Thinking of Shell structure modeling and calculation, Shell elemement modeling using FEM is coming to my mind (maybe BEM is also sufficient). However focusing onto Vibration Shell elements might not be sufficient as you Need to consider bending momenta within the elements or you use several layers of FE elements over the wall thickness. This will allow you to Analyze the possible vibration modes of your Shell structure. But for an active Vibration control this Approach will not be suitable, because the extensive numerical calculation will be much too slow for it. If you can identify the eigenfrequencies of the vibration modes of your structure, you can use them elsewhere (e.g. SIMULINK) in simplified models / mathematical descriptions for more efficiency.

Usage of FEM with complex model is possible for such application in very efficient way. The correct method is so called modal dynamics. In general, it works in the same way as Ruediger said.

In short:

1. The complex structural model (solid, a few element on the thickness) is build

2. Given number of the eigenmodes (shapes and frequencies), lets say 100, is calculated.

3. The dynamics of the structure is converted from cartesian coordinates to so called modal coordinates, so instead millions of DOF you have only 100 DOF represented by calculated eigenmodes.

4. Transient dynamics of the structure, in your case response to the active damping, is simulated in extremely efficient way (only 100 DOF to calculate). In fact, it is superposition of all calculated eigenmodes, DOFs are just amplitudes of these modes. You can apply transient load or displacement to resemble the active damping

Number of modes taken into consideration depend on complexity of the structure vibrations, sometimes only 10 is enough. Results are very accurate, because of perfect stiffness estimation on full structural, solid model.

Because after calculation, the modal DOFs are transformed back into cartesian DOFs, postprocesing and visualization can be done on full structural model. You will get transient response of your structure.

Whole process is fully automatic, and available in commercial software, like Patran-Nastran. Its enough to change solver type from Transient Dynamics to Modal Dynamics.

Mr Kapil,

You should have mentioned its a Isotropic or orthotropic shell,

Mindlin- Resiner plate theory with transverse shear inclusion will be a good theory to model shells and to capture slopes .Control schemes are important for AVC.

you can have look into these papers too

1.S. Raja, T. Ikeda, and D. Dwarakanathan, “Deflection and

vibration control of laminated plates using extension and

shear actuated fiber composites,” Smart Materials Research,

vol. 2011, Article ID 515942, 15 pages, 2011.

2.V. Balamurugan and S. Narayanan, “Active vibration control

of smart shells using distributed piezoelectric sensors and

actuators,” Smart Materials and Structures, vol. 10, no. 2, pp.

173–180, 2001.

3.M. K. Kwak, S. Heo, and M. Jeong, “Dynamic modelling

and active vibration controller design for a cylindrical shell

equipped with piezoelectric sensors and actuators,” Journal of

Sound and Vibration, vol. 321, no. 3-5, pp. 510–524,2009

http://downloads.hindawi.com/journals/smr/2012/513271.pdf‎

The best way to handle active layers applied onto flat shell structures is using layerwise models (you can use partial layerwise models - where through thickness deformation is not considered, otherwise you shall use full layerwise theories).

For curves shells and through deformation of the shells you can used special solid-shells (also adopting "layerwise" formulations).

For the first use you can see:

R. A. S. Moreira, J. Dias Rodrigues, A. J. M. Ferreira

A generalized layerwise finite element for multi-layer damping treatments

Computational Mechanics, Vol.37, pp.426-444, 2006

For the second use check:

R.A.S. Moreira, R.J. Alves de Sousa, R.A.F. Valente, A solid-shell layerwise finite element for non-linear geometric and material analysis, Composite Structures 92(6):1517-1523, 2010

it depends on the fact that the shell is thin (Naghdi's model takes in account the transverse shear ) or very thin and regular, which mean that we can neglect the transvese shear energy, so the Koiter's model is preferable