Vibration Suppression of MR Sandwich Beams Based On Fuzzy Logic

Abstract

In this paper, the vibration suppression capabilities of magnetorheological (MR) layer in smart beams is investigated. A three-layered beam including MR elastomer layer sandwiched between two elastic layers is considered. By assuming the properties of MR layer in the pre-yield region as viscoelastic materials behavior, the governing equations of motion as well as the corresponding boundary conditions are derived using Hamilton’s principle. Due to field-dependent shear modulus of MR layer, the stiffness and damping properties of the smart beam can be changed by the application of magnetic field. This feature is utilized to suppress the unwanted vibration of the system. The appropriate magnetic field applied over the beam is chosen through a fuzzy controller for improving the transient response. The designed fuzzy controller uses the modal displacement and modal velocity of the beam as its inputs. Free and forced vibration of smart sandwich beam is investigated using numerical simulations. The results show that the magnetorheological layer along with the designed fuzzy controller can be effectively used to suppress the unwanted vibration of the system. The qualitative and quantitative knowledge resulting from this research is expected to enable the analysis, design and synthesis of smart beams for improving the dynamic performance of smart engineering structures.

Appendix

In this section, we present the free response of a sandwiched beam with arbitrary boundary conditions. The characteristic Eq. (24.26) is a sixth-order polynomial which can be rewritten as

$$ {\delta}^3+{G}_0\left(I-1\right){\delta}^2-{\omega}_n^2\delta -{G}_0I{\omega}_n^2=0 $$

(24.32)

where

$$ \delta ={\lambda}^2 $$

(24.33)

The roots of the Eq. (24.32) is described in [25]. The general solution of the system is expressed as

$$ \left\{\begin{array}{c}\hfill w\left(x,t\right)={e}^{i{\omega}_nt}\sum_{i=1}^6{c}_i{\mathrm{e}}^{\lambda_ix}\kern0.75em \hfill \\ {}\hfill u\left(x,t\right)={e}^{i{\omega}_nt}\sum_{i=1}^6{R}_i{c}_i{\mathrm{e}}^{\lambda_ix}\hfill \end{array}\right. $$

(24.34)

where for i = 1,2,···,6, λ _i are the roots of the characteristic equation. Substituting Eq. (24.34) into the boundary conditions Eqs. (24.17), (24.18), and (24.19) yields a matrix equation

$$ \sum_{j=1}^6{a}_{ij}{c}_j=0\kern3em i=1,\dots, 6 $$

(24.35)

where

$$ \begin{array}{l}{a}_{1j}=1\kern0.5em ,\kern0.75em {a}_{2j}={\lambda}_j\kern0.5em ,\kern0.5em {a}_{3j}={R}_j\kern0.5em ,\kern0.75em {a}_{4j}={\lambda}_j^2{e}^{\lambda_j},\kern0.5em \\ {}{a}_{5j}={R}_j{\lambda}_j{e}^{\lambda_j}\kern0.5em ,\kern1em {a}_{6j}=G\left({R}_j+{\lambda}_j\right)-{\lambda}_j^3\end{array} $$

(24.36)

for j = 1,…,6, and R _n is given in Eq. (24.25). The natural frequencies ω _n and the solutions of the characteristic equation Eq. (24.26) can be obtained by equating the coefficient matrix zero in Eq. (24.35).