Bernhard, G. Washington, F. VanKeuls, F.
Miranda, and C. Baumann, B. Washington, and G. Silverberg, L. Meirovitch, 21 , pp. Washington , G. Intelligent Material Systems and Structures, 10 , pp. Guidance, Control, and Dynamics , 22 , pp. Kiely, E. Washington, and J. Smart Materials and Structures , 7 , pp. Smart Materials and Structures, 7 , pp. Dynamic Systems, Measurement, and Control, , pp. Smart Materials and Structures, 5 Refereed Conferences B. Kim, G. Headings, G. Washington, S. Midlam-Mohler, and J. Kim B. Headings L. Conference , Kwak, S. SPIE 7th Int.
Symposium on Smart Structures and Materials , Brahma A. Glenn, Y. Guezennec, T. Miller, G. The average value of the state estimate should be equal to the average value of the true states. Also the expected value of the state estimate should be equal to the expected value of the true state. The variation between the state estimate and the true state should be as small as possible. So, we require an estimator with smallest possible error variance. The above two requirements of the estimator are well satisfied by the Kalman filter. The assumptions about noise that affects the performance of our system are as follows.
The mean of w and z should be zero and they are independent random variables, i. We define the noise covariance matrices S w and S z for process noise and measurement noise covariance as. The control law developed for the LQG controller is based on the estimation of the states x of the system rather than the actual states of the system, which is given by.
Where, G k is the gain of the Kalman estimator and is obtained by.
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- Finite element analysis of actively controlled smart plate with patched actuators and sensors.
Where M is the again the solution of another steady state matrix Riccati equation. The dynamic behavior of the Kalman estimator is given by the following first order linear differential equations. Due to the presence of w and z , the estimation error is not converges to zero, but it would remains as small as possible by proper selection of G k. Thus the controlled LQG closed loop can take the final form.
For the validation of the MATAB code developed for the finite element analysis of the piezoelectric plate, we solved the simple benchmark static problem of Hwang and Park  and compared the results obtained with the published results. The length of the width of the laminate is mm and 5 mm respectively and the thickness of each lamina is 0. The configuration of elastic PVDF bimorph is labeled in figure 2 , which is fixed on one side and free at the other side.
The physical dimensions and material properties of the bimorph are listed in the table 1. The beam is discretized into five rectangular elements exactly as in reference . The deflection of the beam along the length for electric potential of 1 Volt applied across the bimorph is calculated and compared with the published results given in reference . The variation of tip deflection for different applied potentials is studied and the results and presented in figure 3 along with Hwang's work. Figure 4 shows the variation of tip deflection for different applied potentials across the bimorph.
It can be observed from these plots that the results obtained are in excellent agreement with those presented in the reference. Therefore it can be used for further analysis with confidence. A rectangular aluminum cantilever plate with four rectangular PZT patches, bonded to the top surface of the plate, are used as actuators and four other PZT patches bonded symmetrically to bottom surface are to be used as sensor, forming four sets of collocated actuator-sensor pairs.
The configuration is depicted in figure 5. The material properties and geometrical dimensions of the host plate and PZT layers are listed in the table 1. The problem domain has been descritized in rectangular identical elements as shown in figure 6. Case 1. The piezoelectric patches are short circuited thereby rendering ineffective the piezoelectric coupling effect that enhances the stiffness of otherwise passive structure. This case includes the pure structural stiffness of the PZT patches and modal analysis involves the solution of following Eigen value problem.
Since the patches are short circuited, no electrical potential across them will build up due to deformation that reduces the coupling terms to zero. Case 2. In this case, all 8 PZT patches are acting as sensor hence modifying the eigen value problem as,. The electrical degree of freedom has been condensed out.
These eigen value problems has been solved using MATLAB's eig subroutine and the lowest 20 natural frequencies for the cantilever plate is given in table 2 and the corresponding first eight mode shapes are shown in figure 7. The FEM model of the problem considered here consists of DOF whose state space model will be of linear differential equation that is computationally expensive. Model reduction technique is employed to reduce the size of problem.
The Hankel singular values present the measure of energy content in various states and herein are being used to truncate the model. In the figure 8 Hankel singular values of the FEM model after transforming it into modal space modes considered and then recasting into state space format is presented.
It can be seen that lowest 20 modes contains most of the system energy thus it is reasonable to consider only 10 normal modes for further investigation of system dynamics. The ultimate aim of a feedback control system is to achieve the maximum control over the system dynamics keeping in view of hardware limitations. The 5 inputs, 4 piezoelectric actuators and one mechanical point force and 4 outputs, the 4 piezoelectric sensors constitute the MIMO system shown in figure 5.
The LQG regulator consists of two parts an optimal controller and a state estimator Kalman observer. The controller gain is calculated by optimizing the functional of equation The optimal gain matrix which is obtained from the solution of the matrix Riccati equation for a choice of state weighing matrix Q as diagonal matrix whose first element is 10 12 and the remaining elements are unity.
This choice is to give priority to control the first mode. This value is chosen in such a way to keep the actuator voltage under the specified limit. The linear quadratic optimal gain matrix is presented in table 3 which shows that the entries of the optimal gains matrix for columns 1 and 11 which correspond to first mode for modal displacements and velocities are larger than other entries correspond to remaining modes, which indicate that the designed controller is more effective for controlling the vibrations for first mode but can effectively control the vibrations for other modes also.
The entries of optimal gain matrix in rows 1 and 3 which correspond inputs through actuators 1 and 3 are larger about 10 times for all modes as compared with the entries for inputs through actuator 2 and 4. This indicates that more control effort is needed at actuator 1 and 3 for controlling the vibration of the plate.
The Kalman filter design is based on 4 noisy measured outputs of the sensors and five inputs to the system that includes 4 actuators and one mechanical point force input on tip of cantilever plate. Steady state Kalman gain matrix is given in table 4. After obtaining the controller and observer gain matrix now system is exited with given initial condition. Initial condition vector is derived by deforming the plate by a 0.
Deflection thereby obtained is transformed into modal space using weighted modal matrix. That in turn has been used as initial condition modal displacement vector with conjunction of zero modal velocities. The various parameters of system response are presented in figures Points are shown on grid whose time history is presented in figure 6. Since the plate was excited in such a way that the first bending mode was dominating the system behavior and for that reason, a large state weight was attributed to the element that corresponds to the 1st mode in state weighing matrix Q.
From Figure 12 one can infer that first mode is decaying faster than other modes. In this work a numerical analysis of active vibration control of smart flexible structures is presented. Linear Quadratic Gaussian LQG controller was designed for controlling the lateral vibrations of the plate which is based on the optimal control technique. The control model assumes that four piezoelectric patches out of eight acts as distributed sensors, the other four acts as distributed actuators, and the signals generated through was used as a feed back reference in the closed loop control system.
The designed model provides a means to accurately model the dynamic behavior and control strategies for vibration of smart structures with piezoelectric actuators and sensors. The natural frequencies of vibration are obtained with and without electromechanical coupling. It is observed that electromechanical coupling effect is more effective for lower frequencies. Since most of the energy is associated with the first few modes, therefore these modes only need to be controlled.
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As observed from plots, the control model is quite effective. Alkhatib and M. Active structural vibration control: a review. Bailey and J. Distributed piezoelectric-polymer active vibration control of a cantilever beam. Baz and S. Performance of an active control system with piezoelectric actuators. Caruso, S. Galeani, and L. Active vibration control of an elastic plate using multiple piezoelectric sensors and actuators. Crawley and E. Detailed models of piezoceramic actuation of beams.
Crawley and J. Use of piezoelectric actuators as elements of intelligent structures. One idea is to place capsules or hollow internal gauges, spaced about 2 m apart. The sensors serve as a data col- Research Center, are looking into smart paints which lector as well as a wireless transmitter.
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Optical fibers which change in light trans- 3. Reliability and safety of structures using stability- mission due to stress are useful sensors. They can be based hybrid controls, Professor B. Spencer embedded in concrete or attached to existing structures. University of Notre Dame NSF-supported researchers at Rutgers University studied optical fiber sensor systems for on-line and real-time MR fluid dampers are one of the most promising monitoring of critical components of structural systems smart damping intelligent isolation systems according to such as bridges for detection and warning of imminent Professor B.
Spencer, Jr, due to proven technology— structural systems failure. NSF grantees at Brown Uni- reliable and robust; low cost; insensitivity to tempera- versity and the University of Rhode Island investigated ture; low power; and scalable to full-scale civil engineer- the fundamentals and dynamics of embedded optical ing applications. Japanese researchers recently dampers in suppressing earthquake excitation in the lab- developed glass and carbon fiber reinforced concrete oratory.
Under NSF 3. Flexible wings for uninhabited air vehicles, P. University of Florida researchers at the University of California—Berkeley recently completed a study of the application of ER Ifju and colleagues are conducting research on design fluids for the vibration control of structures. Courtesy of the late W. Courtesy of B. Spencer, Notre Dame University.
There is tional lifting body paradigm. The flexible wing design currently a trend toward thicker materials of the order consists of a carbon fiber skeleton and a latex rubber of hundreds of micrometers in MEMS because a larger skin with a thin under-cambered shape based on those aspect ratio is needed for a mechanical device to be able of biological counterparts.
The flight characteristics of to transmit usable forces and torques. The mechanical the flexible wings have many superior qualities, adapting testing techniques are being extended to MEMS. The air frame and wings of the aircraft are 3. Passive and active damping control for large made from carbon fiber. These micro aerial vehicles civil structures, N. Wereley University of MAVs have been configured to optimize stability and Maryland flight performance. The control surfaces elevons are mounted under the wings so that they are continuously The objective of the research activities is to augment washed by the propeller.
This allows the aircraft to be damping in large civil structures applications via both controlled, even when the airspeed is low. Separating the passive and active means, to reduce structural response. A good RC sis, and experimental demonstration of passive, semi- pilot can hover the MAV and use the elevons as thrust active, and active structural damping control for civil directors.
The MAV is also capable of stable flight structures using smart materials and structures tech- between 10 mph and 40 or 50 mph. It can perform loops nology. The research includes consideration of stability and barrel rolls. Damping strategies are being tested on 3. Mechanical testing and microstructural studies of dynamically scaled three-story civil structures building MEMS materials, Professors Sharpe and Hemker models using dampers such as depicted in Fig. Johns Hopkins University. Design, modeling, and development of active New test methods have been developed to measure aperture antennas, G.
While engin- eers today are able to design MEMS and predict their The primary goal of this work is the development of overall response, they cannot yet optimize the design to a novel class of aperture antennas capable of variable predict the allowable load and life of a component directivity beam steering and power density variable because the mechanical properties of the material are not focusing or beam shaping.
The actuation for these available. This research is attempting to measure and antennas is employed using a distinct mechanism.
The provide such data. Furthermore, they are performing mechanism employs polyvinylidene fluoride PVDF comprehensive microstructural studies of MEMS film bonded to a lightweight metalized mylar structure. Since the film is piezoelectric, any electrical voltage drop across the film will cause the film to lengthen in its stretch direction. The resulting lateral deflection from a length change causes the production of a moment which causes the antenna surface deflection. This is illustrated in Fig.
Ultra-precision shape-controlled smart structures, J. Main University of Kentucky Engineering systems that require ultra-precision con- trol of positions and shapes often use active materials as extremely precise actuators. Many active materials respond to applied electric fields, which are conven- tionally applied by distributed electrodes. In this investi- gation active material actuator technology is combined with an electron gun charge deposition system and the Fig. Palm-sized micro aerial vehicle.
Courtesy of P. Ifju, Univer- characteristics of the resulting shape and position control sity of Florida. Lower plots show displacement left and acceleration right responses to A no control, B skyhook controller, and C continuous sliding mode controller. Courtesy of N. Wereley, University of Maryland. Use of discrete high deflection piezoelectric picomotor actuators to produce deformation for an active aperture antenna, allowing expanded ground coverage through beam steering.
Courtesy of G. Washington, The Ohio State University.
Dynamics and Control of Advanced Structures and Machines
Actuation sys- active material, thus making it possible to command tems such as this have the potential to be controllable at strains and displacements at extremely precise locations the nanometer level. The overall technical objec- 3. Hybrid magnetostrictive composite material for tive is the development of a thorough understanding of active control, G. Carman University of California electron gun shape control of piezoelectric materials, at Los Angeles including possible shape-control limits, resolution, and dynamic behavior.
Piezoelectric bimorph configuration left that allows actuation via a non-contact electron gun source that can target multiple locations of the distributed active surface right. Courtesy of J. Main, University of Kentucky.