Hybrid optimization of a vibration isolation system considering layout of structure and locations of components

Abstract

A vibration isolation system is designed using novel hybrid optimization techniques, where locations of machines, locations of isolators and layout of supporting structure are all taken as design variables. Instead of conventional parametric optimization model, the 0-1 programming model is established to optimize the locations of machines and isolators so that the time-consuming remeshing procedure and the complicated sensitivity analysis with respect to position parameters can be circumvented. The 0-1 sequence for position design variables is treated as binary bits so as to reduce the actual number of design variables to a great extent. This way the 0-1 programming can be solved in a quite efficient manner using a special version of genetic algorithm(GA) that has been published by the authors. The layout of supporting structure is optimized using SIMP based topology optimization method, where the fictitious elemental densities are taken as design variables ranging from 0 to 1. Influence of different design variables is firstly investigated by numerical examples. Then a hybrid multilevel optimization method is proposed and implemented to simultaneously take all design variables into account.

Appendix: Sensitivity information

In this section we give the sensitivity information which is needed when solving (30). Keep in mind that only ρ _i is taken as design variable in (30).

1.1 A.1 The sensitivity of U w.r.t. $\bar {\boldsymbol \rho }$

According to the motion equation (11), U can be expressed as

$$ \boldsymbol{U} = \boldsymbol{S}^{-1}\boldsymbol{F} $$

(38)

Differentiating the above equation w.r.t. ρ _i gives

$$ \frac{\partial \boldsymbol{U}}{\partial \bar\rho_{i}} = -\boldsymbol{S}^{-1}\frac{\partial \boldsymbol{S}}{\partial \bar\rho_{i}}\boldsymbol{U} $$

(39)

and $\frac {\partial {\boldsymbol {S}}}{\partial {\rho _{i}}}$ is given by:

$$ \frac{\partial \boldsymbol{S}}{\partial \bar\rho_{i}} = 3{\bar\rho_{i}^{2}}\left( -\omega^{2}\boldsymbol{M}_{i}^{\text{sh,e}}+ \boldsymbol{K}_{i}^{\text{sh,e}} \right) $$

(40)

1.2 A.2 The sensitivity of h w.r.t. $\bar {\boldsymbol \rho }$

Differentiate h = U ^⊤ F F ^⊤ U w.r.t. $\bar \rho _{i}$ gives

$$ \frac{\partial {h}}{\partial \bar\rho_{i}} = 2\boldsymbol{U}^{\top}\boldsymbol{F}\boldsymbol{F}^{\top}\frac{\partial \boldsymbol{U}}{\partial \bar\rho_{i}} $$

(41)

Substitute (38–40) into (41) gives

$$ \begin{array}{ll} \frac{\partial {h}}{\partial \bar\rho_{i}} &= -2\boldsymbol{U}^{\top}\boldsymbol{F}\boldsymbol{U}^{\top} \frac{\partial \boldsymbol{S}}{\partial \bar\rho_{i}}\boldsymbol{U} \\ &= -6{x}_{i}^{2} \left[ \boldsymbol{U}^{\top}\boldsymbol{F}\boldsymbol{U}^{\top} \left( -\omega^{2}\boldsymbol{M}_{i}^{\text{sh,e}}+\boldsymbol{K}_{i}^{\text{sh,e}}\right) \boldsymbol{U} \right] \end{array} $$

(42)

Note that (42) can be effectively evaluated in the elemental level. Now $\frac {\partial {h_{\text {E}}}}{\partial {\bar \rho _{i}}}$ and $\frac {h_{0}}{\bar \rho _{i}}$ can be easily given:

$$\begin{array}{@{}rcl@{}} \frac{\partial h_{\text{E}}}{\partial \bar\rho_{i}} &=& -6{x}_{i}^{2} \left[ \boldsymbol{U}^{\top}\boldsymbol{F}\boldsymbol{U}^{\top} \left( -\omega_{\text{E}}^{2}\boldsymbol{M}_{i}^{\text{sh,e}}+\boldsymbol{K}_{i}^{\text{sh,e}}\right) \boldsymbol{U} \right] \\ \frac{\partial h_{0}}{\partial \bar\rho_{i}} &=& -6{x}_{i}^{2} \left[ \boldsymbol{U}^{\top}\boldsymbol{F}^{\text{sta}}\boldsymbol{U}^{\top} \boldsymbol{K}_{i}^{\text{sh,e}} \boldsymbol{U} \right] \end{array} $$

(43)

1.3 A.3 The sensitivity of J _E w.r.t. $\bar {\boldsymbol \rho }$

Differentiating $J_{\text {E}} = \boldsymbol {U}^{\top }\boldsymbol {G}_{\text {I}}^{\top }\boldsymbol {G}_{\text {I}}\boldsymbol {U}$ we have:

$$ \frac{\partial J_{\text{E}}}{\partial \bar{\rho}_{i}} = -2\boldsymbol{U}^{\top}\boldsymbol{G}_{\text{I}}^{\top}\boldsymbol{G}_{\text{I}}\boldsymbol{S}^{-1}\frac{\partial \boldsymbol{S}}{\partial \bar{\rho}_{i}}\boldsymbol{U} $$

(44)

1.4 A.4 Density filter and projection operator

Now we consider how to transform $\frac {\partial J_{\text {E}}}{\partial \bar {\rho }_{i}}$, $\frac {\partial h_{\text {E}}}{\partial \bar {\rho }_{i}}$, $\frac {\partial h_{0}}{\partial \bar {\rho }_{i}}$ into $\frac {J_{\text {E}}}{\rho _{i}}$, $\frac {h_{\text {E}}}{\rho _{i}}$, $\frac {h_{0}}{\rho _{i}}$. By using the chain rule, according to (31) and (33) we have:

$$ \frac{\partial h_{\text{E}}}{\partial {\rho}_{i}} = \sum\limits_{j=1}^{n_{\text{sh}}}{\frac{\partial \tilde\rho_{j}}{\partial \rho_{i}}\frac{\text{d} \bar{\rho_{j}}}{\text{d} \tilde{\rho_{j}}}\frac{\partial h_{\text{E}}}{\partial \bar\rho_{j}}} $$

(45)

In matrix form we have:

$$ \frac{\text{d} h_{\text{E}}}{\text{d} \boldsymbol\rho} = \boldsymbol{H}^{\top} \left[\begin{array}{lll} \frac{\partial \bar{\rho}_{1}}{\partial \tilde{\rho}_{1}} & & \\ & {\ddots} & \\ & & \frac{\partial \bar{\rho}_{n_{\text{sh}}}}{\partial \tilde{\rho}_{n_{\text{sh}}}} \end{array}\right] \left[\begin{array}{c} \frac{\partial h_{\text{E}}}{\partial \bar{\rho}_{1}} \\ {\vdots} \\ \frac{\partial h_{\text{E}}}{\partial \bar{\rho}_{n_{\text{sh}}}} \end{array}\right] $$

(46)

By replacing h _E with h ₀ or J _E, similar equations can be obtained and are omited here.