Study on stability of cable-anchored underwater platform by parametric excitation

Abstract

Aiming at the stability of the heaving–pitching coupling motion of the cable-anchored underwater platform in the nonlinear flow field, considering the change of its initial equilibrium position under the flow field, a coupled heaving–pitching motion equation is established; the transition curve of stability at frequency ratios of 0, 0.5, and 1 was obtained by using the virtual mass method and deformation parameter method. According to the Floquet theory, the stability characteristics under weak parameter excitation are studied. The stability and instability regions were identified and verified by time history response. The results show that the range of the stability region can be changed by adjusting the damping of the system. With the increase of damping, the range of stability gradually increases. Therefore, it is possible to take measures such as adding the helical damping side plate, changing the cross-sectional shape of the platform floating body, and increasing the cross-sectional diameter to suppress the heaving displacement and pitching range of the cable-anchored underwater platform.

Keywords

Stability parametric excitation Mathieu equation cable-anchored under platform

Introduction

The cable-anchored underwater platform is mainly used for supporting platforms of river, lake security monitoring devices, and other equipment. It is installed parallel to the cross section of the river channel and the sonar and other monitoring devices installed on it which can detect and scan the security inspection area. It needs to work under a certain depth of water for a long time and continuously perform real-time detection and scanning of the ship bottom passing by to monitor the abnormal characteristics of the ship bottom to determine whether the bottom of the ship is carrying dangerous goods. In order to obtain stable monitoring signals, it is necessary that the cable-anchored underwater platform itself has stability under the action of the flow field. Therefore, this topic studies this issue. The cable-anchored underwater platform is a new type of the monitoring platform. Its own stability is very important. During the working process, it will be subjected to uncertain fluid loads from the external flow field, and the working environment is complicated. Under the influence of external complex environmental loads, large coupling motions of heaving and pitching will occur. Such large coupling motion will make it difficult to obtain accurate monitoring results and even cause the platform to overturn and instability. Therefore, it has great engineering and theoretical significance to study the stability of cable-anchored underwater platforms by the external complex environment.

Existing literature works have few studies on the new type of device. Considering the similarities between its structure and underwater suspension tunnels, various types of the offshore platforms, and wind power floating foundations in mooring methods, in the course of the subject research, similar research results were used as a reference. Literature works^1-4 flexibly dealt with the offshore wind power floating foundations, studied the coupling dynamics of leaf blades and towers, and further studied the variation of wind power efficiency with structural dynamic response. Literature works^5-10 studied the chattering of wind power foundations by stochastic dynamics methods and mainly studied the dynamic characteristics of structures under the action of random wind and pulsating wind. Literature works^11-13 studied transient response analysis for large-scale wind power foundations and obtained the change trend of kinematic characteristics of the tower under instantaneous wind load. The literature works^14-17 studied the inherent characteristics of the tower by changing the tower structure parameters and connection methods and obtained the relationship between the natural frequency order and the impeller operating speed when the tower is not resonant and optimized the tower structure parameters. Literature works^18-21 introduced the foliar theory to study the dynamic response of the tower under different wind speeds by numerical analysis and obtained the law of wind pressure change with tower height. The literature works ^22-25 studied the mutual coupling between the fan blades and the airflow field and studied the effect of the airflow field on the lift, resistance, and torque of the fan blades. Literature works^{26, 27} proposed a new method for dealing with gusts and grid coupling of fan towers. The changes in the boundary layer of the fan after loading were systematically analyzed. Literature²⁸ studied the dynamic characteristics of the wind turbine tower when considering both the hydrodynamic and aerodynamic damping and obtained the relationship between the system frequency and the damping coefficient. Literature²⁹ proposed a new tension leg wind power floating foundation. The improved gradient method was used to optimize the weight of the wind turbine tower, and the structural parameters were systematically designed. In literature³⁰, the finite element method and boundary element method were used to simulate the dynamic response of the upper rigid body of the wind turbine tower and the dynamic characteristics of the wind turbine under transient wind. The above research results are mainly related to the dynamic response and optimization design of the wind turbine tower structure under different external loads. At present, the research about the floating foundation of large-scale offshore wind power is mainly developed in a nonlinear direction. When studying the dynamic characteristics of the wind turbine tower, it is necessary to simultaneously study the nonlinear characteristics of the floating foundation under the combined action of wind force and flow field force; then study the causes of its period, period doubling and chaotic motion, and the stability of the fan structure caused by the change of nonlinear vibration displacement; and further study the change law of other kinetic parameters. This kind of literature articles has great significance for studying the dynamic characteristics and stability analysis of cable-anchored underwater platform. Literature³¹ studied the dynamic response and stability of the ocean spar platform model under different wave excitation through pool experiments. The research results show that when the wave excitation period and the platform heaving period are approximately close, the platform will appear as an instability phenomenon of heaving and pitching; the experimental results and the standard Matthew equation stability map have good stability. In literature³², a Chinese scholar, Fan Zheliang also conducted a study on the instability of the spar platform under the action of the mooring system tension and determined the stability and instability decomposition region. Literature³³ conducted experimental research on a new spar platform; the results show that when the platform pitching motion period and heaving period ratio is 2 times, there will be instability of pitching motion; during the instability process, the heaving mode energy is converted to the pitching mode energy, and the stability of the platform is improved by adding a damping device. Considering the structure, mooring method and flow field conditions of the cable-anchored underwater platform are similar to the spar platforms which are currently being studied; there are not lots of research results of the cable-anchored underwater platform. Therefore, using the research results of the abovementioned literature articles as reference, studying the stability of the cable-anchored underwater platform under nonlinear conditions and special conditions is of great significance for design and engineering application of such structures.

Heaving and pitching coupled motion equations

As shown in Figure 1, the composition of the cable-anchored underwater platform is composed of a floating body, some accessories, and mooring lines. The dynamic characteristics of the platform are affected not only by the water environment but also by its structural shape and materials. The environmental loads that have the greatest impact on the dynamic characteristics of the platform are the flow field velocity, direction, and the installation depth of the floating body. The main forces on the floating body in a certain depth of water are fluid force and resistance of flow. Among them, the flow resistance parallel to the floating body axis direction is the smallest, and the fluid force and flow resistance perpendicular to the floating body axis direction are the largest.³⁴ The midpoint of the floating body axis is the coordinate origin, the floating body axis is the X axis, the Y axis is perpendicular to the floating body axis and points to the water surface, and the positive direction of the flow field velocity is the Z axis direction. The rotation of the platform around the X axis is the pitching motion, and the Y direction motion is the heaving motion; when the platform heaves and pitches under the action of the flow field, the body coordinate system changes from O-XYZ to O^′-X^′Y^′Z^′. Its mechanical model is shown in Figure 2.

Figure 1.

Composition diagram of the cable-anchored underwater platform.

Figure 2.

Mechanical model diagram of the cable-anchored underwater platform.

For the cable-anchored underwater platform, the heaving motion and pitching motion are simplified to a function with only one harmonic term,³⁵ which can be expressed as

ξ_{3} (t) = ξ_{3} cos ω_{3} t

(1)

ξ_{5} (t) = ξ_{5} cos ω_{5} t

(2)

where

ξ_{3} (t)

is the heaving motion displacement,

ξ_{5} (t)

is the pitching angle displacement,

ξ_{3}

is the heaving displacement amplitude of the cable-anchored underwater platform,

ξ_{5}

is the heaving angle amplitude of the cable-anchored underwater platform,

ω_{3}

is the heaving motion frequency, and

ω_{5}

is the pitching motion frequency; the coupled governing equation of the heaving–pitching motion in the time domain can be established as below

(I + Δ I) {\ddot{ξ}}_{5} (t) + C_{55} {\dot{ξ}}_{5} (t) + [ρ g \nabla \bar{G M} - \frac{1}{2} ρ g (\nabla + 2 A_{W} \bar{G M}) ξ_{3} (t)] ξ_{5} (t) = 0

(3)

where I is the moment of inertia of pitching motion,

Δ I

is the added moment of inertia of pitching motion,

C_{55}

is the moment coefficient of resistance of pitching motion,

ρ

is the fluid density of the flow field,

\nabla

is the drainage volume of the platform,

\bar{G M}

is the distance from the initial equilibrium position of the platform to the surface of the flow field, and

A_{W}

is the longitudinal cross-sectional area of the platform. Equation (3) can clearly show the coupling of pitching motion and heaving motion, and

- 1 / 2 ρ g \nabla ξ_{3} (t)

shows the effect of heaving motion on pitching, which is equivalent to adding parametric excitation based on the pitching moment. Due to the existence of the additional excitation force, the energy generated by the heaving is input to the pitching motion, which may cause the pitching to vibrate greatly and even reach an instability state. Variable

τ = ω_{3} t

was introduced, and equation (3) can be simplified as

{\ddot{ξ}}_{5} + δ {\dot{ξ}}_{5} + (γ - β cos t) ξ_{5} = 0

(4)

where

γ = ρ g \nabla \bar{G M} / (I + Δ I) ω_{3}^{2} = ω_{5}^{2} / ω_{3}^{2}

;

δ = C_{55} / (I + Δ I) ω_{3}

and

β = ρ g (\nabla + 2 A_{w} \bar{G M}) ξ_{3} / (I + Δ I) ω_{3}^{2}

Mathieu equation with weak parameters

Because the floating body of the cable-anchored underwater platform is located at a certain depth of water, when the change of the heaving excitation amplitude $ξ_{3}$ is small, the approximate solution of equation (4) can be solved by the deformation parameter method,³⁶ assuming $β$ is approximately a small parameter, $γ$ can be expressed as follows

γ = m^{2} + β δ_{1} + β^{2} δ_{2} + \dots

(5)

Suppose that $δ$ can be expanded into the following algebraic form

δ = β δ_{1} + β^{2} δ_{2} + \dots

(6)

The solution of equation (4) with small parameters can be expressed as follows

ξ_{5} = ξ_{50} + β ξ_{51} + β^{2} ξ_{52} + \dots

(7)

Substituting equations (5), (6), and (7) into (4) and comparing the exponential of $β$ to get

{\ddot{ξ}}_{50} + m^{2} ξ_{50} = 0

(8)

{\ddot{ξ}}_{51} + m^{2} ξ_{51} = - δ_{1} {\ddot{ξ}}_{50} - δ_{1} ξ_{50} + ξ_{50} cos τ

(9)

{\ddot{ξ}}_{52} + m^{2} ξ_{52} = - δ_{2} {\ddot{ξ}}_{50} - δ_{1} {\dot{ξ}}_{51} - δ_{1} ξ_{51} - δ_{2} ξ_{50} + ξ_{51} cos τ

(10)

By calculating the values of $β$ and $γ$ and drawing them as plane curves with horizontal and vertical coordinates, the standard Mathieu equation has periodic solutions on the curve. The $γ - β$ plane is divided into two parts by the Mathieu equation curves, which are the stable region and the unstable region. The following three cases of $m = 0, 1, 1 / 2$ are analyzed.

When $m = 0$ , solving equations (6)–(8), we get

ξ_{50} = a (a is constant)

(11)

ξ_{51} = - a cos τ

(12)

{\dot{ξ}}_{52} = - δ_{1} a sin τ - σ_{2} a - \frac{a}{2} (1 + cos 2 τ)

(13)

For equation (13), let $σ_{2} = - (1 / 2)$ ; the long term can be eliminated. The periodic solution of equation (13) is

ξ_{52} = a (δ_{1} sin τ + \frac{1}{8} cos 2 τ)

(14)

The periodic solution $ξ_{5}$ of the pitching angle can be solved as

ξ_{5} = a - a ε cos τ + a (δ_{1} sin τ + \frac{1}{8} cos 2 τ) γ^{2}

(15)

The transition curve equation is

β = - \frac{1}{2} γ^{2}

(16)

2. When $m = 1$ , solving equation (8), we get

ξ_{50} = a cos τ + b sin τ (a, b is constant)

(17)

Substituting equation (17) into (9), we get

{\ddot{ξ}}_{51} + m^{2} ξ_{51} = (δ_{1} a - σ_{1} b) sin τ - (σ_{1} a - δ_{1} b) cos τ + \frac{a}{2} + \frac{1}{2} (a cos 2 τ + b sin 2 τ)

(18)

Eliminating the long term of equation (18), the formula can be obtained as $δ_{1}^{2} + σ_{1}^{2} = 0$ , $δ_{1} = σ_{1} = 0$ , and solving equation (18), we get

ξ_{51} = \frac{a}{2} - \frac{1}{6} (a cos 2 τ + b sin 2 τ)

(19)

Substituting equations (18) and (19) into (10), we get

ξ_{5} = a cos τ + b sin τ + γ [\frac{a}{2} - \frac{1}{6} (cos 2 τ + b sin 2 τ)] - γ^{2} {\begin{cases} \frac{1}{18} [(σ_{1} a + 2 δ_{1} b) cos 2 τ + (2 δ_{1} a - σ_{1} b) sin 2 τ] \\ + \frac{σ_{1}}{2} a - \frac{1}{96} (a cos 3 τ + b sin 3 τ) \end{cases}}

(20)

The transition curve is

\frac{5}{12} γ^{2} = - 2 (β - 1) \pm \sqrt{9 {(β - 1)}^{2} + 5 δ^{2}}

(21)

3. When $m = 1 / 2$ , solving equation (8), we get

ξ_{50} = a cos \frac{τ}{2} + b sin \frac{τ}{2} (a, b is constant)

(22)

Solving equation (9) yields

ξ_{51} = - \frac{1}{4} (a cos \frac{3}{2} τ + b sin \frac{3}{2} τ)

(23)

Solving equation (10), eliminating the long term, and achieving the precision of $γ^{2}$ , the boundary equation can be obtained as

γ^{2} = (10 - 8 β) \pm 8 \sqrt{\frac{3}{2} - \frac{1}{4} δ^{2} - 2 β}

(24)

ξ_{5} = a cos τ + b sin τ - \frac{1}{4} γ (a cos \frac{3}{2} τ + b sin \frac{3}{2} τ) - \frac{1}{16} γ^{2} [\begin{array}{l} (2 σ_{1} a + 3 δ_{1} b) cos \frac{3}{2} τ - \frac{1}{16} (3 δ_{1} a + 2 σ_{1} b) sin \frac{3}{2} τ \\ + \frac{1}{48} (a cos \frac{5}{2} τ + b sin \frac{5}{2} τ) \end{array}]

(25)

Design parameters

The design parameters of the cable-anchored underwater platform are shown in Table 1.

Table 1.

Design variables.

Item	Value
Floating body length $(m)$	50
Floating body diameter $(m)$	0.2
Mooring diameter $(m)$	0.006
Inertial force coefficient	1
Flow field depth $(m)$	63.2
Distance from the center of float to fluid surface $(m)$	10
Viscous drag coefficient	0.4
Flow field density $(kg / m^{3})$	$1 \times 10^{3}$
Average velocity of flow field at floating body $(m / s)$	2.52
Lift coefficient	0.07
Floating weight $(kg)$	365.75
Floating body elastic modulus $(Mpa)$	17500
Dimensionless damping ratio	0.01

Natural frequencies of heaving and pitching

Model establishment

In order to carry out the stability analysis and time history response research under the weak parameter excitation of the cable-anchored underwater platform, it is necessary to solve its natural frequency of heaving and pitching to determine the value of the parameter. Using the fluid–structure interaction analysis method, the force of the flow field is loaded onto the model by the principle of the virtual mass method. Considering the difficulty of solving, a finite element line surface model of the cable-anchored underwater platform is established. According to the design parameters and requirements, the mooring buoy adopts plate and shell elements; considering the mechanical properties of the mooring cables under tension and torsion, the cable elements are used for simulation. The fluid–structure interaction finite element line surface model is shown in Figure 3.

Figure 3.

Finite element line surface model of cable-anchored underwater platform.

Fluid–structure interaction calculation

Only when the mutual coupling between the platform and the flow field is defined, the platform motion and the flow field motion can be transferred to each other. The surface of the platform buoy is taken as the coupling surface, and the coupling surface is the transmission medium of the interaction force between the flow field and the platform. In the finite element software solution process, it is necessary to ensure that the coupling surface is closed. The normal direction of the coupling surface points to the outside of the flow field, and the platform coupling surface is located in the flow field grid.³⁷ The definition of the coupling surface is shown in Figure 4. The virtual mass method is used to solve the eigenvalues of the cable-anchored underwater platform at a certain water depth; the calculation results and mode shapes of the platform’s heaving and pitching modes under 10 m depth are shown in Figures 5 and 6.

Figure 4.

Coupling surface normal direction.

Figure 5.

Heaving modes of the cable-anchored underwater platform.

Figure 6.

Pitching modes of the cable-anchored underwater platform.

The natural frequency of heaving for the cable-anchored underwater platform in 10 m depth flow field is 0.0235 Hz, and the natural frequency of pitching is 0.0079 Hz.

Stability analysis

Stable region solution

Equation (4) is a linear differential equation with the periodic coefficients. According to the Floquet theory of linear differential equations with periodic coefficients, the boundary equation curve of the function divides the function plane into two regions, namely, the stable region and the unstable region.³⁸ According to the excessive curve equation, equation (4) is a standard damped Mathieu equation. The stability of the Mathieu equation is determined by the parameters $γ$ and $β$ .³⁹ When $δ = 0$ , equation (4) is an undamped Mathieu equation. The undamped and different damped transition curves are plotted, respectively, and the undamped stable and unstable regions are obtained. Figure 7 is the undamped transition curve, and Figure 8 is the damping transition curve with different values.

Figure 7.

Undamped stability map.

Figure 8.

Different damping stability map.

It can be seen from Figure 7 that the stability map of the undamped Mathieu equation of the cable-anchored underwater platform is surrounded by two curves. The inside of the curve is the unstable region, and the outside of the curve is the stable region. When the system damping ratio is changed, it can be seen from Figure 8 that with the increase of the damping, the instability range of the area surrounded by the transition equation curve gradually decreases, and the stability range gradually increases. Therefore, the range of heaving displacement and pitching angle of the cable-anchored underwater platform can be controlled by changing the damping, such as adding a spiral damping side plate outside the cable-anchored underwater platform, changing the cross-sectional shape of the platform floating body, and increasing the cross-sectional diameter.

Time history analysis

In order to verify the stability analysis, points A and B were taken in the stable area and the unstable area, respectively, to analyze the pitching time history of the cable-anchored underwater platform.

It can be seen from Figure 9 and Figure 10 that the stability analysis method of the Mathieu equation is adopted to analyze the stability of the cable-anchored underwater platform under weak parameter excitation, and the stable region and unstable region can be obtained. The time history analysis of different points in the stable region and unstable region shows that the motion response of the cable-anchored underwater platform in the stable region under the action of a random flow field can obtain a stable periodic solution, and the motion response under the flow field cannot obtain a stable periodic solution. Therefore, the stability of the platform parameter excited pitching motion can be studied under a large number of parameter combinations, and the stability solution is also verified.

Figure 9.

Point A’s stability time history.

Figure 10.

Point B’s instability time history.

Conclusion

Through the deformation parameter method, the transition curve and the second-order approximate solution of the cable-anchored underwater platform can be obtained when the frequency ratio is 0, 1, and 0.5, respectively; then, to study its stability under the weak parameter excitation by the Floquet theory, the stability region and instability region can be determined.

Based on the finite element method and virtual mass method of structural vibration modes in the flow field, the fluid–solid coupling finite element model by considering the action of the flow field is established, and the heaving and pitching model calculations under the action of the flow field are realized.

The range of the stability area can be adjusted by changing the damping of the system. With the increase of damping, the instability range of the area enclosed by the transition equation curve gradually decreases and the stability range gradually increases. The damping is used to control the range of heaving displacement and pitching angle of the cable-anchored underwater platform, such as adding a spiral damping side plate outside the cable-anchored underwater platform, changing the cross-sectional shape of the platform floating body, and increasing the cross-sectional diameter.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study is support by National Natural Science Foundation of China (51775307; 51875314) and Yichang Key Laboratory of Robot and Intelligent System, China Three Gorges University, Yichang 443002, China (JXYC00003). Hubei Key Laboratory of Hydroelectric Machinery Design & Maintenance, China Three Gorges University, Yichang 443002, China: (2019KJX04, 2018KJX09).

ORCID iD

Kongde He

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