## X. Transient analysis of an in-vacuum Piezoelectric or Magnetostrictive Structure Driven by External Forces (TRA2)

In this analysis, known loads (forces F, electric charges q,magnetic fluxes fb) are applied on thepiezoelectric or magnetostrictive structure. The system of equations becomes:

where × and >×× denotethe first and second time derivative respectively. w0 is the pulsation at which the materials losses are defined. This differentialequation is solved by an iterative method, taking a constant time step D t. Three methods areimplemented: the Central Difference Method, the Newmark Method and the Wilson-q Method. The method, the time step and the method’sparameters are defined with the TRANSIENT command. Thecode computes the displacement field U, theelectrical potential F, the reducedmagnetic potential f, and the currentsin magnetic sources I for the requested timesteps. The problem can be solved only in double precision.

Important notice: the Central Difference Method algorithm does not accept losses for electric or magnetic degrees of freedom, only for mechanical degrees of freedom. This means that the matrices [K"F], [K"uf], [K"uI], [K"FF], [K"fI], [K"ff] and [K"II] must be zero.