解析的

Based on the Betti-Rayleigh dynamic reciprocal theorem, the reciprocal property of the Green function is demonstrated; and the analytical solution of soil vibration subjected to moving loads is deduced. By means of double Fourier integral transform, the problem is firstly transformed into wave number-frequency domain, which makes it very convenient to derive the solution and furthermore to analyze the frequency characteristics of soil vibration.

基于Betti-Rayleigh动力互易定理，论证了地基土Green函数的互易性，推导了移动荷载引起的地基土振动解析解，进而推证出移动常力荷载和移动简谐荷载作用下地基于表面振动在波数－频率域内和空间－时间域内的解析表达式，并采用Matlab对其进行数值计算。

Use of analytical solution for predicting the ground settlement induced by tunneling in free soil and Plasticity Yield Criterion get the calculation process of analytical solution for predicting the plastic zone induced by tunneling in free soil.

利用预测自由地层中隧道开挖引起土体沉降的解析解，引入理想塑性屈服准则，得到了自由地层中隧道开挖产生塑性区范围的解析解的计算过程。

Based on the method that a rectangular function can be expanded into a sum of complex Gaussian functions with finite numbers, the propagation characteristics of a Gaussian beam passing through apertured spatial filter systems are analysed and the corresponding analytical formulae are obtained respectively.

根据矩形函数可以展开为有限复高斯函数之和的方法，从柯林斯衍射积分公式出发，对高斯光束通过受限空间滤波器传输特性进行分析，得到了相应的近似解析式。利用Matlab对其解析式进行数值模拟，得出一组直观的结果。

According to the relationship between hyperbola and its normal, the analytic equation is discussed.

根据双曲线与其法线的几何特征关系建立了求解法线的解析方程，分析了法线问题解析求解的繁杂性。

Three elements should be calculated in first-order circuit, which can synthesise analytical solution. Second-order series RLC and parallel RLC both can be calculated by the general function, which uses the Matlab to calculate differential equations. Generally, second-order and high-order circuits use Laplace transformation to formulate circuit equation, and get analytical solution by using Laplace inverse transform.

一阶电路先计算3要素，后合成解析结果；RLC串联和并联的二阶电路采用自编的通用函数计算，自编函数采用了Matlab求解微分方程的符号运算方法；一般的二阶电路和高阶电路采用拉氏变换列写电路方程，再用拉氏反变换得到解析结果。

Firstly, the function sinx that appears in the simple pendulum equation is replaced with a polynomial of degree 3 using Maclaurin series expansion and Chebyshev polynomial approximation. Subsequently, the resulting equation which is a Duffing type equation is approximately solved with combing Newton's method and the harmonic balance method.

首先，利用Maclaurin展开和Chebyshev多项式加速技术将单摆振动方程中的正弦型恢复力用三次多项式近似代替，得到一个Duffing型方程；然后，将牛顿法与谐波平衡法结合起来建立Duffing方程的解析逼近周期及周期解，从而给出单摆振动的解析逼近解。

This method is helpful for us to find the way of the analytic solution of scalar field equation.

他们的解析解法对我们寻求标量场方程的解析解有着重要的启发意义。

This paper presents analytical solutions of internal forces of a solid three hinged arch subjected to complex loadings by a singular function method.

用奇异函数法导出了实体三铰拱在复杂载荷作用下内力的解析解，与传统的数解法相比，由于用统一的解析式表达内力，便于编程计算，也便于根据已知外载情况合理地设计拱轴

The first-order of the modal flexibility for a real symmetric undamped structure with distinct eigenvalues was derived based on an efficient algebraic method. The formulae are simple and compact, and it is easy to program. Moreover, it only requires one eigenvector to be known to calculate the sensitivity of modal flexibility.

代数方法解析推导出无阻尼线性系统模态柔度一阶灵敏度的解析表达式，该表达式简洁紧凑，便于编程，而且在计算阶模态柔度的灵敏度时，只需一阶模态信息，具有显著的优点。

In accordance with the functions of analytical solution and numerical solution for solving algebraical equation group,a design approach was introduced to conduct the pole placement of PI and PID controllers for first and second order controlled models using the symbolic operation.

利用符号运算，针对代数方程组求解的解析解功能和数值解功能，提出了1种对1阶被控对象和2阶被控对象模型进行PI控制器和PID控制器极点配置的设计方法，给出了相应控制器结构参数的解析解，并在算例中进行了数值仿真验证。

Breath, muscle contraction of the buttocks; arch body, as far as possible to hold his head, right leg straight towards the ceiling (peg-leg knee in order to avoid muscle tension).

呼气，收缩臀部肌肉；拱起身体，尽量抬起头来，右腿伸直朝向天花板（膝微屈，以避免肌肉紧张）。

The cost of moving grain food products was unchanged from May, but year over year are up 8%.

粮食产品的运输费用与5月份相比没有变化，但却比去年同期高8％。