First, an efficient and accurate finite element model for smart composite beams is presented. The developed model is based on layerwise theory and includes the electromechanical coupling effects. Then, an efficient design optimization algorithm is developed which combines the layerwise finite element analysis model for the smart laminated beam, sensitivity analysis based on analytical gradients and sequential quadratic programming (SQP). Optimal size/location of sensors/actuators is determined for dynamic displacement measurement purposes and for vibration control applications. For static and eigenvalue problems, the objective is to minimize the mass of the beam under various constraints including interlaminar stresses, displacements, and frequencies. For transient vibration problems, the objective is the minimization of the actuation control effort to suppress the vibration in a controlled manner. Illustrative examples are provided to validate the formulation and to demonstrate the capabilities of the present methodology.

Laminated composite structures with sensing and actuation capabilities so called as smart laminated structures potentially have numerous applications in modern industries including aerospace and automotive industries. Piezoelectric materials are the most appropriate sensing/actuating elements utilized in smart laminated structures. These novel smart structures combine the superior mechanical properties of conventional composite materials and incorporate the additional inherent capability of piezoelectric layers to sense and adapt their static and dynamic response.

Due to the presence of large number of material and geometrical parameters as well as loading conditions, the determination of the optimal design of the system becomes a crucial issue in smart laminated structures. The conventional design methodology leads to very long and expensive procedures which sometimes make the design infeasible, thus requiring a robust design optimization algorithm. Design optimization of smart isotropic structures has been tackled for variety of purposes and objectives such as the determination of optimal location of actuators/sensors for vibration control [

Structural modeling of the smart system plays an essential role in the efficiency and accuracy of the design optimization algorithm. Most of the structural models of smart laminated structures have been developed based on the equivalent single-layer (ESL) theories [

Application of different materials (sensor, actuator, composite, etc.) and geometries (thickness, size, and location) to construct a smart laminated structure causes strong inhomogeneities through the thickness. Therefore, a robust electromechanical model is required to account for the material and electromechanical inhomogeneities in these hybrid laminates and to provide accurate prediction of the active response of the structures.

Robbins and Reddy [

Most of available works in the field of design optimization of smart laminated structures are developed based on the structural modeling using classical laminate and 3 first-order shear deformation theories [

Another dominant factor in any gradient-based design optimization procedures is the evaluation of functions gradients which computationally influence the accuracy and efficiency of the procedure. A coupled layerwise finite element model has been recently developed by the authors [

Optimal size/location of sensors/actuators is determined for dynamic displacement measurement purposes and for vibration control applications. For static and eigenvalue problems, the objective is to minimize the mass of the beam under various constraints including interlaminar stresses, displacements, and frequencies. For transient vibration problems, the objective is the minimization of the actuation control effort to suppress the vibration in a controlled manner. Illustrative examples are given to validate the formulation and to demonstrate the capabilities of the present model.

The displacement and electric potential fields in a laminated beam based on the layerwise theory are given by

Finite element formulation of the smart laminated beam has been obtained by incorporating local in-plane approximations for the state variables introduced in (

Considering the electric potential vector as

Static problems in smart laminated structures usually referred to problems dealing with controlling the shape due to applied external load where the system basically works in actuating mode. For static problems, (

Dynamic problems in smart laminated structures typically referred to the problems concerning with vibration control and measurements. For dynamic problems considering structural damping, (

Sequential quadratic programming (SQP) is a widely used method for most complex nonlinear optimization problems owing to its robustness and high efficiency in searching for the optimum point. Considering a general problem of minimizing the objective function

The matrix

To improve the chances of obtaining a minimum closer to global minimum, a trial and error or heuristic approach is used together with the SQP method. A simple heuristic method involves randomly selecting a set of starting points in the hope that one of these starting points is close to the global minimum. While global minimum is not assured, the probability of obtaining better minimum increases with the number of starting points. For more details of the SQP procedure, one may consult the book written by Arora [

In the present work, the analytical gradient of objective and constraint functions have been derived based on the layerwise finite element formulation [

This section presents the results for several representative problems. The effects of various design variables on the design objective and constraints are also investigated. All the applications focus on a laminated cantilever beam with embedded or surface-bonded piezoelectric sensors and/or actuators. The properties of the material used for all the example applications can be found in Table

Material properties used in numerical examples.

Properties | NCT/301 | Adhesive | PZT |
---|---|---|---|

144.3 | 6.9 | 63.0 | |

9.85 | 6.9 | 63.0 | |

0.28 | .4 | 0.28 | |

4.34 | 2.46 | 24.8 | |

— | — | ||

— | — | ||

^{3}) | 1385 | 1662 | 7600 |

Laminated structures integrated with piezoceramic elements are widely used in aerospace applications where the mass of the structures is a major concern. However, the density of piezoceramic materials is roughly four times higher than graphite/epoxy material (commonly used composite material for aerospace applications); thus, minimizing the mass while maintaining the functionality of the system is one of the main issues in designing smart laminated structures. In this section mass minimization of smart laminated beam is studied for different design constraints, including deflection, interlaminar stresses, and natural frequencies.

In this problem, the optimal thicknesses of both piezoelectric sensor and actuator patches have been determined in order to achieve the minimum mass of the beam. The beam consists of six layers of 0.125 mm thick graphite/epoxy and is configured as ^{2}) piezoelectric patches are mounted at the first two elements on top surface as sensing device, and another two similar patches are mounted at the bottom surface of the laminate as actuating devices as shown in Figure

Schematic illustration of piezo-laminated cantilever beam.

The objective of this problem is to obtain the thicknesses of both sensors and actuators to minimize the mass of the beam where the tip deflection, tip

Considering the constant thickness for composite layers, the second term in (

The thicknesses of composite and PZT layers as well as the applied voltage are determined for laminate configurations,

Optimal layer thickness (mm) and PZT voltage (volt).

Laminate | Layer thickness | |||||

0.1 | 0.1 | 0.888 | 0.889 | 0.889 | 0.887 | |

0.1 | 0.1 | 0.913 | 1.110 | 0.963 | 0.8973 | |

0.1 | 0.1 | 0.1 | 0.810 | 0.790 | 0.887 |

PZT thickness (mm) and voltage (volt).

Laminate | Top PZT (mm) | Bottom PZT (mm) | Voltage (V) |
---|---|---|---|

.1 | .887 | 72.2 | |

.1 | .887 | 66.33 | |

.1 | .887 | 91.5 |

Iteration procedure leading to the optimal mass for different laminate configurations is provided in Figure

Iteration history to minimize the mass.

As it was expected from the conclusion derived from the results provided in Tables

Optimal interlaminar stresses.

This result is of great importance in designing smart laminated beam since the force applied by the PZT actuators usually is very small. In terms of manufacturing, the bonding between piezoelectric elements and host structure should be well treated in order to prevent detaching of actuators from the composite layers

In addition, the applied voltage should be kept in a safe region for PZT ceramics. It is observed that the optimal thickness of the upper and lower PZT patches greatly depends on the lamination orientation and applied voltage. It should be noted that monitoring interlaminar stresses in optimization procedure when using ESL theories is not straightforward and also it is not accurate. It requires indirect and not-so-accurate computations, and also it is very time consuming. However, the finite element model based on layerwise theory provides interlaminar stresses that are very close to the results obtained using three-dimensional elements [

The smart laminated beam described in Section

Considering the constant mass of the PZT elements, the first term is determined as 27.36 gr. The optimal design for layer thickness is obtained considering with and without electromechanical coupling effect in the finite element model and given in Table

Optimal layer thickness with frequency constraint.

Layer thickness | Total mass | Reduces mass | |
---|---|---|---|

Initial design | 1.0 (mm) | 82.0 (gr) | 0.0 (percent) |

Optimal (no coupling) | 0.59 (mm) | 58.55 (gr) | 28.65 (percent) |

Optimal (coupling) | 0.54 (mm) | 56.0 (gr) | 34.14 (percent) |

A cantilever laminated beam with 4 layers ^{3}) is attached on the top surface at 0.05 m from the fixed end of the beam. The free end of the beam is initially displaced by 10 mm. It is desired to determine the minimum electric potential to suppress the transient vibration in a controlled manner. It is required to reduce the settling time to 0.5 sec by applying electric potential to the actuators. The optimization problem is minimization of the control effort in time interval 0 to

Figure ^{2}

Vibration suppression using optimal electric potential.

Optimal voltages applied at actuators.

Performance history for the optimal control problem.

Initial and optimal control forces control problem.

Further, the effects of laminate configuration and location of actuator are investigated. The smart laminated beam of the previous example is considered except that the laminate configuration and location of the actuator are changed for different cases.

In the first case, the actuator,

In Figures

Effects of laminate configuration and location of actuator on vibration suppression using optimal electric potential.

Effects of laminate configuration and location of actuator on optimal voltages applied at actuators.

As it is observed from Figure

Smart laminated composite beams with surface-bonded/embedded piezoelectric layers as sensors and/or actuators have been investigated based on the layerwise displacement theory. A finite element formulation with considering electromechanical effects has been developed to investigate the effects of the size and location of piezoelectric patches on the design parameters. The developed finite element formulation for the coupled analysis of smart laminated beams along with analytical function gradients have been combined with sequential quadratic programming to develop an efficient design optimization algorithm. Utilizing the developed algorithm, the optimal design of smart laminated beams for various constraints and objectives has been investigated. In static case, nodal displacement and interlaminar stresses are considered as constraints to minimize the mass. In dynamic problems, natural frequencies are monitored as design constraints for eigenvalue problem, and the minimization of control effort is considered for transient vibration. Due to the higher accuracy of displacement and stresses obtained from layerwise theory, the optimal designs of smart laminated structures based on the layerwise approach are more reliable than that of the equivalent single layer theories. In addition, utilizing analytical gradients in sequential quadratic programming leads to faster convergence to the optimal design.

The authors wish to acknowledge the support provided by Sharif University of Technology, International Campus, and Kish Water and Power Company on Kish Island.