Fig. 1

The schematic diagram of the coupled system (color online)"

Fig. 2

Bifurcation diagrams of the equilibrium positions: (a) k1=k2=1 100 N/m and b2=40 mm; (b) b1=b2=40 mm and k2=1 100 N/m (color online)"

Fig. 3

Asymmetric potential energy: (a) distance asymmetry when k1=k2=1 100 N/m, b1=39 mm, and b2=41 mm; (b) stiffness asymmetry when k1=1 050 N/m, k2=1 100 N/m, and b1=b2=40 mm (color online)"

Table 1

The parameters of the coupled system"

Fig. 4

Comparison results of polynomial fitting: (a) symmetric TNES; (b) ATNES (color online)"

Table 2

The fitting coefficients of the nonlinear restoring force as shown in Fig. 4"

Fig. 5

(a) Ⅰ-TNES, where the middle barrier depth is greater than the side barrier depth; (b) Ⅱ-TNES, where the middle barrier depth is equal to the side barrier depth; (c) Ⅲ-TNES, where the middle barrier depth is less than the side barrier depth; (d) the potential barrier depth varies with the left horizontal distance b1 (color online)"

Fig. 6

(a) The potential energy; (b) the LO amplitude-frequency response when F0=0.3 N; (c) the LO amplitude-frequency response when F0=1 N; (d) the vibration reduction efficiency with various values of the left horizontal distance b1 when b2=41 mm (color online)"

Fig. 7

The phase trajectories of the ATNES in the resonance region when k1=k2=1 100 N/m and b2=41 mm with various values of the excitation amplitude F0 and the left horizontal distance b1 (color online)"

Fig. 8

(a) The potential energy; (b) the vibration reduction efficiency with the change of the left horizontal distance b1 when b2=40.38 mm (color online)"

Fig. 9

The phase trajectories of the ATNES in the resonance region when k1=k2=1 100 N/m and b2=40.38 mm with the change of the excitation amplitude F0 and the left horizontal distance b1 (color online)"

Fig. 10

(a) The potential energy; (b) the vibration reduction efficiency with the change of b1 when b2=40 mm (color online)"

Fig. 11

The phase trajectories of the ATNES in the resonance region when k1=k2=1 100 N/m and b2=40 mm with the change of the excitation amplitude F0 and the left horizontal distance b1 (color online)"

Fig. 12

(a) The potential energy; (b) the vibration reduction efficiency with the change of the left oblique spring stiffness k1 when k2=1 100 N/m and b1=b2=41 mm (color online)"

Fig. 13

The phase trajectories of the ATNES in the resonance region when b1=b2=41 mm and k2=1 100 N/m with the change of the excitation amplitude F0 and the left oblique spring stiffness k1 (color online)"

Fig. 14

(a) The potential energy; (b) the vibration reduction efficiency with the change of the oblique spring stiffness k1 when k2=1 100 N/m and b1=b2=40.38 mm (color online)"

Fig. 15

The phase trajectories of the ATNES in the resonance region when b1=b2=40.38 mm and k2=1 100 N/m with the change of the excitation amplitude F0 and the left oblique spring stiffness k1 (color online)"

Fig. 16

(a) The potential energy; (b) the vibration reduction efficiency with the change of the left oblique spring stiffness k1 when k2=1 100 N/m and b1=b2=40 mm (color online)"

Fig. 17

The phase trajectories of the ATNES in the resonance region when b1=b2=40 mm and k2=1 100 N/m with the change of the excitation amplitude F0 and the oblique spring stiffness k1 (color online)"