Modeling method for boundary conditions of the rectangular thin plate by using the power series polynomial

There exist complex deformation characteristics and torque transmission mechanisms of the structure whose boundary is constrained locally. This paper proposes the power series polynomial constraining method to establish the dynamical modeling of a rectangular thin plate with a localized constraint. The boundary condition can be constructed by applying the power series polynomial with undetermined coefficients to the free boundary directly. This means that the derivation of the admissible function no longer relies on the first term associated with a specific constraint. The undetermined coefficient of the power series polynomial can be obtained while calculating the weight coefficient of the admissible function by using the Rayleigh–Ritz method. Then natural frequencies are calculated and polynomial coefficients can be further obtained. The influence of variations in boundary length, constraint length, constraint position and multiple discontinuous localized constraints on natural frequencies of the plate is studied. Convergence verification is performed for the truncated number of orthogonal polynomials and power series multipliers. Then the appropriate number of the term for the power series multiplier is determined. Natural frequencies of the cantilever plate and the opposite sides simply supported plate obtained by using the proposed method are compared with those obtained using the traditional method. Then natural frequencies of the plate with a local boundary constrained are compared with those obtained from the finite element software MSC.Patran. The fairly low relative error demonstrates the validity of the proposed method. The dynamical response analysis shows the superiority of the proposed method for the boundary locally constrained boundaries, which cannot be adequately handled by the traditional method.. The power series polynomial overcomes the limitation that the traditional Lagrange multiplier method can only construct point constraints.