Natural vibration analysis of tensioned risers by segmentation method

Résumé — Analyse des vibrations propres des risers tendus par la méthode de segmentation — Dans le but de l’évaluation des vibrations des risers induites par vortex (VIV), les vibrations modales d’un riser uniforme tendu en eau profonde sont considérées. Du fait de la faible rigidité des risers, les vibrations du type câble et poutre sont analysées. Les solutions exactes et asymptotiques de l’équation différentielle de câble sont présentées. Les vibrations de la poutre sont déterminées en modifiant la solution câble pour chaque demi-longueur d’onde du mode propre. La procédure analytique simplifiée est vérifiée par la méthode des éléments finis. Les conclusions utiles sur le comportement dynamique des risers dans le domaine des hautes et basses fréquences sont présentées. Abstract — Natural Vibration Analysis of Tensioned Risers by Segmentation Method — The natural vibration of deepwater tensioned uniform risers is analysed. The riser is considered as both a cable and a beam due to the fairly weak flexural stiffness. The exact and asymptotic solutions of the cable differential equation are presented. The beam vibration is determined by modifying the cable solution for each half-wave of the natural mode which are called segments . The simplified analytical procedure is verified by the finite element method. Very valuable conclusions are drawn on riser dynamic behaviour in the lower and higher frequency domain. The main application of the study is the assessment of forced vortex-induced riser vibration.


INTRODUCTION
Risers are one of the basic elements of offshore installations, which are used for drilling, production and intervention.They are exposed to the strong influences of environmental conditions in unshielded deep waters.Risers are very sensitive to external excitation due to their slenderness (Patel et al., 1995;Park et al., 2002;Hong and Koterayama, 2003;Chatjigeorgiou et al., 2003).
The main problem in design and operation of deepwater risers nowadays is vortex-induced vibration (VIV) in a severe current.According to full scale measurements, the vortexshedding frequency is adopted to the one of natural frequencies of the riser.This phenomenon is known as lock-in, which means that the vortex shedding frequency is locked into the riser's natural frequency.The amplitude of such resonant elastic response is of the order of magnitude of the riser diameter.This excessive vibration may cause serious fatigue damage, especially in the higher frequency domain (Bowman and Howells, 1998;Howells and Lim, 1998;Khalak and Williamson 1999;Goverdhan and Williamson, 2000;Bell et al., 2002;Yamamoto et al., 2004.).
The theoretical consideration shows that the envelope of riser natural modes is damped from the sea bottom to the riser top.On the other hand, the vortex energy distribution is reduced from the sea surface to the bottom.Thus, the envelope of the forced vibration amplitude is almost constant per riser height.
Besides the stream-wise motion, the cross-stream vibration is also excited due to alternate vortex generation at the cylinder's sides.Thus, the vortex-induced riser vibration is analysed as a single or two degrees of freedom system with natural modes.Time solution is achieved by the time integration of the rather complex governing equations of motion.Therefore, for the vortex-induced forced vibration a reliable analysis of the riser natural vibration is very important.
Since a deepwater riser is a slender structure, the influence of flexural stiffness on the few lowest natural modes is very low and rises for higher modes.Therefore, in order to simplify the procedure the riser vibration analysis is usually based on cable vibration theory and its analogical extension to the beam vibration problems.Thus, a rather simplified analytical approach is described in Sparks (2002) in order to explain the physics of riser dynamics.However, it is possible to solve the problem by the same approach but in a more sophisticated way.This is done in this paper, where only minor simplifications are introduced.The influence of vibration parameters based on their relations in governing equations is analysed in detail and some additional explanations of riser dynamic behaviour are given.
The procedure is illustrated in a case study of a uniform deepwater riser with simply supported edges.The riser is considered as a cable and beam with zero and small flexural stiffness respectively.The differential equation of cable vibration is solved exactly by Bessel's functions and approximately by the asymptotic approach.
In order to analyse the beam vibration, the cable solution is modified following an idea elaborated in Sparks (2002).However, the procedure is formulated in a somewhat different way.The concept of an equivalent tension force is introduced as substitution of linearly varying real force.The obtained results by the simplified procedure called the segmentation method are validated by the finite element solution.

Exact Solution
The differential equation of natural transverse vibration of a tensioned cable with neglecting flexural stiffness yields (Timoshenko, 1955;Clough and Penzien, 1975): (1) where T x is tension force, w is distributed weight, m is specific mass, y is displacement, x is vertical coordinate and t is time.The tension force consists of a constant and varying part (Spark, 2002): (2) where T b and T t are the bottom and the top value respectively.Natural vibration is of a harmonic nature: where Y is the mode function and ω n is the natural frequency.By substituting (3) into (1) the modal equation is obtained: (4) By following Sparks (2002), (4) may be written in the form: (5) where: (6) Substitution: (7) transforms (5) into the standard form of Bessel's equation: (8) where according to (2), ( 6) and ( 7): (9) The solution of (8) yields: (10) where J 0 (z) and Y 0 (z) are Bessel functions of the first and second kind of order zero (Kreyszig, 1993;Abramowitz and Stegun, 1968).The relative value of the integration constants A and B, and natural frequency ω n contained in the argument of Bessel functions (9) depend on the given boundary conditions.

Asymptotic Solution
In the case that the minimum value of argument z is large enough, what depends on the value of the bottom tension force T b in (9), differential equation ( 8) may be solved approximately.By substituting: (11) into (8) yields: (12) If the variable term 1 /4z 2 in (12) is small with respect to unity it may be replaced with its average value 2ε in the cable domain: (13) Thus, Equation ( 12) is transformed into a differential equation with constant coefficients: (14) and its solution reads: (15) where: (16) If the value of ε is very small, then it may be neglected and the asymptotic solution of the differential equation (8) according to (11) and (15) takes the form: The boundary conditions for the simply supported cable read Y = 0 at z = z t and z = z b .This leads to the frequency equation: (18) and its eigenvalues: By employing ( 9) and ( 19) one finds the formula for the natural frequency: (20) After determining the relative values of the integration constants A and B based on the boundary conditions, the expression for the vibration mode (17) may be presented in the normalized form: (21) By employing (9), the mode equation ( 21) takes the form: It is interesting that the mode envelope does not depend on the mode number n, i.e. it is presented by the same function for all the natural modes.On the other hand, the argument z is the mode-dependent function (9) which, taking into account (20) reads: If the above determination of the natural frequencies is performed by employing more accurate solution of the differential equation, represented by formulae ( 11) and ( 15), then the corrected natural frequency reads: (24)

Coordinates of Vibration Nodes and Anti-Nodes
The n-th natural mode is shown in Figure 1, as well as the numbering of vibration nodes.Since at the k-th node the deflection Y k = 0, according to (21) it must be: Cable vibration mode.
By employing (2) and ( 23) one obtains for the k-th node the coordinate: The length of the k-th half-wave of n-th mode is: Anti-nodes are points of maximum deflection and their coordinates are determined from condition dY / dx = 0. Referring to (21) yields: (28) The application of the fundamental identities of trigonometric functions of two arguments (Bronshtejn and Semendjajev, 1975) transforms the condition dY / dx = 0 into: where: (30) Since γ is very small it may be neglected and ( 29) is approximately satisfied if: Thus, the coordinate of the j-th anti-node similarly to (26) takes an approximate value:

Definition of Equivalent Tension Force
Instead of variable tension force T x and specific weight w let us express the differential equation of cable vibration (4) with equivalent constant force T e for simplicity reason, which gives the same natural frequencies.In that case: (39) and natural frequency for the simply supported ends takes the value: (40) The natural frequency for the case with a variable tension force is represented by (20).A formula for the equivalent tension force is determined by equalising ( 20) and ( 40) that leads to: (41)

Mode Half-Wave Characteristics
Since each k-th half-wave of a vibration mode has the same natural frequency, the formula (41) is generally valid for all half-waves, Figure 1, i.e.: (42) The equivalent forces are related to the vibration antinodes (Sparks, 2002).According to (19) for the mode adjacent nodes yields: (43) By substituting ( 23) into (42) one finds: (44) Furthermore, by inserting (44) into (42) and employing (41), a direct relation between half-wave equivalent force and its length is obtained: Finally, by summing up (45) for all the mode half-waves, and taking into account that , the condition for the cable equivalent forces yields: (46) Thus, the average value of the square root half-wave equivalent forces is equal to the square root of the cable equivalent force.

Differential Equation
Differential equation of beam natural vibration, taking into account flexural stiffness, according to Timoshenko (1955) and Clough and Penzien (1975)  The beam equation ( 49) is of the fourth order with a variable coefficient.The intension is to transform it into a second order equation and solve it by analogy with the cable solution.For that purpose let us express the term of the fourth derivation by a term of the second derivation: where Q x is an unknown hypothetic tension force.
By substituting (50) into (49) yields: (51) where: (52) In addition (50) may be expressed by the bending moment: (53) Following the way of transformation of the cable equation in Section 1.1 let us introduce substitutions: (54) (55) Thus, Equation ( 51) is transformed into the form: (56) where according to (2) and ( 55): (57) Now, a new argument may be introduced: where T' x is defined by (52).Equation ( 56) takes the form: (59) where coefficients p(z*,z') and q(z*,z') are very complex implicit functions.They include the unknown force Q x which should be determined from (53).
Due to the above reasons it is not possible to transform (59) into the form of a cable Equation ( 8) and apply analogical parameters and solution.Thus, a suggestion concerning this matter, given in Sparks (2002), leads to a rather rough approximate solution especially for higher modes.Therefore, the problem is solved in a more reliable way by the so-called segmentation method, following the idea presented in Sparks, 2002.

Segmentation Solution
In order to simplify the terminology the mode half-waves are called segments for short.Equation (51) with variable tension force T' x may be expressed for the k-th segment by its constant equivalent force T' ek , which ensures the same natural frequency: where: (61) T' ek is the equivalent segment force due to sliding of the nodes and anti-nodes, and Q k is a hypothetic constant force as a result of flexural stiffness effects, Figure 2.
On the other hand, Equation (50) for the k-th segment, with its constant force Q k , after double integration takes the form: (62) Differential equations ( 60) and ( 62) are of the same type with the common mode function Y. Therefore, their solutions have to be identical and that is achieved if their coefficients are proportional.This condition leads to the following formula for the definition of the unknown force Q k : (63) The segment natural frequency according to (40) yields: (64) The beam segment length from (64) is: (65) Since the sum of the segment length has to be equal to the beam length, i.e.
, and since natural frequency ω' n is common for all the segments, summing up (65) results in: (66) The natural frequency of cable vibration ω n Equation ( 40) may be incorporated into (66).In that way (66) takes the following form: (67) Furthermore, by substituting (66) into (65) the segment length yields a more convenient form: (68) Finally, by substituting ( 64) into (63) a simple formula for Q k is obtained: (69) All the beam vibration parameters are non-linearly mutually dependent, and therefore the problem has to be solved by an iteration procedure as elaborated in Appendix.
The beam mode shape is defined separately for each segment as solution of Equation (60), i.e.: (70) where: (71) and x' k is the node coordinate.Formula (70) may be extended to hold the complete beam mode shape including the mode envelope in analogy with the cable solution ( 21): (72) where: (73) Since according to (64) and ( 71): (74) it follows from ( 73): (75) The difference between the coordinate boundary values yields: (76) The unknown bottom value ζ b may be determined empirically if the top value of the normalised mode envelope, C t , is known from another source.Thus, according to (72): (77) and by employing (76) one obtains: (78) The extended formula for the beam's natural mode (72) represents the solution of a hypothetic beam differential equation which is analogical to the cable equation ( 8).

ILLUSTRATIVE EXAMPLE
The application of the previously presented cable and beam vibration theory is illustrated in the case of a drilling riser analysed in Sparks (2002).Its main characteristics are the following: Riser length L = 2000 m Flexural stiffness EI = 318.6 • 10 6 Nm 2 In order to validate the obtained results by the asymptotic integration and segmentation method respectively, the cable vibration is also analysed by the exact method expressed by the Bessel functions, while beam vibration is performed by the finite element method (Zienkiewicz, 1971;Bathe, 1996).The values of natural frequencies are compared in Table 1.The asymptotic solution of cable vibration shows some discrepancies with respect to the exact solution, which are rapidly decreasing for higher modes.The estimated error of the first natural frequency reads 1.69%, which is fairly close to the predicted error ε = 2.05% according to Equation ( 13).
The beam natural frequencies are determined by the iterative segmentation method for the given accuracy of 10 -7 .The iterated parameters converge uniformly.The number of iteration steps depends on the mode number.For instance, the given accuracy is achieved in 7 and 39 steps for n = 10 and 50 respectively.
The beam vibration analysis is also performed by the finite element method employing a special program developed within a Master's Thesis (Parunov, 1996).The FEM model consists of 200 beam elements of equal 10 m length that is lower than the minimum segment length.The value of the first natural frequency is very close to the exactly determined cable frequency, since influence of the flexural stiffness on the first natural mode is negligible.
In the lower frequency domain the beam natural frequencies determined by the segmentation method are somewhat higher than the FEM values.In the higher frequency domain the situation is opposite.Generally, the agreement between the results is very good, since for instance discrepancy between the 50-th natural frequency values is only 0.05%.
The numerical procedure of the segmentation method is illustrated in Table 2 in the case of the 50-th natural mode.
The obtained results are also shown in Figures 3 and 4  Cable (EI = 0): ω 50 = 4.05447 s -1 Beam (EI > 0): ω' 50 = 5.47942 s -1 Eq. ( 20) Eq. ( 66) Eq. ( 27) Eq. ( 42) Eq. ( 69) Eq. ( 61) Eq. ( 68    The coordinates of the cable and beam nodes, x k and x' k respectively, are listed in Table 3 and shown in Figure 5. Due to the sliding of the beam segments with respect to the cable ones, the values of x' k are increased.The variation of its boundary values is zero as consequence of the fixed ends.The cable z k -coordinate (23) varies linearly from the bottom to the top, Figure 6.The difference between the top and the bottom values reads z n+1z 1 = nπ which is in accordance with (19).
The FEM analysis of beam vibration shows that the top value of the mode envelope reads C t = 0.975 and according to ( 76  Beam (EI > 0): ω' 50 = 5.47942 s -1 Eq. ( 20) Eq. ( 68) Eq. ( 26) Eq. ( 23) Eq. ( 83 The 50-th natural mode of cable vibration. Figure 9 The 50-th natural mode of beam vibration.The coordinate coefficient α k = π/l' k , Equation ( 74), is shown in Figure 7.It is proportional to in accordance with (69).The cable and beam vibration modes determined by the asymptotic integration and segmentation method are shown in Figures 8 and 9 respectively.The difference between the mode shapes is evident.In the latter case the mode envelope is expanding, and the bottom and top segment lengths are increasing and decreasing respectively.
The variation of the top value of the beam mode envelope, determined by the FEM analysis as function of mode number, is shown in Figure 10.It starts from the constant cable value, and approaches the unity when n increases to infinity.

CONCLUSION
Analysis of the natural vibration of tensioned risers is an important step in the investigation of vortex-induced forced vibration.The weak influence of the riser flexural stiffness is analysed through correlation of cable and beam vibration.The analytical procedures are employed for consistent analysis of the riser dynamic behaviour.The differential equation of cable vibration is solved exactly by Bessel's functions and approximately by the asymptotic integration.The problem of beam vibration is solved by the modification of the cable solution and its verification is performed by the finite element method.In both cases a fairly good agreement between the obtained results is achieved.
Based on the performed analyses the following conclusions may be drawn: -the cable natural frequencies ω n are linearly dependent on mode number n, -the cable mode wave envelope depends only on distribution of tension force T x and is the same for all vibration modes, -the cable mode half-wave lengths, l k , and square root of equivalent tension force, , are linearly dependent on half-wave number k, -the equivalent tension force T e transfers the variable coefficient in the differential equation into a constant value making in such a way its analytical solution is possible, -the segmentation method for beam vibration analysis, based on the modification of the cable results, is a very effective iteration procedure, -the beam natural frequencies are increased with respect to the cable ones due to flexural stiffness, but the increase is fairly low for the first modes, -the position of the beam vibration nodes compared to those in the cable vibration is changed; mode half-wave lengths are increased and decreased at the bottom and top part of the riser, respectively, -the beam mode wave envelope depends on the mode number; it changes from the cable envelope shape for n = 1 to constant value as n increases to infinity.The obtained analytical results may be used only for risers of uniformly distributed characteristics.For risers of non-uniform characteristics, numerical methods, such as the finite element method, have to be used.In any case the analytical solution is useful for the explanation of the riser dynamic behaviour.

Final Manuscript receveid in March 2006
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Figure 2Cable and beam parameters.
Figure 3Cable and beam segment lengths.
Figure 4Cable and beam segment forces.
Figure 2, the value of is increased with respect to the cable values , Figure 4.The boundary increases are very small since sliding of the bottom and top segment is negligible.The hypothetical tension force due to flexural rigidity, Q k , is of the same order of magnitude as the equivalent forces, Figure 4.It takes the maximum value at the bottom.As a result the increase of with respect to is very large, much larger at the bottom than at the top.
Figure 5Cable and beam node coordinates.

Figure 6
Figure 6Dimensionless coordinates of cable and beam vibration nodes.

Figure 7
Figure 7Coefficient of beam mode argument.
) one finds ζ b = ζ 1 = 3024.28.The ζ coordinate of the vibration nodes by employing (74) is determined in Table 3 and shown in Figure 6.The diagrams of z k and ζ k are parallel straight lines.A considerably increased value of ζ k compared to z k is the result of flexural stiffness.
Figure 10Top value of mode envelope.