﻿ 考虑基础柔性的隔振系统功率流特性分析
 舰船科学技术  2016, Vol. 38 Issue (6): 70-74 PDF

Study on power flow characteristics of isolation system based on flexible foundation
HUANG Wu-de, CHE Chi-dong, CHEN Guang-ye
School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiaotong University, Shanghai 200240, China
Abstract: According to the installation standards and vibration isolation requirements of ship equipments, a mathematical model of vibration isolation system consisting of equipment, isolator and flexible foundation is developed and solved, and the response and power flow formula of the system under harmonic excitation are worked out. The effects that equipment mass, isolator stiffness and damping as well as flexible foundation stiffness and damping have on the performance of the isolation system is discussed. As theoretical results show, it enhances the performance of the system to properly increase equipment mass, isolator damping or flexible foundation damping; larger isolator stiffness is not beneficial to vibration isolation; the effect of flexible foundation stiffness is related to excitation frequency. General principles of the design of vibration isolation based on flexible foundation is put forward.
Key words: vibration isolation     flexible foundation     power flow
0 引言

1 柔性圆板基础上隔振系统的数学模型 1.1 模型描述

 图 1 机械设备安装示意图 Fig. 1 Sketch map of mechanical device installation

 图 2 柔性基础隔振模型 Fig. 2 Model of vibration isolation system with flexible foundation

 $\frac{{E{h^2}}}{{12\rho (1-{\mu ^2})}}{\nabla ^4}w + \frac{{{\partial ^2}w}}{{\partial {t^2}}} = 0 \text{。}$ (1)

 $M\ddot x{\rm{ + }}C(\dot x-\dot w(a,t)) + K(x-w(a,t)) = {F_a} \cdot {e^{j\omega t}} \text{。}$ (2)

 $f = {f_a}{e^{j\omega t}} = C(\dot x-\dot w(a,t)) + K(x-w(a,t)) \text{。}$ (3)
1.2 模型求解

 ${\nabla ^4}{w_a} = {k^4}{w_a} \text{，}$ (4)

 $({\nabla ^2} + {k^2})({\nabla ^2}-{k^2}){w_a} = 0 \text{。}$ (5)

 $({\nabla ^2} + {k^2}){w_a}^{\rm I} = 0 \text{，}$ (6)
 $({\nabla ^2}-{k^2}){w_a}^{{\rm I}{\rm I}} = 0 \text{。}$ (7)

 $({{{d^2}} \over {d{r^2}}} + {1 \over r}{d \over {dr}} + {k^2}){w_a}^{\rm I} = 0 \text{，}$ (8)
 $({{{d^2}} \over {d{r^2}}} + {1 \over r}{d \over {dr}}-{k^2}){w_a}^{{\rm I}{\rm I}} = 0 \text{。}$ (9)

 ${w_a}^{\rm I} = A{J_0}(kr) + B{N_0}(kr) \text{。}$ (10)

 ${w_a}^{ I} = G{I_0}(kr) + H{K_0}(kr) \text{。}$ (11)

wa 可表示为：

 ${w_a}^{} = A{J_0}(kr) + B{N_0}(kr) + G{I_0}(kr) + H{K_0}(kr) \text{。}$ (12)

 ${w_a}(b) = 0 \text{，}$ (13)
 ${M_r}\left| {_{r = b}} \right. =-D({{{d^2}{w_a}} \over {d{r^2}}} + \mu {1 \over r}{{d{w_a}} \over {dr}})\left| {_{r = b}} \right. = 0 \text{。}$ (14)

 ${{\partial {w_a}} \over {\partial r}}\left| {_{r = a}} \right. = 0 \text{，}$ (15)
 ${Q_r} =-D\frac{d}{{dr}}({\nabla ^2}{w_a}) =-{{{f_a}} \mathord{\left/ {\vphantom {{{f_a}} {2{\rm{\pi }}a}}} \right. } {2{\rm{\pi }}a}} \text{。}$ (16)

 $A{J_0}(kb) + B{N_0}(kb) + G{I_0}(kb) + H{K_0}(kb) = 0 \text{，}$ (17)
 $\begin{array}{l} A\left[{-{k^2}{J_0}(kb)-k\displaystyle\frac{{\mu-1}}{b}{J_1}(kb)} \right]+\\ [8pt] B\left[{-{k^2}{N_0}(kb)-k\displaystyle\frac{{\mu-1}}{b}{N_1}(kb)} \right]+\\ [8pt] G\left[{{k^2}{I_0}(kb) + k \displaystyle\frac{{\mu-1}}{b}{I_1}(kb)} \right]+\\ [8pt] H\left[{{k^2}{K_0}(kb)-k\displaystyle\frac{{\mu-1}}{b}{K_1}(kb)} \right] = 0 \text{，} \end{array}$ (18)
 \begin{aligned} A\left[{-k{J_1}(ka)} \right] + & B\left[{-k{N_1}(ka)} \right] + G\left[{k{I_1}(ka)} \right] + \\ & H\left[{-k{K_1}(ka)} \right] \!\!=\!\! 0 \text{。} \end{aligned} (19)
 \begin{aligned} A\left[{{k^3}{J_1}(ka)} \right] + & B\left[{{k^3}{N_1}(ka)} \right] + G\left[{{k^3}{I_1}(ka)} \right] + \\ & H\left[{-{k^3}{K_1}(ka)} \right] = {f \mathord{\left/ {\vphantom {f {2\pi aD}}} \right. } {2\pi aD}} \text{。} \end{aligned} (20)

 $Z = {{{f_a}} \over {{w_a}(a)}} \text{。}$ (21)

 ${x_a} = {{{F_a}} \over {{Z_x}}} \text{，}$ ${Z_x} =-M{\omega ^2} + Z{{K + jC\omega } \over {Z + K + jC\omega }} \text{，}$ (22)

 ${Z_x} =-M{\omega ^2} + Z{{K + jC\omega } \over {Z + K + jC\omega }} \text{。}$ (24)

2 功率流分析

2.1 传递到基础板的功率流

 ${P_i} = f \cdot \dot w(a,t) \text{，}$ (25)

 ${\bar P_i} = {1 \over T}\int_0^T {{P_i}} {\text{d}}t \text{。}$ (26)

2.2 基础板平均总动能

 $d{E_k} = {1 \over T}\int_0^T {{1 \over 2}} (2\pi rdr \cdot h\rho ){({{dw(r,t)} \over {{\text{d}}t}})^2}{\text{d}}t \text{，}$ (27)

 ${E_k} = \int_a^b {{\text{d}}{E_k}} \text{。}$ (28)

3 频谱分析

3.1 设备质量的影响

 图 3 不同设备质量下的功率流曲线 Fig. 3 Power flow curves with different equipment mass
3.2 减振器刚度的影响

 图 4 不同隔振器刚度下的功率流曲线 Fig. 4 Power flow curves with different isolator stiffness
3.3 减振器阻尼的影响

 图 5 不同隔振器阻尼下的功率流曲线 Fig. 5 Power flow curves with different isolator damping
3.4 基础板刚度的影响

 图 6 不同基础刚度下的功率流曲线 Fig. 6 Power flow curves with different foundation stiffness
3.5 基础板阻尼的影响

 图 7 不同基础阻尼下的功率流曲线 Fig. 7 Power flow curves with different foundation damping
4 结语

1）机械设备的质量越大，传递到柔性基础的能量越小，对系统隔振越有利；

2）隔振器刚度越大，传递到柔性基础的能量越大，对隔振设计不利，应在保证静力要求的前提下合理降低隔振器的刚度；

3）隔振器阻尼越大，传递到柔性基础的能量越小，对系统隔振越有利，且隔振器阻尼主要在系统固有频率附近起作用，能有效抑制共振处峰值；

4）在大部分频段，基础板刚度越大，传递到柔性基础上的能量越小，适当增加基础板刚度有利于隔振设计，但并非在所有频段上都如此，因此在隔振设计时，应充分考虑外激励频率与基础板柔性的影响；

5）基础板阻尼越大，其总振动能量越小，越有利于隔振设计。

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