基于椭圆积分的单支点半柔壁喷管大挠度成型半解析求解与验证

为提升跨超声速风洞单支点半柔壁（SJSF）喷管分析精度和效率，发展了基于椭圆积分的柔性壁板大挠度变形半解析求解方法。以0.3 m低温风洞单支点半柔壁喷管为研究对象，建立型面组件力学模型，基于欧拉梁理论推导柔性壁板型面曲线的微分方程，给出了椭圆积分的表达形式。利用Maple软件编制了计算程序，并以此针对马赫数为1.15和1.3两种工况，开展柔性壁板大挠度变形的半解析求解，得到了变形型面坐标和结构等效应力曲线。通过对比有限元（FEM）仿真结果以及实验测量数据，结果表明:基于椭圆积分的求解结果与其他两种结果均能良好吻合，型面坐标最大偏差分别为喷管出口高度尺寸的0.18‰和0.32‰，结构等效应力最大相对误差小于10%，椭圆积分求解耗时仅为有限元仿真的11.2%。 

Semi-analytic solution and verification for large deflection forming of single-jack semi-flexible nozzle based on elliptic integral

To improve the analysis accuracy and efficiency of single-jack semi-flexible（SJSF） nozzles in transonic and supersonic wind tunnels，a semi-analytical solution method based on ellipse integral for the flexible plate large deflection was developed.Taking the 0.3 m cryogenic wind tunnel SJSF nozzle as the research object，the mechanical model was established.Based on the Euler beam theory，the differential formulas of the flexible plate deflection were derived，and the elliptic integral form was obtained.Based on the program compiled using Maple software，the semi-analytic solution was carried out under Mach number of 1.15 and 1.3 conditions.The deformation contour and structural equivalent stress were obtained.Compared with the finite element method （FEM） results and experimental measurement，it showed that the elliptic integral results were in good agreement with the other two results.The maximum error of the contour coordinates was 0.18‰ and 0.32‰ of the nozzle outlet height size，respectively.The maximum relative error of the equivalent stress was less than 10%，and the elliptic integral solution time was only 11.2% of the FEM.