Graphical abstract The system bifurcation diagrams for x 1 are shown for harmonic excitation force F ˜ = (a) 1.14 × 10−3, (b) 0.1, (c) 1.0 and (d) 2.0, respectively. There are two intervals of nonlinear motion at 0 < η < 0.006 and 1.248 ≦ η ≦ 2.0, respectively when harmonic excitation force ( F ˜ ) increases to 2.0. Image, graphical abstract 
Numerical computation and nonlinear dynamic analysis of ultrasonic cutting system