Several solutions for multimodal vibration damping of thin mechanical structures based on piezoelectric coupling have been developed over the years. Among them, piezoelectric network damping consists in using piezoelectric transducers to couple a structure to an electrical network, where the transferred electrical energy can be dissipated. In particular, the effectiveness of coupling rods, beams and plates to their analogous electrical networks has been proven. This work is the first step going towards more complex structures. After defining and experimentally validating a fully passive electrical analogous network of a simply-supported plate, the study is extended to the damping of a non-periodic plate. The non-periodicities here studied include the addition of a local mass and a variable thickness. Numerical simulations and experiments show that in these cases, a broadband damping is achieved once the piezoelectric transducers are coupled to an adequate analogous network. A finite element model of the structure coupled to a 2D non-periodic electrical network is concurrently developed and validated, which is another contribution of the present work.