According to the Mindlin plate theory and the first-order piston theory, this work obtains accurate closed-form eigensolutions for the flutter problem of three-dimensional (3D) rectangular laminated panels. The governing differential equations are derived by the Hamilton’s variational principle, and then solved by the iterative Separation-of-Variable (iSOV) method, which are applicable to arbitrary combinations of homogeneous Boundary Conditions (BCs). However, only the simply-support, clamped and cantilever panels are considered in this work for the sake of clarity. With the closed-form eigensolutions, the flutter frequency, flutter mode and flutter boundary are presented, and the effect of shear deformation and aerodynamic damping on flutter frequencies is investigated. Besides, the relation between panel energy and the work of aerodynamic load is discussed. The numerical comparisons reveal the following. (A) The flutter eigenvalues obtained by the present method are accurate, validated by the Finite Element Method (FEM) and the Galerkin method. (B) When the span-chord ratio is larger than 3, simplifying a 3D panel to 2D (two-dimensional) panel is reasonable and the relative differences of the flutter points predicted by the two models are less than one percent. (C) The reciprocal relationship between the mechanical energy of the panel and the work done by aerodynamic load is verified by using the present flutter eigenvalues and modes, further indicating the high accuracy of the present solutions. (D) The coupling of shear deformation and aerodynamic damping prevents frequency coalescing.

Stiffened plate structures are widely used in aerospace, shipbuilding and other engineering fields, so the research on its vibration characteristics has significant academic and practical value. Based on the Rayleigh quotient and Rayleigh-Ritz method, this paper presents a semi-analytical method to solve the natural modes of rectangular stiffened plates. In this method, the closed-form mode functions of thin plates are used as the basis functions of the mode functions of stiffened plates, and the natural frequency equation and the mode functions of rectangular stiffened plates are derived according to the Rayleigh quotient. In addition, the governing differential equation of the simply supported stiffened rectangular plate with single stiffener is derived according to the Rayleigh quotient, and the exact solution is obtained using the separation-of-variable method. The effectiveness and accuracy of the proposed method are verified by comparing the present results with those of the finite element method and literature. The proposed method in this work can be used for theoretical analysis and parametric design of stiffened plates.

The Wilson-ρ_∞ technique is formed by establishing a relationship between ρ_∞ and θ, as the high-frequency dissipation of the Wilson-θ method cannot be properly controlled by ρ_∞ (spectral radius at infinite frequency). The numerical performances of this approach are compared with those of the Generalized-α method. In the Wilson-ρ_∞ method, there are two different θ for a given ρ_∞. The characteristic roots of the Jacobi matrix corresponding to both θ are different, and the corresponding Wilson-ρ_∞ method has different numerical performances. A better θ is recommended according to the properties of the spectral radius. In addition, an analog system of a single degree-of-freedom forced vibration system is constructed with the dissipation and frequency of the Wilson-ρ_∞ method, and the initial conditions of which and the forces acting on the analog system are the same with those of the original system. It is evident that the steady state responses have no cumulative amplitude errors and phase errors, and the results of the Wilson-ρ_∞ method match the analytical solutions of the analog system.

Highly accurate closed-form eigensolutions for flutter of three-dimensional (3D) panel with arbitrary combinations of simply supported (S), glide (G), clamped (C) and free (F) boundary conditions (BCs), such as cantilever panels, are achieved according to the linear thin plate theory and the first-order piston theory as well as the complex modal analysis, and all solutions are in a simple and explicit form. The iterative Separation-of-Variable (iSOV) method proposed by the present authors is employed to obtain the highly accurate eigensolutions. The flutter mechanism is studied with the benefit of eigenvalue properties from mathematical senses. The effects of boundary conditions, chord-thickness ratios, aerodynamic damping, aspect ratios and in-plane loads on flutter properties are examined. The results are compared with those of Kantorovich method and Galerkin method, and also coincide well with analytical solutions in literature, verifying the accuracy of the present closed-form results. It is revealed that, (A) the flutter characteristics are dominated by the cross section properties of panels in the direction of stream flow; (B) two types of flutter, called coupled-mode flutter and zero-frequency flutter which includes zero-frequency single-mode flutter and buckling, are observed; (C) boundary conditions and in-plane loads can affect both flutter boundary and flutter type; (D) the flutter behavior of 3D panel is similar to that of the two-dimensional (2D) panel if the aspect ratio is up to a certain value; (E) four to six modes should be used in the Galerkin method for accurate eigensolutions, and the results converge to that of Kantorovich method which uses the same mode functions in the direction perpendicular to the stream flow. The present analysis method can be used as a reference for other stability issues characterized by complex eigenvalues, and the highly closed-form solutions are useful in parameter designs and can also be taken as benchmarks for the validation of numerical methods.