Repeated Eigenvalues For The Beam Optimization Problems Of Thermal Vibration And Buckling

Thermal problems such as heat conduction, heat radiation and friction heat are often happened in our life and industry. The structure under thermal will produce thermal stress with the restriction of thermal deformation caused by the property expanding with heat and contracting with cold. Thermal load acting on a beam-column, which is fixed at its both ends to restrict its thermal expansion generates axial compressive force. The axial compressive force could reduce the fundamental frequency, even thermal buckling happens when thermal load is up to big enough. Through changing the cross sectional distribution, it can change the bending stiffness and compressive force in addition to change the mass and stiffness distribution, thus change the frequency and critical buckling temperature.Consider the beam with geometrically similar cross-sections of variable size, with the given volume and length constraint, it is necessary to maximize the fundamental frequency or critical buckling temperature through changing the cross-sections distribution. In consideration of repeated eigenvalues, the necessary optimality condition for a maximum refer to a single and bimodal problem is studied, and the sensitivities of single and repeated eigenvalues are solved. Applying optimality criterion method introduced by Olhoff, we undergo the optimization procedure for maximizing the thermal fundamental frequency and critical buckling temperature.Numerical examples show that thermal fundamental frequency and critical buckling temperature both increase by optimization. And bimodal optimum design is related to the cross sectional lower bound. When maximize fundamental frequency, the frequency increment is bigger with a higher thermal load, and in addition to cross sectional area lower bound, the bimodal optimum design also bases on thermal load. At last, we compare the cross sectional area distribution obtained by maximizing the thermal fundamental frequency, critical buckling temperature and critical compressive load, they are quite similar. What’s more, we discuss the similarity reason through studying the optimization problem of minimizing the compressive force caused by the thermal load.