## H. Modal Analysis of a Piezoelectric Structure (MOD2)

In this analysis, the system of equations is reduced to:

For different electrical boundary conditions, the code computes the eigenvalues and eigenvectors of the linear system that are thestructure's eigenvalues and eigenvectors. The computation is done using real values with no internal losses. Eigenvectors are normalized such that UT [M] U= 1, [M] acting for the meshed structure, that is including the axisymmetryfactor 2p for axisymmetric structures but notincluding symmetry factors induced by explicit boundary conditions.

The first equation system in this section can be rewritten, isolating the electrical potentials linked to the electrodes in the vector Fe and the internalelectrical potentials in the vector Fi:

The short-circuit modal analysis, obtained by taking Fe = 0 for the potential of the electrodes, leads tocomputation of the resonance modes. The open-circuit modal analysis, obtainedby taking q = 0,leads to the computation of the antiresonance modes. Calculating the resonanceand antiresonance modes can be carried out in one step (see Section III.D, ANALYSIS entry). In this case the excitation electrodes arereferenced in the EXCITATIONS entry.

The matrices are assembled and stored to file by columns. Lanczos' algorithm is used to solve the problem. It can be done only in doubleprecision. It has to be noted that:

- The retained algorithm provides the rigid body modes when theyexist
- This algorithm computes the eigenvalues very accurately even whenthe resonance modes are very close or when there is an ill-conditioned matrix.

Note:

The electrical degrees-of-freedom are placed at thebeginning of the assembled matrix. The numerical system can differ in sizefrom a harmonic analysis. Also, any provided mechanicalexcitation will be treated as blocked.