 (13)
式中 和 。这样就得到 个包含个未知参数的二次非线性代数方程组。数值经验表明，  可以取为等效线性化法或高斯闭合法求解得到的系统响应概率密度函数，即
 (14)
2 算例
考虑如下带位移偶次方项的非线性随机振子：
 (15)
式中 是零均值高斯白噪声，其中 和 是相互独立的建筑工程论文，相关函数为 。该系统的参数值设为  。同时为了验证以上求解过程的有效性，对该系统进行了蒙特卡洛模拟，模拟产生了 个样本来计算系统响应的概率密度函数。
图3.1位移响应的概率密度函数比较
Fig.3.1 Comparison for PDF ofdisplacement
图3.2位移响应的对数概率密度函数比较
Fig.3.2 Comparison for Log(PDF) ofdisplacement
图.3.3速度响应的概率密度函数比较
Fig.3.3 Comparison for PDF of velocity
图3.4速度响应的对数概率密度函数比较
Fig.3.4 Comparison for Log(PDF) ofvelocity
用EPC法且多项式的阶数分别为2(EPC n=2）和6(EPC n=6)时求得的位移概率密度函数值由图3.1所示，同时用EPC法得到的结果与蒙特卡洛模拟（MCS）的结果进行了比较。从图3.1中可以看出，当用EPC法n=6时，两种方法得到的结果符合较好。但是当用EPC法且n=2时，EPC法得到的结果与模拟的结果偏离较大，此时的EPC结果与等效线性化结果一致。为了考察EPC所得位移概率密度函数值在尾部的情况，图3.2中列出了位移概率密度函数的对数值。可以看出用EPC法n=6所得结果比EPC法且n=2所得结果改善很多。图3.3给出了速度的概率密度函数值及其比较。可以看出，当n=6时EPC法得到的结果与模拟的结果同样符合较好，尤其在如图3.4所示的尾部区域论文开题报告。从图中也可以看出位移和速度的概率密度函数均是关于均值不对称的。
3 结论
对于响应均值非零的带位移偶次项和参数激励的非线性随机振子建筑工程论文，给出了EPC法求解Fokker-Planck方程的过程。同时用EPC法得到的结果与蒙特卡洛模拟结果进行了比较。当用EPC法且取二阶多项式时，EPC方法得到的结果与等效线性化法得到的结果一致，但是与模拟的结果有着明显的不同。当用EPC法取六阶多项式时，EPC方法得到的结果与模拟的结果符合较好，尤其是在系统响应概率密度函数的尾部区域。数值结果同时显示，在此情况下，位移和速度的概率密度函数均是关于均值不对称的。这说明EPC法同样适用于响应均值非零的带位移偶次项的非线性随机振子在参数激励下的响应概率密度函数求解问题。
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