Free vibration analysis of spherical caps by the pseudospectral method

Journal of Mechanical Science and Technology (009) ~8 Journal of Mechanical Science and Technology www.springerlin.com/content/78-9x DOI 0.007/s06-008-0906-7 Free vibration analysis of spherical caps by the pseudospectral method Jinhee Lee * Department of Mechano-Informatics Hongi University hoongnam 9-70 Korea (Manuscript Received July 8 008; Revised September 008; Accepted September 7 008) -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Abstract The pseudospectral method is applied to the axisymmetric and asymmetric free vibration analysis of spherical caps. The displacements and the rotations are expressed by hebyshev polynomials and Fourier series and the collocated equations of motion are obtained in terms of the circumferential wave number. Numerical examples are provided for clamped hinged and free boundary conditions. The results show good agreement with those of existing literature. Keywords: hebyshev polynomials; Free vibration; Pseudospectral method; Spherical shell --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------. Introduction This paper was recommended for publication in revised form by Associate Editor Eung-Soo Shin * orresponding author. 8 860 89 Fax.: 8 86 66 E-mail address: jinhlee@hongi.ac.r KSME & Springer 009 Research on the free vibration of spherical caps can largely be divided into two groups: the analytical approach and the application of the discretization methods. Hoppmann [] proposed employing Bessel functions in the analysis of axisymmetric free vibration of shallow spherical shells. Legendre functions were widely used in the axisymmetric free vibration analyses of spherical caps [-]. Prasad [] obtained the solution of the reduced equations of motion of spherical caps by the use of associated Legendre functions. Zarghamee and Robinson [6] applied the Holzer method to the axisymmetric and asymmetric free vibration analysis of clamped spherical caps. Wu and Heyliger [7] approximated the axisymmetric and asymmetric modes of spherical cap vibrations by Hermite polynomials and Fourier series. The finite element method has been one of the most favored discretization methods in the vibration analyses of spherical shells. Singh and Mirza [8] applied it to the analysis of the asymmetric vibration of spherical caps where clamped and hinged boundary conditions were considered. Tessler and Spiridigliozzi [9] developed a two-node axisymmetric shell element to compute the natural frequencies of clamped hemispherical shells. Gautham and Ganesan [0] studied the axisymmetric and asymmetric free vibrations of clamped spherical caps using shear deformable semianalytic shell finite elements. Buchanan and Rich [] computed the natural frequencies of spherical shells with simply supported free and fixed boundary conditions using a Lagrangian finite element in spherical coordinates. Fan and Luah [] used the spline finite element method to compute the natural frequencies of clamped hemispherical domes. In this study the pseudospectral method is applied to the symmetric and asymmetric modes of free vibration analysis of spherical caps where hebyshev polynomials and Fourier series are selected as the basis functions in the colatitudinal direction and in the circumferential direction respectively. The pseudospectral method can be considered to be a spectral method that performs a collocation process. A brief explanation of the pseudospectral method can be found in the free vibration analysis of helical springs [].. Pseudospectal formulations The equations of motion of spherical shells with the

J. Lee / Journal of Mechanical Science and Technology (009) ~8 effects of transverse shear and rotary inertia taen into account were derived by Soedel [] as N ( Nsin ) N cos Q = R hu sin ω sin ρ N ( N sin ) N cos Q sin = ω Rsin ρhv Q Q sin N N Rsin ρhw ( ) ( ) M ( Msin ) M cos QR = Ψ M ( M sin ) M cos ρh QR sin = ω Rsin Ψ ρh sin ω Rsin sin (a) (b) (c) (d) (e) for harmonic motions at natural frequency ω in radian/second. U V and W are the displacements in the colatitudinal circumferential and normal directions respectively as shown in Fig.. Ψ and Ψ are the rotations in the colatitudinal and circumferential directions. R h and ρ are the radius the wall thicness and the density of the spherical shell. Φ is the spherical cap angle. The stress resultants N N N Q Q M M and M are defined as U W N = R R V U W ν Rsin Rtan R Fig.. Geometry of a spherical cap. (a) V U N = R sin R tan W U W ν R R R ν V N = N = R V U Rtan Rsin (b) (c) Ψ M = D R Ψ Ψ ν Rsin Rtan (d) M D Ψ Ψ ν Ψ = R sin R tan (e) ν Ψ M = M = D R Ψ Ψ Rtan Rsin (f) U W Q = Ψ R R (g) V W Q = Ψ R Rsin. (h) and D are defined as Eh ( ν ) ( ) = and D = Eh ν. E G ν and κ represent Young s modulus the shear modulus Poisson s ratio and the shear correction factor respectively. When the stress resultants of Eqs. (a)- (h) are substituted into the equations of motion (a)- (e) we have U U ν U sin cos sin ν V ν sin U tan ν V W ν sin tan ρh Rsin Ψ = ω R sin U ν U ν U tan ν V V V sin cos sin ν sin V tan W ν R sin Ψ ρh R sin V (a) (b)

J. Lee / Journal of Mechanical Science and Technology (009) ~8 U V ν sin cos U W W W sin cos sin ( ν) sin W Ψ R Ψ sin cos Ψ ρh R sin W W Ψ Ψ Rsin U sin cos D Ψ ν R cos (c) ν sin Ψ (d) sin tan D ν Ψ ν Ψ ρh = ω R sin Ψ tan D W ν Ψ ν Ψ R sin V D tan ν Ψ Ψ Ψ sin sin ν sin R Ψ tan D ρh R sin Ψ. D Typical boundary conditions are clamped: (e) U = 0 V = 0 W = 0 Ψ = 0 Ψ = 0 (a) that the component ( ) f n does not change sign as the pole is crossed when n is even while Fig. (b) indicates that f n ( ) must always change sign as the pole is crossed when n is odd. Fig. (a) and Fig. (b) imply that f n ( ) is an odd function when n is an odd number and that f n ( ) is an even function when n is an even number. The hebyshev polynomials of the first ind are defined recursively as ( x) = ( ) = ( ) ( ) ( ) ( ) ( ) T0 T x x x T x = xt x T x which maes the terms 0( ) ( ) ( ) even functions of x while ( ) ( ) ( ) (6) T x T x T x be T x T x T x are odd functions of x. The colatitude ranges from 0 to Φ and it is convenient to use the normalized form ξ = [ 0 ]. (7) Φ The displacements and the rotations are then approximated as follows: ( ξ ) = U af ( ξ) cos n (8a) = ( ) ( ) ξ = V bf ξ sin n (8b) = hinged: U = 0 V = 0 W = 0 Ψ = 0 M =0 (b) free: N = 0 N = 0 Q = 0 M = 0 M = 0. (c) Assume a function ( ) f which is periodic in the circumferential direction f ( ) = fn ( ) cosn () n = 0 and follow the term ( ) fn cosn along a meridian over the pole. The polar projection of Fig. (a) shows Fig. Polar projections in the vicinity of the pole showing the positive (H) and negative (L) regions for a term f ( ) cosn. n

J. Lee / Journal of Mechanical Science and Technology (009) ~8 ( ) ( ) ξ = W cf ξ cos n (8c) = Ψ ( ξ ) = df ( ξ) cos n (8d) = Ψ ( ξ ) = ef ( ξ) sin n (8e) = where a b c d and e are the expansion coefficients. K is the number of collocation points in the colatitudinal direction. F ( ξ ) is the basis function selected to be an odd function for odd n and an even function for even n. Such a basis function can be realized by use of hebyshev polynomials as follows: F ( ξ ) ( ξ ) ( n ) ( ξ ) ( n = ) T = 0 = T. (9) Expansions of (8a)-(8e) are substituted into Eqs. (a)-(e) which are then collocated at the Gauss- Lobatto collocation points ( i ) π ξi = cos ( i = K) (0) K to yield the collocated equations as follows: sini cosi a F F Φ Φ ( ν ) n ν sin if sin i tan i K ( ν) n ( ν) n b F F Φ tani K sini c ν F Φ K d RsiniF R sin i ρh ω a F ( ν) n ( ν) n a F F Φ tani cn ν F ( ν) sini ( ν) cosi b F F Φ Φ n ν sin if sin i tan e Rsin i F R sin i ρh ω b F = = (a) (b) sini a ν F cos i F Φ bn ν F sini cosi c F F Φ Φ κ Gh n ( ν) sinif (c) sini sini d R F cos i F Φ en RF R sin i ρh c F K K Rsini a RsiniF ( ξi) c F ( ξi) D D Φ K sini cosi d F ( ξi) F ( ξi) Φ Φ ( ν) n ν sin R i F ( ξi) (d) sin i tan D K ( ν) n ( ν) n e F ( ξi) F ( ξi) Φ tani R sin i ρh d F ( ξi) D K K b RsiniF c RnF D D ( ν) n ( ν) n d F F Φ tani ( ν) sini ( ν) cosi e F F Φ Φ (e) n ν sin R i F sin i tan i D R sin i ρh e F D ( i= K) with i =Φξi. The notation (' ) stands for the differentiation with respect to ξ. Eqs. (a-e) are rearranged as A( n) { δ} B( n) { χ} () = ω ( ( n) { δ} D( n) { χ} ) where matrices A ( n ) B ( n ) ( n ) and D( n ) correspond to the circumferential wave number n. The vectors in Eq. () are defined as { δ } = { a a ak b b bk c c ck T d d dk e e ek} () T { } = { a b c d e }. χ

J. Lee / Journal of Mechanical Science and Technology (009) ~8 The total number of equations in Eqs. (a)-(e) is K whereas the total number of expansion coefficients is (K). The remaining five equations are obtained from the boundary conditions. The boundary conditions (a)- (c) at ξ = are expressed by use of the expansions of Eqs. (8a)- (8e) as follows: clamped: K K K af () = 0 bf () = 0 cf () = 0 K K df () = 0 ef () = 0 hinged: K K af () = 0 bf () = 0 K K cf () = 0 ef () = 0 F () νf() νnf() d e Φ tan Φ sin Φ = 0 free: (a) (b) K F () νf() νnf () a b c( ν) F( ) = 0 Φ tanφ sinφ nf() F () F() a b = 0 sinφ Φ tanφ F () νf() νnf() d e = 0 (c) Φ tanφ sinφ nf() F () F() d e = 0 sinφ Φ tanφ F () af () c drf () = 0. Φ A boundary condition set out of Eqs. (a)-(c) can be rewritten in the matrix form F( n) { δ} G ( n) { χ} = { 0} () where { 0 } is a zero vector. Since { } can be expressed as χ in Eq. () { χ} = G( n) F ( n) { δ} (6) Eq. () can be reformulated as ( A( n) B( n) G( n) F( n) ){ δ} = ω ( ( n) D( n) G( n) F( n) ){ δ}. (7) The solution of Eq. (7) yields the estimates for the natural frequencies and corresponding expansion coefficients. Easy implementation of the boundary conditions is one of the merits of the present study. The procedure outlined in Eqs. ()-(7) can be applied to the vibration analysis of spherical caps with the clamped hinged and free boundary conditions.. Numerical examples The algebraic problem of Eq. (7) is solved for the natural frequencies using Matlab. A convergence chec of the natural frequencies of a clamped hemispherical shell for the axisymmetric mode ( n =0) is performed and the computed frequency parameter Ω= ωr ρ E is given in Table. The number of collocation points K ranges from 0 to 80. It is shown it requires less than K =60 for the lowest five natural frequencies to converge to three significant digits. The convergence is considerably slower than the Timosheno beams [] that had required less than K =0 for the lowest six natural frequencies to converge to six significant digits. The computed frequencies are in excellent agreement with those of Kunieda []. Poisson s ratio and the shear correction factor are ν =0. and κ =/6 throughout the present study. The present method is also proved for the asymmetric cases. The frequency parameter Ω= ωr ρ( ν ) E of clamped and hinged hemispherical shells for the asymmetric modes ( n = Table. onvergence test of frequency parameter Ω= ωr ρe of axisymmetric modes of hemispherical shell vibrations ( Φ =π/) with clamped boundary conditions for Rh = 00 ν =0. and κ =/6. present study n mode K =0 K =0 K =60 K =80 0 0.76 0.908 0.9898.0.09 0.768 0.99 0.987.07.086 0.76 0.987 0.98.08.076 0.76 0.98 0.98.00.070 Kunieda [] 0.760 0.98 0.98.00.07

6 J. Lee / Journal of Mechanical Science and Technology (009) ~8 Table. omparison of frequency parameter Ω= ωr ρ( ν ) E of asymmetric modes of hemispherical shell vibrations ( Φ =π/) for Rh = 00 ν =0. κ =/6 and K =80. n mode clamped boundary conditi hinged boundary conditi ons ons present study Singh Mirza [7] present study Singh Mirz a [7] 0. 0.87 0.9 0.986.00 0. 0.86 0.9 0.96 0.996 0. 0.89 0.97 0.97 0.997 0. 0.86 0.96 0.900 0.9870 0.898 0.99 0.99 0.987.00 0.900 0.99 0.977.07.07 0.97 0.989 0.996.0.0 0.996 0.970.066.078. 0.898 0.98 0.97 0.98.007 0.90 0.99 0.977.077.07 0.98 0.990 0.9966.0.7 0.997 0.97.070.07.8 0.890 0.98 0.9 0.9787.080 0.90 0.96 0.976.067.060 0.9 0.977 0.998.0.0990 0.987 0.97.066.070.8 0.890 0.97 0.9 0.978.08 0.90 0.9 0.976.07.06 0.9 0.977 0.9960.0.0 0.987 0.97.069.07. Table. Frequency parameter Ω= ωr ρ E of axisymmetric modes of hemispherical shells ( Φ =π/) with free boundary conditions for Rh = 00 ν =0. κ =/6 and K =80. n mode present study hao et al [] 0 0.8707 0.9 0.989 0.870 0.97 0.986 ) is given in Table where the finite element solutions of Singh and Mirza [8] are also given for comparison and where it is shown that the computed frequency parameters are practically identical to those of the finite element analysis in the lower modes and deviate slightly for the higher modes. While there are numerous reports on the subject of the spherical shell vibrations with clamped and hinged boundary conditions only a limited number of solutions are available for the spherical shell vibrations with free boundary conditions. Studies on the asymmetric vibrations of spherical caps [6 7 8 0 Table. Frequency parameter Ω= ωr ρ E for spherical caps with free boundary conditions for Rh = 00 ν =0. κ =/6 and K =80. Φ n=0 n= n= n= n= n= n=6 n=7 π/6 π/ π/ π/ 0.9766.070.6.9.709 0.98 0.9879.060.0706.7 0.99 0.97 0.99.0.08 0.870 0.90 0.9766 0.9908.000 0.09 0.990.07.0.967 0.006 0.968 0.999.08. 0.007 0.9 0.987.008.09 0.008 0.876 0.9 0.9767 0.996 0.088.00.0.7.679 0.087 0.980.0.06.6 0.06 0.967 0.990.00.088 0.0070 0.96 0.96 0.98 0.997 0.08.07.768..887 0.0 0.996.0.00.6 0.07 0.979 0.9996.07.07 0.08 0.9 0.97 0.990.008 0.7.077.79.6. 0.07.0060.06.0.07 0.089 0.9860.00.07.0 0.06 0.960 0.98 0.997.0 0.6..00.8.88 0.7.0.089.09.9 0.076 0.99.07.068.9 0.0 0.979 0.989.007.06 0...9.00.676 0.69.0..80.970 0.087.000.00.096.8 0.0799 0.9808 0.9967.0.08 0.0.09.79.8.9769 0.8.0676.8.60.60 0.60.06.0608.0. 0.08 0.988.00.08.00 Table. Frequency parameter Ω= ωr ρ E for hemispherical shells with free boundary conditions for Rh =00 ν =0. κ =/6 and K =80. Φ n=0 n= n= n= n= n= n=6 n=7 π/6 π/ π/ π/ 0.989.087.07.068.898 0.999.008.089.68.70 0.90 0.9780.00.088.079 0.8707 0.9 0.989.009.0 0.06.06.09.679.9 0.0 0.979.07.8.996 0.0070 0.90 0.99.09.6 0.007 0.8768 0.96 0.988.0 0.076.068.70.96.89 0.069 0.990.07.89.98 0.09 0.9666.008.0776.977 0.08 0.97 0.968 0.9967.098 0.69.86.90.0.078 0.08.00.8.697.7 0.0 0.989.06.8.779 0.060 0.9 0.98.07.0 0.80.9.868.70.8 0.8.066.9.6.99 0.099.00.0700.87.77 0.067 0.96 0.9960.00.0798 0..7.880.69.708 0.90.07.0.6868.7 0.66.09..69.88 0.066 0.9798.06.08.8 0.90.697.8.6.90 0.067.7.609.880.7 0.07.06.700.9.609 0.8 0.99.00.0808.8 0.786.96.9.09.609 0.06.66.69.097.690 0.760.097.86.7.7 0.0.0099.09.0.0 ] do not show how the vibrations of spherical shells with free boundary conditions can be dealt with. In the pseudospectral method the free boundary condition of Eq. (c) is merged into the collocated equations () so that the system Eq. (7) can be tailored

J. Lee / Journal of Mechanical Science and Technology (009) ~8 7 Table 6. Frequency parameter Ω= ωr ρ E for hemispherical shells with free boundary conditions for Rh = 0 ν =0. κ =/6 and K =80. Φ n=0 n= n= n= n= n= n=6 n=7 π/6 π/ π/ π/.008.7.887.7.9 0.96.06.00.8.767 0.90 0.99.098.6.676 0.87 0.969 0.997.009. 0.0.0.6868.8090.07 0.0 0.990.7.8.78 0.09 0.997.0.89.76 0.0070 0.8780 0.967.00.098 0.8.70.0.869.7 0.07.000.9.790.989 0.00 0.988.088.00.70 0.07 0.907 0.988.078.90 0.6..69.68.89 0.89.9.86.007.906 0.0.09.69.7.868 0.0700 0.9.0.098.0 0.80.908.00.986.76 0.77.6.79.08.867 0.86.070.77.6.0 0.9 0.98.087.09.08 0.800.8.89.799 6.99 0..90.007.870.86 0.77.97..888.778 0.08.009.096.7..06.8660.6 6.689 8.688 0.76.97.8..679 0.88..76.09.6676 0.86.06....986.9.78 7.606 9.8 0.768.86.67.70.8977 0..96.766..978 0.8.097.60.69.6690 for the analysis with the free boundary conditions. Table shows the frequency parameter Ω= ωr ρ E for hemispherical shells with free boundary conditions for axisymmetric modes which are compared with those of hao et al. []. Tables -6 show the frequency parameter Ω= ωr ρ E for spherical cap vibrations with free boundary conditions for the circumferential wave number ranging from n =0 to n =7 and for the spherical cap angle Φ =π/6 π/ π/ and π/. The thicness ratio varies from Rh=0 to Rh=00. The case with n =0 stands for the axisymmetric vibration and the cases with n represent asymmetric vibration. As a whole the frequency parameter of the asymmetric modes increases as n increase. The frequency parameter of the first mode of the asymmetric modes however is surprisingly lower than that of higher modes and the corresponding axisymmetric mode. It is also apparent that the frequency parameter tends to increase as the thicness of the shell increases.. onclusions The pseudospectral method is applied to the axisymmetric and asymmetric free vibration analysis of spherical caps. The equations of motion tae the effects of transverse shear and rotary inertia into account. The displacements and the rotations are represented by hebyshev polynomials in the colatitudinal direction and Fourier series expansions in the circumferential direction. onsidering the symmetry and the antisymmetry of the solutions odd functions and even functions of hebyshev polynomials are used alternatively as basis functions according to the circumferential wave number. The equations of motion are collocated to yield the system of equations that correspond to the circumferential wave number. Numerical examples are provided for clamped hinged and free boundary conditions. The results show good agreement with those of the existing literature Acnowledgment This wor was supported by 008 Hongi University Research fund. References [] W. H. Hoppmann Frequencies of vibration of shallow spherical shells Journal of Applied Mechanics (96) 0-07. [] A. Kalnins Effects of bending on vibrations of spherical shells Journal of Acoustical Society of America 6 () (96) 7-8. [] H. Kunieda Flexural axisymmetric free vibrations of a spherical dome: exact results and approximate solutions Journal of Sound and Vibration 9 () (98) -0. [].. hao T. P. Tung and Y.. hern Axisymmetric free vibration of thic orthotropic hemispherical shells under various edge conditions Journal of Vibration and Acoustics () (99) -9. []. Prasad On vibration of spherical shells Journal of Acoustical Society of America 6 () (96) 89-9. [6] M. S. Zarghamee and A. R. Robinson A numerical method for analysis of free vibration of spherical shells AIAA Journal (7) (967) 6-6. [7] Y.. Wu and P. Heyliger Free vibration of layered piezoelectric spherical caps Journal of Sound and Vibration () (00) 7-. [8] A. V. Singh and S. Mirza Asymmetric modes and associated eigenvalues for spherical shells Journal of Pressure Vessel Technology 07 () (98) 77-8. [9] A. Tessler and L. Spiridigliozzi Resolving membrane and shear locing phenomena in curved

8 J. Lee / Journal of Mechanical Science and Technology (009) ~8 shear-deformable axisymmetric shell elements International Journal for numerical methods in engineering 6 (988) 97-086. [0] B. P. Gautham and N. Ganesan Free vibration characteristics of isotropic and laminated orthotropic spherical shells Journal of Sound and Vibration 0 () (997) 7-0. [] G. R. Buchanan and B. S. Rich Effect of boundary conditions on free vibration of thic isotropic spherical shells Journal of Vibration and ontrol 8 () (00) 89-0. [] S.. Fan and M. H. Luah Free vibration analysis of arbitrary thin shell structures by using spline finite element Journal of Sound and Vibration 79 () (99) 76-776. [] J. Lee Free vibration analysis of cylindrical helical springs by the pseudospectral method Journal of Sound and Vibration 0 (-) (007) 8-96. [] W. Soedel On the vibration of shells with Ti- mosheno-mindlin type shear deflections and rotary inertia Journal of Sound and Vibration 8 () (98) 67-79. [] J. Lee and W. W. Schultz Eigenvalue analysis of Timosheno beams and axisymmetric Mindlin plates by the pseudospectral method Journal of Sound and Vibration 69 (-) (00) 609-6. Jinhee Lee received B.S. and M.S. degrees from Seoul National University and KAIST in 98 and 98 respectively. He received his Ph.D. degree from University of Michigan in 99 and joined Dept. of Mechano-Informatics of Hongi University in hoongnam Korea. His research interests include inverse problems pseudospectral method vibration and dynamic systems.