Dynamic and steady-state performance analysis of a linear solenoid parallel elastic actuator with nonlinear stiffness

Abstract

The peak time, overshoot and steady-state displacement are taken as the performance indicators to predict and adjust the steady-state and dynamic performance of a large-stroke, high-speed linear solenoid parallel elastic actuator (LSPEA) with nonlinear stiffness in the process of the overshoot, damped oscillation and stabilization. The analytical, numerical and finite element methods are used to obtain those indicators under different stiffness and damping. First, the analytical functions of the electromagnetic force and the magnetic field distribution are presented. The nonlinear vibration equation is obtained by dynamic modeling. The averaging method and the KBM method are employed to obtain analytical results of the nonlinear undamped vibration equation. The equivalent linearization of the damped nonlinear system is performed to obtain the approximate analytical results of the performance indicators including equivalent stiffness, damping ratio, and damping frequency. Second, the approximate analytical model and numerical model are constructed with the Simulink software to solve the nonlinear equation. ANSYS Maxwell and Adams software are adopted for the transient analysis of the LSPEA. The displacement curves and performance indicators obtained by the analytical model, numerical model, finite element and experiment are compared. The maximum errors of the peak time, overshoot and steady displacement through the analytical model and experiment are 4.54 ms, 8.84% and 0.44 mm, respectively. The maximum errors through the numerical model and experiment are 5.04 ms, 8.57% and 0.42 mm, respectively. The established analytical and numerical models can be used to determine the static and dynamic performance of the LSEA within a certain error, which is helpful for the solution of nonlinear systems caused by multi-physics coupling.

Appendices

Appendix A

$$ \left\{ \begin{aligned} x_{1} = \frac{1}{{w_{0}^{2} }}\left( {\frac{{3A_{2} a^{2} }}{2} + \frac{{5A_{3} a^{3} }}{2} + \frac{{35A_{4} a^{4} }}{8}} \right) - \frac{1}{{3w_{0}^{2} }}\left( {\frac{{A_{2} a^{2} }}{2} + \frac{{3A_{3} a^{3} }}{2} + \frac{{7A_{4} a^{4} }}{2}} \right)\cos 2\psi + \hfill \\ \quad \quad \frac{1}{{8w_{0}^{2} }}\left( {\frac{{A_{3} a^{3} }}{4} + \frac{{A_{4} a^{4} }}{2}} \right)\cos 3\psi - \frac{{A_{4} a^{4} }}{{120w_{0}^{2} }}\cos 4\psi \hfill \\ w_{1} = - \frac{1}{{2aw_{0} }}\left( {2A_{2} {{a}}^{2} + \frac{15}{4}A_{3} {{a}}^{3} + 7A_{4} {{a}}^{4} } \right) \hfill \\ a_{1} = 0 \hfill \\ \end{aligned} \right. $$

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$$ \left\{ \begin{aligned} x_{2} = \frac{1}{{w_{0}^{4} }}\left( {3A_{2}^{2} a^{3} + \frac{{93A_{2} A_{3} a^{4} }}{8} + \frac{{87A_{3}^{2} a^{5} }}{8} + \frac{{93A_{2} A_{4} a^{5} }}{4} + \frac{{2707A_{3} A_{4} a^{6} }}{64} + \frac{{643A_{4}^{2} a^{7} }}{16}} \right) \hfill \\ - \frac{\cos 2\psi }{{3w_{0}^{4} }}\left[ \begin{aligned} - \frac{{A_{2}^{2} a^{3} }}{3} + \frac{{15A_{2} A_{3} a^{4} }}{32} + \frac{{45A_{3}^{2} a^{5} }}{32} + \frac{{39A_{2} A_{4} a^{5} }}{8} + \frac{{3423A_{3} A_{4} a^{6} }}{32} + \hfill \\ \frac{{3269A_{4}^{2} a^{7} }}{240} + \frac{4}{3}\left( {2A_{2} {{a + }}\frac{{15A_{3} {{a}}^{2} }}{4} + 7A_{4} a^{3} } \right)\left( {\frac{{A_{2} {{a}}^{2} }}{2} + \frac{{3A_{3} {{a}}^{3} }}{2} + \frac{{7A_{4} {{a}}^{4} }}{2}} \right) \hfill \\ \end{aligned} \right] \hfill \\ - \frac{\cos 3\psi }{{8w_{0}^{4} }}\left[ \begin{aligned} \frac{{A_{2}^{2} a^{3} }}{6} + \frac{{17A_{2} A_{3} a^{4} }}{16} + \frac{{105A_{3}^{2} a^{5} }}{64} + \frac{{47A_{2} A_{4} a^{5} }}{40} + \frac{{113A_{3} A_{4} a^{6} }}{20} + \hfill \\ \frac{{91A_{4}^{2} a^{7} }}{16} + \frac{9}{8}\left( {2A_{2} {{a + }}\frac{{15A_{3} {{a}}^{2} }}{4} + 7A_{4} a^{3} } \right)\left( {\frac{{A_{3} {{a}}^{3} }}{4} + A_{4} {{a}}^{4} } \right) \hfill \\ \end{aligned} \right] \hfill \\ + \frac{\cos 4\psi }{{15w_{0}^{4} }}\left[ \begin{aligned} \frac{{5A_{2} A_{3} a^{4} }}{32} + \frac{{15A_{3}^{2} a^{5} }}{32} + \frac{{77A_{2} A_{4} a^{5} }}{120} + \frac{{967A_{3} A_{4} a^{6} }}{320} + \hfill \\ \frac{{217A_{4}^{2} a^{7} }}{48} - \frac{{2A_{4} {{a}}^{4} }}{15}\left( {2A_{2} {{a + }}\frac{{15A_{3} {{a}}^{2} }}{4} + 7A_{4} a^{3} } \right) \hfill \\ \end{aligned} \right] \hfill \\ - \frac{\cos 5\psi }{{24w_{0}^{4} }}\left( {\frac{{3A_{3}^{2} a^{5} }}{128} + \frac{{11A_{2} A_{4} a^{5} }}{120} + \frac{{37A_{3} A_{4} a^{6} }}{80} + \frac{{49A_{4}^{2} a^{7} }}{48}} \right) \hfill \\ + \frac{\cos 6\psi }{{35w_{0}^{4} }}\left( {\frac{{7A_{3} A_{4} a^{6} }}{320} + \frac{{7A_{4}^{2} a^{7} }}{80}} \right) - \frac{{A_{4}^{2} a^{7} \cos 7\psi }}{{11520w_{0}^{4} }} \hfill \\ w_{2} = - \frac{1}{{2aw_{0} }}\left( \begin{aligned} \frac{{17A_{2}^{2} a^{3} }}{{6w_{0}^{2} }} + \frac{{13A_{2} A_{3} a^{4} }}{{w_{0}^{2} }} + \frac{{1725A_{3}^{2} a^{5} }}{{128w_{0}^{2} }} + \frac{{115A_{2} A_{4} a^{5} }}{{4w_{0}^{2} }} + \hfill \\ \frac{{897A_{3} A_{4} a^{6} }}{{16w_{0}^{2} }} + \frac{{4473A_{4}^{2} a^{7} }}{{80w_{0}^{2} }} + aw_{1}^{2} \hfill \\ \end{aligned} \right) \hfill \\ a_{2} = 0 \hfill \\ \end{aligned} \right. $$

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Appendix B

See Table 4.

Table 4 Comparison of dynamic indicators under different stiffness and damping

Appendix C

See Figs.

Fig. 17

Five groups of transient curves of Actuator #2 under the voltage of 0.8–2.2 V

17 and

Fig. 18

Five groups of transient curves of Actuator #3 under the voltage of 2–6 V

18.