Forced Vibration Analysis of Euler-Bernoulli Double-Beam Systems by Means of Green’s Functions
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摘要:
双曲梁系统通常出现在许多工程领域。与双直梁系统相比，该系统在噪声和振动控制问题上的效率更高。该文采用经典的Euler-Bernoulli曲梁模型来模拟双曲梁系统，通过Green函数和Laplace变换方法得到双曲梁系统稳态受迫振动的闭合形式解，该解可用于任何边界条件。在数值部分，通过与参考文献中的一些结果进行比较来验证本方案的解。讨论了一些重要的几何和物理参数对振动响应的影响以及弹性层刚度与双曲梁系统之间的相互作用。结果表明，梁的半径趋于无穷大时，双曲梁系统退化为双直梁系统，此外，双曲梁系统也可以简化为一个直梁和一个曲梁的组合形式。
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关键词:
- Green函数 /
- Euler-Bernoulli梁模型 /
- 双曲梁系统 /
- Winkler型弹性层 /
- Laplace变换
Abstract:Double-curved-beam (DCB) systems are usually seen in many engineering fields. Compared to straight double-beam systems, DCB systems are more efficient in noise and vibration control problems. To obtain closed-form solutions of steady-state forced vibrations of DCB systems, the classical Euler-Bernoulli curved beam (ECB) model was employed to model vibration equations for the DCB systems. Green’s functions and the Laplace transform methods were used to get the closed-form solutions to the vibration equations for the DCB systems. These solutions apply to arbitrary boundary conditions. Numerical tests were conducted to verify the present solutions with related results from previous literatures. Effects of some important geometric and physical parameters on vibration responses and the interaction between the elastic layer stiffness and the DCB system, were discussed. The results show that, the DCB system will degenerate to a straight double-beam system when the 2 radii approach infinity, moreover, the DCB system can be simplified as one comprising a straight beam and a curved beam.
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图 1 任意边界条件下的双曲梁系统在
$x = {x_0}$ 处受到外激励作用Figure 1. The DCB system with arbitrary boundary conditions subjected to a load
图 2 退化简支曲梁在外部动力作用下的无量纲挠度
Figure 2. Dimensionless displacement
$g( \xi$ , 0.5)(n=1, m=1, 2) of the degenerated ECB as a function of dimensionless coordinate$\xi $ 
图 3 退化简支双曲梁在外部动力作用下的无量纲挠度
Figure 3. Dimensionless displacement
$g( \xi$ , 0.5)(n=1, m=1, 2) of the degenerated DCB as a function of dimensionless coordinate$\xi $ 
图 4 相同半径不同弹性层弹性模量下双曲梁的无量纲化位移
$g_{nm}( \xi$ , 0.5)(n=1, m=1, 2)$(R = 100$ ,$\varOmega ' = 0.5)$ Figure 4. Dimensionless Green’s function
$g_{nm}( \xi$ , 0.5)(n=1, m=1, 2) as a function of dimensionless coordinate$x$ for different values of stiffness modulus K$(R = 100$ ,$\varOmega ' = 0.5)$ 
图 5 不同半径相同弹性层弹性模量和相同外激励下双曲梁的无量纲化位移
$g_{nm}( \xi$ , 0.5)(n=1, m=1, 2)$({K_0} = 0.1$ ,$\varOmega ' = {\text{0}}.5)$ Figure 5. Dimensionless Green’s function
$g_{nm}( \xi$ , 0.5)(n=1, m=1, 2) as a function of dimensionless coordinate$x$ for different radius values$({K_0} = 0.1$ ,$\varOmega ' = {\text{0}}.5)$ 
图 6 相同半径相同弹性层弹性模量不同外激励下双曲梁的无量纲化位移
$({K_0} = 0.1$ ,$\varOmega ' = 5$ ,$10)$ Figure 6. Dimensionless Green’s function
$g_{nm}( \xi$ , 0.5)(n=1, m=1, 2) as a function of dimensionless coordinate$x$ for different external frequency values$({K_0} = 0.1$ ,$\varOmega ' = 5$ ,$10)$ 表 1 双曲梁的不同边界条件
Table 1. Boundary conditions (BCs) of the DCB
BC beam ECB pinned upper beam ${W_1}\left| {_{x = 0,L} } \right. = 0,{\text{ } }{\lambda _{15} }W_1^{\left( 4 \right)} + {\lambda _{ {\text{16} } } }W_1^{\prime \prime }\left| {_{x = 0,L} } \right.{\text{ = 0, } }{\lambda _{ {\text{11} } } }W_1^{\left( 5 \right)} + {\lambda _{ {\text{12} } } }W_1^{\prime \prime \prime } + {\lambda _{ {\text{13} } } }W_1^\prime + {\lambda _{ {\text{14} } } }W_2^\prime \left| {_{x = 0,L} } \right. = 0$ bottom beam ${W_{\text{2} } }\left| {_{x = 0,L} } \right. = 0,{\text{ } }{\lambda _{ {\text{25} } } }W_{\text{2} } ^{\left( 4 \right)} + {\lambda _{ {\text{26} } } }W_{\text{2} }^{\prime \prime }\left| {_{x = 0,L} } \right.{\text{ = 0, } }{\lambda _{ {\text{21} } } }W_{\text{2} }^{\left( 5 \right)} + {\lambda _{ {\text{22} } } }W_{\text{2} }^{\prime \prime \prime } + {\lambda _{ {\text{23} } } }W_{\text{2} }^\prime + {\lambda _{ {\text{24} } } }W_{\text{1} } ^\prime \left| {_{x = 0,L} } \right. = 0$ fixed upper beam ${W_1}\left| {_{x = 0,L} } \right. = 0,{\text{ } }W_1^\prime \left| {_{x = 0,L} } \right.{\text{ = 0, } }{\lambda _{ {\text{11} } } }W_1^{\left( 5 \right)} + {\lambda _{ {\text{12} } } }W_1^{\prime \prime \prime }\left| {_{x = 0,L} } \right. = 0$ bottom beam $W_2\left| {_{x = 0,L} } \right. = 0,{\text{ } }W_2^\prime \left| {_{x = 0,L} } \right.{\text{ = 0, } }{\lambda _{ {\text{21} } } }W_1^{\left( 5 \right)} + {\lambda _{ {\text{22} } } }W_1^{\prime \prime \prime }\left| {_{x = 0,L} } \right. = 0$ free upper beam $W_1^{\prime \text{}\prime }|{}_{x=0,L}=0,\text{ }W_1^{\prime \text{}\prime \text{}\prime }|{}_{x=0,L}\text{=0, }{\lambda }_{11}W_1^{\left( 5 \right)} + {\lambda }_{13}W_1^{\prime } + {\lambda }_{14}W_2^{\prime }|{}_{x=0,L}=0$ bottom beam $W_2^{\prime \text{}\prime }|{}_{x=0,L}=0,\text{ }W_2^{\prime \text{}\prime \text{}\prime }|{}_{x=0,L}\text{=0, }{\lambda }_{21}W_2^{\left( 5 \right)} + {\lambda }_{23}{W}_2^{\prime } + {\lambda }_{24}{W}_1^{\prime }|{}_{x=0,L}=0$ 
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