0.2 m. The boundary condition is a completely simply supported boundary. When the temperature is constant，the modal frequency decreases with the increase of the ratio ofRa/Rb. When the ratio ofRa/Rbis constant，the modal frequency decreases with the increasing temperature，which is also consistent with that in Fig.3.

Fig.7 shows the first natural frequencies for the doubly curved shell with different ratios ofa/bunder different temperatures. The material properties and geometrical dimensions used for the doubly curved shell are given as follows：ρ=2 700 kg/m3，ν=0.3，Ra=Rb=4 m andh=0.002 m. The boundary condi?tion is a completely simply supported boundary. It is clearly seen that the modal frequency decreases with the increase ratio ofa/bat the same temperature.This is because the first natural frequency of the shell decreases with increasing DCS size.

Fig.7 Comparison of the first natural frequencies for DCS with different ratio of a/b

3 Conclusions

In this paper，a theoretical model of the doubly curved shell under a steady thermal environment is established，and the natural frequency and mode shape of the doubly curved shell are solved by the Rayleigh-Ritz method. By setting several classical boundaries and different structural parameters，the modal frequencies of the doubly curved shell at sev?eral temperatures are analyzed. This paper mainly draws the following conclusions：

（1）The detailed solution process of the natu?ral frequency and mode shape of the doubly curved shell in a thermal environment is established. It is verified that the results are stable when the trunca?tion coefficientM=N=14，and the results show a good agreement with the finite element method un?der completely clamped and simply supported boundary conditions.

（2）Under the completely clamped and sim?ply supported boundary conditions，the modal fre?quency decreases with increasing temperature. With increasing shell thickness，the first modal frequency decreases，which is opposite to that whenRa/Rbanda/bincrease.

AcknowledgementsThis work was supported by the Na?tional Natural Science Foundation of China（No. 51805341）and the Natural Science Foundation of Jiangsu Province（No.BK20180843）.

AuthorsMr. ZHANG Yongfeng received the B.S. degree from School of Mechanical and Electrical Engineering，So?ochow University in 2020，and is currently pursuing MA.Eng in the School of Mechanical and Electrical Engineering，Soochow University.

Prof. WANG Gang received the B.E. and Ph.D. degrees from Harbin Engineering University，Harbin，China，in 2010 and 2016，respectively. From 2013 to 2015，he was pursuing a co?training Ph.D. at Wayne State University. His research interests include dynamics analysis，vibration and noise control，etc. At present，he has presided over one na?tional Natural Science Foundation Project，three provincial and ministerial projects.

Author contributionsMr. ZHANG Yongfeng completed

the calculation of data results and simulation comparison，and wrote the third and fourth parts of the manuscript. Mr.ZHU Ziyuan integrated the theoretical model and completed the first and second parts of the manuscript. Prof. WANG Gang designed the study， complied the models， and reviewed the manuscript. All authors commented on the manuscript draft and approved the submission.

Competing interestsThe authors declare no competing interests.