Effectiveness of the segment method in absolute and joint coordinates when modelling risers
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Abstract
This paper presents two formulations of the segment method: one with absolute coordinates and the second with joint coordinates. The nonlinear equations of motion of slender links are derived from the Lagrange equations by means of the methods used in multibody systems. Values of forces and moments acting in the connections between the segments are defined using a new and unique procedure which enables the mutual interaction of bending and torsion to be considered. The models take into account the influence of the velocity of the internal fluid flow on the riser’s dynamics. The dynamic analysis of a riser with fluid flow requires calculation of the curvature by approximation of the Euler angles with polynomials of the second order. The influence of the sea environment, such as added mass of water, drag and buoyancy forces as well as sea current, is considered. In addition, the influence of torsion is discussed. Validation is carried out for both models by comparing the authors’ own results with those obtained from experimental measurements presented in the literature and from COMSOL, Riflex and Abaqus software. The validation is concerned with vibrations of cables and the riser with internal fluid flow as well as with frequencies of free and forced vibrations of a riser fully or partially submerged in water. The numerical effectiveness of both formulations is examined for dynamic analysis of the riser, whose top end is moving in a horizontal plane. Conclusions concerned with the effectiveness of both formulations of the segment method and the influence of torsional vibrations on numerical results are formulated.
1 Introduction
The segment method is, in addition to the yet more popular finite element method, one of the discretisation methods for continuous systems used very widely in modelling systems with slender links. Initially, the segment method (SM) was developed by Huston and coauthors for modelling lines and cables and for analysing large deflections of slender links. Winget and Huston in [1] present a nonlinear, spatial model of a cable or chain, in which the continuous system is replaced by a series of links connected to each other by ballandsocket joints. Closed kinematic subchains are left out of consideration. The position of each link is defined in relation to the preceding link. Hydrodynamic forces are included in the equations of motion. The calculations are carried out for a rotary crane with a payload hanging on a cable partly submerged in water. The extension of the method to systems with treelike structures, still without closed subchains, is presented in [2]. In addition to relative angles, Euler parameters with relative angular velocities are also introduced for the description of system kinematics. The equations of motion are developed on the basis of the Lagrange form of d’Alembert’s principle. The method in these formulations does not account for the flexibility of cables. Flexibility of the system is taken into account in [3] by introducing three rotational and three translational springs at the connection point between the segments. The author considers small relative displacements which enable the interacting forces in the connection between two segments to be replaced by one force and one torque.
Paper [4] presents the generalisation and recapitulation of the results presented in [1, 2, 3], namely calculation of stiffness coefficients of rotary springs as well as derivation of the equations of motion, while the continuity of displacements is ensured. An extensive reference list (158 items) concerned with modelling of rigid and flexible multibody systems is included; however, no calculation example, as in [2, 3], is given.
The validation of the elaborated models is presented in [5] by using as an example a multilink pendulum with a lumped mass at its end. The authors compare their own results with an analytical solution for vibrations with small amplitudes and frequencies of free vibrations. In addition, results are compared with experimental measurements and with SEADYN computer simulation for the displacements of the end of the cable with a spherical body attached and submerged in water. All examples deal with planar models.
A computer program based on the formulations of the segment method was developed for spatial dynamic analysis of bodies and towing or hoisting cables partially or fully submerged in water [1, 2, 3, 4, 5].
Analysis of the influence of the water environment with special attention to hydrodynamic forces such as buoyancy, inertial forces and drag forces is discussed in [6]. Two cases are considered: buoy release and anchor drop. The discussion is limited to planar cases.
The application of the segment method to modelling of flexible systems is presented in [7, 8]. The first paper is concerned with theoretical aspects of the method and introduces springs connecting the segments together with a description of stiffness coefficient calculations. The second presents numerical examples and describes planar dynamics of a clamped and rotating beam. A large difference between the results obtained with those obtained using FEM can be seen; for the rotating beam, the differences are about 14%.
Problems connected with winding and unwinding cables with variable lengths are analysed in [9, 10]. Payout/reelin phenomena occurring in many offshore systems are a difficult problem because they require changeable mass and length of segments to be taken into account. In order to eliminate problems with calculation of moments of inertia of the segments with varied lengths, the authors apply the lumped mass method and consider the motion of a cable towed by a vessel. Amirouch [11] describes a procedure for creating a structural matrix of connections of bodies in a treelike structure when the segment method for dynamic analysis of systems undergoing large base motion is applied. The author uses Kane’s equations presented by Huston and others and describes the advantages of the method for computer implementation especially when the equations of motion are formulated for flexible systems with small relative deflections, yet no numerical examples are presented.
The segment method, due to its simple physical interpretation and easy computer implementation, has also been applied in modelling beams [12, 13], cranes [14], robot manipulators [15], railroad track structures [16, 17] and compliant mechanisms [18]. A variant of the SM called the flexible segment method (FSM) is presented in [19], and its application to dynamic analysis of risers is described in [20, 21].
It is important to note that also a different definition of cables is presented in the literature. Huston and coauthors in papers [1, 2, 3, 4, 5, 6, 7, 8, 9, 10] treat cables as elements with bending and torsional flexibility. Shear and longitudinal flexibility are very rarely considered. A different definition of cables is used in the review paper [22], where cables are treated as elements consisting of helically twisted strands around the core wire. Cables considered in [22] are categorised according to their geometry, the method of considering friction among the wires and the way of formulating equilibrium equations. Three basic groups of models are distinguished. The first group consists of thinrod models, initially without bending and torsional flexibilities [23, 24], which finally have been included [25, 26] by modelling individual wires as thin rods. Semicontinuous models [27, 28, 29], in which each layer of a twisted wire is modelled as an orthotropic cylinder, form the second group of models. The third includes beam models [30, 31, 32], in which solid beam theory is used in order to formulate the equilibrium equations. A review of research concerned with this subject can be found in [33, 34]. Large deflections of cables are considered in [35, 36]. The formulations of the segment method presented in our paper are closest to the third group of models discussed in [22].
Along with the development of the finite segment method, the rigid finite element method (RFEM) similar in approach has been developed in Poland [37, 38]. In both methods (SM and RFEM), a continuous system is replaced by a discrete system of rigid bodies, yet the methods differ above all in their origin and formalism. According to [1, 2], the segment method was initially used for analysis of large deflections of cables and ropes without analysis of flexibility effects. By contrast, the rigid finite element method from the very beginning was formulated for flexible systems with spring–damping elements, although the first research was concerned with small vibrations about the static equilibrium. The classical formulation assumes that each rigid element has six degrees of freedom described by absolute displacements (translational and rotational). The rigid elements are connected by means of three translational springs (limiting longitudinal and transverse displacements) and three rotational springs (limiting relative rotations). Thus, unlike in SM, RFEM does not ensure the continuity of displacements.
Large base motion is introduced in [39, 40] but only for planar systems, while the respective formulation for spatial systems is presented in [41, 42, 43, 44, 45]. The formulations described in these papers as well as in [46] also include a modification of the method which ensures the continuity of displacements. This modification can be considered as another formulation of the segment method different in formalism from that presented by Huston and his coworkers, since it uses Lagrange equations of the second order and homogenous transformations.
The rigid finite element method and its modified formulations are applied to dynamic analysis of a wide range of machines and mechanisms including manipulators [41, 42, 45, 47], beams [43, 44], cranes [48, 49], vehicles [50] and band saws [51].
RFEM formulated in joint coordinates is also applied by Szczotka [52, 53, 54, 55] for dynamic analysis of offshore pipeline installations with special emphasis on the following installation techniques: Slay, Jlay and reel lay.
Modification of the RFEM formulated in absolute coordinates, which takes into account, apart from bending and torsional flexibilities (like SM), also longitudinal flexibility, is applied to analysis of forced vibrations of risers in [56, 57, 58, 59, 60]. Paper [61] presents applications of the method to dynamic analysis of risers and a rotating flexible beam.
3D formulation of the segment method in joint coordinates, with consideration of bending flexibility when torsion is omitted, is presented in [62]. Numerical effectiveness of this formalism enabled the method to be applied to the solution of a dynamic optimisation problem of stabilisation of forces in the connection of a riser with a wellhead.

Both formulations of the SM consistently use ZYX Euler angles measured in the global coordinate system. Formulations presented by other authors usually use relative angles defining the deflections with respect to the preceding segment.

The equations of motion are derived from the Lagrange equations of the second order. The kinetic energy of the flexible link when absolute coordinates are used is formulated by homogenous transformations, similarly to the RFEM [46].

The formulation of an original method for calculating moments in the connection of segments reflecting bending and torsion differs from previously published approaches. A special expression for the potential energy of deformation of rotary springs is formulated, which enables mutual interaction of torsion and bending to be considered. The presented formulation enables static and dynamic analysis to be carried out when large differences between the angles of segments occur. It should also be noted that the formulae defining bending and torsional moments are identical for both joint and absolute coordinates.

Formulae for consideration of the influence of the sea environment based on Morison equations are given in a compact form for both formulations of SM.

An algorithm enabling consideration of the influence of internal fluid flow on equations of motion and vibration of a riser is developed. Because of the step change of inclination angles of segments with respect to local or global reference systems (lack of continuity of the angles’ first derivative), special approximation curves are proposed in order to define the curvature of the riser.
Moreover, the numerical effectiveness of both formulations of the segment method in riser modelling is compared, and conclusions are formulated.
2 Segment method—geometry of the system
The main idea of the segment method consists in replacing the continuous system with a finite number of rigid bodies—segments connected by means of rotary joints. Mass (inertial) features of the discretised link are assumed by rigid segments, while spring–damping elements are responsible for flexibility and damping. Depending on how the position of a chosen segment is described and thus on the choice of the generalised coordinates, two approaches to the description of the motion of the segment can be distinguished: based on absolute or joint coordinates.
The angles \(\psi _i =\varphi _{i,3},\theta _i =\varphi _{i,2},\gamma _i =\varphi _{i,1} \) called yaw, pitch and roll [63] describe rotations of the ith segment about axes of system \({x}^{\prime }{y}^{\prime }{z}^{\prime }\) as follows. The segment i rotates by the angle \(\psi _i \) about the axis \({z}^{\prime }\), and thus, the system assigned to the segment assumes the position \({x^{\prime \prime }}{y}^{\prime \prime }{z}^{\prime }\). Such a system is rotated about the axis \({y}^{\prime \prime }\) by the angle \(\theta _i \) assuming the position \({x}^{\prime \prime \prime }{y}^{\prime \prime }{z}^{\prime \prime \prime }\). The last rotation by the angle \(\gamma _i \) about the axis \({x}^{\prime \prime \prime }={x}_{i}^{\prime }\) leads to the system associated with segment i with axes \({x}_i^{\prime } {y}_i^{\prime } {z}_i^{\prime }\). The position of the beginning of the segment, according to the choice of the generalised coordinates, is defined either by the coordinates \(x_i,y_i,z_i\) of the origin of the system \({x}_i^{\prime } {y}_i^{\prime } {z}_i^{\prime } \) with respect to the global coordinate system (for absolute coordinates), or depends on coordinates of the preceding segments when joint coordinates are chosen. Continuity of displacements at connecting points of the segments is required, and thus, when absolute coordinates are used, reactions in these connections are externalised and respective constraint equations are formulated. This is unnecessary when joint coordinates are used.
2.1 Absolute coordinates
2.2 Joint coordinates
3 Lagrange operators, kinetic energy
In order to formulate the equations of motion, the kinetic and potential energies of the flexible link have to be defined, the dissipation function when damping is considered has to be formulated, and generalised forces resulting from gravity and external forces together with those reflecting the sea environment when the link is fully or partially submerged in the sea have to be introduced.
Form (10a) will be used when the kinetic energy is calculated in joint coordinates and (10b) when absolute coordinates are used.
Thus, the respective transformations are presented separately for each type of generalised coordinates.
3.1 Absolute coordinates
The calculation of elements of the matrices \(\mathbf{A}_i \left( {\mathbf{q}_{A,i} } \right) \) and vectors \(\mathbf{h}_{A,i} \left( \mathbf{q}_{A,i} ,{\dot{\mathbf{q}}}_{A,i} \right) \) according to formula (13) can be easily algorithmised. Many of the elements are equal to zero.
3.2 Joint coordinates
4 Energy of spring deformation
It is important to stress that in both formulations of the segment method the expressions for the derivatives of energy of rotational springs have identical form (26) although their assignment to the respective equations of motion depends on the choice of the generalised coordinates.
 A.Segment 0$$\begin{aligned} \mathbf {F}_{s,0}^{\left( A \right) } =\left[ {{\begin{array}{c} 0 \\ {{\overline{\mathbf {D}}}_1 {\overline{\mathbf {M}}}_1 } \\ \end{array} }} \right] \end{aligned}$$(27a)
 B.Segments \(1\ldots n1\)$$\begin{aligned} \mathbf {F}_{s,i}^{\left( A \right) } =\left[ {{\begin{array}{c} 0 \\ {\overline{\overline{\mathbf {D}}}_i {\overline{\mathbf {M}}}_i +{\overline{\mathbf {D}}}_{i+1} {\overline{\mathbf {M}}}_{i+1} } \\ \end{array} }} \right] \end{aligned}$$(27b)
 C.
Segment n
 A.Segment 0$$\begin{aligned} \mathbf {F}_{s,0}^{(J)} ={\overline{\mathbf {D}}}_1 {\overline{\mathbf {M}}}_1 \end{aligned}$$(28a)
 B.Segments \(1\ldots n1\)$$\begin{aligned} \mathbf {F}_{s,i}^{(J)} ={\overline{\overline{\mathbf {D}}}}_i {\overline{\mathbf {M}}}_i+{\overline{\mathbf {D}}}_{i+1} {\overline{\mathbf {M}}}_{i+1} \end{aligned}$$(28b)
 C.Segment n$$\begin{aligned} \mathbf {F}_{s,n}^{(J)} = {\overline{\overline{\mathbf {D}}}}_n {\overline{\mathbf {M}}}_n \end{aligned}$$(28c)
5 Gravity and buoyancy forces
 A.when absolute coordinates are used:where \(\mathbf{G}_{c,i} =\left[ {{\begin{array}{lll} {{{\varvec{\updelta }}}_2^T \mathbf{R}_{i,1} \mathbf{r}_{c,i}^{\prime } } \\ {{{\varvec{\updelta }}}_2^T \mathbf{R}_{i,2} \mathbf{r}_{c,i}^{\prime } } \\ {{{\varvec{\updelta }}}_2^T \mathbf{R}_{i,3} \mathbf{r}_{c,i}^{\prime } } \\ \end{array} }} \right] \);$$\begin{aligned} \frac{\partial V_{g,i} }{\partial \mathbf{q}_{A,i} }={\overline{m}}_i g\left[ {{\begin{array}{l} {\mathbf{\delta }_2 } \\ {\mathbf{G}_{c,i} } \\ , \end{array} }} \right] =\mathbf{Q}_{A,i}^{(g)}, \end{aligned}$$(32)
 B.
when joint coordinates are used:
Thus, in case A due to gravity and buoyancy forces, the vector \(\mathbf{Q}_{A,i}^{(g)}\) occurs on the right side of the Lagrange equations of segment i, while in the case of joint coordinates on the right side of the equations corresponding to \(\mathbf{r}_0 \) there is a component \(\mathbf{Q}_{J,i,\mathbf{r}_0 }^{(g)}\), but in the equations corresponding to the segments \(0\ldots i\), there are vectors \(\mathbf{Q}_{J,i,{\varvec{\upvarphi }}_0 }^{(g)} \),..., \(\mathbf{Q}_{J,i,{\varvec{\upvarphi }}_j }^{(g)} \),..., \(\mathbf{Q}_{J,i,{\varvec{\upvarphi }}_i }^{(g)} \), respectively.
6 Drag forces
6.1 Hydrodynamic drag in absolute coordinates
This vector will be included in the right side of the equations of motion of segment i.
6.2 Hydrodynamic drag in joint coordinates
 A.
Vector \(\mathbf{Q}_{i,\mathbf{r}_0 }^{(H)} \) from (42a) in equations corresponding to \(\mathbf{r}_0 \).
 B.
Vectors \(\mathbf{Q}_{i,{\varvec{\upvarphi }}_k }^{(H)} \), whose components are defined by (42b), in equations corresponding to \({\varvec{\upvarphi }}_k\) (\(k=0,{\ldots },i1\)).
 B.
Vectors \(\mathbf{Q}_{i,{\varvec{\upvarphi }}_i }^{(H)} \), defined by (42c), in equations corresponding to \({\varvec{\upvarphi }}_i \).
6.3 Internal fluid flow in the riser
7 Constraints and synthesis of equations
Applications of the SM which will be presented in further sections require constraint equations reflecting the connection of both ends of the riser with the environment to be formulated. Moreover, when absolute coordinates are used, constraints have to be imposed on the connections between segments. The procedure that introduces constraint reactions into equations and formulates constraint equations for both types of coordinates is described below.
7.1 Absolute coordinates
7.2 Joint coordinates
In this case, constraint equations refer only to points \(A_0\) and \(A_{n+1} \) because the choice of generalised coordinates ensures continuity of displacements at points \(A_1,{\ldots },A_n\).
8 Model validation
Both formulations of the segment method should give identical numerical results for the same data. The numerical experiments carried out confirm this compatibility. Some small differences may occur when coefficients for the constraint stabilisation method are not chosen carefully enough (for absolute coordinates). Calculation times are different for simulations of identical problems of dynamics of risers, and the next section is devoted to these problems.
Validation is carried out in three steps: first, the results of the authors’ own simulations dealing with forced vibrations of a cable (beam) without consideration of the sea environment are compared with the results obtained by means of the software Abaqus. The next two steps of validation concerned with free vibrations of the riser with internal fluid flow are compared with those obtained by means of the finite element method (COMSOL program) presented in [64], and subsequently, the frequencies of free and forced vibrations are compared with the experimental measurements and calculations by means of the finite element method (RIFLEX program) presented in [65].
8.1 Vibrations of a cable
Parameters of the beam
Notation  Unit  Value  

Length  L  m  20 
Outer diameter  \(D_\mathrm{out} \)  m  0.1 
Inner diameter  \(D_\mathrm{inn} \)  m  0.09 
Material density  \(\rho \)  kg/m\(^{3}\)  7850 
Young’s modulus  E  N/m\(^{2}\)  \(2.07\times 10^{1}\) 
The results presented in Fig. 12 show the influence of the number of segments into which the beam is discretised. It is important to note that already for \(n=10\) the results obtained differ from those obtained for \(n=50\) only by 1%. Differences between both formulations of SM are negligible.
Parameters of the line
Notation  Unit  Value  

Length  L  m  200 
Outer diameter  \(D_\mathrm{out} \)  m  0.03 
Material density  \(\rho \)  kg/m\(^{3}\)  6500 
Young’s modulus  E  N/m\(^{2}\)  \(1.0\times 10^{11}\) 
Comparison of times of calculations
n  J  A  

3  2  3  2  
10  \(0'10''\)  \(0'10''\)  \(0'10''\)  \(0'10''\) 
20  \(0'35''\)  \(0'25''\)  \(0'40''\)  \(0'35''\) 
30  \(1'10''\)  \(0'50''\)  \(1'15''\)  \(0'55''\) 
40  \(2'10''\)  \(1'30''\)  \(2'00''\)  \( 1'45''\) 
50  \(3'25''\)  \(2'30''\)  \(2'50''\)  \(2'40''\) 
60  \(5'00''\)  \(3'50''\)  \(4'00''\)  \(3'45''\) 
70  \(7'00''\)  \(5'35''\)  \(5'30''\)  \(5'10''\) 
80  \(9'35''\)  \(7'40''\)  \(7'10''\)  \(6'40''\) 
90  \(12'35''\)  \(10'20''\)  \(9'20''\)  \(8'35''\) 
100  \(16'15''\)  \(13'35''\)  \(10'30''\)  \(9'35''\) 
Times of calculations for SM in absolute coordinates (“A”) for \(n\le 50\) are longer than in joint coordinates (“J”). However, for \(n>50\) there is a reverse dependency, which means that then SM in absolute coordinates is more effective numerically.
8.2 Frequencies of free vibrations with consideration of internal fluid flow
Parameters of the riser with internal fluid
Notation  Unit  Value  

Young’s modulus  E  N/m\(^{2}\)  \(2.1\times 10^{11}\) 
Length  L  m  150 
Outer diameter  \(D_\mathrm{out}\)  m  0.250 
Inner diameter  \(D_\mathrm{inn}\)  m  0.125 
Pipe density  \(\rho _r \)  kg/m\(^{3}\)  7800 
Water density  \(\rho _w \)  kg/m\(^{3}\)  1020 
Internal fluid density  \(\rho _f \)  kg/m\(^{3}\)  870 
Added mass coefficient  \(C_M \)  –  2.0 
Tangential drag coefficient  \(D_x \)  –  0.3 
Internal flow velocity  u  m/s  125.36 
Normal drag coefficient  \(D_{yz} \)  –  1.2 
Top tension coefficient  \(i_T \)  –  1–3 
Results presented in [49] are obtained for the riser divided into 1500 elements (the authors do not give information about the type of elements, but presumably either shell or solid elements are used), and our results are obtained by means of the segment method (SM) with discretisation into \(n=50\) elements. It is important to note that the differences do not exceed 0.1%.
Influence of n on first two frequencies of free vibrations
n  First frequency  Second frequency 

10  0.451303  1.083908 
25  0.449724  1.074758 
50  0.449483  1.073415 
75  0.449438  1.073165 
100  0.449422  1.073078 
125  0.449415  1.073038 
150  0.449411  1.073016 
[49]  0.449  1.073 
8.3 Comparison of the results with those from experimental measurements and FEM calculations
Parameters of the riser
Notation  Unit  Value  

Mass per unit length  \(m^{*}\)  kg/m  0.668 
Bending stiffness  EI  Nm\(^{2}\)  120 
Top tension  T  N  212 
Upper spring coefficient  \(k_u\)  N/m  1e6 
Bottom spring coefficient  \(k_b\)  N/m  3020 
Water and mud density  \(\rho _w,\rho _F \)  kg/m\(^{3}\)  1025 
Drag coefficient  \(C_D =D_{yz} \)  –  1.0 
Added mass coefficient  \(C_M \)  –  2.1 
Frequencies of free vibrations of the riser (Fig. 16a)
i  E  R  A  S  \(\varepsilon ({R})[\%]\)  \(\varepsilon ({A})[\%]\)  \(\varepsilon ({S})[\%]\) 

\(\omega _1 \)  0.646  0.648  0.662  0.642  0.3  2.5  0.6 
\(\omega _2 \)  1.477  1.445  1.448  1.434  2.2  2.0  2.9 
\(\omega _3 \)  2.619  2.503  2.448  2.488  4.4  6.5  5.0 
Differences between values \(\omega _1\), \(\omega _2\), \(\omega _3\) from experimental measurements and calculated with the segment method are not larger than 5%.
Frequencies of free vibrations of the riser from Fig. 16c
i  E  R  S  \(\varepsilon ({R}) [\%]\)  \(\varepsilon ({S}) [\%]\) 

\(\omega _1 \)  0.681  0.668  0.664  1.9  2.5 
\(\omega _2 \)  1.635  1.472  1.468  10.0  10.2 
\(\omega _3 \)  2.725  2.505  2.503  8.1  8.1 
It can be seen that the differences, as in the case of using the Riflex package, are smaller than 10.5%.
In summary, the validation process has proved that the results obtained during the analysis of both free and forced vibrations are compatible with experimental measurements and the calculation results presented in [66] and [67]. Formulation of the segment method in joint coordinates has been used, and in order to calculate frequencies of free vibrations a procedure presented in [60] has been applied. Calculations were carried out on a midlevel PC with a 3.6 GHz processor, and the programs were written in Delphi.
9 Numerical simulations and effectiveness of SM
Parameters of the riser
Parameter  Notation  Value  Unit 

Riser length  L  1500  m 
Outer diameter  \(D_\mathrm{out} \)  0.430  m 
Inner diameter  \(D_\mathrm{inn} \)  0.390  m 
Young’s modulus  E  2.07e11  N/m\(^{2}\) 
Kirchhoff modulus  G  0.83e11  N/m\(^{2}\) 
Riser density  \(\rho \)  7850  kg/m\(^{3}\) 
Water density  \(\rho _w \)  1025  kg/m\(^{3}\) 
Tangential drag coefficient  \(D_x \)  0.1  – 
Normal drag coefficient  \(D_{yz} \)  1.0  – 
Added mass coefficient  \(C_M \)  2.0  – 
Horizontal displacement  d  505  m 
Calculation times
n  J  A  

3  2  3  2  
10  \(0^{\prime }21^{\prime \prime }\)  \(0^{\prime }17^{\prime \prime }\)  \(0^{\prime }18^{\prime \prime }\)  \(0^{\prime }16^{\prime \prime }\) 
20  \(0^{\prime }58^{\prime \prime }\)  \(0^{\prime }45^{\prime \prime }\)  \(0^{\prime }47^{\prime \prime }\)  \(0^{\prime }41^{\prime \prime }\) 
40  \(3^{\prime }22^{\prime \prime }\)  \(2^{\prime }25^{\prime \prime }\)  \(2^{\prime }24^{\prime \prime }\)  \(1^{\prime }59^{\prime \prime }\) 
60  \(7^{\prime }48^{\prime \prime }\)  \(5^{\prime }19^{\prime \prime }\)  \(4^{\prime }59^{\prime \prime }\)  \(4^{\prime }01^{\prime \prime }\) 
80  \(14^{\prime }47^{\prime \prime }\)  \(9^{\prime }49^{\prime \prime }\)  \(10^{\prime }06^{\prime \prime }\)  \(6^{\prime }46^{\prime \prime }\) 
100  \(24^{\prime }49^{\prime \prime }\)  \(15^{\prime }58^{\prime \prime }\)  \(13^{\prime }33^{\prime \prime }\)  \(10^{\prime }24^{\prime \prime }\) 
Having analysed the results presented in Table 11 and Fig. 25, it can be seen that shorter calculation times for \(n\ge 20\) are obtained when absolute coordinates are chosen. This effect is even larger when torsion is omitted.
9.1 Final remarks
The paper presents two formulations of the segment method: by means of joint and absolute coordinates. Analysis of published research shows that the formulation which uses joint coordinates is more frequent; however, formulation of the segment method for dynamic analysis of risers is simpler when absolute coordinates are applied.
Validation of the methods presented in Sect. 8 confirms the correctness of the formulated algorithms. The frequencies of free vibrations of the riser with consideration of the internal fluid flow calculated by means of the segment method are closely compatible with those obtained using the finite element method. Even for \(n\ge 25\), the differences are less than 0.1%. Also, a comparison of the authors’ own results with those obtained from experimental measurement and calculations carried out with the Riflex software package shows that the differences in frequencies of free vibrations are less than 5% for the riser of 9 m and 10.5% for the riser of 9.5 m with a spring at its bottom end. Equally good compatibility of results is obtained when vibrations forced by the harmonic motion of the top end are analysed.
Both formulations differ in numerical effectiveness. The calculations presented in Sect. 9 show that shorter calculation times are required when absolute coordinates are used. When the number of segments \(n\ge 40\), the calculation time for joint coordinates is almost 50% longer than when absolute coordinates are used.
Differences in formulations of the SM (absolute vs joint coordinates)
Feature  Absolute coordinates  Joint coordinates 

Generalised coordinate vector of the segment  \(\mathbf{q}_{A,i} =\left[ {{\begin{array}{l} {\mathbf{r}_i } \\ {{\varvec{\upvarphi }}_i } \\ \end{array} }} \right] \)  \(\mathbf{q}_{J,i} ={\varvec{\upvarphi }}_i \) 
Generalised coordinate vector of the link  \(\mathbf{q}_A =\left[ {\mathbf{q}_{A,0}^T }\quad \ldots {\mathbf{q}_{A,i}^T }\quad \ldots \quad {\mathbf{q}_{A,n}^T } \right] ^\mathrm{T}\)  \(\mathbf{q}_J =\left[ {\mathbf{r}_0^T }\quad {\mathbf{q}_{J,0}^T }\quad \ldots {\mathbf{q}_{J,i}^T }\quad \ldots \quad {\mathbf{q}_{J,n}^T }\right] ^\mathrm{T}\) 
Number of generalised coordinates  \(N_A =6\left( {n+1} \right) \)  \(N_J =3+3\left( {n+1} \right) \) 
Transformation of coordinates to global coordinate system{ }  \(\mathbf{r}=\mathbf{r}_i +\mathbf{R}_i \mathbf{r}^{{\prime }}\)  \(\mathbf{r}_i =\mathbf{r}_0 +\mathop \sum \limits _{j=0}^{i1} \mathbf{R}_j \left[ {{\begin{array}{l} {l_j } \\ 0 \\ 0 \\ \end{array} }} \right] =\mathbf{r}_0 +\mathop \sum \limits _{j=0}^{i1} l_j {\varvec{\upphi }} _j \) 
Pseudoinertial matrix  Blockdiagonal  Full 
Equations of translational constraints (connection of the segments)  Superfluous  Necessary 
Numerical effectiveness (calculation time)  \(n\le 20\)  
+  +  
\(n>20\)  
−  + 
An important conclusion from the analysis of the numerical results is that when displacements and bending stress are analysed for typical risers, it is possible to leave torsion out of consideration. Neglecting torsion would result in further considerable shortening of calculation time. However, this can be done only when torsional vibrations are not important for fatigue analysis.

dimension of the mass matrix \(N_A =6\left( {n+1} \right) \) for SM/A and \(N_J =3+3\left( {n+1} \right) \) for SM/J. In order to calculate the vector \({{\ddot{\mathbf{q}}}}\), it is necessary to solve either \(N_A +3\left( {n+1} \right) \) or \(N_J \) linear algebraic equations at each integration step. Therefore, calculation times are considerably shorter when the smaller number of equations in SM/J is solved.

number of arithmetic operations connected with generation of mass matrices, constraint equations and the vector of right hand sides of equations. In this case, SM/A is superior especially when the sea environment is considered. This is due to the fact that forces acting at segment i (also inertial forces) in SM/J also influence all preceding segments and thus lengthen the time of generation of the equations of motion.
Whenever possible, it is more reasonable to apply formulation of the segment method in absolute coordinates and neglect torsion. A further reduction in calculation time can be expected if the specific form of the matrix \(\mathbf{D}\) (of reactions and constraints) is used in generating the equations of motion in absolute coordinates.
Notes
Compliance with ethical standards
Conflict of interest
The authors declare that they have no conflict of interest.
References
 1.Winget, J.M., Huston, R.L.: Cable dynamics—a finite segment approach. Comput. Struct. 6, 475–480 (1976)CrossRefGoogle Scholar
 2.Huston, R.L., Passarello, C.E., Harlow, M.W.: Dynamics of multirigidbody systems. J. Appl. Mech. 45, 889–894 (1978)CrossRefGoogle Scholar
 3.Huston, R.L.: Flexibility effects in multibody systems. Mech. Res. Commun. 7, 261–268 (1980)zbMATHCrossRefGoogle Scholar
 4.Huston, R.L.: Multibody dynamics including the effect of flexibility and compliance. Comput. Struct. 14, 443–451 (1981)CrossRefGoogle Scholar
 5.Huston, R.L., Kamman, J.W.: Validation of finite segment cable models. Comput. Struct. 15(6), 653–660 (1982)zbMATHCrossRefGoogle Scholar
 6.Kamman, J.W., Huston, R.L.: Advanced structural applications. Modelling of submerged cable dynamics. Comput. Struct. 20(1–2), 623–629 (1985)CrossRefGoogle Scholar
 7.Connely, J.D., Huston, R.L.: The dynamics of flexible multibody systems: a finite segment approach—I. Theoretical aspects. Comput. Struct. 50(2), 252–258 (1994)CrossRefGoogle Scholar
 8.Connely, J.D., Huston, R.L.: The dynamics of flexible multibody systems: a finite segment approach—II. Example problems. Comput. Struct. 50(2), 259–262 (1994)CrossRefGoogle Scholar
 9.Kamman, J.W., Huston, R.L.: Modeling of variable length towed and tethered cable structures. J. Guid. Control Dyn. 22(4), 602–608 (1999)CrossRefGoogle Scholar
 10.Kamman, J.W., Huston, R.L.: Multibody dynamics modelling of variable length cable systems. Multibody Syst. Dyn. 5, 21–221 (2001)zbMATHCrossRefGoogle Scholar
 11.Amirouche, M.L.: Dynamic analysis of ‘treelike’ structures undergoing large motions–a finite segment approach. Eng. Anal. 3(2), 111–117 (1986)CrossRefGoogle Scholar
 12.Yin, X., Du, S., Hu, J., Ding, J.: Analysis of dynamical buckling and post buckling for beams by finite segment method. Appl. Math. Mech. 26(9), 1181–1187 (2005)zbMATHCrossRefGoogle Scholar
 13.Neves, A.C., Simoes, F.M.F., Pinto da Costa, A.: Vibrations of cracked beams: discrete mass and stiffness models. Comput. Struct. 168, 68–77 (2016)CrossRefGoogle Scholar
 14.Bak, M.K., Hansen, M.R.: Analysis of offshore knuckle boom crane–part one: modeling and parameter identification. Model. Identif. Control 34(4), 157–174 (2013)CrossRefGoogle Scholar
 15.Li, H., Zhang, X.: A method for modelling flexible manipulators: transfer matrix method with finite segments. Int. J. Comput. Inf. Eng. 10(6), 1086–1093 (2016)MathSciNetGoogle Scholar
 16.Hamper, B.M., Recuero, M.A., Escalona, L.J., Shabana, A.A.: Use of finite element and finite segment methods in modeling rail flexibility: a comparative study. J. Comput. Nonlinear Dyn. 7, 0410071–04100711 (2012)Google Scholar
 17.Hamper, B.M., Zaazaa, K.E., Shabana, A.A.: Modeling Railroad track structures using finite segment method. Acta Mech. 223(8), 1707–1721 (2012)zbMATHCrossRefGoogle Scholar
 18.Turkkan, O.A., Su, H.J.: A general and efficient multiple segment method for kinetostatic analysis of planar compliant mechanisms. Mech. Mach. Theory 112, 205–217 (2017)CrossRefGoogle Scholar
 19.Xu, X.S., Wang, S.W.: A flexible segment model based dynamics calculation method for free handing marine risers in reentry. China Ocean Eng. 26(1), 139–152 (2012)CrossRefGoogle Scholar
 20.Xu, X.S., Wang, S.W., Lian, I.: Dynamic motion and tension of marine cables being laid during velocity change of mother vessel. China Ocean Eng. 27(5), 629–644 (2013)CrossRefGoogle Scholar
 21.Xu, X.S., Yao, B.H., Ren, P.: Dynamic calculation for underwater moving slander bodies based of flexible segment model. Ocean Eng. 57, 111–127 (2013)CrossRefGoogle Scholar
 22.Spak, K., Agnes, G., Inman, D.: Cable modeling and internal damping developments. Appl. Mech. Rev. 65, 010801118 (2013)CrossRefGoogle Scholar
 23.Triantafyllou, M.S.: Linear dynamics of cables and chains. Shock Vib. Dig. 16(3), 9–17 (1984)CrossRefGoogle Scholar
 24.Triantafyllou, M.S.: Dynamics of cables and chains. Shock Vib. Dig. 19(12), 3–5 (1987)CrossRefGoogle Scholar
 25.Velinsky, S.A.: On the design of wire rope. ASME J. Mech. Trans. 111(3), 382–388 (1989)CrossRefGoogle Scholar
 26.Sathikh, S., Moorthy, M.B.K., Krishnan, M.: A symmetric linear elastic model for helical wire strands under axisymmetric loads. J. Strain Anal. 31(5), 389–399 (1996)CrossRefGoogle Scholar
 27.Raoof, M., Hobbs, R.E.: Analysis of multilayered structural strands. J. Eng. Mech. 114(7), 1166–1182 (1988)CrossRefGoogle Scholar
 28.Jolicoeur, C., Cardou, A.: Semicontinuous mathematical model for bending of multilayered wire strands. J. Eng. Mech. 122(7), 643–650 (1996)CrossRefGoogle Scholar
 29.Crossley, J.A., Spencer, A.J.M., England, A.H.: Analytical solutions for bending and flexure of helically reinforced cylinders. Int. J. Solids Struct. 40, 777–806 (2003)zbMATHCrossRefGoogle Scholar
 30.Ashkenazi, R., Weiss, M.P., Elata, D.: Torsion and bending stresses in wires of nonrotating tower crane ropes. OIPECC Bull. 87, 1157–1172 (2004)Google Scholar
 31.Elata, D., Eshkenazy, R., Weiss, M.P.: The mechanical behavior of a wire rope with an independent wire rope core. Int. J. Solids Struct. 41, 1157–1172 (2004)zbMATHCrossRefGoogle Scholar
 32.Usabiaga, H., Pagalday, J.M.: Analytical procedure for modeling recursively and wire by wire stranded ropes subjected to traction and torsion loads. Int. J. Solids Struct. 45, 5503–5520 (2008)zbMATHCrossRefGoogle Scholar
 33.Rega, G.: Nonlinear vibrations of suspended cables—Part I: modeling and analysis. ASME Appl. Mech. Rev. 57(6), 443–478 (2004)CrossRefGoogle Scholar
 34.Rega, G.: Nonlinear vibrations of suspended cables—Part II: deterministic phenomena. ASME Appl. Mech. Rev. 57(6), 479–514 (2004)CrossRefGoogle Scholar
 35.Srinil, N., Rega, G., Chucheepsakul, S.: Threedimensional nonlinear coupling and dynamic tension in the largeamplitude free vibrations of arbitrarily sagged cables. J. Sound Vib. 269, 823–852 (2004)CrossRefGoogle Scholar
 36.Lacarbonara, W., Paolone, A., Vestroni, F.: Nonlinear modal properties of nonshallow cables. Int. J. Nonlinear Mech. 42(3), 542–554 (2007)zbMATHCrossRefGoogle Scholar
 37.Kruszewski, J.: Application of finite element method to calculations of ship structure vibrations (theory). European shipbuilding. J. Ship Tech. Soc. 3, 38–42 (1975)Google Scholar
 38.Kruszewski, J., Gawroński, W., Wittbrodt, E., Najba,r F., Grabowski, S.: Metoda sztywnych elementów skończonych. (The rigid finite element method). Arkady, Warszawa (1975) (in Polish)Google Scholar
 39.Wittbrodt, E.: Dynamika układów o zmiennej w czasie konfiguracji z zastosowaniem metody elementów skończonych. (Dynamics of systems with changing in time configuration analysed by the finite element method). Gdańsk University Press, No 354, Gdansk (1983) (in Polish)Google Scholar
 40.Wojciech, S.: Dynamika płaskich mechanizmów dźwigniowych z uwzględnieniem podatności ogniw oraz tarcia i luzów w węzłach. (Dynamics of planar linkage mechanisms with consideration of both flexible links and friction as well as clearance in joints). Lodz Technical University Press, Monographs No 66, Lodz (1984) (in Polish)Google Scholar
 41.Wojciech, S.: Dynamic analysis of manipulators with flexible links. Arch. Mech. Eng. XXXVII(3), 169–188 (1990)Google Scholar
 42.AdamiecWójcik, I.: Dynamic analysis of manipulators with flexible links. PhD thesis, Strathclyde University, Glasgow (1992)Google Scholar
 43.Wojciech, S., AdamiecWójcik, I.: Nonlinear vibrations of spatial viscoelastic beams. Acta Mech. 98, 15–25 (1993)zbMATHCrossRefGoogle Scholar
 44.Wojciech, S., AdamiecWójcik, I.: Experimental and computational analysis of large amplitude vibrations of spatial viscoelastic beams. Acta Mech. 106, 127–136 (1994)CrossRefGoogle Scholar
 45.Wittbrodt, E., Wojciech, S.: Application of rigid finite element method to dynamic analysis of spatial systems. J. Guid. Control Dyn. 18(4), 891–898 (1995)CrossRefGoogle Scholar
 46.Wittbrodt, E., AdamiecWójcik, I., Wojciech, S.: Dynamics of Flexible Multibody Systems. Rigid Finite Element Method. Springer, Berlin (2006)zbMATHGoogle Scholar
 47.Płosa, J., Wojciech, S.: Dynamics of systems with changing configuration and with flexible beamlike links. Mech. Mach. Theory 35, 1515–1534 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
 48.Osiński, M., Wojciech, S.: Application of nonlinear optimisation methods to input shaping of the hoist drive of an offshore crane. Nonlinear Dyn. 17, 369–386 (1998)zbMATHCrossRefGoogle Scholar
 49.AdamiecWójcik, I., Drąg, Ł., Wojciech, S., Metelski, M., Nadratowski, K.: A 3D model for static and dynamic analysis of an offshore knuckle boom crane. Appl. Math. Model. 66, 256–274 (2019)MathSciNetCrossRefGoogle Scholar
 50.Szczotka, M., Wojciech, S.: Application of joint coordinates and homogeneous transformations to modeling of vehicle dynamics. Nonlinear Dyn. 52(4), 377–393 (2008)zbMATHCrossRefGoogle Scholar
 51.Wojnarowski, J., AdamiecWójcik, I.: Application of the rigid finite element method to modelling of free vibrations of band saw frame. Mech. Mach. Theory 40, 241–258 (2005)zbMATHCrossRefGoogle Scholar
 52.Szczotka, M.: Pipe laying simulation with an active reel drive. Ocean Eng. 37, 539–548 (2010)CrossRefGoogle Scholar
 53.Szczotka, M.: Dynamic analysis of an offshore pipe laying operation using the reel method. Acta Mech. Sin. 27(1), 44–55 (2011)zbMATHCrossRefGoogle Scholar
 54.Szczotka, M.: A modification n of the rigid finite element method and its application to the Jlay problem. Acta Mech. 220(1–4), 183–198 (2011)zbMATHCrossRefGoogle Scholar
 55.Szczotka, M., Wojciech, S., Maczyński, A.: Mathematical model of a pipelay spread. Arch. Mech. Eng. 54(1), 27–46 (2007)Google Scholar
 56.AdamiecWójcik, I., Wojciech, S., Wittbrodt, E.: Rigid finite element method in modelling of bending and longitudinal vibrations of ropes. Int. J. Appl. Mech. Eng. 17(3), 665–676 (2012)Google Scholar
 57.AdamiecWójcik, I., Brzozowska, L., Wojciech, S.: Modification of the rigid finite element method in modeling dynamics of lines and risers. Arch. Mech. Eng. 60(3), 409–429 (2013)CrossRefGoogle Scholar
 58.Drag, Ł.: Model of an artificial neural network for payload positioning in sea waves. Ocean Eng. 115, 123–134 (2016)CrossRefGoogle Scholar
 59.Drąg, Ł.: Application of dynamic optimization to the trajectory of cablesuspended load. Nonlinear Dyn. 84(3), 1637–1653 (2016)MathSciNetCrossRefGoogle Scholar
 60.Drąg, Ł.: Application of dynamic optimisation to stabilise bending moments and top tension forces in risers. Nonlinear Dyn. 88(3), 2225–2239 (2017)MathSciNetCrossRefGoogle Scholar
 61.AdamiecWójcik, I., Drąg, Ł., Wojciech, S.: A new approach to the rigid finite element method in modeling spatial slender systems. Int. J. Struct. Stabil. Dyn. 18(2), 18500171–185001727 (2018)MathSciNetCrossRefGoogle Scholar
 62.AdamiecWójcik, I., Wojciech, S.: Application of the finite segment method to stabilisation of the force in a riser connection with a wellhead. Nonlinear Dyn. 93, 1853–1874 (2018)CrossRefGoogle Scholar
 63.Bauchau, O.A.: Flexible Multibody Dynamics. Springer, Heidelberg (2011)zbMATHCrossRefGoogle Scholar
 64.Zhang, X., Gou, R., Yang, W., Chang, X.: Vortexinduced vibration dynamics of a flexible fluid convening marine riser subjected to axial harmonic tension. J. Braz. Soc. Mech. Sci. Eng. 40, 365 (2016)CrossRefGoogle Scholar
 65.Kaewunruen, S., Chiravatchradej, J., Chucheepsakols, S.: Nonlinear free vibrations of marine risers/pipes transporting fluid. Ocean Eng. 32, 417–440 (2005)CrossRefGoogle Scholar
 66.Yin, D., Lie, H., Russo, M., Grytoyr, G.: Drilling riser model tests for software verification. In: ASME 35th International Conference on Offshore Mechanical and Arctic Engineering, vol. 2: CFD and VIV():V002T08A032 (2016)Google Scholar
 67.Yin, D., Lie, H., Russo, M., Grytoyr, G.: Drilling riser model tests for software verification. J. Offshore Mech. Arct. Eng. 140(1), 011701–01170115 (2017)CrossRefGoogle Scholar
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