METHOD FOR DYNAMIC CHARACTERISTIC ANALYSIS OF BEAMS WITH VARYING CROSS-SECTIONS BY USING PERIDYNAMIC DIFFERENTIAL OPERATOR
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摘要: 变截面梁式构件广泛应用于工程结构中，其动力特性更是结构设计和状态评估中的重要考虑因素之一。基于新兴的近场动力学微分算子(Peridynamic differential operator, PDDO)，尝试提出了一种用于变截面梁动力特性分析的非局部方法。将变截面梁的动力学微分控制方程与边界条件通过PDDO由局部微分形式转化为对应的非局部积分形式，再结合拉格朗日乘数法与变分原理，将非局部积分形式的控制方程与边界条件转化为标准特征值问题表达形式，从而求得自振频率与振型。通过对等截面梁的自由振动分析并与解析解对比，验证了该方法良好的收敛性与准确性。进一步通过求解下边界一次、二次变化的连续变截面梁，证明了该方法对于任意变截面梁自由振动分析的适用性与可靠性。开展含孔变截面梁的自由振动分析，体现了该文的非局部方法在含缺陷构件振动分析和损伤识别问题方面的潜力，可为含缺陷变截面构件的动力分析问题提供新思路。Abstract: Beam-type components with non-uniform cross-sections are widely used in various engineering structures, the dynamic characteristic of which is one of the most important factors in structural design and state evaluation. A non-local numerical model for dynamic characteristic analysis of beams with varying cross-sections is presented by using a peridynamic differential operator (PDDO). The dynamic differential equations and boundary conditions of beams with variable cross-sections are reformulated to a non-local integral form based on PDDO. The natural frequency and vibration mode can be solved by employing the variational analysis and Lagrange multiplier method. A comparison study of beams with constant cross-sections is conducted to validate the accuracy and convergence of this non-local method by comparing the non-local analysis results with analytical solutions in pertinent literatures. The vibrations of two beams with linearly varying and parabolic convex lower surfaces are analyzed, respectively, to verify the applicability and reliability of the method proposed in the free vibration analysis of beams with arbitrarily varying cross-sections. The free vibration of a beam with a hole and parabolic convex lower surface is investigated further to demonstrate the potential of this presented approach in the vibration analysis and damage identification of defective components with non-uniform cross-sections, and which provides a new perspective for the free vibration analysis of defective components with non-uniform cross-sections.
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图 1 任意变截面简支梁
Figure 1. Simply supported beam with arbitrarily and continuously varying cross-section
图 5 各离散间距下前3阶无量纲自振频率的相对误差
Figure 5. Relative error of first 3 non-dimensional frequencies for different dense mesh
图 8 下边界一次、二次变化梁的前3阶振型(Ha/H = Hb/H = 2, y = 0.4H)
Figure 8. First 3 mode shapes in x and y directions at y = 0.4H for the beam with linearly varying and parabolic convex lower surface (Ha/H = Hb/H = 2, y = 0.4H)
图 10 下边界二次变化含孔梁的前4阶轴向应力模态 (r/H = 0.6)
Figure 10. First 4 stress modes in x direction for the beam with a hole and parabolic convex lower surface (r/H = 0.6)
表 1 等截面梁的无量纲自振频率
Table 1. Non-dimensional frequencies of the beam with constant cross-section
阶次 本文解 解析解[10] 相对误差/(%) 1 9.7332 9.6420 0.946 2 37.240 36.893 0.940 3 78.588 77.860 0.935 4 108.77 108.00 0.706 5 129.62 128.42 0.929 6 187.02 185.32 0.919 7 217.26 215.76 0.699 8 248.53 246.26 0.921 
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表 2 下边界一次变化梁的无量纲自振频率
Table 2. Non-dimensional frequencies of the beam with linearly varying lower surface
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表 3 下边界二次变化梁的无量纲自振频率
Table 3. Non-dimensional frequencies of the beam with parabolic convex lower surface
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表 4 下边界二次变化含孔梁的无量纲自振频率
Table 4. Non-dimensional frequencies of the beam with a hole and parabolic convex lower surface
阶次 r/H = 0.4 r/H =0.6 r/H =0.8 1 10.235 10.335 10.153 2 37.035 36.675 34.956 3 79.879 80.177 79.080 4 130.74 124.14 100.50 5 141.13 129.60 108.26 6 197.15 198.78 197.61 7 254.23 233.76 199.19 8 275.65 279.85 285.51
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