Active vibration control of flexible structures by optimally placed sensors and actuators

Abstract

The active vibration reduction of plane and stiffened plates was investigated using a genetic algorithm based on finite element modelling to optimise the location of sensors and actuators. The main aspects of this work were:  Development of a finite element model for a plate stiffened by beams with discrete sensors and actuators bonded to its surface.  Development of a finite element program for steel plates with various symmetrical and asymmetrical stiffening and edge conditions.  Development of a genetic algorithm program based on the finite element modelling for the optimisation of the location and number of sensor/actuator pairs and feedback gain.  Determination of optimum locations and feedback gain for collocated piezoelectric sensors and actuators on steel plates with various symmetrical and asymmetrical stiffening and edge conditions.  Development of fitness and objective functions to locate sensors and actuators.  Development of fitness and objective functions to determine the optimal number of sensors and actuators.  Development of a reduced search space technique for symmetrical problems.  Optimisation of vibration reduction control scheme parameters using the genetic algorithm.  Optimisation of the number and location of sensor/actuator pairs and feedback gain to reduce material costs and structural weight and to achieve effective vibration reduction. The modelling was validated by comparison with conventional finite element analysis using ANSYS, and by experiment. The modelling was developed using a quadrilateral isoparametric finite element, based on first order shear deformation theory and Hamilton’s principle, which may be arbitrarily stiffened by beams on its edges. The model can be applied to flat plates with or without stiffening, with discrete piezoelectric sensors and actuators bonded to its surfaces. The finite element modelling was tested for flat and stiffened plates with different boundary conditions and geometries, and the results of the first six natural frequencies were validated with the ANSYS package and experimentally. A genetic algorithm placement strategy is proposed to find the global optimal distribution of two, four, six and ten sensor/actuator pairs and feedback gain based on the minimisation of optimal linear quadratic index as an objective function, and applied to a cantilever plate to attenuate the first six modes of vibration. The configuration of this global optimum was found to be symmetrically distributed about the dynamic axes of symmetry and gave higher vibration attenuation than previously published results with an asymmetrical distribution which was claimed to be optimal. Another genetic algorithm placement strategy is proposed to optimise sensor/actuator locations using new fitness and objective functions based on . This is applied to the same cantilever plate, and was also found to give a symmetrical optimal sensor/actuator configuration. As before, it was found that the optimal transducer locations are distributed with the same axes of symmetry and in agreement with the ANSYS results. A program to simulate the active vibration reduction of stiffened plates with piezoelectric sensors and actuators was written in the ANSYS Parametric Design Language (APDL). This makes use of the finite element capability of ANSYS and incorporates an estimator based on optimal linear quadratic and proportional differential control schemes to investigate the open and closed loop time responses. The complexity of the genetic algorithm problem is represented by the number of finite elements, sensor/actuator pairs and modes required to be suppressed giving a very large search space. In this study, this problem was reduced by the development of a new half and quarter chromosomes technique exploiting the symmetries of the structure. This greatly reduces the number of generations, and hence the computing time, required for the genetic algorithm to converge on the global optimal solution. This could be significant when the technique is applied to large and complex structures. Finally, new fitness and objective functions were proposed to optimise the number of sensor/actuator pairs required for effective active vibration reduction in order to reduce the added cost and weight. The number, location and feedback gain were optimised for the same cantilever plate and it was found that two sensor/actuator pairs in optimal locations could be made to give almost as much vibration reduction as ten pairs.