Free vibrations of a flat frame partially resting on a Winkler elastic foundation in terms of uneven distribution of flexural stiffness

This work was devoted to the issue of free vibrations and loss of stability of a Γ-type flat frame made of a prismatic beam and a column with a variable cross-section, partially supported on a Winkler elastic foundation. The physical model of the system was subjected to the Euler’s force. The problem was formulated on the basis of the Bernoulli-Euler theory. Based on the Hamilton’s principle, differential equations of displacements and boundary conditions were determined. The numerical algorithm was used to find the maximum critical force being a function of many variables. Within the kinetic criterion of loss of stability, the changes in the natural frequency of optimized frame as a function of external load was determined. On the basis of the obtained results it was concluded that it is possible to control (improve) the dynamic properties while improving the stability of the system through elastic base support and appropriate shaping.