Fig. 1. Multi-stable cantilever beam with two magnets: (a) The state concluding three stable points; (b) the state concluding four stable points.

Fig. 2. Schematic diagram of there-dimension coordinate system and magnetizing currents on the surface of the circular magnet.

Fig. 3. The curves of B_i and B_j varying with x: (a) The curves of B_i varying with x, y = 6.0 mm; (b) the curves of B_j varying with x, y = 6.0 mm; (c) the curves of B_i varying with x, y = 10.0 mm; (d) the curves of B_j varying with x, y = 10.0 mm.

Fig. 4. Magnetic induction intensity measurement system: (a) The measurement of B_i; (b) the measurement of B_j.

Fig. 5. The position of the rectangular magnet in coordinate system when the cantilever beam is bent.

Fig. 6. Two kinds of calculation of the position of the magnet at the free end of the beam.

Fig. 7. Displacement measuring device.

Fig. 8. Schematic diagram of the size of the rectangular magnet and the magnetizing currents on the surface of the rectangular magnet.

Fig. 9. The curves of F_i and F_j varying with x_C: (a) The curves of F_i varying with x_C, d = 5.8 mm; (b) the curves of F_j varying with x_C, d = 5.8 mm; (c) the curves of F_i varying with x_C, d = 8.0 mm; (d) the curves of F_j varying with x_C, d = 8.0 mm.

Fig. 10. Magnetic force measurement system: (a) The measurement of F_i; (b) the measurement of F_j.

Fig. 11. The system potential function varying with d when the size of the rectangular magnet is 10 mm × 10 mm × 3 mm and the ring magnet is 40 mm (φ₁) × 20 mm (φ₂) × 3 mm: (a) Three dimensional diagram of system potential function; (b) two dimensional diagram of system potential function when d = 3 mm, d = 6 mm and d = 20 mm.

Fig. 12. The system potential function varying with d: (a) The size of the rectangular magnet is 20 mm × 20 mm × 3 mm; (b) the size of the rectangular magnet is 30 mm × 30 mm × 3 mm.

Fig. 13. The system potential function varying with l_A when d = 6 mm and the size of the ring magnet is 40 mm(φ₁) × 20 mm(φ₂) × 3 mm: (a) Three dimensional diagram of system potential function; (b) two dimensional diagram of system potential function when l_A = 3 mm, l_A = 10 mm, l_A = 20 mm, l_A = 30 mm and l_A = 45 mm.

Fig. 14. (a) The curves of W₂ and W₃ varying with x_C when l_A = 20 mm and l_A = 30 mm; (b) the curves of F_i varying with x_C when l_A = 20 mm and l_A = 30 mm.

Fig. 15. The structure concluding three stable points: (a) The middle state point; (b) the upper stable point.

Fig. 16. The structure concluding four stable points: (a) The upper stable point 1; (b) the upper state point 2.

Fig. 17. The vibration response of the tri-stable cantilever beam: (a) The time domain chart; (b) the phase chart.

Fig. 18. The vibration response of the quad-stable cantilever beam: (a) The time domain chart; (b) the phase chart.