Design and Experiments of a Galfenol Composite Cantilever Beam-Driven Magnetostrictive Micro-gripper

Abstract

The work presents a novel magnetostrictive micro-gripper driven by iron–gallium alloy (Galfenol) composite cantilever beam. The design and operation principles of the given micro-gripper are first introduced. Then, we analyze the magnetic distribution and the jaw displacement of the micro-gripper by theoretical simulations. The performance of magnetostrictive micro-gripper has been tested using the multilevel driving strategy to eliminate the vibration. The proposed micro-gripper can grasp micro-target with a maximum gap of 250 μm, and the maximum exciting current of 1 A. The experimental results demonstrate that the proposed micro-gripper has a simple structure and fast response, suitable for micromanipulation and micro-assembly applications.

Appendices

Appendix 1

See Fig. 10.

$$\begin{aligned} \sigma_{x,s} = \frac{{E_{s} }}{{1 - \nu_{s}^{2} }}(\varepsilon_{x,s} + \nu_{s} \varepsilon_{y,s} ),\;\sigma_{x,f} = \frac{{E_{f} }}{{1 - \nu_{f}^{2} }}[(\varepsilon_{x,f} - \lambda_{s} ) + \nu_{f} (\varepsilon_{y,f} + \lambda_{f} /2)] \hfill \\ \sigma_{x,s} = \frac{{E_{s} }}{{1 - \nu_{s}^{2} }}(\varepsilon_{y,s} + \nu_{s} \varepsilon_{x,s} ),\;\sigma_{y,f} = \frac{{E_{f} }}{{1 - \nu_{f}^{2} }}[(\varepsilon_{y,f} + \lambda_{s} /2) + \nu_{f} (\varepsilon_{x,f} - \lambda_{s} )] \hfill \\ \end{aligned}$$

(8)

With

$$\varepsilon_{x,i} = \varepsilon_{x}^{0} - \frac{z}{{R_{x} }},\quad \varepsilon_{y,i} = \varepsilon_{y}^{0} - \frac{z}{{R_{y} }},\quad i = s,f$$

(9)

For the convenience of analysis, several dimensionless parameters are defined in (10). Where β, κ are the non-dimensional parameters; a_x and a_y are the reduced curvature.

$$\beta = {{\frac{{E_{f} }}{{1 - \nu_{f}^{2} }}} \mathord{\left/ {\vphantom {{\frac{{E_{f} }}{{1 - \nu_{f}^{2} }}} {\frac{{E_{s} }}{{1 - \nu_{s}^{2} }}}}} \right. \kern-0pt} {\frac{{E_{s} }}{{1 - \nu_{s}^{2} }}}},\quad \kappa = {{t_{f} } \mathord{\left/ {\vphantom {{t_{f} } {t_{s} }}} \right. \kern-0pt} {t_{s} }},\quad \alpha_{x} = t_{f} /R_{x} ,\quad \alpha_{y} = t_{s} /R_{y}$$

(10)

The equilibrium state of the system is described by the following equations:

$$\begin{aligned} \int {\sigma_{x,f} } {\text{d}}y{\text{d}}z + \int {\sigma_{x,s} } {\text{d}}y{\text{d}}z = 0,\;\int {\sigma_{x,f} } z{\text{d}}y{\text{d}}z + \int {\sigma_{x,s} z} {\text{d}}y{\text{d}}z = 0 \hfill \\ \int {\sigma_{y,f} } {\text{d}}x{\text{d}}z + \int {\sigma_{y,s} } {\text{d}}x{\text{d}}z = 0,\;\int {\sigma_{y,f} z} {\text{d}}y{\text{d}}z + \int {\sigma_{y,s} z} {\text{d}}x{\text{d}}z = 0 \hfill \\ \end{aligned}$$

(11)

The R_x,R_y,$\varepsilon_{x}^{0}$, and $\varepsilon_{y}^{0}$ can be solved by this equations set, and the reduced curvature α_x and α_y can be expressed as:

$$\alpha_{x} = - \frac{3}{2}\beta \lambda_{s} k(1 + \kappa )[(1 + v_{f} )(1 + v_{s} )A + 3(1 - \nu_{f} )(1 - \nu_{s} )B]$$

(12)

$$\alpha_{y} = - \frac{3}{2}\beta \lambda_{s} k(1 + \kappa )[(1 + v_{f} )(1 + v_{s} )A{ - }3(1 - \nu_{f} )(1 - \nu_{s} )B]$$

(13)

with A and B being

$$A = [(1 + \nu_{s} )^{2} + 2\beta (1 + \nu_{f} )(1 + \nu_{s} )(2\kappa + 3\kappa^{2} + 2\kappa^{3} ) + \beta^{2} \kappa^{4} (1 + \nu_{f} )^{2} ]^{ - 1}$$

(14)

$$B = [(1{ - }\nu_{s} )^{2} + 2\beta (1{ - }\nu_{f} )(1{ - }\nu_{s} )(2\kappa + 3\kappa^{2} + 2\kappa^{3} ) + \beta^{2} \kappa^{4} (1{ - }\nu_{f} )^{2} ]^{ - 1}$$

(15)

Assuming the y-direction deflection is zero, α_y= 0, the deflection of cantilever beam can be expressed as:

$$\Delta_{x} = \frac{{3L^{2} \beta \lambda \kappa (1 + \kappa )^{2} }}{{4(t_{s} + t_{f} )}}[(1 + \nu_{f} )(1 + \nu_{s} )A + 3(1 - \nu_{f} )(1 - \nu_{f} )B]$$

(16)

Fig. 10

Sketch of a composite cantilever beam. a The coordinate system, where the x–y plane coincides with the mid-plane; b the anticlastic deformation of a magnetized cantilever

Appendix 2

The D–H model is defined as follows:

$$\varepsilon = - \frac{\sigma }{E} + \frac{{\lambda_{s} }}{{M_{s}^{2} }}M^{2}$$

(17)

$$H = \frac{1}{k}f^{ - 1} \left( {\frac{M}{{M_{s} }}} \right) - \frac{{2\lambda_{s} \sigma }}{{\mu_{0} M_{s}^{2} }}M$$

(18)

With the Langevin equation,

$$M = M_{s} [\coth (kH) - 1/kH]$$

(19)

The magnetic susceptibility χ can be expressed as:

$$\chi=\frac{M_{s}}{k}\left(1-\coth^{2}\left(\frac{H}{k}\right)+\left(\frac{k}{H}\right)^{2}\right)$$

(20)

where k = 1/3χ_mM_s, χ_m is the initial susceptibility.