工程科学学报，第39卷，第4期：593603,2017年4月 Chinese Journal of Engineering,Vol.39,No.4:593-603,April 2017 DOI:10.13374/j.issn2095-9389.2017.04.015:http://journals.ustb.edu.cn SI一FLAT板形仪检测原理的流固耦合振动分析 李 秾”，孔宁)四，李洪波”，张杰”，贾生晖”，褚玉刚，刘海军》 1)北京科技大学机械工程学院，北京1000832)武汉钢铁（集团）公司，武汉430083 ☒通信作者，E-mail:kongning(@usth.cdu.cm 摘要为从力学本质上揭示S一LAT非接触式板形仪的检测原理，基于薄板流固耦合振动理论，建立了薄板振幅与残余 应力关系的数学模型.在非协调Foppl-von Karman方程组的平衡方程中引入惯性项与流体压强项，利用气动载荷在时间上 的周期性将流体速度函数、流体压强函数、薄板挠度函数和薄板应力势函数的时间变量分离出来，得到描述S一FLAT板形仪 稳定工作状态的偏微分方程组。进一步利用分离变量法求解该方程组，最终建立起薄板振幅与残余应力的数学关系。同时 结合实测残余应力数据，利用Siemens提出的振幅一残余应力模型反算得到实际薄板振幅分布，并将其与流固耦合振动模型 计算的振幅进行对比，验证了提出的数学模型的可靠性.进一步利用流固耦合振动模型分析了气泵进风口流体速度、检测距 离和激振频率对振幅的影响，为S-FLAT板形仪科学合理的利用提供了理论依据. 关键词板形仪：残余应力：振动：薄板：流固耦合 分类号TG335.56 Analysis of fluid-structure interaction vibration based on the detection principle of SI-FLAT flatness measurement systems LI Nong'》,KONG Ning》a,LI Hong-bo',ZHANG Jie”,JIA Sheng-hui,CHU Yu-gang2,LN Hai-jun》 1)School of Mechanical Engineering,University of Science and Technology Beijing,Beijing 100083,China 2)Wuhan Iron and Steel (Group)Corporation,Wuhan 430083,China Corresponding author,E-mail:kongning@ustb.edu.cn ABSTRACT In order to reveal the mechanical essence of the detecting principle of SI-FLAT flatness measurement systems,the mathematical model of the relationship between amplitude and residual stress was established,based on the theory of fluid-structure interaction vibration of thin plates.The terms of inertia and fluid pressure were introduced to the equilibrium equation in incompatible Foppl-von Karman equations.The time variables were separated out from the velocity function of fluid,pressure function of fluid, deflection function of thin plates and stress potential function of thin plates with consideration of periodic aerodynamic load.Therefore, the partial differential equations aiming at steady state of SI-FLAT flatness measurement systems was obtained.Solving the equations by using the method of separation of variables,the mathematical relationship between amplitude and residual stress was established. Combined with measured residual stress,the distribution of actual amplitude of thin plates could be calculated by the Siemens'ampli- tude-residual stress model,which coincided with the amplitude calculated by the fluid-structure interaction vibration model.The influences of fluid velocity at air pump's inlet,detecting distance and excitation frequency on amplitude were analyzed by using the fluid-structure interaction vibration model,which provides a theoretical basis for application of SI-FLAT flatness measurement systems. KEY WORDS flatness measurement systems:residual stress:vibration:thin plates:fluid-structure interaction 收稿日期：2016-09-14 基金项目：中央高校基本科研业务费专项资金资助项目(FRF-TP-15O16A3:“十二五”国家科技支撑计划资助项目(2015BAF30B01)工程科学学报，第 39 卷，第 4 期: 593--603，2017 年 4 月 Chinese Journal of Engineering，Vol． 39，No． 4: 593--603，April 2017 DOI: 10． 13374 /j． issn2095--9389． 2017． 04． 015; http: / /journals． ustb． edu． cn SI--FLAT 板形仪检测原理的流固耦合振动分析 李 秾1) ，孔 宁1) ，李洪波1) ，张 杰1) ，贾生晖2) ，褚玉刚2) ，刘海军2) 1) 北京科技大学机械工程学院，北京 100083 2) 武汉钢铁( 集团) 公司，武汉 430083 通信作者，E-mail: kongning@ ustb． edu． cn 摘 要 为从力学本质上揭示 SI--FLAT 非接触式板形仪的检测原理，基于薄板流固耦合振动理论，建立了薄板振幅与残余 应力关系的数学模型． 在非协调 Fppl--von Kármán 方程组的平衡方程中引入惯性项与流体压强项，利用气动载荷在时间上 的周期性将流体速度函数、流体压强函数、薄板挠度函数和薄板应力势函数的时间变量分离出来，得到描述 SI--FLAT 板形仪 稳定工作状态的偏微分方程组． 进一步利用分离变量法求解该方程组，最终建立起薄板振幅与残余应力的数学关系． 同时 结合实测残余应力数据，利用 Siemens 提出的振幅--残余应力模型反算得到实际薄板振幅分布，并将其与流固耦合振动模型 计算的振幅进行对比，验证了提出的数学模型的可靠性． 进一步利用流固耦合振动模型分析了气泵进风口流体速度、检测距 离和激振频率对振幅的影响，为 SI--FLAT 板形仪科学合理的利用提供了理论依据． 关键词 板形仪; 残余应力; 振动; 薄板; 流固耦合 分类号 TG335. 5 + 6 收稿日期: 2016--09--14 基金项目: 中央高校基本科研业务费专项资金资助项目 ( FＲF--TP--15--016A3) ; “十二五”国家科技支撑计划资助项目( 2015BAF30B01) Analysis of fluid--structure interaction vibration based on the detection principle of SI--FLAT flatness measurement systems LI Nong1) ，KONG Ning1)  ，LI Hong-bo1) ，ZHANG Jie1) ，JIA Sheng-hui2) ，CHU Yu-gang2) ，LIU Hai-jun2) 1) School of Mechanical Engineering，University of Science and Technology Beijing，Beijing 100083，China 2) Wuhan Iron and Steel ( Group) Corporation，Wuhan 430083，China Corresponding author，E-mail: kongning@ ustb． edu． cn ABSTＲACT In order to reveal the mechanical essence of the detecting principle of SI--FLAT flatness measurement systems，the mathematical model of the relationship between amplitude and residual stress was established，based on the theory of fluid--structure interaction vibration of thin plates． The terms of inertia and fluid pressure were introduced to the equilibrium equation in incompatible Fppl--von Kármán equations． The time variables were separated out from the velocity function of fluid，pressure function of fluid， deflection function of thin plates and stress potential function of thin plates with consideration of periodic aerodynamic load． Therefore， the partial differential equations aiming at steady state of SI--FLAT flatness measurement systems was obtained． Solving the equations by using the method of separation of variables，the mathematical relationship between amplitude and residual stress was established． Combined with measured residual stress，the distribution of actual amplitude of thin plates could be calculated by the Siemens’ampli￾tude--residual stress model，which coincided with the amplitude calculated by the fluid--structure interaction vibration model． The influences of fluid velocity at air pump’s inlet，detecting distance and excitation frequency on amplitude were analyzed by using the fluid--structure interaction vibration model，which provides a theoretical basis for application of SI--FLAT flatness measurement systems． KEY WOＲDS flatness measurement systems; residual stress; vibration; thin plates; fluid--structure interaction