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基于深度学习的深海天然气水合物开采用采矿隔水管振动预测模型及振动特性
基于深度学习的深海天然气水合物开采立管振动预测模型及振动特性

引用as：Phys.Fluids 37， 012011 （2025）;d oi：10.1063/5.0245296
引文名称：物理．流体 37，012011 （2025）；道：10．1063／5．0245296 提交日期：2024 年 10 月 25 日 • 接受日期：2024 年 12 月 15 日 •
提交时间： 2024 年 10 月 25 日 - 接受时间： 2024 年 12 月 15 日 - 发表日期： 2024 年 12 月 15 日
在线发布日期：2025 年 1 月 7 日
在线出版： 2025 年 1 月 7 日
郭晓强 1 , 2 , a ) 1 , 2 , a ) ^(1,2,a)){ }^{1,2, a)}（D）李莹伟 2 2 ^(2){ }^{2}（DD Qi Li（李琦）， 2 2 ^(2){ }^{2}（D）宋宇轩（Yuxuan Song）， 3 3 ^(3){ }^{3}（D）徐捷（Jie Xu）， 7 7 ^(7){ }^{7}（D） 及 Kelun Yang（杨恪伦）（D）
背景
1 1 ^(1){ }^{1}河北工业大学 机械工程学院，天津 300401
2 2 ^(2){ }^{2}河北工业大学 电子与信息工程学院，天津 300401
3 3 ^(3){ }^{3}山东大学 信息科学与工程学院，山东 青岛 266237
a） a） ^("a）"){ }^{\text {a）}}）应向其 addressed:786526101@qq.com 通信的作者
2 2 ^(2){ }^{2} 河北工业大学电子与信息工程学院，天津 300401，中国
3 3 ^(3){ }^{3} 信息科学与工程学院，山东大学，青岛 266237，中国
a） a） ^("a）"){ }^{\text {a）}}） 通讯作者：786526101＠qq．com  联盟
1 1 ^(1){ }^{1} 机械工程学院，河北工业大学，天津 300401，中国

抽象  摘要

在深海天然气水合物开采立水管作业过程中，采用简化的理论模型获得振动预测，其准确性无法得到有效保证，深度学习方法可以有效解决这一问题，因此，利用基于深度学习的长短期记忆网络建立了深海天然气水合物开采立管的三维振动预测模型，并可以借助获得的采矿立管振动数据进行训练在现场，并实现对矿用隔水管振动响应的提前预测，以期在后期 period.In 顺序有效地验证模型的正确性，采用类似的原理开发了一种模拟内外流激励下采矿隔水管振动的实验台，将实验试验结果与模型预测结果进行比较，并得出决策系数 ( R 2 ) R 2 (R^(2))\left(R^{2}\right)达到 99 % 99 % 99%99 \%进一步验证了该模型能够实现深海水合物开采隔水管振动预测，采用能量法和汉密尔顿原理建立了深海水合物开采隔水管气-液-固三相流诱导振动的理论模型，将后期预测结果与理论模型计算结果进行了对比，发现 culations.It测定 ( R 2 ) R 2 (R^(2))\left(R^{2}\right)达到 94.59 % 94.59 % 94.59%94.59 \%在此基础上，预测了不同剪切流、平台升沉运动参数、提升流速、水合物丰度和水合物粒径下采矿隔水管的振动响应，研究了作和结构参数对采矿隔水管振动响应的影响，揭示了采矿隔水管的振动特性，研究结果提供了水合物开采立管安全的理论基础和预测建模工具。
在深海天然气水合物开采立管运行过程中，其振动预测采用简化的理论模型，其精度无法得到有效保证。为了有效验证模型的正确性，利用类似原理开发了实验平台，用于模拟采矿立管在内外流激励下的振动。此外，为了进一步验证该模型能够实现对深井采矿立管的振动预测，还对该模型的能量方法进行了验证。为进一步验证模型能够实现深海水合物开采立管的振动预测，利用能量法和汉密尔顿原理建立了深海水合物开采立管气液固三相流诱导振动的理论模型。在此基础上，预测了采矿立管在不同剪切流、平台波浪运动参数、提升流速、水合物丰度和水合物粒径条件下的振动响应。 研究了操作和结构参数对采矿立管振动响应的影响，揭示了采矿立管的振动特性。 研究结果为水合物采矿立管的安全提供了理论基础和预测模型工具。

根据 AIP Publishing 的独家许可发布.https：//doi.org/10.1063/5.0245296
由 AIP Publishing 独家授权出版．https：／／doi．org／10．1063／5．0245296

一、引言  I． 引 言

天然气水合物被认为是一种低碳、不合时宜、有潜力的替代能源，主要位于深水区（水深 500 − 1500 m 500 − 1500 m 500-1500m500-1500 \mathrm{~m}）和超深水（水深1500 m以上），其中大部分具有埋深浅、矿床松散、地层承载力低等特点，提出了井口吸锚井固态流态化开采的创新方法。 1 , 2 1 , 2 ^(1,2){ }^{1,2}主要思想是将水合矿体转移到封闭的多相气体开采冒口中，利用提升过程中水温升高和静水压力降低的自然规律，在提升过程中水合物
天然气水合物被认为是一种低碳、非传统、潜在的替代能源，主要分布在深水（水深 500 − 1500 m 500 − 1500 m 500-1500m500-1500 \mathrm{~m} ）和超深水（水深 1500 米以上），大多具有埋藏浅、矿床松散、地层承载力低的特点。 1 , 2 1 , 2 ^(1,2){ }^{1,2} 其主要思路是将水合物矿体移入封闭的多相气开采立管中，利用海水温度升高、静水压力降低的自然规律，使水合物在提升过程中逐渐蒸发，并逐渐气化。
逐渐蒸发[如图1（a）所示].作为核心设备，采矿提升管受内部气-液-固多相流（水合矿机体受环境因素自然分解成天然气）、结构接触和碰撞（立管-连续管和套管-连续管）、外部海洋随机载荷和平台的运动[如图1（b）所示].振动机理异常复杂，容易导致摩擦射孔、疲劳断裂、井口失稳等严重事故。 3 , 4 3 , 4 ^(3,4){ }^{3,4}然而，目前深海采矿隔水管的振动预测仅限于建立理论和仿真模型，需要对实际工况进行大量简化，导致精度下降
在提升过程中［如图 1（a）所示］。 作为核心设备，采矿立管受到内部气-液-固多相流的共同作用（水合物矿体因环境因素自然分解为天然气），结构接触和碰撞（立管-连续管和套管-连续管），以及永恒海洋的塑料装载，和套管-连续管的碰撞。环境因素自然分解为天然气）、结构接触和碰撞（隔水管-连续管、套管-连续管）、外部海洋随机载荷和平台运动等因素的影响［如图所示 1（b）］．振动机理异常复杂，极易引发严重事故，如摩擦穿孔、疲劳断裂、井口失稳等。然而，目前对深海采矿立管的振动预测仅限于建立理论模型和模拟模型，需要对实际工况进行大量简化，导致预测精度下降。
图 1.深海天然气水合物的固态流态化萃取。
图 1.深海天然气水合物的固态流化提取。
的预测结果。随着远程传输技术的发展，可以将采矿隔水管的振动数据实时传输到平台。亟需建立基于深度学习的深海采矿隔水管预测模型。利用现场实测数据，可预测开采隔水管在作业后期的振动响应，指导作业参数的实时调整，确保采矿隔水管在深海天然气水合物开采中的安全。
的预测结果。随着远程传输技术的发展，采矿立管的振动数据可以实时传输到平台。建立基于深度学习的深海采矿立管预测模型迫在眉睫。利用现场实测数据，预测采矿立管在后期作业中的振动响应，指导作业参数的实时调整，确保深海天然气水合物开采中采矿立管的安全。
对于矿用隔水管的振动预测，早期学者主要针对不同工况进行振动理论建模研究，利用传统的动力学理论，建立相应的控制方程，并实现其数值解，得到隔水管在不同工况下的振动响应数据。最初，重点是研究管柱中的单相流诱导振动。艾特肯 5 5 ^(5){ }^{5}研究了无外励磁管中流体诱导柱振动的问题，并确定了管中流体诱导柱振动的现象。之后，学者们主要集中在流体感应冒口振动模型的建立和求解上， 6 6 ^(6){ }^{6}在不考虑非线性因素的情况下建立了振动模型，得到了立管的解析解，由于工况复杂，模型的计算精度无法满足现场要求。Dou 等人开始在立管的流体诱导振动中考虑非线性因素，并建立了柱子的非线性振动模型，但建立的动力学模型只考虑了一个方向的振动。之后，Li 等人。 8 8 ^(8){ }^{8}发现大长径比管状柱的纵横向耦合效应明显，不容忽视。出于这个原因，Xing 等人。 9 9 ^(9){ }^{9}建立了梁的纵向和横向耦合振动模型。近年来，随着中国油气向深海开发，钻完井作业的风险和效率不断遇到新的瓶颈。Gao 等人。 10 10 ^(10){ }^{10}指出，深水钻井是深水条件下海上油气工程的关键环节之一，需要特殊的深水管道和立管系统（包括深水导管、钻井分水器、套管等）。深水舱壁作为
对于矿用立管的振动预测，早期学者主要开展了不同工况下的振动理论建模研究，利用传统的动力学理论，建立相应的控制方程，并实现其数值求解，得到不同工况下立管的振动响应数据。最初的重点是研究单相流引起的油管串振动。Aitken 5 5 ^(5){ }^{5} 在没有外部激励的情况下研究了管内流体诱发的管柱振动问题，并确定了管内流体诱发的管柱振动现象。之后，学者们主要关注流体诱导立管振动模型的建立和求解， 6 6 ^(6){ }^{6} 建立的振动模型没有考虑非线性因素，得到了立管的解析解，随着工况的复杂化，模型的计算精度不能满足现场要求。Dou 等人开始考虑立管流体诱导振动中的非线性因素，建立了立管柱的非线性振动模型，但建立的动力学模型只考虑了单向振动。之后，Li 等人 8 8 ^(8){ }^{8} 发现大长径比管柱的纵向和横向耦合效应非常明显，不容忽视。为此，Xing 等 9 9 ^(9){ }^{9} 建立了梁的纵横向耦合振动模型。近年来，随着我国油气向深海发展，钻完井作业的风险和效率不断遇到新的瓶颈。Gao 等人 10 10 ^(10){ }^{10} 指出，深水钻井是深水条件下海洋油气工程的关键环节之一，需要特殊的深水管材和隔水管系统（包括深水导管、钻井水分离器、套管等）。由于海上油气开采的核心设备，受内部流体冲击载荷和外部海洋随机载荷和平台运动的影响，其非线性振动机构较为复杂。为了有效解决这个难题，Li et al. 11 11 ^(11){ }^{11}和 胡 等人。 12 12 ^(12){ }^{12}开展了随机波作用下海洋平台的动力响应特性研究，为海洋立管振动分析奠定了上边界基础。Zhang 等人。 13 13 ^(13){ }^{13}采用数值仿真技术模拟深海中的波浪运动，为立管的外部载荷边界奠定了基础。Zhang 等人。 14 14 ^(14){ }^{14}长期致力于竖井的力学行为，建立了深水钻井平台-竖管耦合系统的动力学模型和波激疲劳损伤评估方法。Zhang 等人。 15 15 ^(15){ }^{15}和 Lu 等人。 16 16 ^(16){ }^{16}借助各机构建立的大型海洋深水实验室平台，开展了海洋立管涡振的实验研究，通过运行参数和疲劳失效机理阐明了立管涡振的特点，为立管的设计奠定了实验基础。在近期的工作中，作者长期致力于海洋立管的非线性振动理论模型、数值求解和仿真实验的研究，先后建立了深水试验管-立管系统的耦合振动模型 17 17 ^(17){ }^{17}以及高温、高压、高产量的冒口的非线性流变振动模型 ( 3 H ) ( 3 H ) (3H)(3 \mathrm{H})气井 18 18 ^(18){ }^{18}研究了不同条件下隔水管和 3 H 气井的非线性振动特性，揭示了隔水管的摩擦磨损和振动疲劳失效机理。 19 19 ^(19){ }^{19}
作为海洋油气开采的核心设备，深水舱壁受内部流体冲击载荷和外部海洋随机载荷及平台运动的影响，其非线性振动机理较为复杂。为了有效解决这一难题，Li 等 11 11 ^(11){ }^{11} 和 Hu 等 12 12 ^(12){ }^{12} 对海洋平台在随机波作用下的动态响应特性进行了研究，为海洋立管的振动分析奠定了上边界基础。Zhang 等 13 13 ^(13){ }^{13} 利用数值模拟技术模拟了深海中的波浪运动，为立管外荷载边界奠定了基础。Zhang 等 14 14 ^(14){ }^{14} 长期致力于隔水管的力学行为研究，建立了深水钻井平台-隔水管耦合系统动态模型和波激疲劳损伤评估方法。Zhang 等 15 15 ^(15){ }^{15} 和 Lu 等 16 16 ^(16){ }^{16} 借助各单位建立的大型海洋深水实验平台，开展了海洋立管涡振实验研究，阐明了立管涡振的运行参数特征和疲劳失效机理，为立管设计奠定了实验基础。 在最近的工作中，作者长期致力于海洋隔水管的非线性振动理论模型、数值求解和模拟实验研究，先后建立了深水 17 17 ^(17){ }^{17} 试验管-隔水管系统耦合振动模型和高温、高压、高产 ( 3 H ) ( 3 H ) (3H)(3 \mathrm{H}) 气井隔水管非线性流变振动模型，研究了隔水管和 3 H 气井在不同工况下的非线性振动特性，揭示了隔水管和 3 H 气井在不同工况下的非线性振动特性、 ( 3 H ) ( 3 H ) (3H)(3 \mathrm{H}) 气井隔水管的非线性流变振动模型， 18 18 ^(18){ }^{18} 研究了隔水管和 3 H 气井在不同条件下的非线性振动特性，揭示了隔水管的摩擦磨损和振动疲劳失效机理。 19 19 ^(19){ }^{19}
关于冒口多相流致振动的研究主要集中在研究气-液和固-液两相流对冒口的振动影响上，缺乏对气-液-固三相流对冒口振动影响的分析。首先，学者们为刚性管道的两相流振动建立了水平管道气液两相流振动模型。 20 20 ^(20){ }^{20}他们发现，管道的振动响应与两相流参数（密度、压力和流速）密切相关。随着实验仪器性能的
立管多相流诱导振动的研究主要集中于研究气液、固液两相流对立管的振动影响，缺乏气液固三相流对立管振动影响的分析。最初，学者们针对刚性管道两相流振动建立了水平管道气液两相流振动模型。 20 20 ^(20){ }^{20} 他们发现管道的振动响应与两相流参数（密度、压力和流速）密切相关。随着提高和计算机技术的发展，一些学者采用理论方法结合数值模拟技术，建立了蒸汽发生器传热管气液两相流振动模型 21 21 ^(21){ }^{21} and a gas-liquid two-phase flow vibration model of marine flexible risers 22 22 ^(22){ }^{22} and put forward a fatigue life prediction model based on the prediction of neural network under the effect of random load 23 , 24 23 , 24 ^(23,24){ }^{23,24} An experimental method was used to investigate the gas-liquid two-phase flow-induced vibration study of marine flexible risers, and it was found that the gas-liquid ratio and the gas-liquid twophase flow rate are the main factors affecting the vibration of flexible risers. Some scholars 25 , 26 25 , 26 ^(25,26){ }^{25,26} investigated the solid-liquid two-phase flow vibration problem of deep-sea lift pipe, established the particle flow model inside the pipe and the solid-liquid two-phase flow vibration model of the lift pipe, and found that the particle concentration and the transport velocity were the main influencing factors of the vibration response of the lift pipe, and Guo et al. 27 27 ^(27){ }^{27} used the energy method and the Hamilton’s principle to establish a solid-liquid two-phase flow vibration model of a hydraulic lift pipe for deep-sea mining and predicted the excitation of the inner and outer flow of the lift pipe and the solid-liquid two-phase flow vibration model of a deep-sea mining hydraulic lift pipe.
关于冒口多相流致振动的研究主要集中在研究气-液和固-液两相流对冒口的振动影响上，缺乏对气-液-的分析。固三相流对冒口振动影响的分析。首先，学者们为刚性管道的两相流振动建立了水平管道气液两相流振动模型。 22 22 ^(22){ }^{22} ，提出了随机载荷作用下基于神经网络预测的疲劳寿命预测模型 23 , 24 23 , 24 ^(23,24){ }^{23,24} ，并采用实验方法对船用柔性立管进行了气液两相流振动研究、发现气液比和气液两相流流速是影响柔性立管振动的主要因素。有学者 25 , 26 25 , 26 ^(25,26){ }^{25,26} 研究了深海升降管的固液两相流振动问题，建立了管内颗粒流动模型和升降管的固液两相流振动模型，发现颗粒浓度和输运速度是升降管振动响应的主要影响因素，Guo 等 27 27 ^(27){ }^{27} 研究了深海升降管的固液两相流振动问题，建立了管内颗粒流动模型和升降管的固液两相流振动模型，发现颗粒浓度和输运速度是升降管振动响应的主要影响因素。 27 27 ^(27){ }^{27} 利用能量法和汉密尔顿原理建立了深海采矿液压提升管的固液两相流振动模型，并预测了提升管内外流的激振情况和深海采矿液压提升管的固液两相流振动模型。
With the development of intelligent technology, structural vibration prediction research has also evolved from theoretical modeling to data-driven prediction based on field measurement data. In recent years, deep learning has been widely applied in computer vision, 28 28 ^(28){ }^{28} medical images, 29 , 30 29 , 30 ^(29,30){ }^{29,30} and time series. 31 , 32 31 , 32 ^(31,32){ }^{31,32} The hierarchical neural network model with reinforcement learning capability was proposed, such as a deep belief network (DBN), which accelerated the development of deep learning. 33 , 34 33 , 34 ^(33,34){ }^{33,34} Deep learning can fully exploit the feature information in big data, and the massive data can offset the complexity increase brought by deep learning and improve its generalization ability. Long short-term memory (LSTM) network 35 − 38 35 − 38 ^(35-38){ }^{35-38} is a variant of recurrent neural network 39 , 40 39 , 40 ^(39,40){ }^{39,40} (RNN), which is particularly suitable for dealing with long sequence prediction problems. LSTM has been widely used in long-sequence prediction. Zha et al. 41 41 ^(41){ }^{41} proposed a convolutional neural network (CNN)-LSTM model to predict gas field production. Lin et al. 42 42 ^(42){ }^{42} proposed a two-stage attention LSTM model to predict power loadings. Cho et al. 43 43 ^(43){ }^{43} integrated weather research with a forecast hydrological modeling system (WRF-Hydro) and an LSTM model to improve runoff simulation. Zhang et al. 44 44 ^(44){ }^{44} proposed a CNNLSTM model to predict future air quality conditions. Huang et al. 45 45 ^(45){ }^{45} proposed an LSTM model considering the gas injection effect to predict the production of carbonate reservoirs. Stefenon et al. 46 46 ^(46){ }^{46} proposed a sequence-to-sequence-LSTM model to predict the 1 h reservoir water level accurately. Shen et al. 47 47 ^(47){ }^{47} proposed a CNN-LSTM model to accurately predict wind speed and ensure the safety of traveling unmanned sailing boats. Zhang et al. 48 48 ^(48){ }^{48} proposed the CNN-LSTM model, which considers the spatiotemporal characteristics of soil temperature field (STF) and accurately predicts effluent temperature. Huang et al. 49 49 ^(49){ }^{49} proposed a CNN-LSTM model with a conditional generative adversarial network-CNN and bi-directional short-term and long-term memory (Bi-LSTM) to improve the accuracy of hourly photovoltaic (PV) power prediction and Siłka et al. 50 50 ^(50){ }^{50} proposed an RNN-LSTM model to predict the vibration of high-speed trains with an accuracy of more than 99 % 99 % 99%99 \%. Li et al. 51 51 ^(51){ }^{51} proposed a hybrid model combining a Bayesian neural network and an impedance model to predict the vibration response of
随着智能技术的发展，结构振动预测研究也从理论建模发展到基于现场测量数据的数据驱动预测。近年来，深度学习已广泛应用于计算机视觉、 28 28 ^(28){ }^{28} 医学图像、 29 , 30 29 , 30 ^(29,30){ }^{29,30} 和时间序列等领域。 31 , 32 31 , 32 ^(31,32){ }^{31,32} 具有强化学习能力的分层神经网络模型被提出，如深度信念网络（DBN），加速了深度学习的发展。 33 , 34 33 , 34 ^(33,34){ }^{33,34} 深度学习可以充分利用大数据中的特征信息，海量数据可以抵消深度学习带来的复杂性增加，提高深度学习的泛化能力。长短期记忆（LSTM）网络 35 − 38 35 − 38 ^(35-38){ }^{35-38} 是递归神经网络 39 , 40 39 , 40 ^(39,40){ }^{39,40} （RNN）的一种变体，特别适合处理长序列预测问题。LSTM 已广泛应用于长序列预测。Zha 等人 41 41 ^(41){ }^{41} 提出了一种卷积神经网络（CNN）-LSTM 模型来预测气田产量。Lin 等人 42 42 ^(42){ }^{42} 提出了一种两阶段注意 LSTM 模型来预测电力负荷。Cho 等 43 43 ^(43){ }^{43} 将气象研究与预报水文建模系统（WRF-Hydro）和 LSTM 模型相结合，改进了径流模拟。Zhang 等人 44 44 ^(44){ }^{44} 提出了一种 CNNLSTM 模型来预测未来的空气质量状况。Huang 等 45 45 ^(45){ }^{45} 提出了一种考虑注气效应的 LSTM 模型，用于预测碳酸盐岩储层的产量。Stefenon 等人 46 46 ^(46){ }^{46} 提出了一种序列到序列-LSTM 模型，用于准确预测 1 h 储层水位。Shen 等人 47 47 ^(47){ }^{47} 提出了一种 CNN-LSTM 模型，用于准确预测风速，确保无人驾驶帆船的行驶安全。 Zhang 等人 48 48 ^(48){ }^{48} 提出的 CNN-LSTM 模型考虑了土壤温度场（STF）的时空特征，能准确预测污水温度。Huang 等人 49 49 ^(49){ }^{49} 提出了一种具有条件生成对抗网络-CNN 和双向短时与长时记忆（Bi-LSTM）的 CNN-LSTM 模型，以提高每小时光伏（PV）功率预测的精度；Siłka 等人 50 50 ^(50){ }^{50} 提出了一种 RNN-LSTM 模型，用于预测高速列车的振动，精度超过 99 % 99 % 99%99 \% 。Li 等人 51 51 ^(51){ }^{51} 提出了一种结合贝叶斯神经网络和阻抗模型的混合模型，用于预测高速列车的振动响应。
buildings. LSTM models are widely used in power, water, industrial, and manufacturing fields to improve early warning and safety in the field.
建筑物 LSTM 模型被广泛应用于电力、水利、工业和制造领域，以改善现场的预警和安全。
It can be seen that the existing marine riser vibration models mainly focus on the traditional theoretical modeling, using the classical kinetic theory (D’Alembert’s principle, Hamilton’s principle, Lagrange’s equation) and realizing the prediction of riser vibration data through numerical solving. However, traditional theoretical modeling always needs to simplify some field conditions, and it cannot predict the vibration changes of the riser induced by sudden changes in the environmental conditions during the operation. With the development of remote transmission technology, the real-time acquisition of deep-sea riser vibration data can be realized, but the prediction research of late vibration conditions on the acquired real-time data have not been reported. For this reason, a prediction model of deepsea hydrate mining riser vibrational displacement based on deep learning with the help of the LSTM network is established. It obtains vibrational displacement data through the vibration simulation experiments of the mining riser and the theoretical vibration model, and part of the data is used as the training model. The vibrational displacement data obtained from the vibration simulation experiment and the vibration theoretical model of the mining riser, part of which is used as the training set and part as the test set, are used to validate the vibration prediction model of the mining riser based on deep learning, based on which the vibration response of the riser pipe is predicted under different operating parameters. The vibration characteristics are not disclosed, so a model basis is provided for the vibration monitoring of the deepsea hydrate mining riser.
可以看出，现有的海洋立管振动模型主要以传统的理论建模为主，采用经典的动力学理论（达朗贝尔原理、汉密尔顿原理、拉格朗日方程），通过数值求解实现对立管振动数据的预测。然而，传统的理论建模总是需要简化一些现场条件，无法预测运行过程中环境条件突变所诱发的立管振动变化。随着远程传输技术的发展，深海立管振动数据的实时采集得以实现，但对采集到的实时数据进行后期振动状况预测研究的报道却寥寥无几。为此，本文借助 LSTM 网络，建立了基于深度学习的深海水合物开采立管振动位移预测模型。该模型通过采矿立管振动模拟实验和理论振动模型获取振动位移数据，并将部分数据作为训练模型。通过振动仿真实验和采矿立管振动理论模型获得的振动位移数据，一部分作为训练集，一部分作为测试集，用于验证基于深度学习的采矿立管振动预测模型，并在此基础上预测立管在不同运行参数下的振动响应。振动特性未公开，因此为深海水合物采矿立管的振动监测提供了模型基础。

II. PREDICTION MODEL FOR MINING RISER VIBRATION BASED ON DEEP LEARNING
II.基于深度学习的采矿立管振动预测模型

A. LSTM model for prediction vibration of mining riser
A.用于预测采矿立管振动的 LSTM 模型

In deep learning, the design of neural networks has evolved to address different challenges. Traditional neural networks process information through fully connected layers, but they face the problems of gradient vanishing and gradient explosion when dealing with time series data. These problems make it difficult for the networks to learn long-term dependencies, limiting their performance when dealing with long sequence data. To address these problems, recurrent neural networks (RNNs) were designed specifically for processing sequence data. However, RNNs still have challenges in dealing with long-term dependencies. LSTM was introduced to overcome these challenges. LSTM improve the performance of RNNs by introducing three key compo-nents-forgetting gates, input and output gates, and a memory unit. The forgetting gate is responsible for deciding which historical information needs to be retained, the input gate controls the addition of new information, and the output gate determines the final output content. This design allows the LSTM to memorize or forget information selectively, efficiently process long sequential data, and capture key features. The LSTM structure is schematically shown in Fig. 2.
在深度学习中，神经网络的设计不断发展，以应对不同的挑战。传统的神经网络通过全连接层处理信息，但在处理时间序列数据时面临梯度消失和梯度爆炸的问题。这些问题使得神经网络难以学习长期依赖关系，从而限制了其处理长序列数据时的性能。为了解决这些问题，人们专门设计了用于处理序列数据的递归神经网络（RNN）。然而，RNN 在处理长期依赖关系时仍面临挑战。LSTM 的出现就是为了克服这些难题。LSTM 通过引入三个关键部件（遗忘门、输入和输出门以及存储单元）来提高 RNN 的性能。遗忘门负责决定哪些历史信息需要保留，输入门控制新信息的添加，输出门决定最终的输出内容。这种设计使 LSTM 能够有选择地记忆或遗忘信息，高效处理长序列数据，并捕捉关键特征。LSTM 结构示意图如图 2 所示。
f t = σ ( W f ⋅ ( h t − 1 , x t ) + b f ) , i t = σ ( W i ⋅ ( h t − 1 , x t ) + b i ) , C ~ t = tanh ⁡ ( W C ⋅ ( h t − 1 , x t ) + b C ) , C t = f t ⋅ C t − 1 + i t ⋅ C ~ t o t = σ ( W o ⋅ ( h t − 1 , x t ) + b o ) , f t = σ W f ⋅ h t − 1 , x t + b f , i t = σ W i ⋅ h t − 1 , x t + b i , C ~ t = tanh ⁡ W C ⋅ h t − 1 , x t + b C , C t = f t ⋅ C t − 1 + i t ⋅ C ~ t o t = σ W o ⋅ h t − 1 , x t + b o , {:[f_(t)=sigma(W_(f)*(h_(t-1),x_(t))+b_(f))","],[i_(t)=sigma(W_(i)*(h_(t-1),x_(t))+b_(i))","],[ tilde(C)_(t)=tanh(W_(C)*(h_(t-1),x_(t))+b_(C))","],[C_(t)=f_(t)*C_(t-1)+i_(t)* tilde(C)_(t)],[o_(t)=sigma(W_(o)*(h_(t-1),x_(t))+b_(o))","]:}\begin{gathered} f_{t}=\sigma\left(W_{f} \cdot\left(h_{t-1}, x_{t}\right)+b_{f}\right), \\ i_{t}=\sigma\left(W_{i} \cdot\left(h_{t-1}, x_{t}\right)+b_{i}\right), \\ \tilde{C}_{t}=\tanh \left(W_{C} \cdot\left(h_{t-1}, x_{t}\right)+b_{C}\right), \\ C_{t}=f_{t} \cdot C_{t-1}+i_{t} \cdot \tilde{C}_{t} \\ o_{t}=\sigma\left(W_{o} \cdot\left(h_{t-1}, x_{t}\right)+b_{o}\right), \end{gathered}
FIG. 2. Schematic diagram of LSTM structure.
图 2.LSTM 结构示意图。
h t = o t ⋅ tanh ⁡ ( C t ) , h t = o t ⋅ tanh ⁡ C t , h_(t)=o_(t)*tanh(C_(t)),h_{t}=o_{t} \cdot \tanh \left(C_{t}\right),
where f t , i t , C t , o t , h t f t , i t , C t , o t , h t f_(t),i_(t),C_(t),o_(t),h_(t)f_{t}, i_{t}, C_{t}, o_{t}, h_{t}, and C ~ t C ~ t tilde(C)_(t)\tilde{C}_{t} are the oblivion gate state, the input gate state, the cell state, the output gate state, the hidden layer output vector, and the input state of memory unit, respectively; W f , W i , W C W f , W i , W C W_(f),W_(i),W_(C)W_{f}, W_{i}, W_{C}, and W o W o W_(o)W_{o} are the matrix of weights; b f , b i , b C b f , b i , b C b_(f),b_(i),b_(C)b_{f}, b_{i}, \mathrm{~b}_{C}, and b o b o b_(o)\mathrm{b}_{o} are the bias terms; and σ σ sigma\sigma is sigmoid activation function.
其中， f t , i t , C t , o t , h t f t , i t , C t , o t , h t f_(t),i_(t),C_(t),o_(t),h_(t)f_{t}, i_{t}, C_{t}, o_{t}, h_{t} 和 C ~ t C ~ t tilde(C)_(t)\tilde{C}_{t} 分别为遗忘门状态、输入门状态、单元状态、输出门状态、隐层输出向量和存储单元的输入状态； W f , W i , W C W f , W i , W C W_(f),W_(i),W_(C)W_{f}, W_{i}, W_{C} 和 W o W o W_(o)W_{o} 为权值矩阵； b f , b i , b C b f , b i , b C b_(f),b_(i),b_(C)b_{f}, b_{i}, \mathrm{~b}_{C} 和 b o b o b_(o)\mathrm{b}_{o} 为偏置项； σ σ sigma\sigma 为西格码激活函数。
LSTM 模型，由门参与，这些门是忘记门、输入门、输出门。所有门均包括 f i , i t , o t ∈ { 0 , 1 } , 0 f i , i t , o t ∈ { 0 , 1 } , 0 f_(i),i_(t),o_(t)in{0,1},0f_{i}, i_{t}, o_{t} \in\{0,1\}, 0表示门已关闭且未传递任何序列，1 表示门已打开且序列通过。失忆门 f t ∈ { 0 , 1 } f t ∈ { 0 , 1 } f_(t)in{0,1}f_{t} \in\{0,1\}由一个 sigmoid 函数组成，该函数确定是保留还是丢弃单元格状态中的序列，其中 0 表示遗忘，1 表示记住序列。输入门由两层组成， i t ∈ { 0 , 1 } i t ∈ { 0 , 1 } i_(t)in{0,1}i_{t} \in\{0,1\}确定是否更新单元格的输入，以及 tanh 函数 ∈ { − 1 , 1 } ∈ { − 1 , 1 } in{-1,1}\in\{-1,1\}为单元创建新的向量，最后将这些层组合在一起，以使用 Eqs 中显示的公式更新 LSTM 模型单元。（4） 和 （6）。输出门由 sigmoid 和 tanh 函数组成，用于确定 LSTM 模型对序列输出的贡献，而双曲割线函数最终在 LSTM 单元之间映射，如 Eqs 所示。（1）-（3） 和 （5）。 σ ( ) σ ( ) sigma()\sigma()和 tanh ⁡ ( ) tanh ⁡ ( ) tanh()\tanh ()是激活函数，公式如下所示：
σ ( x ) = 1 1 + e − x tanh ⁡ ( x ) = e x − e − x e x + e − x σ ( x ) = 1 1 + e − x tanh ⁡ ( x ) = e x − e − x e x + e − x {:[sigma(x)=(1)/(1+e^(-x))],[tanh(x)=(e^(x)-e^(-x))/(e^(x)+e^(-x))]:}\begin{aligned} \sigma(x) & =\frac{1}{1+e^{-x}} \\ \tanh (x) & =\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}} \end{aligned}
与大多数神经网络模型类似，LSTM 的参数初始化是随机的。在训练过程中，训练数据用于通过时间反向传播 （BPTT） 算法计算损失函数和梯度，随后根据这些梯度更新模型的参数，直到达到优化的稳态。平方误差通常用作在 t t tt第 th 个时间步长。
E t = ( y t − Y t ) 2 E t = y t − Y t 2 E_(t)=(y_(t)-Y_(t))^(2)E_{\mathrm{t}}=\left(y_{t}-Y_{t}\right)^{2}
哪里 E t E t E_(t)E_{t}是 t t tt第 time step 和 y t y t y_(t)y_{t}和 Y t Y t Y_(t)Y_{t}是 t t tt第 th 个时间步长。在反向传播中，权重矩阵 W f , W i , W C , W o W f , W i , W C , W o W_(f),W_(i),W_(C),W_(o)W_{f}, W_{i}, W_{C}, W_{\mathrm{o}}分为两部分 W f h , W f x W f h , W f x W_(fh),W_(fx)W_{f h}, W_{f x},
W i h , W i x , W C h , W C x , W o h , W o x W i h , W i x , W C h , W C x , W o h , W o x W_(ih),W_(ix),W_(Ch),W_(Cx),W_(oh),W_(ox)W_{i h}, W_{i x}, W_{C h}, W_{C x}, W_{o h}, W_{o x}.LSTM 有四个加权输入，分别对应于 f t , i t , C t , o t f t , i t , C t , o t f_(t),i_(t),C_(t),o_(t)f_{t}, i_{t}, C_{t}, o_{t}中，重新定义四个加权输入项和误差项，如下所示：
net f , t = W f [ h t − 1 , x t ] + b f = W f h h t − 1 + W f x x t + b f , net i , t = W i [ h t − 1 , x t ] + b i = W i h h t − 1 + W i x x t + b i , net C ~ , t = W C [ h t − 1 , x t ] + b C = W C h h t − 1 + W C x x t + b C , net o , t = W o [ h t − 1 , x t ] + b o = W o h h t − 1 + W o x x t + b o δ f , t = def ∂ E t ∂ n e t f , t , δ i , t = def ∂ E t ∂ n e t i , t , δ C ~ , t = def ∂ E t ∂ n e t C ~ , t δ o , t = = def ∂ E t ∂ n e t o , t .  net  f , t = W f h t − 1 , x t + b f = W f h h t − 1 + W f x x t + b f ,  net  i , t = W i h t − 1 , x t + b i = W i h h t − 1 + W i x x t + b i ,  net  C ~ , t = W C h t − 1 , x t + b C = W C h h t − 1 + W C x x t + b C ,  net  o , t = W o h t − 1 , x t + b o = W o h h t − 1 + W o x x t + b o δ f , t =  def  ∂ E t ∂ n e t f , t , δ i , t =  def  ∂ E t ∂ n e t i , t , δ C ~ , t =  def  ∂ E t ∂ n e t C ~ , t δ o , t = =  def  ∂ E t ∂ n e t o , t . {:[" net "_(f,t)=W_(f)[h_(t-1),x_(t)]+b_(f)=W_(fh)h_(t-1)+W_(fx)x_(t)+b_(f)","],[" net "_(i,t)=W_(i)[h_(t-1),x_(t)]+b_(i)=W_(ih)h_(t-1)+W_(ix)x_(t)+b_(i)","],[" net "_( tilde(C),t)=W_(C)[h_(t-1),x_(t)]+b_(C)=W_(Ch)h_(t-1)+W_(Cx)x_(t)+b_(C)","],[" net "_(o,t)=W_(o)[h_(t-1),x_(t)]+b_(o)=W_(oh)h_(t-1)+W_(ox)x_(t)+b_(o)],[delta_(f,t)="" def ""(delE_(t))/(del net_(f,t))","],[delta_(i,t)="" def ""(delE_(t))/(del net_(i,t))","],[delta_( tilde(C),t)="" def ""(delE_(t))/(del net_( tilde(C),t))],[delta_(o,t)=_("=")^("" def "")(delE_(t))/(del net_(o,t)).]:}\begin{aligned} & \text { net }_{f, t}=W_{f}\left[h_{t-1}, x_{t}\right]+b_{f}=W_{f h} \mathrm{~h}_{t-1}+W_{f x} x_{t}+b_{f}, \\ & \text { net }_{i, t}=W_{i}\left[h_{t-1}, x_{t}\right]+b_{i}=W_{i h} \mathrm{~h}_{t-1}+W_{i x} x_{t}+b_{i}, \\ & \text { net }_{\tilde{C}, t}=W_{C}\left[h_{t-1}, x_{t}\right]+b_{C}=W_{C h} \mathrm{~h}_{t-1}+W_{C x} x_{t}+b_{C}, \\ & \text { net }_{o, t}=W_{o}\left[h_{t-1}, x_{t}\right]+b_{o}=W_{o h} \mathrm{~h}_{t-1}+W_{o x} x_{t}+b_{o} \\ & \delta_{f, t} \xlongequal{\text { def }} \frac{\partial E_{t}}{\partial n e t_{f, t}}, \\ & \delta_{i, t} \xlongequal{\text { def }} \frac{\partial E_{t}}{\partial n e t_{i, t}}, \\ & \delta_{\tilde{C}, t} \xlongequal{\text { def }} \frac{\partial E_{t}}{\partial n e t_{\tilde{C}, t}} \\ & \delta_{o, t} \xlongequal[=]{\text { def }} \frac{\partial E_{t}}{\partial n e t_{o, t}} . \end{aligned}
沿时间倒推以计算 的误差项 t − 1 t − 1 t-1t-1时间步长 δ t − 1 T δ t − 1 T delta_(t-1)^(T)\delta_{t-1}^{T},
δ t − 1 T = ∂ E t ∂ h t − 1 = ∂ E t ∂ h t ∂ h t ∂ h t − 1 = δ t T ∂ h t ∂ h t − 1 = δ f , t T ∂ n e t f , t ∂ h t − 1 + δ i , t T ∂ n e t i , t ∂ h t − 1 + δ C ~ , t T ∂ n e t C ~ , t ∂ h t − 1 + δ o , t T ∂ n e t o , t ∂ h t − 1 = δ f , t T W f h + δ i , t T W i h + δ C ~ , t T W C h + δ o , t T W o h . δ t − 1 T = ∂ E t ∂ h t − 1 = ∂ E t ∂ h t ∂ h t ∂ h t − 1 = δ t T ∂ h t ∂ h t − 1 = δ f , t T ∂ n e t f , t ∂ h t − 1 + δ i , t T ∂ n e t i , t ∂ h t − 1 + δ C ~ , t T ∂ n e t C ~ , t ∂ h t − 1 + δ o , t T ∂ n e t o , t ∂ h t − 1 = δ f , t T W f h + δ i , t T W i h + δ C ~ , t T W C h + δ o , t T W o h . {:[delta_(t-1)^(T)=(delE_(t))/(delh_(t-1))=(delE_(t))/(delh_(t))(delh_(t))/(delh_(t-1))=delta_(t)^(T)(delh_(t))/(delh_(t-1))],[=delta_(f,t)^(T)(del net_(f,t))/(delh_(t-1))+delta_(i,t)^(T)(del net_(i,t))/(delh_(t-1))+delta_( tilde(C),t)^(T)(del net_( tilde(C),t))/(delh_(t-1))+delta_(o,t)^(T)(del net_(o,t))/(delh_(t-1))],[=delta_(f,t)^(T)W_(fh)+delta_(i,t)^(T)W_(ih)+delta_( tilde(C),t)^(T)W_(Ch)+delta_(o,t)^(T)W_(oh).]:}\begin{aligned} \delta_{t-1}^{T} & =\frac{\partial E_{\mathrm{t}}}{\partial \mathrm{~h}_{t-1}}=\frac{\partial E_{t}}{\partial h_{t}} \frac{\partial h_{t}}{\partial h_{t-1}}=\delta_{t}^{T} \frac{\partial h_{t}}{\partial h_{t-1}} \\ & =\delta_{f, t}^{T} \frac{\partial n e t_{f, t}}{\partial h_{t-1}}+\delta_{i, t}^{T} \frac{\partial n e t_{i, t}}{\partial h_{t-1}}+\delta_{\tilde{C}, t}^{T} \frac{\partial n e t_{\tilde{C}, t}}{\partial h_{t-1}}+\delta_{o, t}^{T} \frac{\partial n e t_{o, t}}{\partial h_{t-1}} \\ & =\delta_{f, t}^{T} W_{f h}+\delta_{i, t}^{T} W_{i h}+\delta_{\tilde{C}, t}^{T} W_{C h}+\delta_{o, t}^{T} W_{o h} . \end{aligned}
基于定义的 δ f , t , δ i , t , δ C ~ , t , δ o , t δ f , t , δ i , t , δ C ~ , t , δ o , t delta_(f,t),delta_(i,t),delta_( tilde(C),t),delta_(o,t)\delta_{f, t}, \delta_{i, t}, \delta_{\tilde{C}, t}, \delta_{o, t}，可以推断出
δ f , t T = δ t T ⋅ o t ⋅ ( 1 − tanh ⁡ ( C t ) 2 ) ⋅ C t − 1 ⋅ f t ⋅ ( 1 − f t ) δ i , t T = δ t T ⋅ o t ⋅ ( 1 − tanh ⁡ ( C t ) 2 ) ⋅ C ~ t ⋅ i t ⋅ ( 1 − i t ) δ C ~ , t T = δ t T ⋅ o t ⋅ ( 1 − tanh ⁡ ( C t ) 2 ) ⋅ i t ⋅ ( 1 − C ~ 2 ) δ o , t T = δ t T ⋅ tanh ⁡ ( C t ) ⋅ o t ⋅ ( 1 − o t ) δ f , t T = δ t T ⋅ o t ⋅ 1 − tanh ⁡ C t 2 ⋅ C t − 1 ⋅ f t ⋅ 1 − f t δ i , t T = δ t T ⋅ o t ⋅ 1 − tanh ⁡ C t 2 ⋅ C ~ t ⋅ i t ⋅ 1 − i t δ C ~ , t T = δ t T ⋅ o t ⋅ 1 − tanh ⁡ C t 2 ⋅ i t ⋅ 1 − C ~ 2 δ o , t T = δ t T ⋅ tanh ⁡ C t ⋅ o t ⋅ 1 − o t {:[delta_(f,t)^(T)=delta_(t)^(T)*o_(t)*(1-tanh (C_(t))^(2))*C_(t-1)*f_(t)*(1-f_(t))],[delta_(i,t)^(T)=delta_(t)^(T)*o_(t)*(1-tanh (C_(t))^(2))* tilde(C)_(t)*i_(t)*(1-i_(t))],[delta_( tilde(C),t)^(T)=delta_(t)^(T)*o_(t)*(1-tanh (C_(t))^(2))*i_(t)*(1- tilde(C)^(2))],[delta_(o,t)^(T)=delta_(t)^(T)*tanh(C_(t))*o_(t)*(1-o_(t))]:}\begin{gathered} \delta_{f, t}^{T}=\delta_{t}^{T} \cdot o_{t} \cdot\left(1-\tanh \left(C_{t}\right)^{2}\right) \cdot C_{t-1} \cdot f_{t} \cdot\left(1-f_{t}\right) \\ \delta_{i, t}^{T}=\delta_{t}^{T} \cdot o_{t} \cdot\left(1-\tanh \left(C_{t}\right)^{2}\right) \cdot \tilde{C}_{t} \cdot i_{t} \cdot\left(1-i_{t}\right) \\ \delta_{\tilde{C}, \mathrm{t}}^{T}=\delta_{t}^{T} \cdot o_{t} \cdot\left(1-\tanh \left(C_{t}\right)^{2}\right) \cdot i_{t} \cdot\left(1-\tilde{C}^{2}\right) \\ \delta_{o, t}^{T}=\delta_{t}^{T} \cdot \tanh \left(C_{t}\right) \cdot o_{t} \cdot\left(1-o_{t}\right) \end{gathered}
方程 （18）-（22） 表示在一段时间内将误差传回的过程，现在它可以将误差传回以下方程的任何时刻 K 矩：
δ k T = ∏ j = k t − 1 δ f , j T W f h + δ i , j T W i h + δ C ~ , j T W C h + δ o , j T W o h δ k T = ∏ j = k t − 1   δ f , j T W f h + δ i , j T W i h + δ C ~ , j T W C h + δ o , j T W o h delta_(k)^(T)=prod_(j=k)^(t-1)delta_(f,j)^(T)W_(fh)+delta_(i,j)^(T)W_(ih)+delta_( tilde(C),j)^(T)W_(Ch)+delta_(o,j)^(T)W_(oh)\delta_{k}^{T}=\prod_{j=k}^{t-1} \delta_{f, j}^{T} W_{f h}+\delta_{i, j}^{T} W_{i h}+\delta_{\tilde{C}, j}^{T} W_{C h}+\delta_{o, j}^{T} W_{o h}
对于 W f h , W i h , W C h , W o h , b f , b i , b C , b o W f h , W i h , W C h , W o h , b f , b i , b C , b o W_(fh),W_(ih),W_(Ch),W_(oh),b_(f),b_(i),b_(C),b_(o)W_{f h}, W_{i h}, W_{C h}, W_{o h}, b_{f}, b_{i}, b_{C}, b_{o}权重梯度，梯度 ∂ E t ∂ W oh , t ∂ E t ∂ W oh  , t (delE_(t))/(delW_("oh "),t)\frac{\partial E_{\mathrm{t}}}{\partial W_{\text {oh }}, t}在 t t tt首先计算时间步长，然后将每个 Time Step 的梯度相加以获得最终的梯度 ∂ E t ∂ W oh ∂ E t ∂ W oh  (delE_(t))/(delW_("oh "))\frac{\partial E_{t}}{\partial W_{\text {oh }}}，这可以用以下方程表示：
∂ E t ∂ W o h , t = ∂ E t ∂ n e t o , t ∂ n e t o , t ∂ W o h , t = δ o , t h j − 1 T ∂ E t ∂ W o h = ∑ j = 1 t δ o , j h j − 1 T ∂ E t ∂ W o h , t = ∂ E t ∂ n e t o , t ∂ n e t o , t ∂ W o h , t = δ o , t h j − 1 T ∂ E t ∂ W o h = ∑ j = 1 t   δ o , j h j − 1 T {:[(delE_(t))/(delW_(oh,t))=(delE_(t))/(del net_(o,t))(del net_(o,t))/(delW_(oh,t))=delta_(o,t)h_(j-1)^(T)],[(delE_(t))/(delW_(oh))=sum_(j=1)^(t)delta_(o,j)h_(j-1)^(T)]:}\begin{gathered} \frac{\partial E_{\mathrm{t}}}{\partial W_{o h, t}}=\frac{\partial E_{t}}{\partial n e t_{o, t}} \frac{\partial n e t_{o, t}}{\partial W_{o h, t}}=\delta_{o, t} h_{j-1}^{T} \\ \frac{\partial E_{\mathrm{t}}}{\partial W_{o h}}=\sum_{j=1}^{t} \delta_{o, j} h_{j-1}^{T} \end{gathered}
同样，的 W f x , W i x , W C x W f x , W i x , W C x W_(fx),W_(ix),W_(Cx)W_{f x}, W_{i x}, W_{C x}和 W o x W o x W_(ox)W_{o x}可以直接根据相应的误差项进行计算。利用上述公式，深度学习中的训练算法可以优化权重，最小化损失函数，提高模型预测精度。
模型训练后，均方误差 （RMSE）、平均绝对百分比误差 （MAE） 和决定系数 ( R 2 ) R 2 (R^(2))\left(R^{2}\right)用作 LSTM 模型的性能指标。越接近 R 2 R 2 R^(2)R^{2}为 1 ，则预测性能越好，而 R 2 R 2 R^(2)R^{2}范围介于 [ 0 , 1 ] [ 0 , 1 ] [0,1][0,1].上述指标的计算方式如下：
MAE = 1 n ∑ i = 1 n | y i − y ^ i | , RMSE = 1 n ∑ i = 1 n ( y i − y ^ i ) 2 , R 2 = 1 − ∑ i = 1 n ( y ^ i − y i ) 2 ∑ i = 1 n ( y ¯ i − y i ) 2 ,  MAE  = 1 n ∑ i = 1 n   y i − y ^ i , RMSE = 1 n ∑ i = 1 n   y i − y ^ i 2 , R 2 = 1 − ∑ i = 1 n   y ^ i − y i 2 ∑ i = 1 n   y ¯ i − y i 2 , {:[" MAE "=(1)/(n)sum_(i=1)^(n)|y_(i)- hat(y)_(i)|","],[RMSE=sqrt((1)/(n)sum_(i=1)^(n)(y_(i)- hat(y)_(i))^(2),)],[R^(2)=1-(sum_(i=1)^(n)( hat(y)_(i)-y_(i))^(2))/(sum_(i=1)^(n)( bar(y)_(i)-y_(i))^(2))","]:}\begin{gathered} \text { MAE }=\frac{1}{\mathrm{n}} \sum_{i=1}^{n}\left|y_{i}-\hat{y}_{i}\right|, \\ \mathrm{RMSE}=\sqrt{\frac{1}{n} \sum_{i=1}^{n}\left(y_{i}-\hat{y}_{i}\right)^{2},} \\ R^{2}=1-\frac{\sum_{i=1}^{n}\left(\hat{y}_{i}-y_{i}\right)^{2}}{\sum_{i=1}^{n}\left(\bar{y}_{i}-y_{i}\right)^{2}}, \end{gathered}
哪里 y ^ i y ^ i hat(y)_(i)\hat{y}_{i}是众数的预测值， y i y i y_(i)y_{i}是样本的实际值，而 y ¯ y ¯ bar(y)\bar{y}是样本的实际平均值。MAE 和 RMSE 的值越小，模型的预测结果越好。

B. 使用模拟实验结果进行验证

海洋工程结构是巨大的，尤其是深海水合物开采立管的气-液-固三相流诱导振动建模研究中的立管，其长度长达数公里。为保证物理模型能够准确反映实际情况，在实验中有效再现真实情况，物理模型必须满足与真实结构相似的三个条件，即几何相似性、运动学相似性和动态相似性。 52 52 ^(52){ }^{52}
(1) Geometric similarity:
(1) 几何相似性：
Geometric similarity means that the structural prototype has the same dimensional proportions as the physical model in all directions in three-dimensional (3D) space as follows:
几何相似性是指结构原型在三维（3D）空间的各个方向上与实体模型具有相同的尺寸比例，具体如下：
X X ′ = Y Y ′ = Z Z ′ = λ , λ A = A A ′ = λ 2 , λ V = V V ′ = λ 3 , X X ′ = Y Y ′ = Z Z ′ = λ , λ A = A A ′ = λ 2 , λ V = V V ′ = λ 3 , {:[(X)/(X^('))=(Y)/(Y^('))=(Z)/(Z^('))=lambda","],[lambda_(A)=(A)/(A^('))=lambda^(2)","],[lambda_(V)=(V)/(V^('))=lambda^(3)","]:}\begin{gathered} \frac{X}{X^{\prime}}=\frac{Y}{Y^{\prime}}=\frac{Z}{Z^{\prime}}=\lambda, \\ \lambda_{A}=\frac{A}{A^{\prime}}=\lambda^{2}, \\ \lambda_{V}=\frac{V}{V^{\prime}}=\lambda^{3}, \end{gathered}
where X , Y , Z X , Y , Z X,Y,ZX, Y, Z are the 3 D coordinates of the prototype ( m ); X ′ , Y ′ , Z ′ X ′ , Y ′ , Z ′ X^('),Y^('),Z^(')X^{\prime}, Y^{\prime}, Z^{\prime} are the 3D coordinates of the model ( m ); A A AA and A ′ A ′ A^(')A^{\prime} are the areas of the prototype and model, respectively ( m 2 ) ; V m 2 ; V (m^(2));V\left(\mathrm{m}^{2}\right) ; V and V ′ V ′ V^(')V^{\prime} are the volumes of the prototype and model, respectively ( m 3 ) m 3 (m^(3))\left(\mathrm{m}^{3}\right); and λ , λ A , λ V λ , λ A , λ V lambda,lambda_(A),lambda_(V)\lambda, \lambda_{A}, \lambda_{V} are length similarity ratio, area similarity ratio, and volume similarity ratio, respectively.
其中， X , Y , Z X , Y , Z X,Y,ZX, Y, Z 是原型的三维坐标（米）； X ′ , Y ′ , Z ′ X ′ , Y ′ , Z ′ X^('),Y^('),Z^(')X^{\prime}, Y^{\prime}, Z^{\prime} 是模型的三维坐标（米）； A A AA 和 A ′ A ′ A^(')A^{\prime} 分别是原型和模型的面积； ( m 2 ) ; V m 2 ; V (m^(2));V\left(\mathrm{m}^{2}\right) ; V 和 V ′ V ′ V^(')V^{\prime} 分别是原型和模型的体积； ( m 3 ) m 3 (m^(3))\left(\mathrm{m}^{3}\right) 和 λ , λ A , λ V λ , λ A , λ V lambda,lambda_(A),lambda_(V)\lambda, \lambda_{A}, \lambda_{V} 分别是长度相似比、面积相似比和体积相似比。
In some practical engineering problems, due to the significant differences in the scales of the three directions, if engineering insists on using a completely geometrically similar model for the study, we may face problems, such as excessive site requirements or observation difficulties. Therefore, the metamorphosis
在一些实际工程问题中，由于三个方向的尺度差异较大，如果工程上坚持使用完全几何相似的模型进行研究，可能会面临场地要求过高或观测困难等问题。因此，变形
similarity model must be used in some cases (different scale constants in each direction in three dimensions are metamorphic similarity), although this approach cannot fully simulate real physical phenomena, it can still provide some valuable research results. Therefore, deep-sea risers with very large length-to-diameter ratios generally use metamorphic similarity, and the geometric similarity ratio of the risers can be set as radial similarity ratio λ d λ d lambda_(d)\lambda_{d} and longitudinal similarity ratio λ l λ l lambda_(l)\lambda_{l}.
虽然这种方法不能完全模拟真实的物理现象，但仍能提供一些有价值的研究成果。因此，长径比非常大的深海立管一般采用变质相似模型，立管的几何相似比可设定为径向相似比 λ d λ d lambda_(d)\lambda_{d} 和纵向相似比 λ l λ l lambda_(l)\lambda_{l} 。
(2) Motion similarity:  (2) 运动相似性：
Motion similarity describes that in two similar individuals or systems, the displacement, velocity, and acceleration of particles or objects in corresponding time and space are not only in the same direction but also in a fixed proportion. According to the discussion in the literature, 53 53 ^(53){ }^{53} to achieve motion similarity, the density and elastic modulus of the experimental material should conform to the specific relationship as follows:
运动相似性是指在两个相似的个体或系统中，粒子或物体在相应的时间和空间中的位移、速度和加速度不仅方向相同，而且比例固定。根据文献中的讨论， 53 53 ^(53){ }^{53} 要实现运动相似性，实验材料的密度和弹性模量应符合如下特定关系：
E m ρ m = 1 λ E s ρ s , E m ρ m = 1 λ E s ρ s , (E_(m))/(rho_(m))=(1)/(lambda)(E_(s))/(rho_(s)),\frac{E_{m}}{\rho_{m}}=\frac{1}{\lambda} \frac{E_{s}}{\rho_{s}},
where ρ m , ρ s ρ m , ρ s rho_(m),rho_(s)\rho_{m}, \rho_{s} are the densities of models and entities, respectively ( kg / m 3 ) kg / m 3 (kg//m^(3))\left(\mathrm{kg} / \mathrm{m}^{3}\right) and E m E m E_(m)E_{m} and E s E s E_(s)E_{s} are the elastic modulus of the model and the solid, respectively ( N / m 2 ) N / m 2 (N//m^(2))\left(\mathrm{N} / \mathrm{m}^{2}\right).
其中， ρ m , ρ s ρ m , ρ s rho_(m),rho_(s)\rho_{m}, \rho_{s} 分别为模型和实体的密度 ( kg / m 3 ) kg / m 3 (kg//m^(3))\left(\mathrm{kg} / \mathrm{m}^{3}\right) ， E m E m E_(m)E_{m} 和 E s E s E_(s)E_{s} 分别为模型和实体的弹性模量 ( N / m 2 ) N / m 2 (N//m^(2))\left(\mathrm{N} / \mathrm{m}^{2}\right) 。
(3) Dynamic similarity:  (3) 动态相似性：
Froude’s similar focus on the balance between gravity and inertial forces allows the experimental flow to be consistent with the prototype in terms of wave motion and flow transitions denoted as
弗劳德同样关注重力和惯性力之间的平衡，这使得实验水流在波浪运动和水流转换方面与原型保持一致。
v m g L m = v s g L s . v m g L m = v s g L s . (v_(m))/(sqrt(gL_(m)))=(v_(s))/(sqrt(gL_(s))).\frac{v_{m}}{\sqrt{g L_{m}}}=\frac{v_{s}}{\sqrt{g L_{s}}} .
The velocity similarity ratio and the geometric similarity ratio should satisfy
速度相似比和几何相似比应满足
λ v = λ l 1 2 , λ v = λ l 1 2 , lambda_(v)=lambda_(l^((1)/(2))),\lambda_{v}=\lambda_{l^{\frac{1}{2}}},
where v m , v s v m , v s v_(m),v_(s)v_{m}, v_{s} are the characteristic velocities of the model and entity ( m / s ) , L m , L s ( m / s ) , L m , L s (m//s),L_(m),L_(s)(\mathrm{m} / \mathrm{s}), L_{m}, L_{s} are the feature line scales of the model and entity ( m ), g g gg is the acceleration of gravity ( m / s 2 ) m / s 2 (m//s^(2))\left(\mathrm{m} / \mathrm{s}^{2}\right), and λ ν , λ l λ ν , λ l lambda_(nu),lambda_(l)\lambda_{\nu}, \lambda_{l} are the velocity similarity ratio and geometric similarity ratio, respectively.
其中， v m , v s v m , v s v_(m),v_(s)v_{m}, v_{s} 是模型和实体的特征速度 ( m / s ) , L m , L s ( m / s ) , L m , L s (m//s),L_(m),L_(s)(\mathrm{m} / \mathrm{s}), L_{m}, L_{s} 是模型和实体的特征线尺度 ( m )， g g gg 是重力加速度 ( m / s 2 ) m / s 2 (m//s^(2))\left(\mathrm{m} / \mathrm{s}^{2}\right) 和 λ ν , λ l λ ν , λ l lambda_(nu),lambda_(l)\lambda_{\nu}, \lambda_{l} 分别是速度相似比和几何相似比。
The Strouhal similarity focuses on the similarity of vortex shedding or periodic phenomena in the flow and ensures that the key parameters, such as the frequency and intensity of vortex shedding in the experiment, are similar to the prototype, denoted as
Strouhal 相似性侧重于流动中涡流脱落或周期现象的相似性，并确保实验中涡流脱落的频率和强度等关键参数与原型相似，表示为
v m T m L m = v s T s L s , v m T m L m = v s T s L s , (v_(m)T_(m))/(L_(m))=(v_(s)T_(s))/(L_(s)),\frac{v_{m} T_{m}}{L_{m}}=\frac{v_{s} T_{s}}{L_{s}},
where T m , T s T m , T s T_(m),T_(s)T_{m}, T_{s} are the cycles of the model and the entity (s).
其中 T m , T s T m , T s T_(m),T_(s)T_{m}, T_{s} 是模型和实体（s）的周期。
The geometric similarity used is metamorphic similarity, the radial similarity ratio is set to λ d = 20 λ d = 20 lambda_(d)=20\lambda_{d}=20, the axial similarity ratio is set to λ l = 667 λ l = 667 lambda_(l)=667\lambda_{l}=667, and the radial similarity ratio λ d λ d lambda_(d)\lambda_{d} is selected as the main similarity ratio. The length, outer diameter, and wall thickness of the experimental riser are determined according to the geometric similarity, and the PVC material with smaller elastic modulus and stiffness is selected according to the motion similarity. The actual and experimental parameters of the riser are shown in Table I, the actual and experimental parameters of variables are shown in Table II, and the actual and experimental parameters of hydrate particles are shown in Table III.
采用的几何相似度为变质相似度，径向相似比设为 λ d = 20 λ d = 20 lambda_(d)=20\lambda_{d}=20 ，轴向相似比设为 λ l = 667 λ l = 667 lambda_(l)=667\lambda_{l}=667 ，选择径向相似比 λ d λ d lambda_(d)\lambda_{d} 作为主要相似比。根据几何相似性确定实验立管的长度、外径和壁厚，根据运动相似性选择弹性模量和刚度较小的 PVC 材料。立管的实际参数和实验参数见表 I，变量的实际参数和实验参数见表 II，水合物颗粒的实际参数和实验参数见表 III。
TABLE I. Comparison of actual and experimental parameters of risers.
表 I.立管实际参数与实验参数的比较。
Parameter name  参数名称 Length (m)  长度（米） External diameter (m)  外径（米） Internal diameter (m)  内径（米） Material type  材料类型 Density (kg/m 3 3 ^(3){ }^{3} )
密度（千克/米 3 3 ^(3){ }^{3} )		 Modulus of elastic (GPa)
弹性模量（GPa）
Actual value  实际价值 2000 0.44 0.4 13Cr-L80 7850 207
Experimental value  实验值 3 0.022 0.02 PVC 1600 2.1
Parameter name Length (m) External diameter (m) Internal diameter (m) Material type Density (kg/m ^(3) ) Modulus of elastic (GPa) Actual value 2000 0.44 0.4 13Cr-L80 7850 207 Experimental value 3 0.022 0.02 PVC 1600 2.1| Parameter name | Length (m) | External diameter (m) | Internal diameter (m) | Material type | Density (kg/m ${ }^{3}$ ) | Modulus of elastic (GPa) | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | Actual value | 2000 | 0.44 | 0.4 | 13Cr-L80 | 7850 | 207 | | Experimental value | 3 | 0.022 | 0.02 | PVC | 1600 | 2.1 |
TABLE II. Actual and experimental parameters for variables.
表 II.变量的实际参数和实验参数。
Variant  变体 Actual value 54 54 ^(54){ }^{54}  实际值 54 54 ^(54){ }^{54} Experimental value  实验值 Similarity ratio 55 55 ^(55){ }^{55}
相似比 55 55 ^(55){ }^{55}
Uniform current speed ( m / s m / s m//s\mathrm{m} / \mathrm{s} )
均匀流速 ( m / s m / s m//s\mathrm{m} / \mathrm{s} )		 0-2.4 0-0.5 λ d λ d sqrt(lambda_(d))\sqrt{\lambda_{d}}
Top tension (N)  顶部张力（N） 1.0 × 10 5 − 3.0 × 10 5 1.0 × 10 5 − 3.0 × 10 5 1.0 xx10^(5)-3.0 xx10^(5)1.0 \times 10^{5}-3.0 \times 10^{5} 10-40 λ d 3 λ d 3 lambda_(d)^(3)\lambda_{d}^{3}
Instream flow rate ( m / s m / s m//s\mathrm{m} / \mathrm{s} )
溪流流速 ( m / s m / s m//s\mathrm{m} / \mathrm{s} )		 0-10.0 0-2.2 λ d λ d sqrt(lambda_(d))\sqrt{\lambda_{d}}
Internal flow three-phase content (gas: liquid: solid)
内流三相含量（气体：液体：固体）
55%: 42%: 3%
50%: 45%: 5%
55%: 42%: 3% 50%: 45%: 5%| 55%: 42%: 3% | | :--- | | 50%: 45%: 5% |
Variant Actual value ^(54) Experimental value Similarity ratio ^(55) Uniform current speed ( m//s ) 0-2.4 0-0.5 sqrt(lambda_(d)) Top tension (N) 1.0 xx10^(5)-3.0 xx10^(5) 10-40 lambda_(d)^(3) Instream flow rate ( m//s ) 0-10.0 0-2.2 sqrt(lambda_(d)) Internal flow three-phase content (gas: liquid: solid) "55%: 42%: 3% 50%: 45%: 5%" | Variant | Actual value ${ }^{54}$ | Experimental value | Similarity ratio ${ }^{55}$ | | :--- | :--- | :--- | :--- | | Uniform current speed ( $\mathrm{m} / \mathrm{s}$ ) | 0-2.4 | 0-0.5 | $\sqrt{\lambda_{d}}$ | | Top tension (N) | $1.0 \times 10^{5}-3.0 \times 10^{5}$ | 10-40 | $\lambda_{d}^{3}$ | | Instream flow rate ( $\mathrm{m} / \mathrm{s}$ ) | 0-10.0 | 0-2.2 | $\sqrt{\lambda_{d}}$ | | Internal flow three-phase content (gas: liquid: solid) | | 55%: 42%: 3% <br> 50%: 45%: 5% | | | | | | |
In order to verify the correctness of the gas-liquid-solid threephase flow-induced vibration model of the deep-sea hydrate mining riser, the experiment focuses on the coupling effect of internal and external flows. Therefore, the experimental system must be equipped with devices capable of generating internal and external flows and should be able to simulate the actual working environment of the deep-sea hydrate mining riser. Based on these requirements, the structure of the experimental system shown in Fig. 3 is designed, including the experimental rig, three-phase flow generation device, riser, and strain measurement acquisition system.
为了验证深海水合物开采立管的气-液-固三相流诱导振动模型的正确性，实验的重点是内外流的耦合效应。因此，实验系统必须配备能够产生内外流动的装置，并且能够模拟深海水合物开采立管的实际工作环境。根据这些要求，设计了图 3 所示的实验系统结构，包括实验台、三相流生成装置、立管和应变测量采集系统。
The experimental bench is primarily composed of a truss and an experimental table. This setup enables the parallel movement of the experimental bench, ensuring the simulation of a uniform force similar to that of a uniform sea current. The three-phase flow generator is designed to produce gas-liquid-solid three-phase flow. It uses rubber particles as solid components, air as gas, and water as liquid. The water pump extracts the pre-mixed solid-liquid two-phase flow from the mixing tank, and the solid-liquid two-phase flow passes through the electromagnetic flow meter into the three-way pipe as illustrated in Fig. 4. At the same time, the air compressor will push the gas into the gas relief valve and damping valve. After desiccant drying, the gas flows through the gas flow meter into the three-way pipe, where it mixes with the solid-liquid two-phase flow to form a gas-liquid-solid three-phase flow. After the three-phase flow passes through the experimental riser, it enters the mixing tank, completing the entire pipeline circulation process.
实验台主要由桁架和实验桌组成。这种设置可使实验台平行移动，确保模拟出类似于均匀海流的均匀力。三相流发生器旨在产生气-液-固三相流。它使用橡胶颗粒作为固体成分，空气作为气体，水作为液体。如图 4 所示，水泵从混合罐中抽取预混合的固液两相流，固液两相流通过电磁流量计进入三通管。同时，空气压缩机将气体推入气体溢流阀和阻尼阀。干燥剂干燥后，气体通过气体流量计流入三通管，与固液两相流混合形成气液固三相流。三相流通过实验立管后，进入混合罐，完成整个管道循环过程。
This device maintains stable gas pressure through the gas relief valve. It controls the gas flow by adjusting the gas damping valve opening and monitoring the flow meter. Similarly, it regulates the solid-liquid two-phase flow by adjusting the return valve opening and observing the flow meter. This design not only ensures pressure
该装置通过气体减压阀保持稳定的气体压力。它通过调节气体阻尼阀的开度和监测流量计来控制气体流量。同样，它通过调节回流阀的开度和观察流量计来调节固液两相流。这种设计不仅能确保压力
TABLE III. Actual and experimental parameters of hydrate particles.
表 III.水合物颗粒的实际参数和实验参数。
Parameter name  参数名称 Diameter ( mm ) ( mm ) (mm)(\mathrm{mm})  直径 ( mm ) ( mm ) (mm)(\mathrm{mm}) Material type  材料类型 Density ( kg / m 3 ) kg / m 3 (kg//m^(3))\left(\mathrm{kg} / \mathrm{m}^{3}\right)  密度 ( kg / m 3 ) kg / m 3 (kg//m^(3))\left(\mathrm{kg} / \mathrm{m}^{3}\right)
Actual value  实际价值 3 − 12 3 − 12 3-123-12 Gas hydrate  天然气水合物 880 − 1350 880 − 1350 880-1350880-1350
Experimental value  实验值 0.6 ( 30 0.6 ( 30 0.6(300.6(30 mesh ) ) ))   0.6 ( 30 0.6 ( 30 0.6(300.6(30 网格 ) ) )) Rubber granule  橡胶颗粒 900
Parameter name Diameter (mm) Material type Density (kg//m^(3)) Actual value 3-12 Gas hydrate 880-1350 Experimental value 0.6(30 mesh ) Rubber granule 900| Parameter name | Diameter $(\mathrm{mm})$ | Material type | Density $\left(\mathrm{kg} / \mathrm{m}^{3}\right)$ | | :--- | :---: | :---: | :---: | | Actual value | $3-12$ | Gas hydrate | $880-1350$ | | Experimental value | $0.6(30$ mesh $)$ | Rubber granule | 900 |
balance in the pipeline, creating a stable gas-liquid-solid three-phase flow, but also allows for controlling the flow rate and proportion of each phase.
在管道中实现平衡，形成稳定的气-液-固三相流，同时还能控制每相的流速和比例。
The experimental riser is a straight PVC pipe with a length of 3 m , an external diameter of 20 mm , and a wall thickness of 1 mm , with the top tension fixture connected at the upper end and the universal joint connected at the lower end. Strain gauges are pasted every 0.5 m according to the sensor arrangement schematic, and since the experimental riser is located in water, it needs to be further waterproofed using AB adhesive, and the strain gauges are numbered sequentially from top to bottom, as shown in Fig. 5.
实验立管为 PVC 直管，长度为 3 米，外径为 20 毫米，壁厚为 1 毫米，上端连接顶部拉伸夹具，下端连接万向接头。根据传感器布置示意图，每隔 0.5 米粘贴一个应变片，由于实验立管位于水中，需要使用 AB 胶进一步防水，应变片从上到下依次编号，如图 5 所示。
First, the experimental equipment should be installed and the experimental equipment and data acquisition equipment should be debugged, including all the equipment, valves, sensors, and flow meters. Second, the air tightness of each circuit is checked to avoid errors in the experimental results caused by the flow loss of each phase. Then, a certain proportion of solid particles and water are added to the mixing tank. Finally, the 18 sets strain response datasets of the mining riser collected by experiment using the strain gauge, which include strain response along in-line flow (IL) and cross-flow (CF) directions. The experimental conditions as shown in Table IV. The specific testing point is 380 mm away from the water surface. The stable vibration time of the riser is selected for training, and the time of each case is 20 s . The sampling frequency of strain gauge is 2000 Hz . The model of strain gauge is 120 − 5 AA 120 − 5 AA 120-5AA120-5 \mathrm{AA}, with a sensitivity factor of 2.11 ± 1 % 2.11 ± 1 % 2.11+-1%2.11 \pm 1 \% and a resistance of 120 ± 0.5 Ω 120 ± 0.5 Ω 120+-0.5 Omega120 \pm 0.5 \Omega. Due to the nonlinear nature of the vibration signal prediction process, a time-dependent correlation exists between the data points, which can be viewed as a time-series prediction result. Therefore, the first 80 % 80 % 80%80 \% of dataset is taken as the training set, and the last 20 % 20 % 20%20 \% of dataset is selected as the test set. Meanwhile, the LSTM model is used to predict the last 20 % 20 % 20%20 \% of vibration dataset of the mining rise. Finally, the correctness of the deep-learning model is verified by comparing the last 20 % 20 % 20%20 \% of experimental data with model prediction data.
首先，安装实验设备，调试实验设备和数据采集设备，包括所有设备、阀门、传感器和流量计。其次，检查各回路的气密性，避免各相流量损失造成实验结果误差。然后，在混合罐中加入一定比例的固体颗粒和水。最后，使用应变仪对采矿立管进行实验，收集了 18 组应变响应数据集，包括沿线流（IL）和横流（CF）方向的应变响应。实验条件如表 IV 所示。具体测试点距离水面 380 毫米。选择立管的稳定振动时间进行训练，每例时间为 20 s，应变计的采样频率为 2000 Hz。应变计模型为 120 − 5 AA 120 − 5 AA 120-5AA120-5 \mathrm{AA} ，灵敏度系数为 2.11 ± 1 % 2.11 ± 1 % 2.11+-1%2.11 \pm 1 \% ，电阻为 120 ± 0.5 Ω 120 ± 0.5 Ω 120+-0.5 Omega120 \pm 0.5 \Omega 。由于振动信号预测过程的非线性性质，数据点之间存在随时间变化的相关性，可将其视为时间序列预测结果。因此，将数据集的第一个 80 % 80 % 80%80 \% 作为训练集，选择数据集的最后一个 20 % 20 % 20%20 \% 作为测试集。同时，使用 LSTM 模型预测矿升振动数据集的最后 20 % 20 % 20%20 \% 。最后，通过比较实验数据的最后 20 % 20 % 20%20 \% 与模型预测数据，验证深度学习模型的正确性。
The model is trained to learn by moving the window, and in the second training iteration, the window movement continues until the training set is completely exhausted. Meanwhile, the test set the model performance, and this prediction model performs several iterations and finally completes the prediction of vibration signals. The model
通过移动窗口对模型进行学习训练，在第二次训练迭代中，窗口继续移动，直到训练集完全耗尽。同时，通过测试集来检验模型的性能，该预测模型经过多次迭代，最终完成振动信号的预测。模型
FIG. 3. Schematic diagram of the experimental system structure.
图 3.实验系统结构示意图。
accuracy can be further improved by varying the number of LSTM layers and the number of hidden neurons, which are different in the LSTM model, and the parameters are adjusted several times to get the lowest average RMSE value. Therefore, these parameters are finally used for LSTM model training.
通过改变 LSTM 模型中不同的 LSTM 层数和隐藏神经元数，可以进一步提高精度。因此，这些参数最终被用于 LSTM 模型训练。
Model training starts with training the model by forward testing strategy and reducing the model error by backpropagation. The loss function of the training and testing sets is minimized during model training, and the model automatically stops iterating to obtain the best model parameters. The 18 sets of experimental datasets data are
模型训练首先采用前向测试策略训练模型，并通过反向传播减少模型误差。在模型训练过程中，训练集和测试集的损失函数最小，模型自动停止迭代，以获得最佳模型参数。18 组实验数据集数据为
FIG. 4. Physical diagram of gas-liquidsolid three-phase flow system.
图 4.气液固三相流系统物理图。

(a) High-speed camera  (a) 高速摄像机

(e) Top tension fixtures
(e) 顶部拉伸固定装置

(b) Top tension gauge
(b) 顶部拉力计

(f) Strain gauges  (f) 应变计

© Gas flowmeter  气体流量计

(g) Sensor arrangement  (g) 传感器布置

(d) Acquisition instrument
(d) 购置文书

(h)Riser installation  (h) 立管安装
FIG. 5. Strain gauge encapsulation process and riser installation diagrams.
图 5.应变计封装过程和立管安装图。
trained to obtain the average RMSE value of the model, as shown in Table V. It shows that the LSTM model has high accuracy and can be applied to mining riser vibration signal prediction. The parameters of LSTM prediction model are shown in Table VI.
表明 LSTM 模型具有较高的精度，可应用于采矿立管振动信号预测。LSTM 预测模型的参数如表 VI 所示。
After the training, the deep learning model is tested, all the cases are relabeled, and the detailed predictions are shown in Table VII. The model predicts relatively accurately under the test set, with an RMSE of no more than 43 % 43 % 43%43 \% and an MAE of less than 37 % 37 % 37%37 \%. The coefficient of determination R 2 R 2 R^(2)R^{2} is more than 99 % 99 % 99%99 \%, and the prediction accuracy is high. The 18 sets of experimental datasets were categorized, and the prediction results were analyzed according to the different internal flow rates, external flow rates, gas-liquid-solid ratios, and particle
训练完成后，对深度学习模型进行测试，重新标记所有案例，详细预测结果如表 VII 所示。该模型在测试集下的预测结果相对准确，RMSE 不大于 43 % 43 % 43%43 \% ，MAE 小于 37 % 37 % 37%37 \% 。判定系数 R 2 R 2 R^(2)R^{2} 大于 99 % 99 % 99%99 \% ，预测精度较高。对 18 组实验数据集进行了分类，并根据不同的内部流速、外部流速、气液固比和颗粒大小对预测结果进行了分析。
TABLE IV. Data sets under different experimental conditions.
表 IV.不同实验条件下的数据集
Working condition  工作状态 Solid particle size (mesh)
固体颗粒大小（目）		 Gas liquid solid ratio
气液固比率		 External flow rate ( m / min m / min m//min\mathrm{m} / \mathrm{min} )
外部流速 ( m / min m / min m//min\mathrm{m} / \mathrm{min} )		 Internal flow rate ( m / s m / s m//s\mathrm{m} / \mathrm{s} )
内部流速 ( m / s m / s m//s\mathrm{m} / \mathrm{s} )
1 10 2%:95%:3% 30 1.3
2 10 2%:95%:3% 30 1.5
3 10 2%:95%:3% 30 1.7
4 10 2%:95%:3% 10 1.7
5 10 2%:95%:3% 20 1.7
6 10 6%:85%:9% 30 1.7
7 10 10%:75%:15% 30 1.7
8 20 2%:95%:3% 30 1.7
9 30 2%:95%:3% 30 1.7
Working condition Solid particle size (mesh) Gas liquid solid ratio External flow rate ( m//min ) Internal flow rate ( m//s ) 1 10 2%:95%:3% 30 1.3 2 10 2%:95%:3% 30 1.5 3 10 2%:95%:3% 30 1.7 4 10 2%:95%:3% 10 1.7 5 10 2%:95%:3% 20 1.7 6 10 6%:85%:9% 30 1.7 7 10 10%:75%:15% 30 1.7 8 20 2%:95%:3% 30 1.7 9 30 2%:95%:3% 30 1.7| Working condition | Solid particle size (mesh) | Gas liquid solid ratio | External flow rate ( $\mathrm{m} / \mathrm{min}$ ) | Internal flow rate ( $\mathrm{m} / \mathrm{s}$ ) | | :--- | :--- | :--- | :--- | :--- | | 1 | 10 | 2%:95%:3% | 30 | 1.3 | | 2 | 10 | 2%:95%:3% | 30 | 1.5 | | 3 | 10 | 2%:95%:3% | 30 | 1.7 | | 4 | 10 | 2%:95%:3% | 10 | 1.7 | | 5 | 10 | 2%:95%:3% | 20 | 1.7 | | 6 | 10 | 6%:85%:9% | 30 | 1.7 | | 7 | 10 | 10%:75%:15% | 30 | 1.7 | | 8 | 20 | 2%:95%:3% | 30 | 1.7 | | 9 | 30 | 2%:95%:3% | 30 | 1.7 |
mesh, with each category corresponding to the IL vibrational displacement and the cross-flow vibrational displacement.
网格，每一类分别对应 IL 振动位移和横流振动位移。
The solid lines of the prediction plots indicate the actual values and the solid dots indicate the predicted values in Figs. 6-9 and 12.
预测图中的实线表示实际值，实心点表示图 6-9 和图 12 中的预测值。
TABLE V. Average RMSE under different structures.
表 V.不同结构下的平均 RMSE
LSTM layers  LSTM 层 100 neurons  100 个神经元 200 neurons  200 个神经元 300 neurons  300 个神经元
1 1.707782796 0.838184788 0.917497296
2 1.107782796 0.227251225 0.513787193
3 1.353659478 0.315092532 0.714235539
LSTM layers 100 neurons 200 neurons 300 neurons 1 1.707782796 0.838184788 0.917497296 2 1.107782796 0.227251225 0.513787193 3 1.353659478 0.315092532 0.714235539| LSTM layers | 100 neurons | 200 neurons | 300 neurons | | :--- | :---: | :---: | :---: | | 1 | 1.707782796 | 0.838184788 | 0.917497296 | | 2 | 1.107782796 | 0.227251225 | 0.513787193 | | 3 | 1.353659478 | 0.315092532 | 0.714235539 |
TABLE VI. LSTM prediction model parameters.
表 VI.LSTM 预测模型参数
Parameter name  参数名称 Numerical value  数值
LSTM layers  LSTM 层 2
Number of hidden neuron nodes
隐藏神经元节点数		 200
Input feature dimension  输入特征维度 3
Output feature dimension
输出特征维度		 2
Maximum number of iterations
最大迭代次数		 100
Optimizer  优化器 Adam  亚当
Activation function  激活功能 Relu
Loss function  损失函数 RMSE
Dropout layer  疏水层 0.2
Parameter name Numerical value LSTM layers 2 Number of hidden neuron nodes 200 Input feature dimension 3 Output feature dimension 2 Maximum number of iterations 100 Optimizer Adam Activation function Relu Loss function RMSE Dropout layer 0.2| Parameter name | Numerical value | | :--- | :--- | | LSTM layers | 2 | | Number of hidden neuron nodes | 200 | | Input feature dimension | 3 | | Output feature dimension | 2 | | Maximum number of iterations | 100 | | Optimizer | Adam | | Activation function | Relu | | Loss function | RMSE | | Dropout layer | 0.2 |
TABLE VII. Prediction evaluation of deep learning models.
表 VII.深度学习模型的预测评估
Number of groups  组数 Datasets under different experimental conditions (Particle size - Gas liquid solid ratio External flow rate - Internal flow rate)
不同实验条件下的数据集（粒度 - 气液固比率 外部流速 - 内部流速）		 Direction  方向 RMSE MAE R 2 R 2 R^(2)R^{2}
1 10-2%:95%:3%-10-1.7 IL 2.82% 1.20% 98.36%
2 10-2%:95%:3%-20-1.7 IL 4.14% 3.65% 99.97%
3 10-2%:95%:3%-30-1.3 IL 3.80% 3.34% 99.91%
4 10-2%:95%:3%-30-1.5 IL 6.03% 5.51% 99.84%
5 10-2%:95%:3%-30-1.7 IL 3.38% 2.21% 99.90%
6 10-6%:85%:9%-30-1.7 IL 4.77% 4.06% 99.96%
7 10-10%:75%:15%-30-1.7 IL 5.91% 5.25% 99.91%
8 20-2%:95%:3%-30-1.7 IL 3.85% 3.13% 99.97%
9 30-2%:95%:3%-30-1.7 IL 5.97% 5.03% 99.94%
10 10-2%:95%:3%-10-1.7 CF 1.30% 1.03% 99.75%
11 10-2%:95%:3%-20-1.7 CF 42.05% 36.52% 99.73%
12 10-2%:95%:3%-30-1.3 CF 14.63% 12.30% 99.73%
13 10-2%:95%:3%-30-1.5 CF 38.04% 36.08% 98.53%
14 10-2%:95%:3%-30-1.7 CF 19.24% 14.07% 99.10%
15 10-6%:85%:9%-30-1.7 CF 16.25% 13.14% 99.93%
16 10-10%:75%:15%-30-1.7 CF 25.06% 22.23% 99.69%
17 20-2%:95%:3%-30-1.7 CF 19.31% 13.54% 99.87%
18 30-2%:95%:3%-30-1.7 CF 13.31% 10.78% 99.95%
Number of groups Datasets under different experimental conditions (Particle size - Gas liquid solid ratio External flow rate - Internal flow rate) Direction RMSE MAE R^(2) 1 10-2%:95%:3%-10-1.7 IL 2.82% 1.20% 98.36% 2 10-2%:95%:3%-20-1.7 IL 4.14% 3.65% 99.97% 3 10-2%:95%:3%-30-1.3 IL 3.80% 3.34% 99.91% 4 10-2%:95%:3%-30-1.5 IL 6.03% 5.51% 99.84% 5 10-2%:95%:3%-30-1.7 IL 3.38% 2.21% 99.90% 6 10-6%:85%:9%-30-1.7 IL 4.77% 4.06% 99.96% 7 10-10%:75%:15%-30-1.7 IL 5.91% 5.25% 99.91% 8 20-2%:95%:3%-30-1.7 IL 3.85% 3.13% 99.97% 9 30-2%:95%:3%-30-1.7 IL 5.97% 5.03% 99.94% 10 10-2%:95%:3%-10-1.7 CF 1.30% 1.03% 99.75% 11 10-2%:95%:3%-20-1.7 CF 42.05% 36.52% 99.73% 12 10-2%:95%:3%-30-1.3 CF 14.63% 12.30% 99.73% 13 10-2%:95%:3%-30-1.5 CF 38.04% 36.08% 98.53% 14 10-2%:95%:3%-30-1.7 CF 19.24% 14.07% 99.10% 15 10-6%:85%:9%-30-1.7 CF 16.25% 13.14% 99.93% 16 10-10%:75%:15%-30-1.7 CF 25.06% 22.23% 99.69% 17 20-2%:95%:3%-30-1.7 CF 19.31% 13.54% 99.87% 18 30-2%:95%:3%-30-1.7 CF 13.31% 10.78% 99.95%| Number of groups | Datasets under different experimental conditions (Particle size - Gas liquid solid ratio External flow rate - Internal flow rate) | Direction | RMSE | MAE | $R^{2}$ | | :--- | :--- | :--- | :--- | :--- | :--- | | 1 | 10-2%:95%:3%-10-1.7 | IL | 2.82% | 1.20% | 98.36% | | 2 | 10-2%:95%:3%-20-1.7 | IL | 4.14% | 3.65% | 99.97% | | 3 | 10-2%:95%:3%-30-1.3 | IL | 3.80% | 3.34% | 99.91% | | 4 | 10-2%:95%:3%-30-1.5 | IL | 6.03% | 5.51% | 99.84% | | 5 | 10-2%:95%:3%-30-1.7 | IL | 3.38% | 2.21% | 99.90% | | 6 | 10-6%:85%:9%-30-1.7 | IL | 4.77% | 4.06% | 99.96% | | 7 | 10-10%:75%:15%-30-1.7 | IL | 5.91% | 5.25% | 99.91% | | 8 | 20-2%:95%:3%-30-1.7 | IL | 3.85% | 3.13% | 99.97% | | 9 | 30-2%:95%:3%-30-1.7 | IL | 5.97% | 5.03% | 99.94% | | 10 | 10-2%:95%:3%-10-1.7 | CF | 1.30% | 1.03% | 99.75% | | 11 | 10-2%:95%:3%-20-1.7 | CF | 42.05% | 36.52% | 99.73% | | 12 | 10-2%:95%:3%-30-1.3 | CF | 14.63% | 12.30% | 99.73% | | 13 | 10-2%:95%:3%-30-1.5 | CF | 38.04% | 36.08% | 98.53% | | 14 | 10-2%:95%:3%-30-1.7 | CF | 19.24% | 14.07% | 99.10% | | 15 | 10-6%:85%:9%-30-1.7 | CF | 16.25% | 13.14% | 99.93% | | 16 | 10-10%:75%:15%-30-1.7 | CF | 25.06% | 22.23% | 99.69% | | 17 | 20-2%:95%:3%-30-1.7 | CF | 19.31% | 13.54% | 99.87% | | 18 | 30-2%:95%:3%-30-1.7 | CF | 13.31% | 10.78% | 99.95% |
Setting the solid particles of 10 mesh, the gas-solid-liquid ratio of 2 % : 3 % : 95 % 2 % : 3 % : 95 % 2%:3%:95%2 \%: 3 \%: 95 \%, the external flow rate of 30 m / min 30 m / min 30m//min30 \mathrm{~m} / \mathrm{min}, and the internal flow rate of 1.3 , 1.5 1.3 , 1.5 1.3,1.51.3,1.5, and 1.7 m / s 1.7 m / s 1.7m//s1.7 \mathrm{~m} / \mathrm{s}, respectively. The LSTM model is trained to predict the vibrational displacement curves of the mining riser in both the IL and CF directions. In Fig. 6(a), the IL flow direction vibrational displacement amplitude is small, the fluctuation range is small, and the maximum magnitude is more than 5 mm . From the prediction results, the three prediction curves have high prediction accuracy, with
设定固体颗粒为 10 目，气固液比为 2 % : 3 % : 95 % 2 % : 3 % : 95 % 2%:3%:95%2 \%: 3 \%: 95 \% ，外部流速为 30 m / min 30 m / min 30m//min30 \mathrm{~m} / \mathrm{min} ，内部流速分别为 1.3 , 1.5 1.3 , 1.5 1.3,1.51.3,1.5 和 1.7 m / s 1.7 m / s 1.7m//s1.7 \mathrm{~m} / \mathrm{s} 。通过训练 LSTM 模型，可以预测采矿立管在 IL 和 CF 方向上的振动位移曲线。图 6（a）中，IL 流向振动位移振幅较小，波动范围较小，最大振幅大于 5 mm。从预测结果来看，三条预测曲线的预测精度较高，其中
an average RMSE of 4.40%, an average MAE of 3.69%, and an average R 2 R 2 R^(2)\mathrm{R}^{2} of 99.88 % 99.88 % 99.88%99.88 \%. In Fig. 6(b), with the increase in the internal flow rate, the vibrational displacement of the riser also increases and then decreases, the vibration trend of the riser is the similar when the internal flow rate is different. The prediction curves show a significant deviation at the wave’s peak when the internal flow rate is 1.5 m / s 1.5 m / s 1.5m//s1.5 \mathrm{~m} / \mathrm{s}. The RMSE error is 38.04 % 38.04 % 38.04%38.04 \%, the MAE error is 36.08 % 36.08 % 36.08%36.08 \%, and the R 2 R 2 R^(2)\mathrm{R}^{2} is 98.53%.
平均 RMSE 为 4.40%，平均 MAE 为 3.69%，平均 R 2 R 2 R^(2)\mathrm{R}^{2} 为 99.88 % 99.88 % 99.88%99.88 \% 。在图 6（b）中，随着内部流速的增加，立管的振动位移也先增大后减小，当内部流速不同时，立管的振动趋势相似。当内部流速为 1.5 m / s 1.5 m / s 1.5m//s1.5 \mathrm{~m} / \mathrm{s} 时，预测曲线在波峰处出现明显偏差。RMSE 误差为 38.04 % 38.04 % 38.04%38.04 \% ，MAE 误差为 36.08 % 36.08 % 36.08%36.08 \% ， R 2 R 2 R^(2)\mathrm{R}^{2} 为 98.53%。

(a) Displacement of IL direction
(a) IL 方向的位移

(b) Displacement of CF direction
(b) CF 方向的位移
FIG. 6. Vibrational displacement of riser under different internal flow rates.
图 6.不同内部流速下的立管振动位移。
FIG. 7. Vibrational displacement of riser under different external flow velocities.
图 7.不同外部流速下的立管振动位移。
When the solid particles are 10 mesh, the gas-solid-liquid ratio is 2 % : 3 % : 95 % 2 % : 3 % : 95 % 2%:3%:95%2 \%: 3 \%: 95 \%, the external flow rate is 10,20 , and 30 m / min 30 m / min 30m//min30 \mathrm{~m} / \mathrm{min}, and the internal flow rate is 1.7 m / s 1.7 m / s 1.7m//s1.7 \mathrm{~m} / \mathrm{s}, the parameters of the LSTM model are set, and the LSTM model is trained to predict the vibrational displacement curves of the mining riser in the IL direction and the CF direction. In Fig. 7(a), the overall amplitude of the IL vibrational displacement is small. The vibrational amplitude is not more than 1 mm when the external flow rate is 10 m / min 10 m / min 10m//min10 \mathrm{~m} / \mathrm{min}. With the increase in the external flow rate, the vibrational amplitude shows a tendency of increasing and then decreasing, and the maximum vibrational amplitude is more than 4 mm when the external flow rate is 20 m / min 20 m / min 20m//min20 \mathrm{~m} / \mathrm{min}. Moreover, at the external flow rate of 10 m / min 10 m / min 10m//min10 \mathrm{~m} / \mathrm{min}, the prediction deviation of the wave occurs, the maximum value of RMSE is 2.82 % 2.82 % 2.82%2.82 \%, the maximum value of MAE is 1.20 % 1.20 % 1.20%1.20 \%, and the R 2 R 2 R^(2)\mathrm{R}^{2} is 98.36 % 98.36 % 98.36%98.36 \%. The average RMSE of the three curves is 3.45 % 3.45 % 3.45%3.45 \%, the average MAE is 2.35 % 2.35 % 2.35%2.35 \%, and the average R 2 R 2 R^(2)R^{2} is 99.41 % 99.41 % 99.41%99.41 \%. In
当固体颗粒为 10 目，气固液比为 2 % : 3 % : 95 % 2 % : 3 % : 95 % 2%:3%:95%2 \%: 3 \%: 95 \% ，外部流速为 10、20 和 30 m / min 30 m / min 30m//min30 \mathrm{~m} / \mathrm{min} ，内部流速为 1.7 m / s 1.7 m / s 1.7m//s1.7 \mathrm{~m} / \mathrm{s} 时，设置 LSTM 模型的参数，训练 LSTM 模型预测采矿立管在 IL 方向和 CF 方向的振动位移曲线。在图 7（a）中，IL 振动位移的整体振幅较小。当外部流速为 10 m / min 10 m / min 10m//min10 \mathrm{~m} / \mathrm{min} 时，振动幅度不超过 1 mm。随着外部流速的增加，振幅呈先增大后减小的趋势，当外部流速为 20 m / min 20 m / min 20m//min20 \mathrm{~m} / \mathrm{min} 时，最大振幅超过 4 mm。此外，当外部流速为 10 m / min 10 m / min 10m//min10 \mathrm{~m} / \mathrm{min} 时，波的预测偏差出现，RMSE 的最大值为 2.82 % 2.82 % 2.82%2.82 \% ，MAE 的最大值为 1.20 % 1.20 % 1.20%1.20 \% ， R 2 R 2 R^(2)\mathrm{R}^{2} 为 98.36 % 98.36 % 98.36%98.36 \% 。三条曲线的平均 RMSE 为 3.45 % 3.45 % 3.45%3.45 \% ，平均 MAE 为 2.35 % 2.35 % 2.35%2.35 \% ，平均 R 2 R 2 R^(2)R^{2} 为 99.41 % 99.41 % 99.41%99.41 \% 。在
Fig. 7(b), the variation of vibrational amplitude is relatively small at an external flow rate of 10 m / min 10 m / min 10m//min10 \mathrm{~m} / \mathrm{min}. With an external flow rate of 20 m / 20 m / 20m//20 \mathrm{~m} / min, the amplitude exceeds 19 mm . The experiment shows that the vibration waveform amplitude increases significantly with the increase in the external flow rate. The vibration waveform amplitude reaches the maximum when the external flow rate is 20 m / min 20 m / min 20m//min20 \mathrm{~m} / \mathrm{min}.
图 7(b) 中，当外部流速为 10 m / min 10 m / min 10m//min10 \mathrm{~m} / \mathrm{min} 时，振动振幅的变化相对较小。当外部流速为 20 m / 20 m / 20m//20 \mathrm{~m} / min 时，振幅超过 19 mm。实验表明，随着外部流速的增加，振动波形振幅也显著增加。当外部流速为 20 m / min 20 m / min 20m//min20 \mathrm{~m} / \mathrm{min} 时，振动波形振幅达到最大值。
当固体颗粒为10目时，气液固比不同（2%：95%：3%，6%：85%：9%，10%：75%：15%），外流速 30 m / min 30 m / min 30m//min30 \mathrm{~m} / \mathrm{min}，内部流速为 1.7 m / s 1.7 m / s 1.7m//s1.7 \mathrm{~m} / \mathrm{s}，设置 LSTM 模型参数，并训练 LSTM 模型，从而可以预测采矿隔水管在 IL 方向和 CF 方向的振动位移曲线。在图 8（a） 中，随着气体比的增加，振动的幅度趋于增加，然后减小。当气-液-固比 10 % : 75 % : 15 % 10 % : 75 % : 15 % 10%:75%:15%10 \%: 75 \%: 15 \%.平均值
图 8.不同气液固比下立管的振动位移。
图 9.不同固体粒径下冒口的振动位移。
三条曲线的 RMSE 为 4.69 % 4.69 % 4.69%4.69 \%，则平均 MAE 为 3.84 % 3.84 % 3.84%3.84 \%和平均值 R 2 R 2 R^(2)\mathrm{R}^{2}是 99.92 % 99.92 % 99.92%99.92 \%.在图 . 8 ( b ) 8 ( b ) 8(b)8(\mathrm{~b})，随着瓦斯比的增加，振动幅值呈现先增大后减小的趋势。当气液固比为 6%：85%：9% 时，振动位移幅值大于 14 mm。3 条自然振动位移曲线具有相同的变化趋势，预测精度很高。当气液固比 6 % : 85 % : 9 % 6 % : 85 % : 9 % 6%:85%:9%6 \%: 85 \%: 9 \%，则预测精度最好。三条曲线的平均 RMSE 为 20.18 % 20.18 % 20.18%20.18 \%，则平均 MAE 为 16.48 % 16.48 % 16.48%16.48 \%、平均值 R 2 R 2 R^(2)\mathrm{R}^{2}是 99.57 % 99.57 % 99.57%99.57 \%，并且整体预测精度更高。
对于不同的固体颗粒（10、20 和 30 目），气液比 2 % : 3 % : 95 % 2 % : 3 % : 95 % 2%:3%:95%2 \%: 3 \%: 95 \%，则外部流速为 30 m / min 30 m / min 30m//min30 \mathrm{~m} / \mathrm{min}，并且内部流速达到最大值 1.7 m / s 1.7 m / s 1.7m//s1.7 \mathrm{~m} / \mathrm{s}，设置 LSTM 模型的参数来训练 LSTM 模型。预测提取的组带在 IL 和 CF 方向上的振动位移曲线。在图 9（a） 中，当固体颗粒为 20 目时，振动位移的最大值大于 6 mm，当固体颗粒为 10 目和 30 目时，振动位移呈小范围波动，当固体颗粒为 30 目时，振动波形波峰的预测较差。原因是 LSTM 模型现在可能在后期阶段被过度拟合。但是，总体预测趋势良好，RMSE 为 5.97 % 5.97 % 5.97%5.97 \%，MAE 为 5.03 % 5.03 % 5.03%5.03 \%和 R 2 R 2 R^(2)\mathrm{R}^{2}之 99.94 % 99.94 % 99.94%99.94 \%.在图 9（b） 中，当固体颗粒为 30 目时，振动位移达到最大值超过 17 mm。平均值
FIG. 10. Schematic diagram of external fluid forces acting on the risers.
图 10.作用在立管上的外部流体力示意图。
RMSE of the three curves is 17.29 % 17.29 % 17.29%17.29 \%, the average MAE is 12.80 % 12.80 % 12.80%12.80 \%, the average R 2 R 2 R^(2)\mathrm{R}^{2} is 99.64 % 99.64 % 99.64%99.64 \%. Moreover, the predicted results of the vibrational displacement in the IL and CF directions have a good agreement with the experimental measurements under any particle size. Through the analysis of the coefficient of determination, the displacements of the mining riser are all distributed in the vicinity of the baseline of the zero-error displacement. The prediction accuracy is good, and the coefficient of determination reaches more than 99 % 99 % 99%99 \%. It shows that the model has high accuracy and can effectively predict the vibration response data of the deep-sea gas hydrate mining riser, providing a model basis for real-time on-site warning.
三条曲线的 RMSE 为 17.29 % 17.29 % 17.29%17.29 \% ，平均 MAE 为 12.80 % 12.80 % 12.80%12.80 \% ，平均 R 2 R 2 R^(2)\mathrm{R}^{2} 为 99.64 % 99.64 % 99.64%99.64 \% 。此外，在任何粒度条件下，IL 和 CF 方向振动位移的预测结果与实验测量结果都有很好的一致性。通过测定系数分析，采矿立管的位移均分布在零误差位移基线附近。预测精度较好，判定系数达到 99 % 99 % 99%99 \% 以上。表明该模型精度较高，能有效预测深海天然气水合物开采立管的振动响应数据，为现场实时预警提供了模型依据。

C. Verification with theoretical model calculation results
C.与理论模型计算结果的验证

In order to further verify the correctness and effectiveness of the prediction model, the Hamiltonian principle and finite element method were used to establish a nonlinear vibration theoretical model of the deep-sea hydrate mining riser under internal and external flow excitation, and on-site parameters are set to calculate the vibration response of the mining riser. The correctness and effectiveness of deep learning prediction model can be verified by comparing with the calculation results. Hamilton’s principle demonstrates that the trajectory of an object at any given time and interval makes the action quantity obtain the extreme value, linking the kinetic energy and potential energy of the object, which can be expressed as follows:
为了进一步验证预测模型的正确性和有效性，利用哈密顿原理和有限元法建立了深海水合物采矿立管在内外流激励下的非线性振动理论模型，并设置了现场参数，计算了采矿立管的振动响应。通过与计算结果对比，验证了深度学习预测模型的正确性和有效性。汉密尔顿原理表明，物体在任何给定时间和间隔内的运动轨迹都会使作用量获得极值，将物体的动能和势能联系起来，可表示如下：
∫ t i t f [ δ ( W K − U P ) + δ W ] d t = 0 ∫ t i t f   δ W K − U P + δ W d t = 0 int_(t_(i))^(t_(f))[delta(W_(K)-U_(P))+delta W]dt=0\int_{t_{i}}^{t_{f}}\left[\delta\left(W_{K}-U_{P}\right)+\delta W\right] \mathrm{d} t=0
where W K W K W_(K)W_{K} is the total kinetic energy of the riser system ( J ), U P U P U_(P)U_{P} is the total potential energy of the riser system (J), and W W WW is the work done on the riser system by the non-conservative force (J).
其中， W K W K W_(K)W_{K} 是立管系统的总动能（J）， U P U P U_(P)U_{P} 是立管系统的总势能（J）， W W WW 是非守恒力对立管系统所做的功（J）。
According to the second Kirchhoff stress, after ignoring the Poisson effect, variational division of the potential energy generated by the structural deformation of the riser, the total kinetic energy of the riser system, and the variation of the work done by the nonconservative force on the riser system can be expressed as follows:
根据第二基尔霍夫应力，在忽略泊松效应后，立管结构变形产生的势能、立管系统的总动能以及非守恒力对立管系统做功的变化的变分可表示如下：
{ { ∫ t i ( W K − U P ) d t = ∫ t i t f ∫ 0 L { [ − ( m v + m a ) v ¨ x + ( ρ v I + ρ a C m I o ) x ¨ x ′ ′ − m i ( 0 ¨ x + 2 U v ˙ x ′ + U 2 v x ′ ′ ) − ( E I v x ′ ′ ) ′ ′ + F T v x ′ ′ + E A ( ( v z ′ + 1 2 ( v x ′ 2 + v y ′ 2 ) ) v x ′ ) ′ ] δ v x + − ( m v + m a ) i ¨ y + ( ρ v I + ρ a C m I o ) i ¨ y ′ ′ − m i ( b ¨ y + 2 U v ˙ y ′ + U 2 v y ′ ′ ′ ) − ( E I v y ′ ′ ) ′ ′ + F T v y ′ ′ + E A ( ( v z ′ + 1 2 ( v x ′ 2 + v y ′ 2 ) ) v y ′ ) ′ ] δ v y + [ − ( m v + m a ) v ¨ z − m i ( 0 ¨ z + 2 U v ˙ z ′ + U 2 v z ′ ′ ) + E A ( v z ′ + 1 2 ( v x ′ 2 + v y ′ 2 ) ) ′ } δ v z } d z d t ∫ t i   W K − U P d t = ∫ t i t f   ∫ 0 L   − m v + m a v ¨ x + ρ v I + ρ a C m I o x ¨ x ′ ′ − m i 0 ¨ x + 2 U v ˙ x ′ + U 2 v x ′ ′ − E I v x ′ ′ ′ ′ + F T v x ′ ′ + E A v z ′ + 1 2 v x ′ 2 + v y ′ 2 v x ′ ′ δ v x + − m v + m a i ¨ y + ρ v I + ρ a C m I o i ¨ y ′ ′ − m i b ¨ y + 2 U v ˙ y ′ + U 2 v y ′ ′ ′ − E I v y ′ ′ ′ ′ + F T v y ′ ′ + E A v z ′ + 1 2 v x ′ 2 + v y ′ 2 v y ′ ′ δ v y + − m v + m a v ¨ z − m i 0 ¨ z + 2 U v ˙ z ′ + U 2 v z ′ ′ + E A v z ′ + 1 2 v x ′ 2 + v y ′ 2 ′ δ v z d z d t {[{int_(t_(i))(W_(K)-U_(P))dt=int_(t_(i))^(t_(f))int_(0)^(L){[{:[[-(m_(v)+m_(a))v^(¨)_(x)+(rho_(v)I+rho_(a)C_(m)I_(o))x^(¨)_(x)^('')-m_(i)(0^(¨)_(x)+2Uv^(˙)_(x)^(')+U^(2)v_(x)^(''))],[-(EIv_(x)^(''))^('')+F_(T)v_(x)^('')+EA((v_(z)^(')+(1)/(2)(v_(x)^('2)+v_(y)^('2)))v_(x)^('))^(')]]deltav_(x)+:}],[-(m_(v)+m_(a))i^(¨)_(y)+(rho_(v)I+rho_(a)C_(m)I_(o))i^(¨)_(y)^('')-m_(i)(b^(¨)_(y)+2Uv^(˙)_(y)^(')+U^(2)v_(y)^('''))],[-(EIv_(y)^(''))^('')+F_(T)v_(y)^('')+EA((v_(z)^(')+(1)/(2)(v_(x)^('2)+v_(y)^('2)))v_(y)^('))^(')]]deltav_(y):}],[+[[-(m_(v)+m_(a))v^(¨)_(z)-m_(i)(0^(¨)_(z)+2Uv^(˙)_(z)^(')+U^(2)v_(z)^(''))],[+EA(v_(z)^(')+(1)/(2)(v_(x)^('2)+v_(y)^('2)))^(')]}deltav_(z)]}dzdt\left\{\begin{array}{l} \left\{\int_{t_{i}}\left(W_{K}-U_{P}\right) \mathrm{d} t=\int_{t_{i}}^{t_{f}} \int_{0}^{L}\left\{\begin{array}{l} {\left[\begin{array}{l} -\left(m_{v}+m_{a}\right) \ddot{v}_{x}+\left(\rho_{v} I+\rho_{a} C_{m} I_{o}\right) \ddot{x}_{x}^{\prime \prime}-m_{i}\left(\ddot{0}_{x}+2 U \dot{v}_{x}^{\prime}+U^{2} v_{x}^{\prime \prime}\right) \\ -\left(E I v_{x}^{\prime \prime}\right)^{\prime \prime}+F_{T} v_{x}^{\prime \prime}+E A\left(\left(v_{z}^{\prime}+\frac{1}{2}\left(v_{x}^{\prime 2}+v_{y}^{\prime 2}\right)\right) v_{x}^{\prime}\right)^{\prime} \end{array}\right] \delta v_{x}+} \\ -\left(m_{v}+m_{a}\right) \ddot{i}_{y}+\left(\rho_{v} I+\rho_{a} C_{m} I_{o}\right) \ddot{i}_{y}^{\prime \prime}-m_{i}\left(\ddot{b}_{y}+2 U \dot{v}_{y}^{\prime}+U^{2} v_{y}^{\prime \prime \prime}\right) \\ -\left(E I v_{y}^{\prime \prime}\right)^{\prime \prime}+F_{T} v_{y}^{\prime \prime}+E A\left(\left(v_{z}^{\prime}+\frac{1}{2}\left(v_{x}^{\prime 2}+v_{y}^{\prime 2}\right)\right) v_{y}^{\prime}\right)^{\prime} \end{array}\right] \delta v_{y}\right. \\ +\left[\begin{array}{l} -\left(m_{v}+m_{a}\right) \ddot{v}_{z}-m_{i}\left(\ddot{0}_{z}+2 U \dot{v}_{z}^{\prime}+U^{2} v_{z}^{\prime \prime}\right) \\ +E A\left(v_{z}^{\prime}+\frac{1}{2}\left(v_{x}^{\prime 2}+v_{y}^{\prime 2}\right)\right)^{\prime} \end{array}\right\} \delta v_{z} \end{array}\right\} \mathrm{d} z \mathrm{~d} t
哪里 ∫ A x d A = ∫ A y d A = ∫ A x y d A = 0 , ∫ A x 2 d A = ∫ A y 2 d A = I , A ∫ A   x d A = ∫ A   y d A = ∫ A   x y d A = 0 , ∫ A   x 2 d A = ∫ A   y 2 d A = I , A int_(A)xdA=int_(A)ydA=int_(A)xydA=0,int_(A)x^(2)dA=int_(A)y^(2)dA=I,A\int_{A} x \mathrm{~d} A=\int_{A} y \mathrm{~d} A=\int_{A} x y \mathrm{~d} A=0, \int_{A} x^{2} \mathrm{~d} A=\int_{A} y^{2} \mathrm{~d} A=I, A是立管的横截面积 ( m 2 ) m 2 (m^(2))\left(\mathrm{m}^{2}\right)和 A = π ( D o 2 − D i 2 ) 4 A = π D o 2 − D i 2 4 A=(pi(D_(o)^(2)-D_(i)^(2)))/(4)A=\frac{\pi\left(D_{o}^{2}-D_{i}^{2}\right)}{4}，其中 D o D o D_(o)D_{o}和 D i D i D_(i)D_{i}分别是踢面的外径和内径 ( m ) , v x , v y , v z ( m ) , v x , v y , v z (m),v_(x),v_(y),v_(z)(\mathrm{m}), v_{x}, v_{y}, v_{z}是 x , y , z x , y , z x,y,zx, y, zdirections 分别 ( m ) , v ˙ x ( m ) , v ˙ x (m),v^(˙)_(x)(\mathrm{m}), \dot{v}_{x}, v ˙ y , v ˙ z v ˙ y , v ˙ z v^(˙)_(y),v^(˙)_(z)\dot{v}_{y}, \dot{v}_{z}是位移相对于 Riser 时间的一阶导数 x , y , z x , y , z x,y,zx, y, zdirections 分别 ( m / s ) , i ¨ x , b ¨ y , b ¨ z ( m / s ) , i ¨ x , b ¨ y , b ¨ z (m//s),i^(¨)_(x),b^(¨)_(y),b^(¨)_(z)(\mathrm{m} / \mathrm{s}), \ddot{i}_{x}, \ddot{b}_{y}, \ddot{b}_{z}是位移相对于 Riser 时间的二阶导数。 x , y , z x , y , z x,y,zx, y, zdirections 分别 ( m / s 2 ) , v ′ x , v y ′ , v z ′ m / s 2 , v ′ x , v y ′ , v z ′ (m//s^(2)),v^(')x,v_(y)^('),v_(z)^(')\left(\mathrm{m} / \mathrm{s}^{2}\right), v^{\prime} x, v_{y}^{\prime}, v_{z}^{\prime}是 位移在 z z zz-Axis 中 x , y , z x , y , z x,y,zx, y, zdirections 分别 ( rad ) , v ′ ′ x , v ′ ′ y , v ′ ′ z ( rad ) , v ′ ′ x , v ′ ′ y , v ′ ′ z (rad),v^('')x,v^('')y,v^('')z(\mathrm{rad}), v^{\prime \prime} x, v^{\prime \prime} y, v^{\prime \prime} z是 z z zz-Axis 中 x , y , z x , y , z x,y,zx, y, z方向分别为 （ ms ）、 L L LL是踢面的长度 （ m ）， E E EE是冒口材料的弹性模量 （Pa）， I I II是立管截面的惯性矩 ( m 4 ) , F T m 4 , F T (m^(4)),F_(T)\left(\mathrm{m}^{4}\right), F_{T}是踢面张力 ( N ) . U ( N ) . U (N).U(\mathrm{N}) . U是冒口中的流速 ( m / s ) ( m / s ) (m//s)(\mathrm{m} / \mathrm{s}), v x , v y , v z v x , v y , v z v_(x),v_(y),v_(z)v_{x}, v_{y}, v_{z}是 x , y x , y x,yx, y和 z z zzdirections 分别 ( m / s ) , A 0 ( m / s ) , A 0 (m//s),A_(0)(\mathrm{m} / \mathrm{s}), A_{0}是踢面外圆的横截面积 ( m 2 ) , A i m 2 , A i (m^(2)),A_(i)\left(\mathrm{m}^{2}\right), A_{i}是踢面内圆的横截面积 ( m 2 ) , I o m 2 , I o (m^(2)),I_(o)\left(\mathrm{m}^{2}\right), I_{o}是立管圆柱截面的惯性矩 ( m 4 ) , ρ v m 4 , ρ v (m^(4)),rho_(v)\left(\mathrm{m}^{4}\right), \rho_{v}是踢面的材料密度 ( kg / m 3 ) , ρ i kg / m 3 , ρ i (kg//m^(3)),rho_(i)\left(\mathrm{kg} / \mathrm{m}^{3}\right), \rho_{i}是流入密度 ( kg / m 3 ) , ρ a kg / m 3 , ρ a (kg//m^(3)),rho_(a)\left(\mathrm{kg} / \mathrm{m}^{3}\right), \rho_{a}是海水的密度 ( kg / m 3 ) , C m kg / m 3 , C m (kg//m^(3)),C_(m)\left(\mathrm{kg} / \mathrm{m}^{3}\right), C_{m}是附加质量系数，对于具有圆形截面的立管，通常为 1.0， m v m v m_(v)m_{v}是踢面的单位质量 ( kg / m ) , m a ( kg / m ) , m a (kg//m),m_(a)(\mathrm{kg} / \mathrm{m}), m_{a}是附加的单位质量 ( kg / m ) ( kg / m ) (kg//m)(\mathrm{kg} / \mathrm{m}), F D , F L F D , F L F_(D),F_(L)F_{D}, F_{L}，分别表示外部流体 IL 方向的阻力和 CF 方向的升力 （ N ）， F c x , F c y , F c z F c x , F c y , F c z F_(cx),F_(cy),F_(cz)F_{c x}, F_{c y}, F_{c z}是结构阻尼力 ( N ) , w g ( N ) , w g (N),w_(g)(\mathrm{N}), w_{g}是 Recter 纵向力 ( N ) ( N ) (N)(\mathrm{N})和 W l W l W_(l)W_{l}是内部流动传输过程中的能量损失 [与气-液-固三相流 （J） 的流动特性有关]。
其中 ∫ A x d A = ∫ A y d A = ∫ A x y d A = 0 , ∫ A x 2 d A = ∫ A y 2 d A = I , A ∫ A   x d A = ∫ A   y d A = ∫ A   x y d A = 0 , ∫ A   x 2 d A = ∫ A   y 2 d A = I , A int_(A)xdA=int_(A)ydA=int_(A)xydA=0,int_(A)x^(2)dA=int_(A)y^(2)dA=I,A\int_{A} x \mathrm{~d} A=\int_{A} y \mathrm{~d} A=\int_{A} x y \mathrm{~d} A=0, \int_{A} x^{2} \mathrm{~d} A=\int_{A} y^{2} \mathrm{~d} A=I, A 是立管的横截面积 ( m 2 ) m 2 (m^(2))\left(\mathrm{m}^{2}\right) 和 A = π ( D o 2 − D i 2 ) 4 A = π D o 2 − D i 2 4 A=(pi(D_(o)^(2)-D_(i)^(2)))/(4)A=\frac{\pi\left(D_{o}^{2}-D_{i}^{2}\right)}{4} ，其中 D o D o D_(o)D_{o} 和 D i D i D_(i)D_{i} 是立管的外径和内径，分别为 ( m ) , v x , v y , v z ( m ) , v x , v y , v z (m),v_(x),v_(y),v_(z)(\mathrm{m}), v_{x}, v_{y}, v_{z} 是立管在 x , y , z x , y , z x,y,zx, y, z 方向上的位移，分别为 ( m ) , v ˙ x ( m ) , v ˙ x (m),v^(˙)_(x)(\mathrm{m}), \dot{v}_{x} ， v ˙ y , v ˙ z v ˙ y , v ˙ z v^(˙)_(y),v^(˙)_(z)\dot{v}_{y}, \dot{v}_{z} 是立管在 x , y , z x , y , z x,y,zx, y, z 方向上的位移相对于时间的一阶导数，分别为 ( m / s ) , i ¨ x , b ¨ y , b ¨ z ( m / s ) , i ¨ x , b ¨ y , b ¨ z (m//s),i^(¨)_(x),b^(¨)_(y),b^(¨)_(z)(\mathrm{m} / \mathrm{s}), \ddot{i}_{x}, \ddot{b}_{y}, \ddot{b}_{z} 是立管在 x , y , z x , y , z x,y,zx, y, z 方向上的位移相对于时间的二阶导数，分别为 ( m / s 2 ) , v ′ x , v y ′ , v z ′ m / s 2 , v ′ x , v y ′ , v z ′ (m//s^(2)),v^(')x,v_(y)^('),v_(z)^(')\left(\mathrm{m} / \mathrm{s}^{2}\right), v^{\prime} x, v_{y}^{\prime}, v_{z}^{\prime} 是立管在 x , y , z x , y , z x,y,zx, y, z 方向上的位移相对于时间的一阶导数、 ( m / s ) , i ¨ x , b ¨ y , b ¨ z ( m / s ) , i ¨ x , b ¨ y , b ¨ z (m//s),i^(¨)_(x),b^(¨)_(y),b^(¨)_(z)(\mathrm{m} / \mathrm{s}), \ddot{i}_{x}, \ddot{b}_{y}, \ddot{b}_{z} 分别为 x , y , z x , y , z x,y,zx, y, z 方向上立管位移随时间的二阶导数， ( m / s 2 ) , v ′ x , v y ′ , v z ′ m / s 2 , v ′ x , v y ′ , v z ′ (m//s^(2)),v^(')x,v_(y)^('),v_(z)^(')\left(\mathrm{m} / \mathrm{s}^{2}\right), v^{\prime} x, v_{y}^{\prime}, v_{z}^{\prime} 分别为 x , y , z x , y , z x,y,zx, y, z 方向上立管位移在 z z zz 轴上的一阶导数， ( rad ) , v ′ ′ x , v ′ ′ y , v ′ ′ z ( rad ) , v ′ ′ x , v ′ ′ y , v ′ ′ z (rad),v^('')x,v^('')y,v^('')z(\mathrm{rad}), v^{\prime \prime} x, v^{\prime \prime} y, v^{\prime \prime} z 分别为 x , y , z x , y , z x,y,zx, y, z 方向上立管位移在 z z zz 轴上的二阶导数（毫秒）、 L L LL 是立管长度（m）， E E EE 是立管材料的弹性模量（pa）、 I I II 是立管断面的惯性矩 ( m 4 ) , F T m 4 , F T (m^(4)),F_(T)\left(\mathrm{m}^{4}\right), F_{T} 是立管张力 ( N ) . U ( N ) . U (N).U(\mathrm{N}) . U 是立管中的流速 ( m / s ) ( m / s ) (m//s)(\mathrm{m} / \mathrm{s}) 、 v x , v y , v z v x , v y , v z v_(x),v_(y),v_(z)v_{x}, v_{y}, v_{z} 是水流在 x , y x , y x,yx, y 和 z z zz 方向的绝对速度、 ( m / s ) , A 0 ( m / s ) , A 0 (m//s),A_(0)(\mathrm{m} / \mathrm{s}), A_{0} 分别为隔水管外圆的横截面积 ( m 2 ) , A i m 2 , A i (m^(2)),A_(i)\left(\mathrm{m}^{2}\right), A_{i} 为隔水管内圆的横截面积 ( m 2 ) , I o m 2 , I o (m^(2)),I_(o)\left(\mathrm{m}^{2}\right), I_{o} 为隔水管圆柱截面的转动惯量 ( m 4 ) , ρ v m 4 , ρ v (m^(4)),rho_(v)\left(\mathrm{m}^{4}\right), \rho_{v} 为隔水管的材料密度 ( kg / m 3 ) , ρ i kg / m 3 , ρ i (kg//m^(3)),rho_(i)\left(\mathrm{kg} / \mathrm{m}^{3}\right), \rho_{i} 为流入密度 ( kg / m 3 ) , ρ a kg / m 3 , ρ a (kg//m^(3)),rho_(a)\left(\mathrm{kg} / \mathrm{m}^{3}\right), \rho_{a} 为海水密度 ( kg / m 3 ) , C m kg / m 3 , C m (kg//m^(3)),C_(m)\left(\mathrm{kg} / \mathrm{m}^{3}\right), C_{m} 为附加质量系数，一般为 1. m v m v m_(v)m_{v} 是立管的单位质量 ( kg / m ) , m a ( kg / m ) , m a (kg//m),m_(a)(\mathrm{kg} / \mathrm{m}), m_{a} 是附加单位质量 ( kg / m ) ( kg / m ) (kg//m)(\mathrm{kg} / \mathrm{m}) 、 F D , F L F D , F L F_(D),F_(L)F_{D}, F_{L} 分别表示 IL 方向的阻力和 CF 方向的升力。 F c x , F c y , F c z F c x , F c y , F c z F_(cx),F_(cy),F_(cz)F_{c x}, F_{c y}, F_{c z} 是结构阻尼力 ( N ) , w g ( N ) , w g (N),w_(g)(\mathrm{N}), w_{g} 是立管纵向力 ( N ) ( N ) (N)(\mathrm{N}) ， W l W l W_(l)W_{l} 是内部流动传输过程中的能量损失 [与气液固三相流的流动特性有关 (J)]。
将方程（37）代入方程（36），得到 x , y , z x , y , z x,y,zx, y, z可以按如下方式获得：
{ ( m v + m a + m i ) v ¨ x − ( ρ v I + ρ a C m I o ) v ′ ′ ¨ x + c v ˙ x + 2 m i U v ˙ ′ x + m i U 2 v ′ ′ x + E I v x ( 4 ) − F T v ′ ′ x − E A ( ( v z ′ + 1 2 ( v x ′ 2 + v y ′ 2 ) ) v x ′ ) ′ = F D , ( m v + m a + m i ) v ¨ y − ( ρ v I + ρ a C m I o ) v ′ ′ ˙ y + v y ˙ + 2 m i U v ˙ y ′ y + m i U 2 v ′ ′ ′ y + E I v y ( 4 ) − F T v ′ ′ y − E A ( ( ( v z ′ + 1 2 ( v x ′ 2 + v y ′ 2 ) ) v y ′ ) ′ = F L , ( m v + m a + m i ) v ¨ z + v z ˙ − E A ( v z ′ ′ + 1 2 ( v x ′ 2 + v y ′ 2 ) ′ ) = w g . m v + m a + m i v ¨ x − ρ v I + ρ a C m I o v ′ ′ ¨ x + c v ˙ x + 2 m i U v ˙ ′ x + m i U 2 v ′ ′ x + E I v x ( 4 ) − F T v ′ ′ x − E A v z ′ + 1 2 v x ′ 2 + v y ′ 2 v x ′ ′ = F D , m v + m a + m i v ¨ y − ρ v I + ρ a C m I o v ′ ′ ˙ y + v y ˙ + 2 m i U v ˙ y ′ y + m i U 2 v ′ ′ ′ y + E I v y ( 4 ) − F T v ′ ′ y − E A v z ′ + 1 2 v x ′ 2 + v y ′ 2 v y ′ ′ = F L , m v + m a + m i v ¨ z + v z ˙ − E A v z ′ ′ + 1 2 v x ′ 2 + v y ′ 2 ′ = w g . {[(m_(v)+m_(a)+m_(i))v^(¨)_(x)-(rho_(v)I+rho_(a)C_(m)I_(o))v^('')^(¨)_(x)+cv^(˙)_(x)+2m_(i)Uv^(˙)^(')_(x)],[quad+m_(i)U^(2)v^('')_(x)+EIv_(x)^((4))-F_(T)v^('')_(x)-EA((v_(z)^(')+(1)/(2)(v_(x)^('2)+v_(y)^('2)))v_(x)^('))^(')=F_(D)","],[(m_(v)+m_(a)+m_(i))v^(¨)_(y)-(rho_(v)I+rho_(a)C_(m)I_(o))v^('')^(˙)_(y)+v_(y)^(˙)+2m_(i)Uv^(˙)_(y)^(')_(y)],[quad+m_(i)U^(2)v^(''')_(y)+EIv_(y)^((4))-F_(T)v^('')_(y)-EA(((v_(z)^(')+(1)/(2)(v_(x)^('2)+v_(y)^('2)))v_(y)^('))^(')=F_(L),:}],[(m_(v)+m_(a)+m_(i))v^(¨)_(z)+v_(z)^(˙)-EA(v_(z)^('')+(1)/(2)(v_(x)^('2)+v_(y)^('2))^('))=w_(g).]:}\left\{\begin{array}{l} \left(m_{v}+m_{a}+m_{i}\right) \ddot{v}_{x}-\left(\rho_{v} I+\rho_{a} C_{m} I_{o}\right){\ddot{v^{\prime \prime}}}_{x}+c \dot{v}_{x}+2 m_{i} U \dot{v}^{\prime}{ }_{x} \\ \quad+m_{i} U^{2} v^{\prime \prime}{ }_{x}+E I v_{x}{ }^{(4)}-F_{T} v^{\prime \prime}{ }_{x}-E A\left(\left(v_{z}^{\prime}+\frac{1}{2}\left(v_{x}^{\prime 2}+v_{y}^{\prime 2}\right)\right) v_{x}^{\prime}\right)^{\prime}=F_{D}, \\ \left(m_{v}+m_{a}+m_{i}\right) \ddot{v}_{y}-\left(\rho_{v} I+\rho_{a} C_{m} I_{o}\right){\dot{v^{\prime \prime}}}_{y}+\dot{v_{y}}+2 m_{i} U \dot{v}_{y}^{\prime}{ }_{y} \\ \quad+m_{i} U^{2} v^{\prime \prime \prime}{ }_{y}+E I v_{y}{ }^{(4)}-F_{T} v^{\prime \prime}{ }_{y}-E A\left(\left(\left(v_{z}^{\prime}+\frac{1}{2}\left(v_{x}^{\prime 2}+v_{y}^{\prime 2}\right)\right) v_{y}^{\prime}\right)^{\prime}=F_{L},\right. \\ \left(m_{v}+m_{a}+m_{i}\right) \ddot{v}_{z}+\dot{v_{z}}-E A\left(v_{z}^{\prime \prime}+\frac{1}{2}\left(v_{x}^{\prime 2}+v_{y}^{\prime 2}\right)^{\prime}\right)=w_{g} . \end{array}\right.
假设流体和柱之间的相对流速为 V r V r V_(r)V_{r}，字符串的流出率为 U c U c U_(c)U_{c}.基于稳态流动，作用在柱上的稳态阻力和升力如图 10 所示。
根据 Morison 方程，作用在采矿柱上的阻力和升力如下：
{ F D = 1 2 ρ a U c 2 D o C ¯ d − ρ a U c D o C ¯ d ∂ v x ∂ t + 1 2 ρ a U c 2 D o C D , F L = − 1 2 ρ a U c D o C ¯ d ∂ v y ∂ t + 1 2 ρ a U c 2 D o C L , F D = 1 2 ρ a U c 2 D o C ¯ d − ρ a U c D o C ¯ d ∂ v x ∂ t + 1 2 ρ a U c 2 D o C D , F L = − 1 2 ρ a U c D o C ¯ d ∂ v y ∂ t + 1 2 ρ a U c 2 D o C L , {[F_(D)=(1)/(2)rho_(a)U_(c)^(2)D_(o) bar(C)_(d)-rho_(a)U_(c)D_(o) bar(C)_(d)(delv_(x))/(del t)+(1)/(2)rho_(a)U_(c)^(2)D_(o)C_(D)","],[F_(L)=-(1)/(2)rho_(a)U_(c)D_(o) bar(C)_(d)(delv_(y))/(del t)+(1)/(2)rho_(a)U_(c)^(2)D_(o)C_(L)","]:}\left\{\begin{array}{l} F_{D}=\frac{1}{2} \rho_{a} U_{c}^{2} D_{o} \bar{C}_{d}-\rho_{a} U_{c} D_{o} \bar{C}_{d} \frac{\partial v_{x}}{\partial t}+\frac{1}{2} \rho_{a} U_{c}^{2} D_{o} C_{D}, \\ F_{L}=-\frac{1}{2} \rho_{a} U_{c} D_{o} \bar{C}_{d} \frac{\partial v_{y}}{\partial t}+\frac{1}{2} \rho_{a} U_{c}^{2} D_{o} C_{L}, \end{array}\right.
哪里 C ¯ d C ¯ d bar(C)_(d)\bar{C}_{d}是稳态阻力系数， C D , C L C D , C L C_(D),C_(L)C_{D}, C_{L}分别是脉动阻力系数和脉动升力系数。
流体力变化和管振动之间的耦合可以用尾流振荡器模型来描述。van der Pol 非线性振动方程用于描述流体涡流的脱落特性，尾流振荡器的控制方程如下：
{ q ¨ x + ε x ω s ( q x 2 − 1 ) q ˙ x + ( 2 ω s ) 2 q x = A x D i x ¨ ¨ x , q ¨ y + ε y ω s ( q y 2 − 1 ) q ˙ y + ( ω s ) 2 q y = A y D i i ¨ y , q ¨ x + ε x ω s q x 2 − 1 q ˙ x + 2 ω s 2 q x = A x D i x ¨ ¨ x , q ¨ y + ε y ω s q y 2 − 1 q ˙ y + ω s 2 q y = A y D i i ¨ y , {[q^(¨)_(x)+epsi_(x)omega_(s)(q_(x)^(2)-1)q^(˙)_(x)+(2omega_(s))^(2)q_(x)=(A_(x))/(D_(i))x^(¨)^(¨)_(x)","],[q^(¨)_(y)+epsi_(y)omega_(s)(q_(y)^(2)-1)q^(˙)_(y)+(omega_(s))^(2)q_(y)=(A_(y))/(D_(i))i^(¨)_(y)","]:}\left\{\begin{array}{l} \ddot{q}_{x}+\varepsilon_{x} \omega_{s}\left(q_{x}^{2}-1\right) \dot{q}_{x}+\left(2 \omega_{s}\right)^{2} q_{x}=\frac{A_{x}}{D_{i}} \ddot{\ddot{x}}_{x}, \\ \ddot{q}_{y}+\varepsilon_{y} \omega_{s}\left(q_{y}^{2}-1\right) \dot{q}_{y}+\left(\omega_{s}\right)^{2} q_{y}=\frac{A_{y}}{D_{i}} \ddot{i}_{y}, \end{array}\right.
哪里 q x , q y q x , q y q_(x),q_(y)q_{x}, q_{y}是 IL 和 CF 方向的无量纲唤醒振荡器变量， ω s ω s omega_(s)\omega_{s}是涡旋脱落频率，并且 ε x , ε y , A x , A y ε x , ε y , A x , A y epsi_(x),epsi_(y),A_(x),A_(y)\varepsilon_{x}, \varepsilon_{y}, A_{x}, A_{y}是实验确定的无量纲协变量。
通过引入 IL 和 CF 无量纲唤醒振荡器变量 q x q x q_(x)q_{x}和 q y q y q_(y)q_{y}到脉动阻力系数 C D ( = C d ′ q x 2 ) C D = C d ′ q x 2 C_(D)(=C_(d)^(')(q_(x))/(2))C_{D}\left(=C_{d}^{\prime} \frac{q_{x}}{2}\right)和脉动升力系数 C L ( = C l ′ , q ν 2 ) C L = C l ′ , q ν 2 C_(L)(=C_(l)^('),(q_(nu))/(2))C_{L}\left(=C_{l}^{\prime}, \frac{q_{\nu}}{2}\right)并通过使 c ′ = 1 2 ρ a U c D o C ¯ d c ′ = 1 2 ρ a U c D o C ¯ d c^(')=(1)/(2)rho_(a)U_(c)D_(o) bar(C)_(d)c^{\prime}=\frac{1}{2} \rho_{a} U_{c} D_{o} \bar{C}_{d}，作用在立管上的外部流体力的最终形式可以得到如下：
{ F D = 1 2 ρ a U c 2 D o C ¯ d − 2 c ′ ∂ v x ∂ t + 1 2 ρ a U c 2 D o C d ′ q x 2 , F L = − c ′ ∂ v y ∂ t + 1 2 ρ a U c 2 D o C l ′ q y 2 . F D = 1 2 ρ a U c 2 D o C ¯ d − 2 c ′ ∂ v x ∂ t + 1 2 ρ a U c 2 D o C d ′ q x 2 , F L = − c ′ ∂ v y ∂ t + 1 2 ρ a U c 2 D o C l ′ q y 2 . {[F_(D)=(1)/(2)rho_(a)U_(c)^(2)D_(o) bar(C)_(d)-2c^(')(delv_(x))/(del t)+(1)/(2)rho_(a)U_(c)^(2)D_(o)C_(d)^(')(q_(x))/(2)","],[F_(L)=-c^(')(delv_(y))/(del t)+(1)/(2)rho_(a)U_(c)^(2)D_(o)C_(l)^(')(q_(y))/(2).]:}\left\{\begin{array}{l} F_{D}=\frac{1}{2} \rho_{a} U_{c}^{2} D_{o} \bar{C}_{d}-2 c^{\prime} \frac{\partial v_{x}}{\partial t}+\frac{1}{2} \rho_{a} U_{c}^{2} D_{o} C_{d}{ }^{\prime} \frac{q_{x}}{2}, \\ F_{L}=-c^{\prime} \frac{\partial v_{y}}{\partial t}+\frac{1}{2} \rho_{a} U_{c}^{2} D_{o} C_{l}^{\prime} \frac{q_{y}}{2} . \end{array}\right.
参考 Refs 的结论。56 和 57 中，van der Pol 方程的参数取如下： A x = 48 , A y = 12 A x = 48 , A y = 12 A_(x)=48,A_(y)=12A_{x}=48, A_{y}=12, ε x = 1.2 , ε y = 0.3 , C d = 1.2 , C ′ d = 0.3 , C ′ l = 0.4 ε x = 1.2 , ε y = 0.3 , C d = 1.2 , C ′ d = 0.3 , C ′ l = 0.4 epsi_(x)=1.2,epsi_(y)=0.3,C_(d)=1.2,C^(')_(d)=0.3,C^(')_(l)=0.4\varepsilon_{x}=1.2, \varepsilon_{y}=0.3, C_{d}=1.2, C^{\prime}{ }_{d}=0.3, C^{\prime}{ }_{l}=0.4和 S t = 0.2 S t = 0.2 S_(t)=0.2S_{t}=0.2.
图 11.计算流程图。
浮平台的升降和沉沉运动类似于简谐运动，其升降和沉没位移由波高和船舶或平台的类型、结构、规模等因素决定，一般升降和沉井位移远小于波高，运动规律可以表示为
u boat ( t ) = μ H 2 sin ⁡ ( 2 π T W t ) , u boat  ( t ) = μ H 2 sin ⁡ 2 π T W t , u_("boat ")(t)=(mu H)/(2)sin((2pi)/(T_(W))t),u_{\text {boat }}(t)=\frac{\mu H}{2} \sin \left(\frac{2 \pi}{T_{W}} t\right),
哪里 u boat ( t ) u boat  ( t ) u_("boat ")(t)u_{\text {boat }}(t)是平台位移 （ m ）， H H HH是波高 （ m ）， T W T W T_(W)T_{W}是波周期 （ s ），而 μ μ mu\mu是升力位移与波高的比率。
多相流对管道的激励效应会有所不同。流体密度的变化是立管中多相流激发管道振动的主要原因。 58 , 59 58 , 59 ^(58,59){ }^{58,59}由于水合物运移过程中井筒温度和压力的变化，水合物会分解，导致沿采矿隔水管长度产生不同的气-液-固三相含量和密度，进而影响采矿隔水管中内部三相流体的附加质量。因此，需要获取采矿隔水管中内部三相流体沿水深的密度，以分析流型对采矿隔水管的激励效应。
根据我们最近工作中的水合物分解模型、立管温度分布模型和立管压力分布模型， 60 60 ^(60){ }^{60}内容 E g , E l , E s E g , E l , E s E_(g),E_(l),E_(s)E_{g}, E_{l}, E_{s}和流速 v g , v l , v s v g , v l , v s v_(g),v_(l),v_(s)v_{g}, v_{l}, v_{s}可以分别获得油管中流体沿井深的气相、液相和固相，流体在油管中的气相、液相和固相在三个方向上的绝对速度可以表示如下：
{ v g x = v ˙ x + v g v x ′ , v g y = v ˙ y + v g v y ′ , v g z = v ˙ z + v g ( 1 + v z ′ ) v l x = v ˙ x + v l v x ′ , v l y = v ˙ y + v l v y ′ , v l z = v ˙ z + v l ( 1 + v z ′ ) v s x = v ˙ x + v s v x ′ , v s y = v ˙ y + v s v y ′ , v s z = v ˙ z + v s ( 1 + v z ′ ) , v g x = v ˙ x + v g v x ′ , v g y = v ˙ y + v g v y ′ , v g z = v ˙ z + v g 1 + v z ′ v l x = v ˙ x + v l v x ′ , v l y = v ˙ y + v l v y ′ , v l z = v ˙ z + v l 1 + v z ′ v s x = v ˙ x + v s v x ′ , v s y = v ˙ y + v s v y ′ , v s z = v ˙ z + v s 1 + v z ′ , {[v_(gx)=v^(˙)_(x)+v_(g)v_(x)^(')",",v_(gy)=v^(˙)_(y)+v_(g)v_(y)^(')",",v_(gz)=v^(˙)_(z)+v_(g)(1+v_(z)^('))],[v_(lx)=v^(˙)_(x)+v_(l)v_(x)^(')",",v_(ly)=v^(˙)_(y)+v_(lv_(y)^('))",",v_(lz)=v^(˙)_(z)+v_(l)(1+v_(z)^('))],[v_(sx)=v^(˙)_(x)+v_(s)v_(x)^(')",",v_(sy)=v^(˙)_(y)+v_(s)v_(y)^(')",",v_(sz)=v^(˙)_(z)+v_(s)(1+v_(z)^('))","]:}\left\{\begin{array}{lll} v_{g x}=\dot{v}_{x}+v_{g} v_{x}^{\prime}, & v_{g y}=\dot{v}_{y}+v_{g} v_{y}^{\prime}, & v_{g z}=\dot{v}_{z}+v_{g}\left(1+v_{z}^{\prime}\right) \\ v_{l x}=\dot{v}_{x}+v_{l} v_{x}^{\prime}, & v_{l y}=\dot{v}_{y}+v_{l v_{y}^{\prime}}, & v_{l z}=\dot{v}_{z}+v_{l}\left(1+v_{z}^{\prime}\right) \\ v_{s x}=\dot{v}_{x}+v_{s} v_{x}^{\prime}, & v_{s y}=\dot{v}_{y}+v_{s} v_{y}^{\prime}, & v_{s z}=\dot{v}_{z}+v_{s}\left(1+v_{z}^{\prime}\right), \end{array}\right.
哪里 v g x , v g y , v g z v g x , v g y , v g z v_(gx),v_(gy),v_(gz)v_{g x}, v_{g y}, v_{g z}是气相在 x , y , z x , y , z x,y,zx, y, zdirections （ m / s m / s m//s\mathrm{m} / \mathrm{s}), v l x , v l y , v l z v l x , v l y , v l z v_(lx),v_(ly),v_(lz)v_{l x}, v_{l y}, v_{l z}是液相在 x , y , z x , y , z x,y,zx, y, zdirections 分别 ( m / s ) ( m / s ) (m//s)(\mathrm{m} / \mathrm{s})和 v s x , v s y , v s z v s x , v s y , v s z v_(sx),v_(sy),v_(sz)v_{s x}, v_{s y}, v_{s z}




（a） IL 方向


的位移 （b） CF 方向的位移
图 12.不同实际工况下立管方向的振动预测效果。是固相的绝对速度 x , y , z x , y , z x,y,zx, y, zdirections （ m / s m / s m//s\mathrm{m} / \mathrm{s}).
根据汉密尔顿原理，油管中流体的质量和速率会影响油管中流体的动能。当分别考虑气相、液相和固相的含量以及气相、液相和固相沿井深变化的速率时，油管中流体的动能可以表示如下：
©轴向位移
图 12.（续。
W in = 1 2 ∫ 0 L { A i E g ρ g ( v g x 2 + v g y 2 + v g z 2 ) + A i E l ρ l ( v l x 2 + v l y 2 + v l z 2 ) + A i E s ρ s ( v s x 2 + v s y 2 + v s z 2 ) } d z = 1 2 ∫ 0 L { A i E g ρ g [ ( v ˙ x + v g v x ′ ) 2 + ( v ˙ y + v g v y ′ ) 2 + ( v ˙ z + v g ( 1 + v z ′ ) ) 2 ] + A i E l ρ l [ ( v ˙ x + v l v x ′ ) 2 + ( v ˙ y + v l v y ′ ) 2 + ( v ˙ z + v l ( 1 + v z ′ ) ) 2 ] + A i E s ρ s [ ( v ˙ x + v s v x ′ ) 2 + ( v ˙ y + v s v y ′ ) 2 + ( v ˙ z + v s ( 1 + v z ′ ) ) 2 ] } d z W in  = 1 2 ∫ 0 L   A i E g ρ g v g x 2 + v g y 2 + v g z 2 + A i E l ρ l v l x 2 + v l y 2 + v l z 2 + A i E s ρ s v s x 2 + v s y 2 + v s z 2 d z = 1 2 ∫ 0 L   A i E g ρ g v ˙ x + v g v x ′ 2 + v ˙ y + v g v y ′ 2 + v ˙ z + v g 1 + v z ′ 2 + A i E l ρ l v ˙ x + v l v x ′ 2 + v ˙ y + v l v y ′ 2 + v ˙ z + v l 1 + v z ′ 2 + A i E s ρ s v ˙ x + v s v x ′ 2 + v ˙ y + v s v y ′ 2 + v ˙ z + v s 1 + v z ′ 2 d z {:[W_("in ")=(1)/(2)int_(0)^(L){[A_(i)E_(g)rho_(g)(v_(gx)^(2)+v_(gy)^(2)+v_(gz)^(2))+A_(i)E_(l)rho_(l)(v_(lx)^(2)+v_(ly)^(2)+v_(lz)^(2))],[+A_(i)E_(s)rho_(s)(v_(sx)^(2)+v_(sy)^(2)+v_(sz)^(2))]}dz],[=(1)/(2)int_(0)^(L){[A_(i)E_(g)rho_(g)[(v^(˙)_(x)+v_(g)v_(x)^('))^(2)+(v^(˙)_(y)+v_(g)v_(y)^('))^(2)+(v^(˙)_(z)+v_(g)(1+v_(z)^(')))^(2)]],[+A_(i)E_(l)rho_(l)[(v^(˙)_(x)+v_(l)v_(x)^('))^(2)+(v^(˙)_(y)+v_(l)v_(y)^('))^(2)+(v^(˙)_(z)+v_(l)(1+v_(z)^(')))^(2)]],[+A_(i)E_(s)rho_(s)[(v^(˙)_(x)+v_(s)v_(x)^('))^(2)+(v^(˙)_(y)+v_(s)v_(y)^('))^(2)+(v^(˙)_(z)+v_(s)(1+v_(z)^(')))^(2)]]}dz]:}\begin{aligned} & W_{\text {in }}= \frac{1}{2} \int_{0}^{L}\left\{\begin{array}{l} A_{i} E_{g} \rho_{g}\left(v_{g x}^{2}+v_{g y}^{2}+v_{g z}^{2}\right)+A_{i} E_{l} \rho_{l}\left(v_{l x}^{2}+v_{l y}^{2}+v_{l z}^{2}\right) \\ +A_{i} E_{s} \rho_{s}\left(v_{s x}^{2}+v_{s y}^{2}+v_{s z}^{2}\right) \end{array}\right\} \mathrm{d} z \\ &=\frac{1}{2} \int_{0}^{L}\left\{\begin{array}{l} A_{i} E_{g} \rho_{g}\left[\left(\dot{v}_{x}+v_{g} v_{x}^{\prime}\right)^{2}+\left(\dot{v}_{y}+v_{g} v_{y}^{\prime}\right)^{2}+\left(\dot{v}_{z}+v_{g}\left(1+v_{z}^{\prime}\right)\right)^{2}\right] \\ +A_{i} E_{l} \rho_{l}\left[\left(\dot{v}_{x}+v_{l} v_{x}^{\prime}\right)^{2}+\left(\dot{v}_{y}+v_{l} v_{y}^{\prime}\right)^{2}+\left(\dot{v}_{z}+v_{l}\left(1+v_{z}^{\prime}\right)\right)^{2}\right] \\ +A_{i} E_{s} \rho_{s}\left[\left(\dot{v}_{x}+v_{s} v_{x}^{\prime}\right)^{2}+\left(\dot{v}_{y}+v_{s} v_{y}^{\prime}\right)^{2}+\left(\dot{v}_{z}+v_{s}\left(1+v_{z}^{\prime}\right)\right)^{2}\right] \end{array}\right\} \mathrm{d} z \end{aligned}
水合物气-液-液-固三相流在提升过程中会存在能量损失，包括内流与管壁之间的摩擦损失，以及粒子碰撞引起的能量损失
W l = W f + W c W l = W f + W c W_(l)=W_(f)+W_(c)W_{l}=W_{f}+W_{c}
哪里 W l W l W_(l)W_{l}是向内流动的能量损失， W f W f W_(f)W_{f}是内部流动与管壁之间的摩擦损失，以及 W c W c W_(c)W_{c}是粒子碰撞造成的能量损失，可以表示如下：
W f = − π D i ∫ 0 L [ α 0 ρ s ( ρ m v m ) − 0.36 v s 2 + 1 2 f ρ l v l 2 ] ( 1 + 2 v z ′ + v z ′ 2 ) d z W f = − π D i ∫ 0 L   α 0 ρ s ρ m v m − 0.36 v s 2 + 1 2 f ρ l v l 2 1 + 2 v z ′ + v z ′ 2 d z W_(f)=-piD_(i)int_(0)^(L)[alpha_(0)rho_(s)(rho_(m)v_(m))^(-0.36)v_(s)^(2)+(1)/(2)frho_(l)v_(l)^(2)](1+2v_(z)^(')+v_(z)^('2))dzW_{f}=-\pi D_{i} \int_{0}^{L}\left[\alpha_{0} \rho_{s}\left(\rho_{m} v_{m}\right)^{-0.36} v_{s}^{2}+\frac{1}{2} f \rho_{l} v_{l}^{2}\right]\left(1+2 v_{z}^{\prime}+v_{z}^{\prime 2}\right) d z
根据文献的理论分析 61 61 ^(61){ }^{61}以及文献的实验分析 62 62 ^(62){ }^{62}为 W c W c W_(c)W_{c}，粒子碰撞引起的能量损失的拟合方程可以表示如下：
W c = − ∫ 0 L β 0 ( 1 − ρ s ρ m ) [ ( v ˙ x + v s v x ′ ) 2 + ( v ˙ y + v s v y ′ ) 2 + ( v ˙ z + v s ( 1 + v z ′ ) ) 2 ] d z W c = − ∫ 0 L   β 0 1 − ρ s ρ m v ˙ x + v s v x ′ 2 + v ˙ y + v s v y ′ 2 + v ˙ z + v s 1 + v z ′ 2 d z {:[W_(c)=-int_(0)^(L)beta_(0)(1-(rho_(s))/(rho_(m)))[(v^(˙)_(x)+v_(s)v_(x)^('))^(2)+(v^(˙)_(y)+v_(s)v_(y)^('))^(2):}],[{:+(v^(˙)_(z)+v_(s)(1+v_(z)^(')))^(2)]dz]:}\begin{aligned} W_{c}= & -\int_{0}^{L} \beta_{0}\left(1-\frac{\rho_{s}}{\rho_{m}}\right)\left[\left(\dot{v}_{x}+v_{s} v_{x}^{\prime}\right)^{2}+\left(\dot{v}_{y}+v_{s} v_{y}^{\prime}\right)^{2}\right. \\ & \left.+\left(\dot{v}_{z}+v_{s}\left(1+v_{z}^{\prime}\right)\right)^{2}\right] \mathrm{d} z \end{aligned}
代入 eqs.（46） 和 （47） 放入方程 （45） 中，得到
W l = − ∫ 0 L { π D i [ α 0 ρ s ( ρ m v m ) − 0.36 v s 2 + 1 2 f ρ l v l 2 ] ( 1 + 2 v z ′ + v z ′ 2 ) + β 0 ( 1 − ρ s ρ m ) [ ( v ˙ x + v s v x ′ ) 2 + ( v ˙ y + v s v y ′ ) 2 + ( v ˙ z + v s ( 1 + v z ′ ) ) 2 ] } d z W l = − ∫ 0 L   π D i α 0 ρ s ρ m v m − 0.36 v s 2 + 1 2 f ρ l v l 2 1 + 2 v z ′ + v z ′ 2 + β 0 1 − ρ s ρ m v ˙ x + v s v x ′ 2 + v ˙ y + v s v y ′ 2 + v ˙ z + v s 1 + v z ′ 2 d z W_(l)=-int_(0)^(L){[piD_(i)[alpha_(0)rho_(s)(rho_(m)v_(m))^(-0.36)v_(s)^(2)+(1)/(2)frho_(l)v_(l)^(2)](1+2v_(z)^(')+v_(z)^('2))],[+beta_(0)(1-(rho_(s))/(rho_(m)))[(v^(˙)_(x)+v_(s)v_(x)^('))^(2)+(v^(˙)_(y)+v_(s)v_(y)^('))^(2):}],[{:+(v^(˙)_(z)+v_(s)(1+v_(z)^(')))^(2)]]}dzW_{l}=-\int_{0}^{L}\left\{\begin{array}{l} \pi D_{i}\left[\alpha_{0} \rho_{s}\left(\rho_{m} v_{m}\right)^{-0.36} v_{s}^{2}+\frac{1}{2} f \rho_{l} v_{l}^{2}\right]\left(1+2 v_{z}^{\prime}+v_{z}^{\prime 2}\right) \\ +\beta_{0}\left(1-\frac{\rho_{s}}{\rho_{m}}\right)\left[\left(\dot{v}_{x}+v_{s} v_{x}^{\prime}\right)^{2}+\left(\dot{v}_{y}+v_{s} v_{y}^{\prime}\right)^{2}\right. \\ \left.+\left(\dot{v}_{z}+v_{s}\left(1+v_{z}^{\prime}\right)\right)^{2}\right] \end{array}\right\} \mathrm{d} z
将外部流体力、随井深变化的内流混合密度、沿井深变化的内流混合速率和内流能量损失引入方程（38），得到深海水合物开采隔水管中气-液-固三相流的振动控制方程，如下所示：
{ { m v + m a + [ A i ( E g ρ g + E l ρ l + E s ρ s ) + β 0 ( 1 − ρ s ρ m ) ] } D ˙ ¨ x − ( ρ v I + ρ a C m I o ) v ′ ′ ¨ x + ( c + 2 c ′ ) v ˙ ˙ x + E I v x ( 4 ) + 2 [ A i E g ρ g v g + A i E l ρ l v l + A i E s ρ s v s + β 0 ( 1 − ρ s ρ m ) v s ] v ˙ ′ x + [ A i E g ρ g v g 2 + A i E l ρ l v l 2 + A i E s ρ s v s 2 + β 0 ( 1 − ρ s ρ m ) v s 2 ] v ′ ′ x − F T v ′ ′ x − E A ( ( v z ′ + 1 2 ( v x ′ 2 + v y ′ 2 ) ) v x ′ ) ′ = 1 2 ρ a U c 2 D o C ¯ d + 1 2 ρ a U c 2 D o C d ′ q x 2 , { { m v + m a + [ A i ( E g ρ g + E l ρ l + E s ρ s ) + β 0 ( 1 − ρ s ρ m ) ] } b ¨ y − ( ρ v I + ρ a C m I o ) v ¨ ′ ′ y + ( c + c ′ ) v ˙ y + E I v y ( 4 ) + 2 [ A i E g ρ g v g + A i E l ρ l v l + A i E s ρ s v s + β 0 ( 1 − ρ s ρ m ) v s ] v ˙ ′ y + [ A i E g ρ g v g 2 + A i E l ρ l v l 2 + A i E s ρ s v s 2 + β 0 ( 1 − ρ s ρ m ) v s 2 ] v ′ ′ y − F T v ′ ′ y − E A ( ( v z ′ + 1 2 ( v x ′ 2 + v y ′ 2 ) ) v y ′ ) ′ = 1 2 ρ a U c 2 D o C l ′ q y 2 , { m v + m a + [ A i ( E g ρ g + E l ρ l + E s ρ s ) + β 0 ( 1 − ρ s ρ m ) ] } ∂ ¨ z + c v ˙ z ˙ − E A ( v z ′ ′ + 1 2 ( v x ′ 2 + v y ′ 2 ) ′ ) + π D i [ α 0 ρ s ( ρ m v m ) − 0.36 v s 2 + 1 2 f ρ l v l 2 ] v ′ ′ z = m v g + ρ m A i g − ρ a π D o 2 4 g − F T , q ¨ x + ε x ω s ( q x 2 − 1 ) q ˙ x + ( 2 ω s ) 2 q x = A x D 0 x ¨ , q ¨ y + ε y ω s ( q y 2 − 1 ) q ˙ y + ( ω s ) 2 q y = A y D 0 y ¨ . m v + m a + A i E g ρ g + E l ρ l + E s ρ s + β 0 1 − ρ s ρ m D ˙ ¨ x − ρ v I + ρ a C m I o v ′ ′ ¨ x + c + 2 c ′ v ˙ ˙ x + E I v x ( 4 ) + 2 A i E g ρ g v g + A i E l ρ l v l + A i E s ρ s v s + β 0 1 − ρ s ρ m v s v ˙ ′ x + A i E g ρ g v g 2 + A i E l ρ l v l 2 + A i E s ρ s v s 2 + β 0 1 − ρ s ρ m v s 2 v ′ ′ x − F T v ′ ′ x − E A v z ′ + 1 2 v x ′ 2 + v y ′ 2 v x ′ ′ = 1 2 ρ a U c 2 D o C ¯ d + 1 2 ρ a U c 2 D o C d ′ q x 2 , m v + m a + A i E g ρ g + E l ρ l + E s ρ s + β 0 1 − ρ s ρ m b ¨ y − ρ v I + ρ a C m I o v ¨ ′ ′ y + c + c ′ v ˙ y + E I v y ( 4 ) + 2 A i E g ρ g v g + A i E l ρ l v l + A i E s ρ s v s + β 0 1 − ρ s ρ m v s v ˙ ′ y + A i E g ρ g v g 2 + A i E l ρ l v l 2 + A i E s ρ s v s 2 + β 0 1 − ρ s ρ m v s 2 v ′ ′ y − F T v ′ ′ y − E A v z ′ + 1 2 v x ′ 2 + v y ′ 2 v y ′ ′ = 1 2 ρ a U c 2 D o C l ′ q y 2 , m v + m a + A i E g ρ g + E l ρ l + E s ρ s + β 0 1 − ρ s ρ m ∂ ¨ z + c v ˙ z ˙ − E A v z ′ ′ + 1 2 v x ′ 2 + v y ′ 2 ′ + π D i α 0 ρ s ρ m v m − 0.36 v s 2 + 1 2 f ρ l v l 2 v ′ ′ z = m v g + ρ m A i g − ρ a π D o 2 4 g − F T , q ¨ x + ε x ω s q x 2 − 1 q ˙ x + 2 ω s 2 q x = A x D 0 x ¨ , q ¨ y + ε y ω s q y 2 − 1 q ˙ y + ω s 2 q y = A y D 0 y ¨ . {:[{{m_(v)+m_(a)+[A_(i)(E_(g)rho_(g)+E_(l)rho_(l)+E_(s)rho_(s))+beta_(0)(1-(rho_(s))/(rho_(m)))]}D^(˙)^(¨)_(x)-(rho_(v)I+rho_(a)C_(m)I_(o))v^('')^(¨)_(x)+(c+2c^('))v^(˙)^(˙)_(x):}],[+EIv_(x)^((4))+2[A_(i)E_(g)rho_(g)v_(g)+A_(i)E_(l)rho_(l)v_(l)+A_(i)E_(s)rho_(s)v_(s)+beta_(0)(1-(rho_(s))/(rho_(m)))v_(s)]v^(˙)^(')_(x)+[A_(i)E_(g)rho_(g)v_(g)^(2)+A_(i)E_(l)rho_(l)v_(l)^(2)+A_(i)E_(s)rho_(s)v_(s)^(2)+beta_(0)(1-(rho_(s))/(rho_(m)))v_(s)^(2)]v^('')_(x)],[-F_(T)v^('')^(x)-EA((v_(z)^(')+(1)/(2)(v_(x)^('2)+v_(y)^('2)))v_(x)^('))^(')=(1)/(2)rho_(a)U_(c)^(2)D_(o) bar(C)_(d)+(1)/(2)rho_(a)U_(c)^(2)D_(o)C_(d)^(')(q_(x))/(2)","],[{[{m_(v)+m_(a)+[A_(i)(E_(g)rho_(g)+E_(l)rho_(l)+E_(s)rho_(s))+beta_(0)(1-(rho_(s))/(rho_(m)))]}b^(¨)_(y)-(rho_(v)I+rho_(a)C_(m)I_(o))v^(¨)^('')_(y)+(c+c^('))v^(˙)_(y)],[quad+EIv_(y)^((4))+2[A_(i)E_(g)rho_(g)v_(g)+A_(i)E_(l)rho_(l)v_(l)+A_(i)E_(s)rho_(s)v_(s)+beta_(0)(1-(rho_(s))/(rho_(m)))v_(s)]v^(˙)^(')_(y)+[A_(i)E_(g)rho_(g)v_(g)^(2)+A_(i)E_(l)rho_(l)v_(l)^(2)+A_(i)E_(s)rho_(s)v_(s)^(2)+beta_(0)(1-(rho_(s))/(rho_(m)))v_(s)^(2)]v^('')_(y)-F_(T)v^('')y]:}],[-EA((v_(z)^(')+(1)/(2)(v_(x)^('2)+v_(y)^('2)))v_(y)^('))^(')=(1)/(2)rho_(a)U_(c)^(2)D_(o)C_(l)^(')(q_(y))/(2)","{m_(v)+m_(a)+[A_(i)(E_(g)rho_(g)+E_(l)rho_(l)+E_(s)rho_(s))+beta_(0)(1-(rho_(s))/(rho_(m)))]}del^(¨)_(z)+cv^(˙)_(z)^(˙)],[-EA(v_(z)^('')+(1)/(2)(v_(x)^('2)+v_(y)^('2))^('))+piD_(i)[alpha_(0)rho_(s)(rho_(m)v_(m))^(-0.36)v_(s)^(2)+(1)/(2)frho_(l)v_(l)^(2)]v^('')_(z)=m_(v)g+rho_(m)A_(i)g-rho_(a)pi(D_(o)^(2))/(4)g-F_(T)","],[q^(¨)_(x)+epsi_(x)omega_(s)(q_(x)^(2)-1)q^(˙)_(x)+(2omega_(s))^(2)q_(x)=(A_(x))/(D_(0))x^(¨)","q^(¨)_(y)+epsi_(y)omega_(s)(q_(y)^(2)-1)q^(˙)_(y)+(omega_(s))^(2)q_(y)=(A_(y))/(D_(0))y^(¨).]:}\begin{aligned} & \left\{\left\{m_{v}+m_{a}+\left[A_{i}\left(E_{g} \rho_{g}+E_{l} \rho_{l}+E_{s} \rho_{s}\right)+\beta_{0}\left(1-\frac{\rho_{s}}{\rho_{m}}\right)\right]\right\} \ddot{\dot{D}}_{x}-\left(\rho_{v} I+\rho_{a} C_{m} I_{o}\right){\ddot{v^{\prime \prime}}}_{x}+\left(c+2 c^{\prime}\right) \dot{\dot{v}}_{x}\right. \\ & +E I v_{x}{ }^{(4)}+2\left[A_{i} E_{g} \rho_{g} v_{g}+A_{i} E_{l} \rho_{l} v_{l}+A_{i} E_{s} \rho_{s} v_{s}+\beta_{0}\left(1-\frac{\rho_{s}}{\rho_{m}}\right) v_{s}\right] \dot{v}^{\prime}{ }_{x}+\left[A_{i} E_{g} \rho_{g} v_{g}^{2}+A_{i} E_{l} \rho_{l} v_{l}^{2}+A_{i} E_{s} \rho_{s} v_{s}^{2}+\beta_{0}\left(1-\frac{\rho_{s}}{\rho_{m}}\right) v_{s}^{2}\right] v^{\prime \prime}{ }_{x} \\ & -F_{T}{v^{\prime \prime}}^{x}-E A\left(\left(v_{z}^{\prime}+\frac{1}{2}\left(v_{x}^{\prime 2}+v_{y}^{\prime 2}\right)\right) v_{x}^{\prime}\right)^{\prime}=\frac{1}{2} \rho_{a} U_{c}^{2} D_{o} \bar{C}_{d}+\frac{1}{2} \rho_{a} U_{c}^{2} D_{o} C_{d}{ }^{\prime} \frac{q_{x}}{2}, \\ & \left\{\begin{array}{l} \left\{m_{v}+m_{a}+\left[A_{i}\left(E_{g} \rho_{g}+E_{l} \rho_{l}+E_{s} \rho_{s}\right)+\beta_{0}\left(1-\frac{\rho_{s}}{\rho_{m}}\right)\right]\right\} \ddot{b}_{y}-\left(\rho_{v} I+\rho_{a} C_{m} I_{o}\right) \ddot{v}^{\prime \prime}{ }_{y}+\left(c+c^{\prime}\right) \dot{v}_{y} \\ \quad+E I v_{y}{ }^{(4)}+2\left[A_{i} E_{g} \rho_{g} v_{g}+A_{i} E_{l} \rho_{l} v_{l}+A_{i} E_{s} \rho_{s} v_{s}+\beta_{0}\left(1-\frac{\rho_{s}}{\rho_{m}}\right) v_{s}\right] \dot{v}^{\prime}{ }_{y}+\left[A_{i} E_{g} \rho_{g} v_{g}^{2}+A_{i} E_{l} \rho_{l} v_{l}^{2}+A_{i} E_{s} \rho_{s} v_{s}^{2}+\beta_{0}\left(1-\frac{\rho_{s}}{\rho_{m}}\right) v_{s}^{2}\right] v^{\prime \prime}{ }_{y}-F_{T} v^{\prime \prime} y \end{array}\right. \\ & -E A\left(\left(v_{z}^{\prime}+\frac{1}{2}\left(v_{x}^{\prime 2}+v_{y}^{\prime 2}\right)\right) v_{y}^{\prime}\right)^{\prime}=\frac{1}{2} \rho_{a} U_{c}^{2} D_{o} C_{l}^{\prime} \frac{q_{y}}{2},\left\{m_{v}+m_{a}+\left[A_{i}\left(E_{g} \rho_{g}+E_{l} \rho_{l}+E_{s} \rho_{s}\right)+\beta_{0}\left(1-\frac{\rho_{s}}{\rho_{m}}\right)\right]\right\} \ddot{\partial}_{z}+\dot{c \dot{v}_{z}} \\ & -E A\left(v_{z}^{\prime \prime}+\frac{1}{2}\left(v_{x}^{\prime 2}+v_{y}^{\prime 2}\right)^{\prime}\right)+\pi D_{i}\left[\alpha_{0} \rho_{s}\left(\rho_{m} v_{m}\right)^{-0.36} v_{s}^{2}+\frac{1}{2} f \rho_{l} v_{l}^{2}\right] v^{\prime \prime}{ }_{z}=m_{v} g+\rho_{m} A_{i} g-\rho_{a} \pi \frac{D_{o}^{2}}{4} g-F_{T}, \\ & \ddot{q}_{x}+\varepsilon_{x} \omega_{s}\left(q_{x}^{2}-1\right) \dot{q}_{x}+\left(2 \omega_{s}\right)^{2} q_{x}=\frac{A_{x}}{D_{0}} \ddot{x}, \ddot{q}_{y}+\varepsilon_{y} \omega_{s}\left(q_{y}^{2}-1\right) \dot{q}_{y}+\left(\omega_{s}\right)^{2} q_{y}=\frac{A_{y}}{D_{0}} \ddot{y} . \end{aligned}