Analysis of fluid-structure interaction vibration based on the detection principle of SI-FLAT flatness measurement systems
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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Föppl-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.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 Föppl-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'amplitude-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.
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