Journal of Mechanical Science and Technology 6 (5) (0) 9~5 www.springerlink.com/content/738-9x DOI 0.007/s06-0-0336- Experimental and numerical investigation of modal properties for liquid-containing structures Hassan Jalali,* and Fardin Parvizi Department of Mechanical Engineering, Arak University of Technology, Arak 3835-77, Iran School of Mechanical Engineering, Iran University of science and Technology, Tehran 68, Iran (Manuscript Received August 9, 0; Revised January 0, 0; Accepted January 30, 0) ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Abstract The dynamic response of liquid-containing structures is governed by their modal properties, which are affected by the mass of liquid and other fluid structure interaction mechanisms. Therefore, knowledge of the effects of different parameters on modal properties is helpful in conducting a precise dynamic response analysis. In this paper, the effects of liquid on the modal properties of two structures, i.e., a pipe structure and a cylindrical storage tank, are investigated experimentally. The experimental results are then used to construct accurate analytical/numerical models for these structures. The models are capable of regenerating the experimental dynamic characteristics of the structures with an acceptable accuracy, indicating a proper modeling of the effects of liquid and the corresponding interaction mechanisms. Keywords: Analytical/numerical modeling; Experimental results; Fluid-structure interaction; Modal properties ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------. Introduction The response of mechanical structures under dynamic loading conditions is governed by their modal properties (i.e., natural frequencies, mode shapes and damping ratios). Therefore, knowledge of the effects of different parameters on the modal properties of structures is helpful in conducting a precise dynamic response analysis. The modal characteristics of liquid-containing structures (e.g., pipes or cylindrical storage tanks) are affected by the presence of liquid. Liquid changes the natural frequencies or mode shapes of the structure either by adding mass (i.e., the mass effect of liquid) or through other fluid structure interaction (FSI) mechanisms. FSI problems have received much attention in the past decade, as they can be found in almost all engineering fields and applied sciences. FSI is important in studying the dynamic behavior of offshore structures, dams, ship motion, aircraft, and satellites, among others. The effects of FSI on the modal properties of structures have been studied by many researchers in the past, and a wide range of papers has been published on this subject. Later on, a brief literature survey has been presented. Research in the field of FSI can be classified into two categories: experimental * Corresponding author. Tel.: +98 86 36700, Fax.: +98 86 367000 E-mail address: jalali@iust.ac.ir Recommended by Associate Editor Cheolung Cheong. KSME & Springer 0 investigation of the modal properties of liquid-containing structures and analytical modeling of FSI mechanisms. Graham and Rodrigues [] studied the modal properties of a cylindrical fluid-filled storage. They obtained the natural frequencies by solving the corresponding governing equations. Gorbunov [] and Budak and Ovcharenko [3] used an experimental investigation to study the effect of a liquid on the natural frequencies of shells of revolution. They studied the dependencies of the resonant frequencies and vibration modes on the level of liquid filling. Chiba et al. [] obtained experimental results on the free vibration of a clamp-free cylindrical shell partially filled with liquid and compared them with analytically calculated results. Curadelli et al. [5] studied the dynamic response of elevated spherical tanks. He used horizontal base motion to excite the structure. Mazuch et al. [6] investigated the natural frequencies and modes of vibration for a vertical cylindrical shell by increasing the water level. He used both the finite element (FE) method and the experimental modal analysis. Goncalves and Batista [7] performed a theoretical analysis on the effects of a variable height of fluid on the natural frequencies of simply supported fluid-filled vertical cylindrical shells. They used the Rayleigh-Ritz technique to obtain an approximate solution. Sivak and Telalov [8] experimentally investigated the free vibrations of a clamped titanium cylindrical shell partially filled with water. YongLiang et al. [9] considered the finite element formulation of asymmetric cylindrical shells containing flowing fluid. They
50 H. Jalali and F. Parvizi / Journal of Mechanical Science and Technology 6 (5) (0) 9~5 used the FE model for the vibration analysis of cylindrical shells. Finite Fourier transform and Fourier series expansions were adopted by Kyeong and Seong [0] to develop an analytical approach to estimate the natural frequencies of a circular cylindrical shell partially filled with liquid. Chiba [] experimentally investigated the nonlinear dynamic response of cantilever circular cylindrical tanks. Eberle et al. [] considered the eigen-solution of a spherical steel tank. They compared experimental and analytical natural frequencies and mode shapes and obtained a good agreement. A comprehensive literature review on the problem of FSI, especially the sloshing phenomenon for partially filled liquid containers, was made by Ribouillat and Liksonov [3]. In the current paper, the effects of liquid on natural frequencies and mode shapes of a pipe structure and a cylindrical storage tank are experimentally investigated. These structures are partially filled with water, and experimental modal analysis is performed to extract their modal properties. The level of liquid is increased, and its effects on the natural frequencies and mode shapes are studied. The effect of liquid on the dynamic properties of the pipe structure is modeled analytically using the Euler- Bernoulli beam theory. The behavior of the cylindrical storage tank can be described using an FE model. Good agreements are obtained between the experimental and analytical/numerical natural frequencies and mode shapes. Note that, in this paper, the modal parameters of liquid storage structures under the no flow-rate are considered. In addition, in the literature, FSI has been usually studied using water. The same strategy is used in this paper. The aim of this paper is to investigate FSI mechanisms, and the effect of viscosity is not a primary concern. In the next section, the experimental modal analysis is presented.. Experimental modal analysis In this section, experimental modal testing is used, and the effects of fluid on the dynamic properties of a pipe structure and a cylindrical storage tank are characterized. In the performed experiments, the structures were excited using a B&K00 mini shaker attached to the structures through a stinger. A B&K800 force transducer was placed between the stinger and the structure to measure the excitation force. The response of the structures was measured through a set of DJB A/0/V accelerometers. The excitation force and response signals were transferred to a PULSE analyzer, and the FRFs were calculated. As the boundary conditions affect the dynamic properties of the pipe and the cylindrical storage tank and to remove the effects of the boundary conditions from the experimental modal results, a free-boundary condition provided by suspending the structures using flexible strings was used. This boundary condition can be considered in analytical and numerical models without uncertainty. First, the modal testing on the pipe structure was considered.. Pipe structure An aluminum pipe having the following dimensions was Table. Natural frequencies (Hz) for different water levels (mm). h 0 7 35 5 0.0 0.00 39.68 38.87 36.50 33.5 0.8 09.87 08.8 06.56 00.8 9. 3.68.00.00 07.6 95. 77.68 Fig.. Experimental test setup (pipe). Log. Mag. 6 0 6 used: di = 0.05 m as the internal diameter, do = 0.060 m as the external diameter, and L = 3 m as the length. Fig. shows the test setup and a picture of the cross section of the pipe. The shaker was positioned at one end of the pipe. The structure was excited using a pseudo-random forcing function. The frequency response functions (FRFs) between the shaker and accelerometers placed on the structure were measured. The experiments were repeated for different levels of liquid (i.e., water). Using the measured FRFs at each water level, the natural frequencies and mode shapes were extracted. Table presents the natural frequencies for the different water levels. The results shown in Table indicate that by increasing the level of water, the natural frequencies decrease. To investigate the effects of liquid on the modal properties of the pipe, the direct FRFs for the different water levels are compared in Fig.. Two conclusions can be made from the results shown in Fig.. First, the direct FRFs at different liquid levels are the same. In other words, by increasing the water level, the shape of the direct FRFs does not change. Second, by increasing the level 35.8 350.68 37.3 30.8 39.87 9.37 5 59.87 58.3 53.50 50.87 7.93 3.37 6 76.56 7.37 707.8 693. 65.06 59.93 8 0 50 00 50 00 50 300 350 00 50 500 550 Frequency (Hz) Fig.. Comparison of the direct FRFs.
H. Jalali and F. Parvizi / Journal of Mechanical Science and Technology 6 (5) (0) 9~5 5 Natural Frequency 0 3 0 0 0.0 0.0 0.03 0.0 0.05 h (m) Fig. 3. Changes in the natural frequencies with variation in the water level. h= cm h= cm h=0 0 5 0 5 0 5 0 5 0 5 0 5 00 00 300 00 500 600 700 800 900 00 00 300 00 500 600 700 800 900 00 00 300 00 500 600 700 800 900 Frequency (Hz) Fig. 5. Measured FRFs at different water levels. Fig.. Experimental test setup (cylindrical tank). of the water, all natural frequencies decrease at the same rate (Fig. 3). The findings show that the only effect of the liquid on the dynamics of the pipe is the addition of mass. In other words, no other fluid structure interaction mechanism can affect the dynamic response of the pipe. Fluid structure interaction targets the structural transfer function by showing new or losing existing modes when the liquid level is changed. The mass effect of the liquid on the dynamics of the pipe will be analyzed theoretically in the next sections.. Cylindrical storage tank Second, the modal testing of a cylindrical storage tank was considered. The tank had a diameter of 0.6 m and a height of 0.5 m. It was made of an aluminum sheet 0.00 m in thickness. The tank was suspended vertically using flexible strings, and an electromagnetic shaker was used to excite it. Fig. shows the test setup. The structural response of the tank to a pseudo-random forcing function was measured using 9 accelerometers. After measuring the excitation force and response signals, the FRFs were calculated. In these experiments, three different water levels were used, i.e., h = 0, h = 0.0 m, and h = 0.0 m. Fig. shows the direct FRFs for three different liquid levels. The results presented in Fig. 5 show that by increasing the level of water, the FRFs changed. This finding indicates that the dynamics of the structure is affected by the fluid structure interaction. Natural frequencies and mode shapes for the different liquid levels will be presented in the subsequent sections. Based on the experimental results of both pipe and cylindrical storage tank structures, the effect of liquid on the dynamics of these structures is a function of their structural dimensions. In other words, if the ratio of length to diameter is large, the liquid affects the mass of the structure. Otherwise, the modal properties, i.e., mode shapes and natural frequencies, are affected by other fluid structure interactions. The next section considers the investigation of the behavior of the pipe structure and cylindrical storage tank using analytical and FE models. 3. Analytical and FE modeling 3. Pipe structure In this section, an analytical approach is used to describe the behavior of the pipe at different water levels. The dimensions of the pipe structure (i.e., L/d o = 50) are such that the dominant modes in its dynamic response are bending modes, which can be described by the Euler-Bernoulli beam theory. The equation governing the free vibration of a beam structure in a free boundary condition is derived as follows []: wxt (, ) wxt (, ) EI + m = 0. x t The governing Eq. () is subjected to the following boundary conditions: 3 3 w(0,) t w(0,) t w( L,) t w( L,) t = = = = 0 3 3 x x x x () (-5) where w(x,t), EI, m, and L are the lateral movement, flexural rigidity, mass of unit length, and length of the beam, respectively. Using the separation variable method, i.e., wxt (, ) = ϕ( xtt ) ( ), the equation governing the natural fre-
5 H. Jalali and F. Parvizi / Journal of Mechanical Science and Technology 6 (5) (0) 9~5 Table. Comparison between the experimental and analytical natural frequencies for h = 0, E = 68 GPa, m =.9 Kg/m. Ana. 0.0 39. -.7 0.8 08.63 -.0 3.68.95-0.80 35.8 35.05 0.07 5 59.87 55.87.5 6 76.56 73.53.50 Table 3. Comparison of and FE natural frequencies (Hz) @ h = 0. FEM 8. 8.09 -.8 7. 6.5-0.53 3 30.9 3.98 0.7 89.95 7.6-3.73 5 59.5 598.9 0.75 Fig. 6. Mode shapes of the pipe for the different liquid levels. quencies of the free beam can be obtained as follows []: cos( λl)cosh( λl) 0, λ m EI = = (6, 7) where is the natural frequency of the beam. Solving Eq. (6) results in the analytical natural frequencies. In Table, the experimental natural frequencies in the case of h = 0 are compared with those obtained using Eq. (6). The results presented in Table show that the analytical model is capable of regenerating the experimental results. Therefore, this model can be used for investigating the effect of liquid on the dynamics of the pipe as described below. Based on Eq. (7), as λ is constant, all natural frequencies decrease by the same coefficient (m /m ) 0.5 when the mass of the unit length of the beam increases from m to m. m may be the mass of the unit length of the beam when h = 0, and m may be the mass of the unit length of the beam for h = 0.007 m. In other words, m and m are the mass of the unit length of the beam for two different water levels. This finding shows that by adding liquid into the pipe and because the liquid affects the mass properties of the pipe, its natural frequencies decrease by the same coefficient. Such condition is shown in Fig. 3. For example, the ratio of the natural frequencies for h = 0.007 m to the natural frequencies for h = 0 is 0.997. In other words, by multiplying the natural frequencies corresponding to h = 0 by 0.997, the natural frequencies corresponding to h = 0.007 m are obtained. This ratio for h = 0.0 m and h = 0.05 m is 0.839. Note that adding liquid to the pipe changes the natural frequencies and not the mode shapes. In Fig. 6, the first two experimental mode shapes of the pipe are compared for the different liquid levels. 3. Cylindrical storage tank The geometry and boundary conditions of the cylindrical shell used in the experiments described in the previous section prevent the analytical investigation of its dynamic characteristics. In the following, the capability of the FE analysis in the Fig. 7. The FE model for h = 0. m. dynamic response prediction of a cylindrical shell is examined. FE analyses using the commercial computer code ANSYS 0 [5] are performed to obtain the modal characteristics of the water storage tank. The tank under consideration is an elastic, thin, and circular cylindrical shell with constant thickness. The shell material of the tank is assumed homogeneous, isotropic, and linearly elastic. The tank s wall is modeled by the quadrilateral shell element SHELL63 from the ANSYS element library. In the final version of the model, 0 elements in the axial direction and 5 elements in the circumferential direction, for a total of 60 uniform elements, are used to mesh the shell. The ANSYS fluid element FLUID80 is adopted to model the fluid in the tank. This element has eight nodes with three degrees of freedom at each node. The fluid element is used to model fluids contained within vessels having no net flow rate. FLUID80 is particularly well suited for calculating hydrostatic pressures and fluid solid interactions. Acceleration effects, such as in sloshing problems, and temperature effects may be included. The FE model is shown in Fig. 7. The material properties of the shell elements are considered as follows: Young s modulus E = 69.73 GPa, Poisson s ratio υ = 0.33, and mass density ρ = 683 (Kg/m 3 ). Water, as the contained fluid, has a density of 997 (Kg/m 3 ). The sound speed in water is 86 m/sec, which is equivalent to a bulk modulus of elasticity of. GPa. μ = 0.00089 is used as water viscosity [6]. The Block Lanczos method is used, and the eigenvalues and eigenvectors are extracted for each water level. Note that the extracted modes contain both the fluid modes and the structure modes. The natural frequencies of the structure modes obtained by the FE model are compared with the experimental results in the case of h = 0 in Table 3.
H. Jalali and F. Parvizi / Journal of Mechanical Science and Technology 6 (5) (0) 9~5 53 Table. Comparison of and FE natural frequencies (Hz) @ h = 0. m. 3 5 79.77 8.0 390.5 5.60 67.6 FEM 78. 7.65 386.86 50.33 688.6 -.33-0.5-0.9-3.88.7 Table 5. Comparison of and FE natural frequencies (Hz) @ h = 0. m. 3 5 67.95 79.9 76.85 356.09 6.0 FEM 65.99 76.86 63.79 359. 5.59 -.88 -.7 -.7 0.87. Fig. 9. Comparison between the experimental results and FE mode shapes @ h = 0. m. Fig. 8. Comparison of and FE mode shapes @ h = 0. Fig. 0. Comparison between the experimental results and FE mode shapes @ h = 0. m. Table 3 shows good agreement between FE and the experimental natural frequencies. In Fig. 8, the experimental and FE mode shapes are compared as well. The first row includes the experimental mode shapes, and second and third rows are the mode shapes obtained from the FE model. In the third row, the displacements of the FE model in the same locations as those measured in the experiment are shown, making a better comparison between the experimental and FE results possible. The modes shown in Fig. 8 indicate the accuracy of the FE model of the cylindrical storage tank. The capability of the FE model containing the fluid elements in regenerating the test results is then investigated. In Tables and 5, the experimental natural frequencies and FE results are compared for cases h = 0. m and h = 0. m, respectively. Figs. 9 and 0 show the corresponding mode shapes. The results presented in Figs. 8-0 show that the FE model could regenerate the experimental results with acceptable accuracy. These results indicate that by adding liquid to the storage tank, the natural frequencies can be decreased (e.g., first and second modes in Figs. 8-0). Moreover, owing to the effect of fluid structure interaction, new modes appear while the existing modes disappear. Conclusions In this paper, the effects of liquid on the modal properties of liquid-containing structures were investigated experimentally and analytically/numerically. Two structures, a pipe and a cylindrical storage tank, were considered, and modal testing was performed to obtain their natural frequencies and mode shapes under different liquid levels. The effect of liquid on the dynamics of the pipe was the addition of mass, and a semi-
5 H. Jalali and F. Parvizi / Journal of Mechanical Science and Technology 6 (5) (0) 9~5 analytical model was able to describe the dynamics of this structure. The experimental results for the cylindrical storage tank indicate that the effect of liquid on this structure is more than a mass effect. By increasing the level of the liquid, new modes appear while the existing modes disappear. The FE model developed for the tank was able to regenerate the dynamic properties of this structure. Nomenclature------------------------------------------------------------------------ E h I L m w ν ρ μ References : Young s modulus : Water level : Cross section moment of inertia : Length : Mass of unit length : Lateral movement : Poisson's ratio : Density : Viscosity [] E. W. Graham and A. M. Rodriguez, The characteristics of fuel motion which affect airplane dynamics, ASME Journal of Applied Mechanics, 9 (95) 38-388. [] Yu. A. Gorbunov, Experimental investigation of the vibrations of spherical and cylindrical shells with a liquid in the presence of internal friction, Proceeding of 5 th All-Union Conference on the Theory of Plates and Shells, Nauka, Moscow (965) 3-5. [3] V. D. Budak and A. V. Ovcharenko, Experimental determination of the effect of a liquid on the natural vibrations of a thin-walled cylindrical shell, Zb. Nauk. Prats Ukr. Derzh. Mykolaiv. Ped. Univ., (999) 3-39. [] M. Chiba, N. Yamaki and J. Tani, Free vibration of a clamped-free circular cylindrical shell partially filled with liquid part III: Experimental results, Thin-Walled Structures, 3 (985) -. [5] O. Curadelli, D. Ambrosini, A. Mirasso and M. Amani, Resonant frequencies in an elevated spherical container partially filled with water: FEM and measurement, Journal of Fluids and Structures, 6 (00) 8-59. [6] T. Mazuch, Natural modes and frequencies of a thin clamped free steel cylindrical storage tank partially filled with water: fem and measurement, Journal of Sound and Vibration, 93 (996) 669-690. [7] P. B. Goncalves and R. C. Batista, Frequency response of cylindrical shells partially submerged or filled with liquid, Journal of Sound and Vibration, 3 (987) 59-70. [8] V. F. Sivak and A. I. Telalov, Experimental investigation of vibration of cylindrical shell in contact with liquid, International Journal of Applied Mechanics, 7 (99) 8-88. [9] Z. YongLiang, M. R. Jason and G. G. Daniel, A comparative study of axisymmetric finite elements for the vibration of thin cylindrical shells conveying fluid, International Journal for Numerical Methods in Engineering, 5 (00) 89-0. [0] J. Kyeong-Hoon and L. Seong-Cheol, Hydroelastic vibration of a liquid-filled circular cylindrical shell, Computers and Structures, 66 (998) 73-85. [] H. Chiba, Nonlinear hydrostatic vibration of a cantilever cylindrical tank-ii: experiment (Liquid-filled case), International Journal of Nonlinear Mechanics, 8 (993) 60-6. [] F. Eberle, B. Goller and R. Krieg, Comparison between calculated and measured eigen-frequencies for spherical steel containment shells, Nuclear Engineering and Design, 0 (990) 5-3. [3] S. Rebouillat and D. Liksonov, Fluid structure interaction in partially filled liquid containers: A comparative review of numerical approaches, Computer and Fluids, 39 (00) 739-76. [] L. Meirovitch, Fundamentals of vibrations, McGraw-Hill, Boston (00). [5] ANSYS, ANSYS structural analysis guide, ANSYS, Inc., Houston (00). [6] M. J. Jhung, J. C. Jo and S. J. Jeong, Impact analysis of a water storage tank, Nuclear Engineering and Technology, 38 (006) 68-688. Hassan Jalali is an assistant professor at Arak University of Technology. He received his Ph.D. degree from Iran University of Science and Technology in 007. His main area of research is modeling and identification of nonlinear mechanical systems with a focus on mechanical joints and interfaces.