Experimental investigation of nonlinear dynamic tension In mooring lines

Transcription

1 J Mar Sci Technol (2012) 17: DOI /S ORIGINAL ARTICLE Experimental investigation of nonlinear dynamic tension In mooring lines Su-xia Zhang You-gang Tang Xi-jun Liu Received: 20 October 2010/Accepted: 27 December 2011 /Published online: 2 February 2012 JASNAOB 2012 Abstract Mooring lines are one of the most important parts of spar platforms. In this work, experimental investigation was canied out to obtain the dynamic tension and discover the nonlinear characters while a mooring hne transfers from taut to taut-slack. Results show that, when system parameters change, the mooring line system may transform from one steady state to another, accompanied by tension skip that is 5 times the former steady tension and twice the latter steady tension. Such skip tension may result in platform breakage. For the entire cable structure, the effects of system parameters are very important and need further theoretical research. The presented research lays the foundation for further studies on this transforming mechanism and for optimization of engineering designs. Keywords Taut-slack Mooring lines Nonlinear dynamic tension Chaos Experimental investigation 1 Introduction Mooring systems are widely used in ocean engineering. The moored buoyancy moves with large amplitude when S. Zhang (El) X. Liu School of Mechanical Engineering, Tianjin University, Tianjin, China zhangsux@tju.edu.cn S. Zhang X. Liu Tianjin Key Laboratory of Nonlinear Dynamics and Chaos Control, Tianjin University, Tianjin, China Y. Tang School of Civil Engineering, Tianjin University, Tianjin, China the ocean environment is extreme, and mooring lines quickly transfer from a slack state to a taut state, resulting in a snapping action that produces a large tensile force known as a snap load, posing great danger to such ocean structures. Hennessey [1] investigated and developed snap loads on synthetic fiber ropes based on experimental data collected in the master's thesis of Pearson [2], evaluating the behavior of ropes during the snapping action. Additional tests were also conducted under more controlled conditions to better understand how ropes change throughout a sequence of similar snap loadings and also to determine the amount of energy dissipated. Goeller [3] reviewed experimental and analytical investigations dealing with the dynamic response of vertically hanging segmented cable systems where the upper portion is a stranded steel cable and the lower segment is a nylon rope. A generalized distributed mass model has previously been developed by the authors for a segmented cable made up of two viscoelastic materials, including internal damping and linear external damping of the payload and cable. In that model, the viscoelastic behavior is simulated by a two-parameter Voigt model and external linear viscous damping is used for the payload and the cable, but an approximate method takes into account nonlinear damping effects due to the solid-fluid interaction. In general, good agreement was obtained between this mathematical model and experimental results. It was also shown that this agreement is better if one approximates the viscoelastic behavior of the nylon rope by a lumped three-parameter system. The behavior of single nylon cables was also studied analytically and experimentally under dynamic conditions in air and in water media. Fylling [4] presented a comparison of experimentally and theoretically determined tension in a horizontally ^

2 182 J Mar Sci Technol (2012) 17: suspended cable excited by forced end-point motion. The finite-element program LINDYN was based on a linear stiffness representation and includes nonlinear hydrodynamic forces according to the Morison equation. The comparison indicates that the theoretical results coirelate spring " crown block cable of length Fig. 1 Scheme of experimental setup Fig. 2 Experimental photo in tank excitation at a variety of Frequencies and amplitudes tension measurement well with experiments for moderate nonlinearities, provided that the dynamic force does not exceed the static force. Vassalos and Huang [5] canied out a series of model tests in a towing tank, using a horizontally suspended submerged cable to examine snap loading. The results obtained from monitoring the dynamic tension variation are presented in the time and frequency domains, together with a description of the dynamic behaviors of the cable. Sundaravadivelu [6] canied out experimental studies on a single-point taut moored buoy cable system in a 2-mwide regular wave flume for different wave heights and wave periods. Surge and heave accelerations of the buoy were measured using accelerometers inside the buoy in the respective directions. Two ring gages, one at the top and another at the bottom, were used to determine the top and bottom tensions in the cable. The results were processed to plot the variation of acceleration and tension for various wave steepnesses and relative water depths, finding application in study of the behavior of mooring systems. Halliwell [7] presented an extensive model test program on the behavior of single-point mooring (SPM) systems to investigate the influence on system response, as measured by motions and mooring forces. The environmental parameters varied were the frequency and height of regular waves; tests were also made using iixegular waves for comparison with the regular wave results. The influence of the hawser was examined by varying its length and elastic properties. Earth Berntsena [8] presented a conceptually new controller for position mooring operations. By using a structural reliability irieasure for the mooring lines, the new controller protects the mooring system whenever needed as a result of severe weather conditions and high environmental loads by maintaining Fig. 3 Tension history curves: a excited end, b fixed end time (s)

3 J Mar Sci Teclinol (2012) 17: Fig. 4 Tension spectrum at the excited end vortex-induced transverse loading at the expense of increased pitch motions. Zhang [10] investigated the factors affecting marine cable tension in taut-slack conditions. The results showed that, with the same pretension and excitation frequency, snap tension increases with increasing excitation amplitude; with the same excitation amplitude, the snap tension in the line increases with increasing excitation frequency. The tension increases with increasing excitation amplitude, stiffness, and modulus. Snap tension decreases with increases in the length of the line. In this work, experimental investigation was earned out to obtain the dynamic tension while the mooring line transfers from taut to taut-slack and to discover the nonlinear characters, laying the foundation for further studies of this transforming mechanism and optimization of engineering designs. New results are observed in the experiments, being of great significance for research and controllers of snap tension in mooring lines. 2 Model test setup Fig. 5 Tension spectrum at the fixed end The experiments were canied out in a ship model test tank in Tianjin University; the model test setup is shown in Fig. 1 [5]. The spring is used to change the stiffness of the cable system, and the value of the spring constant is N/m. An oscillator is constructed to excite harmonically the upper end of the cable with amplitude up to a maximum of 15 cm. Loading cells were attached to both ends of the cable to record the dynamic tension. The function of the crown block is to immerse the cable in the water, because the density of the cable is less than that of water, or out of the water for convenience during testing. The crown block is fixed on the bottom of the truss, shown as Fig. 2. ' In the experiment, load cells of dynamic strain gage type and resistance strain sensor were used. the probabihty of mooring hne failure below a preset value. In particular, excessive use of thrusters caused by conservatively defined safety regions in conventional PM systems is avoided, yielding fuel-optimal operation. The feasibility of the controller was successfully verified in laboratory experiments. Fitzgerald [9] verified that the dominant behavior can be simulated by a relatively simple mathematical model, allowing the critical parameters of peak anchor loads and pitch angles to be calculated and extrapolated to full scale. The experimental and simulation results demonstrated that the mass characteristics of a non-surface-piercing tower can be used to offset some of the challenges of moving to shallow water. If done correctly, it is possible to keep horizontal anchor loads under control and reduce 3 Results and discussion For cable length of 28 m, pretension of 164 N, frequency of 1.2 Hz, and excitation amplitude of 8 cm, the snap tension was recorded and observed. Tension history curves of the excited and fixed ends are shown in Fig. 3, from which we can see that there is a spike in the curve of the excited end, while the curve at the fixed end is harmonic. The frequency spectra are shown in Figs. 4 and 5. It is evident that, when snap tension occurs, the frequency content at the fixed end is composed of odd harmonics, while the harmonic content of the dynamic tension at the excited end changes from comprising only odd harmonics to including all harmonic orders, which is the same as in ^

4 184 J Mar Sci Teclinol (2012) 17: Fig. 6 Tension history curves at the mooring line end in the initial stage: a excited end, b fixed end tiine(s) time(s) tinie(s) Fig. 7 Tension history curves at the mooring line end in the middle stage: a excited end, b fixed end IOS tiine(s) [4]. The differences between reference [4] and this paper are that the top of the tension curve is relatively flat and the minimum tension is negative in this paper, while in [4] the minimum tension was very small or zero, and even some negative tension appears. These differences are due to the cable material. When the elastic stiffness is large and the bending stiffness of the line is not zero, the relatively large tension lasts some minutes, and the minimum tension is negative. The above results indicate that snap tension appears at the excited end. So, breakage may occur near the chain jack for practical applications in spar platforms. Given cable length of 30 m, excitation frequency of 1.2 Hz, and amplitude 8 cm, the tension in the cable appears to skip. In the first 100 s, the tension curves are shown in Fig. 6, but at 109 s, there is a skip in the tension curve, as shown as Fig. 7, after which the tension curve enters a different state, shown as Fig. 8. In the progress of transforming, there is no human influence. The frequency spectra are shown in Figs. 9 and 10. The tension curve in Fig. 6 is approximately harmonic, and there is no snap tension in the line. Comparison shows that Fig. 8 is similar to Fig. 3, both showing snap tensions. The transforming process from no snap tension to the <Ö Spr inger

5 J Mar Sci Teclmol (2012) 17: Fig. 8 Tension history curves at the mooring line end in the latter stage: a excited end, b fixed end time(s) time(s) _ Fig. 9 Tension spectrum at the excited end Fig. 10 Tension spectrum at the fixed end appearance of snap loading is recorded in Fig. 7, being accompanied with the tension skip. From the experimental results it can be seen that the first maximum tension at the initial stage is N, and at the middle stage the maximum tension is N, but the maximum tension is N during the skip. The transformation from taut to taut-slack condition evidently amplifies the tension amplitude. To reveal the mechanism of tension skip, the tension spectra at the excited and fixed ends were obtained. It is evident from the tension spectrum at the excited end that the harmonic content of the dynamic tension, besides primary frequency multiplication, such as ƒ, 2f, 3f, 4f, 5f, 6f, If, which is similar to Fig. 4, also includes other highfrequency components, which are similar to random vibration. Such random components in a determinate system may represent chaos [11]. Chaos often appears when system parameters are changed. In the process of testing, the length of the cable will become long after several periodic loadings because of plastic deformation, and the elastic modulus will change at the same time, with no other artificial parameters changes. Even at the same excitation frequency and amplitude, the system may be in the taut or taut-slack condition because of different initial conditions. The tension skip indicates that the mooring system is sensitive to the system parameters. When the parameters alter, the system may transform from one steady state, such as that shown in Fig. 6, to

6 186 J Mar Sci Technol (2012) 17: another, such as that- shown in Fig. 8, accompanied by a tension skip that is 5 times the former steady tension and double the latter steady tension. There is chaotic motion in the mooring line. The skip tension may cause platform breakage. For the entire cable structure, the effects of system parameters are very important and need further theoretical research. 4 Conclusions Based upon the test results, the following points can be concluded: 1. In the nonslack condition, tension variation is primarily at the excitation frequency. 2. Once the taut-slack condition sets in, harmonics of higher-frequency components appear in the tension. 3. The taut-slack condition significantly amplifies the maximum tension amplitude. 4. Once the transformation from the taut to taut-slack condition occurs, chaos may appear, the tension amplitude is evidently enlarged, and there is a complex dynamic response, which needs further theoretical research. Acknowledgments Gratefiü thanks are due to the National Natural Science Foundation of China (grant no ), Tianjin Natural Science Foundation (grant no. 09JCZDJC26800), and Seed Foundation of Tianjin University. References 1. Hennessey CM (2003) Analysis and modeling of snap loads on synthetic fiber ropes, master dissertation, Virginia Polytechnic Institute and State University 2. Pearson NJ (2002) Experimental snap loading of synthetic fiber ropes, master dissertation, Virginia Polytechnic Institute and State University 3. Goeller JE, Laura PA (1971) Analytical and experimental study of the dynamic response of segmented cable systems. J Sound Vib 18(3): Fylling IJ, Wold PT (1979) Cable dynamics-comparison of experimental and analytic results, report R (3RD ED). The Ship Research Institute of Norway R 5. Vassalos D, Huang S (2004) Experimental investigation of snap loading of marine cables. In: Proceedings of the 14th international offshore and polar engineering conference, Toulon, France 6. Sundaravadivelu R, Harikrishna Babu M, Murugaganesh R (1991) Experimental investigation on a single point buoy mooring system. Ocean Eng 18(5): Halliwell AR, HaiTis RE (1988) A parametric experimental study of low-frequency motions of single point mooring systems in waves. Appl Ocean Res 10(2): Barth Berntsena PI, Aamob OM, Leira BJ (2009) Ensuring mooring line integrity by dynamic positioning: controller design and experimental tests. Automatica Fitzgerald J, Bergdahl L (2009) Rigid moorings in shallow water: a wave power application. Part I: expeiimental verification of methods. Mar Struct 22: Zhang S-X, Tang Y-G (2009) Experimental investigation into factors affecting marine cable snap tension in taut-slack conditions. J Harbin Eng Univ 30(10): Chen Y-S (2002) Nonlinear vibrations. Higher Education Press, Beijing