## Introduction

### Smart materials

*Shape memory alloys**Electro / magneto-strictive materials**Electro / magneto-rheological fluids**Fiber optics*

Magnetostriction is the process by which a ferromagnetic material changes from one shape to another in the presence of a magnetic field. In the absence of an applied field, controllable fluids approximate Newtonian fluids reasonably well.

### Piezoelectric materials

Engineering design problems are often of a discrete nature (eg number of actuators), so the above methods described in the previous subsection are not applicable or tend to get stuck in local optima. Coefficient of thermal expansion along the main directions { }α ik =[α α1 2 0]kT. The structural mass matrix is given as. A.49).

Design analysis of smart FRP composite structures

*Analysis of smart FRP structures**Optimal placement of sensors and actuators**Control of smart structures**Feedforward control**Feedback control*

### Optimization algorithms

Over many generations, natural populations evolve according to the principles of natural selection and survival of the fittest. The least fit members of the population are less likely to be selected for reproduction and therefore die out.

### Scope of the present work

This new generation contains a greater proportion of the qualities possessed by the good members of the previous generation. By favoring the mating of the more fit individuals, the most promising areas of the search space are explored.

### Organization of the thesis

In the late 1970s, the application of PVDF was used in many transducer applications, e.g. ultrasonic equipment, sonar, strain gauges, etc. Only thermal loading has been applied to the top and bottom of the plate.

## Literature Review

### Electro-mechanical modeling

The finite element formulation was developed based on the QUAD 4 plate finite element used in COSMIC/NASTRAN and ASTROS. 31] presented a finite element model of an eight-node shell element for modeling thin-shell structure containing integrated distributed.

### Piezo-thermo-elastic analysis

A combined thermo-piezoelectric-mechanical model of composite laminates with surface-bonded piezoelectric actuators was developed by Chattopadhyay et al. 77] developed the combined thermo-piezoelectric-mechanical theory, based on layer displacement field and higher-order electric and temperature fields, to study the dynamic response and control of smart cylindrical composite shells.

### Optimal placement of piezoelectric sensors and actuators

*Parameter variation**Stochastic methods*

Kang et al [90] optimized the placement of piezoelectric localized sensor/actuator pairs for active vibration control of laminated beams by maximizing the structural damping index, a weighted sum of the achieved modal damping of each vibration mode. The dissipation energy of the active controller is maximized for a fixed number of three actuators.

### Control schemes for active vibration control

139] presented a new smart structure optimal design strategy applied to robust vibration control of a piezoelectric laminated beam. LQR control strategy was considered and was observed to be effective for the vibration control of the structure.

### Motivation and objectives of the present work

In this chapter, the GA-based LQR optimal control scheme (described in Chapter 4) is used for the optimal vibration control of various types of smart shell structures. The optimal vibration control of this Panel with Placement2 under thermo-mechanical loading using the GA-LQR control scheme is also discussed in the next subsection.

## Finite Element Formulation of Smart FRP Composite Shell Structures

### Classical shell theories (CST)

The behavior of deformations of shell structures in general is quite complex due to the interaction between bending and stretching/diaphragm in carrying the applied loads. Later, the accuracy of the so-called Kirchhoff-Love shell theory was investigated by a large group of scientists, and its current state has been greatly influenced by ideas of Vlasov [161], Galimov [162], Koiter [163] and Mushtari and Galimov [164].

### Classical lamination theory (CLT) and Mindlin’s hypothesis

Due to the Mindlin hypothesis, the shear stress is shown as constant across the thickness. But according to the elementary theory, the transverse shear stress varies parabolically in the thickness direction.

### Shell finite element for piezo-thermo-elastic analysis

*Assumptions made in present formulation**For mechanical analysis**For electro-mechanical analysis**For piezothermoelastic analysis**Geometry of shell midsurface**Isoparametric mapping**Strain displacement relations**In-plane/bending strain-displacement matrix**Transverse strain displacement matrix**Piezothermoelastic constitutive relation**Direct and converse piezoelectric relations**Electrical potential in the piezoelectric layers/ patches**Temperature field**Finite element equations**Static finite equations**Dynamic finite element equations**Finite element equations for electromechanical analysis*

Two independent coordinates( , )α α1 2 in the parametric space are considered as the curvilinear coordinates of the middle surface of. This chapter presents the important conclusions based on finite element analysis and optimal active vibration control of different types of smart FRP composite shell structures using the methodologies developed in this work.

## Genetic Algorithms for Optimal Actuators Placement and Control

### Encoding

In binary coded GAs, the string length must be chosen a priori to allow GAs to achieve a certain accuracy in the solution. To avoid the problems associated with binary coded GAs for continuous space search, many truly coded GAs have been developed.

### Reproduction operator

Apart from this, simulated binary crossover and probability-based mutation real coded genetic algorithm can completely overcome the problems associated with binary coded genetic algorithm. One of the main advantages of this type of real coded GA is that it can search in continuous space with or without variable boundaries by automatically creating expanded or contracted child solutions in space.

Mutation Operator

### State-Space Representation

The decoupled dynamic equations (Eq. 4.7)) taking into account the modal damping can be written as. 4.9) can be represented in the state space form as is the disturbance matrix, { }ud is the disturbance input vector, { }φa is the control input, and. 4.11). Two types of sensor output equations are considered for mechanical and thermo-mechanical loading.

Controllability index for actuator location

LQR Optimal feedback control

*Determination of weighting matrices*

### GA for optimal placement

*Uniform crossover**Mutation*

A positive integer acj∈[1, ]m is randomly generated, which replaces the old one when it changes. If acjis is equal to the old one, then a new positive integer is selected again until they are different in the chromosome.

Optimal placement using ICGA

### GA for LQR control scheme

*Simulated binary crossover (SBX)**Parameter-based mutation operator*

In most literature, small and large values of ηc are taken as 2 and 5, respectively [172]. A polynomial probability distribution was used to generate a solution b near the parent solution using the following procedure.

### The GA approach to optimal LQR

The maximum actuator voltage variations for the GA-LQR control scheme without and with consideration of the pyroelectric effect are shown in Figure 7.22. The maximum actuator voltage variations for the GA-LQR control scheme without and with consideration of the pyroelectric effect are shown in Figure 7.29.

## Validation and Thermo-Electro-Mechanical Responses of Smart FRP

### Structural validation

*Spherical laminated composite shell**Ellipsoidal laminated composite shell*

It can be observed that the results obtained from the current finite element code are very close to those obtained from published analytical solutions [165,. The dimensionless fundamental frequencies obtained from this code were compared with the available FE solution [61] and listed in Table 5.5 for different R/a and a/h ratios.

### Electro-mechanical validation

A uniform stress was applied through the thickness and the calculated transverse deflections at the five nodes obtained from this code were also compared. It can be seen from the table that the results from the current finite element code are in good agreement with the already published results.

### Validation for piezo-thermo-elastic analysis

*Static displacement of piezo-laminated plate**Thermo-electric analysis*

The piezo-thermoelastic analysis of simply supported piezo-laminated composite plate was performed to validate the present shell finite element code. The induced sensor voltage was calculated for both thermal voltage effect and pyroelectric effect and was validated with the available results [86].

### Validation for optimal actuators placement

Figure 5.7 shows the convergence plot with generations for the integer-coded and binary-coded GA for this problem, and it can be observed that while the integer-coded GA converges after 11 generations, the binary-coded GA converges only after 286 generations, clearly showing the computational advantage of integer. coded GA over the binary coded GA for this problem as discussed in Section 4. Therefore, the integer coded GA was used to determine the optimal placement of sensors and actuators.

### Validation for optimal LQR gain

This result is expected since the bending of the first vibration mode reaches its maximum value at the fixed end of the cantilever beam, and a similar observation was also reported by Wang and Wang [97]. The optimal control gain vector calculated by the present code is observed to be in good agreement with that given in Meirovitch [176].

### Thermo-electro-mechanical analysis of smart FRP composite shell structures

*Piezo-laminated spherical shell under thermal load**Comparative study on thermo-electro-mechanical responses of different shells*

The comparison of the central displacements neglecting and taking into account the pyroelectric effect with the R/a ratios due to the bottom surface temperature 50ºC and the top surface temperature 0ºC for a/h =10 is shown in Fig. Comparison of center displacements neglecting and taking into account the pyroelectric effect with R/ a ratio due to the lower surface temperature of 50ºC and the upper surface temperature Fig.

### Summary

It has been observed that the pyroelectric effect is significant in both mechanical and electrical responses for piezolaminated shell panels under different thermal loads. It has also been observed that double-curved shell panels are stiffer against thermally induced static displacement compared to other shell panels.

### Problem definition

Based on the developed methodology (as discussed in Chapter 4), the optimal placements of GA-based PZT actuators coded with integers for different types of smart shell structures are obtained and the superiority of the present method over existing methods is also presented . The mechanical, electrical and related material properties [141] used in this study are listed in Table 5.11.

### Optimal placement of collocated sensors and actuators

First eight natural frequencies corresponding to mode-shape-based, placement 1 and placement 2 composite sensors and actuator placement considered in the present work are shown in the table 6.2. It was observed from Table 6.2 that the placement of actuators has some effect on the system eigenvalues.

### Optimal placement of collocated sensors and actuators

It was also observed from Table 6.1 that the conventional mode-shaped placement gives much less controllability index, considering control overflow than that of Placement 2. It can be made clear from Table 6.3 that some of the natural frequencies have increased significantly in the case of Post 2 than that of post 1.

Optimal placement of collocated sensors and actuators on

Optimal placement of collocated sensors and actuators on

### Summary

In contrast, with the proposed GA-LQR control scheme, full control of the combined thermomechanical displacement is possible. Narayanan, Active vibration control of smart shells using distributed piezoelectric sensors and actuators, Smart Mater.

## Genetic Algorithm (GA) based Optimal Vibration Control of Smart FRP

### Optimal vibration control of laminated spherical shell panel

*Optimal vibration control under impulse load**Optimal vibration control under combined impulse and thermal load*

The maximum variation of actuator voltage using layout based on mode shape, Layout1 and Layout2 for the GA-LQR control scheme is. The maximum actuator voltage variations using Placement2 for the LQR and GA-LQR control scheme are also shown in Fig.

### Optimal vibration control of laminated ellipsoidal shell panel

It can be observed that although the actuator voltage requirement in the case of the GA-LQR control scheme is higher than that in the simple LQR control scheme, the simple LQR could not achieve full control in the case of thermal loading - mechanic. From Fig 7.22, it can be observed that although the voltage requirement is higher in the case of considering the pyroelectric effect, this voltage leads to better control of the combined thermo-mechanical displacement than that without considering the pyroelectric effect.

### Optimal vibration control of laminated doubly curved shell panel

The maximum actuator voltage variations for GA-LQR control scheme without and considering pyroelectric effect are shown in Fig. From this study it could be concluded that optimal actuator placement using Placement2 and subsequent GA-LQR control scheme leads to the maximization of closed loop damping ratio for both mechanical as well as thermomechanical loading within the limit of input/actuator voltage.

### Optimal vibration control of laminated cylindrical shell panel

It could also be observed that the achieved damping ratio is higher in the case of without pyroelectric effect. However, a better control of the combined thermomechanical displacement has been achieved by considering the pyroelectric effect, which is clearly shown in Figure 1.

### Summary

Lee, Optimal placement of piezoelectric sensors and actuators for composite plate vibration control using genetic algorithms, Smart Mater. Chandrawat, Multivariable adaptive vibration control of smart structures using iterative (LQG) control strategies, Smart Mater.

## Conclusion and Scope of Further Work

Scope of Further Work

### Smart structure schematic diagram

As a result of this change in temperature, positive and negative charges move to opposite ends through migration (ie the material becomes polarized) and therefore an electric potential is established. It could be observed that.. the damping ratio obtained is more in the case of without pyroelectric effect.

PZT patches bonded laminated composite plate with feedback control

Principle of feedforward control

Principle of feedback control

Geometry of layered composite shell panel in Cartesian coordinate system

A framework of simple genetic algorithm

Flowchart of GA based LQR

Geometry of layered composite shell panel

Schematic view of a bimorph beam

Piezolaminated beam structure

Sensor voltage due to thermal strain effect

Sensor voltage due to pyroelectric effect

Optimal location of four actuators on the cantilever beam based on

Variation of controllability index with generation using integer and

Two degree of freedom system

Variation of central displacements with R/a ratios due to bottom

Variation of central displacements with R/a ratios due to bottom

Variation of central displacements with R/a ratios due to bottom

Variation of central displacements with R/a ratios due to bottom

Material properties for Gr/Epoxy lamina

Transverse deflections of piezoelectric bimorph actuator

Material properties of Gr/Epoxy and PVDF

Comparison of center deflection of piezo-laminated plate

Material properties of Gr/Epoxy and PZT

Several important parameters for integer and binary coded GA

Material properties of Gr/Epoxy lamina and PZT

Value of maximum controllability index for different placement schemes

First eight natural frequencies of the different pizolaminated

First eight natural frequencies of the different pizolaminated

First eight natural frequencies of the different pizolaminated

First eight natural frequencies of the different pizolaminated

Coupled material properties for Gr/Epoxy lamina and PZT

Different actuator placement schemes used