A kind of magnetic shape memory alloy (MSMA) microgripper is proposed in this paper, and its nonlinear dynamic characteristics are studied when the stochastic perturbation is considered. Nonlinear differential items are introduced to explain the hysteretic phenomena of MSMA, and the constructive relationships among strain, stress, and magnetic field intensity are obtained by the partial least-square regression method. The nonlinear dynamic model of a MSMA microgripper subjected to in-plane stochastic excitation is developed. The stationary probability density function of the system’s response is obtained, the transition sets of the system are determined, and the conditions of stochastic bifurcation are obtained. The homoclinic and heteroclinic orbits of the system are given, and the boundary of the system’s safe basin is obtained by stochastic Melnikov integral method. The numerical and experimental results show that the system’s motion depends on its parameters, and stochastic Hopf bifurcation appears in the variation of the parameters; the area of the safe basin decreases with the increase of the stochastic excitation, and the boundary of the safe basin becomes fractal. The results of this paper are helpful for the application of MSMA microgripper in engineering fields.
INTRODUCTION
Microgripper is a basic kind of mechanical structure and used widely in industries of electronics, information technology, optics, medicine, and bio-technology. Shape memory alloy (SMA) microgripper has high work outputs, higher power-to-weight ratio and corrosion resistance, which make it to be attractive solutions for handling micro-parts.1 However, the traditional SMA, such as Ti-Ni alloy, can only be control through heating and cooling. The low response speed limited its application in industry fields.2
Recently, magnetic shape memory alloy (MSMA), such as Ni–Mn–Ga alloy, has gained considerable attention, as they exhibit special properties such as magnetic shape memory effect and magnetostrictive effect.3 Based on these properties, MSMA smart structure can be designed to achieve complex action. MSMA thin film is a basic kind of MSMA smart structure, which usually used in microelectro-mechanical system (MEMS) such as microvalves and microscanners.4 In this paper, MSMA thin film is introduced to improve the response speed and energy efficiency of microgripper.
There is always random perturbation in MEMS. To the MSMA microgripper, random perturbation can reduce the precision of the actuator and cause the damage to the system. Based on the magnetostrictive effect, MSMA thin films can be controlled effectively to reduce the stochastic vibration of the microgripper, which is helpful in improving the precision of actuator. For controlling the system vibration efficiently, it is necessary to obtain the dynamic characteristics of MSMA microgripper.
Several scholars have studied the response of SMA microgripper and the characteristics of MSMA.1–3,5–8 Kohl et al. proposed a SMA microgripper with integrated antagonism with integrated actuation units for antagonistic control of integrated gripping jaws,1 and studied the shape memory effect and magnetostriction in polycrystalline Ni–Mn–Ga thin film microactuators.3 Kyung et al. designed a microgripper for micromanipulation of microcomponents using SMA wires and flexible hinges.2 Karaca et al. analyzed the shape memory and pseudoelasticity response of NiMnCoIn MSMA single crystals.5 Gauthier et al. set up a kind of nonlinear Hamiltonian model of MSMA-based actuators.6 Kiefer et al. developed a model of the coupled strain and magnetization response of MSMAs under magneto-mechanical loading.7 Wang et al. obtained the magneto-mechanical response of an MSMA sample in finite element simulation.8 Although much progress has been obtained, the theoretical results about the dynamic characteristics of MSMA microgripper are limited.
Due to the hysteretic nonlinear characteristics of MSMA materials, MSMA microgripper has complex nonlinear dynamic characteristics, which cause the difficulty to control. The current paper aims to provide a kind of method to solve the nonlinear dynamical characteristics of a MSMA microgripper subjected to in-plane stochastic excitation. Van der Pol item is improved to explain the hysteretic loop of MSMA. The nonlinear dynamic model of a MSMA microgripper subjected to in-plane stochastic excitation is developed, and the stochastic bifurcation characteristics of the system are analyzed. Finally, the theoretical results are proved by experiments.
HYSTERETIC NONLINEAR MODEL OF MSMA
The experimental results of Ni-Mn-Ga MSMA at room temperature 25 °C are shown in Fig. 1. Evidently, hysteretic phenomena are present in MSMA thin films.
To obtain the response and the dynamic characteristics of MSMA thin films in theory, it is necessary to develop the nonlinear dynamic model of MSMA thin films, which is dependent on the constructive model of MSMA. Given the hysteretic characteristics of MSMA, most of the MSMA models are shown as equations with subsection functions or double integral functions, which are difficult to analyze theoretically. Most of the results could only be obtained by numerical or experimental method.9–13 In this paper, a new nonlinear differential item, which is developed from Van der Pol differential item, is introduced to describe the hysteretic nonlinear characteristics of the MSMA’s strain–stress curves.
Van der Pol equation is a kind of nonlinear differential equation. It can be shown as follows:
The item is called Van der Pol item. The essences of Van der Pol item are two parabolic lines when is constant. It means that the initial Van der Pol model can only be used in some basic parabolic hysteresis loop. However, the strain–stress curves of MSMA are not parabolic lines. In this paper, Van der Pol hysteretic model is developed as follows to describe the strain–stress curves of MSMA:
where is the stress, ε is the strain, f1(ε) = a1ε + a2ε3 + a3ε5 is the skeleton curve of the hysteretic loop; and is the improved Van der Pol item, which describes the difference between the skeleton curve and the hysteretic loop, ai(i = 1 ∼ 7) are coefficients, which can be obtained in fitting method.
In Eq. (2), only the hysteretic characteristic of strain–stress curves of MSMA is considered. MSMA is driven by the magnetic field, and magnetic field intensity (MFI) can also influence the strain and stress of MSMA, which is also shown in Fig. 1. Coupling items about MFI and strain are also found. Supposing that all of the coupling items about strain and MFI appear in the model, the basic model of MSMA can be shown as follows:
where HM is MFI.
Equation (3) is very complex, and there should be some items whose effects are not significant in the basic model. In this paper, the partial least-square regression software SIMCA-P is used to find the relationships among strain, stress, and MFI. Partial least-square regression method minimizes the sum of squared errors to find the best match of data, and is usually used for curve fitting. The variable importance (VIP) and the coefficient values of each item are shown in Figs. 2 and 3.
According to variable importance (VIP), the most significant variables (V IP > 0.8) in the basic model are chosen as follows: ε5, , ε5HM, , ε3, , , , ε, . Thus, the final relationship among stress, strain, and MFI are as follows:
The variable importance (VIP) and the coefficient values of each item in Eq. (4) are shown in Figs. 4 and 5. We can see that all the items in Eq. (4) are significant.
After the coefficient value of each item in Eq. (4) is determined, we can test the fitting effect of Eq. (4) to experimental data. The results of DModX and DModY tests are shown in Figs. 6 and 7. DModX and DModY tests can show the residual standard deviations (RSD) of samples in X space and Y space, where X space is the data space of the independent variable and Y space is the data space of the dependent variable. Only little points are singular, which means that most of the forecasted data are effective.
The result of the forecast test to Eq. (4) when HM = 0.2T is shown in Fig. 8, where the black line represents the real data and the red line represents the forecast value. Equation (4) can describe the real curve well.
BIFURCATION CHARACTERISTICS OF A MSMA MICROGRIPPER SUBJECTED TO IN-PLANE STOCHASTIC EXCITATION
The classic SMA microgripper is propose by Lee et al. in 1996.14 In this paper, SMA film is substituted by MSMA film to improve the response speed of the microgripper. The structure of MSMA microgripper is shown in Fig. 9. The MSMA microgripper is made up of MSMA thin film, Si3N4 layer, and Si substrate. Gripping jaws are fabricated in Si substrate by erosion, Si3N4 layer is fabricated in Si substrate by nitriding, and MSMA thin film is fabricated in Si3N4 layer by magnetron sputtering. The MSMA microgripper can be regarded as a composite cantilever plate, where a is the length of the MSMA microgripper; is the in-plane stochastic excitation, which is induced by stochastic magnetic field H, ς(t) is Gauss white noise whose mean is zero and intensity is 2D, D > 0.
The boundary conditions of the MSMA microgripper can be written as follows:
where
Thus, the vibration mode u(x, y, t) can be assumed as follows:
where
Considering the complex characteristics of composite materials, we introduce Hamilton’s principle to dynamic modeling of the system. The Hamilton function can be presented as:
where the effect of the Si3N4 layer to the dynamic characteristics of the system is ignored since the thickness of the Si3N4 layer is only 10 nm; T1 is kinetic energy of the MSMA thin films, ; T2 is the kinetic energy of the Si substrate, ; MH is the magnetic field energy of the MSMA thin films, ; U1 is the potential energy of the MSMA thin films, ; U2 is the potential energy of the Si substrate, ; W is the power of the external force, ; ρj(j = 1, 2) is the density of the materials (1:MSMA, 2: Si substrate), V is the volume of the MSMA thin film, Ej(j = 1, 2) is the elastic modulus; Aj(j = 1, 2) is the area of cross sections.
According to the Hamilton principle,
Thus, the nonlinear dynamic model of the MSMA microgripper can be shown as follows:
According to Eq. (5), the dynamic equation of the system’s response can be solved from Eq. (8) by the Galerkin method as follows:
When a > > b, the coupling coefficient γ is close to zero, the composite plate can be simplified as a composite beam, and the dynamic equation of system response can be shown as follows:
where η1 is the damping coefficient; , , , , , , , , a is the length of the beam, b is the width, h is the thickness.
Equation (10) is a stochastic nonlinear differential equation. The deterministic system corresponding to Eq. (10) can be shown as follows:
Let u1 = q and , Eq. (11) can be rewritten as follows:
The non-damping autonomous system from Eq. (12) is shown as follows:
Its Hamilton function is
According to the quasi-nonintegrable Hamiltonian system theory, the Hamiltonian function H(t) converges weakly in probability to an one-dimensional Ito diffusion process.15,16 The averaged Ito equation about the Hamiltonian function can be shown as follows:
where B(t) is standard Wiener process, m(H) and σ(H) are the drift and diffusion coefficients of Ito stochastic process, which can be obtained by the stochastic averaging method:
The averaged FPK equation of Eq. (10) is:
where f = f(H) is the stationary probability density (SPD) function of the system’s response. Thus,
where is a normalization constant.
Equation (19) can also be rewritten as follows:
where , , . Evidently, decreases monotonically since η > 0, c1 > 0, D > 0 and e2 > 0. Thus, the bifurcation characteristics of f(H) is only determined by the item H2 + υH + λ.
Now we can calculate the transition sets of the system response. Let and H = x2, we obtain . To , we obtain:
Since G(−x, λ, υ) = − G(x, λ, υ), we obtain:
where μ = x2.
Bifurcation sets: G = Gx = Gλ = 0
Thus, the bifurcation set of the system is ;
Hysteretic sets: G = Gx = Gxx = 0
Thus, the hysteretic sets of the system are and , which means ;
Double limited sets: G = Gx = 0
Thus, the double limited sets of the system are and , which means λ = x4.
The transition sets of the system response are shown in Fig. 10. These transfer sets divide the parameter space to three parts, and the corresponding stationary probability density functions of the system response in the three parts are also shown in Fig. 10, where c1 = 2500, c2 = − 100, c3 = 20, D = 0.5, η = 0.02, l = 1.
Stationary probability density (SPD) of the system response in different parameters.
Stationary probability density (SPD) of the system response in different parameters.
From Fig. 10, we can see that:
when , the steady-state probability density of H = 0 is the maximum, and there is a crest in the SPD of the system response;
when , there is a crest and a loop in the SPD of the system response. Although the steady-state probability density of H = 0 is also the maximum, it decreases with the variation of the parameters;
when , there is a loop in the SPD of the system response, the loop has the maximum SPD. It means that the system’s motion is the stochastic vibration near a periodic orbit.
In sum, stochastic Hopf bifurcation appears in the variation of the parameters;
The experimental results of a MSMA microgripper subject to in-plane stochastic excitation are shown in Figs. 11 and 12. Two kinds of Ni-Mn-Ga alloy is chosen as MSMA film: Ni-Co-Mn-Ga-In and Ni-Mn-Fe-Ga. The length of the microgripper is 10 cm, and its width is 1 cm. The MSMA film is prepared on the beam in magnetron sputtering method. The length of Ni–Mn–Ga thin film is 2 cm, its width is 1 cm, and its thickness is 100 nm. The lengths of Ni–Mn–Ga thin film are less than the one of microgripper because Ni-Mn-Ga thin film is brittle. The vibration amplitude of system is shown as output voltage of sensor. From the experimental results, we can see that the system’s motion is stochastic, while its vibration amplitude is determined by materials’ properties. The response of MSMA microgripper with Ni–Co-Mn–Ga-In film is near a periodic motion, which maybe has relationship with its large hysteretic loop.
SAFE BASIN OF A MSMA MICROGRIPPER SUBJECTED TO IN-PLANE STOCHASTIC EXCITATION
To the non-damping autonomous system Eq. (13), the boundaries of safe basin are the system’s homoclinic and heteroclinic orbits. There exists five fixed points in the system: u = q1 and u = − q1 are saddles; u = q2, u = 0, and u = − q2 are centers, where
There is a homoclinic orbit Γ+ho to and from +q1 and another homoclinic orbit Γ−ho to and from -q1. And there is a heteroclinic orbit Γ+he from +q1 to −q1, together with another heteroclinic orbit Γ−he from −q1 to +q1. The phase orbits of system (13) are shown in Fig. 13.
The homoclinic trajectories Γ+ho and Γ−ho are given by
and the heteroclinic orbits Γ+he and Γ−he are defined as:
where , , .
To the stochastic system Eq. (10), the boundary of its safe basin can be determined by the stochastic Melnikov integration as follows:17
where and mean the homoclinic orbits and , and the heteroclinic orbits and ; the item −I represents the mean of the Melnikov process due to damping force, ; z1(t1) denotes the random portion of the Melnikov process due to the bounded noise ξ(t), ; z2(t1) denotes the random portion of the Melnikov process due to the bounded noise ς(t), .
As for a deterministic system, the Melnikov function M(t) = 0 means the chaotic motion of the system. The chaotic motion is regarded as unsafe, so the Melnikov function determines the boundaries of safe basin, which are known as the deterministic system’s homoclinic and heteroclinic orbits. Both of the orbits are smooth.
As for a stochastic system, the stochastic Melnikov integration M(t1) = 0 also means that the motion system is unsafe.18 However, the boundary of the safe basin is not smooth. Certain points in the safe basin will leave the safe basin due to the stochastic excitation, which causes the boundary of the safe basin become fractal. The fractal phenomena of the safe basin’s boundary are called basin erosion.19 According to Eq. (24), we can judge whether the trajectory of the phase points leave the safe basin. The motion of the phase point will be regarded as unsafe when M(t1) = 0, and then the phase point will be considered as leaving the safe basin and deleted. The variation in the safe basin of the normalized system Eq. (12) subjected to stochastic excitation is shown in Fig. 14, where η = 0.02, c1 = 2500, c2 = − 100, c3 = 20, c4 = 50, c5 = − 8, c6 = 2.5, and c7 = − 0.06. The area of the safe basin decreases when the stochastic excitation increases, and the boundary of safe basin becomes fractal.
Safe basin of the system subjected to stochastic excitation: (a) f1 = 0.1; (b) f1 = 0.2; (c) f1 = 0.3.
Safe basin of the system subjected to stochastic excitation: (a) f1 = 0.1; (b) f1 = 0.2; (c) f1 = 0.3.
CONCLUSIONS
A kind of magnetic shape memory alloy (MSMA) microgripper is proposed in this paper, and its nonlinear dynamic characteristics are studied when the stochastic perturbation is considered. Nonlinear differential items are introduced to explain the hysteretic phenomena of MSMA, and the constructive relationships among strain, stress, and magnetic field intensity are obtained by the partial least-square regression method. The nonlinear dynamic model of a MSMA microgripper subjected to in-plane stochastic excitation is developed. The stationary probability density function of the system’s response is obtained, the transition sets of the system are determined, and the conditions of stochastic bifurcation are obtained. The homoclinic and heteroclinic orbits of the system are given, and the boundary of the system’s safe basin is obtained by stochastic Melnikov integral method. The numerical and experimental results show that the system’s motion depends on its parameters, and stochastic Hopf bifurcation appears in the variation of the parameters; the area of the safe basin decreases with the increase of the stochastic excitation, and the boundary of the safe basin becomes fractal. The results of this paper are helpful for the application of MSMA microgripper in engineering fields.
ACKNOWLEDGMENTS
The authors gratefully acknowledge the support of the Natural Science Foundation of China (NSFC) through Grant Nos. 11272229, 11302144, and 11402168, the Ph.D. Programs Foundation of the Ministry of Education of China through Grant No. 20120032120006, and the Tianjin Research Program of Application Foundation and Advanced Technology through Grant No. 13JCYBJC17900 and 14JCQNJC05300.