This paper analyzed the fundamental limitations of previous work and developed a new method to optimally locate actuators and sensors for structures with close modes. Optimization criteria were defined based on the distinguishing modal controllability and observability measures of close modes. An appropriate genetic algorism was adopted as the optimization algorism. Solving the high order Lyapunov functions was avoided by means of the closed-form expressions for controllability and observability Grammians. Since structure with widely separated natural frequencies is approximately balanced, computational efficiency was improved by grouping close modes together and dealing with the resulting subsystems independently. Finally, the effectiveness and optimality of the novel placement scheme were verified on a model structure with close modes. close modes, controllability/observability, sensor/actuator, optimization criterion, optimization algorithm Citation:Liu X X, Hu J. On the placement of actuators and sensors for flexible structures with closely spaced modes.

This article investigates topology optimization for piezoelectric thin-shell structures under the linear quadratic regulator optimal control. In the optimization model, the structural dynamic compliance is taken as the measure of control performance, and the relative densities describing the distribution of the piezoelectric material are considered as design variables. An artificial material model with penalization on both mechanical and piezoelectric properties is employed. For the purpose of improving computational efficiency of the sensitivity and response analysis, modal superposition method is adopted. The derivative of the Riccati equation governing the linear quadratic regulator control with respect to the design variables is shown to be a Lyapunov equation. In conjunction with the adjoint variable method, the design sensitivities of the dynamic compliance are obtained using the solution of the Lyapunov equation. Numerical examples demonstrate the validity of the proposed method and show the significance of layout design of piezoelectric sensors/actuators. The influences of some key factors on the optimization solutions are discussed. It is shown that the optimized layout of the piezoelectric patches may be significantly influenced by the excitation frequency, but only slightly affected by the choice of the weighting matrix in the linear quadratic regulator control. This work aims to provide an efficient gradient-based mathematical programming method for guiding the layout design of actuators and sensors in smart structures under optimal vibration control. However, the considered model is a purely mathematical one without consideration of engineering realization, thus the optimization result may only serve as an upper bound for practically realizable control performance.

The purpose of this work is to develop a topology optimization scheme for thin-shell piezoelectric smart structures under the linear quadratic regulator optimal control for suppressing transient vibrations. The transient responses of the dynamic system are found by the finite element method based on the classical plate theory and a time-integration algorithm. A mode superposition method is employed to improve the computational efficiency. The pseudodensities indicating the layout of piezoelectric actuators are taken as the design variables. The sensitivity analysis for a general time-integral function of transient structure response under optimal control is derived with the adjoint-variable method. A Lyapunov equation is solved to determine the derivative of the feedback gain in the sensitivity analysis. In the numerical examples, the integral of the structural response over a specified time interval is chosen as the objective function. Then, the optimization problem is solved with a gradient-based programming algorithm. The optimization results illustrate the validity of the proposed method. It is shown that the effects of active control have been substantially improved through optimization. The influences of some key factors on the optimized design are also discussed.

A new extruded octagonal multi-cell tubular energy absorber is developed for improving and enhancing the vehicle crashworthiness. A combined experimental and numerical study is conducted to further assess its performance. In this study, a sled impact test is first performed to investigate the structural response of an octagonal multi-cell tube, and the test data are then used to validate the FE tube model. Using this FE tube model, the effect of key geometric parameters on the structural crashworthiness is analysed in detail. It is found that the number of webs along the circumferential direction, the size of inner tube and wall thicknesses of outer tube, web and inner tube have significant effects on crashworthiness of the energy absorber mentioned above. Based on the parametric study, multi-objective optimisation is subsequently performed to further improve its energy absorption efficiency. These findings can provide valuable guidelines for designing such new types of energy absorber under dynamic axial impact.

For the problem posed by closely spaced modes, this paper defined the MCC (modal correlation criterion) to measure the degree of correlation between close modes. It was proved that structures with certain features tend to have closely clustered modes and the corresponding mode shapes highly correlated. With this understanding, the closed-form expressions for controllability and observability Grammians were adopted to analyze the impacts of actuator/sensor placement on the controllability/observability of highly correlated close modes. On this basis, the problem of actuator/sensor placement, when the optimization criterion is based on modal controllability/observability, was simplified. Moreover, the dimension of the control/measurement vector in independent modal space control for highly correlated close modes was proved to have the potential to be reduced, therefore fewer actuators and sensors were required in this dimension-reduced control strategy. Finally, the desirable vibration suppression for an example structure showed that the theory and methods of this paper were accurate and effective. close modes, correlation, controllability/observability, sensor/actuator placement, low-order control Citation:Liu X X, Hu J. Low order vibration control for structures with highly correlated close modes.