Study of the Centrifugal Stiffening Effect Using the Finite Element Absolute Nodal Coordinate Formulation

The finite element absolute nodal coordinate formulation is usedin this investigation to study the centrifugal stiffening effect onrotating two-dimensional beams. It is demonstrated that the geometricstiffening effect can be automatically accounted for in the above mentionedfinite element formulation by using an expression for the elastic forcesobtained with a general continuum mechanics approach. The Hill equation thatgoverns the vibration of the rotating beam is obtained in terms of a set ofgeneralized coordinates that describe the beam displacements and slopes.Under the assumption of small deformation, the Hill equation is linearized,and the complete solution is obtained and used to demonstrate analyticallythat such a solution does not exhibit instabilities as the angular velocityof the beam increases. The results obtained using this finite elementprocedure are compared with the results reported in the literature.     

Definition of the Elastic Forces in the Finite-Element Absolute Nodal Coordinate Formulation and the Floating Frame of Reference Formulation

The equivalence of the finite-element formulations used inflexible multibody dynamics is the focus of this investigation. Thisequivalence will be used to address several fundamental issues related tothe deformations, flexible body coordinate systems, and the geometriccentrifugal stiffening effect. Two conceptually different finite-elementformulations that lead to exact modeling of the rigid body dynamics will beused. The first one is the absolute nodal coordinateformulation in which beams and plates can be treated as isoparametricelements. This formulation leads to a constant and symmetric mass matrix andhighly nonlinear elastic forces. In this study, it is demonstrated thatdifferent element coordinate systems which are used for the convenience ofdescribing the element deformations lead to similar results as the elementsize is reduced. In particular, two element frames are used;the pinned and the tangent frames. The pinned frame has one ofits axes passing through two nodes of the element, while the tangent frame isrigidly attached to one of the ends of the element. Numerical resultsobtained using these two different frames are found tobe in good agreement as the element size decreases. The relationshipbetween the coordinates used in the absolute nodal coordinate formulationand the floating frame of reference formulation is presented. Thisrelationship can be used to obtain the highly nonlinear expression of thestrain energy used in the absolute nodal coordinate formulation from thesimple energy expression used in the floating frame of referenceformulation. It is also shown that the source of the nonlinearityis due to the finite rotation of the element. The result of the analysispresented clearly demonstrates that the instability observedin high-speed rotor analytical models due to the neglect of the geometriccentrifugal stiffening is not a problem inherent to a particular finite-element formulation. Such a problem can only be avoided by considering the known linear effect of the geometric centrifugal stiffening or by using a nonlinear elastic model as recently demonstrated. Fourier analysis of the solutions obtained in this investigation also sheds new light on the fundamental problem of the choice of the deformable body coordinate system in the floating frame of reference formulation. Another method forformulating the elastic forces in the absolute nodal coordinate formulationbased on a continuum mechanics approach is also presented.