Geometric stiffness and stability of rigid body modes

Abstract The objective of this study is to examine the effect of geometric stiffness forces on the stability of elastic and rigid body modes. A simple rotating beam model is used to demonstrate the effect of axial forces and dynamic coupling between the modes of displacement on the rigid body motion. The effect of longitudinal deformation due to bending is systematically introduced to the dynamic equations using the principle of virtual work. The effect of higher order terms in the inertia forces as the result of including longitudinal displacement caused by bending deformation is examined using several models. One of these models is a linear model is which the effect of longitudinal displacement due to bending is neglected in formulating the inertia forces, but this effect is considered when the elastic forces are formulated. This model shows unstable behavior at high values of the angular velocity of the beam. Three different beam models are then developed in order to examine the effect of geometric stiffness forces. In the first model, called theconsistent complete model(CCM), the effect of longitudinal displacement caused by bending is included in formulating both the inertia and elastic forces. In the second model, called theconsistent incomplete model(CIM), the effect of longitudinal displacement due to bending is neglected in formulating both the elastic and inertia forces. In the third model, thesecond inconsistent model(SIM), the effect of longitudinal displacement due to bending is included in formulating the inertia forces, but this effect is neglected when the elastic forces are formulated. Numerical results obtained in this investigation demonstrate that the three models lead to a stable solution at high values of angular velocities. These results also demonstrate that including the effect of longitudinal displacement due to bending in the inertia forces is not the only approach that can be used to maintain the beam stability at high values of angular velocity. The effect of geometric stiffness forces on the stability of rigid body modes of a translating and rotating beam model is also examined in this paper.