Self-adjoint systems and non self-adjoint systems are two types of systems arising from structural dynamics. Most systems encountered in structural dynamics are self-adjoint in character. This character is due to their conservative nature of these systems. However, in the presence of some non-conservative forces, for example, the circulatory forces or forces associated with flutter, the systems become non self-adjoint. In this thesis, both self-adjoint and non self-adjoint spinning Timoshenko beam systems are investigated thoroughly. Fundamental mathematical concepts on adjoints of operators and matrix operators are defined and described. General analytical approaches based on modal analysis to solve these two types of systems are presented.

As an application of self-adjoint problems, dynamic analysis involving free vibration and forced vibration is carried out for a spinning Timoshenko beam system with general boundary con- ditions. System equations of motion are formulated from both the inertial coordinate frame approach and the rotating coordinate frame approach. In the free vibration analysis, frequency equations and mode shapes are derived for the six complete sets of general classical boundary conditions. A simplified solution for free vibration is achieved for a simply-supported spinning Timoshenko beam and a linearized expression to calculate the natural frequencies is also pre- sented in this case. The advantage of this linearized expression is that it is very easy to use and yet sufficiently accurate for engineering purposes. In the forced vibration analysis, dynamic response under a moving load is calculated analytically for four boundary conditions which do not involve rigid body motion. Both in plane and out of plane deflections are presented.

For a spinning beam, it is necessary to study the relationship between the frequencies and mode shapes calculated from the inertial frame approach and the rotating frame approach. In this thesis, it is proved that the frequencies obtained from these two formulations are different by a spin-rate while the mode shapes are identical. In addition, the work clarifies some concepts on critical speeds for a spinning Timoshenko beam system. Critical speeds are calculated for the six boundary conditions.

As an application of non self-adjoint problems, the free and forced vibration analyses are carried out for a cantilever spinning Timoshenko beam system subjected to both concentrated follower forces and distributed follower forces. Due to the presence of the follower forces, the system becomes non self-adjoint. The eigenpairs, comprising the eigenvalues and eigenvectors of both the original and the adjoint systems are computed via a series-type solution for the most general case. In the free vibration analysis, only the eigenpairs associated with the original system are required to obtain the system's natural frequencies and mode shapes. On the other hand, in performing the forced vibration analysis, it is necessary to use the eigenpairs of the adjoint system as well. In response calculation, an exponentially decreasing type of transverse force is considered. With the presence of the follower forces, the system may exhibit a flutter instability if the magnitudes of the follower forces are sufficiently large. The critical points pertaining to the onset of instability are presented.