The present study is concerned with the "Free Flexural Vibrations Response of Composite Mindlin Plates or Panels with a Centrally Bonded Symmetric Double Lap Joint (or Symmetric Double Doubler Joint). The plate "adherends" and the plate "doublers" are considered as dissimilar, orthotropic "Mindlin Plates" with the transverse and the rotary moments of inertia. The relatively, very thin adhesive layers are taken into account in terms of their transverse normal and shear stresses. The mid-center of the bonded region of the joint is at the mid-center of the entire system. In order to facilitate the present solution technique, the dynamic equations of the plate "adherends" and the plate "doublers" with those of the adhesive layers are reduced to a set of the "Governing System of First Order ordinary Differential Equations" in terms of the "state vectors" of the problem. This reduced set establishes a "Two-Point Boundary Value Problem" which can be numerically integrated by making use of the "Modified Transfer Matrix Method (MTMM) (with Interpolation Polynomials)". In the adhesive layers, the "hard" and the "soft" adhesive cases are accounted for. It was found that the adhesive elastic constants drastically influence the mode shapes and their natural frequencies. Also, the numerical results of some parametric studies regarding the effects of the "Position Ratio" and the "Joint Length Ratio" on the natural frequencies for various sets of support conditions are presented.