Nonlinear normal modes and bifurcations of geometrically nonlinear vibrations of beams with breathing cracks

Two types of partial differential equations, which describe geometrically nonlinear vibrations of flexible beams with breathing cracks, are considered. The mechanical vibrations with two kinds of nonlinearities are considered. The first type of the partial differential equation uses crack function to describe the vibrations of the beams with one crank. The second model can be used to describe the vibrations of beams with several cranks. This approach uses delta function to simulate every crack. A contact parameter is used to describe nonlinearity due to crack breathing. The Galerkin technique is applied to derive the system of the ordinary differential equations with polynomial nonlinearity and piecewise-linear functions of the generalized coordinates. The combination of the collocation method and arc length continuation technique is applied to analyze numerically the nonlinear vibrations, their stability and bifurcations. The nonlinear normal modes of the geometrically nonlinear free vibrations of the beam with the breathing crack are analyzed numerically. The backbone curve of these nonlinear modes contains two loops, saddle-node bifurcations and Neimark–Sacker bifurcations. As follows from the numerical analysis, the nonlinear normal modes modal lines are curved essentially in configuration space. The frequency responses of the forced vibrations contain loops and saddle-node bifurcations. Moreover, the frequency responses of the forced vibrations contain Neimark–Sacker bifurcations, which result in steady quasi-periodic vibrations. The properties of the quasi-periodic vibrations are analyzed numerically.