Statička i dinamička analiza elastičnog štapa promenljivog preseka metodom diskretizacije na krute segmente

Abstract

A new approach to the discretization of the non-homogeneous flexible beam with variable cross-section into the rigid segments is proposed in this dissertation. The Euler-Bernoulli beam model was considered. Absolute coordinates relative to the inertial coordinate system were used to describe the position of the rigid segments. The differential equations of motion of the considered system of rigid segments were formed into the two steps. In the first step, the differential equations of motion of the system of three rigid segments, by which the one flexible segment of constant parameters is discretized, were formed. The Lagrange's equations of the first kind were used for this purpose due to the presence of redundant coordinates. After the elimination of the Lagrange multipliers, the differential equations of motion of the flexible segment of constant parameters in independent coordinates were obtained. In the second step, the differential equations of motion of the entire variable-parameter flexible beam were formed by using the Lagrange equations of the second kind. Differential equations of motion of the discretized model of axially compressed flexible beam with arbitrarily variable parameters in the form of the system of rigid segments were obtained. On the basis of the obtained differential equations of motion, the characteristic problem is formed from which it is possible to analyze the modal characteristics and the value of critical buckling force of the considered beam. The proposed method is verified through numerical examples. The proposed method of discretization of the flexible beam is extended to the dynamic analysis of the compliant mechanisms and the rotational flexible beam. Compliant mechanisms in which the rigid members and flexible joints are serially connected in the form of an open kinematic chain without branching were considered. The proposed discretized model of the compliant joint takes into account the shear effect. By appropriate selection of coordinates of the compliant members points it is possible to determine their displacements in an efficient manner. Also, by using the proposed approach it is possible to analyze the modal characteristics of this type of mechanisms. The members that describe the influence of the inertial forces on the beam during the beam rotation are identified in the differential equations. As intensity of the beam angular velocity increases, the some members of the stiffness matrix decrease. This phenomenon is usually called the effect of dynamic softening of beam during the rotation and it is characteristic of the linear models. The efficiency of the formed discretized models of the compliant mechanisms and the rotational flexible beam was verified in numerical examples.