Development of Vibration Annihilation of Sandwiched Beam with MROF DTSMC: A Novel Approach

In the present chapter, an analytical model of a flexible beam fixed at an end with embedded shear sensors and actuators is developed. The smart cantilever beam model is evolved using a piezoelectric sandwich beam element, which accommodates sensor and actuator embedded at distinct locations and a regular sandwiched beam element, having rigid foam at the core. A FE model of a piezoelectric sandwich beam is evolved using laminated beam theory in MATLAB®. Each layer behaves as a Timoshenko beam and the cross-section of the beam remains plane and rotates about the neutral axis of the beam, but it does not remain normal to the deformed longitudinal axis. Keeping the sensor and actuator location fixed in a MIMO system; state space models of the smart cantilever beam is obtained. The proper selection of control strategy is very much crucial in order to obtain the better control. Here a DSM controller is designed to control first three modes of vibration of a flexible cantilever beam modelled based on Timoshenko beam theory. New active vibration control scheme to suppress the vibrations of MIMO model has developed. The actuators are located at 2nd and 5th FE positions while sensors are set at 6th and 10th FE position to form the embedded smart cantilever beam with 10 finite elements. The piezo crystals are located in the central core at prescribed locations, rest core filled with rigid foam, and this central core is sandwiched between two regular steel beams. Modelling a smart structure inclusive of sensor/actuator mass and stiffness and by altering its orientation in the beam from the free end to the fixed end acquaint an ample modification in the system’s structural response attributes. Sensor voltage is lower when the piezo patch is imposed at the free end due to the exiguous strain rate and hence demand more control endeavor. MIMO control is superior over SISO control due to its multifarious interactions of input and output and all-inclusive control endeavor needed by MIMO controller is less than SISO controller and also placing the piezo at two distinct FE locations on the beam establishing the significant modification in the system structural traits than placing it lonely at a location. The multirate output feedback dependent DSMC strategy are more harmonious as compared to the other control approaches viz. periodic output feedback (POF) and fast output sampling (FOS) controllers. The multirate output feedback based DSMC policy are more episodic as compared to the other control techniques. In discrete quasi sliding mode control (DQSMC) with output samples, there is a necessity of switching function for control and hence engendering some chattering phenomenon, while control strategy presented in the present article is the MROF based DSMC technique obtained from Bartoszewicz’s law does not demand any use of switching function and provides control input directly in form of past control data and past samples. The system responds well in closed loop and does not manifest inexpedient chattering phenomenon. MROF based DSMC employ the signum function in the control input and the control is computed from the immediate past control value and the past control output samples. The fractious system takes an extended time to damp out the oscillations in contrast to the system with the designed sliding mode control input means without control the transient response was preeminent and with control, the vibrations are quashed. From simulation results, it can be inferred that sensor output at FE 6 is more than sensor output at FE 10 by approximately 10 times due to its high strain rate at FE 6 as compared at FE 10 and also the control input are approximately 10 times smaller in case of MROF as compared to SISO case. In case of MROF technique, the states of the system are needed neither for switching function assessment nor for the feedback denotation. DSMC algorithm are computationally unpretentious, ensures better enduringness, brisk convergence and exalted steady state authenticity of the system. The technique used is more feasible as the output being used rather than states. Hence, it can be concluded that the multivariable control is best among all the models due to its multilevel interactions on both input and output. A MIMO model furnishes excellent energy distribution and even good administration of actuation forces and minimal requirement of control forces as compared to SISO model for the case of smart cantilever beam with embedded sensors and actuators.