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A four-node composite facet-shell element is developed, accounting for electromechanical coupling of Macrofiber Composite (MFC) and conventional PZT patches. Further a warping correction is included in order to capture correctly the induced strain of conformable MFC, surface bonded on a cylindrical shell. The element performance to model the relations between in-plane electric field to normal strains is examined with the help of experiment and ANSYS analysis. In ANSYS, a simple modeling scheme is proposed for MFC using a parallel capacitors concept. The independent modal space control technique has been revisited to address the control of combination resonances through a selective modal space control scheme, where two or more modes can be combined to form the vibrating system or plant in modal domain. The developed control schemes are implemented in a digital processor using DS1104 and the closed-loop vibration control experiments are conducted on a CFRP shell structure. The influence of directionally induced actuation of MFC actuators on elastic couplings of composite shell is studied theoretically and is subsequently demonstrated in experiments. MFC actuators provide the much needed optimization domain for achieving the vibration control of combination resonances of elastically coupled deep-shell structure.

Active control techniques are becoming more popular in recent years due to the emergence of a field called “

Shell structures are commonly adopted in aerospace vehicles [

Vibration control of thin-shell structures involving modeling, analysis by piezoelectric materials has gained importance due to extension actuation, which couples the membrane strain effectively with an applied electric field [

The review of relevant literatures has brought out the following observations.

Use of MFC actuators for directional actuation has been found efficient.

Studies on isotropic beams, plates, and shells with piezoelectric patches are successfully conducted.

Beam, plate, and triangular shell finite elements are proposed with MFC actuators.

However, the study on composite shells with MFC actuators is very much limited; in particular the vibration control of a deep-shell configuration (like a fuselage panel) is not yet attempted.

A simple and efficient electro mechanically coupled composite shell element, validated with experiment, may be required to analyze the curved composite skin/panel of aircraft structure with MFCs. The commercial codes such as ANSYS/ABAQUS have solid elements, which account for piezoelectric coupling; however aspect ratio of the solid element may be a constraint, dealing with the thin MFC’s, when we employ them along with plate bending elements in a large structure.

Influence of in-plane actuation on the vibration control of coupled membrane-bending/membrane-twisting of deep shell modes has to be evaluated to build an efficient AVC system to address resonant and combination resonances.

By keeping these objectives, a shear flexible, field consistent four-node facet-shell element has been proposed in the present work. Further a deep cylindrical shell (

A four-node composite plate element, integrated with active layers has been formulated with five mechanical degrees of freedom

Electromechanically coupled composite facet element.

The formulation includes three piezoelectric layers, which can be placed anywhere along the thickness direction of the laminated composite shell. The total electric potential in each active layer is then given by

In (_{33} actuation and _{31} actuation. The material coordinate transformation is done for MFC actuation to accommodate the in-plane electric field in transverse direction [

The kinematic relations are described for membrane, bending, and shear strain fields as follows:

Similarly the electric fields both in-plane and transverse direction are defined as

In (

The displacements and electric potentials are then approximated within the element using the linear shape functions [

The final mechanical and electrical degrees of freedoms are presented below:

The work done in the piezoelectric continuum acting as an actuator and a sensor is expressed, following the virtual work principle as

Using the kinematics (

Similarly the sensor equation is given by

The force and moment corrections are done on this four-node element, following Naganarayana and Prathap [

Six global mechanical degrees of freedom are considered, that is,

If the set of assembled equilibrium equations in local coordinates is considered at each nodal point, then we have six equations of which the last one (corresponding to

The membrane in-plane shear locking (

In this section, the control procedure is briefly explained. A linear quadratic regulator is designed in modal domain using the system matrices, obtained from finite element analysis. The stiffness, mass, actuator, and sensor matrices of the shell are obtained in uncoupled form in modal domain by transforming them from physical coordinates

The dynamic equation is then written in modal form as

Further, the closed-loop system is built using the dynamic equation in state space form as follows:

The control design has taken into account the process disturbance (

Therefore the final closed-loop system is built with a liner control law as

Both LQR gain (

The developed shell4 element has been validated first for its capability to solve static, dynamic, and piezoelectric problems. For this purpose, we have adopted both numerical and experimental approaches.

A clamped circular cylinder with an applied potential on the whole upper (

Deflection in

Radius (cm) | 20 | 30 | 40 | 50 |
---|---|---|---|---|

Reference [ | 3.42 | 4.22 | 4.73 | 4.91 |

Present FEM | 3.41 | 4.25 | 4.67 | 4.95 |

In order to validate the dynamic behavior of the developed element, a simply supported laminated cylindrical shell is considered with

Nondimensional fundamental frequencies of cylindrical shell under uniformly distributed load.

Reddy [ | Present | ||

5 | 28.825 | 29.335 | 28.948 |

10 | 16.706 | 16.945 | 16.765 |

20 | 11.841 | 11.945 | 11.868 |

50 | 10.063 | 10.118 | 10.076 |

100 | 9.782 | 9.829 | 9.794 |

10^{30} | 9.687 | 9.730 | 9.697 |

For an illustration of combination resonance control, a deep cylindrical shell is considered (see Figure _{33}) of MFC, by adopting an equivalent electric field ideology (refer to Figure

Material properties of CFRP, MFC, PZT-5A.

Material Property | CFRP (BDC) | MFC | PZT-5A |
---|---|---|---|

Young’s modulus | 34.29 GPa | 30.25 GPa | 48 GPa |

Young’s modulus | 34.29 GPa | 15.99 GPa | 48 GPa |

Shear modulus | 2.0 GPa | 5.5 GPa | 18.32 GPa |

Poisson’s ratio _{12} | 0.2 | 0.306 | 0.31 |

Poisson’s ratio _{ 13} | 0.2 | 0.306 | 0.31 |

Density | 1333 Kg/m^{3} | 5020 Kg/m^{3} | 7500 Kg/m^{3} |

Piezoelectric Stress constant ( | — | 13.954 C/m^{2} | — |

Piezoelectric Stress constant ( | — | −5.868 C/m^{2} | 13.152 C/m^{2} |

Dielectric constant | — | 1.42 × 10^{−8} F/m | 3.01 × 10^{−8} F/m |

Cylindrical composite shell.

MFC actuator modeling scheme in ANSYS.

The finger electrode width and the required electric field are simulated by assuming a suitable element width and its associated element potential to be imposed as boundary conditions.

The actual electrode area is

Before generating the system matrices for controller design, the modeling scheme of MFC in finite element procedure is validated with experiment and ANSYS analysis. For this purpose we have used only two actuators

Induced deflection of shell with two MFC actuators.

Actuator state | Experiment | Shell facet element | ANSYS | |||

Position A Disp (mm) | Position B Disp (mm) | Position A Disp (mm) | Position B Disp (mm) | Position A Disp (mm) | Position B Disp (mm) | |

0.025 | 0.028 | 0.027 | 0.027 | 0.026 | 0.026 | |

— | 0.085 | −0.052 | 0.079 | −0.053 | 0.077 | |

0.085 | — | 0.079 | −0.052 | 0.077 | −0.053 |

Position A (

The free vibration analysis is carried out to compute the frequencies and mode shapes. The results are presented in Table

Dynamics of composite shell.

Mode number | Frequency (Hz) | Damping | |
---|---|---|---|

Experiment | FEM (Present) | ||

Shell mode 1 | 53.4 | 52.12 | 1.24 |

Shell mode 2 | 95.3 | 103.46 | 0.77 |

Shell mode 3 | 170.2 | 171.43 | 0.38 |

Shell mode 4 | 186.1 | 193.05 | 0.36 |

Shell mode shapes.

The experimentally measured frequencies are listed in Table

The simulation studies are performed under sine disturbances, before implementing the designed controller in hardware-in-loop experiments. Resonance mode control (IMSC), combination resonance control (SMSC) concepts have been attempted on the composite shell structure. The computed control gains are presented in terms of actuator voltages in Table

Voltages applied to actuators for control of various modes.

Mode(s) | Actuator voltage in (volts) | ||

1 | 44 | 44 | — |

2 | 96 | 88 | — |

3 | — | — | 48 |

4 | — | — | 96 |

1 and 3 | 35 | 35 | 60 |

2 and 4 | 60 | 60 | 60 |

Frequency response plots of LQG (IMSC) scheme.

Bode response of the first mode (52.12 Hz)

Bode response of the third mode (171.43 Hz)

Bode response of selective modes 1 and 3 using LQG (SMSC) scheme.

Development of active vibration control scheme consists of controller design and its implementation with analog or digital instrumentation. In most of the industrial applications, PID control finds a major place for its simplicity; however tuning a PID controller for particular environment may not work or be feasible for every loading conditions. Therefore, modern control concepts are considered to address the system performance in a wider spectrum of loadings, including the system uncertainties. Linear Quadratic Regulator (LQR) is a simple and effective control algorithm; but in order to implement this algorithm for real-time application, there is a need to have the full state information of the vibrating system. A full state feedback based on LQR is practically feasible with Kalman filter as a system estimator. Therefore, in the present work, we have adopted modal Kalman filter along with the modal LQR controller in order to implement both independent (IMSC) and selective modal control (SMSC) schemes. System information is read in the form of displacement through PZT patches, surface bonded below the composite shell, and velocity information is subsequently estimated through Kalman filter.

The composite cylindrical shell is fabricated using CFRP bidirectional cloth, subsequently instrumented with three MFC actuators on the top. The following actuators of Smart Materials are used: M4010 P1, active area ^{2} and M8528 F1, active area ^{2}. In order to have a collocated actuator and sensor configuration, three PZT-5A patches are surface bonded, right below every MFC actuator. The location for the actuators are chosen based on the mode shape such that the actuators

Instrumentation for applying the disturbance,

Sensor electronics,

Actuator electronics,

DSP.

Experimental setup.

Three PZT patches are employed as feedback sensors; in addition, an accelerometer is used at the tip of the shell for observation purpose. A charge conditioner is used for collecting the output of accelerometer and force transducer. The disturbance loop consists of a signal generator, power amplifier, and a force transducer. The charges induced from the PZT patches are received by sensor electronics and are fed into the ADCs of DS 1104 board. The LQG controller is implemented with the help of real-time workshop/RTI software as SIMULINK block set; refer to Figure

Simulink model of LQG controller.

A virtual instrumentation panel is created in control desk (dSPACE product) to monitor the disturbance force, sensors’ output, and actuators’ input. It facilitates to fine-tune the control gains of the actuators in real time (see Table

A thorough modal survey is performed on the shell to establish its frequencies, damping for the first four modes (see Table

The Linear Quadratic Gaussian (LQG) controller has been designed such that each mode is significantly controlled through independent modal space control (IMSC) scheme. The effectiveness of modal control concept for multiple modes is examined through a combination resonance or selective modal space control (SMSC) technique. For this purpose, two modal sets are selected, namely, 1, 3 and 2, 4. The state-space matrices of selective modes are combined to form a single system with multiple target modes. Furthermore, LQG control is designed for these combination resonances. Modal coupling is a normal phenomenon in thin-walled composite shell structures; with the presence of aerodynamics, noise, and so forth, the coupled problems such as aeroelasticity, vibroacoustics may pose serious challenges for the designers. Therefore, in such situations, the selective modal control may certainly improve the structural performances and reduce the control spillover and number of actuators.

In order to examine the usefulness of the developed SMSC technique, we have conducted active control experiments on the cylindrical CFRP shell. The AVC experiments are conducted by implementing the designed LQG (IMSC) and LQG (SMSC) controllers in DS1104, an R & D DSP board of dSPACE. The DSP has got multichannel ADCs and DACs, which are used for setting the multi-channel closed-loop configuration. The actuators

Comparison of control performances by IMSC and SMSC schemes.

Mode(s) | Open-loop response in g (m/sec^{2}) | Closed-loop response in g (m/sec^{2}) | % Control |
---|---|---|---|

1 | 3.6 | 1.0 | 83.15 |

2 | 2.8 | 0.4 | 96.86 |

3 | 2.9 | 1.2 | 73.33 |

4 | 2.0 | 1.0 | 72.97 |

1 and 3 | 4.6 | 2.3 | 66 and 82 |

2 and 4 | 4.6 | 2.4 | 85 and 69 |

Resonant mode control (IMSC).

Shell Mode 1

Shell mode 2

Shell mode 3

Shell mode 4

Selective modal space control technique (SMSC).

Selective shell modes control (1 and 3)

Selective shell modes control (2 and 4)

The experimental structure considered is actually a deep-shell category, and therefore its elastic couplings appear to be very significant. Hence the first four modes are targeted to demonstrate the ability of in-plane actuation of MFC’s in controlling the out-of-plane elastic couplings. The PZT fibers are oriented in actuators

The outcome of the experiments is examined and the following observations are made.

Directionally efficient MFC actuators can be effectively employed in IMSC scheme of composite shell or panel vibrations for example, cases like a tonal-noise induced vibro-acoustic response control (refer Figures

In-plane actuation strain of MFC can be effectively tailored with elastic couplings of shell structures, namely, membrane-bending, membrane-shear, and membrane-twisting. It is shown by the actuator performance of

SMSC technique is observed to be very useful for combination resonances especially for the 2nd and 4th modes, where one can notice that independently the actuators

In the 1st and 3rd modes’ control, though the trend appears to be same, only marginal benefit is seen from power saving point of view; however from control spill over consideration, combination resonance shows a promising feature because both modes are simultaneously controlled.

The developed facet-shell element is efficient in modeling composite shell with MFC and is incorporated in ANSYS for large smart structural analysis.

The user programmable feature (UPF) available in ANSYS is utilized (USER300 subroutine) to implement the shell4 piezoelectric composite element. We have employed the developed coupled FE procedure with our own degrees of freedoms (mechanical and electrical) to compute the element matrices (stiffness, mass, and piezoelectric). The MATLAB codes are rewritten in Visual Intel FORTRAN for this purpose. The other features (preprocessor, solver, and postprocessor) of ANSYS are then effectively used.

A four-node composite facet-shell element is developed based on first-order shear deformation theory through a co-ordinate transformation of a plate element. The element warping is included further to correct its out-of-plane deformation to capture the shell behavior properly. Electro-mechanical coupling is introduced through a linear piezoelectric theory to idealize the macrocharacteristics of MFC actuators and PZT-5A sensors. The developed element is subsequently validated with quasi-static piezoelectric coupling experiment conducted on a cylindrical shell structure.

Modal control technique, IMSC, is applied to LQG control procedures and it has been extended to address the combination resonances control (SMSC) of shell vibrations. A cylindrical composite shell vibration control has been illustrated, where both simple resonance and combination resonances controls are demonstrated. It is observed that IMSC is best suited for modes that are quite separated from other neighborhood elastic modes, whereas SMSC technique appears to be more appropriate for modes, which may develop a strong coupling with adjacent/closely placed poles or modes of almost similar nature (2nd and 4th modes in the present case). The efficiency of directionally active MFC actuators is thus evaluated for their application in the geometrically curved aircraft composite panels.