AXIAL INSTABILITY OF ROTATING RODS REVISITED

For strain sufficiently small such that Hooke's Law is valid, it is sh own that only a linear model for axial deformation of rotating rods ca n be derived. As discussed in the literature, this linear model exhibi ts an instability when the angular speed reaches a certain critical va lue. However, unless this linear model is valid for large strain, it i s impossible to determine whether this instability really exists; beca use, as the angular speed is increased, the strain becomes large well short of the critical speed. Next, axial deformation of rotating rods is analyzed using two strain energy functions to model non-linear elas tic behavior. The first of these functions is the usual quadratic stra in energy function augmented with a cubic term. With this model it is shown that no instability exists if the non-linearity is stiffening (i .e. if the coefficient of the cubic term is positive), although the st rain can become large. If the non-linearity is of the softening variet y, then the critical angular speed drops as the degree of softening in creases. Still, the strains are large enough that, except for rubber-l ike materials, a non-linear elastic model is not likely to be appropri ate. The second strain energy function is based on the square of the l ogarithmic strain and yields a softening model. It quite accurately mo dels the behavior of certain rubber rods which exhibit the instability within the validated range of elongation.