A bistable nonlinear electromagnetic actuator with elastic boundary for actuation performance improvement

Abstract

This study proposes a novel bistable nonlinear electromagnetic actuator with elastic boundary (BEMA-EB) to enhance the actuation performance when controlled by a harmonic input signal. The structure of the BEMA-EB has an inclined spring, one end of which is supported by an elastic boundary. The inclined spring produces bistable nonlinearity to realize the large-amplitude inter-well actuation responses, and the elastic boundary brings additional dynamic coupling to enhance the inter-well actuation performance. The governing equations of the BEMA-EB controlled by a harmonic input signal are formulated. To show the advantages of the BEMA-EB, the study performs comparison between the BEMA-EB, a bistable electromagnetic actuator which does not have elastic boundary, and an equivalent linear electromagnetic actuator. The results show that the BEMA-EB has a much broader inter-well actuation bandwidth and smaller input-signal-amplitude threshold of activating the favorable inter-well actuation, which verify the merits of both the bistable nonlinearity and elastic boundary. To develop insights into the nonlinear dynamic behaviors of the BEMA-EB, the bifurcation features are investigated in terms of the inclined spring stiffness, the input signal frequency and amplitude. Phase portraits and Poincare maps are presented to illustrate how the actuation responses of the BEMA-EB evolve corresponding to the changes of the above parameters. Finally, basin-of-attraction maps are given to uncover the occurring probabilities of the different types of the actuation responses with respect to the different distributions of the initial conditions. The results quantitatively corroborate that the likelihood of the favorable inter-well actuation is significantly increased by using the BEMA-EB.

Appendix: Model validation using the MSC Adams software

The MSC Adams software is used to validate the mathematical model of the BEMA-EB. In the software, the 3D virtual prototype of the BEMA-EB is established, as shown in Fig. 22. The simulation parameters are listed in Table 1 (as same as Fig. 9e).

Fig. 22

The 3D virtual prototype of the BEMA-EB in Adams software. a Isometric graph; b planar graph

Table 1 The simulation parameters

The comparative results between the numerical results of the governing equations and Adams software are presented in Fig. 23. Figure 23 shows the acceleration responses obtained by the Adams for the BEMA-EB under forward swept harmonic input signal from 1 Hz to 10 Hz with the amplitude $I = 2 {\text{A}}$. The sweeping rate is 0.025 Hz/s. It is seen that the two results (Figs. 9e and 23) are in good agreement.

Fig. 23

a Acceleration responses obtained by the Adams for the BEMA-EB under forward swept harmonic input signal when $k_{1} = 2000 {\text{N}}/{\text{m}}$, $k_{2} = 2000 {\text{N}}/{\text{m}}$, $I = 2 {\text{A}}$. b Result of Fig. 9e

In addition, another forward swept response results of both Adams software and the governing equations are presented. $I = 1.5 {\text{A}}$, $k_{1} = 2000 {\text{N}}/{\text{m}}$, $k_{2} = 2000 {\text{N}}/{\text{m}}$ are employed in this calculation. Figure 24 shows the displacement, velocity and the acceleration responses of both the methods. It is seen that the numerical results are in good agreement with the Adams software results.

Fig. 24

Comparison between the numerical method and the Adams. The blue curves indicate the responses obtained by numerical method, while the red curves represent the Adams. a, b Displacement response; c, d velocity response; e, f acceleration response