Asymptotic analysis of the vibrations of variable length cantilever beams

We investigate the behavior of a uniform cantilever beam with a small mass at its tip. The beam can move out of a rigid body; thus, its span varies with time. The body can have no motion or can be spinning; both simulations are considered. The reasons for such investigations are: (1) it is the subject of current interest in connection with the recent developments in the space and robotics fields; and (2) to study the behavior analytically because very few analytical solutions are available. The interest here is in the linear vibratory motions of the beam with the objective of developing approximate asymptotic expressions for the beam displacements. A formulation of the problem is made which accounts for the time mass. Two ways to eject the beam out of the system are considered: root force ejection and time dependent acceleration field ejection. The assumed modes method is adopted to derive the corresponding modal equations that describe the in-plane lateral deflections of the beam. The case when there is no tip mass is also included. Approximate asymptotic expressions for the beam displacement are presented for three types of length variations: uniform, exponential, and a linear variation of the square of the length. In the course of the analysis, the system is allowed to spin about an axis normal to the ejection direction, and modal equations governing vibrations in and out of the plane of rotation are derived. Approximate expressions for the beam displacement are presented in the absence of the tip mass for the three length variations mentioned previously. An approximate solution for the beam loaded by a tip mass is presented only for uniform length variations. The results from the investigation are discussed.