An eigenvalue analysis of Timoshenko beam undergoing coupled elastic-rigid body motion
Rotating beam is a common element of many mechanical systems such as propellers, helicopter rotors, and machines. In designing these mechanical systems, knowledge of the natural frequency of the rotating beam is crucial to avoid resonance. In this study, a method for calculating the natural frequency of a rotating beam including gyroscopic effect is presented. The method uses coupled elastic and rigid body (shadow beam method) and finite element dicretization to derive a set of second -order differential equations. This set of equations is transformed to a set of first-order differential equations of the form Ix + Gx - Q where / and G are symmetric and antisymmetric, respectively. A procedure which transforms this equation to an eigenvalue problem with a symmetric matrix is discussed. It is shown that this matrix is of the form Ar A where A is an antisymmetric matrix. The property that singular values and vectors of A are the eigenvalues and eigenvectors of AT A is used to calculate the natural frequencies. The results of a numerical study to show the effects of tip mass, spin softening, material properties, and speed on the natural frequencies are presented.