Steady-state dynamic analysis for piezoelectric materials

In the steady-state dynamics procedure the frequency of the sinusoidal load is user-defined, with a complex-valued solution for various quantities such as stresses and displacements that occur when steady state is reached. The accuracy of the modal procedure depends on the representation contained in the eigendata extracted previously, whereas the direct procedure utilizes all the available degrees of freedom. For the modal procedure the vertical deflection at the center of the top surface for the sinusoidally applied pressure load for the two models with no damping is 1.56 × 10⁻⁷ for the model that uses CAX4E elements and 1.55 × 10⁻⁷ for the model with CAX8E elements. For the direct procedure the deflection is 1.80 × 10⁻⁷ for the model with CAX4E elements and 1.79 × 10⁻⁷ for the model with CAX8E elements. If all the possible modes are used in the frequency extraction, the modal procedures give the same results as the direct procedure.

These results are in the same range as that reported in Mercer et al. (1987) for similar models. Since no damping is included in the model, the phase angles for all variables are either 0 or 180°. The phase angle for the 2-displacement at the center of the top surface is 180°. This phase shift is evident from Figure 1. The total force on the restrained bottom edge of the CAX8E model is output; the total force in the vertical direction matches the sum of the reaction forces on the edge.

The deflection at the center of the top surface for the analysis including Rayleigh damping using CAX4E elements (input file ppzossd3.inp) is 1.21 × 10⁻⁷ with a phase angle of 133°. The reduction in magnitude and the phase shift are evident from Figure 2. The problem including Rayleigh damping is also analyzed using CAX3E, CAX6E, and CAX8RE elements in input files ppzossd7.inp, ppzossd8.inp, and ppzossd9.inp, respectively. In these analyses the deflection and phase angle at the center of the top surface match with those obtained using CAX4E elements. In addition, the results from direct and modal dynamic solutions in each case are in good agreement.

The analyses using input files ppzossd4.inp and ppzossd4a.inp have a sinusoidally varying surface charge. Because of the inability of the modal procedures to transform these loads adequately into modal loads, only the direct procedure is used. The first step uses the REAL ONLY feature of the direct calculation analysis procedure. The second step includes both real and imaginary terms, with the load applied in the imaginary plane using the IMAGINARY parameter. The vertical deflection at the center of the top surface is 5.5 × 10⁻⁸. The phase shift for the vertical displacement at this location is 180° (no damping involved) for the case when the charges are applied in the real plane and −90.0° when the charges are applied in the imaginary plane. The potential at that nodal location is 158., with a phase shift of 0° for the first step and a phase shift of 90° in the second step.