变截面阻抗桩纵向振动问题积分变换解

利用拉普拉斯（Laplace)变换，将变截面阻抗桩振动的时域定解问题转换到频域的形式，求得了关于桩顶位移响应及速度响应的传递函数，然后利用求留数的方法求得了桩土系统的脉冲响应函数，利用脉冲响应函数求得了多种不同激振波形条件下（稳态正弦激励，瞬态半正弦激振及叠加有高频干扰波的瞬态半正弦激振等）的桩顶速度响应的表达式，并对桩土系统的频域特性作了深入的研究，得到了许多重要的结论。此外还将稳态正弦激振、态半正弦激振条件下的解与有关文献所采用分离变量法得到的解进行了对比，为两者的正确性提供了重要的证明。

A STUDY ON VERTICAL VIBRATION OF PILE WITH INHOMOGENEOUS CROSS SECTION ACOUSTIC IMPEDANCE WITH LAPLACE TRANSFORM

In this paper, a mathematical model of pile-soil system vibration was firstly established under any vertical excitation, which consists of three pile-segments with different cross-section acoustic impedance embedded in a homogeneous soil. The model was then turn from time-domain into frequency-domain with Laplace transform, after that a relevant analytical solution to the model was presented. According to the solution, impulse response function and frequency response function as well as transmit function of the pile-soil system about displacement or velocity response of the pile top were developed. With a convolution of the impulse response function and any kind of excitation, the corresponding velocity response of pile top can be easily derived, which just is the response (usually in a form of curve) that may be used to analyze in pile integrity testing. As a special example, when the pile top was imposed with stable sine excitation, two kinds of solutions to the vertical vibration obtained respectively with Laplace transform (in this paper) or with separation of variables (in Ref.6]) were compared and proved to be identical, based on the aforementioned solution, some diagrams were prepared to illustrate influences of damping and rigidity of the soil surrounding the pile and rigidity of soil under the pile tip as well as ratios of cross section acoustic impedance of two sides of interface between the pile segments on the velocity response of the pile top. In addition, influences of frequency and amplitude of highs superimposed on the low frequency excitation were also discussed. It can be shown that there no difference between the influence of the high frequency excitation and that of low one at the condition of ideal elastic pile. On the other hand, the influences of the highs superimposed on the low frequency excitation on the velocity response curves of the pile top only act at the moment of the excitation and arriving of reflected wave at the pile top by the interfaces or the pile tip so that the influence on analysis of pile integrity may be neglected.