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Dynamics of small-sagged taut-slack marine cables
Compuwr
0045-7949(95)00159-x
DYNAMICS
d Srrucrures Vol. 58. No. 3. pp. 557-562. 1996 Elsevier Science Ltd Printed in Great Britain 00457949/96 19.50 + 0.00
OF SMALL-SAGGED TAUT-SLACK CABLES
MARINE
D. Vassalos and S. Huangt Marine Technology Centre, University of Strathclyde, 100 Montrose Street, Glasgow G4 OLZ, Scotland, U.K. (Received 21 October
1994)
Ah&art-The snap tension of small-sagged marine cables operating in alternating taut-slack conditions is considered in this paper. The marine cable is suspended at the two ends on the same level. One of the two end8 is fixed, while the other is subjected to horizontal excitation. A single non-linear equation of vertical motion of the cable is derived and solved by the Galerkin method. It is found that only the symmetric modes with respect to the mid-span need to be considered. Numerical examples are given and comparisons are made between numerical and experimental results which help to demonstrate the validity of the present method.
1. INTRODUCTION Marine cables are widely used for offshore and subsea engineering. They provide an economic means for various offshore activities. Although cable dynamics in air is an old subject with analytic solutions dating back to the 18th century [I], it is only during the
second half of this century that marine cables have been subjected to systematic research as the increasing activities involved in offshore gas and oil exploration require a good understanding of the statics and dynamics of marine cables. Due to the fluid drag, the dynamics of marine cables are quite different from the cable dynamics in air. In air the geometric configuration is the only source of nonlinearity, assuming that the strain-tension relationship is linear and most cables respond to an applied load through a combination of stretching and line deformation. The natural frequencies for cable dynamics in air are dominated by the ratio of the elastic stiffness to the catenary stiffness. However, when the cable moves in water, there is another source of nonlinearity due to fluid drag. In this case cable dynamics is more influenced by the elastic stiffness, unless the cable vibrates very slowly, as the drag forces cause the cable to respond more by stretching and less by line deformation. In applications such as mooring a buoy, towing a ship or tethering a subsea unit, marine cables can become slack if the tension temporarily falls to a level which is comparable to the distributed drag force along the cable, and thus operate in alternating tautslack conditions under periodic environmental excitation. In the slack state, the motion of the cable is t To whom all correspondence should be addressed. Present address: 2H Offshore Engineering Ltd, 103 Mayford Centre, Mayford Green, Woking GU22 OPP, U.K. 557
dominated by its own inertia force and the fluid drag force. Depending upon the rate at which the cable becomes taut during the motion cycle, the transition from the slack state to the taut state may cause high tension in the cable which can have detrimental effects and may even cause cable breakage. In this case, there is an additional marked nonlinearity due to the discontinuity in the elastic stiffness. A substantial amount of research has been conducted in pursuing solutions of problems associated with marine cable dynamics. The prevalent methods include (a) Finite difference method [2,3]. The non-linear hyperbolic equations, which govern the motions of the cable, are approximated by finite differences and solved in the time domain. (b) Galerkin method [4,5]. This method takes advantage of the smooth configuration of the cable, assuming the solution to be a sum of basic modes. (c) Lumped mass method [6,7]. The governing equations become ordinary differential equations after lumping the distributed mass of the cable. Numerical integration is then performed. (d) Analytic method [8,9]. Due to the non-linear and coupled nature of the governing equations, analytic solutions are only available in simplified cases. It is not intended to give a comprehensive list of references here. A common feature of most investigations cited above is that they only deal with the taut marine cable dynamics where tension does not fall to a near-zero level. In spite of the practical significance of taut-slack marine cables and the associated snap loading, only a few investigations have been conducted addressing the problem. In Ref. [lo], a single-degree-of-freedom model is proposed for the preliminary assessment of snap loading occurrence for a one-dimensional marine cable-body system, and
558
D. Vassalos and S. Huang
a more complex multi-degree-of-freedom model for predicting the snap loading qualitatively. In Ref. [1 11, a two-dimensional case is considered which is similar to the concern of this paper. However, a different formulation and numerical method are used. The present study investigates the non-linear dynamics of small-sagged marine cables operating in alternating taut-slack conditions. Only the in-plane motions are considered here as the parametrically excited large out-of-plane motions are unlikely to happen. After introducing simplifications, a single equation is derived for the vertical in-plane motion. This non-linear equation is under a combination of parametric and external excitations. Time-domain solutions are performed through numerical integration which employs the Galerkin procedure with sinusoids as deflection modes. It is found that only the symmetric
modes
with respect
to the mid-span
need to be considered. The approach adopted here is capable of tackling the pertinent nonlinearities of marine cable dynamics including damping, geometric nonlinearity and the discontinuity in the axial stiffness. Numerical results are presented and a good agreement is achieved with available experimental results. Results of further parametric investigations are also presented to illustrate the effects of various factors. 2. GOVERNING
EQUATIONS
Consider a uniform small-sagged cable suspended at the two ends on the same level, as depicted in Fig. 1. One end is fixed, while the other is subjected to horizontal excitation u0 sin(o,t). The non-linear equations of motion describing the in-plane dynamics can be written as
+;pC,DJPcosa-dsinaj x(ticosa
--d sina)sina
-tpC,Dltisina+dcosal
x (ti sin a + ti cos a)cos a,
(2)
where u and v are the horizontal and vertical components of the displacement, s is the Lagrangian co-ordinate, T, and T are the static and dynamic tensions, p is the fluid density, D is the cable diameter, A is the cross-sectional area, C, and C, are the tangential and normal drag coefficients, m is the mass distribution, m, and m, are the added masses, g is the gravitational acceleration, and x(s) and y(s) define the initial static profile of the cable. In this model, the fluid drag on the cable is approximated under the assumption that it can be resolved into normal and tangential components, each a function of the relative velocity in that direction. The above equations are valid for taut cables where T0 + T > 0. For T, + T < 0, the first terms on the right-hand sides of eqns (1) and (2) should be set to zero, i.e. T=
-T,.
If a linear strain-tension relation is adopted, the dynamic tension T of the taut cable is given by the following expression: T=EAc,=EA(c,-c,)xEA
;; { (3)
where E is Young’s modulus of the cable, and cd is the strain due to dynamic tension, x jti cos c( - d sin al(ti cos a - d sin a)
xcosa-fpC,Dltisina+dcosa( x(ti sina fd cosa)sina
Fig 1. Co-ordinate system and cable configuration.
is the total strain, and (1)
is the strain due to the static tension. All strains are assumed small with respect to unity. Equations (1) and (2) give the coupled governing equations for both the vertical and horizontal inplane motions. Simplifications can be achieved based upon the following assumptions: (a) dx/ds x 1 and a e 1. These derive from the assumption of small sag. (b) The dynamic tension T of the taut cable is a function of time alone. This assumption is based upon the consideration that the dynamic tension propagates along the cable at the speed of m.
559
Dynamics of small-sagged taut-slack marine cables while the lateral motion examined in this paper propagates along the cable at the speed of
(6) where if if
Since EA >>To + T, it is valid to assume that T is independent of s. (c) The gradient of the horizontal component of the dynamic displacement is small, and hence
AT>--T, AT<-To
and AT= EA
2au as"as'
u,sinr
au
0
(mg--;A)'(~_2x)
+
Under the above assumptions,
0
eqn (3) leads to
3.NUMERICALSOLUTIONS
Equation (6) is not amenable to analytic study. For numerical analysis, the solution is assumed to be in the form
Integration of this equation with respect to x and application of the boundary conditions at the two ends, i.e. ~(0, t) = 0, u(l, t) = u. sin(o,t), lead to
N v(x, t) = C v,(t)sin(irrx), ,=I
(7)
where v,(t) (i = 1,2, . , N) are unknown functions of time, and N is the number of modes to be taken into account. In this approximation, the modes are either symmetric or antisymmetric with respect to the mid-span. Applying Galerkin’s method yields the following N equations:
(4) where 1 is the span. It is apparent from eqns (2) and (4) and the assumption CICC1 that the vertical motion is now decoupled from the horizontal one. On the
0
4 T
---
(8)
mg-pA
ia To (m + m,)la$
basis of this, only the vertical motion will be considered in the following. For a small-sagged cable, the initial static profile can be approximated by the following parabolic profile [l] Y(X) = y
where
r.= - J' 1 DpC,l
m+m,
.f =x/l;
N
(I - x)x. ix
the following normalized v’=v//;
sin(inx) 5 d sin(krrx) k=l
x 1 d sin(krrx)dx k=,
0
Introducing
o
12,= Uo/l;
T
variables:
cm,*
and substituting eqns (4) and (5) into eqn (2) and utilizing the assumption u c 1, i.e. cos c( z 1 and sin c( zz 0, we have the following non-dimensional equation of the vertical motion (for brevity of notation, all tildes are omitted)
ol=T
if if
AT>-To AT< -To
2(mg - PA )I Trr 0
in which [N/2] denotes the truncation largest whole number.
of N/2 to the
560
D.
Vassalos and S. Huang
It can be seen from the above equations that the coupling effects of different modes are represented through non-linear terms. The modes corresponding to even i are under parametric excitation alone whereas the modes corresponding to odd i are under a combination of parametric and external excitations. Furthermore, it can be shown that v,(t) = 0 (i = 2,4, . ,2[N/2]) satisfy the governing equations since in this case r, = 0, for i = 2,4, . . ,2[N/2]. Therefore, it is only necessary to solve the governing equations of u,(t), i = 1, 3, . . ,2[N/2] - (- l)N. Numerical integration in the time domain is carried out using the Rung+Kutta method. As the focus of this study is on the steady-state vibrations of the cable, the numerical integration is performed with initial conditions given by static equilibrium. The simulations are carried out for a sufficiently long time to ensure that transient effects are not included in the final output.
04
Model tests on a small-sagged marine cable were performed in the Ship Research Institute of Norway with consideration of the snap tension [12]. The tests consisted of restraining one end of a horizontally placed cable and exciting the other end sinusoidally in the horizontal direction with known amplitudes and frequencies. The test results are used here to verify the validity and accuracy of the present method. Figures 2-5 show the maximum and minimum values of tension in the cable under excitations at different amplitudes and frequencies for both the numerical and the experimental results. The agreement is good. It is clear from these figures that when the excitations are not severe, i.e. the excitation frequency is low or the amplitude is small or both, the first term in eqn (7) alone can suffice for an adequate description of the dynamics. However, when the excitation becomes severe, more terms are needed. As far as the tension is concerned, it can be seen that when the excitation is not severe, the maximum and minimum tensions are almost symmetrical with a mean value which is equal to the initial static tension. This symmetry can no longer hold when the excitation is increased as shown in Fig. 5. As the minimum tension approaches zero, that is the cable starts to operate in an alternating taut-slack condition, the maximum tension increases significantly. The parameters used in the above computation are based upon the original system as given in Table 1. To analyse the effects of the parameters shown, some of these parameters are chosen to vary. These include the drag coefficient, the initial tension and Young’s modulus. In all the following numerical computations, N is set to Il. Figures 69 show the effects of the drag coefficient. The normal drag coefficient of a marine cable is usually set at around 1.5. However, a range of values
: : 02
I
:
04
EXCITATION
: 06
:
:
:
:
0.6
FREQUENCY
:
1
4 12
(Hz)
Fig. 2. Comparison between the numerical and experimental results of both the maximum and the minimum tensions with the excitation amplitude 0.025 m.
200
150 z p
4. NUMERICAL RESULTS AND COMMENTS
:
0
100
P 50
0-l::::: 0
02
04
EXCITATION
06
06
FREQUENCY
1
12
(Hz)
Fig. 3. Comparison between the numerical and experimental results of both the maximum and the minimum tensions with the excitation amplitude 0.05 m.
300 ,
I
250 g
200
p
150
; +
100 50 0 0
02
04
EXCITATION
06
06
FREQUENCY
1
12
(Hz)
Fig. 4. Comparison between the numerical and experimental results of both the maximum and the minimum tensions with the excitation amplitude 0.075 m.
0
02
04
EXCITATION
06
06
FREWENCY
1
12
(Hz)
Fig. 5. Comparison between the numerical and experimental results of both the maximum and the minimum tensions with the excitation amplitude 0.1 m.
Dynamics of small-sagged taut-slack marine cables
561
Table 1. The nrincinal narameters of the cable and test Values
Parameters diameter length span sag-span ratio
0.01 m
10.9774 m 10.792 m 0.0812 10” N mm2 0.61 kg m-’ 88 N 0.02550.1 m (tl.2 Hz 0.55 Hz
Young’s modulus mass distribution initial tension range of amplitude range of frequency WI
has been quoted in the literature, the uncertainty being dependent upon the strumming effect. Generally, the value in any given case is within the range from 1.0 to 2.4. In Figs 6 and 7, numerical results of both the tension and motion are given for a very small drag coefficient at 0.2. As expected, these results bear the features of vibrating cables in air when one can find pronounced resonances. As the drag coefficient increases, shown in Fig 8 and 9, the resonance becomes unimportant and the motion is reduced considerably. In Fig. 8, the sharp increase in tension is due to the snap loading. Figures 10 and 11 show the effects of the initial tension on both the total tension and the magnitude
1
0
2
3
Nondimensional
frequency
4
Fig. 8. Non-dimensional maximum and minimum tensions vs the non-dimensional excitation frequency 0,/w, with U, equal to 0.075 m.
0
2
3
Nondimensional
frequency
1
4
Fig. 9. Non-dimensional maximum displacements of different modes vs the non-dimensional excitation frequency w,/w, with C, equal to 1.6 and no equal to 0.075 m.
60-
Fig. 6. Non-dimensional maximum and minimum tensions vs the non-dimensional excitation frequency wr/w, with C, equal to 0.2 and u,, equal to 0.075 m. 0
02
04
06
08
1
12
Excitation frequency (Hz)
Fig. 10. Effect of initial tension on both the maximum and minimum tensions in the cable. The excitation amplitude is fixed at 0.075 m.
0 0
2
3
NondimensIonal
1
fmquency
4
Fig. 7. Non-dimensional maximum displacements of different modes vs the non-dimensional excitation frequency or/w,
with C, equal to 0.2 and u,, equal to 0.075 m.
of the first three modes. These figures also show the effect of the sag-span ratio as it is given by (mg - pA)1/(8T,,). As the initial tension increases, the sag decreases. In this case, cable dynamics is influenced mainly by the elastic stiffness, resisting the applied load through stretching. As demonstrated in these two figures, both the maximum tension and the magnitude of modes increase as the initial tension increases.
562
D. Vassalos and S. Huang
employs mainly the catenary stiffness to resist the applied excitation. As the excitation frequency becomes greater, the elastic stiffness becomes more important.
5. CONCLUSIONS
0 0
02
04
06
06
1
12
Excitation frequency (Hz)
Fig. II. Effect of initial tension on the non-dimensional magnitudes of the first three symmetric modes. The excitation amplitude is fixed at 0.075 m.
A non-linear model is developed for small-sagged marine cables subjected to horizontal excitations at one end and operating in taut-slack conditions. A numerical analysis based upon the Galerkin method is conducted to investigate a system of non-linear equations under both parametric and external excitations. Numerical results are compared with available experimental results and a good agreement is achieved. The influence of various factors, such as excitation amplitude, excitation frequency, drag coefficient, initial tension and Young’s modulus, are also examined.
6. REFERENCES
0 0
02
04
0.6
0.6
Excitation frequency
1
1.2
(Hz)
Fig. 12. Effect of Young’s modulus on both the maximum and minimum non-dimensional tensions in the cable. Excitation amplitude is fixed at 0.1 m.
Mooring Dynamics for Offshore Application. Part 1, Theory. MITSG86-1, MIT, Cambridge, MA (1986).
5. J. H. Milgram, M. S. Triantafyllou, F. C. Frimm and G. Anagnostou, Seakeeping and extreme tensions in offshore towing. SNAME Trans. 96, 35-70 (1988). 6. H. Polacheck, T. S. Walton, R. Mejia and C. Dawson, Transient motion of an elastic cable immersed in a fluid. Math. Tables Other Aids Compur. 17, 6043
005
E
E 004 B
d 5P
1. H. M. Irvine, Cable structures. The MIT Press, Cambridge, MA (1981). 2. C. M. Ablow and S. Schechter, Numerical simulation of undersea cable dynamics. Ocean Engng 10, 443457 (1983). 3. J. J. Burgess, Modelling of undersea cable installation with a finite difference method. In: Proc. Int. Ofihore and Polar Engineering Conf., Edinburgh (1991). 4. M. S. Triantafyllou, A. Bliek, J. Burgess and H. Shin,
003
5 6
(1963). 7. C. M. Larsen and I. J. Fylling, Dynamic behaviour of anchor lines. Norwegian Maritime Res. 3 (1982). 8. R. M. Kennedy and E. S. Strahan, A linear theory of
E B c
O.O'
1
0020 L 0
02
0.4
0.6
0.6
1
12
9.
Excitatfon frequency (Hz)
Fig. 13. Effect of Young’s modulus on the non-dimensional magnitudes of the first three symmetrical modes. Excitation amplitude is fixed at 0.1 m.
In Figs 12 and 13, the effects of Young’s modulus
are examined. When the excitation frequency is low, the fluid drag force is small and the system
10. 11.
12.
transverse cable dynamics at low frequencies. NUSC Technical Report 6463, Naval Underwater Systems Centre, Connecticut (1981). M. S. Triantafyllou, Preliminary design of mooring systems. J. Ship Res. 26, 25-35 (1982). J. M. Niedzwecki and S. K. Thampi, Snap loading of marine cable systems. Appl. Ocean Res. 13,2-l 1 (1991). V. J. Papazoglou, S. A. Mavrakos and M. S. TriantafylIOU, Non-linear cable response and model testing in water, J. Sound Vibr. 140, 103-I 15 (1990). I. J. Fylling and P. T. Weld, Cable dynamics--comparison of experimental and analytic results. Report R-89.79, The Ship Research Institute of Norway (1979).
0045-7949(95)00159-x
DYNAMICS
d Srrucrures Vol. 58. No. 3. pp. 557-562. 1996 Elsevier Science Ltd Printed in Great Britain 00457949/96 19.50 + 0.00
OF SMALL-SAGGED TAUT-SLACK CABLES
MARINE
D. Vassalos and S. Huangt Marine Technology Centre, University of Strathclyde, 100 Montrose Street, Glasgow G4 OLZ, Scotland, U.K. (Received 21 October
1994)
Ah&art-The snap tension of small-sagged marine cables operating in alternating taut-slack conditions is considered in this paper. The marine cable is suspended at the two ends on the same level. One of the two end8 is fixed, while the other is subjected to horizontal excitation. A single non-linear equation of vertical motion of the cable is derived and solved by the Galerkin method. It is found that only the symmetric modes with respect to the mid-span need to be considered. Numerical examples are given and comparisons are made between numerical and experimental results which help to demonstrate the validity of the present method.
1. INTRODUCTION Marine cables are widely used for offshore and subsea engineering. They provide an economic means for various offshore activities. Although cable dynamics in air is an old subject with analytic solutions dating back to the 18th century [I], it is only during the
second half of this century that marine cables have been subjected to systematic research as the increasing activities involved in offshore gas and oil exploration require a good understanding of the statics and dynamics of marine cables. Due to the fluid drag, the dynamics of marine cables are quite different from the cable dynamics in air. In air the geometric configuration is the only source of nonlinearity, assuming that the strain-tension relationship is linear and most cables respond to an applied load through a combination of stretching and line deformation. The natural frequencies for cable dynamics in air are dominated by the ratio of the elastic stiffness to the catenary stiffness. However, when the cable moves in water, there is another source of nonlinearity due to fluid drag. In this case cable dynamics is more influenced by the elastic stiffness, unless the cable vibrates very slowly, as the drag forces cause the cable to respond more by stretching and less by line deformation. In applications such as mooring a buoy, towing a ship or tethering a subsea unit, marine cables can become slack if the tension temporarily falls to a level which is comparable to the distributed drag force along the cable, and thus operate in alternating tautslack conditions under periodic environmental excitation. In the slack state, the motion of the cable is t To whom all correspondence should be addressed. Present address: 2H Offshore Engineering Ltd, 103 Mayford Centre, Mayford Green, Woking GU22 OPP, U.K. 557
dominated by its own inertia force and the fluid drag force. Depending upon the rate at which the cable becomes taut during the motion cycle, the transition from the slack state to the taut state may cause high tension in the cable which can have detrimental effects and may even cause cable breakage. In this case, there is an additional marked nonlinearity due to the discontinuity in the elastic stiffness. A substantial amount of research has been conducted in pursuing solutions of problems associated with marine cable dynamics. The prevalent methods include (a) Finite difference method [2,3]. The non-linear hyperbolic equations, which govern the motions of the cable, are approximated by finite differences and solved in the time domain. (b) Galerkin method [4,5]. This method takes advantage of the smooth configuration of the cable, assuming the solution to be a sum of basic modes. (c) Lumped mass method [6,7]. The governing equations become ordinary differential equations after lumping the distributed mass of the cable. Numerical integration is then performed. (d) Analytic method [8,9]. Due to the non-linear and coupled nature of the governing equations, analytic solutions are only available in simplified cases. It is not intended to give a comprehensive list of references here. A common feature of most investigations cited above is that they only deal with the taut marine cable dynamics where tension does not fall to a near-zero level. In spite of the practical significance of taut-slack marine cables and the associated snap loading, only a few investigations have been conducted addressing the problem. In Ref. [lo], a single-degree-of-freedom model is proposed for the preliminary assessment of snap loading occurrence for a one-dimensional marine cable-body system, and
558
D. Vassalos and S. Huang
a more complex multi-degree-of-freedom model for predicting the snap loading qualitatively. In Ref. [1 11, a two-dimensional case is considered which is similar to the concern of this paper. However, a different formulation and numerical method are used. The present study investigates the non-linear dynamics of small-sagged marine cables operating in alternating taut-slack conditions. Only the in-plane motions are considered here as the parametrically excited large out-of-plane motions are unlikely to happen. After introducing simplifications, a single equation is derived for the vertical in-plane motion. This non-linear equation is under a combination of parametric and external excitations. Time-domain solutions are performed through numerical integration which employs the Galerkin procedure with sinusoids as deflection modes. It is found that only the symmetric
modes
with respect
to the mid-span
need to be considered. The approach adopted here is capable of tackling the pertinent nonlinearities of marine cable dynamics including damping, geometric nonlinearity and the discontinuity in the axial stiffness. Numerical results are presented and a good agreement is achieved with available experimental results. Results of further parametric investigations are also presented to illustrate the effects of various factors. 2. GOVERNING
EQUATIONS
Consider a uniform small-sagged cable suspended at the two ends on the same level, as depicted in Fig. 1. One end is fixed, while the other is subjected to horizontal excitation u0 sin(o,t). The non-linear equations of motion describing the in-plane dynamics can be written as
+;pC,DJPcosa-dsinaj x(ticosa
--d sina)sina
-tpC,Dltisina+dcosal
x (ti sin a + ti cos a)cos a,
(2)
where u and v are the horizontal and vertical components of the displacement, s is the Lagrangian co-ordinate, T, and T are the static and dynamic tensions, p is the fluid density, D is the cable diameter, A is the cross-sectional area, C, and C, are the tangential and normal drag coefficients, m is the mass distribution, m, and m, are the added masses, g is the gravitational acceleration, and x(s) and y(s) define the initial static profile of the cable. In this model, the fluid drag on the cable is approximated under the assumption that it can be resolved into normal and tangential components, each a function of the relative velocity in that direction. The above equations are valid for taut cables where T0 + T > 0. For T, + T < 0, the first terms on the right-hand sides of eqns (1) and (2) should be set to zero, i.e. T=
-T,.
If a linear strain-tension relation is adopted, the dynamic tension T of the taut cable is given by the following expression: T=EAc,=EA(c,-c,)xEA
;; { (3)
where E is Young’s modulus of the cable, and cd is the strain due to dynamic tension, x jti cos c( - d sin al(ti cos a - d sin a)
xcosa-fpC,Dltisina+dcosa( x(ti sina fd cosa)sina
Fig 1. Co-ordinate system and cable configuration.
is the total strain, and (1)
is the strain due to the static tension. All strains are assumed small with respect to unity. Equations (1) and (2) give the coupled governing equations for both the vertical and horizontal inplane motions. Simplifications can be achieved based upon the following assumptions: (a) dx/ds x 1 and a e 1. These derive from the assumption of small sag. (b) The dynamic tension T of the taut cable is a function of time alone. This assumption is based upon the consideration that the dynamic tension propagates along the cable at the speed of m.
559
Dynamics of small-sagged taut-slack marine cables while the lateral motion examined in this paper propagates along the cable at the speed of
(6) where if if
Since EA >>To + T, it is valid to assume that T is independent of s. (c) The gradient of the horizontal component of the dynamic displacement is small, and hence
AT>--T, AT<-To
and AT= EA
2au as"as'
u,sinr
au
0
(mg--;A)'(~_2x)
+
Under the above assumptions,
0
eqn (3) leads to
3.NUMERICALSOLUTIONS
Equation (6) is not amenable to analytic study. For numerical analysis, the solution is assumed to be in the form
Integration of this equation with respect to x and application of the boundary conditions at the two ends, i.e. ~(0, t) = 0, u(l, t) = u. sin(o,t), lead to
N v(x, t) = C v,(t)sin(irrx), ,=I
(7)
where v,(t) (i = 1,2, . , N) are unknown functions of time, and N is the number of modes to be taken into account. In this approximation, the modes are either symmetric or antisymmetric with respect to the mid-span. Applying Galerkin’s method yields the following N equations:
(4) where 1 is the span. It is apparent from eqns (2) and (4) and the assumption CICC1 that the vertical motion is now decoupled from the horizontal one. On the
0
4 T
---
(8)
mg-pA
ia To (m + m,)la$
basis of this, only the vertical motion will be considered in the following. For a small-sagged cable, the initial static profile can be approximated by the following parabolic profile [l] Y(X) = y
where
r.= - J' 1 DpC,l
m+m,
.f =x/l;
N
(I - x)x. ix
the following normalized v’=v//;
sin(inx) 5 d sin(krrx) k=l
x 1 d sin(krrx)dx k=,
0
Introducing
o
12,= Uo/l;
T
variables:
cm,*
and substituting eqns (4) and (5) into eqn (2) and utilizing the assumption u c 1, i.e. cos c( z 1 and sin c( zz 0, we have the following non-dimensional equation of the vertical motion (for brevity of notation, all tildes are omitted)
ol=T
if if
AT>-To AT< -To
2(mg - PA )I Trr 0
in which [N/2] denotes the truncation largest whole number.
of N/2 to the
560
D.
Vassalos and S. Huang
It can be seen from the above equations that the coupling effects of different modes are represented through non-linear terms. The modes corresponding to even i are under parametric excitation alone whereas the modes corresponding to odd i are under a combination of parametric and external excitations. Furthermore, it can be shown that v,(t) = 0 (i = 2,4, . ,2[N/2]) satisfy the governing equations since in this case r, = 0, for i = 2,4, . . ,2[N/2]. Therefore, it is only necessary to solve the governing equations of u,(t), i = 1, 3, . . ,2[N/2] - (- l)N. Numerical integration in the time domain is carried out using the Rung+Kutta method. As the focus of this study is on the steady-state vibrations of the cable, the numerical integration is performed with initial conditions given by static equilibrium. The simulations are carried out for a sufficiently long time to ensure that transient effects are not included in the final output.
04
Model tests on a small-sagged marine cable were performed in the Ship Research Institute of Norway with consideration of the snap tension [12]. The tests consisted of restraining one end of a horizontally placed cable and exciting the other end sinusoidally in the horizontal direction with known amplitudes and frequencies. The test results are used here to verify the validity and accuracy of the present method. Figures 2-5 show the maximum and minimum values of tension in the cable under excitations at different amplitudes and frequencies for both the numerical and the experimental results. The agreement is good. It is clear from these figures that when the excitations are not severe, i.e. the excitation frequency is low or the amplitude is small or both, the first term in eqn (7) alone can suffice for an adequate description of the dynamics. However, when the excitation becomes severe, more terms are needed. As far as the tension is concerned, it can be seen that when the excitation is not severe, the maximum and minimum tensions are almost symmetrical with a mean value which is equal to the initial static tension. This symmetry can no longer hold when the excitation is increased as shown in Fig. 5. As the minimum tension approaches zero, that is the cable starts to operate in an alternating taut-slack condition, the maximum tension increases significantly. The parameters used in the above computation are based upon the original system as given in Table 1. To analyse the effects of the parameters shown, some of these parameters are chosen to vary. These include the drag coefficient, the initial tension and Young’s modulus. In all the following numerical computations, N is set to Il. Figures 69 show the effects of the drag coefficient. The normal drag coefficient of a marine cable is usually set at around 1.5. However, a range of values
: : 02
I
:
04
EXCITATION
: 06
:
:
:
:
0.6
FREQUENCY
:
1
4 12
(Hz)
Fig. 2. Comparison between the numerical and experimental results of both the maximum and the minimum tensions with the excitation amplitude 0.025 m.
200
150 z p
4. NUMERICAL RESULTS AND COMMENTS
:
0
100
P 50
0-l::::: 0
02
04
EXCITATION
06
06
FREQUENCY
1
12
(Hz)
Fig. 3. Comparison between the numerical and experimental results of both the maximum and the minimum tensions with the excitation amplitude 0.05 m.
300 ,
I
250 g
200
p
150
; +
100 50 0 0
02
04
EXCITATION
06
06
FREQUENCY
1
12
(Hz)
Fig. 4. Comparison between the numerical and experimental results of both the maximum and the minimum tensions with the excitation amplitude 0.075 m.
0
02
04
EXCITATION
06
06
FREWENCY
1
12
(Hz)
Fig. 5. Comparison between the numerical and experimental results of both the maximum and the minimum tensions with the excitation amplitude 0.1 m.
Dynamics of small-sagged taut-slack marine cables
561
Table 1. The nrincinal narameters of the cable and test Values
Parameters diameter length span sag-span ratio
0.01 m
10.9774 m 10.792 m 0.0812 10” N mm2 0.61 kg m-’ 88 N 0.02550.1 m (tl.2 Hz 0.55 Hz
Young’s modulus mass distribution initial tension range of amplitude range of frequency WI
has been quoted in the literature, the uncertainty being dependent upon the strumming effect. Generally, the value in any given case is within the range from 1.0 to 2.4. In Figs 6 and 7, numerical results of both the tension and motion are given for a very small drag coefficient at 0.2. As expected, these results bear the features of vibrating cables in air when one can find pronounced resonances. As the drag coefficient increases, shown in Fig 8 and 9, the resonance becomes unimportant and the motion is reduced considerably. In Fig. 8, the sharp increase in tension is due to the snap loading. Figures 10 and 11 show the effects of the initial tension on both the total tension and the magnitude
1
0
2
3
Nondimensional
frequency
4
Fig. 8. Non-dimensional maximum and minimum tensions vs the non-dimensional excitation frequency 0,/w, with U, equal to 0.075 m.
0
2
3
Nondimensional
frequency
1
4
Fig. 9. Non-dimensional maximum displacements of different modes vs the non-dimensional excitation frequency w,/w, with C, equal to 1.6 and no equal to 0.075 m.
60-
Fig. 6. Non-dimensional maximum and minimum tensions vs the non-dimensional excitation frequency wr/w, with C, equal to 0.2 and u,, equal to 0.075 m. 0
02
04
06
08
1
12
Excitation frequency (Hz)
Fig. 10. Effect of initial tension on both the maximum and minimum tensions in the cable. The excitation amplitude is fixed at 0.075 m.
0 0
2
3
NondimensIonal
1
fmquency
4
Fig. 7. Non-dimensional maximum displacements of different modes vs the non-dimensional excitation frequency or/w,
with C, equal to 0.2 and u,, equal to 0.075 m.
of the first three modes. These figures also show the effect of the sag-span ratio as it is given by (mg - pA)1/(8T,,). As the initial tension increases, the sag decreases. In this case, cable dynamics is influenced mainly by the elastic stiffness, resisting the applied load through stretching. As demonstrated in these two figures, both the maximum tension and the magnitude of modes increase as the initial tension increases.
562
D. Vassalos and S. Huang
employs mainly the catenary stiffness to resist the applied excitation. As the excitation frequency becomes greater, the elastic stiffness becomes more important.
5. CONCLUSIONS
0 0
02
04
06
06
1
12
Excitation frequency (Hz)
Fig. II. Effect of initial tension on the non-dimensional magnitudes of the first three symmetric modes. The excitation amplitude is fixed at 0.075 m.
A non-linear model is developed for small-sagged marine cables subjected to horizontal excitations at one end and operating in taut-slack conditions. A numerical analysis based upon the Galerkin method is conducted to investigate a system of non-linear equations under both parametric and external excitations. Numerical results are compared with available experimental results and a good agreement is achieved. The influence of various factors, such as excitation amplitude, excitation frequency, drag coefficient, initial tension and Young’s modulus, are also examined.
6. REFERENCES
0 0
02
04
0.6
0.6
Excitation frequency
1
1.2
(Hz)
Fig. 12. Effect of Young’s modulus on both the maximum and minimum non-dimensional tensions in the cable. Excitation amplitude is fixed at 0.1 m.
Mooring Dynamics for Offshore Application. Part 1, Theory. MITSG86-1, MIT, Cambridge, MA (1986).
5. J. H. Milgram, M. S. Triantafyllou, F. C. Frimm and G. Anagnostou, Seakeeping and extreme tensions in offshore towing. SNAME Trans. 96, 35-70 (1988). 6. H. Polacheck, T. S. Walton, R. Mejia and C. Dawson, Transient motion of an elastic cable immersed in a fluid. Math. Tables Other Aids Compur. 17, 6043
005
E
E 004 B
d 5P
1. H. M. Irvine, Cable structures. The MIT Press, Cambridge, MA (1981). 2. C. M. Ablow and S. Schechter, Numerical simulation of undersea cable dynamics. Ocean Engng 10, 443457 (1983). 3. J. J. Burgess, Modelling of undersea cable installation with a finite difference method. In: Proc. Int. Ofihore and Polar Engineering Conf., Edinburgh (1991). 4. M. S. Triantafyllou, A. Bliek, J. Burgess and H. Shin,
003
5 6
(1963). 7. C. M. Larsen and I. J. Fylling, Dynamic behaviour of anchor lines. Norwegian Maritime Res. 3 (1982). 8. R. M. Kennedy and E. S. Strahan, A linear theory of
E B c
O.O'
1
0020 L 0
02
0.4
0.6
0.6
1
12
9.
Excitatfon frequency (Hz)
Fig. 13. Effect of Young’s modulus on the non-dimensional magnitudes of the first three symmetrical modes. Excitation amplitude is fixed at 0.1 m.
In Figs 12 and 13, the effects of Young’s modulus
are examined. When the excitation frequency is low, the fluid drag force is small and the system
10. 11.
12.
transverse cable dynamics at low frequencies. NUSC Technical Report 6463, Naval Underwater Systems Centre, Connecticut (1981). M. S. Triantafyllou, Preliminary design of mooring systems. J. Ship Res. 26, 25-35 (1982). J. M. Niedzwecki and S. K. Thampi, Snap loading of marine cable systems. Appl. Ocean Res. 13,2-l 1 (1991). V. J. Papazoglou, S. A. Mavrakos and M. S. TriantafylIOU, Non-linear cable response and model testing in water, J. Sound Vibr. 140, 103-I 15 (1990). I. J. Fylling and P. T. Weld, Cable dynamics--comparison of experimental and analytic results. Report R-89.79, The Ship Research Institute of Norway (1979).