Vibration of an Offshore Structure Having the Form of a Hollow Column Partially Filled with Multiple Fluids and Immersed in Water

Abstract

This paper employs the numerical assembly method (NAM) to determine the exact frequency-response amplitudes of an offshore structure such as piles or towers having the form of a hollow column filled with multiple fluids, immersed in water, carrying an eccentric tip mass supported by a translational spring and/or a rotational spring, and subjected to a harmonic force. The hollow column is modeled as a Bernoulli-Euler cantilever beam fixed at the bottom. For the case of zero harmonic force, the simultaneous equations of the vibration system reduce to an eigenvalue problem so that the natural frequencies and mode shapes of the beam can also be obtained. The effect of height of filled fluids on the characteristics of free vibration is also presented.

1. Introduction

In offshore engineering, structures such as towers or piles immersed in water can be predicted with rational accuracy from a fixedly supported beam with a tip mass. Many important papers [1–5] have been published in the field. Those researches was assumed that the beam was solid or empty hollow. Chan and Zhang [6] presented the natural frequency of a cantilever tube filled with only one liquid by using the continuity and equilibrium conditions at the liquid level position. Amabili [7] studied the free vibration of circular cylindrical shells and tubes completely filled with one dense fluid and partially immersed in one different fluid. Wu at el. [8–10] used three different methods to present the natural frequencies and mode shapes of an immersed solid beam carrying an eccentric tip mass with rotary inertia. Lin [11–13] presented the exact natural frequencies and mode shape of a beam carrying multiple concentrated elements.

From the foregoing literature review one finds that the literature regarding determination of the natural frequencies and mode shapes of a hollow beam (partially or completely) filled with fluids of different densities and (partially or completely) immersed in different fluids is little. In this paper an accurate method is presented to determine the natural frequencies and mode shapes of an immersed hollow beam filled with fluids of different densities and carrying an eccentric tip mass supported by a rotational spring and a translational spring. The vibration characteristics of the beam system subjected to a harmonic force are also presented.

In the exploratory offshore drilling operations, the long slender vertical cylindrical pipe placed between the sea surface and the ocean floor for conveying various fluids is called “riser.” In the existing literature concerning the analyses of marine risers [14], the fluids in the “riser” are assumed to be uniform with constant mass density for simplicity. In practice, the last assumption may be different from the actual situations to some degree, because the riser pipe may be filled with several fluids with different mass densities simultaneously, such as water, gas, oil, or clay. For this reason, this paper tries to present a technique to study the dynamic behaviors of a pipe filled with multiple fluids and immersed in water. Because the presented mathematical model is closer to the marine risers, it is believed that more satisfactory results will be obtained.

In order to confirm the reliability of the presented approach, the numerical example illustrated in this paper is also conducted by using the FEM and good agreement is achieved. One of the predominant merits for the presented approach is that its numerical results belong to the “exact solutions” and may be the benchmark for evaluating the accuracy of the other approximate methods (such as FEM).

2. Theoretical Model

Figure 1 shows the sketch of a immersed hollow beam filled with fluids of different densities and carrying an eccentric tip mass with rotary inertia, rotational spring, and translational spring supports. The points corresponding to the locations of fluid interfaces and/or applied concentrated forces are referred to as “stations.” The positions of stations are defined by 𝑥_𝑛(𝑛=1,…,𝑛) and the subscript of 𝑥_𝑛 refer to the numbering of the stations. Symbol “⊗” denotes the center of mass of the tip mass. 𝐸 is Young’s modulus of the beam, 𝐼 is moment of inertia of the cross-sectional area of the beam, 𝑑_oand𝑑_𝑠 are outer and inner diameters of the hollow beam, respectively, 𝑚 is mass per unit length of the beam, 𝜌_𝑡, 𝜌_𝑜, and 𝜌_𝑠 are the density of the stainless steel beam, density of the outer fluid (water) of the hollow beam, and density of the filled fluids of the hollow beam, respectively. 𝑀_𝑒 is the tip mass, 𝐽_𝑒 is its moment of inertia, 𝑘_𝑇𝑒 is the translational spring constant of supporting tip mass, 𝑘_Re is the rotational spring constant of supporting tip mass, 𝑒_𝑚 is the distance between the upper end of beam and center of gravity of the tip mass, and 𝑒_𝑘 is the distance between the upper end of beam and the translational spring support.

Figure 1 

The sketch of an immersed hollow beam filled with fluids of different densities, subjected to a force, and carrying an eccentric tip mass supported by a translational and a rotational spring.

2.1. Equation of Motion and Displacement Function

The differential equation of motion for the 𝑖th beam segment of a uniform immersed hollow beam filled with fluids of different densities, subjected to a harmonic force, and carrying an eccentric tip mass supported by a rotational spring and a translational spring (cf. Figure 1) with small deflections is given by 𝐸𝐼_𝑖𝜕⁴𝑦_𝑖(𝑥,𝑡)𝜕𝑥⁴+𝑚_𝑖𝜕²𝑦_𝑖(𝑥,𝑡)𝜕²𝑡=𝐹(𝑡)⋅𝛿𝑥−𝑥_𝑖,𝑚_𝑖=𝑚+𝑚_𝑜𝑖+𝑚_𝑠𝑖 (2.1)(2.2) with 𝑚, 𝑚_𝑜𝑖, and 𝑚_𝑠𝑖 given by 𝜋𝑑𝑚=^2𝑜−𝑑²_𝑠𝜌_𝑡4,𝑚_𝑜𝑖=𝜋𝑑²_𝑜𝜌_𝑜𝑖4,𝑚_𝑠𝑖=𝜋𝑑²_𝑠𝜌_𝑠𝑖4, (2.3)(2.4)(2.5) where 𝑚, 𝑚_𝑜𝑖, and 𝑚_𝑠𝑖 are the mass per unit length of the 𝑖th beam segment, outer fluid (added) mass per unit length of the 𝑖th beam segment, and filled fluid (added) mass per unit length of the 𝑖th beam segment, respectively, 𝑦_𝑖(𝑥,𝑡) is the transverse displacement at position 𝑥 and time 𝑡 for the 𝑖th beam segment, and 𝐹(𝑡) is a force at time 𝑡. Besides, 𝛿(𝑥−𝑥_𝑖) is the Dirac delta with 𝑥_𝑖 denoting the coordinate at which the concentrated force 𝐹(𝑡) is applied.

If the applied concentrated force takes the form 𝐹(𝑡)=𝐹(𝑥)𝑒^{𝑗𝜔𝑒𝑡}, (2.6) then, in the steady state, one has 𝑦_𝑖(𝑥,𝑡)=𝑌_𝑖(𝑥)𝑒^{𝑗𝜔𝑒𝑡}, (2.7) where 𝑌_𝑖(𝑥) is the amplitude of 𝑦_𝑖(𝑥,𝑡), 𝜔_𝑒 is the exciting frequency of the applied harmonic forces, 𝐹 is amplitude of 𝐹(𝑡), and 𝑗=√−1.

Substitution of (2.6), (2.7) into (2.1) gives 𝑌_𝑖^′′′′(𝑥)−𝛽⁴_𝑖𝑌_𝑖(𝑥)=𝐹𝐸𝐼⋅𝛿𝑥−𝑥_𝑖, (2.8) where 𝜔_𝑣=𝛽_𝑖𝐿²𝐸𝐼_𝑖𝑚_𝑖𝐿^41/2=Ω²_𝑖𝐸𝐼_𝑖𝑚_𝑖𝐿^41/2. (2.9)

Equation (2.8) is a nonhomogeneous equation, and its “complete” solution takes the form 𝑌_𝑖(𝑥)=𝐶_𝑖,1sin𝛽_𝑖𝑥+𝐶_𝑖,2cos𝛽_𝑖𝑥+𝐶_𝑖,3sinh𝛽_𝑖𝑥+𝐶_𝑖,4cosh𝛽_𝑖𝑥−𝐹𝛽⁴_𝑖𝐸𝐼⋅𝛿𝑥−𝑥_𝑖, (2.10) in which 𝐶₁,𝐶₂,𝐶₃, and 𝐶₄ are the unknown integration constants. From (2.10), one sees that, for any beam segment on which no concentrated force 𝐹(𝑡) being applied, its displacement function will take the form 𝑌_𝑖(𝑥)=𝐶_𝑖,1sin𝛽_𝑖𝑥+𝐶_𝑖,2cos𝛽_𝑖𝑥+𝐶_𝑖,3sinh𝛽_𝑖𝑥+𝐶_𝑖,4cosh𝛽_𝑖𝑥. (2.11)

2.2. Coefficient Matrix [𝐵_𝑢] for an Intermediate Fluid Interface

If the station numbering of an intermediate fluid interface is 𝑢, then the continuity of deformations and the equilibrium of moments and forces at station 𝑢 require that 𝑌_𝑢𝜉_𝑢=𝑌_𝑢+1𝜉_𝑢𝑌^′_𝑢𝜉_𝑢=𝑌^′_𝑢+1𝜉_𝑢𝑌_𝑢^′′𝜉_𝑢=𝑌^′′_𝑢+1𝜉_𝑢𝑌_𝑢^′′𝜉_𝑢=𝑌^′′′_𝑢+1𝜉_𝑢𝜉_𝑢=𝑥_𝑢𝐿. (2.12)(2.13)(2.14)(2.15)(2.16)

Substitution of (2.11) into (2.12)–(2.15) leads to 𝐶_𝑢,1sinΩ_𝑢𝜉_𝑢−𝐶_𝑢,2cosΩ_𝑢𝜉_𝑢+𝐶_𝑢,3sinhΩ_𝑢𝜉_𝑢+𝐶_𝑢,4coshΩ_𝑢𝜉_𝑢−𝐶_𝑢+1,1sinΩ_𝑢+1𝜉_𝑢−𝐶_𝑢+1,2cosΩ_𝑢+1𝜉_𝑢+𝐶_𝑢+1,3sinhΩ_𝑢+1𝜉_𝑢+𝐶_𝑢+1,4coshΩ_𝑢+1𝜉_𝑢Ω=0,_𝑢𝐶_𝑢,1cosΩ_𝑢𝜉_𝑢−𝐶_𝑢,2sinΩ_𝑢𝜉_𝑢+𝐶_𝑢,3coshΩ_𝑢𝜉_𝑢+𝐶_𝑢,4sinhΩ_𝑢𝜉_𝑢−Ω_𝑢+1𝐶_𝑢+1,1cosΩ_𝑢+1𝜉_𝑢−𝐶_𝑢+1,2sinΩ_𝑢+1𝜉_𝑢+𝐶_𝑢+1,3coshΩ_𝑢+1𝜉_𝑢+𝐶_𝑢+1,4sinhΩ_𝑢+1𝜉_𝑢Ω=0,_2𝑢−𝐶_𝑢,1sinΩ_𝑢𝜉_𝑢−𝐶_𝑢,2cosΩ_𝑢𝜉_𝑢+𝐶_𝑢,3sinhΩ_𝑢𝜉_𝑢−𝐶_𝑢,4coshΩ_𝑢𝜉_𝑢−Ω²_𝑢+1−𝐶_𝑢+1,1sinΩ_𝑢+1𝜉_𝑢−𝐶_𝑢+1,2cosΩ_𝑢+1𝜉_𝑢+𝐶_𝑢+1,3sinhΩ_𝑢+1𝜉_𝑢−𝐶_𝑢+1,4coshΩ_𝑢+1𝜉_𝑢Ω=0,_3𝑢−𝐶_𝑢,1cosΩ_𝑢𝜉_𝑢−𝐶_𝑢,2sinΩ_𝑢𝜉_𝑢+𝐶_𝑢,3coshΩ_𝑢𝜉_𝑢−𝐶_𝑢,4sinhΩ_𝑢𝜉_𝑢−Ω³_𝑢+1−𝐶_𝑢+1,1cosΩ_𝑢+1𝜉_𝑢−𝐶_𝑢+1,2sinΩ_𝑢+1𝜉_𝑢+𝐶_𝑢+1,3coshΩ_𝑢+1𝜉_𝑢−𝐶_𝑢+1,4sinhΩ_𝑢+1𝜉_𝑢Ω=0,_𝑢=𝜔²𝑚_𝑢𝐿⁴𝐸𝐼^1/4,Ω_𝑢+1=𝜔²𝑚_𝑢+1𝐿⁴𝐸𝐼^1/4,𝑚_𝑢=𝑚+𝑚_𝑜𝑢+𝑚_𝑠𝑢,𝑚_𝑢+1=𝑚+𝑚_𝑜(𝑢+1)+𝑚_𝑠(𝑢+1). (2.17)(2.18)(2.19)(2.20)(2.21)(2.22)(2.23)(2.24)

Writing (2.17)–(2.20) in matrix form, one has 𝐵_𝑢𝐶_𝑢={0}, (2.25) where 𝐶_𝑢=𝐶_𝑢,1𝐶_𝑢,2𝐶_𝑢,3𝐶_𝑢,4𝐶_𝑢+1,1𝐶_𝑢+1,2𝐶_𝑢+1,3𝐶_𝑢+1,4 (2.26)

In the above equations (2.25) and (2.26), the symbols, [] and {}, denote the rectangular matrix and column vector, respectively. The coefficient matrix [𝐵_𝑢] is placed in (A.1) of Appendix A.

2.3. Coefficient Matrix [𝐵_𝑣] for an Intermediate Applied Force

If the station numbering for the intermediate harmonic concentrated force normal to the beam is 𝐯, then the continuity of deformations and equilibrium of moments and forces require that 𝑌_𝑣𝜉_𝑣=𝑌_𝑣+1𝜉_𝑣𝑌^′_𝑣𝜉_𝑣=𝑌^′_𝑣+1𝜉_𝑣𝑌_𝑣^′′𝜉_𝑣=𝑌^′′_𝑣+1𝜉_𝑣1𝐸𝐼𝐿³𝑌_𝑣^′′′𝜉_𝑣+𝐹_𝑣1=𝐸𝐼𝐿³𝑌^′′′_𝑣+1𝜉_𝑣. (2.27)(2.28)(2.29)(2.30)

From (2.11), (2.27)–(2.30), one obtains 𝐶_𝑣,1sinΩ_𝑣𝜉_𝑣+𝐶_𝑣,2cosΩ_𝑣𝜉_𝑣+𝐶_𝑣,3sinhΩ_𝑣𝜉_𝑣+𝐶_𝑣,4coshΩ_𝑣𝜉_𝑣−𝐶_𝑣+1,1sinΩ_𝑣+1𝜉_𝑣+𝐶_𝑣+1,2cosΩ_𝑣+1𝜉_𝑣+𝐶_𝑣+1,3sinhΩ_𝑣+1𝜉_𝑣+𝐶_𝑣+1,4coshΩ_𝑣+1𝜉_𝑣Ω=0,_𝑣𝐶_𝑣,1cosΩ_𝑣𝜉_𝑣−𝐶_𝑣,2sinΩ_𝑣𝜉_𝑣+𝐶_𝑣,3coshΩ_𝑣𝜉_𝑣+𝐶_𝑣,4sinhΩ_𝑣𝜉_𝑣−Ω_𝑣+1𝐶_𝑣+1,1cosΩ_𝑣+1𝜉_𝑣−𝐶_𝑣+1,2sinΩ_𝑣+1𝜉_𝑣+𝐶_𝑣+1,3coshΩ_𝑣+1𝜉_𝑣+𝐶_𝑣+1,4sinhΩ_𝑣+1𝜉_𝑣Ω=0,_2𝑣−𝐶_𝑣,1sinΩ_𝑣𝜉_𝑣−𝐶_𝑣,2cosΩ_𝑣𝜉_𝑣+𝐶_𝑣,3sinhΩ_𝑣𝜉_𝑣+𝐶_𝑣,4coshΩ_𝑣𝜉_𝑣−Ω²_𝑣+1−𝐶_𝑣+1,1sinΩ_𝑣+1𝜉_𝑣−𝐶_𝑣+1,2cosΩ_𝑣+1𝜉_𝑣+𝐶_𝑣+1,3sinhΩ_𝑣+1𝜉_𝑣+𝐶_𝑣+1,4coshΩ_𝑣+1𝜉_𝑣Ω=0,_3𝑣−𝐶_𝑣,1cosΩ_𝑣𝜉_𝑣+𝐶_𝑣,2sinΩ_𝑣𝜉_𝑣+𝐶_𝑣,3coshΩ_𝑣𝜉_𝑣+𝐶_𝑣,4sinhΩ_𝑣𝜉_𝑣−Ω³_𝑣+1−𝐶_𝑣+1,1cosΩ_𝑣+1𝜉_𝑣+𝐶_𝑣+1,2sinΩ_𝑣+1𝜉_𝑣+𝐶_𝑣+1,3coshΩ_𝑣+1𝜉_𝑣+𝐶_𝑣+1,4sinhΩ_𝑣+1𝜉_𝑣=−𝐹_𝑣𝐿³.𝐸𝐼 (2.31)(2.32)(2.33)(2.34)

Writing (2.31)–(2.34) in matrix form, one has 𝐵_𝑣𝐶_𝑣=𝐷_𝑣,𝐶_𝑣=𝐶_𝑣,1𝐶_𝑣,2𝐶_𝑣,3𝐶_𝑣,4𝐶_𝑣+1,1𝐶_𝑣+1,2𝐶_𝑣+1,3𝐶_𝑣+1,4𝐷_𝑣,=000−𝐹_𝑣𝐿³𝐸𝐼. (2.35)(2.36)(2.37)

And the coefficient matrix [𝐵_𝑣] is placed in (B.1) of Appendix B.

2.4. Coefficient Matrices [𝐵₀] and [𝐵_𝑁] for the Two Ends of the Entire Beam

The lower end of hollow beam is fixed as shown in Figure 1, then the boundary condition is 𝑌₀(0)=𝑌^′₀(0)=0. (2.38)

From (2.11), (2.38), one obtains 𝐵₀𝐶₀={0}, (2.39) where 𝐵₀=12,𝐶123401011010₀=𝐶_0,1𝐶_0,2𝐶_0,3𝐶_0,4. (2.40)(2.41)

If the tip of the beam is carrying an eccentric mass with rotary inertia, a rotational spring, and translational spring as shown in Figure 1, then the boundary conditions are 1𝐸𝐼𝐿²𝑌_𝑁^′′𝜉_𝑁−𝐽_𝑒𝜔²+𝑀_𝑒𝑒²_𝑚𝜔²−𝑘_Re−𝑒²_𝑘𝑘_𝑇𝑒1𝐿𝑌^′_𝑁𝜉_𝑁−𝑀_𝑒𝑒_𝑚𝜔²−𝑒_𝑘𝑘_𝑇𝑒𝑌_𝑁𝜉_𝑁1=0𝐸𝐼𝐿³𝑌_𝑁^′′′𝜉_𝑁+𝑀_𝑒𝑒_𝑚𝜔²−𝑒_𝑘𝑘_𝑇𝑒1𝐿𝑌^′_𝑁𝜉_𝑁+𝑀_𝑒𝜔²−𝑘_𝑇𝑒𝑌_𝑁𝜉_𝑁=0, (2.42)(2.43) where 𝑁=𝑛+1, (2.44)𝑛 is the total number of intermediate stations.

From (2.11) and (2.42)–(2.43), one obtains −Ω²_𝑁−𝑀^∗_𝑒𝑒^∗_𝑚Ω⁴_𝑁+𝑘^∗_𝑇𝑒𝑒^∗_𝑘𝐶_𝑁,1sinΩ_𝑁+𝐶_𝑁,2cosΩ_𝑁+Ω²_𝑁−𝑀^∗_𝑒𝑒^∗_𝑚Ω⁴_𝑁+𝑘^∗_𝑇𝑒𝑒^∗_𝑘𝐶_𝑁,3sinhΩ_𝑁+𝐶_𝑁,4−𝐽coshΩ_∗𝑒Ω⁵_𝑁−𝑘^∗_ReΩ_𝑁+𝑀^∗_𝑒𝑒^∗_𝑚²Ω⁵_𝑁−𝑘^∗_𝑇𝑒𝑒^∗_𝑘²Ω_𝑁𝐶_𝑁,1cosΩ_𝑁−𝐶_𝑁,2sinΩ_𝑁+𝐶_𝑁,3coshΩ_𝑁+𝐶_𝑁,4sinhΩ_𝑁𝑀=0,_∗𝑒Ω⁴_𝑁−𝑘^∗_Te𝐶_𝑁,1sinΩ_𝑁+𝐶_𝑁,2cosΩ_𝑁+𝐶_𝑁,3sinhΩ_𝑁+𝐶_𝑁,4coshΩ_𝑁+𝑀^∗_𝑒𝑒^∗_𝑚Ω⁵_𝑁−𝑘^∗_Te𝑒^∗_𝑘Ω_𝑁−Ω³_𝑁𝐶_𝑁,1cosΩ_𝑁−𝐶_𝑁,2sinΩ_𝑁+𝑀^∗_𝑒𝑒^∗_𝑚Ω⁵_𝑁−𝑘^∗_Te𝑒^∗_𝑘Ω_𝑁+Ω³_𝑁𝐶_𝑁,3coshΩ_𝑁+𝐶_𝑁,4sinhΩ_𝑁𝑀=0,_∗𝑒=𝑀_𝑒,𝐽𝑚𝐿_∗𝑒=𝐽_𝑒𝑚𝐿³,𝑘^∗_R𝑒=𝑘_Re𝐿,𝑘𝐸𝐼_∗Te=𝑘_Te𝐿³,𝑒𝐸𝐼_∗𝑚=𝑒_𝑚𝐿,𝑒^∗_𝑘=𝑒_𝑘𝐿. (2.45)(2.46)(2.47)(2.48)(2.49)(2.50)(2.51)(2.52)

Writing (2.45)–(2.46) in matrix form, one has 𝐵_𝑁𝐶_𝑁={0}, (2.53) where 𝐶_𝑁=𝐶_𝑁,1𝐶_𝑁,2𝐶_𝑁,2𝐶_𝑁,2. (2.54)

And the coefficient matrix [𝐵_𝑁] is placed in (C.1) of Appendix C.

2.5. Determination of Natural Frequencies and Mode Shapes of the Beam

The integration constants relating to the lower-end and upper-end supports of the beam are defined by (2.41) and (2.54), respectively, while those relating to the intermediate stations are defined by (2.26) and (2.36) depending upon the interface of fluid and/or the concentrated force being located there. The associated coefficient matrices are given by [𝐵₀] (cf. (2.40)), [𝐵_𝑢](cf. (A.1) of Appendix A), [𝐵_𝑣] (cf. (B.1) of Appendix B) and [𝐵_𝑁] (cf. (C.1) of Appendix C). From the last equations concerned, one may see that the identification number for each element of the last coefficient matrices is shown on the top side and right side of each matrix. Therefore, using the numerical assembly technique, one may obtain a matrix equation for all the integration constants of the entire beam 𝐵𝐶=𝐷. (2.55)

For the case of free vibrations, the applied force amplitude 𝐹 is zero and (2.55) reduces to 𝐵𝐶={0}. (2.56)

Nontrivial solution of (2.56) requires that its coefficient determinant is equal to zero, that is, 𝐵=0, (2.57) which is the frequency equation for the present problem.

In this paper, the incremental search method is used to find the natural frequencies of the vibrating system, 𝜔_𝑣 (𝑣=1,2,…). With respect to each natural frequency 𝜔_𝑣, one may obtain the corresponding integration constants from (2.56). The substitution of the last integration constants into the displacement functions of the associated beam segments will determine the corresponding mode shape of the entire beam, 𝑌^(𝑣)(𝜉).

2.6. Determination of Forced Vibration Response of the Beam

For the case of forced vibrations, from (2.55), one has 𝐶=𝐵⁻¹𝐷. (2.58) Thus, if the exciting frequency 𝜔_𝑒 (or the associated dimensionless frequency parameter Ω_𝑒) of the harmonic forces is given, then one may obtain the corresponding integration constants from (2.58). The substitution of last integration constants into the displacement functions of associated beam segments will determine the corresponding vibration amplitude |𝑌(𝜉)|.

3. Numerical Results and Discussions

Before the vibration analysis of an immersed hollow beam filled with fluids of different densities, subjected to a force, and carrying an eccentric tip mass supported by a translational and a rotational spring is performed, the reliability of the theory and the computer program developed for this paper are confirmed by comparing the present results with those obtained from the existing literature.

3.1. Reliability of the Developed Computer Program

The first example studied is a fixedly supported immersed beam carrying tip mass 𝑀^∗_𝑒=𝑀_𝑒/(𝜌𝐴𝐿)=0.1 with rotary inertia 𝐽^∗_𝑒=𝐽_𝑒/(𝜌𝐴𝐿³)=0.1 and eccentricity 𝑒=0.5m and two intermediate lumped masses 𝑀^∗₂=𝑀^∗₃=0.5(𝜌𝐴𝐿)kg located at 𝑥₂=13m and 𝑥₃=14m, respectively, for the case of draft ratio 𝐿^∗₁=𝐿₁/𝐿=0.5 and water density 𝜌=1000Kg/m³. The dimensions of the solid beam are as follows: Young’s modulus =2.068×10¹¹N/m², diameter 𝑑= 0.3 m, mass density 𝜌=7850Kg/m³, and total length 𝐿=15m. The lowest four natural frequencies of the immersed beam are shown in Table 1. From Table 1, one sees that the results of this paper are in good agreement with those of [9].

Table 1 

The lowest four natural frequencies of a fixedly supported immersed beam carrying an eccentric tip mass and two intermediate lumped masses.

3.2. Natural Frequencies and Mode Shapes of the Immersed Hollow Beam Filled with Two Kinds of Fluids Carrying an Eccentric Tip Mass with Rotary Inertia, Translational and Rotational Spring Supports

The second example studied is a immersed hollow beam filled with two kinds of fluids and carrying an eccentric tip mass with rotary inertia, translational and rotational spring supports (cf. Figure 2). The parameters of the steel hollow beam are as follows: total length 𝐿=15m, outer diameter 𝑑_𝑜=0.6 m, inner diameter 𝑑_𝑠=0.54 m, Young’s modulus 𝐸=2.068×10¹¹N/m², density 𝜌_𝑡=7850 kg/m³, density of the inner upper fluids 𝜌_𝑠2=1000 kg/m³, and density of the inner lower fluid 𝜌_𝑠1=1400 kg/m³. The nondimensional parameters of the eccentric tip mass are as follows: mass 𝑀^∗_𝑒=𝑀_𝑒/𝑚𝐿=1; rotary inertia of the tip mass 𝐽^∗_𝑢=0.01; constants of the rotational spring support 𝑘^∗_Re=10; constants of the translational spring support 𝑘^∗_Te=30; distance between the fixed point of the tip mass and its center of gravity 𝑒^∗_𝑚=𝑒_𝑚/𝐿=1/15; distance between the fixed point of the tip mass and the translational spring support 𝑒^∗_𝑘=𝑒_𝑘/𝐿=0.1; the nondimensional height of the beam immersed in water 𝜉^∗₂=9/15=0.6; height of filled lower fluid 𝜉^∗₁=𝑥₁/𝐿=0.3; height of filled with upper fluid 𝜉^∗_𝑖=𝑥_𝑖/𝐿=0.4to0.8, respectively. The lowest four natural frequencies and non-dimensional parameters for the hollow beam are shown in Table 2. In order to confirm the reliability of the presented results, the current example is also conducted by using the FEM with the lowest four natural frequencies shown in Table 2. From the table one sees that the lowest four natural frequencies of the hollow beam obtained from FEM are very close to those obtained from the presented approach; furthermore, they decrease with increasing the height of filled upper fluid due to its added mass. Corresponding to the four natural frequencies listed in Table 2, the lowest four mode shapes of the hollow beam are shown in Figures 3(a)–3(c), respectively, where (a), (b), and (c) refer to the 1st,2nd, and 3rd mode shapes of the hollow beam. Besides, the curves − − − −, — — — —, – – – –, …………, and −⋅−⋅− denote the mode shapes of the hollow beam filled with the upper fluid height 𝜉^∗_𝑖=𝑥_𝑖/𝐿=0.4to0.8, respectively.

Table 2 

The lowest four natural frequencies of the immersed hollow beam filled with two kinds of fluids and carrying an eccentric tip mass with a translational and a rotational spring supports (cf. Figure 2).

Figure 2 

The sketch of a immersed hollow beam filled with two kinds of fluids and carrying an eccentric tip mass with rotary inertia, translational and rotational spring supports.

(a)

(b)

(c)

Figure 3 

The lowest four mode shapes of the hollow beam are shown in Figures 3(a)–3(c), respectively, where (a), (b), and (c) refer to the 1st, 2nd, and 3rd mode shapes of the hollow beam. Besides, the curves − − − −, — — — —, – – – –, …………, and −⋅−⋅− denote the mode shapes of the hollow beam filled with the upper fluid height 𝜉∗𝑖=𝑥𝑖/𝐿=0.4to0.8, respectively.

3.3. Forced Vibration Responses of the Hollow Beam Filled with Two kinds of Fluids, Immersed in Water and Carrying an Eccentric Tip Mass

The beam system of the present example is the same as the last one but the beam subjected to a harmonic force 𝐹^∗_𝑣=1 is located at 𝜉=0.8. For each value of Ω (from 0 to 13.0 with interval ΔΩ=0.001), one may obtain the integration constants from (2.58). Then we compute the vibration amplitudes when it is placed in different locations on the beam. Figure 4 shows the relationship between the dimensionless frequency parameters (Ω) and the dimensionless response amplitudes (|𝑌(𝜉)|_max=|𝑌(𝜉)|_max/(𝐹𝐿³/𝐸𝐼)) for the points located at 𝜉=0.4, 0.6, and 0.8, where the horizontal axis is the dimensionless frequency parameter (Ω) and the vertical axis is the dimensionless vibration amplitude (|𝑌(𝜉)|_max). The blue, black, and red curves are for points located at locations 𝜉=0.4, 0.6, and 0.8, respectively. The vibration amplitudes for the point located at either 𝜉=0.4, 0.6 or 0.8 have peaks when the dimensionless frequency parameter Ω≈ 2.9, 4.9, 7.4, 9.6, and 12.7. This is because when the dimensionless frequency parameter Ω is near any of the natural frequencies of the beam, as shown in the parenthesis of line 1 of Table 2, resonance appears.

Figure 4 

The relationship for dimensionless frequency parameters (Ω) versus dimensionless amplitudes (|𝑌(𝜉)|max) for the points located at 𝜉=0.4, 0.6, and 0.8 of the beam subjected to a harmonic force applied at 𝜉=0.8.

4. Conclusion

The results presented in this work have found that one can obtain the lowest several “exact” natural frequencies and corresponding mode shapes of an immersed hollow beam partially filled with two kinds of fluids and carrying an eccentric tip mass with rotary inertia, rotational and translation spring supports. For the beam subjected to a harmonic force, one can determine the frequency-response curve for any point of the beam using this method. Because a peak will appear in each curve when the exciting frequency of the harmonic force is near any of natural frequencies of the beam, one can determine natural frequencies of the beam based on the peaks of any frequency-response curve.

Appendices

A.

Consider the following: 𝐵_𝑢=𝔅ℭ𝔇𝔈𝔊ℌℑ𝔎𝔄𝔉𝔐𝔑𝔒𝔓𝔔ℜ𝔖𝔗𝔘𝔙𝔚𝔛𝔜ℨ𝔞𝔟𝔠𝔡𝔢𝔣𝔤𝔥𝜃4𝑢−34𝑢−24𝑢−14𝑢4𝑢+14𝑢+24𝑢+34𝑢+44𝑢−14𝑢4𝑢+14𝑢+2_𝑢=Ω_𝑢𝜉_𝑢,𝜃_𝑢+1=Ω_𝑢+1𝜉_𝑢. (A.1)(A.2)(A.3)where 𝔅 denotes sin𝜃_𝑢, ℭ denotes Ω_𝑢cos𝜃_𝑢, 𝔇 denotes −Ω²_𝑢sin𝜃_𝑢, 𝔈 denotes −Ω³_𝑢cos𝜃_𝑢, 𝔊 denotes cos𝜃_𝑢, ℌ denotes −Ω_𝑢sin𝜃_𝑢, ℑ denotes −Ω²_𝑢cos𝜃_𝑢, 𝔎 denotes Ω³_𝑢sin𝜃_𝑢, 𝔄 denotes sinh𝜃_𝑢, 𝔉 denotes Ω_𝑢cosh𝜃_𝑢, 𝔐 denotes Ω²_𝑢sinh𝜃_𝑢, 𝔑 denotes Ω³_𝑢cosh𝜃_𝑢, 𝔒 denotes cosh𝜃_𝑢, 𝔓 denotes Ω_𝑢sinh𝜃_𝑢, 𝔔 denotes Ω²_𝑢cosh𝜃_𝑢, ℜ denotes Ω³_𝑢sinh𝜃_𝑢, 𝔖 denotes −sin𝜃_𝑢+1, 𝔗 denotes −Ω_𝑢+1cos𝜃_𝑢+1, 𝔘 denotes Ω²_𝑢+1sin𝜃_𝑢+1, 𝔙 denotes Ω³_𝑢+1cos𝜃_𝑢+1, 𝔚 denotes −cos𝜃_𝑢+1, 𝔛 denotes Ω_𝑢+1sin𝜃_𝑢+1, 𝔜 denotes Ω²_𝑢+1cos𝜃_𝑢+1, ℨ denotes −Ω³_𝑢+1sin𝜃_𝑢+1, 𝔞 denotes −sinh𝜃_𝑢+1, 𝔟 denotes −Ω_𝑢+1cosh𝜃_𝑢+1, 𝔠 denotes −Ω²_𝑢+1sinh𝜃_𝑢+1, 𝔡 denotes −Ω³_𝑢+1cosh𝜃_𝑢+1, 𝔢 denotes −cosh𝜃_𝑢+1, 𝔣 denotes −Ω_𝑢+1sinh𝜃_𝑢+1, 𝔤 denotes −Ω²_𝑢+1cosh𝜃_𝑢+1, and 𝔥 denotes −Ω³_𝑢+1sinh𝜃_𝑢+1.

B.

Consider the following:𝐵_𝑣=𝔦𝔧𝔨𝔩𝔪𝔫𝔬𝔭𝔮𝔯𝔰𝔱𝔲𝔳𝔴𝔵𝔶𝔷𝕫𝕪𝕩𝕨𝕧𝕦𝕥𝔰𝕣𝕢𝕡𝕠𝕟𝕞𝜃4𝑣−34𝑣−24𝑣−14𝑣4𝑣+14𝑣+24𝑣+34𝑣+44𝑣−14𝑣4𝑣+14𝑣+2_𝑣=Ω_𝑣𝜉_𝑣,𝜃_𝑣+1=Ω_𝑣+1𝜉_𝑣. (B.1)(B.2)(B.3)where 𝔦 denotes sin𝜃_𝑣, 𝔧 denotes Ω_𝑣cos𝜃_𝑣, 𝔨 denotes −Ω²_𝑣sin𝜃_𝑣, 𝔩 denotes −Ω³_𝑣cos𝜃_𝑣, 𝔪 denotes cos𝜃_𝑣, 𝔫 denotes −Ω_𝑣sin𝜃_𝑣, 𝔬 denotes −Ω²_𝑣cos𝜃_𝑣, 𝔭 denotes Ω³_𝑣sin𝜃_𝑣, 𝔮 denotes sinh𝜃_𝑣, 𝔯 denotes Ω_𝑣cosh𝜃_𝑣, 𝔰 denotes Ω²_𝑣sinh𝜃_𝑣, 𝔱 denotes Ω³_𝑣cosh𝜃_𝑣, 𝔲 denotes cosh𝜃_𝑣, 𝔳 denotes Ω_𝑣sinh𝜃_𝑣, 𝔴 denotes Ω²_𝑣cosh𝜃_𝑣, 𝔵 denotes Ω³_𝑣sinh𝜃_𝑣, 𝔶 denotes −sin𝜃_𝑣+1, 𝔷 denotes −Ω_𝑣+1cos𝜃_𝑣+1, 𝕫 denotes Ω²_𝑣+1sin𝜃_𝑣+1, 𝕪 denotes Ω³_𝑣+1cos𝜃_𝑣+1, 𝕩 denotes −cos𝜃_𝑣+1, 𝕨 denotes Ω_𝑣+1sin𝜃_𝑣+1, 𝕧 denotes Ω²_𝑣+1cos𝜃_𝑣+1, 𝕦 denotes −Ω³_𝑣+1sin𝜃_𝑣+1, 𝕥 denotes −sinh𝜃_𝑣+1, 𝔰 denotes −Ω_𝑣+1cosh𝜃_𝑣+1, 𝕣 denotes −Ω²_𝑣+1sinh𝜃_𝑣+1, 𝕢 denotes −Ω³_𝑣+1cosh𝜃_𝑣+1, 𝕡 denotes −cosh𝜃_𝑣+1, 𝕠 denotes −Ω_𝑣+1sinh𝜃_𝑣+1, 𝕟 denotes −Ω²_𝑣+1cosh𝜃_𝑣+1, and 𝕞 denotes −Ω³_𝑣+1sinh𝜃_𝑣+1.

C.

Consider the following: 𝐵_𝑁=𝔸𝔹ℂ𝔻𝔼𝔽𝔾ℍ4𝑁−34𝑁−24𝑁−14𝑁_𝑞𝑞−1𝛿1=−Ω²_𝑁−𝑀^∗_𝑒𝑒^∗_𝑚Ω⁴_𝑁+𝑘^∗_Te𝑒^∗_𝑘,𝛿2=Ω²_𝑁−𝑀^∗_𝑒𝑒^∗_𝑚Ω⁴_𝑁+𝑘^∗_Te𝑒^∗_𝑘,𝑘=𝑀^∗_𝑒Ω⁴_𝑁−𝑘^∗_Te,𝜂=𝐽^∗_𝑒Ω⁵_𝑁−𝐾^∗_ReΩ_𝑁+𝑀^∗_𝑒𝑒^∗2_𝑚Ω⁵_𝑁−𝑘^∗_Te𝑒^∗2_𝑘Ω₂,𝜓1=𝑀^∗_𝑒𝑒^∗_𝑚Ω⁵_𝑁−𝑘^∗_Te𝑒^∗_𝑘Ω_𝑁−Ω³_𝑁,𝜓2=𝑀^∗_𝑒𝑒^∗_𝑚Ω⁵_𝑁−𝑘^∗_Te𝑒^∗_𝑘Ω_𝑁+Ω³_𝑁, (C.1)(C.2)(C.3)(C.4)(C.5)(C.6)where 𝔸 denotes (𝛿1sin𝜃_𝑁+𝜂cos𝜃_𝑁), 𝔹 denotes (𝜅sin𝜃_𝑁+𝜓1cos𝜃_𝑁), ℂ denotes (𝛿1cos𝜃_𝑁−𝜂sin𝜃_𝑁), 𝔻 denotes (𝜅cos𝜃_𝑁−𝜓1sin𝜃_𝑁), 𝔼 denotes (𝛿2sinh𝜃_𝑁+𝜂cosh𝜃_𝑁), 𝔽 denotes (𝜅sinh𝜃_𝑁+𝜓2cosh𝜃_𝑁), 𝔾 denotes (𝛿2cosh𝜃_𝑁+𝜂sinh𝜃_𝑁), and ℍ denotes (𝜅cosh𝜃_𝑁+𝜓2sinh𝜃_𝑁).