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 frst 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 frst 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 varia...ble-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 effcient
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 identifed 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
effciency of the formed discretized models of the compliant mechanisms and the rotational
flexible beam was verifed in numerical examples.
Faculty:
University of Kragujevac, Faculty of Mechanical and Civil Engineering, Kraljevo
