CN112113695A - 基于五栅式应变花消除残余应力测试钻孔偏心误差的方法 - Google Patents
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Abstract
本发明公开了一种基于五栅式应变花消除残余应力测试钻孔偏心误差的方法，包括S1、根据钻孔应变法，设计五栅式残余应力测试应变花；S2、基于线弹性力学叠加原理，推导得到五栅式应变花测试的应变求解应力状态与钻孔偏心量的非线性方程组；S3、基于牛顿迭代法求解非线性方程组，计算得到应力状态和钻孔偏心量。
Description
技术领域
本发明属于钻孔应变法测试残余应力的技术领域，具体涉及一种基于五栅式应变花消除残余应力测试钻孔偏心误差的方法。
背景技术
钻孔应变法是常用的残余应力测试方法，属于半破坏性测试方法，它具有对工件的损伤小，操作简单，测试费用低等优点。钻孔应变法在工件表面钻一小孔(直径0.8mm～4.8mm)，通过应变计测量钻孔所释放的应变来推算残余应力。简单地讲，钻孔应变法先在工件表面粘贴应变花，然后在应变花中心处钻半径为a、深h的小孔，先前粘贴的应变计会测得钻孔所释放的应变，最后根据所测的应变推算残余应力。
钻孔偏心引起的误差是钻孔应变法残余应力测试中一种常见的误差。钻孔偏心对残余应力测试结果影响很大，且很难避免。研究表明当钻孔偏心为0.1倍孔径时，测试误差在特殊应力状态下可高达70％。然而影响钻孔偏心的因素较多，其大小与测试人员技能、测试仪器及操作环境有关。一般来讲，在水平板面上钻孔，对中精度容易保证，但待测表面若是弧面或斜面就很容易出现偏心，如焊缝表面、圆管表面、斜向角焊缝表面。此外操作空间对对中精度也有一定的影响，狭小的、受限的操作空间往往更容易出现偏心。例如，在对正交异性钢桥面板焊接残余应力的测量中，共测试253个测点，由偏心过大而判定为无效的测点就达到75个，占比近30％。
由于钻孔应变法是一种有损测量方法，钻孔在一定范围内会改变原有的应力状态。若待测点钻孔出现偏心，根据规范规定，补测测点也需要与原测点间隔15孔径以上。这对于应力梯度较大的应力状态，由测点位置的偏差将导致不可忽略的误差，因此需要对钻孔偏心的大小进行尽可能的限制，若出现偏心也需要对结果进行校正。对此，我国船舶行业标准CB/T 3395-2013[i]在附录中给出了依据偏心大小与方向进行修正的方法，然而该方法实际应用比较困难，因为在高倍放大镜下，钻孔边缘存在很多毛刺和凹凸，难以清楚地确定钻孔的边界，也就难以测量偏心的大小与方向。
综上所述，钻孔应变法由于其经济性、便捷性已成为当前广泛应用的残余应力测试方法，测试时钻孔偏心导致的误差不可忽略且难以避免，而现有的根据偏心位置修正的方法因钻孔边界的难以界定限制了其广泛的应用。
发明内容
本发明的目的在于针对现有技术中的上述不足，提供一种基于五栅式应变花消除残余应力测试钻孔偏心误差的方法，以解决现有测试时钻孔偏心导致误差的问题。
为达到上述目的，本发明采取的技术方案是：
一种基于五栅式应变花消除残余应力测试钻孔偏心误差的方法，其包括：
S1、根据钻孔应变法，设计五栅式残余应力测试应变花；
S2、基于线弹性力学叠加原理，推导得到五栅式应变花测试的应变求解应力状态与钻孔偏心量的非线性方程组；
S3、基于牛顿迭代法求解非线性方程组，计算得到应力状态和钻孔偏心量。
本发明提供的基于五栅式应变花消除残余应力测试钻孔偏心误差的方法，具有以下有益效果：
本发明为了剔除钻孔应变释放法中钻孔偏心带来的测试误差，提供了涉及产品设计、理论创新、程序实现的一整套方法体系与实现途径。
1、从产品设计上，本发明提供了一种五栅式残余应力测试应变花。根据该应变花测试的应变，结合本发明的求解方法，可以剔除钻孔应变释放法中钻孔偏心带来的测试误差。
2、从理论创新上，本发明根据五栅式残余应力测试应变的设计布局，基于弹性力学，将钻孔偏心视为待解的未知量，推导出了求解一点应力状态的非线性方程组，即式(18)。
3、从程序实现上，本发明根据非线性方程组，式(18)，基于牛顿迭代法构建了求解钻孔偏心量(x,y)及一点的应力状态(σ1,σ2,β)，并提供了关键性参数的表达式(见附录A，附录B)。
4、由于本发明可自动剔除钻孔偏心带来的测试误差，因此基于本发明开展残余应力测试对钻孔对中精度的要求更低，由国标要求的±0.004D降低到±0.1D，D＝2R。针对三栅式应变花，国标GBT31310-2014的规定钻孔对中精度需要控制在±0.004D以内，而本发明钻孔对中精度仅要求钻孔不影响应变栅正常工作即可，一般控制在±0.1D就可满足要求。
5、为了能较好地保证国标规定的钻孔对中精度，目前常采用带有定位套筒的气动涡轮钻机钻孔，该装置的价格在10万～35万不等。由于本发明对钻孔对中精度要求更低，采用常见的手持电钻即可满足精度要求。所以，基于本发明进行残余应力测试，在保证测试精度的前提下，可极大地降低测试成本。
6、由于本发明对钻孔对中精度要求更低，相应地对测试姿态、操作空间限制更少。当前若采用倾斜姿态钻孔更容易出现偏心；相对于平面，在斜面或弧面上钻孔对中更加困难，对操作空间的要求更高。基于本发明进行残余应力测试适应测试姿态更多，对操作空间的限制更少。
7、基于本发明进行残余应力测试，由于本发明已经考虑钻孔偏心的影响，所以即便钻孔时出现偏心，也可以求解正常的残余应力状态，从而降低了无效测点出现的概率，减少补测测点，减少对待测物的损伤。
附图说明
图1为三栅式残余应力测试应变花布局。
图2为商用残余应力测试用应变花。
图3为五栅式残余应力测试应变花。
图4为带圆孔薄板受均布荷图。
图5为五栅式应变花残余应力求解示意图。
图6为应力分量的坐标变化示意。
具体实施方式
下面对本发明的具体实施方式进行描述，以便于本技术领域的技术人员理解本发明，但应该清楚，本发明不限于具体实施方式的范围，对本技术领域的普通技术人员来讲，只要各种变化在所附的权利要求限定和确定的本发明的精神和范围内，这些变化是显而易见的，一切利用本发明构思的发明创造均在保护之列。
根据本申请的一个实施例，本方案的基于五栅式应变花消除残余应力测试钻孔偏心误差的方法，以下将对本方法进行详细描述。
首先，由弹性力学可知，表征一点的应力状态需要9个应力分量，但由于垂直于待测表面方向上(z向)的应力分量(σxz、σyz、σzx、σzy、σzz)均为零，所以实际上这9个分量仅有3个(σxx、σxy、σyy)独立的变量是未知的。若将待测点的应力状态表示为主应力的形式，只需要2个主应力参数σ1、σ2和1个角度参数θ就可以表征待测点的应力状态。为了求解这3个未知量需要建立3个方程，因此钻孔应变法常采用如图1所示的三栅式应变花来测试残余应力。这类应变花具有3个敏感栅，按顺时针编号为1#、2#、3#，分别位于0°、45°、90°角上。实际应用中为尽可能降低钻孔偏心引起的测试误差，2#敏感栅常设计在225°角上，当前商用的残余应力测试用应变花均是按该设计制造的三栅式应变花，如图2所示。
本方法具体包括：
步骤S1、根据钻孔应变法，设计五栅式残余应力测试应变花，其具体包括：
参考图3，若考虑偏心的影响，对于待测应力状态，实际上有σ1、σ2、θ及偏心坐标x、y这5个未知量。因此，至少需要建立5个方程才能唯一确定这5个未知量。对此，设计出一种五栅式应变花。该应变花含有5个敏感栅，按顺时针编号为1#、2#、3#、4#及5#，分别位于0°、45°、90°、180°、270°。图3所示P1～P5是敏感栅的中心，O为钻孔圆心，a为孔径，R代表敏感栅中心P1～P5至O的距离。
步骤S2、基于线弹性力学叠加原理，推导得到五栅式应变花测试的应变求解应力状态与钻孔偏心量的非线性方程组，其具体步骤包括：
步骤S2.1、设有均值线弹性材料的薄板，在均布荷载σx作用下，任意一点P(R,θ)应力状态在极坐标系下可表示为：
步骤S2.2、现以O为圆心钻一半径为a的圆孔，并假设开孔前后材料均处于线弹性范围，则钻孔后P点的应力状态可表示为：
步骤S2.3、在P点，由钻孔而导致的应力变化可表示为：
步骤S2.4、设由钻孔而释放的径向应变、环向应变及剪应变分别为
根据胡克定律，开孔引起的应力变化可表示为：
其中，E为弹性模量；μ为泊松系数；G为剪切弹性模量，G＝E/2(1+μ)。
步骤S2.5、将式(4)代入式(3)整理得：
步骤S2.6、现令
步骤S2.7、式(6)给出了单向荷载作用下钻孔释放应变的表达式，在双向荷载作用下的释放应变可根据叠加原理求出；显然，在沿Y轴荷载的单向荷载σy作用下式(6)仍然成立，此时只需将式中的2θ改为2(θ+90°)就可得到P(R,θ)点在σy作用下的释放应变：
步骤S2.8、将式(6)与式(7)相加，得到在双向荷载σx与σy作用下钻孔所释放的应变。
步骤S3、基于牛顿迭代法求解非线性方程组，计算得到应力状态和钻孔偏心量，其具体步骤包括：
步骤S3.1、基于设计的五栅式应变花构建坐标系OXY，参考图5，其中，O为未偏心时的孔心，Y轴沿1#敏感栅布置，X轴与3#敏感栅重叠；O离各敏感栅中心(P1～P5)的距离均为R；σy、σx是主应力；θ是应力方向角，定义为从应力轴到1#敏感栅纵轴的角度，以顺时针旋转为正，逆时针为负；O'为钻孔出现偏心时的孔心，设O'的坐标为(x,y)。
定义O'离i号敏感栅中心Pi的距离为Ri，则Ri可表示为：
步骤S3.2、定义向量
与向量
的夹角为βi。规定βi以
以顺时针转向
为正，反之为负，则βi可表示为：
其中，sgn(n)表示符号函数，根据n的符号取值：
图5所示θi表示从σy应力轴到
向量的转角，可表示为θ与βi的函数：
步骤S3.3、将式(12)代入式(8)可得到钻孔偏心情况下，各应变片中心Pi的应变状态：
其中，
值得指出的是式(13)中
是在钻孔发生偏心时，沿偏心O'指向各敏感栅中心的应变，即
为沿
方向的径向应变。但实际上应变片所测得的应变是沿
方向，即图5中所示的
应变。因此需要将
表示为
及
的形式，涉及应力分量的坐标转换。
步骤S3.4、所示为应力分量的坐标变换示意图。如图6所示，现有包含x面、y面和径向坐标面的微元三角板I、II，并假设微元厚度为1，则由微元体沿x和y两个方向的静力平衡ΣFx＝0，ΣFy＝0可得应力分量的变换式：
根据胡克定律，在直角坐标系与极坐标下的应变分量与应力分量有关系式：
将式(15)代入式(14)，联合求解得到以εr、εθ及γrθ表示的εx形式：
步骤S3.5、将图6中的x轴与图5中的
向量方向对应，改写式(16)得到与敏感栅读数对应的应变：
步骤S3.6、式(17)就是考虑钻孔偏心影响的、与应变计读数对应的应变
现假设i号敏感栅实测的应变为εi，那么
与εi两者的差值就可以表示为σx、σy、θ、x及y的函数；定义
代入式(17)，得：
步骤S3.7、式(18)为含有5个未知数的非线性方程组，该方程组无显式的解析解；对此可以先按链式法则展开，然后根据牛顿迭代法求解未知量。
为求解式(18)所示方程组，定义F(X)＝[f1 f2 f3 f4 f5]T，X＝[σx σy θ x y]T。F(X)各元素fi的具体表达式见附录A，F(X)对各未知数的偏导数可表示为式(19)的形式，式(19)中fij具体表达式参见附录B。
原非线性方程组F(X)＝0可以根据牛顿迭代法，按式(20)通过迭代求得近似解：
X(k+1)＝X(k)-[F′(X(k))]-1F(X(k)) (20)
其中，X(k)为方程组的k组近似值，
当k＝0时，X(0)表示迭代初始值，若假设
则可按式(21)取值作为迭代初始值。
步骤S3.8、获得迭代公式及迭代初始值后，再给定精度水平ε和最大迭代次数N，对于k＝0,1,2,…,N，按式(20)求解钻孔偏心下残余应力的步骤为：
步骤S3.8.1、将X(k)代入式(18)、式(19)分别求得F(X(k))、F′(X(k))；
步骤S3.8.2、将X(k)、F(X(k))及F′(X(k))代入式(20)求解X(k+1)；
步骤S3.8.3、判断
是否小于精度水平ε，若是，X(k)≈X*(X*为真实解)，迭代终止；否则k＝k+1，转入步骤S3.8.1步进行下一次迭代，直至k＝N或计算结果满足精度要求。附录A
f1＝
cos(acos((R-y)/(x^2+(R-y)^2)^(1/2))*sgn(x))^2*((cos(2*t-2*acos((R-y)/(x^2+(R-y)^2)^(1/2))*sgn(x))*(μ/2+1/2)*(S1-S2)*((3*a^4)/(x^2+(R-y)^2)^2-(4*a^2)/((x^2+(R-y)^2)*(μ+1))))/E-(a^2*(S1+S2)*(μ/2+1/2))/(E*(x^2+(R-y)^2)))-Em1+sin(acos((R-y)/(x^2+(R-y)^2)^(1/2))*sgn(x))^2*((a^2*(S1+S2)*(μ/2+1/2))/(E*(x^2+(R-y)^2))-(cos(2*t-2*acos((R-y)/(x^2+(R-y)^2)^(1/2))*sgn(x))*(μ/2+1/2)*((3*a^4)/(x^2+(R-y)^2)^2-(4*a^2*μ)/((x^2+(R-y)^2)*(μ+1)))*(S1-S2))/E)-(sin(2*acos((R-y)/(x^2+(R-y)^2)^(1/2))*sgn(x))*sin(2*t-2*acos((R-y)/(x^2+(R-y)^2)^(1/2))*sgn(x))*(S1-S2)*((2*a^2)/(x^2+(R-y)^2)-(3*a^4)/(x^2+(R-y)^2)^2)*(μ+1))/(2*E)
f2＝
-Em2-sin(sgn(x-y)*acos((R-(2^(1/2)*(x+y))/2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2)))^2*((cos(2*t+pi/2-2*sgn(x-y)*acos((R-(2^(1/2)*(x+y))/2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2)))*(μ/2+1/2)*(S1-S2)*((3*a^4)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^2-(4*a^2*μ)/((μ+1)*((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2))))/E-(a^2*(S1+S2)*(μ/2+1/2))/(E*((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)))-cos(sgn(x-y)*acos((R-(2^(1/2)*(x+y))/2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2)))^2*((a^2*(S1+S2)*(μ/2+1/2))/(E*((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2))-(cos(2*t+pi/2-2*sgn(x-y)*acos((R-(2^(1/2)*(x+y))/2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2)))*(μ/2+1/2)*(S1-S2)*((3*a^4)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^2-(4*a^2)/((μ+1)*((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2))))/E)-(sin(2*t+pi/2-2*sgn(x-y)*acos((R-(2^(1/2)*(x+y))/2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2)))*sin(2*sgn(x-y)*acos((R-(2^(1/2)*(x+y))/2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2)))*((2*a^2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)-(3*a^4)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^2)*(S1-S2)*(μ+1))/(2*E)
f3＝
sin(acos((R-x)/(y^2+(R-x)^2)^(1/2))*sgn(y))^2*((a^2*(S1+S2)*(μ/2+1/2))/(E*(y^2+(R-x)^2))+(cos(2*t+2*acos((R-x)/(y^2+(R-x)^2)^(1/2))*sgn(y))*(μ/2+1/2)*((3*a^4)/(y^2+(R-x)^2)^2-(4*a^2*μ)/((y^2+(R-x)^2)*(μ+1)))*(S1-S2))/E)-cos(acos((R-x)/(y^2+(R-x)^2)^(1/2))*sgn(y))^2*((cos(2*t+2*acos((R-x)/(y^2+(R-x)^2)^(1/2))*sgn(y))*(μ/2+1/2)*(S1-S2)*((3*a^4)/(y^2+(R-x)^2)^2-(4*a^2)/((y^2+(R-x)^2)*(μ+1))))/E+(a^2*(S1+S2)*(μ/2+1/2))/(E*(y^2+(R-x)^2)))-Em3-(sin(2*acos((R-x)/(y^2+(R-x)^2)^(1/2))*sgn(y))*sin(2*t+2*acos((R-x)/(y^2+(R-x)^2)^(1/2))*sgn(y))*(S1-S2)*((2*a^2)/(y^2+(R-x)^2)-(3*a^4)/(y^2+(R-x)^2)^2)*(μ+1))/(2*E)
f4＝
(sin(2*acos((R+y)/((R+y)^2+x^2)^(1/2))*sgn(x))*sin(2*t+2*acos((R+y)/((R+y)^2+x^2)^(1/2))*sgn(x))*(S1-S2)*(μ+1)*((2*a^2)/((R+y)^2+x^2)-(3*a^4)/((R+y)^2+x^2)^2))/(2*E)-sin(acos((R+y)/((R+y)^2+x^2)^(1/2))*sgn(x))^2*((cos(2*t+2*acos((R+y)/((R+y)^2+x^2)^(1/2))*sgn(x))*(μ/2+1/2)*(S1-S2)*((3*a^4)/((R+y)^2+x^2)^2-(4*a^2*μ)/(((R+y)^2+x^2)*(μ+1))))/E-(a^2*(S1+S2)*(μ/2+1/2))/(E*((R+y)^2+x^2)))-cos(acos((R+y)/((R+y)^2+x^2)^(1/2))*sgn(x))^2*((a^2*(S1+S2)*(μ/2+1/2))/(E*((R+y)^2+x^2))-(cos(2*t+2*acos((R+y)/((R+y)^2+x^2)^(1/2))*sgn(x))*((3*a^4)/((R+y)^2+x^2)^2-(4*a^2)/(((R+y)^2+x^2)*(μ+1)))*(μ/2+1/2)*(S1-S2))/E)-Em4
f5＝
sin(acos((R+x)/((R+x)^2+y^2)^(1/2))*sgn(y))^2*((cos(2*t-2*acos((R+x)/((R+x)^2+y^2)^(1/2))*sgn(y))*(μ/2+1/2)*(S1-S2)*((3*a^4)/((R+x)^2+y^2)^2-(4*a^2*μ)/(((R+x)^2+y^2)*(μ+1))))/E+(a^2*(S1+S2)*(μ/2+1/2))/(E*((R+x)^2+y^2)))-Em5-cos(acos((R+x)/((R+x)^2+y^2)^(1/2))*sgn(y))^2*((a^2*(S1+S2)*(μ/2+1/2))/(E*((R+x)^2+y^2))+(cos(2*t-2*acos((R+x)/((R+x)^2+y^2)^(1/2))*sgn(y))*((3*a^4)/((R+x)^2+y^2)^2-(4*a^2)/(((R+x)^2+y^2)*(μ+1)))*(μ/2+1/2)*(S1-S2))/E)+(sin(2*acos((R+x)/((R+x)^2+y^2)^(1/2))*sgn(y))*sin(2*t-2*acos((R+x)/((R+x)^2+y^2)^(1/2))*sgn(y))*(S1-S2)*(μ+1)*((2*a^2)/((R+x)^2+y^2)-(3*a^4)/((R+x)^2+y^2)^2))/(2*E)
附录B
f11＝
-sin(acos((R-y)/(x^2+(R-y)^2)^(1/2))*sgn(x))^2*((cos(2*t-2*acos((R-y)/(x^2+(R-y)^2)^(1/2))*sgn(x))*(μ/2+1/2)*((3*a^4)/(x^2+(R-y)^2)^2-(4*a^2*μ)/((x^2+(R-y)^2)*(μ+1))))/E-(a^2*(μ/2+1/2))/(E*(x^2+(R-y)^2)))-cos(acos((R-y)/(x^2+(R-y)^2)^(1/2))*sgn(x))^2*((a^2*(μ/2+1/2))/(E*(x^2+(R-y)^2))-(cos(2*t-2*acos((R-y)/(x^2+(R-y)^2)^(1/2))*sgn(x))*(μ/2+1/2)*((3*a^4)/(x^2+(R-y)^2)^2-(4*a^2)/((x^2+(R-y)^2)*(μ+1))))/E)-(sin(2*acos((R-y)/(x^2+(R-y)^2)^(1/2))*sgn(x))*sin(2*t-2*acos((R-y)/(x^2+(R-y)^2)^(1/2))*sgn(x))*((2*a^2)/(x^2+(R-y)^2)-(3*a^4)/(x^2+(R-y)^2)^2)*(μ+1))/(2*E)
f12＝
sin(acos((R-y)/(x^2+(R-y)^2)^(1/2))*sgn(x))^2*((cos(2*t-2*acos((R-y)/(x^2+(R-y)^2)^(1/2))*sgn(x))*(μ/2+1/2)*((3*a^4)/(x^2+(R-y)^2)^2-(4*a^2*μ)/((x^2+(R-y)^2)*(μ+1))))/E+(a^2*(μ/2+1/2))/(E*(x^2+(R-y)^2)))-cos(acos((R-y)/(x^2+(R-y)^2)^(1/2))*sgn(x))^2*((a^2*(μ/2+1/2))/(E*(x^2+(R-y)^2))+(cos(2*t-2*acos((R-y)/(x^2+(R-y)^2)^(1/2))*sgn(x))*(μ/2+1/2)*((3*a^4)/(x^2+(R-y)^2)^2-(4*a^2)/((x^2+(R-y)^2)*(μ+1))))/E)+(sin(2*acos((R-y)/(x^2+(R-y)^2)^(1/2))*sgn(x))*sin(2*t-2*acos((R-y)/(x^2+(R-y)^2)^(1/2))*sgn(x))*((2*a^2)/(x^2+(R-y)^2)-(3*a^4)/(x^2+(R-y)^2)^2)*(μ+1))/(2*E)
f13＝
(2*sin(acos((R-y)/(x^2+(R-y)^2)^(1/2))*sgn(x))^2*sin(2*t-2*acos((R-y)/(x^2+(R-y)^2)^(1/2))*sgn(x))*(μ/2+1/2)*((3*a^4)/(x^2+(R-y)^2)^2-(4*a^2*μ)/((x^2+(R-y)^2)*(μ+1)))*(S1-S2))/E-(sin(2*acos((R-y)/(x^2+(R-y)^2)^(1/2))*sgn(x))*cos(2*t-2*acos((R-y)/(x^2+(R-y)^2)^(1/2))*sgn(x))*(S1-S2)*((2*a^2)/(x^2+(R-y)^2)-(3*a^4)/(x^2+(R-y)^2)^2)*(μ+1))/E-(2*cos(acos((R-y)/(x^2+(R-y)^2)^(1/2))*sgn(x))^2*sin(2*t-2*acos((R-y)/(x^2+(R-y)^2)^(1/2))*sgn(x))*(μ/2+1/2)*(S1-S2)*((3*a^4)/(x^2+(R-y)^2)^2-(4*a^2)/((x^2+(R-y)^2)*(μ+1))))/E
f14＝
cos(acos((R-y)/(x^2+(R-y)^2)^(1/2))*sgn(x))^2*((sin(2*t-2*acos((R-y)/(x^2+(R-y)^2)^(1/2))*sgn(x))*(μ/2+1/2)*(S1-S2)*(4*acos((R-y)/(x^2+(R-y)^2)^(1/2))*dirac(x)+(2*x*sgn(x)*(R-y))/((x^2+(R-y)^2)^(3/2)*(1-(R-y)^2/(x^2+(R-y)^2))^(1/2)))*((3*a^4)/(x^2+(R-y)^2)^2-(4*a^2)/((x^2+(R-y)^2)*(μ+1))))/E-(cos(2*t-2*acos((R-y)/(x^2+(R-y)^2)^(1/2))*sgn(x))*(μ/2+1/2)*(S1-S2)*((12*a^4*x)/(x^2+(R-y)^2)^3-(8*a^2*x)/((x^2+(R-y)^2)^2*(μ+1))))/E+(2*a^2*x*(S1+S2)*(μ/2+1/2))/(E*(x^2+(R-y)^2)^2))-sin(acos((R-y)/(x^2+(R-y)^2)^(1/2))*sgn(x))^2*((sin(2*t-2*acos((R-y)/(x^2+(R-y)^2)^(1/2))*sgn(x))*(μ/2+1/2)*((3*a^4)/(x^2+(R-y)^2)^2-(4*a^2*μ)/((x^2+(R-y)^2)*(μ+1)))*(S1-S2)*(4*acos((R-y)/(x^2+(R-y)^2)^(1/2))*dirac(x)+(2*x*sgn(x)*(R-y))/((x^2+(R-y)^2)^(3/2)*(1-(R-y)^2/(x^2+(R-y)^2))^(1/2))))/E-(cos(2*t-2*acos((R-y)/(x^2+(R-y)^2)^(1/2))*sgn(x))*(μ/2+1/2)*((12*a^4*x)/(x^2+(R-y)^2)^3-(8*a^2*μ*x)/((x^2+(R-y)^2)^2*(μ+1)))*(S1-S2))/E+(2*a^2*x*(S1+S2)*(μ/2+1/2))/(E*(x^2+(R-y)^2)^2))-2*cos(acos((R-y)/(x^2+(R-y)^2)^(1/2))*sgn(x))*sin(acos((R-y)/(x^2+(R-y)^2)^(1/2))*sgn(x))*(2*acos((R-y)/(x^2+(R-y)^2)^(1/2))*dirac(x)+(x*sgn(x)*(R-y))/((x^2+(R-y)^2)^(3/2)*(1-(R-y)^2/(x^2+(R-y)^2))^(1/2)))*((cos(2*t-2*acos((R-y)/(x^2+(R-y)^2)^(1/2))*sgn(x))*(μ/2+1/2)*(S1-S2)*((3*a^4)/(x^2+(R-y)^2)^2-(4*a^2)/((x^2+(R-y)^2)*(μ+1))))/E-(a^2*(S1+S2)*(μ/2+1/2))/(E*(x^2+(R-y)^2)))+2*cos(acos((R-y)/(x^2+(R-y)^2)^(1/2))*sgn(x))*sin(acos((R-y)/(x^2+(R-y)^2)^(1/2))*sgn(x))*((a^2*(S1+S2)*(μ/2+1/2))/(E*(x^2+(R-y)^2))-(cos(2*t-2*acos((R-y)/(x^2+(R-y)^2)^(1/2))*sgn(x))*(μ/2+1/2)*((3*a^4)/(x^2+(R-y)^2)^2-(4*a^2*μ)/((x^2+(R-y)^2)*(μ+1)))*(S1-S2))/E)*(2*acos((R-y)/(x^2+(R-y)^2)^(1/2))*dirac(x)+(x*sgn(x)*(R-y))/((x^2+(R-y)^2)^(3/2)*(1-(R-y)^2/(x^2+(R-y)^2))^(1/2)))+(sin(2*acos((R-y)/(x^2+(R-y)^2)^(1/2))*sgn(x))*sin(2*t-2*acos((R-y)/(x^2+(R-y)^2)^(1/2))*sgn(x))*(S1-S2)*((4*a^2*x)/(x^2+(R-y)^2)^2-(12*a^4*x)/(x^2+(R-y)^2)^3)*(μ+1))/(2*E)-(cos(2*acos((R-y)/(x^2+(R-y)^2)^(1/2))*sgn(x))*sin(2*t-2*acos((R-y)/(x^2+(R-y)^2)^(1/2))*sgn(x))*(S1-S2)*((2*a^2)/(x^2+(R-y)^2)-(3*a^4)/(x^2+(R-y)^2)^2)*(4*acos((R-y)/(x^2+(R-y)^2)^(1/2))*dirac(x)+(2*x*sgn(x)*(R-y))/((x^2+(R-y)^2)^(3/2)*(1-(R-y)^2/(x^2+(R-y)^2))^(1/2)))*(μ+1))/(2*E)+(sin(2*acos((R-y)/(x^2+(R-y)^2)^(1/2))*sgn(x))*cos(2*t-2*acos((R-y)/(x^2+(R-y)^2)^(1/2))*sgn(x))*(S1-S2)*((2*a^2)/(x^2+(R-y)^2)-(3*a^4)/(x^2+(R-y)^2)^2)*(4*acos((R-y)/(x^2+(R-y)^2)^(1/2))*dirac(x)+(2*x*sgn(x)*(R-y))/((x^2+(R-y)^2)^(3/2)*(1-(R-y)^2/(x^2+(R-y)^2))^(1/2)))*(μ+1))/(2*E)
f15＝
cos(acos((R-y)/(x^2+(R-y)^2)^(1/2))*sgn(x))^2*((cos(2*t-2*acos((R-y)/(x^2+(R-y)^2)^(1/2))*sgn(x))*(μ/2+1/2)*(S1-S2)*((6*a^4*(2*R-2*y))/(x^2+(R-y)^2)^3-(4*a^2*(2*R-2*y))/((x^2+(R-y)^2)^2*(μ+1))))/E-(a^2*(S1+S2)*(μ/2+1/2)*(2*R-2*y))/(E*(x^2+(R-y)^2)^2)+(2*sin(2*t-2*acos((R-y)/(x^2+(R-y)^2)^(1/2))*sgn(x))*sgn(x)*(1/(x^2+(R-y)^2)^(1/2)-((R-y)*(2*R-2*y))/(2*(x^2+(R-y)^2)^(3/2)))*(μ/2+1/2)*(S1-S2)*((3*a^4)/(x^2+(R-y)^2)^2-(4*a^2)/((x^2+(R-y)^2)*(μ+1))))/(E*(1-(R-y)^2/(x^2+(R-y)^2))^(1/2)))-sin(acos((R-y)/(x^2+(R-y)^2)^(1/2))*sgn(x))^2*((cos(2*t-2*acos((R-y)/(x^2+(R-y)^2)^(1/2))*sgn(x))*((6*a^4*(2*R-2*y))/(x^2+(R-y)^2)^3-(4*a^2*μ*(2*R-2*y))/((x^2+(R-y)^2)^2*(μ+1)))*(μ/2+1/2)*(S1-S2))/E-(a^2*(S1+S2)*(μ/2+1/2)*(2*R-2*y))/(E*(x^2+(R-y)^2)^2)+(2*sin(2*t-2*acos((R-y)/(x^2+(R-y)^2)^(1/2))*sgn(x))*sgn(x)*(1/(x^2+(R-y)^2)^(1/2)-((R-y)*(2*R-2*y))/(2*(x^2+(R-y)^2)^(3/2)))*(μ/2+1/2)*((3*a^4)/(x^2+(R-y)^2)^2-(4*a^2*μ)/((x^2+(R-y)^2)*(μ+1)))*(S1-S2))/(E*(1-(R-y)^2/(x^2+(R-y)^2))^(1/2)))+(2*cos(acos((R-y)/(x^2+(R-y)^2)^(1/2))*sgn(x))*sin(acos((R-y)/(x^2+(R-y)^2)^(1/2))*sgn(x))*sgn(x)*(1/(x^2+(R-y)^2)^(1/2)-((R-y)*(2*R-2*y))/(2*(x^2+(R-y)^2)^(3/2)))*((a^2*(S1+S2)*(μ/2+1/2))/(E*(x^2+(R-y)^2))-(cos(2*t-2*acos((R-y)/(x^2+(R-y)^2)^(1/2))*sgn(x))*(μ/2+1/2)*((3*a^4)/(x^2+(R-y)^2)^2-(4*a^2*μ)/((x^2+(R-y)^2)*(μ+1)))*(S1-S2))/E))/(1-(R-y)^2/(x^2+(R-y)^2))^(1/2)-(sin(2*acos((R-y)/(x^2+(R-y)^2)^(1/2))*sgn(x))*sin(2*t-2*acos((R-y)/(x^2+(R-y)^2)^(1/2))*sgn(x))*(S1-S2)*((2*a^2*(2*R-2*y))/(x^2+(R-y)^2)^2-(6*a^4*(2*R-2*y))/(x^2+(R-y)^2)^3)*(μ+1))/(2*E)-(2*cos(acos((R-y)/(x^2+(R-y)^2)^(1/2))*sgn(x))*sin(acos((R-y)/(x^2+(R-y)^2)^(1/2))*sgn(x))*sgn(x)*(1/(x^2+(R-y)^2)^(1/2)-((R-y)*(2*R-2*y))/(2*(x^2+(R-y)^2)^(3/2)))*((cos(2*t-2*acos((R-y)/(x^2+(R-y)^2)^(1/2))*sgn(x))*(μ/2+1/2)*(S1-S2)*((3*a^4)/(x^2+(R-y)^2)^2-(4*a^2)/((x^2+(R-y)^2)*(μ+1))))/E-(a^2*(S1+S2)*(μ/2+1/2))/(E*(x^2+(R-y)^2))))/(1-(R-y)^2/(x^2+(R-y)^2))^(1/2)-(cos(2*acos((R-y)/(x^2+(R-y)^2)^(1/2))*sgn(x))*sin(2*t-2*acos((R-y)/(x^2+(R-y)^2)^(1/2))*sgn(x))*sgn(x)*(1/(x^2+(R-y)^2)^(1/2)-((R-y)*(2*R-2*y))/(2*(x^2+(R-y)^2)^(3/2)))*(S1-S2)*((2*a^2)/(x^2+(R-y)^2)-(3*a^4)/(x^2+(R-y)^2)^2)*(μ+1))/(E*(1-(R-y)^2/(x^2+(R-y)^2))^(1/2))+(sin(2*acos((R-y)/(x^2+(R-y)^2)^(1/2))*sgn(x))*cos(2*t-2*acos((R-y)/(x^2+(R-y)^2)^(1/2))*sgn(x))*sgn(x)*(1/(x^2+(R-y)^2)^(1/2)-((R-y)*(2*R-2*y))/(2*(x^2+(R-y)^2)^(3/2)))*(S1-S2)*((2*a^2)/(x^2+(R-y)^2)-(3*a^4)/(x^2+(R-y)^2)^2)*(μ+1))/(E*(1-(R-y)^2/(x^2+(R-y)^2))^(1/2))
f21＝
cos(sgn(x-y)*acos((R-(2^(1/2)*(x+y))/2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2)))^2*((cos(2*t+pi/2-2*sgn(x-y)*acos((R-(2^(1/2)*(x+y))/2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2)))*(μ/2+1/2)*((3*a^4)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^2-(4*a^2)/((μ+1)*((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2))))/E-(a^2*(μ/2+1/2))/(E*((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)))+sin(sgn(x-y)*acos((R-(2^(1/2)*(x+y))/2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2)))^2*((a^2*(μ/2+1/2))/(E*((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2))-(cos(2*t+pi/2-2*sgn(x-y)*acos((R-(2^(1/2)*(x+y))/2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2)))*(μ/2+1/2)*((3*a^4)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^2-(4*a^2*μ)/((μ+1)*((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2))))/E)-(sin(2*t+pi/2-2*sgn(x-y)*acos((R-(2^(1/2)*(x+y))/2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2)))*sin(2*sgn(x-y)*acos((R-(2^(1/2)*(x+y))/2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2)))*((2*a^2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)-(3*a^4)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^2)*(μ+1))/(2*E)
f22＝
sin(sgn(x-y)*acos((R-(2^(1/2)*(x+y))/2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2)))^2*((a^2*(μ/2+1/2))/(E*((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2))+(cos(2*t+pi/2-2*sgn(x-y)*acos((R-(2^(1/2)*(x+y))/2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2)))*(μ/2+1/2)*((3*a^4)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^2-(4*a^2*μ)/((μ+1)*((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2))))/E)-cos(sgn(x-y)*acos((R-(2^(1/2)*(x+y))/2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2)))^2*((cos(2*t+pi/2-2*sgn(x-y)*acos((R-(2^(1/2)*(x+y))/2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2)))*(μ/2+1/2)*((3*a^4)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^2-(4*a^2)/((μ+1)*((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2))))/E+(a^2*(μ/2+1/2))/(E*((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)))+(sin(2*t+pi/2-2*sgn(x-y)*acos((R-(2^(1/2)*(x+y))/2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2)))*sin(2*sgn(x-y)*acos((R-(2^(1/2)*(x+y))/2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2)))*((2*a^2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)-(3*a^4)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^2)*(μ+1))/(2*E)
f23＝
(2*sin(2*t+pi/2-2*sgn(x-y)*acos((R-(2^(1/2)*(x+y))/2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2)))*sin(sgn(x-y)*acos((R-(2^(1/2)*(x+y))/2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2)))^2*(μ/2+1/2)*(S1-S2)*((3*a^4)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^2-(4*a^2*μ)/((μ+1)*((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2))))/E-(2*sin(2*t+pi/2-2*sgn(x-y)*acos((R-(2^(1/2)*(x+y))/2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2)))*cos(sgn(x-y)*acos((R-(2^(1/2)*(x+y))/2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2)))^2*(μ/2+1/2)*(S1-S2)*((3*a^4)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^2-(4*a^2)/((μ+1)*((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2))))/E-(cos(2*t+pi/2-2*sgn(x-y)*acos((R-(2^(1/2)*(x+y))/2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2)))*sin(2*sgn(x-y)*acos((R-(2^(1/2)*(x+y))/2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2)))*((2*a^2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)-(3*a^4)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^2)*(S1-S2)*(μ+1))/E
f24＝
cos(sgn(x-y)*acos((R-(2^(1/2)*(x+y))/2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2)))^2*((sin(2*t+pi/2-2*sgn(x-y)*acos((R-(2^(1/2)*(x+y))/2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2)))*(μ/2+1/2)*(S1-S2)*((3*a^4)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^2-(4*a^2)/((μ+1)*((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)))*(4*dirac(x-y)*acos((R-(2^(1/2)*(x+y))/2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2))+(2*sgn(x-y)*(2^(1/2)/(2*((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2))+((2*x-2^(1/2)*R)*(R-(2^(1/2)*(x+y))/2))/(2*((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(3/2))))/(1-(R-(2^(1/2)*(x+y))/2)^2/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2))^(1/2)))/E-(cos(2*t+pi/2-2*sgn(x-y)*acos((R-(2^(1/2)*(x+y))/2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2)))*((6*a^4*(2*x-2^(1/2)*R))/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^3-(4*a^2*(2*x-2^(1/2)*R))/((μ+1)*((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^2))*(μ/2+1/2)*(S1-S2))/E+(a^2*(S1+S2)*(μ/2+1/2)*(2*x-2^(1/2)*R))/(E*((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^2))-sin(sgn(x-y)*acos((R-(2^(1/2)*(x+y))/2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2)))^2*((sin(2*t+pi/2-2*sgn(x-y)*acos((R-(2^(1/2)*(x+y))/2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2)))*(μ/2+1/2)*(S1-S2)*(4*dirac(x-y)*acos((R-(2^(1/2)*(x+y))/2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2))+(2*sgn(x-y)*(2^(1/2)/(2*((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2))+((2*x-2^(1/2)*R)*(R-(2^(1/2)*(x+y))/2))/(2*((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(3/2))))/(1-(R-(2^(1/2)*(x+y))/2)^2/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2))^(1/2))*((3*a^4)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^2-(4*a^2*μ)/((μ+1)*((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2))))/E-(cos(2*t+pi/2-2*sgn(x-y)*acos((R-(2^(1/2)*(x+y))/2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2)))*(μ/2+1/2)*(S1-S2)*((6*a^4*(2*x-2^(1/2)*R))/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^3-(4*a^2*μ*(2*x-2^(1/2)*R))/((μ+1)*((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^2)))/E+(a^2*(S1+S2)*(μ/2+1/2)*(2*x-2^(1/2)*R))/(E*((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^2))+2*cos(sgn(x-y)*acos((R-(2^(1/2)*(x+y))/2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2)))*sin(sgn(x-y)*acos((R-(2^(1/2)*(x+y))/2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2)))*((a^2*(S1+S2)*(μ/2+1/2))/(E*((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2))-(cos(2*t+pi/2-2*sgn(x-y)*acos((R-(2^(1/2)*(x+y))/2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2)))*(μ/2+1/2)*(S1-S2)*((3*a^4)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^2-(4*a^2)/((μ+1)*((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2))))/E)*(2*dirac(x-y)*acos((R-(2^(1/2)*(x+y))/2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2))+(sgn(x-y)*(2^(1/2)/(2*((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2))+((2*x-2^(1/2)*R)*(R-(2^(1/2)*(x+y))/2))/(2*((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(3/2))))/(1-(R-(2^(1/2)*(x+y))/2)^2/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2))^(1/2))-2*cos(sgn(x-y)*acos((R-(2^(1/2)*(x+y))/2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2)))*sin(sgn(x-y)*acos((R-(2^(1/2)*(x+y))/2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2)))*((cos(2*t+pi/2-2*sgn(x-y)*acos((R-(2^(1/2)*(x+y))/2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2)))*(μ/2+1/2)*(S1-S2)*((3*a^4)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^2-(4*a^2*μ)/((μ+1)*((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2))))/E-(a^2*(S1+S2)*(μ/2+1/2))/(E*((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)))*(2*dirac(x-y)*acos((R-(2^(1/2)*(x+y))/2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2))+(sgn(x-y)*(2^(1/2)/(2*((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2))+((2*x-2^(1/2)*R)*(R-(2^(1/2)*(x+y))/2))/(2*((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(3/2))))/(1-(R-(2^(1/2)*(x+y))/2)^2/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2))^(1/2))+(sin(2*t+pi/2-2*sgn(x-y)*acos((R-(2^(1/2)*(x+y))/2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2)))*sin(2*sgn(x-y)*acos((R-(2^(1/2)*(x+y))/2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2)))*(S1-S2)*((2*a^2*(2*x-2^(1/2)*R))/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^2-(6*a^4*(2*x-2^(1/2)*R))/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^3)*(μ+1))/(2*E)+(cos(2*t+pi/2-2*sgn(x-y)*acos((R-(2^(1/2)*(x+y))/2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2)))*sin(2*sgn(x-y)*acos((R-(2^(1/2)*(x+y))/2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2)))*((2*a^2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)-(3*a^4)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^2)*(S1-S2)*(μ+1)*(4*dirac(x-y)*acos((R-(2^(1/2)*(x+y))/2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2))+(2*sgn(x-y)*(2^(1/2)/(2*((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2))+((2*x-2^(1/2)*R)*(R-(2^(1/2)*(x+y))/2))/(2*((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(3/2))))/(1-(R-(2^(1/2)*(x+y))/2)^2/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2))^(1/2)))/(2*E)-(sin(2*t+pi/2-2*sgn(x-y)*acos((R-(2^(1/2)*(x+y))/2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2)))*cos(2*sgn(x-y)*acos((R-(2^(1/2)*(x+y))/2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2)))*((2*a^2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)-(3*a^4)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^2)*(S1-S2)*(μ+1)*(4*dirac(x-y)*acos((R-(2^(1/2)*(x+y))/2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2))+(2*sgn(x-y)*(2^(1/2)/(2*((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2))+((2*x-2^(1/2)*R)*(R-(2^(1/2)*(x+y))/2))/(2*((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(3/2))))/(1-(R-(2^(1/2)*(x+y))/2)^2/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2))^(1/2)))/(2*E)
f25＝
sin(sgn(x-y)*acos((R-(2^(1/2)*(x+y))/2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2)))^2*((cos(2*t+pi/2-2*sgn(x-y)*acos((R-(2^(1/2)*(x+y))/2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2)))*(μ/2+1/2)*(S1-S2)*((6*a^4*(2*y-2^(1/2)*R))/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^3-(4*a^2*μ*(2*y-2^(1/2)*R))/((μ+1)*((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^2)))/E+(sin(2*t+pi/2-2*sgn(x-y)*acos((R-(2^(1/2)*(x+y))/2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2)))*(μ/2+1/2)*(S1-S2)*(4*dirac(x-y)*acos((R-(2^(1/2)*(x+y))/2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2))-(2*sgn(x-y)*(2^(1/2)/(2*((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2))+((2*y-2^(1/2)*R)*(R-(2^(1/2)*(x+y))/2))/(2*((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(3/2))))/(1-(R-(2^(1/2)*(x+y))/2)^2/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2))^(1/2))*((3*a^4)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^2-(4*a^2*μ)/((μ+1)*((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2))))/E-(a^2*(S1+S2)*(μ/2+1/2)*(2*y-2^(1/2)*R))/(E*((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^2))-cos(sgn(x-y)*acos((R-(2^(1/2)*(x+y))/2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2)))^2*((cos(2*t+pi/2-2*sgn(x-y)*acos((R-(2^(1/2)*(x+y))/2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2)))*((6*a^4*(2*y-2^(1/2)*R))/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^3-(4*a^2*(2*y-2^(1/2)*R))/((μ+1)*((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^2))*(μ/2+1/2)*(S1-S2))/E+(sin(2*t+pi/2-2*sgn(x-y)*acos((R-(2^(1/2)*(x+y))/2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2)))*(μ/2+1/2)*(S1-S2)*((3*a^4)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^2-(4*a^2)/((μ+1)*((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)))*(4*dirac(x-y)*acos((R-(2^(1/2)*(x+y))/2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2))-(2*sgn(x-y)*(2^(1/2)/(2*((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2))+((2*y-2^(1/2)*R)*(R-(2^(1/2)*(x+y))/2))/(2*((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(3/2))))/(1-(R-(2^(1/2)*(x+y))/2)^2/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2))^(1/2)))/E-(a^2*(S1+S2)*(μ/2+1/2)*(2*y-2^(1/2)*R))/(E*((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^2))-2*cos(sgn(x-y)*acos((R-(2^(1/2)*(x+y))/2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2)))*sin(sgn(x-y)*acos((R-(2^(1/2)*(x+y))/2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2)))*((a^2*(S1+S2)*(μ/2+1/2))/(E*((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2))-(cos(2*t+pi/2-2*sgn(x-y)*acos((R-(2^(1/2)*(x+y))/2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2)))*(μ/2+1/2)*(S1-S2)*((3*a^4)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^2-(4*a^2)/((μ+1)*((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2))))/E)*(2*dirac(x-y)*acos((R-(2^(1/2)*(x+y))/2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2))-(sgn(x-y)*(2^(1/2)/(2*((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2))+((2*y-2^(1/2)*R)*(R-(2^(1/2)*(x+y))/2))/(2*((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(3/2))))/(1-(R-(2^(1/2)*(x+y))/2)^2/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2))^(1/2))+2*cos(sgn(x-y)*acos((R-(2^(1/2)*(x+y))/2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2)))*sin(sgn(x-y)*acos((R-(2^(1/2)*(x+y))/2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2)))*((cos(2*t+pi/2-2*sgn(x-y)*acos((R-(2^(1/2)*(x+y))/2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2)))*(μ/2+1/2)*(S1-S2)*((3*a^4)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^2-(4*a^2*μ)/((μ+1)*((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2))))/E-(a^2*(S1+S2)*(μ/2+1/2))/(E*((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)))*(2*dirac(x-y)*acos((R-(2^(1/2)*(x+y))/2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2))-(sgn(x-y)*(2^(1/2)/(2*((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2))+((2*y-2^(1/2)*R)*(R-(2^(1/2)*(x+y))/2))/(2*((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(3/2))))/(1-(R-(2^(1/2)*(x+y))/2)^2/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2))^(1/2))+(sin(2*t+pi/2-2*sgn(x-y)*acos((R-(2^(1/2)*(x+y))/2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2)))*sin(2*sgn(x-y)*acos((R-(2^(1/2)*(x+y))/2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2)))*(S1-S2)*((2*a^2*(2*y-2^(1/2)*R))/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^2-(6*a^4*(2*y-2^(1/2)*R))/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^3)*(μ+1))/(2*E)-(cos(2*t+pi/2-2*sgn(x-y)*acos((R-(2^(1/2)*(x+y))/2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2)))*sin(2*sgn(x-y)*acos((R-(2^(1/2)*(x+y))/2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2)))*((2*a^2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)-(3*a^4)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^2)*(S1-S2)*(μ+1)*(4*dirac(x-y)*acos((R-(2^(1/2)*(x+y))/2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2))-(2*sgn(x-y)*(2^(1/2)/(2*((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2))+((2*y-2^(1/2)*R)*(R-(2^(1/2)*(x+y))/2))/(2*((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(3/2))))/(1-(R-(2^(1/2)*(x+y))/2)^2/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2))^(1/2)))/(2*E)+(sin(2*t+pi/2-2*sgn(x-y)*acos((R-(2^(1/2)*(x+y))/2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2)))*cos(2*sgn(x-y)*acos((R-(2^(1/2)*(x+y))/2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2)))*((2*a^2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)-(3*a^4)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^2)*(S1-S2)*(μ+1)*(4*dirac(x-y)*acos((R-(2^(1/2)*(x+y))/2)/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2))-(2*sgn(x-y)*(2^(1/2)/(2*((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(1/2))+((2*y-2^(1/2)*R)*(R-(2^(1/2)*(x+y))/2))/(2*((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2)^(3/2))))/(1-(R-(2^(1/2)*(x+y))/2)^2/((x-(2^(1/2)*R)/2)^2+(y-(2^(1/2)*R)/2)^2))^(1/2)))/(2*E)
f31＝
sin(acos((R-x)/(y^2+(R-x)^2)^(1/2))*sgn(y))^2*((cos(2*t+2*acos((R-x)/(y^2+(R-x)^2)^(1/2))*sgn(y))*(μ/2+1/2)*((3*a^4)/(y^2+(R-x)^2)^2-(4*a^2*μ)/((y^2+(R-x)^2)*(μ+1))))/E+(a^2*(μ/2+1/2))/(E*(y^2+(R-x)^2)))-cos(acos((R-x)/(y^2+(R-x)^2)^(1/2))*sgn(y))^2*((a^2*(μ/2+1/2))/(E*(y^2+(R-x)^2))+(cos(2*t+2*acos((R-x)/(y^2+(R-x)^2)^(1/2))*sgn(y))*(μ/2+1/2)*((3*a^4)/(y^2+(R-x)^2)^2-(4*a^2)/((y^2+(R-x)^2)*(μ+1))))/E)-(sin(2*acos((R-x)/(y^2+(R-x)^2)^(1/2))*sgn(y))*sin(2*t+2*acos((R-x)/(y^2+(R-x)^2)^(1/2))*sgn(y))*((2*a^2)/(y^2+(R-x)^2)-(3*a^4)/(y^2+(R-x)^2)^2)*(μ+1))/(2*E)
f32＝
(sin(2*acos((R-x)/(y^2+(R-x)^2)^(1/2))*sgn(y))*sin(2*t+2*acos((R-x)/(y^2+(R-x)^2)^(1/2))*sgn(y))*((2*a^2)/(y^2+(R-x)^2)-(3*a^4)/(y^2+(R-x)^2)^2)*(μ+1))/(2*E)-cos(acos((R-x)/(y^2+(R-x)^2)^(1/2))*sgn(y))^2*((a^2*(μ/2+1/2))/(E*(y^2+(R-x)^2))-(cos(2*t+2*acos((R-x)/(y^2+(R-x)^2)^(1/2))*sgn(y))*(μ/2+1/2)*((3*a^4)/(y^2+(R-x)^2)^2-(4*a^2)/((y^2+(R-x)^2)*(μ+1))))/E)-sin(acos((R-x)/(y^2+(R-x)^2)^(1/2))*sgn(y))^2*((cos(2*t+2*acos((R-x)/(y^2+(R-x)^2)^(1/2))*sgn(y))*(μ/2+1/2)*((3*a^4)/(y^2+(R-x)^2)^2-(4*a^2*μ)/((y^2+(R-x)^2)*(μ+1))))/E-(a^2*(μ/2+1/2))/(E*(y^2+(R-x)^2)))
f33＝
(2*cos(acos((R-x)/(y^2+(R-x)^2)^(1/2))*sgn(y))^2*sin(2*t+2*acos((R-x)/(y^2+(R-x)^2)^(1/2))*sgn(y))*(μ/2+1/2)*(S1-S2)*((3*a^4)/(y^2+(R-x)^2)^2-(4*a^2)/((y^2+(R-x)^2)*(μ+1))))/E-(sin(2*acos((R-x)/(y^2+(R-x)^2)^(1/2))*sgn(y))*cos(2*t+2*acos((R-x)/(y^2+(R-x)^2)^(1/2))*sgn(y))*(S1-S2)*((2*a^2)/(y^2+(R-x)^2)-(3*a^4)/(y^2+(R-x)^2)^2)*(μ+1))/E-(2*sin(acos((R-x)/(y^2+(R-x)^2)^(1/2))*sgn(y))^2*sin(2*t+2*acos((R-x)/(y^2+(R-x)^2)^(1/2))*sgn(y))*(μ/2+1/2)*((3*a^4)/(y^2+(R-x)^2)^2-(4*a^2*μ)/((y^2+(R-x)^2)*(μ+1)))*(S1-S2))/E
f34＝
sin(acos((R-x)/(y^2+(R-x)^2)^(1/2))*sgn(y))^2*((cos(2*t+2*acos((R-x)/(y^2+(R-x)^2)^(1/2))*sgn(y))*((6*a^4*(2*R-2*x))/(y^2+(R-x)^2)^3-(4*a^2*μ*(2*R-2*x))/((y^2+(R-x)^2)^2*(μ+1)))*(μ/2+1/2)*(S1-S2))/E+(a^2*(S1+S2)*(μ/2+1/2)*(2*R-2*x))/(E*(y^2+(R-x)^2)^2)-(2*sin(2*t+2*acos((R-x)/(y^2+(R-x)^2)^(1/2))*sgn(y))*sgn(y)*(1/(y^2+(R-x)^2)^(1/2)-((R-x)*(2*R-2*x))/(2*(y^2+(R-x)^2)^(3/2)))*(μ/2+1/2)*((3*a^4)/(y^2+(R-x)^2)^2-(4*a^2*μ)/((y^2+(R-x)^2)*(μ+1)))*(S1-S2))/(E*(1-(R-x)^2/(y^2+(R-x)^2))^(1/2)))-cos(acos((R-x)/(y^2+(R-x)^2)^(1/2))*sgn(y))^2*((cos(2*t+2*acos((R-x)/(y^2+(R-x)^2)^(1/2))*sgn(y))*(μ/2+1/2)*(S1-S2)*((6*a^4*(2*R-2*x))/(y^2+(R-x)^2)^3-(4*a^2*(2*R-2*x))/((y^2+(R-x)^2)^2*(μ+1))))/E+(a^2*(S1+S2)*(μ/2+1/2)*(2*R-2*x))/(E*(y^2+(R-x)^2)^2)-(2*sin(2*t+2*acos((R-x)/(y^2+(R-x)^2)^(1/2))*sgn(y))*sgn(y)*(1/(y^2+(R-x)^2)^(1/2)-((R-x)*(2*R-2*x))/(2*(y^2+(R-x)^2)^(3/2)))*(μ/2+1/2)*(S1-S2)*((3*a^4)/(y^2+(R-x)^2)^2-(4*a^2)/((y^2+(R-x)^2)*(μ+1))))/(E*(1-(R-x)^2/(y^2+(R-x)^2))^(1/2)))+(2*cos(acos((R-x)/(y^2+(R-x)^2)^(1/2))*sgn(y))*sin(acos((R-x)/(y^2+(R-x)^2)^(1/2))*sgn(y))*sgn(y)*(1/(y^2+(R-x)^2)^(1/2)-((R-x)*(2*R-2*x))/(2*(y^2+(R-x)^2)^(3/2)))*((a^2*(S1+S2)*(μ/2+1/2))/(E*(y^2+(R-x)^2))+(cos(2*t+2*acos((R-x)/(y^2+(R-x)^2)^(1/2))*sgn(y))*(μ/2+1/2)*((3*a^4)/(y^2+(R-x)^2)^2-(4*a^2*μ)/((y^2+(R-x)^2)*(μ+1)))*(S1-S2))/E))/(1-(R-x)^2/(y^2+(R-x)^2))^(1/2)-(sin(2*acos((R-x)/(y^2+(R-x)^2)^(1/2))*sgn(y))*sin(2*t+2*acos((R-x)/(y^2+(R-x)^2)^(1/2))*sgn(y))*(S1-S2)*((2*a^2*(2*R-2*x))/(y^2+(R-x)^2)^2-(6*a^4*(2*R-2*x))/(y^2+(R-x)^2)^3)*(μ+1))/(2*E)+(2*cos(acos((R-x)/(y^2+(R-x)^2)^(1/2))*sgn(y))*sin(acos((R-x)/(y^2+(R-x)^2)^(1/2))*sgn(y))*sgn(y)*(1/(y^2+(R-x)^2)^(1/2)-((R-x)*(2*R-2*x))/(2*(y^2+(R-x)^2)^(3/2)))*((cos(2*t+2*acos((R-x)/(y^2+(R-x)^2)^(1/2))*sgn(y))*(μ/2+1/2)*(S1-S2)*((3*a^4)/(y^2+(R-x)^2)^2-(4*a^2)/((y^2+(R-x)^2)*(μ+1))))/E+(a^2*(S1+S2)*(μ/2+1/2))/(E*(y^2+(R-x)^2))))/(1-(R-x)^2/(y^2+(R-x)^2))^(1/2)-(cos(2*acos((R-x)/(y^2+(R-x)^2)^(1/2))*sgn(y))*sin(2*t+2*acos((R-x)/(y^2+(R-x)^2)^(1/2))*sgn(y))*sgn(y)*(1/(y^2+(R-x)^2)^(1/2)-((R-x)*(2*R-2*x))/(2*(y^2+(R-x)^2)^(3/2)))*(S1-S2)*((2*a^2)/(y^2+(R-x)^2)-(3*a^4)/(y^2+(R-x)^2)^2)*(μ+1))/(E*(1-(R-x)^2/(y^2+(R-x)^2))^(1/2))-(sin(2*acos((R-x)/(y^2+(R-x)^2)^(1/2))*sgn(y))*cos(2*t+2*acos((R-x)/(y^2+(R-x)^2)^(1/2))*sgn(y))*sgn(y)*(1/(y^2+(R-x)^2)^(1/2)-((R-x)*(2*R-2*x))/(2*(y^2+(R-x)^2)^(3/2)))*(S1-S2)*((2*a^2)/(y^2+(R-x)^2)-(3*a^4)/(y^2+(R-x)^2)^2)*(μ+1))/(E*(1-(R-x)^2/(y^2+(R-x)^2))^(1/2))
f35＝
cos(acos((R-x)/(y^2+(R-x)^2)^(1/2))*sgn(y))^2*((cos(2*t+2*acos((R-x)/(y^2+(R-x)^2)^(1/2))*sgn(y))*(μ/2+1/2)*(S1-S2)*((12*a^4*y)/(y^2+(R-x)^2)^3-(8*a^2*y)/((y^2+(R-x)^2)^2*(μ+1))))/E+(sin(2*t+2*acos((R-x)/(y^2+(R-x)^2)^(1/2))*sgn(y))*(μ/2+1/2)*(S1-S2)*(4*acos((R-x)/(y^2+(R-x)^2)^(1/2))*dirac(y)+(2*y*sgn(y)*(R-x))/((y^2+(R-x)^2)^(3/2)*(1-(R-x)^2/(y^2+(R-x)^2))^(1/2)))*((3*a^4)/(y^2+(R-x)^2)^2-(4*a^2)/((y^2+(R-x)^2)*(μ+1))))/E+(2*a^2*y*(S1+S2)*(μ/2+1/2))/(E*(y^2+(R-x)^2)^2))-sin(acos((R-x)/(y^2+(R-x)^2)^(1/2))*sgn(y))^2*((cos(2*t+2*acos((R-x)/(y^2+(R-x)^2)^(1/2))*sgn(y))*(μ/2+1/2)*((12*a^4*y)/(y^2+(R-x)^2)^3-(8*a^2*μ*y)/((y^2+(R-x)^2)^2*(μ+1)))*(S1-S2))/E+(sin(2*t+2*acos((R-x)/(y^2+(R-x)^2)^(1/2))*sgn(y))*(μ/2+1/2)*((3*a^4)/(y^2+(R-x)^2)^2-(4*a^2*μ)/((y^2+(R-x)^2)*(μ+1)))*(S1-S2)*(4*acos((R-x)/(y^2+(R-x)^2)^(1/2))*dirac(y)+(2*y*sgn(y)*(R-x))/((y^2+(R-x)^2)^(3/2)*(1-(R-x)^2/(y^2+(R-x)^2))^(1/2))))/E+(2*a^2*y*(S1+S2)*(μ/2+1/2))/(E*(y^2+(R-x)^2)^2))+2*cos(acos((R-x)/(y^2+(R-x)^2)^(1/2))*sgn(y))*sin(acos((R-x)/(y^2+(R-x)^2)^(1/2))*sgn(y))*(2*acos((R-x)/(y^2+(R-x)^2)^(1/2))*dirac(y)+(y*sgn(y)*(R-x))/((y^2+(R-x)^2)^(3/2)*(1-(R-x)^2/(y^2+(R-x)^2))^(1/2)))*((cos(2*t+2*acos((R-x)/(y^2+(R-x)^2)^(1/2))*sgn(y))*(μ/2+1/2)*(S1-S2)*((3*a^4)/(y^2+(R-x)^2)^2-(4*a^2)/((y^2+(R-x)^2)*(μ+1))))/E+(a^2*(S1+S2)*(μ/2+1/2))/(E*(y^2+(R-x)^2)))+2*cos(acos((R-x)/(y^2+(R-x)^2)^(1/2))*sgn(y))*sin(acos((R-x)/(y^2+(R-x)^2)^(1/2))*sgn(y))*((a^2*(S1+S2)*(μ/2+1/2))/(E*(y^2+(R-x)^2))+(cos(2*t+2*acos((R-x)/(y^2+(R-x)^2)^(1/2))*sgn(y))*(μ/2+1/2)*((3*a^4)/(y^2+(R-x)^2)^2-(4*a^2*μ)/((y^2+(R-x)^2)*(μ+1)))*(S1-S2))/E)*(2*acos((R-x)/(y^2+(R-x)^2)^(1/2))*dirac(y)+(y*sgn(y)*(R-x))/((y^2+(R-x)^2)^(3/2)*(1-(R-x)^2/(y^2+(R-x)^2))^(1/2)))+(sin(2*acos((R-x)/(y^2+(R-x)^2)^(1/2))*sgn(y))*sin(2*t+2*acos((R-x)/(y^2+(R-x)^2)^(1/2))*sgn(y))*(S1-S2)*((4*a^2*y)/(y^2+(R-x)^2)^2-(12*a^4*y)/(y^2+(R-x)^2)^3)*(μ+1))/(2*E)-(cos(2*acos((R-x)/(y^2+(R-x)^2)^(1/2))*sgn(y))*sin(2*t+2*acos((R-x)/(y^2+(R-x)^2)^(1/2))*sgn(y))*(S1-S2)*((2*a^2)/(y^2+(R-x)^2)-(3*a^4)/(y^2+(R-x)^2)^2)*(4*acos((R-x)/(y^2+(R-x)^2)^(1/2))*dirac(y)+(2*y*sgn(y)*(R-x))/((y^2+(R-x)^2)^(3/2)*(1-(R-x)^2/(y^2+(R-x)^2))^(1/2)))*(μ+1))/(2*E)-(sin(2*acos((R-x)/(y^2+(R-x)^2)^(1/2))*sgn(y))*cos(2*t+2*acos((R-x)/(y^2+(R-x)^2)^(1/2))*sgn(y))*(S1-S2)*((2*a^2)/(y^2+(R-x)^2)-(3*a^4)/(y^2+(R-x)^2)^2)*(4*acos((R-x)/(y^2+(R-x)^2)^(1/2))*dirac(y)+(2*y*sgn(y)*(R-x))/((y^2+(R-x)^2)^(3/2)*(1-(R-x)^2/(y^2+(R-x)^2))^(1/2)))*(μ+1))/(2*E)
f41＝
sin(acos((R+y)/((R+y)^2+x^2)^(1/2))*sgn(x))^2*((a^2*(μ/2+1/2))/(E*((R+y)^2+x^2))-(cos(2*t+2*acos((R+y)/((R+y)^2+x^2)^(1/2))*sgn(x))*(μ/2+1/2)*((3*a^4)/((R+y)^2+x^2)^2-(4*a^2*μ)/(((R+y)^2+x^2)*(μ+1))))/E)-cos(acos((R+y)/((R+y)^2+x^2)^(1/2))*sgn(x))^2*((a^2*(μ/2+1/2))/(E*((R+y)^2+x^2))-(cos(2*t+2*acos((R+y)/((R+y)^2+x^2)^(1/2))*sgn(x))*((3*a^4)/((R+y)^2+x^2)^2-(4*a^2)/(((R+y)^2+x^2)*(μ+1)))*(μ/2+1/2))/E)+(sin(2*acos((R+y)/((R+y)^2+x^2)^(1/2))*sgn(x))*sin(2*t+2*acos((R+y)/((R+y)^2+x^2)^(1/2))*sgn(x))*(μ+1)*((2*a^2)/((R+y)^2+x^2)-(3*a^4)/((R+y)^2+x^2)^2))/(2*E)
f42＝sin(acos((R+y)/((R+y)^2+x^2)^(1/2))*sgn(x))^2*((a^2*(μ/2+1/2))/(E*((R+y)^2+x^2))+(cos(2*t+2*acos((R+y)/((R+y)^2+x^2)^(1/2))*sgn(x))*(μ/2+1/2)*((3*a^4)/((R+y)^2+x^2)^2-(4*a^2*μ)/(((R+y)^2+x^2)*(μ+1))))/E)-cos(acos((R+y)/((R+y)^2+x^2)^(1/2))*sgn(x))^2*((a^2*(μ/2+1/2))/(E*((R+y)^2+x^2))+(cos(2*t+2*acos((R+y)/((R+y)^2+x^2)^(1/2))*sgn(x))*((3*a^4)/((R+y)^2+x^2)^2-(4*a^2)/(((R+y)^2+x^2)*(μ+1)))*(μ/2+1/2))/E)-(sin(2*acos((R+y)/((R+y)^2+x^2)^(1/2))*sgn(x))*sin(2*t+2*acos((R+y)/((R+y)^2+x^2)^(1/2))*sgn(x))*(μ+1)*((2*a^2)/((R+y)^2+x^2)-(3*a^4)/((R+y)^2+x^2)^2))/(2*E)
f43＝(2*sin(acos((R+y)/((R+y)^2+x^2)^(1/2))*sgn(x))^2*sin(2*t+2*acos((R+y)/((R+y)^2+x^2)^(1/2))*sgn(x))*(μ/2+1/2)*(S1-S2)*((3*a^4)/((R+y)^2+x^2)^2-(4*a^2*μ)/(((R+y)^2+x^2)*(μ+1))))/E-(2*cos(acos((R+y)/((R+y)^2+x^2)^(1/2))*sgn(x))^2*sin(2*t+2*acos((R+y)/((R+y)^2+x^2)^(1/2))*sgn(x))*((3*a^4)/((R+y)^2+x^2)^2-(4*a^2)/(((R+y)^2+x^2)*(μ+1)))*(μ/2+1/2)*(S1-S2))/E+(sin(2*acos((R+y)/((R+y)^2+x^2)^(1/2))*sgn(x))*cos(2*t+2*acos((R+y)/((R+y)^2+x^2)^(1/2))*sgn(x))*(S1-S2)*(μ+1)*((2*a^2)/((R+y)^2+x^2)-(3*a^4)/((R+y)^2+x^2)^2))/E
f44＝sin(acos((R+y)/((R+y)^2+x^2)^(1/2))*sgn(x))^2*((cos(2*t+2*acos((R+y)/((R+y)^2+x^2)^(1/2))*sgn(x))*(μ/2+1/2)*(S1-S2)*((12*a^4*x)/((R+y)^2+x^2)^3-(8*a^2*μ*x)/(((R+y)^2+x^2)^2*(μ+1))))/E-(2*a^2*x*(S1+S2)*(μ/2+1/2))/(E*((R+y)^2+x^2)^2)+(sin(2*t+2*acos((R+y)/((R+y)^2+x^2)^(1/2))*sgn(x))*(μ/2+1/2)*(S1-S2)*(4*acos((R+y)/((R+y)^2+x^2)^(1/2))*dirac(x)+(2*x*sgn(x)*(R+y))/((1-(R+y)^2/((R+y)^2+x^2))^(1/2)*((R+y)^2+x^2)^(3/2)))*((3*a^4)/((R+y)^2+x^2)^2-(4*a^2*μ)/(((R+y)^2+x^2)*(μ+1))))/E)-cos(acos((R+y)/((R+y)^2+x^2)^(1/2))*sgn(x))^2*((cos(2*t+2*acos((R+y)/((R+y)^2+x^2)^(1/2))*sgn(x))*((12*a^4*x)/((R+y)^2+x^2)^3-(8*a^2*x)/(((R+y)^2+x^2)^2*(μ+1)))*(μ/2+1/2)*(S1-S2))/E+(sin(2*t+2*acos((R+y)/((R+y)^2+x^2)^(1/2))*sgn(x))*((3*a^4)/((R+y)^2+x^2)^2-(4*a^2)/(((R+y)^2+x^2)*(μ+1)))*(μ/2+1/2)*(S1-S2)*(4*acos((R+y)/((R+y)^2+x^2)^(1/2))*dirac(x)+(2*x*sgn(x)*(R+y))/((1-(R+y)^2/((R+y)^2+x^2))^(1/2)*((R+y)^2+x^2)^(3/2))))/E-(2*a^2*x*(S1+S2)*(μ/2+1/2))/(E*((R+y)^2+x^2)^2))+2*cos(acos((R+y)/((R+y)^2+x^2)^(1/2))*sgn(x))*sin(acos((R+y)/((R+y)^2+x^2)^(1/2))*sgn(x))*((a^2*(S1+S2)*(μ/2+1/2))/(E*((R+y)^2+x^2))-(cos(2*t+2*acos((R+y)/((R+y)^2+x^2)^(1/2))*sgn(x))*((3*a^4)/((R+y)^2+x^2)^2-(4*a^2)/(((R+y)^2+x^2)*(μ+1)))*(μ/2+1/2)*(S1-S2))/E)*(2*acos((R+y)/((R+y)^2+x^2)^(1/2))*dirac(x)+(x*sgn(x)*(R+y))/((1-(R+y)^2/((R+y)^2+x^2))^(1/2)*((R+y)^2+x^2)^(3/2)))-2*cos(acos((R+y)/((R+y)^2+x^2)^(1/2))*sgn(x))*sin(acos((R+y)/((R+y)^2+x^2)^(1/2))*sgn(x))*(2*acos((R+y)/((R+y)^2+x^2)^(1/2))*dirac(x)+(x*sgn(x)*(R+y))/((1-(R+y)^2/((R+y)^2+x^2))^(1/2)*((R+y)^2+x^2)^(3/2)))*((cos(2*t+2*acos((R+y)/((R+y)^2+x^2)^(1/2))*sgn(x))*(μ/2+1/2)*(S1-S2)*((3*a^4)/((R+y)^2+x^2)^2-(4*a^2*μ)/(((R+y)^2+x^2)*(μ+1))))/E-(a^2*(S1+S2)*(μ/2+1/2))/(E*((R+y)^2+x^2)))-(sin(2*acos((R+y)/((R+y)^2+x^2)^(1/2))*sgn(x))*sin(2*t+2*acos((R+y)/((R+y)^2+x^2)^(1/2))*sgn(x))*(S1-S2)*((4*a^2*x)/((R+y)^2+x^2)^2-(12*a^4*x)/((R+y)^2+x^2)^3)*(μ+1))/(2*E)+(cos(2*acos((R+y)/((R+y)^2+x^2)^(1/2))*sgn(x))*sin(2*t+2*acos((R+y)/((R+y)^2+x^2)^(1/2))*sgn(x))*(S1-S2)*(4*acos((R+y)/((R+y)^2+x^2)^(1/2))*dirac(x)+(2*x*sgn(x)*(R+y))/((1-(R+y)^2/((R+y)^2+x^2))^(1/2)*((R+y)^2+x^2)^(3/2)))*(μ+1)*((2*a^2)/((R+y)^2+x^2)-(3*a^4)/((R+y)^2+x^2)^2))/(2*E)+(sin(2*acos((R+y)/((R+y)^2+x^2)^(1/2))*sgn(x))*cos(2*t+2*acos((R+y)/((R+y)^2+x^2)^(1/2))*sgn(x))*(S1-S2)*(4*acos((R+y)/((R+y)^2+x^2)^(1/2))*dirac(x)+(2*x*sgn(x)*(R+y))/((1-(R+y)^2/((R+y)^2+x^2))^(1/2)*((R+y)^2+x^2)^(3/2)))*(μ+1)*((2*a^2)/((R+y)^2+x^2)-(3*a^4)/((R+y)^2+x^2)^2))/(2*E)
f45＝cos(acos((R+y)/((R+y)^2+x^2)^(1/2))*sgn(x))^2*((a^2*(S1+S2)*(μ/2+1/2)*(2*R+2*y))/(E*((R+y)^2+x^2)^2)-(cos(2*t+2*acos((R+y)/((R+y)^2+x^2)^(1/2))*sgn(x))*(μ/2+1/2)*(S1-S2)*((6*a^4*(2*R+2*y))/((R+y)^2+x^2)^3-(4*a^2*(2*R+2*y))/(((R+y)^2+x^2)^2*(μ+1))))/E+(2*sgn(x)*sin(2*t+2*acos((R+y)/((R+y)^2+x^2)^(1/2))*sgn(x))*(1/((R+y)^2+x^2)^(1/2)-((R+y)*(2*R+2*y))/(2*((R+y)^2+x^2)^(3/2)))*((3*a^4)/((R+y)^2+x^2)^2-(4*a^2)/(((R+y)^2+x^2)*(μ+1)))*(μ/2+1/2)*(S1-S2))/(E*(1-(R+y)^2/((R+y)^2+x^2))^(1/2)))-sin(acos((R+y)/((R+y)^2+x^2)^(1/2))*sgn(x))^2*((a^2*(S1+S2)*(μ/2+1/2)*(2*R+2*y))/(E*((R+y)^2+x^2)^2)-(cos(2*t+2*acos((R+y)/((R+y)^2+x^2)^(1/2))*sgn(x))*(μ/2+1/2)*((6*a^4*(2*R+2*y))/((R+y)^2+x^2)^3-(4*a^2*μ*(2*R+2*y))/(((R+y)^2+x^2)^2*(μ+1)))*(S1-S2))/E+(2*sgn(x)*sin(2*t+2*acos((R+y)/((R+y)^2+x^2)^(1/2))*sgn(x))*(1/((R+y)^2+x^2)^(1/2)-((R+y)*(2*R+2*y))/(2*((R+y)^2+x^2)^(3/2)))*(μ/2+1/2)*(S1-S2)*((3*a^4)/((R+y)^2+x^2)^2-(4*a^2*μ)/(((R+y)^2+x^2)*(μ+1))))/(E*(1-(R+y)^2/((R+y)^2+x^2))^(1/2)))+(2*cos(acos((R+y)/((R+y)^2+x^2)^(1/2))*sgn(x))*sin(acos((R+y)/((R+y)^2+x^2)^(1/2))*sgn(x))*sgn(x)*(1/((R+y)^2+x^2)^(1/2)-((R+y)*(2*R+2*y))/(2*((R+y)^2+x^2)^(3/2)))*((cos(2*t+2*acos((R+y)/((R+y)^2+x^2)^(1/2))*sgn(x))*(μ/2+1/2)*(S1-S2)*((3*a^4)/((R+y)^2+x^2)^2-(4*a^2*μ)/(((R+y)^2+x^2)*(μ+1))))/E-(a^2*(S1+S2)*(μ/2+1/2))/(E*((R+y)^2+x^2))))/(1-(R+y)^2/((R+y)^2+x^2))^(1/2)-(sin(2*acos((R+y)/((R+y)^2+x^2)^(1/2))*sgn(x))*sin(2*t+2*acos((R+y)/((R+y)^2+x^2)^(1/2))*sgn(x))*(S1-S2)*((2*a^2*(2*R+2*y))/((R+y)^2+x^2)^2-(6*a^4*(2*R+2*y))/((R+y)^2+x^2)^3)*(μ+1))/(2*E)-(2*cos(acos((R+y)/((R+y)^2+x^2)^(1/2))*sgn(x))*sin(acos((R+y)/((R+y)^2+x^2)^(1/2))*sgn(x))*sgn(x)*(1/((R+y)^2+x^2)^(1/2)-((R+y)*(2*R+2*y))/(2*((R+y)^2+x^2)^(3/2)))*((a^2*(S1+S2)*(μ/2+1/2))/(E*((R+y)^2+x^2))-(cos(2*t+2*acos((R+y)/((R+y)^2+x^2)^(1/2))*sgn(x))*((3*a^4)/((R+y)^2+x^2)^2-(4*a^2)/(((R+y)^2+x^2)*(μ+1)))*(μ/2+1/2)*(S1-S2))/E))/(1-(R+y)^2/((R+y)^2+x^2))^(1/2)-(cos(2*acos((R+y)/((R+y)^2+x^2)^(1/2))*sgn(x))*sgn(x)*sin(2*t+2*acos((R+y)/((R+y)^2+x^2)^(1/2))*sgn(x))*(1/((R+y)^2+x^2)^(1/2)-((R+y)*(2*R+2*y))/(2*((R+y)^2+x^2)^(3/2)))*(S1-S2)*(μ+1)*((2*a^2)/((R+y)^2+x^2)-(3*a^4)/((R+y)^2+x^2)^2))/(E*(1-(R+y)^2/((R+y)^2+x^2))^(1/2))-(sin(2*acos((R+y)/((R+y)^2+x^2)^(1/2))*sgn(x))*sgn(x)*cos(2*t+2*acos((R+y)/((R+y)^2+x^2)^(1/2))*sgn(x))*(1/((R+y)^2+x^2)^(1/2)-((R+y)*(2*R+2*y))/(2*((R+y)^2+x^2)^(3/2)))*(S1-S2)*(μ+1)*((2*a^2)/((R+y)^2+x^2)-(3*a^4)/((R+y)^2+x^2)^2))/(E*(1-(R+y)^2/((R+y)^2+x^2))^(1/2))
f51＝
sin(acos((R+x)/((R+x)^2+y^2)^(1/2))*sgn(y))^2*((a^2*(μ/2+1/2))/(E*((R+x)^2+y^2))+(cos(2*t-2*acos((R+x)/((R+x)^2+y^2)^(1/2))*sgn(y))*(μ/2+1/2)*((3*a^4)/((R+x)^2+y^2)^2-(4*a^2*μ)/(((R+x)^2+y^2)*(μ+1))))/E)-cos(acos((R+x)/((R+x)^2+y^2)^(1/2))*sgn(y))^2*((a^2*(μ/2+1/2))/(E*((R+x)^2+y^2))+(cos(2*t-2*acos((R+x)/((R+x)^2+y^2)^(1/2))*sgn(y))*((3*a^4)/((R+x)^2+y^2)^2-(4*a^2)/(((R+x)^2+y^2)*(μ+1)))*(μ/2+1/2))/E)+(sin(2*acos((R+x)/((R+x)^2+y^2)^(1/2))*sgn(y))*sin(2*t-2*acos((R+x)/((R+x)^2+y^2)^(1/2))*sgn(y))*(μ+1)*((2*a^2)/((R+x)^2+y^2)-(3*a^4)/((R+x)^2+y^2)^2))/(2*E)
f52＝
sin(acos((R+x)/((R+x)^2+y^2)^(1/2))*sgn(y))^2*((a^2*(μ/2+1/2))/(E*((R+x)^2+y^2))-(cos(2*t-2*acos((R+x)/((R+x)^2+y^2)^(1/2))*sgn(y))*(μ/2+1/2)*((3*a^4)/((R+x)^2+y^2)^2-(4*a^2*μ)/(((R+x)^2+y^2)*(μ+1))))/E)-cos(acos((R+x)/((R+x)^2+y^2)^(1/2))*sgn(y))^2*((a^2*(μ/2+1/2))/(E*((R+x)^2+y^2))-(cos(2*t-2*acos((R+x)/((R+x)^2+y^2)^(1/2))*sgn(y))*((3*a^4)/((R+x)^2+y^2)^2-(4*a^2)/(((R+x)^2+y^2)*(μ+1)))*(μ/2+1/2))/E)-(sin(2*acos((R+x)/((R+x)^2+y^2)^(1/2))*sgn(y))*sin(2*t-2*acos((R+x)/((R+x)^2+y^2)^(1/2))*sgn(y))*(μ+1)*((2*a^2)/((R+x)^2+y^2)-(3*a^4)/((R+x)^2+y^2)^2))/(2*E)
f53＝(2*cos(acos((R+x)/((R+x)^2+y^2)^(1/2))*sgn(y))^2*sin(2*t-2*acos((R+x)/((R+x)^2+y^2)^(1/2))*sgn(y))*((3*a^4)/((R+x)^2+y^2)^2-(4*a^2)/(((R+x)^2+y^2)*(μ+1)))*(μ/2+1/2)*(S1-S2))/E-(2*sin(acos((R+x)/((R+x)^2+y^2)^(1/2))*sgn(y))^2*sin(2*t-2*acos((R+x)/((R+x)^2+y^2)^(1/2))*sgn(y))*(μ/2+1/2)*(S1-S2)*((3*a^4)/((R+x)^2+y^2)^2-(4*a^2*μ)/(((R+x)^2+y^2)*(μ+1))))/E+(sin(2*acos((R+x)/((R+x)^2+y^2)^(1/2))*sgn(y))*cos(2*t-2*acos((R+x)/((R+x)^2+y^2)^(1/2))*sgn(y))*(S1-S2)*(μ+1)*((2*a^2)/((R+x)^2+y^2)-(3*a^4)/((R+x)^2+y^2)^2))/E
f54＝
cos(acos((R+x)/((R+x)^2+y^2)^(1/2))*sgn(y))^2*((cos(2*t-2*acos((R+x)/((R+x)^2+y^2)^(1/2))*sgn(y))*(μ/2+1/2)*(S1-S2)*((6*a^4*(2*R+2*x))/((R+x)^2+y^2)^3-(4*a^2*(2*R+2*x))/(((R+x)^2+y^2)^2*(μ+1))))/E+(a^2*(S1+S2)*(μ/2+1/2)*(2*R+2*x))/(E*((R+x)^2+y^2)^2)+(2*sgn(y)*sin(2*t-2*acos((R+x)/((R+x)^2+y^2)^(1/2))*sgn(y))*(1/((R+x)^2+y^2)^(1/2)-((R+x)*(2*R+2*x))/(2*((R+x)^2+y^2)^(3/2)))*((3*a^4)/((R+x)^2+y^2)^2-(4*a^2)/(((R+x)^2+y^2)*(μ+1)))*(μ/2+1/2)*(S1-S2))/(E*(1-(R+x)^2/((R+x)^2+y^2))^(1/2)))-sin(acos((R+x)/((R+x)^2+y^2)^(1/2))*sgn(y))^2*((cos(2*t-2*acos((R+x)/((R+x)^2+y^2)^(1/2))*sgn(y))*(μ/2+1/2)*((6*a^4*(2*R+2*x))/((R+x)^2+y^2)^3-(4*a^2*μ*(2*R+2*x))/(((R+x)^2+y^2)^2*(μ+1)))*(S1-S2))/E+(a^2*(S1+S2)*(μ/2+1/2)*(2*R+2*x))/(E*((R+x)^2+y^2)^2)+(2*sgn(y)*sin(2*t-2*acos((R+x)/((R+x)^2+y^2)^(1/2))*sgn(y))*(1/((R+x)^2+y^2)^(1/2)-((R+x)*(2*R+2*x))/(2*((R+x)^2+y^2)^(3/2)))*(μ/2+1/2)*(S1-S2)*((3*a^4)/((R+x)^2+y^2)^2-(4*a^2*μ)/(((R+x)^2+y^2)*(μ+1))))/(E*(1-(R+x)^2/((R+x)^2+y^2))^(1/2)))-(2*cos(acos((R+x)/((R+x)^2+y^2)^(1/2))*sgn(y))*sin(acos((R+x)/((R+x)^2+y^2)^(1/2))*sgn(y))*sgn(y)*(1/((R+x)^2+y^2)^(1/2)-((R+x)*(2*R+2*x))/(2*((R+x)^2+y^2)^(3/2)))*((cos(2*t-2*acos((R+x)/((R+x)^2+y^2)^(1/2))*sgn(y))*(μ/2+1/2)*(S1-S2)*((3*a^4)/((R+x)^2+y^2)^2-(4*a^2*μ)/(((R+x)^2+y^2)*(μ+1))))/E+(a^2*(S1+S2)*(μ/2+1/2))/(E*((R+x)^2+y^2))))/(1-(R+x)^2/((R+x)^2+y^2))^(1/2)-(sin(2*acos((R+x)/((R+x)^2+y^2)^(1/2))*sgn(y))*sin(2*t-2*acos((R+x)/((R+x)^2+y^2)^(1/2))*sgn(y))*(S1-S2)*((2*a^2*(2*R+2*x))/((R+x)^2+y^2)^2-(6*a^4*(2*R+2*x))/((R+x)^2+y^2)^3)*(μ+1))/(2*E)-(2*cos(acos((R+x)/((R+x)^2+y^2)^(1/2))*sgn(y))*sin(acos((R+x)/((R+x)^2+y^2)^(1/2))*sgn(y))*sgn(y)*(1/((R+x)^2+y^2)^(1/2)-((R+x)*(2*R+2*x))/(2*((R+x)^2+y^2)^(3/2)))*((a^2*(S1+S2)*(μ/2+1/2))/(E*((R+x)^2+y^2))+(cos(2*t-2*acos((R+x)/((R+x)^2+y^2)^(1/2))*sgn(y))*((3*a^4)/((R+x)^2+y^2)^2-(4*a^2)/(((R+x)^2+y^2)*(μ+1)))*(μ/2+1/2)*(S1-S2))/E))/(1-(R+x)^2/((R+x)^2+y^2))^(1/2)-(cos(2*acos((R+x)/((R+x)^2+y^2)^(1/2))*sgn(y))*sgn(y)*sin(2*t-2*acos((R+x)/((R+x)^2+y^2)^(1/2))*sgn(y))*(1/((R+x)^2+y^2)^(1/2)-((R+x)*(2*R+2*x))/(2*((R+x)^2+y^2)^(3/2)))*(S1-S2)*(μ+1)*((2*a^2)/((R+x)^2+y^2)-(3*a^4)/((R+x)^2+y^2)^2))/(E*(1-(R+x)^2/((R+x)^2+y^2))^(1/2))+(sin(2*acos((R+x)/((R+x)^2+y^2)^(1/2))*sgn(y))*sgn(y)*cos(2*t-2*acos((R+x)/((R+x)^2+y^2)^(1/2))*sgn(y))*(1/((R+x)^2+y^2)^(1/2)-((R+x)*(2*R+2*x))/(2*((R+x)^2+y^2)^(3/2)))*(S1-S2)*(μ+1)*((2*a^2)/((R+x)^2+y^2)-(3*a^4)/((R+x)^2+y^2)^2))/(E*(1-(R+x)^2/((R+x)^2+y^2))^(1/2))
f55＝
cos(acos((R+x)/((R+x)^2+y^2)^(1/2))*sgn(y))^2*((cos(2*t-2*acos((R+x)/((R+x)^2+y^2)^(1/2))*sgn(y))*((12*a^4*y)/((R+x)^2+y^2)^3-(8*a^2*y)/(((R+x)^2+y^2)^2*(μ+1)))*(μ/2+1/2)*(S1-S2))/E-(sin(2*t-2*acos((R+x)/((R+x)^2+y^2)^(1/2))*sgn(y))*((3*a^4)/((R+x)^2+y^2)^2-(4*a^2)/(((R+x)^2+y^2)*(μ+1)))*(μ/2+1/2)*(S1-S2)*(4*acos((R+x)/((R+x)^2+y^2)^(1/2))*dirac(y)+(2*y*sgn(y)*(R+x))/((1-(R+x)^2/((R+x)^2+y^2))^(1/2)*((R+x)^2+y^2)^(3/2))))/E+(2*a^2*y*(S1+S2)*(μ/2+1/2))/(E*((R+x)^2+y^2)^2))-sin(acos((R+x)/((R+x)^2+y^2)^(1/2))*sgn(y))^2*((cos(2*t-2*acos((R+x)/((R+x)^2+y^2)^(1/2))*sgn(y))*(μ/2+1/2)*(S1-S2)*((12*a^4*y)/((R+x)^2+y^2)^3-(8*a^2*μ*y)/(((R+x)^2+y^2)^2*(μ+1))))/E+(2*a^2*y*(S1+S2)*(μ/2+1/2))/(E*((R+x)^2+y^2)^2)-(sin(2*t-2*acos((R+x)/((R+x)^2+y^2)^(1/2))*sgn(y))*(μ/2+1/2)*(S1-S2)*(4*acos((R+x)/((R+x)^2+y^2)^(1/2))*dirac(y)+(2*y*sgn(y)*(R+x))/((1-(R+x)^2/((R+x)^2+y^2))^(1/2)*((R+x)^2+y^2)^(3/2)))*((3*a^4)/((R+x)^2+y^2)^2-(4*a^2*μ)/(((R+x)^2+y^2)*(μ+1))))/E)+2*cos(acos((R+x)/((R+x)^2+y^2)^(1/2))*sgn(y))*sin(acos((R+x)/((R+x)^2+y^2)^(1/2))*sgn(y))*((a^2*(S1+S2)*(μ/2+1/2))/(E*((R+x)^2+y^2))+(cos(2*t-2*acos((R+x)/((R+x)^2+y^2)^(1/2))*sgn(y))*((3*a^4)/((R+x)^2+y^2)^2-(4*a^2)/(((R+x)^2+y^2)*(μ+1)))*(μ/2+1/2)*(S1-S2))/E)*(2*acos((R+x)/((R+x)^2+y^2)^(1/2))*dirac(y)+(y*sgn(y)*(R+x))/((1-(R+x)^2/((R+x)^2+y^2))^(1/2)*((R+x)^2+y^2)^(3/2)))+2*cos(acos((R+x)/((R+x)^2+y^2)^(1/2))*sgn(y))*sin(acos((R+x)/((R+x)^2+y^2)^(1/2))*sgn(y))*(2*acos((R+x)/((R+x)^2+y^2)^(1/2))*dirac(y)+(y*sgn(y)*(R+x))/((1-(R+x)^2/((R+x)^2+y^2))^(1/2)*((R+x)^2+y^2)^(3/2)))*((cos(2*t-2*acos((R+x)/((R+x)^2+y^2)^(1/2))*sgn(y))*(μ/2+1/2)*(S1-S2)*((3*a^4)/((R+x)^2+y^2)^2-(4*a^2*μ)/(((R+x)^2+y^2)*(μ+1))))/E+(a^2*(S1+S2)*(μ/2+1/2))/(E*((R+x)^2+y^2)))-(sin(2*acos((R+x)/((R+x)^2+y^2)^(1/2))*sgn(y))*sin(2*t-2*acos((R+x)/((R+x)^2+y^2)^(1/2))*sgn(y))*(S1-S2)*((4*a^2*y)/((R+x)^2+y^2)^2-(12*a^4*y)/((R+x)^2+y^2)^3)*(μ+1))/(2*E)+(cos(2*acos((R+x)/((R+x)^2+y^2)^(1/2))*sgn(y))*sin(2*t-2*acos((R+x)/((R+x)^2+y^2)^(1/2))*sgn(y))*(S1-S2)*(4*acos((R+x)/((R+x)^2+y^2)^(1/2))*dirac(y)+(2*y*sgn(y)*(R+x))/((1-(R+x)^2/((R+x)^2+y^2))^(1/2)*((R+x)^2+y^2)^(3/2)))*(μ+1)*((2*a^2)/((R+x)^2+y^2)-(3*a^4)/((R+x)^2+y^2)^2))/(2*E)-(sin(2*acos((R+x)/((R+x)^2+y^2)^(1/2))*sgn(y))*cos(2*t-2*acos((R+x)/((R+x)^2+y^2)^(1/2))*sgn(y))*(S1-S2)*(4*acos((R+x)/((R+x)^2+y^2)^(1/2))*dirac(y)+(2*y*sgn(y)*(R+x))/((1-(R+x)^2/((R+x)^2+y^2))^(1/2)*((R+x)^2+y^2)^(3/2)))*(μ+1)*((2*a^2)/((R+x)^2+y^2)-(3*a^4)/((R+x)^2+y^2)^2))/(2*E)
虽然结合附图对发明的具体实施方式进行了详细地描述，但不应理解为对本专利的保护范围的限定。在权利要求书所描述的范围内，本领域技术人员不经创造性劳动即可做出的各种修改和变形仍属本专利的保护范围。
Claims (5)
1.一种基于五栅式应变花消除残余应力测试钻孔偏心误差的方法，其特征在于，包括：
S1、根据钻孔应变法，设计五栅式残余应力测试应变花；
S2、基于线弹性力学叠加原理，推导得到五栅式应变花测试的应变求解应力状态与钻孔偏心量的非线性方程组；
S3、基于牛顿迭代法求解非线性方程组，计算得到应力状态和钻孔偏心量。
2.根据权利要求1所述的基于五栅式应变花消除残余应力测试钻孔偏心误差的方法，其特征在于，所述S1根据钻孔应变法，设计五栅式残余应力测试应变花，包括：
在钻孔外围圆环设置5个敏感栅，将5个敏感栅按顺时针编号为1#、2#、3#、4#及5#，且5个敏感栅分别位于0°、45°、90°、180°、270°对应的圆周上，P1～P5分别为5个敏感栅的中心，O为钻孔圆心，a为孔径，R代表敏感栅中心P1～P5至钻孔圆心O的距离。
3.根据权利要求1所述的基于五栅式应变花消除残余应力测试钻孔偏心误差的方法，其特征在于，S2中基于叠加原理，得到在双向荷载σx与σy作用下钻孔所释放的应变，包括：
S2.1、设存在均值线弹性材料的薄板，则在均布荷载σx作用下，任意一点P(R,θ)应力状态在极坐标系下为：
其中，R为点P距离钻孔圆心的距离，θ为转角，σ′r为P点钻孔前的径向应力，σ′θ为P点钻孔前的周向应力，τ′rθ为P钻孔前的剪应力；
S2.2、现以O为圆心钻一半径为a的圆孔，并假设开孔前后材料均处于线弹性范围，则钻孔后P点的应力状态为：
其中，σ″r为P点钻孔后的径向应力，σ″θ为P点钻孔后的周向应力，τ″rθ为P点钻孔后剪切应力。
S2.3、在P点，由钻孔而导致的应力变化为：
其中，Δσr为P点由钻孔而导致的径向应力变化量，Δσθ为P点由钻孔而导致的周向应力变化量，Δτrθ为点P由钻孔而导致的剪切应力变化量。
S2.4、设由钻孔而释放的径向应变、环向应变及剪应变分别为
根据胡克定律，开孔引起的应力变化可表示为：
其中，E为弹性模量，μ为泊松系数，G为剪切弹性模量，G＝E/2(1+μ)；
S2.5、将式(4)代入式(3)得：
S2.6、令
S2.7、根据叠加原理求出P(R,θ)点在σy作用下的释放应变：
S2.8、将式(6)与式(7)相加，得到在双向荷载σx与σy作用下钻孔所释放的应变：
其中，εr为钻孔释放的径向应变，εθ为钻孔释放的周向应变，γrθ为钻孔释放的剪切应变。
4.根据权利要求1所述的基于五栅式应变花消除残余应力测试钻孔偏心误差的方法，其特征在于，所述S3中基于牛顿迭代法，并根据测试应变，求解应力状态和偏心量，包括：
S3.1、根据设计的五栅式残余应力测试应变花构建坐标系OXY，其中O为未偏心时的孔心，Y轴沿1#敏感栅布置，X轴与3#敏感栅重叠；O离各敏感栅中心P1～P5的距离均为R；O'为钻孔出现偏心时的孔心，设O'的坐标为(x,y)；定义O'离i号敏感栅中心Pi的距离为Ri，则Ri可表示为：
S3.2、定义向量
与向量
的夹角为βi，并定义βi以
以顺时针转向
为正，反之为负，则βi表示为：
其中，sgn(n)表示符号函数，根据n的符号取值：
θi表示从σy应力轴到
向量的转角：
S3.3、将式(12)代入式(8)得到钻孔偏心情况下各应变片中心Pi的应变状态：
其中，
分别是以O'为极心的极坐标系下Pi点的径向应变、轴向应变与剪切应变；
S3.4、根据应力分量的坐标转换，
表示为
及
的形式；
其中，σr为径向应力，σθ为周向应力，τrθ为剪切应力；
根据胡克定律，在直角坐标系与极坐标下的应变分量与应力分量有关系式：
其中，τxy为剪切应力，γxy为剪切应变；
将式(15)代入式(14)，联合求解得到以εr、εθ及γrθ表示的εx形式：
S3.5、改写式(16)得到与敏感栅读数对应的应变：
S3.6、假设i号敏感栅实测的应变为εi，则
与εi两者的差值就可表示为σx、σy、θ、x及y的函数，定义
代入式(17)，得：
S3.7、将式(18)采用链式法则展开，根据牛顿迭代法求解未知量；为求解式(18)所示方程组，定义F(X)＝[f1 f2 f3 f4 f5]T，X＝[σx σy θ x y]T，F(X)对各未知数的偏导数表示为：
原非线性方程组F(X)＝0，根据牛顿迭代法，通过迭代求得近似解：
X(k+1)＝X(k)-[F′(X(k))]-1F(X(k)) (20)
其中，X(k)为方程组的k组近似值，
当k＝0时，X(0)表示迭代初始值，若假设
则得到迭代初始值：
S3.8、给定精度水平ε和最大迭代次数N，对于k＝0,1,2,…,N，按式(20)求解钻孔偏心下残余应力。
5.根据权利要求4所述的基于五栅式应变花消除残余应力测试钻孔偏心误差的方法，其特征在于，所述S3.8中给定精度水平ε和最大迭代次数N，对于k＝0,1,2,…,N，按式(20)求解钻孔偏心下残余应力，包括：
S3.8.1、将X(k)代入式(18)、式(19)分别求得F(X(k))、F′(X(k))；
S3.8.2、将X(k)、F(X(k))及F′(X(k))代入式(20)求解X(k+1)；
S3.8.3、判断
是否小于精度水平ε，若小于，X(k)≈X*，X*为真实解，迭代终止；
否则k＝k+1，转入S3.8.1、进行下一次迭代，直至k＝N或计算结果满足精度要求。
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