Finite element analysis of the geometric stiffening effect. Part 1: A correction in the floating frame of reference formulation

Abstract

The fact that incorrect unstable solutions are obtained for linearly elastic models motivates the analytical study presented in this paper. The increase in the number of finite elements only leads to an increase in the critical speed. Crucial in the analysis presented in this paper is the fact that the mass matrix and the form of the elastic forces obtained using the absolute nodal coordinate formulation remain the same under orthogonal coordinate transformation. The absolute nodal coordinate formulation, in contrast to conventional finite element formulations, does account for the effect of the coupling between bending and extension. Based on the analytical results obtained using the absolute nodal coordinate formulation, a new correction is proposed for the finite element floating frame of reference formulation in order to introduce coupling between the axial and bending displacements. In this two-part paper, two-and three-dimensional finite element models are used to study the problem of rotating beams. The models are developed using the absolute nodal coordinate formulation that allows for accurate representation of the axial strain, thereby avoiding the ill-conditioning problem that arises when classical displacement-based finite element formulations are used. In the first part of the paper, the case of linear elasticity is considered and assumptions used in the finite element floating frame of reference formulation are investigated. In the second part of the paper, non-linear elasticity is considered. A rotating helicopter blade is simulated, and the complexity of the motion suggests the inclusion of rotary inertia, shear deformation, and non-linear elastic forces in order to obtain an accurate solution that does not suffer from the instability problem regardless of the number of finite elements used.

A new locking-free shear deformable finite element based on absolute nodal coordinates

“…Special attention must be paid to Equation (27). As can be seen, the element is able to give the exact value provided by the linear theory of elastic beams.…”

“…It has been extensively studied in the literature (see for example Refs. [27,28]). In this example, a flexible beam is subjected to a prescribed rotation around a perpendicular axis that passes through one of its ends.…”

“…The final angular velocity is now increased to 4 rad/s, which is enough for the set of beam properties used to obtain unstable solutions when the linear theory of elasticity is used and no special measures are taken to account for axial-bending coupling [27]. Results obtained using the same models as in the previous simulation are shown in Fig.…”

Use of the floating frame of reference formulation in large deformation analysis: Experimental and numerical validation

“…The rate of change of the manipulator rotation makes the effect of centrifugal stiffening terms more significant. While for most flexible robotic manipulators, the link axial and torsional vibration modes can be neglected, in solving rotating beam problems using the FFR formulation, an incorrect and unstable solution is obtained if the axial modes are neglected due to lack of the geometric stiffening effect, as previously mentioned [10,11]. In this study, the large transverse deformation of the robot arm is measured, and the measured experimental results are compared with the numerical results obtained using the FFR and ANCF models.…”

Stability analysis of a substructured model of the rotating beam

“…They found that unstable results were obtained with a two elements model. Motivated by their study, García-Vallejo et al [7,8] studied the instability of the ANCF models of the rotating beam. They found that there exists a critical value for the angular velocity of the beam that made unstable the models.…”