A hierarchical generalized formulation for the large-displacement dynamic analysis of rotating plates

Abstract

A hierarchical formulation is presented for the large displacement modal and transient analysis of rotating plates having different thicknesses, sizes, and settings angle with respect to the spinning axis. After derivation of the governing equations of motion by the Principle of Virtual Displacements, the finite element discretization yields a set of nonlinear ordinary-differential equations for the generalized coordinates including gyroscopic terms, centrifugal/Euler accelerations and spin-softening effects. A Total Lagrangian approach is adopted. The modal analysis is performed by a two-step procedure. The static shape of the structure deformed by the centrifugal force is first defined. Subsequently, eigenfrequencies and mode shapes are computed by a classical eigenvalue problem past the static configuration previously computed. The transient analysis of the plates subject to a varying angular velocity is tackled by solving the nonlinear equations of motion by the Generalized-$\alpha$ method coupled to the Newmark’s approximation for the velocity and acceleration fields. It has been numerically proven that accurate results can be obtained by the use of Murakami’s Zig-Zag Function which a-priori enforces the interlaminar discontinuity of the displacements’slopes in the Equivalent Single Layer axiomatic model, thus avoiding higher-order polynomial representations for the displacement fields, or the use of Layer-Wise theories.

A GUF matrices

1.1 Centrifugal forces

The centrifugal acceleration in a finite element formulation generates both an additional stiffness matrix and an apparent force vector. Its contribution to the virtual work is:

(96)

(97)

Performing the spatial discretization of :

(98)

Operating with a similar approach, is written as

(99)

The stiffness and force kernels of the apparent centrifugal force due to the rotation of the body can be identified:

(100)

For example:

(101)

(102)

1.2 Coriolis forces

The Coriolis acceleration provides an additional damping matrix added, if present, to the structural ones. The virtual work contribution is:

(103)

GUF as well the spatial discretization are applied to both the virtual displacement and the velocity. e.g.

$$\begin{aligned}&\,_0\delta u_x(x ,y ,z) = \;{}^x_0F_{\alpha _{u_x} } (z) \;{}^x_0N_I(x,y)\; \,_0\delta U_{x\alpha _{u_x}I } (x ,y) \end{aligned}$$

(104)

$$\begin{aligned}&\,_0\dot{u}_x(x ,y ,z) \;= \;{}^x_0F_{\alpha _{u_x} } (z) \;{}^x_0N_I(x,y)\; \,_0\dot{U}_{x\alpha _{u_x}I } \;\, (x ,y) \end{aligned}$$

(105)

where

$$\begin{aligned} \begin{array}{l} \alpha _{u_x} \!\! = t,l,b \quad l = 2,\dots ,N_{u_x} \quad I = 1,2,\dots M_n \end{array} \end{aligned}$$

(106)

and $M_n$ is the number of nodes of the element.

The explicit expression for the semi-discretized form of Eq. 103 is

(107)

It is possible to recognize the gyroscopic damping matrix:

(108)

where for example the kernel of the matrix relating the $\alpha _{ux}$ term of the polynomial/Legendre function for the x displacement of the node I and the $\alpha _{uy}$ for the y velocity function of node J is

(109)