This paper presents a general approach for the free vibration analysis of curvilinearly stiffened rectangular and quadrilateral plates using the Ritz method by employing classical orthogonal Jacobi polynomials. Both the plate and stiffeners are modeled using first-order shear deformation theory (FSDT). The displacement and rotations of the plate and stiffeners are approximated by separate sets of Jacobi polynomials. The ease of modification of the Jacobi polynomials enables the Jacobi weight function to satisfy geometric boundary conditions without loss of orthogonality. The distinctive advantage of Jacobi polynomials, over other polynomial-based trial functions, lies in that their use eliminates the well-known ill-conditioning issues when a high number of terms are used in the Ritz method, e.g., to obtain higher modes required for vibro-acoustic analysis. In this paper, numerous case studies are undertaken by considering various sets of boundary conditions. The results are verified both with the detailed finite element analysis (FEA) using commercial software msc.nastran and with those available in the open literature. New formulation and results include: (i) exact boundary condition enforcement through Jacobi weight function for FSDT, (ii) formulation of quadrilateral plates with curvilinear stiffeners, and (iii) use of higher order Gauss quadrature scheme for required integral evaluations to obtain higher modes. It is demonstrated that the presented method provides good numerical stability and highly accurate results. The given new numerical results and convergence studies may serve as benchmark solutions for validating the new computational techniques.