Mode veering in weakly coupled systems

The phenomenon of mode veering occurs in systems with a varying parameter. As the parameter varies, so do the natural frequencies. When two natural frequencies approach each other they often veer apart, instead of crossing. The veering is accompanied by rapid variations in the eigenvectors. This phenomenon is analysed in this paper with the focus being on weakly coupled systems of modes or oscillators. The system has any number of degrees of freedom or modes. The system is defined to be uncoupled when the motion in all but one of the modes is blocked. A small stiffness, mass or gyroscopic parameter is assumed to couple the uncoupled-blocked modes. The natural frequencies of the uncoupled system depend on the variable parameter and can cross at certain critical frequencies. The natural frequencies of the coupled system are seen to veer at these critical frequencies in the presence of arbitrarily small coupling, with the eigenvectors rotating and swapping from one branch to another. The separation of the branches around the critical frequencies is seen to depend on the coupling parameter. Examples including a 2 degree of freedom system, a multi-degree of freedom system and a plate with an attached variable oscillator are presented to illustrate the results.