Trans RINA, Vol 161, Part A4, Intl J Maritime Eng, Oct-Dec 2019


+= 


f g A+ i  2


m tx


+ sin ( + ) 0


mi i Uz (in z-direction) (1b)


In terms of the pipe element, similarly, the force equilibrium equations are concluded as follows:


xx   


Ta and


Q z x


+Tap x x x





+ − −m +0  


 i


SF x


 z z


r tt s


p 


Considering Q Mz, where EI is the flexural 


xx = − = −


3 EI 3


stiffness, the governing equation for the vibration of inclined fluid-transporting submarine pipelines is derived based on Equations (1) and (2): 42


EI42 p x


 z


+(miU + −T m m gi ) cos x


2


+2miU xt (


 + + + + + f )


 z p


22 (r r m mi me )


PAi a z


s − t ( e egA sin p + +m mi g) sin   F= L t )  z + + (


 z 2


z x


(3) (in z-direction)


 z 2


w = (2b)    2 + Ta z


+ −f gsin 2


m 2


 +− +


2  2 ff =  q( 1) 22 q txxt C =


qq qF tt


(4) The dimensionless wake variable q (as shown in Figure 3)


is related to the fluctuating lift coefficient CL on the structure, i.e.


forcing term


F az D t


=  


2 2 simulates the effects of the pipe motion on the near wake. 2/f= StV D denotes the


vortex-shedding angular frequency, where St is the Strouhal number. The values of the van der Pol parameter ε and the coupling force scaling parameter A can also be gained through experiments.


( , ) 2C ( , ) /LL0 , where CL0 is the


reference lift coefficient which can be obtained from experiments. On the right-hand side of Equation (4), the


 (in x-direction)


+ +m Sipgcos − = 0 f z (2a)


  


P z  x x


+ AP Si x


 z 2 i 2 +


z x


According to Facchinetti el al. (2004), the structural damping can be calculated by


rm =  s 2 s ratio, and s


EI   mL


s =


simple-supported beam, it can be calculated through 2


4 (Clough & Penzien, 1975). 2.2 WAKE OSCILLATOR MODEL


In the present paper, a nonlinear oscillator equation is adopted to describe the fluid force acted on the structure by the current (Facchinetti, et al, 2004; Iwan, 1981), which is expressed as follows:


, where ζ is the damping  is the angular structural natural frequency. For a


APi


 


m t U z x


i() m U xt  i


x  2


 +


z


 x '


z' x


 


+ 1 2


 2 


zz x xx 


2 fx mgxcos i 


 


+


zz x xx 


2 2


Qx x


+  


AP A xx +


P ii  (a) Fluid element Figure 2: Forces and moments acting on the elements. m g xcos Q p 


Mx x 


+ M 


(b) Pipe element mgxcos i 


() 


r zz sp 2


tt  


2 +m x


Tx x 


a + Ta 


z x


i Sx


M Q Fx w i Sx fx  m g xsin p 


©2019: The Royal Institution of Naval Architects


A-323


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