Instability Regions of a Bernoulli Step Beam Under Controlled Follower Force Using Ritz Method

In this paper a free-free step beam under controlled follower force is studied. The aim is to obtain the critical force which causes the instability of the beam. The follower force is always normal to the beam cross section. As there are no other external forces in this model to control the solid rotation, the follower force is being used as a controller. For this reason, a simple controller was introduced. The effects of controller gain and the location of the sensor on the instability is also studied. An approximation method (Ritz Method) has been used to solve the equation as some parameters in the differential equation are not constant. In this method, the partial differential equation is transformed to a series of ordinary differential equations. First, the equations are solved without considering any non-conservative force (i.e. follower force). In the next step, the exact mode shapes are used as the admissible function to solve the vibration of the step beam considering that the follower force changes the stiffness matrix of the system. It is also showed that the controller gain is appeared in the stiffness matrix. It is also shown in this paper that changing a uniform beam to a step beam affects severely on the instability modes (divergence or flutter), the instability regions and the critical forces.

Instability Regions of a Bernoulli Step Beam Under Controlled Follower Force Using Ritz Method Keywords:

Step Beam , Beam Instability , Ritz Method