Modal analysis for multimodal piezoelectric energy harvester

1) What is the continuity equation between the main beam part 3 and the branched beam?

For single beam, four unknown constants will be there in mode shape function. 

Boundary conditions: 

Fixed end: displacement and slope of the main beam part 1 (2 conditions)

Continuity equation between main beam part 1 and part 2 (4 conditions)

Continuity equation between main beam part 2 and part 3 (4 conditions)   Problem to find Continuity equation between main beam part 3 and Branched beams (4 conditions)   Square matrix  not coming for overall beam to find the frequency parameter (lambda) for each part of the beam. 

2)  How to get the equation for the moving base for branches is the free end of the main beam?

Reference: "Analytical modeling and validation of multi-mode piezoelectric energy harvester"