﻿ 筒式空气弹簧机械阻抗近似解析算法研究
 舰船科学技术  2016, Vol. 38 Issue (5): 44-48 PDF

1. 海军工程大学 振动与噪声研究所, 湖北 武汉 430033 ;
2. 船舶振动噪声重点实验室, 湖北 武汉 430033

Research on the approximate analytic algorithm for mechanical impedance of cylinder type air-spring
LI Peng-hui, SHUAI Chang-geng
1. Institute of Noise and Vibration, Naval University of Engineering, Wuhan 430033, China ;
2. National Key Laboratory on Ship Vibration and Noise, Wuhan 430033, China
Abstract: An idealized cylinder type air-spring mode was created. Based on air wave theory, an approximate analytic algorithm for mechanical impedance of air spring was presented with calculation of air pressure. Finite element method was used to verify the result and it showed that the proposed method is feasible. The effect of the cover plate and the initial pressure to the air spring's mechanical impedance characteristics was done and the performance of isolation was assessed by the force transmissibility. The method provides a new idea for future study on the mechanical impedance characteristics of air spring.
Key words: air spring     air wave     mechanical impedance     approximate analytic algorithm
0 引言

1 数学模型

 图 1 理想化的筒式空气弹簧模型 Fig. 1 Idealized cylinder type air-spring mode
1.1 声压方程的推导

 $\frac{1}{\rho {{c}^{2}}}\frac{{{\partial }^{2}}p}{\partial {{t}^{2}}}+\nabla (-\frac{1}{\rho }(\nabla p-q))=Q。$ (1)

 $\phi =\int{\frac{p}{{{\rho }_{0}}}}\text{d}t。$ (2)

 \begin{align} & \phi ={{e}^{j\omega t}}\sum\limits_{n=0}^{\infty }{\left[ {{A}_{n}}{{J}_{n}}\left( {{k}_{r}}r \right)+{{B}_{n}}{{N}_{n}}\left( {{k}_{r}}r \right) \right]}\times \\ & ({{A}_{1}}\cos h{{k}_{z}}z+{{B}_{1}}\sin h{{k}_{z}}z)\cos \left( n\varphi -{{\varphi }_{n}} \right)。 \\ \end{align} (3)

 $\phi ={{e}^{jwt}}{{J}_{0}}\left( {{k}_{r}}r \right)(A\cos h{{k}_{z}}z+B\sin h{{k}_{z}}z)\text{。}$ (4)

 ${{\dot{V}}_{r}}{{\left( r,z,t \right)}_{r=R}}=-{{\frac{\partial \text{ }\phi }{\partial r}}_{r=R}}=0\text{,}$ (5)

 \begin{align} & \Phi ={{e}^{jwt}}\sum\limits_{m=0}^{\infty }{{{J}_{0}}\left( \frac{{{\tau }_{0m}}}{R}r \right)({{A}_{m}}\cos h\sqrt{{{k}^{2}}-{{(\frac{{{\tau }_{0m}}}{R})}^{2}}}z}+ \\ & {{B}_{m}}\sin h\sqrt{{{k}^{2}}-{{(\frac{{{\tau }_{0m}}}{R})}^{2}}}z)。 \\ \end{align} (6)

 $\phi =(A\cos hkz+B\sin hkz){{e}^{jwt}}。$ (7)

 $P(z,t)=j{{\rho }_{0}}c\frac{{{{\dot{V}}}_{0}}}{\sin hkH}\cos h\left[ k(H-z) \right]{{e}^{jwt}}。$ (8)
1.2 机械阻抗与力传递率的计算

 $\left\{ \begin{matrix} {{Z}_{11}}=j\omega M+{{k}_{1}}/j\omega , \\ {{Z}_{12}}={{k}_{2}}/j\omega , \\ \end{matrix} \right.$ (9)

 $\left\{ \begin{matrix} {{F}_{p}}=j{{e}^{jwt}}{{\rho }_{0}}c{{{\dot{V}}}_{0}}\pi {{R}^{2}}\cos h(kH)/\sin h(kH), \\ F{{'}_{p}}=j{{e}^{jwt}}{{\rho }_{0}}c{{{\dot{V}}}_{0}}\pi {{R}^{2}}/\sin h(kH), \\ \end{matrix} \right.$ (10)

 $\left\{ {\matrix{ {{k_1} = {\rho _0}c\omega \cos h(kH)/\sin h(kH)\pi {R^2},} \cr {{k_2} = {\rho _0}c\omega \pi {R^2}/\sin h(kH).} \cr } } \right.$ (11)

 $\left\{ {\matrix{ {{Z_{11}} = j\omega m - j{\rho _0}c\pi {R^2}\cos h(kH)/\sin h(kH),} \cr {{Z_{12}} = - j{\rho _0}c\pi {R^2}/\sin h(kH).} \cr } } \right.$ (12)

 $\lambda \text{=}\frac{-j{{\rho }_{0}}c\pi {{R}^{2}}/\sin kH}{j\omega m-j{{\rho }_{0}}c\cot (kH)\pi {{R}^{2}}}\text{。}$ (13)
2 仿真验证

 $\left\{ {\matrix{ { - \nabla (a\nabla P) + bP = 0,} \cr {{{\partial P} \over {\partial \vec n}}\left| {\partial \Omega } \right. = f.} \cr } } \right.$ (14)

 图 2 100 Hz 时空气弹簧声压场 Fig. 2 The pressure field of air-spring at 100 Hz

 图 3 机械阻抗计算与仿真对比 Fig. 3 The contrast of mechanical impedance between FEM and analytic algorithm
3 分析与讨论

3.1 上盖板质量的影响

 图 4 不同盖板质量时的输入阻抗 Fig. 4 The mechanical impedance at different cover plate mass

 图 5 不同盖板质量时的力传递率 Fig. 5 The force transmissibility at different cover plate mass
3.2 初始压力的影响

 $P={{\rho }^{n}}\cdot const,$ (15)

 $P={{c}^{2}}\cdot \rho$ (16)

 $\left\{ \begin{array}{*{35}{l}} {{c}_{1}}/{{c}_{0}}={{\left( {{p}_{1}}/{{p}_{0}} \right)}^{\frac{n-1}{2n}}} \\ {{k}_{1}}/{{k}_{0}}={{\left( {{p}_{1}}/{{p}_{0}} \right)}^{-\frac{n-1}{2n}}} \\ \end{array} \right.$ (17)

 图 6 不同初始压力时的机械阻抗 Fig. 6 The mechanical impedance at different initial pressure

 图 7 不同初始压力时的阻抗曲线 Fig. 7 The force transmissibility at different initial pressure
4 结语

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