Hi Kumar,

it depends what you want to model. If you have a small piezoceramic disc without any support you can assume the displacement free boundary conditions. If your tool become bigger or is supported in reality you have to take this into account.

Because you model piezoceramics you will need additional boundary conditions for the electrical potential. This conditions can be short circuit or open. This will influence your stiffness and as a result your eigenfrequencies.

To determine if your model is correct or not, you have to use a reference solution and in the end you have to perform a experiment.

Sir ,my tool model is of 125 mm.Should I analyzed without boundary conditions? How it will affect the result?

I am not used the Abaqus, I prefer to use PASTRAN/NASTRAN. It is possible to perform the modal shape analysis without definition BC.

I suggest you use Multi constraint equation to specify the DOF to be active. If you're analyzing simple beam element structures, I suggest you refer to readily available formulas such as 'Blevins formulas for natural frequencies and mode shapes'.

Cheers, 

Without boundary conditions it is not an structure is a mechanism, then fem analysis is not applicable

I would like to endorse Claudio. Which situation (with or without b.c.) You have to apply to Your analysis depends only on the situation you want to model. Taking real parts, which are mostly attached to some structure, it is appropriate to use b.c.. But the problem is, that rigid b.c. provided by fixed nodes are often to rigid. In reality a connection of parts to structure results in a new bigger structure with different modal properties. Using standard b.c. is only appropriate, when mass and stiffness of the support structure doesn't affects the part to much. So you can use flexible b.c., superelements or model support pars of the bigger structure. The decision is up to You ;)

Boundary conditions on my work are imposed when you sample the structure, That is, the grid that you use. In my work, the sampling of the structure and the data entered into the computer program sat the boundary conditions. The inaccuracies are overcome by removing the outlying points and substituting an average of the data from the 8 points surrounding the outlying point on the grid that you use to sample the structure. Remember, when concerned with the vibrations, it is possible for two vastly different structures to vibrate in the same or similar nodes. 

For example. Suppose we are able to sample a Strativarius violin. The sweet sounds that it produces are a result of its structure and the musician who plays it. The structural vibrations  may in theory also be produced by another structure. The trick is to produce all of the multiple modes of vibration that make the sound that you are hoping to duplicate by building the structure to produce those specific modes of vibration in exactly the same manner as the original sampled instrument vibrates. For a Luthier who produces violins the art of his or her craft is significant. 

The vibrations of large structures must follow the same laws of Physics. However, the boundary conditions of the structure are also affected by the physical conditions of the environment in which the sampling occurs. Sometimes little can be done to change environmental boundary conditions. You are stuck with what nature gives to you unless you want to use an artificially controlled laboratory environment that is removed from the real world. Most scientists have a goal of adjusting laboratory conditions to simulate real world conditions. I hope this helps answer your specific question. 

The question is same as asking:

Can we get a closed form solution of a PDE without boundary conditions; the answer is NO.

Modal analysis is all about stiffness matrix [K] and mass matrix [M];

For a bar problem K = EA (\partial^u / \partialx^2); and M = \rho A (\partial^2 / \partial t^2);

If we solve this equation using separation of variable we will get closed form solution with 4 unknowns; WHOSE SOLUTION IS NOT POSSIBLE WITHOUT BC

However, in FE we use Galerkin approach The same applies here as well..