On the dynamics and unstable region of a class of parametrically excited systems similar to a beam with an axially reciprocating mid-support

By means of Euler–Bernoulli beam theory and the Lagrange equation of the first kind, the dynamic equation of a beam with an axially harmonic reciprocating moving mid-support (beam-AHRMS) is established. Using two modal coordinates, the equation of the beam-AHRMS is reduced to a special form of Hill’s equation, the particularity of which is that its excitation, i.e., time-dependent variable coefficients in front of the state variable of the equation, is a nonlinear function of the excitation in Mathieu’s equation. Using the perturbation method, the approximate solution for the transition curves between the stable and unstable regions of the system is obtained. The most different behaviour of the special form of Hill’s equation is that there exists a knot in the principal unstable region, where the bandwidth of the unstable region becomes zero for any frequency, and the principal parameter resonance is suppressed. A theorem for the occurrence and location of knots in the principal unstable region is proposed and proven. All the results of the analysis and the theorem are verified via a numerical method.