THE MODE AND SCATTERED ENERGY DISTRIBUTION OF GUIDED WAVES PROPAGATING IN TWO COUPLED PLANE-STRAIN LAYERS WITH RIGID BASE
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摘要: 过渡段动力稳定性问题已成为制约400 km/h及以上高铁路基设计的关键难题, 亟需从波动和能量的角度探究由基础非均匀引发的线路系统动力响应放大机理. 文章将轨下基础简化为上表面自由、底端固定的刚性基弹性层, 将高铁过渡段车致弹性波传播问题提炼为非均匀介质刚性基弹性层中波的散射问题, 建立双介质耦合刚性基弹性层平面应变模型, 优化该类波导结构频散方程在复平面求根方法, 并结合岩土类介质特征展开刚性基弹性层频散分析, 以明确其多模式导波特性及散射能量分配, 最后, 围绕弹性层厚度、刚度比等影响因素开展对比分析. 结果表明: 刚性基弹性层各模式导波均具有截止频率, 弹性层厚度越小, 杨氏模量越大, 各阶导波模式的截止频率越高; 入射波在双介质刚性基弹性层发生散射后, 透射场基阶模式导波会占据主体能量, 随着高阶导波模式被逐一激发, 反射场及透射场高阶模式能量占比会在全频率范围呈现“此消彼长”状态; 交换两侧弹性层材料, 改变弹性层厚度及两弹性层刚度比不会显著改变能量分布规律, 但总体来看, 能量更易集中在较软侧弹性层中, 各模式导波在激发初始频段会更为活跃, 可分配到更多能量.Abstract: The dynamic stability of subgrade in transition zones has become a key problem restricting the design of high-speed railway subgrade with a speed of 400 km/h and above. It urgent to explore the amplification mechanism of system dynamic response caused by non-uniform foundation from the perspective of wave and energy. In this paper, the foundation under track is reduced to a elastic layer which has a free surface and rigid bottom. The problem of vehicle induced elastic wave propagation in the transition zones in high-speed railway is refined into the problem of wave scattering in the inhomogeneous elastic layer with rigid base. A plane-strain model of two medium coupling elastic layers with rigid base is established. Facing with the dispersion equation of elastic layer with rigid base, the paper optimizes the method of finding roots in complex plane. Then, the dispersion analyses of the elastic layers that are assigned with geotechnical medium are carried out, and the corresponding multi-mode guided wave characteristics and the distribution of scattered energy are clarified. Furthermore, in terms of the thickness of elastic layer, stiffness ratio of two elastic layers and so on, comparative analyses are carried out at last. The results indicate that all of the guided wave modes in the elastic layer with rigid base have cut-off frequencies. When the thickness of the elastic layer decreases or the Young's modulus of the medium increases, the cut-off frequency of each order guided wave mode becomes higher. In scattering, the fundamental mode of the transmitted field can occupy the main energy. And as modes are excited one by one, the proportion of energy of higher modes of the reflected field and the transmitted field shows a “trade-off” state in the full frequency range. The energy distribution law will not be significantly changed when the elastic layer materials on both sides are exchanged, or the elastic layers thickness and the stiffness ratio is changed. On the whole, the energy is more easily concentrated in the softer elastic layer, and guided wave mode is more active in the initial frequency band after excitation and distributes more energy.
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图 2 双介质耦合刚性基弹性层平面应变模型
Figure 2. Plane-strain model of two medium coupling elastic layers with rigid base
图 3 刚性基弹性层波导结构平面应变模型
Figure 3. Plane-strain model of waveguide structure of elastic layer with rigid base
图 4 频散方程
$ {\text{Det(}}{k_x}) = 0 $ 求解流程图Figure 4. Flow chart of the solution of dispersion equation
$ {\text{Det(}}{k_x}) = 0 $ 
图 6 水平波数
$ {k_x} $ 计算结果及验证Figure 6. Calculation results and verification of horizontal wave number
$ {k_x} $ 
图 7 钢介质刚性基弹性层频散曲线
Figure 7. Dispersion curve of rigid base elastic layer with steel medium
图 8 刚性基弹性层导波模式相速度
Figure 8. Phase velocity of guided wave modes in rigid base elastic layer
图 9 刚性基弹性层导波模式群速度
Figure 9. Group velocity of guided wave modes in rigid base elastic layer
图 10 垂直界面处位移连续性验证
Figure 10. Verification of displacement continuity at vertical interface
图 11 各场传播模式导波分配总能量占比
Figure 11. Proportion of total energy distributed by propagating guided wave modes in each field
图 12 基阶及高阶传播模式导波分配能量占比
Figure 12. Proportion of energy distribution of the fundamental and higher propagating guided wave modes
图 13 材料交换后各场传播模式导波分配能量占比
Figure 13. Proportion of energy distributed by propagating guided wave modes in each field after exchanging material
图 14 不同弹性层厚度下各场传播模式导波分配能量占比
Figure 14. Proportion of energy distributed by propagating guided wave modes in each field under different elastic layer thicknesses
14 不同弹性层厚度下各场传播模式导波分配能量占比 (续)
14. Proportion of energy distributed by propagating guided wave modes in each field under different elastic layer thicknesses (continued)
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