Transactions on Modelling and Simulation vol 10, 1995 WIT Press, ISSN X

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1 The receptance method applied to the free vibration of a circular cylindrical shell filled with fluid and with attached masses M. Amabili Dipartimento di Meccanica, Universita di Ancona, Ancona, Italy Abstract The receptance method is applied to study the free vibrations of a simply supported circular cylindrical shell, empty or filled with a fluid and with masses attached at arbitrary positions. The fluid is assumed to be incompressible and inviscid. The receptance of the filled shell is obtained using the virtual mass approach. Numerical results are given and compared to data obtained by experimental modal analysis performed on a stainless steel tank, empty or filled with water. 1 Introduction Vibrations of combined structures is studied by numerous authors using different techniques. The interest on this topic is surely due to the wide engineering applications of simple elements, joined to construct a system of practical interest. The techniques used in these studies can be divided in numerical, like the finite element method, and analytical Among the last ones, the receptance method, first introduced by Bishop and Johnson [1] has obtained recently a very large success. In particular, the important work of Professor Soedel and his school has given a fundamental stimulus to the diffusion of this technique. The receptance method is illustrated in the book of Soedel [2] with some applications; it was applied, e.g., to ring stiffened cylindrical shells by Wilken and Soedel [3, 4], to rectangular plates by Azimi, Hamilton and Soedel [5], to circular plates by Azimi [6, 7], to ring and tires by Alloei, Soedel and Yang [8, 9, 10] and to plates welded to cylindrical shell by Huang and Soedel [11, 12, 13]. In this paper the receptance method is applied to a simply-supported circular cylindrical shell filled with fluid and with masses attached at arbitrary positions.

2 462 Computational Methods and Experimental Measurements To simulate the inertial effect offluid,the virtual mass approach, introduced by Berry and Reissner [14], was used The aim of this work is the study of a problem of practical interest and the extension of the receptance method to structures in contact with fluid In fact, combined structures like tanks, are often used to store fluids. In this work the effect of lumped masses, simulating the effect of apparatus connected to the shell, on free vibration is investigated either in the case that the cylinder is empty or filled with fluid. A comparison between numerical data and results of experimental modal tests performed on a AISI 304 stainless steel tank empty or filled with water is given. 2 Theoretical developments Two masses attached to a simply supported circular cylindrical shell at arbitrary axial and angular positions are considered. The imposed boundary conditions of the cylinder are equivalent to that one assumed in Soedel [2], see eqns (5.5.1)- (5.5.8), and are called " shear diaphragms" by other authors, see Leissa [15], eqn (2.33). The shell is considered thin and made of linearly elastic, isotropic and homogeneous material. The angle a indicates the angular distance between the two masses and is given by (see Figure 1 ): % (1) where Q\ and 8? are the angular coordinates of the two masses. The shell, A in Figure 1, is connected with the two masses, B and C, in the two points 1 and 2 in the same figure, the displacement XAI of the shell at the point 1, where the mass B is attached, is given by (Soedel [2]): where FAI and FA2 are the amplitudes of harmonic forces applied in 1 and 2, respectively; in fact, the coupling forces FAI can be considered harmonic, ctjj is the receptance of the system A (cylinder) that is defined as the ratio of a displacement response at a certain point i to a harmonic force input at the point j. For the Maxwell's reciprocity theorem otjj = a,,. The displacement %A2 of the shell at the point 2 connected with the mass C is given by: For the mass B the displacement X%, is: %Bl = P,lFB, (4) where Pu is the receptance of the mass B Then, the displacement Xc2 of the mass C is:

3 Computational Methods and Experimental Measurements 463 %C2 =?22 Fc2 (5) where 722 is the receptance of the mass C Due to the connection between A and B at point 1 and between A and C at point 2, we have the following equations: FAI=~FBI (6) F.42 = -Fc2 (7) %A1 = %B1 (8) %A2=%C2 (9) Substituting eqns (4)-(9) in eqns (2) and (3), results in: (a,i+pn)fa, +0^2 = 0 (10) This is a system of homogeneous linear equations; the following determinant must be equal to zero in order to obtain a non trivial solution: 0 (12) From eqn (12) we obtain the frequency equation: (a,,+p,j(a22+y22)-(*,2'==0 (13) The problem is reduced to find roots of eqn (13); this is singular at each natural cylinder frequency and it is zero between the singularities. For this purpose all the receptances involved in this equation are evaluated. The two masses B and C have obviously the following receptances (Soedel [2]): where co is the natural circular frequency of the combined system and MI and M2 are the two masses, respectively. Focusing attention on the shell, the harmonic response at a point of coordinates (x,0) due to a harmonic force applied at (x*,0~) is given by:

4 464 Computational Methods and Experimental Measurements t 00 CO sin(m7cx*/l)sin(m7cx/l)cos[n(9-e*)] W(X,0)= 2^ A 2 2 (16) hlrtt^i^, PmnC^mn ~ ) where h, L, R are the shell thickness, length and radius, respectively; m is the number of longitudinal half-waves, n the number of circumferential waves and conui the natural circular frequency of the shell without masses attached. In eqn (16) only shell modes with radial prevalence (bending modes) are considered and the influence of the modes with n=0 (axisymmetric) are neglected in this expansion. In fact, the natural frequencies of modes with circumferential and longitudinal prevalence (twisting and extension-compression modes) and of the axisymmetric modes are so high that the contribution to the sum in eqn (16) can be neglected in most engineering applications. The shell can be considered either vibrating in vacuum or filled with fluid. The circular frequency CD must be calculated in the vacuum orfluid-filledconfiguration (e.g. Amabili and Dalpiaz [16]). p nm is the virtual density of the shell that is given by (e.g. Amabili and Dalpiaz [16]): PC in vacuum Pmn =1 P L "I CN (") Ps + ^ V i j with fluid m 7i h T, I n K L where ps and pp are the mass density of the shell and the fluid,! and!' the modified Bessel function and its derivative with respect to the argument, respectively. In eqn (17) the fluid is considered incompressible and inviscid. The virtual density (or virtual mass) approach to study vibration of structures in contact with fluid was largely applied to single structural elements, such as plates and shells By using eqn (16), the receptances otjj of the shell can be calculated; they are given by 2 -r «^ 1. j mttxi «..= V V SHT (18) _-i f -j / 2. 2 \ T \ / sin 2:^ (19) z. -r «^ ^ i. m ;i A,. HI ;t *9,.,^^ <*2i =,.Ti» 2. Z- ; 2 T^ sin ^sm ^-cos(na) (20)

5 Computational Methods and Experimental Measurements 465 The shell mode shapes are given by the following expression where the responses due to the harmonic forces FAI and F^ were combined, and the ratio FA2/FA1 is obtained by one of the linear dependent eqns (10)-(11): sin =l n=lpmn(<«>mn ~ k where eok is the natural circular frequency of the coupled shell-mass system. With this method the vibration of the coupled shell-masses system can be studied; the circular frequencies are given by roots ofeqn (13) and mode shapes byeqn(21). However, it is important to remember that also the trivial solution of the linear system given by eqns (10)-(11) exists and is obviously given by FAi=FA2=0. In this case no force couples the masses and the shell, therefore the natural circular frequencies CD of shell-modes that presents nodes at both masses locations are unchanged. Therefore, together with the shell modes resulting from combination with masses, there is the family of typical cylinder modes. In the case of a single mass B attached to the cylinder, eqn (13) is simplified to: <*n+pn=0 (22) Mode shapes of the shell with one mass attached at x=x, and 8=8, are given by: co co. V V m=l ^ n=l Y rn n x. m n x, r, 2 sin sin -icos n(9-9,) (23) ^k / ^ ^ In this case all the natural frequencies of the simply supported cylinder are also solution of the mass-shell system. Corresponding modes present a nodal line at 8=81. By using the same method, k masses can be considered joined to the shell. In this case the frequency equation is obtained by the following determinant: # (24) where 833 represents the receptance of the mass M? connected at point 3 to the shell.

6 466 Computational Methods and Experimental Measurements Natural Frequencies [Hz] Figure 1: Shell-masses system Figure 2: Plot of equation (23); mass of.0967 kg and cylinder filled with water 3. Numerical and experimental results Numerical computations and experimental modal tests were performed on a circular shell having a diameter of 175 mm, a thickness of 1 mm and a length of 664 mm. The material of the cylinder is assumed to be a AISI 304 stainless steel with Young's modulus E = 206 GPa, Poisson's ratio v = 0.3 and density p$ = "' kg nv. The cylinder is tested both empty and filled with water, p? 10'" kg ITT\ and with a kg mass attached at a distance of 262 mm from a shell's edge. Numerical results were computed by using the software Mathematica and experimental data were obtained by using the software CADA-X by LMS The sensor used to measure the shell velocity during tests is a laser Doppler vibrometer Polytec OFV 1102 in order not to alter the mass of the cylinder. The frequency resolution of FRF's is 0.5 Hz. In Table 1 the computed and measured natural frequencies of the cylinder with the attached mass are successfullly compared; the mean error is 1.4 %. Data in this table are presented for both the empty and the water-tilled cylinder with an attached mass Solutions of eqn (22) are graphically given in Figure (2) for the waterfilled shell, 20 terms are considered in the sum that gives the receptances an. Figures (3)-(4) show computed and experimentally detected mode shapes in a cross section at x = x, for both the empty and the water-filled cylinder. A good agreement is shown. It is also interesting to see that the presence of the attached mass has a greater influence on natural frequencies and mode shapes in the case of the empty cylinder. In fact, the water-filled cylinder presents an equivalent mass much greater than the empty one. Therefore, in Figure 2, intersections between an and pn are closer to the natural frequencies of the cylinder without attached mass. In conclusion, the receptance method, already successfully applied to the theoretical study of vibrations of plates and shells, can be also used to study circular cylindrical shells in contact with a fluid, by using the added mass concept. Computed data are in good agreement with experimental results.

7 Computational Methods and Experimental Measurements 467 Figure 3: First three mode shapes of the empty cylinder obtained by roots of eqn (23). Top: computed mode shapes. Bottom: measured mode shapes. Figure 4: First three mode shapes of the water-filled cylinder obtained by roots of eqn (23). Top: computed mode shapes. Bottom: measured mode shapes.

8 468 Computational Methods and Experimental Measurements cylinder empty ecin (23) COnm Th Exp Th Exp Th water-filled cylinder eqn (23) Omn Exp. Th Exp Table 1. Theoretical and experimental natural frequencies of the cylinder empty and water-filled with a 96.7 grams mass attached. The two families of modes obtained by eqn (22), cok, and by the trivial solution FAI=O, co, are shown. References 1. Bishop. RED & Johnson, DC The Mechanics of Vibration, Cambridge University Press. London. I Soedel. W. Vibrations of Shells and Plates. Marcel Dekker. New York. 2nd ed., Wilken. ID & Soedel. W. The receptance method applied to ring-stiffened cylindrical shells: analysis of modal characteristics. Journal of Sound and Vibration (4), Wilken. ID & Soedel. W. Simplified prediction of the modal characteristics of ringstiffened cylindrical shells. Journal of Sound and Vibration, (4), Azimi. S., Hamilton. J.F. & Soedel. W. The receptance method applied to the free vibration of continuous rectangular plates. Journal of Sound and Vibration, 1984, 93(1), Azimi. S. Free vibration of circular plates with elastic edge supports using the receptance method. Journal of Sound and Vibration , Azimi. S. Free vibration of circular plates with elastic or rigid interior support. Journal of Sound and Vibration, 1988, 120, Allaei, D., Soedel. W. & Yang, T.Y. Natural frequencies and modes of rings that deviate form perfect axisymmetry. Journal of Sound and Vibration. 1986, 111(1) Allaei, D.. Soedel, W. & Yang. T.Y. Vibration analysis of non-axisymmctric tires, Journal of Sound and Vibration (1), Allaei, D. Soedel. W. & Yang. T.Y. Eigenvalues of rings with radial spring attachments. Journal of Sound and Vibration. 1987, 121(3) Huang, D.T. & Soedel. W. Natural frequencies and modes of a circular plate welded to a circular cylindrical shell at arbitrary axial positions. Journal of Sound and Vibration, (3) Huang. D.T. & Soedel. W On the free vibrations of multiple plates welded to a cylindrical shell with special attention to mode pairs. Journal of Sound and Vibration, (2) Huang. D.T. & Soedel. W. Study of the forced vibration of shell-plate combinations using the receptancc method. Journal of Sound and I 'ibration (2) Berry, J.G. & Reissner. E. The effect of an internal compressible fluid column on the breathing vibrations of a thin pressurized cylindrical shell. Journal of Aeronautical 6c/c/?cc Leissa. A.W. Vibrations of Shells. NASA SP-160. U.S. Government Printing Office, Washington. D.C Amabili. M. & Dalpiaz, G. Breathing vibrations of a horizontal circular cylindrical tank shell, partially filled with liquid. Journal of Vibration and Acoustics (to be published).