Journal of Mechanical Science and Technology 26 (5) (2012) 1449~1454

www.springerlink.com/content/1738-494x DOI 10.1007/s12206-012-0336-4

Experimental and numerical investigation of modal properties for

liquid-containing structures† Hassan Jalali1,* and Fardin Parvizi2

1Department of Mechanical Engineering, Arak University of Technology, Arak 38135-1177, Iran 2School of Mechanical Engineering, Iran University of science and Technology, Tehran 16844, Iran

(Manuscript Received August 19, 2011; Revised January 20, 2012; Accepted January 30, 2012)

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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 characteris-tics 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 ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- 1. Introduction

The response of mechanical structures under dynamic load-ing conditions is governed by their modal properties (i.e., natural frequencies, mode shapes and damping ratios). There-fore, knowledge of the effects of different parameters on the modal properties of structures is helpful in conducting a pre-cise 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 prob-lems have received much attention in the past decade, as they can be found in almost all engineering fields and applied sci-ences. 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

investigation of the modal properties of liquid-containing structures and analytical modeling of FSI mechanisms. Gra-ham and Rodrigues [1] studied the modal properties of a cy-lindrical fluid-filled storage. They obtained the natural fre-quencies by solving the corresponding governing equations. Gorbunov [2] and Budak and Ovcharenko [3] used an ex-perimental 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. [4] obtained experi-mental results on the free vibration of a clamp-free cylindrical shell partially filled with liquid and compared them with ana-lytically calculated results. Curadelli et al. [5] studied the dy-namic response of elevated spherical tanks. He used horizontal base motion to excite the structure. Mazuch et al. [6] investi-gated 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 theo-retical 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] ex-perimentally investigated the free vibrations of a clamped titanium cylindrical shell partially filled with water. YongLi-ang et al. [9] considered the finite element formulation of asymmetric cylindrical shells containing flowing fluid. They

*Corresponding author. Tel.: +98 861 3670024, Fax.: +98 861 3670020 E-mail address: jalali@iust.ac.ir

† Recommended by Associate Editor Cheolung Cheong. © KSME & Springer 2012

1450 H. Jalali and F. Parvizi / Journal of Mechanical Science and Technology 26 (5) (2012) 1449~1454

used the FE model for the vibration analysis of cylindrical shells. Finite Fourier transform and Fourier series expansions were adopted by Kyeong and Seong [10] to develop an ana-lytical approach to estimate the natural frequencies of a circu-lar cylindrical shell partially filled with liquid. Chiba [11] experimentally investigated the nonlinear dynamic response of cantilever circular cylindrical tanks. Eberle et al. [12] consid-ered the eigen-solution of a spherical steel tank. They com-pared experimental and analytical natural frequencies and mode shapes and obtained a good agreement. A comprehen-sive literature review on the problem of FSI, especially the sloshing phenomenon for partially filled liquid containers, was made by Ribouillat and Liksonov [13].

In the current paper, the effects of liquid on natural frequen-cies and mode shapes of a pipe structure and a cylindrical stor-age 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 proper-ties 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 mecha-nisms, and the effect of viscosity is not a primary concern. In the next section, the experimental modal analysis is presented.

2. 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 per-formed experiments, the structures were excited using a B&K4200 mini shaker attached to the structures through a stinger. A B&K8200 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/120/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 dy-namic 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 pro-vided 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.

2.1 Pipe structure

An aluminum pipe having the following dimensions was

used: di = 0.052 m as the internal diameter, do = 0.060 m as the external diameter, and L = 3 m as the length. Fig. 1 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 struc-ture was excited using a pseudo-random forcing function. The frequency response functions (FRFs) between the shaker and 11 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 1 presents the natural frequencies for the different water levels.

The results shown in Table 1 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. 2.

Two conclusions can be made from the results shown in Fig. 2. 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

Table 1. Natural frequencies (Hz) for different water levels (mm).

6ω 5ω 4ω 3ω 2ω 1ω h

716.56519.87351.81 214.68 110.1840.100 714.37518.31350.68 214.00 109.8740.007

707.81513.50347.43 212.00 108.8139.6814

693.12502.87340.18 207.62 106.5638.8722

652.06472.93319.87 195.12 100.1836.5035

594.93431.37291.37 177.68 91.12 33.2552

Fig. 1. Experimental test setup (pipe).

0 50 100 150 200 250 300 350 400 450 500 550−8

−6

−4

−2

0

2

4

6

Frequency (Hz)

Log.

Mag

.

Fig. 2. Comparison of the direct FRFs.

H. Jalali and F. Parvizi / Journal of Mechanical Science and Technology 26 (5) (2012) 1449~1454 1451

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 inter-action 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.

2.2 Cylindrical storage tank

Second, the modal testing of a cylindrical storage tank was considered. The tank had a diameter of 0.26 m and a height of 0.54 m. It was made of an aluminum sheet 0.002 m in thick-ness. The tank was suspended vertically using flexible strings, and an electromagnetic shaker was used to excite it. Fig. 4 shows the test setup.

The structural response of the tank to a pseudo-random forcing function was measured using 49 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 = 0.014 m, and h = 0.024 m. Fig. 4 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 dif-ferent liquid levels will be presented in the subsequent sec-tions. Based on the experimental results of both pipe and cy-lindrical 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 analyti-cal and FE models.

3. Analytical and FE modeling

3.1 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/do = 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 equa-tion governing the free vibration of a beam structure in a free boundary condition is derived as follows [14]:

4 2

4 2

( , ) ( , ) 0 .w x t w x tEI mx t

∂ ∂+ =

∂ ∂ (1)

The governing Eq. (1) is subjected to the following bound-

ary conditions:

2 3 2 3

2 3 2 3

(0, ) (0, ) ( , ) ( , ) 0w t w t w L t w L tx x x x

∂ ∂ ∂ ∂= = = =

∂ ∂ ∂ ∂ (2-5)

where w(x,t), EI, m, and L are the lateral movement, flexural rigidity, mass of unit length, and length of the beam, respec-tively. Using the separation variable method, i.e.,

( , ) ( ) ( )w x t x T tϕ= , the equation governing the natural fre-

0 0.01 0.02 0.03 0.04 0.05

102

103

h (m)

Natu

ral F

requ

ency

Fig. 3. Changes in the natural frequencies with variation in the waterlevel.

Fig. 4. Experimental test setup (cylindrical tank).

100 200 300 400 500 600 700 800 900−25

−20

−15

−10

h=24

cm

100 200 300 400 500 600 700 800 900−25

−20

−15

−10

h=14

cm

100 200 300 400 500 600 700 800 900−25

−20

−15

−10

Frequency (Hz)

h=0

Fig. 5. Measured FRFs at different water levels.

1452 H. Jalali and F. Parvizi / Journal of Mechanical Science and Technology 26 (5) (2012) 1449~1454

quencies of the free beam can be obtained as follows [14]:

24cos( )cosh( ) 1 0, mL L

EIωλ λ λ− = = (6, 7)

where ω is the natural frequency of the beam. Solving Eq. (6) results in the analytical natural frequencies. In Table 2, the experimental natural frequencies in the case of h = 0 are com-pared with those obtained using Eq. (6).

The results presented in Table 2 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 (m2/m1)0.5 when the mass of the unit length of the beam increases from m1 to m2. m1 may be the mass of the unit length of the beam when h = 0, and m2 may be the mass of the unit length of the beam for h = 0.007 m. In other words, m1 and m2 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 af-fects 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 corre-sponding to h = 0 by 0.997, the natural frequencies corre-sponding to h = 0.007 m are obtained. This ratio for h = 0.014 m and h = 0.052 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.

3.2 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 characteris-tics. In the following, the capability of the FE analysis in the

dynamic response prediction of a cylindrical shell is examined. FE analyses using the commercial computer code ANSYS 10 [15] 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 quadri-lateral shell element SHELL63 from the ANSYS element library. In the final version of the model, 40 elements in the axial direction and 54 elements in the circumferential direction, for a total of 2160 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. Accelera-tion 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 = 2683 (Kg/m3). Water, as the contained fluid, has a density of 997 (Kg/m3). The sound speed in water is 1486 m/sec, which is equivalent to a bulk modulus of elasticity of 2.2 GPa. μ = 0.00089 is used as water viscosity [16].

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 ob-tained by the FE model are compared with the experimental results in the case of h = 0 in Table 3.

Table 2. Comparison between the experimental and analytical natural frequencies for h = 0, E = 68 GPa, m = 1.9 Kg/m.

6ω 5ω 4ω 3ω 2ω 1ω

716.56519.87351.81 214.68 110.18 40.10 Exp.

734.53525.87352.05 212.95 108.63 39.41 Ana.

2.50 1.15 0.07 -0.80 -1.40 -1.71 Err. (%)

Fig. 6. Mode shapes of the pipe for the different liquid levels.

Table 3. Comparison of Exp. and FE natural frequencies (Hz) @ h = 0.

5ω 4ω 3ω 2ω 1ω 594.45 489.95 430.19 227.12 84.44Exp. 598.92 471.64 431.98 226.51 82.09FEM 0.75 -3.73 0.27 -0.53 -2.18 Err. (%)

Fig. 7. The FE model for h = 0.14 m.

H. Jalali and F. Parvizi / Journal of Mechanical Science and Technology 26 (5) (2012) 1449~1454 1453

Table 3 shows good agreement between FE and the ex-perimental 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 ele-ments in regenerating the test results is then investigated. In Tables 4 and 5, the experimental natural frequencies and FE results are compared for cases h = 0.14 m and h = 0.24 m, respectively. Figs. 9 and 10 show the corresponding mode shapes.

The results presented in Figs. 8-10 show that the FE model could regenerate the experimental results with acceptable ac-curacy. These results indicate that by adding liquid to the stor-

age tank, the natural frequencies can be decreased (e.g., first and second modes in Figs. 8-10). Moreover, owing to the effect of fluid–structure interaction, new modes appear while the existing modes disappear

4. 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-

Table 4. Comparison of Exp. and FE natural frequencies (Hz) @ h = 0.14 m.

5ω 4ω 3ω 2ω 1ω 671.26521.60390.52 218.2079.77 Exp. 688.26501.33386.86 217.6578.12 FEM 2.47 -3.88 -0.94 -0.25 -1.33 Err. (%)

Table 5. Comparison of Exp. and FE natural frequencies (Hz) @ h = 0.24 m.

5ω 4ω 3ω 2ω 1ω 446.02356.09276.85 179.9467.95 Exp. 451.59359.21263.79 176.8665.99 FEM 1.24 0.87 -4.71 -1.71 -2.88 Err. (%)

Fig. 8. Comparison of Exp. and FE mode shapes @ h = 0.

Fig. 9. Comparison between the experimental results and FE mode shapes @ h = 0.14 m.

Fig. 10. Comparison between the experimental results and FE mode shapes @ h = 0.24 m.

1454 H. Jalali and F. Parvizi / Journal of Mechanical Science and Technology 26 (5) (2012) 1449~1454

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 dy-namic properties of this structure.

Nomenclature------------------------------------------------------------------------

E : Young’s modulus h : Water level I : Cross section moment of inertia L : Length m : Mass of unit length w : Lateral movement ν : Poisson's ratio ρ : Density μ : Viscosity

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Hassan Jalali is an assistant professor at Arak University of Technology. He re-ceived his Ph.D. degree from Iran Univer-sity of Science and Technology in 2007. His main area of research is modeling and identification of nonlinear mechanical systems with a focus on mechanical joints and interfaces.