Instability Regions of a Flexible Beam Under Controlled Follower Axial Force Using Ritz Method

This paper deals with a free-free beam under a controlled follower force by different locations of its sensor as a realistic simulation of an elastic launch vehicle thrust. A follower axial force is force acting on one end which keeps its direction normal to its free end through its motion. As the exerted axial force is the only force which excites the beam's lateral vibration the added controller is the way to control its solid rotation. The instability regions alter as the sensor location and its gain changes. Effect of the gain of the sensor and its location and their optimum values are discussed in this paper. As the coefficients of the governing equation changes along the structure axis, an approximate solution for the set of developed equations is presented in the paper using Ritz method. First, the natural frequencies of the beam model without its follower force is derived and compared to the one derived from the exact solution and checked the accuracy of the procedure, then after that the follower force is exerted the effect of the controller parameters is inspected. It can be seen the severe effects of non-conservative follower force and the controller on the beam's stiffness matrix which can change instability plots of the structure. Else, the controller optimum parameters as "the best location of sensor from the point of tolerating the maximum of follower force without facing to any instability" are derived in the conclusion section.

Instability Regions of a Flexible Beam Under Controlled Follower Axial Force Using Ritz Method Keywords:

“Beam” , “Instability” , “Ritz Method”