Dynamical nonlinearities in piezoelectric materials have been investigated over different time and frequency scales using four different methods; measurements of displacement and electric polarization of bulk material, measurements of nanometer scale surface structure of the material by an atomic force microscope (AFM), and numerical modelling of ferroelectric materials with quenched randomness. Laser vibrometer measurements of the deformations of piezoelectric materials, d31 type PZT and PMN-PT sheets, have been done under sinusoidal voltage loading with different frequencies. This yields information about the dynamical hysteresis behavior, such as the area of the hysteresis loops as a function of the applied frequency f and voltage amplitude. Similarly the hysteresis loops have been measured for the electric polarization of the same samples. Relaxation behaviors of the same materials have been measured by an AFM. Topography of the piezo sheets was measured after applied DC voltage, indicating slow collective changes in the polarization close and at the sample surface. To investigate the time-dependent hysteresis, we have studied numerically a Ginzburg-Landau-Devonshire (GLD) model for ferroelectric materials including dilution type quenched randomness. Quantities studied include the area of the hysteresis loop, of the polarization in the material, and the coercive electric field Ec as a function of the frequency f, both as a function of the disorder strength.