Research on Shape Memory Alloys and Magnetorheological Fluids for Use in Pneumatic Actuators

Abstract

Adaptive vibrating actuators operating under autovibration regime were developed using materials with shape memory (SM). Mechanical characteristics of such type vibroactuators can be adjusted by changing geometrical parameters of chamber or throttle. It allows to expand the application of pneumatic vibroexciters and use them in different technological processes. Pneumatic vibroactuator elements that consist of magnetorheological (MR) fluids are proposed for micro-throttle and microcompressor design. The main properties of used MR materials and theoretical background of their mechanical and physical parameters are presented, some results of experimental and theoretical research of actuators are introduced and conclusions of the performed research are formulated.

1. Introduction

Smart materials can be used in various technologies, including the structures of pneumatic actuators on air cushions. These actuators may be applied in different technological devices: in vehicle suspensions, rotary mechanisms, vibrating damping systems, various medical devices, etc. Some properties of actuators with elements of shape memory alloy and magnetorheological fluid are analysed in this study.

2. Theoretical background of the use of materials containing shape memory elements in pneumatic actuators

Functional capabilities of pneumatic actuators [1] can be extended via the variations of their ignition chamber volume or the geometrical parameters of their choke [2].

The structure of any pneumatic actuator containing a shape memory (SM) element, i.e. a variable volume chamber, is provided in Figure 1. Such transducers may be used in vibratory machinery, pneumatic shock-absorber structures and regulatory, guidance, positioning and other equipment.

Figure 1.

Schemes of pneumatic vibroexciters constructions, (a) simplified scheme of vibrodrive: 1—oscillatory mass; 2—spring; 3—exciting chamber of vibrodrive; P₁—supplied gas pressure; P_K—pressure inside chamber; x—move of oscillatory mass; r₁—radius of the air supply channel; and r_K—chamber radius. (b) Simplified scheme with element made from smart material: 1—piston; 2—arc-shaped plate with SM material; 3—frame, 4—exciting chamber; 5,6—cooling and heating elements, respectively; 7—spring that pushes piston 1 to the plate 2; r_K—exciting chamber radius; and l_K—chamber height.

Any ignition chamber of a pneumatic actuator consists of transmission (Figure 1b), which is made of plunger 1, SM arch element 2, cooler 5 and heating elements 6. SM transmission element 2 is situated in case 3. Transmission 2 contacts heating 6 and cooling 5 elements. Its plate 2 is being heated, and the heat is being transmitted to the arch element. When the arch material reaches the reverse martensitic transformation temperature, it begins to incline in the opposite direction, and when it reaches the opposite position, the system starts cooling. When the arch material reaches temperature which is lower than its phase transformation temperature, the arch escapes to the backwords direction.

By experimental study of shape memory alloys, it was found that temperature range is about 10–16°C when direct and reverse martensitic transformation happens with SM material (NiTi alloy). So, it is important to mention, in a separate case, the system shown in Figure 1b without complex feedback control elements 5 and 6.

Based on previous research works [3, 4, 5], the plate containing the SM material (nickel and titanium alloy) has been researched as a uniform plate without its decomposition into the core and the outer layer although they could be described as having different deformation properties. Those works offered simplified methods for the calculation of deformations in plates containing SM material. Suppose the plate is situated on a horizontal axis and one or both of its ends are fixed. Its centre is being affected by force which is directed along vertical axis. If the effect of transverse forces that appear during the deformation is ignored, a homogenous bending of the plate in the direction of Y axis can be described using the equation of Euler:

$$\mathit{EJ}\frac{d^{2}y}{{\mathit{dx}}^2}+\mathit{Py}{\left[1+{\left(\frac{\mathit{dy}}{\mathit{dx}}\right)}^2\right]}^{\frac{3}{2}}=0$$ E1

where E refers to the Young’s modulus, J refers to inertia moment and P refers to force which affects the centre of the plate. If both ends of the sample are fixed, marginal conditions of Eq. (1) will be as follows:

$$y\left(0\right)=y\left(l\right)=0$$ E2

where l refers to the plate length.

Properties of SM material used are presented in Table 1.

Martensite start temperature, Ms	18.4°C
Martensite finish temperature, Mf	9°C
Austenite start temperature, As	34.5°C
Austenite finish temperature, Af	49°C
Transformation constant, CM	0.008 GPa/°C
Transformation constant, CA	0.0138 GPa/°C
Transformation constant, σs^cr	0.1 GPa
Transformation constant, σf^cr	0.17 GPa

Table 1.

Properties of NiTi material [6].

The number of experiments led to drawing the curve which shows the correlation between mechanical stress with temperatures in nickel and titanium alloy plate (Figure 2). Based on theoretical assumptions contained in the mentioned works [3, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], such calculations can be also used in chains containing SM elements. While heating the plate up to the austenitic temperature, the sinusoidal bending will be obtained:

Figure 2.

Correlation between mechanical stress and temperature in nickel and titanium alloy plate: σ_r—reaction stress; σ_T—martensitic transformation stress; M₀ and M₁—direct martensitic transformation (A- > M) temperatures, initial and terminal, respectively; and A₀ and A₁—reverse martensitic transformation (M- > A) temperatures, initial and terminal, respectively [3].

$$a\text{sin}\frac{\mathit{\pi \xi }}{l}$$ E3

where a refers to the maximum bending of the plate.

Taking into account Eq. (1), the close solution of bending in the direction of Y axis in the sample whose both ends are fixed can be found [3, 7]:

$$y\left(\xi \right)=\frac{2^{\frac{3}{2}}l}{\pi }\sqrt{\frac{\sigma -{\sigma }_1}{{\sigma }_1}}\text{sin}\frac{\mathit{\pi \xi }}{l}+o\left(\sqrt{\sigma -{\sigma }_1}\right)$$ E4

where the minimum marginal stress is described as

$${\sigma }_1=\frac{{\pi }^2\mathit{EJ}}{l^{2}S}$$ E5

and the arched axis is described by a sinusoid

$$y\left(\xi \right)=\frac{2^{\frac{3}{2}}l}{\pi }\sqrt{\frac{\sigma -{\sigma }_1}{{\sigma }_1}}\text{sin}\frac{\mathit{\pi \xi }}{l}$$ E6

while inertia moment of the plate section in regard to neutral axis is equal to

$$J=\frac{{\mathit{bh}}^3}{12}$$ E7

If S = bh is the plate section area (Figure 3), E is an elasticity module of nickel and titanium alloy (app E = 15.65 × 10¹⁰ Pa).

Figure 3.

Scheme of a nickel and titanium alloy plate.

The force acting upon the plate with SM effect can be defined by the formula

$$F=\left\{\begin{array}{ll}{\mathit{SE}}_{\mathit{ad}}\alpha \left(T-T_0\right),& \mathit{\text{when}}{T}_0\le T<A_0\\ S{\sigma }_r,& \mathit{\text{when}}{A}_0\le T<A_1\\ {\mathit{SE}}_{\mathit{ad}}\alpha \left(T-A_1\right),& \mathit{\text{when}}T>A_1\end{array}\right.$$ E8

where T, α and E_ad are temperature, thermal expansion coefficient and adiabatic Young’s modulus, respectively. By Eqs. (5) and (8), the smallest critical force, when temperature changes between initial and final reverse martensitic transformation temperatures (A₀ < T < A₁), will have the following expression:

$$F_1=\frac{{\pi }^2\mathit{EJ}}{l^2}$$ E9

When the temperature varies between A₀ and A₁, the plate regains its present sinusoidal shape. When the temperature approaches critical temperature T_K, corresponding critical stress σ_K, the plate acquires the linear shape.

Pneumatical vibroexciter with exciting chamber radius r_K = 15 × 10⁻³ m and height l_K = 20 × 10⁻³ m has been analysed in the numerical research. A smart material element in exciting chamber has used a titanium-nickel plate whose length l = 25 × 10⁻³ m, height h = 1.5 × 10⁻³ m and width b = 20 × 10⁻³ m. By the experimental study [3], approximate Young’s modulus for NiTi alloy is E = 15.65 × 10¹⁰ Pa, initial and final, respectively, reverse martensitic transformation temperatures A₀ = 75°C or A₁ = 110°C.

According to Eqs. (3)–(9), a programme simulating the bending of plate has been proposed. The plate axis bending shapes in the direction of Y axis according to the plate length under the various reaction stresses σ_r are shown in Figure 4a.

Figure 4.

Deformation of the NiTi plate with the fastened ends: (a) shapes of bended plate under various stresses, curve y0(ξ), when σ_r = 463.4 MPa, the maximum deflection of bended plate is 0.19 × 10⁻³ m; curve y1(ξ) when σ_r = 464.0 MPa, the maximum deflection of bended plate is 0.91 × 10⁻³ m; and when curve ymax(ξ) when σ_r = 480.0 MPa, the maximum deflection of bended plate is 4.26 × 10⁻³ m; (b) view of bended plate.

The numerical researches have showed that a middle point of plate moves within 0.19 × 10⁻³ m in the direction of Y axis when the reaction stress is equal σ_r = 463.4 MPa. When the reaction stress is σ_r = 464.0 MPa, the maximum displacement of the plate middle point is about 0.91 × 10⁻³ m. According to the alloy characteristics presented in the paper [3], the maximum stress could be σ_r = 480.0 MPa, and maximum deflection could reach the value 4.26 × 10⁻³ m. The critical stress was estimated to be σ_K = 321 MPa.

When stress exceeds critical stress (temperature varies between A₀ and A₁), the sample acquires a sinusoidal shape. It happens when stress exceeds critical margin:

$$\sigma >\left(1+\frac{{\pi }^2{a}^2}{2^3{l}^2}\right){\sigma }_K$$ E10

Inequality (10) comes from Eqs. (3) and (4).

When temperature gets close to its critical margin t_K °C and the stress gets close to its critical level respectfully σ_K, bending of the sample decreases and obtains the shape of a line.

The area of the arched plate will be

$$S_B=\frac{L}{2}b$$ E11

where L refers to the parameter of ellipse which is found having entered the complete elliptic interval of level 2 into Legendre form and e refers to elliptic eccentricity:

$$L=4\frac{l}{2}\underset{0}{\overset{\frac{\pi }{2}}{\int }}\sqrt{1-e^2\text{sin}\psi }\mathit{d\psi }$$ E12

According to literature [4], linear deformations of the plate can be calculated

$$\epsilon =\frac{S_B/2-\mathit{lb}}{\mathit{lb}}$$ E13

Having evaluated methods of identifying bending of the plate under various stresses (Eqs. (4)–(6)) and the deformations under the identified bending (Eqs. (11)–(13)), the authors have developed the software for calculating maximum bending values in plates containing SM materials when their critical stress is known, with additional capability of predicting the shape and the maximum force of bending in the plate. The obtained results are presented in Figure 5.

Figure 5.

Dependence of deflection of NiTi plate with the fastened ends on the various reaction stresses.

SM materials possess two rather distinctive properties: shape memory effect and superelasticity. Further, we will analyse a mathematical model aimed at the calculation of energy absorbed by the plate containing SM elements while it is being bent.

Further on, we will identify the correlation between maximum bending of the sample and the force which affects it in heating and cooling phases. The overall deformation of the sample ε is equal to the sum of elastic deformation ε^e and phase transition deformation ε^tr:

$$\epsilon ={\epsilon }^e+{\epsilon }^{\mathit{tr}}$$ E14

According to the Bernoulli equation, the deformation can be expressed as follows [5]:

$$\epsilon =\frac{2y}{L^2}\left({\delta }_0-\delta \right)$$ E15

where δ₀ refers to initial bending and δ refers to bending after the effect of force F.

Elastic deformation ε^e can be expressed as

$${\epsilon }^e=\frac{\sigma }{E}$$ E16

where σ refers to stress which occurs after the effect of force F, while E refers to Young’s modulus.

The experiments [5] allowed identifying that deformation which occurs in the transition of phases which is directly proportional to martensitic fraction z_σ:

$${\epsilon }^{\mathit{tr}}=\gamma z_{\sigma }$$ E17

where γ refers to maximum superelastic deformation.

The sum bending of the plate can be stated as follows:

$$\delta ={\delta }_0+{\delta }_{\mathit{SMA}}+{\delta }_{\mathit{EXT}}$$ E18

where δ₀ refers to initial bending, δ_SMA refers to bending which occurs in the transition of phases and δ_EXT refers to bending after the effect of force F.

Final expressions of bending for the process of heating and cooling are stated as follows:

$$\delta ={\delta }_0+\frac{3a_2\gamma L^2}{2h}\left(T-A_f^1\right)+\frac{4L^3}{{\mathit{bEh}}^3}F\left(\text{heating}\right)$$ E19

$$\delta ={\delta }_0+\frac{3a_2\gamma L^2}{2h}\left(T-M_f^0-\frac{1}{a_2}\right)+\frac{4L^3}{{\mathit{bEh}}^3}F\left(\text{cooling}\right)$$ E20

where a₂ is a constant (measure unit K⁻¹), A¹_f is temperature of terminating the austenitic phase and M⁰_f is temperature of terminating the martensitic phase.

Equations (19) and (20) make clear the existence of direct correlation between the temperature and the bending of the plate which describes its thermomechanical behaviour.

Eqs. (19) and (20) can be used for the calculation of bending in the plate containing SM elements and for the evaluation of minimum force which is necessary aiming to fix a certain shape of the plate. The numerical research has shown that force which is necessary for the counter-bending of plate varies from 370 to 135 N (it is equal to force being triggered by the plate), when temperature varies from 80 to 100°C.

Numerical results between the maximal displacement of middle point of bended plate and the various reaction stresses are shown in Figure 5.

Based on Eq. (6), it is possible to calculate restricted volume V_PL of bended plate Eq. (21) and find a changed capacity of exciting chamber:

$$V_{\mathit{PL}}\left({\sigma }_r\right)=\underset{0}{\overset{b}{\int }}\underset{0}{\overset{l}{\int }}y\left({\sigma }_r,\xi ,\zeta \right)\mathit{d\xi }\mathit{d\zeta }$$ E21

$$V_M=V_K-V_{\mathit{PL}}\left({\sigma }_{\max }\right)$$ E22

$$V_K=\pi {r_K}^2{l}_K$$ E23

The biggest volume of exciting chamber V_K when the plate has linear shape is expressed in Eq. (23). By using the above set of chamber and plate parameters, it has been found that volume of exciting chamber could be reduced from V_K = 1.767 × 10⁻⁵ m³ to V_M = 1.63 × 10⁻⁵ m³ (about 7.5%).

The volume of exciting chamber influence on vibrations of working body of vibroexciter is illustrated in Figure 6. During the numerical researches [18], after choosing particular geometrical parameters (r_k1, l_k1), initial tightening Hz, value of mass m₁ of working body of vibroexciter and amplitude of vibrations were A₁ = 4.51 × 10⁻³ m and frequency f₁ = 16.81 Hz (autovibration shape X₁ in Figure 6). When the volume of chamber was reduced in 3%, the amplitude of vibrations decreased to a value A₂ = 4.39 × 10⁻³ m, and frequency increased to f₂ = 16.92 Hz. When the volume of camera was reduced in 7.5%, amplitude of vibrations become A₃ = 4.10 × 10⁻³ m and frequency heightened until f₃ = 16.98 Hz (autovibration shape X₂ in Figure 6).

Figure 6.

Autovibrations of mass 1 of vibroexciter in 1 s period: X₁—without element with SMA; X₂—when element with SMA works and decreases the volume of chamber in 7.5%.

2.1 Brinson’s model

One dimensional composite mathematical model of internal variables has been offered by Brinson et al. [19, 20].

This model quantitatively evaluates the absorption of deformation energy and allows predicting marginal load which is necessary for the beginning and the termination of a martensitic phase transformation induced by stress [6, 19, 20]. There, martensitic fracture variable is being decomposed into two components depending on stress and temperature.

In this case, the distribution of deformation within the place is not completely linear: material properties vary from martensitic to austenitic state along the thickness of the plate. While bending, there might be three stages of load defined. In the first stage, the SM sample is in austenitic stage along its thickness. In the second stage, when the load is being increased, the martensitic phase transformation, which is being induced by stress, starts in outer layers. Further increase of load provokes plastic deformations that are not included into the energy absorption calculation model [6, 21].

The deformation energy expression according to Brinson is as follows:

$$U=\mathit{bL}{\sigma }_0{\epsilon }_0\left[\frac{h_e}{6}+\left(h-h_e\right)\left(\frac{E_2}{2E_1}-1\right)+\left(1-\frac{E_2}{E_1}\right)\frac{h^2-h_e^2}{2h_e}+\frac{E_2}{E_1}\frac{h^3-h_e^3}{6h_e^2}\right],$$ E24

while martensitic fraction, which is induced by stress, is stated as

$${\xi }_S=\frac{1}{2}\text{cos}\left\{\frac{\pi }{{\sigma }_s^{\mathit{cr}}-{\sigma }_f^{\mathit{cr}}}\left[{\sigma }_0\left(1+\frac{E_2}{E_1}\left(\frac{2y}{h_e}-1\right)\right)\right]-{\sigma }_f^{\mathit{cr}}-C_M\left(T-M_S\right)\right\}+\frac{1}{2}$$ E25

where σ₀ are ε₀ refer to stress and deformation, respectively, which call for the martensitic transformation; E₁ and E₂ refer to Young’s modulus before and after martensitic phase transformation, which had been induced by stress, respectively; σ_s^cr and σ_f^cr refer to transformation constant; and h_e refers to the thickness of sample core in austenitic stage.

Brinson model evaluates the absorption of deformation energy in SM alloy in both superelasticity and martensitic stages. In addition, it defines the correlation between deformation energy and the thickness of SM plate core.

3. Magnetorheological fluids for use in pneumatic actuators

3.1 Theoretical and experimental background of the micro choke with magnetorheological fluid transmission

Some particularities of the application of magnetorheological (MR) fluid transmissions in microcompressors and pneumatic chokes had been described in scientific works [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33].

Commonly, magnetorheological fluids are used in three flow modes, as shown in Figure 7. Magnetic poles are stationary, and MR fluid is flowing due to a pressure difference perpendicular to magnetic field in valve mode. In the case of shear mode, one of the magnetic poles is moving with respect to the other pole, and the MR fluid is being sheared. MR fluid is placed between two approaching plates and compressed, while magnetic field lines pass through the fluid in squeeze mode [31].

Figure 7.

MR fluid flow modes: (a) valve mode, (b) shear mode, and (c) squeeze mode [31].

The MR fluid application in valve mode is presented in this research. Magnetorheological effect can be used in controlled choke structures which might be characterised as low power and pressure (see Figure 8).

Figure 8.

Micro choke structure containing magnetorheological fluid transmissions: 1—solenoid; 2—constant magnet; 3—tube; 4—magnetic fluid; 5—silicone; and Q₁, Q₂—working medium flow.

This structure consists of the solenoid 1 with a constant magnet of ring shape 2 and bladdery tube 3 coaxially fitted into its cylindrical opening. Tube 3 is filled with 4, which consists of the plug of toric segment shape in a silicone bag, along the constant magnet 2. Under choking, geometrical measures of MR fluid 4 plug and its magnetization force, which is being hold at the internal wall of bladdery tube 3, depend on the gradient of magnetic field created by the constant magnet 2 and solenoid 1. If the solenoid 1 is powered by supply voltage from a control unit (no indication of it in the drawing), the field strength and the shape of free surface of MR fluid plug in a working gap change. The diameter d of the opening also changes under choking. Thus, it proceeds to the work medium flow regulation.

The substantiation of the mathematical model is based on the ferrohydrostatic equation [33, 34]:

$$\mathit{\text{gradp}}=\rho \overline{g}+{\mu }_0\mathit{\text{MgradH}}$$ E26

and the correlation of pressure capillary jump on a free surface

$$p=p_0-\alpha \left(K_1+K_2\right)$$ E27

where K₁ and K₂ refer to the main curvatures of normal sections on a free surface, α refers to surface stress coefficient, p₀ refers to pressure of external environment, p refers to fluid pressure, ρ refers to density, g refers to free fall acceleration, μ₀ refers to vacuum magnetic penetration, M = M(H) refers to fluid magnetization and H refers to magnetic field penetration.

First, let us deduce formulas of main curvatures of normal sections. Suppose

$$x=r\left(s\right)\text{cos}\varphi ,y=r\left(s\right)\text{sin}\varphi ,z=z\left(s\right)$$ E28

It is known that main curvatures of normal sections on surfaces can be expressed through coefficients of the first and the second Gauss form as follows:

$$K_1+K_2=\frac{\mathit{EN}-2\mathit{FM}+\mathit{GL}}{\mathit{EG}-F^2}$$ E29

In this case

$$E={\left(\frac{\partial x}{\partial s}\right)}^2+{\left(\frac{\partial y}{\partial s}\right)}^2+{\left(\frac{\partial z}{\partial s}\right)}^2={r^{\prime }}^2{\text{cos}}^2\varphi +{r^{\prime }}^2{\text{sin}}^2\varphi +{z^{\prime }}^2={r^{\prime }}^2+{z^{\prime }}^2=1,$$ E30

$$F=\frac{\partial x}{\partial s}\frac{\partial x}{\partial \varphi }+\frac{\partial y}{\partial s}\frac{\partial y}{\partial \varphi }+\frac{\partial z}{\partial s}\frac{\partial z}{\partial \varphi }=-{r^{\prime }}^2{\text{cos}}^2\varphi \cdot r\text{sin}\varphi +{r^{\prime }}^2{\text{sin}}^2\varphi \cdot r\text{cos}\varphi =0$$ E31

$$G={\left(\frac{\partial x}{\partial s}\right)}^2+{\left(\frac{\partial y}{\partial s}\right)}^2+{\left(\frac{\partial z}{\partial s}\right)}^2={r^{\prime }}^2{\text{sin}}^2\varphi +{r^{\prime }}^2{\text{cos}}^2\varphi ={r^{\prime }}^2$$ E32

$$L=\frac{{\overline{r}}_{S^2}^{″}\left({\overline{r}}_{S^X}^{\prime }{\overline{r}}_{\varphi }^{\prime }\right)}{\sqrt{\mathit{EG}-F^2}}$$ E33

$$N=\frac{{\overline{r}}_{{\varphi }^2}^{″}\left({\overline{r}}_{S^X}^{\prime }{\overline{r}}_{\varphi }^{\prime }\right)}{\sqrt{\mathit{EG}-F^2}}$$ E34

where

$${\overline{r}}_{S^2}^{″}=\left(r^{″}\text{cos}\varphi ,r^{″}\text{sin}\varphi ,z^{″}\right) {\overline{r}}_{S^X}^{\prime }=\left(r^{\prime }\text{cos}\varphi ,r^{\prime }\text{sin}\varphi \right)$$ E35

$${\overline{r}}_{\varphi }^{\prime }=\left(-r\text{sin}\varphi ,r\text{cos}\varphi ,0\right) {\overline{r}}_{{\varphi }^2}^{″}=\left(-r\text{cos}\varphi ,-r\text{sin}\varphi ,0\right)$$

After integration of Eq. (26) according to z

$$r^{″}=-z^{\prime }\left[\frac{\mathit{\rho gz}}{\vartheta }-\frac{{\mu }_0{M}^{\ast }{H}^{\ast }}{\vartheta }\text{log}\left(\frac{H^{\ast }}{H}\mathit{sh}\frac{H}{H^{\ast }}\right)+C-\frac{z^{\prime }}{r}\right]$$ E36

When s = 0, there are following terms:

$$r\left(0\right)=R_0,r^{\prime }\left(0\right)=-\text{sin}{\alpha }_0,z^{\prime }\left(0\right)=-\text{cos}{\alpha }_0,$$ E37

And when s = s*,

$$z\left(s^{\ast }\right)=0,r^{\prime }\left(s^{\ast }\right)=0,z^{\prime }\left(s^{\ast }\right)=-1$$ E38

The task is solved in the non-asymmetric form:

$${\left(\overline{r}{\overline{z}}^{\prime }\right)}^{\prime }=\overline{r}{\overline{r}}^{\prime }\left[B_0\frac{\overline{z}}{{\overline{r}}^2\left(0\right)}-A_1\varphi \left(\overline{r},\overline{z}\right)+C\right]$$ E39

The digital algorithm with non-inertial variables is investigated:

$$\begin{array}{l}\frac{z_i^{n+1}-z_i^n}{\tau }=\frac{1}{r_i^n}{\left(\frac{r_{i-1}^n+r_i^n}{2}{z}_{s,i}^{n+1}\right)}_{s,i}-\frac{B_0}{{\left(r_0^n\right)}^2}\left[z_i^{n+1}{\left(r_{\tilde{s},i}^n\right)}^{+}+z_i^n{\left(r_{\tilde{s},i}^n\right)}^{-}\right]+ r_{\tilde{s},i}^n\left[A_1\varphi \left(r_i^n,z_i^n\right)-C^n\right]\end{array}$$ E40

where i = 1,…, N − 1

$$z_{s,0}^{n+1}=-\frac{h}{2}\left[\frac{B_0}{{\left(r_0^n\right)}^2}{z}_i^{n+1}-A_1\varphi \left(r_i^n,z_i^n\right)-C^n+\frac{\text{cos}{\alpha }_0}{r_0^2}\right]\text{sin}{\alpha }_0-\text{cos}{\alpha }_0$$ E41

$$z_N^{n+1}=0;$$

$$\frac{r_{i-1}^n+r_i^n}{\tau }=r_{\tilde{s}s,i}^{n+1}+z_{\tilde{s}s,i}^{n+1}\left[\frac{B_0}{{\left(r_0^n\right)}^2}{z}_i^{n+1}-A_1\varphi \left(r_i^n,z_i^n\right)-C^n-\frac{z_{\tilde{s},i}^{n+1}}{r_i^n}\right]$$ E42

There are two sets of items Eⁿ_i and Fⁿ_i such that for each set of elements, C is valid:

$$z_i^{n+1}=E_i^n{z}_{i+1}^{n+1}+F_i^n$$ E43

After the analytical studies, the relationships are obtained:

$$z_i^{n+1}=\frac{A_i^n}{B_i^n-C_i^n{E}_{i-1}^n}{z}_{i+1}^{n+1}+\frac{D_i^n+C_i^n{F}_{i-1}^n}{B_i^n-C_i^n{E}_{i-1}^n}$$ E44

and

$$E_i^n<\frac{A_1^n}{B_1^n-\left(1+2c_0{h}^2\right)C_1^n}<\frac{A_1^n}{A_1^n+1-2c_0{h}^2{C}_1}<1$$ E45

Two sets of ε_i and V_i are determined:

$$r_i^{n+1}={\epsilon }_i{r}_{i+1}^{n+1}+v_i^n$$ E46

where

$${\epsilon }_i=\frac{{\alpha }_i}{{\beta }_i-{\gamma }_i{\epsilon }_{i-1}},i>1$$ E47

$$v_i^n=\frac{{\vartheta }_i^n+{\gamma }_i{v}_{i-1}^n}{{\beta }_i-{\gamma }_i{\epsilon }_{i-1}},i>1$$ E48

After analytical study is found:

$$r_N^{i+1}=\frac{v_{N-1}^n}{1-{\epsilon }_{N-1}}$$ E49

MR fluid magnetic field strength in a working gap has been measured, and the shape of MR fluid plug on free surface has been set [33].

In order to prevent the damage that might be done by pressure transmitted through the micro choke P₁ on the shape of MR fluid plug on free surface under choking, pressure P₁ should not reach the free surface magnetic field force, which has been created.

Experiments allowed identifying [33] magnetic field strength at concentrators (points A and B, Figure 9) moving from the outside of glass tube to the centre in two perpendicular to each other directions, i.e. from point 4 to point 0 and from point 3 to point 0, in steps of 0.5 × 10⁻³ m. Research results are presented in Table 2 and Figure 10.

Figure 9.

Scheme of the identifying magnetic field: (a) measuring directions 3–0 and 4–0; (b) measuring points A and B.

Distance from the outside of glass tube h, ×10^−3 m	Magnetic field strength H, ×10^3 A/m
	Direction 4–0		Direction 3–0
	Point A	Point B	Point A	Point B
0.5	46.871	44.006	47.587	49.179
1.0	36.526	35.651	31.672	34.139
1.5	31.911	24.112	25.942	29.682
2.0	24.987	20.611	19.178	19.417
2.5	20.451	16.870	20.531	15.120
3.0	16.313	10.425	14.801	10.504
3.5	12.812	5.730	11.539	7.639
4.0	11.459	1.751	10.504	4.138
4.5	6.287	1.035	4.218	1.592

Table 2.

MR fluid magnetic field strength in a working gap of the micro choke.

Figure 10.

Correlation between the magnetic field strength H and the distance from a tube wall being measured in two directions: (a) 4–0, 1, measures taken at point A; 2, measures taken at point B; (b) 3–0, 1, measures taken at point A, and 2, measures taken at point B.

3.2 Study of microcompressor with MR fluid

The operation principle of the microcompressor drive with MR liquid under study has been presented in paper [35]. Figure 11 shows part of the general microprocessor view.

Figure 11.

Scheme of an actuator with MR fluid: 1—MR fluid vane; 2—permanent magnet; 3—silicone shell; R_s—stator radius; R_r, rotor radius; e—eccentricity; a, b and c, d—free surface of MR fluid; G—centrifugal force; ω—rotor angular velocity; S and N—poles of permanent magnet; and τ, shear stresses.

In the cylindrical system of axes, when the coordinate axis coincides with the rotor axis, the motion of viscous incompressible MR fluid vane is described by Navier-Stokes equations [36], which are expressed as follows:

$$-\frac{\partial p\text{'}}{\partial r\text{'}}+\frac{1}{{\mathit{Fr}}_m}\frac{\partial S_m^{\text{'}}}{\partial r\text{'}}+\frac{R+r}{{\left(R+1\right)}^2}=0$$ E50

$$-\frac{\partial p^{\text{'}}}{\partial \phi }+\frac{1}{{\mathit{Fr}}_m}\frac{\partial S_m^{\text{'}}}{\partial \phi }+\frac{\mathit{\tau \delta }\left(R+r^{\text{'}}\right)}{\rho u_0^2}=0$$ E51

where ${\mathit{Fr}}_m=\frac{\rho u_0^2}{{\mu }_0{M}_s{H}_{\ast }}$ is magnetic Froude number; $S_m^{\text{'}}=\text{ln}\frac{\mathit{sh\xi }}{\xi }$ is a nondimensional coordinate function of the magnetic field; $r^{\text{'}}=\frac{R+r_a}{R+1}$ is a nondimensional relative coordinate; $\rho$ is the density of the magnetic fluid (kg/m³); $\delta$ is the mean thickness of the working aperture between the rotor and the stator (m); $u_0$ is the vane velocity at the stator (m/s); ${\mu }_0$ is magnetic permittivity (H/m); $M_s={\phi }_{\mathit{dal}}{M}_{\mathit{so}}$ is magnetic saturation of the magnetic fluid (A/m); ${\phi }_{\mathit{dal}}$ is the number of particles in a volume unit; $M_{\mathit{so}}$ is magnetic saturation of a particle (A/m); $H_{\ast }=\frac{\mathit{kT}}{{\mu }_{0}m}$ is the intensity of the magnetic field (A/m); $k$ is Boltzmann constant; $T$ is temperature (K); $m=V_{\mathit{dal}}{M}_{\mathit{so}}$ is the magnetic moment of the particle; $V_{\mathit{dal}}$ is the volume of the magnetic particle (m³); $\xi =\frac{{\mu }_0\mathit{mH}}{\mathit{kT}}$ is the argument of Langevin function, $\tau =\left(R+r^{\text{'}}\right)\left[\frac{{\partial }^{2}v}{\partial r^{\text{'}2}}+\frac{1}{\left(R+r^{\text{'}}\right)}\frac{\partial v}{\partial r^{\text{'}}}-\frac{v}{{\left(R+r^{\text{'}}\right)}^2}\right]$ ; $v=u_0{v}^{\text{'}}$ is the azimuth component of velocity (m/s); and $v^{\text{'}}$ is nondimensional velocity.

When studying the efficiency of the drive, it is important to know MR fluid pressure upon the stator surface. When r’ = 1, the vane pressure upon the stator surface is [35]

$$\begin{array}{l}{p}^{\text{'}}=p_a^{\text{'}}+\frac{1}{{\mathit{Fr}}_m}\left[S_m^{\text{'}}-S_m^{\text{'}}\left(0,{\phi }_a\right)\right]+\frac{1}{2}\left[{\left(\frac{R+r^{\text{'}}}{R+1}\right)}^2-{\left(\frac{R}{R+1}\right)}^2\right]+\\ +\frac{6\left(R+1\right)}{\text{Re}}\left(\phi -{\phi }_a\right)=\sum \left(r^{\text{'}},\phi \right)-\sum \left(0,{\phi }_a\right)+p_a^{\text{'}}\end{array}$$ E52

The performed calculations have shown that the maximum pressure developed by the actuator reaches 10.8 × 10³ Pa, when n = 5000 r/min, q = 0.8 and $\delta$ = 2 × 10⁻³ m.

For experimental study of an actuator with MR fluid, a stand (Figure 12) was made with the following technical parameters: the internal diameter of the stator D_s = 18 × 10⁻³ m, the external diameter of the rotor D_r = 15 × 10⁻³ m, the rotor working length L = 8 × 10⁻³ m, the eccentricity e = 5 × 10⁻⁵ m, the volume of the working chamber V = 6 × 10⁻⁷ m³, the number of permanent magnets is 4, the supply voltage U = 15 V, the supply current I = 0.7 A, the rotation frequency of the rotor n = 0÷5000 r/min, the MR fluid magnetic saturation M_s = 51.7 kA/m, the MR fluid density ρ = 1585 kg/m3 and the diameter of magnetic particles R_dal = 9.01 × 10⁻⁹ m [35].

Figure 12.

Scheme of an experimental stand for the actuator with MR liquid: 1—transducer; 2—control panel; 3, 4—power supply; 5—impulse generator; 6—diaphragm; 7—micromanometer; 8—telltale manometer; and 9, 10—valve.

The distribution of magnetic field in the working aperture of electromagnetic-magnetic systems with MR fluid determines the magnetic-hydrodynamic characteristics of the fluid. Therefore, the magnetic induction in the transducer’s working aperture was measured with teslameter (by inserting a sensitive element with Hall sensor into the aperture). The measurements were made both at the surface of the rotor and the stator at 30°. The results of the measurements are presented in Figure 13. It has been determined that the highest value of magnetic induction is in the narrowest place of the working aperture at the rotor surface, reaching 0.745 T. The dissemination of magnetic induction at separate poles (magnet) does not exceed 1.2%. The experimental stand was also used for determining the values of the pressure and efficiency generated by the transducer. When the transducer rotor with the magnetic fluid vanes is rotating, the pressure difference developed in diaphragm is registered by micromanometer. According to the manometer readings, the pressure and the efficiency developed by the transducer are calculated.

Figure 13.

Dependence of magnetic induction in working aperture upon the rotor turn angle: (a) at the rotor surface; (b) at the stator surface; 1—first magnet; 2—second magnet; 3—third magnet; and 4—fourth magnet.

The dependence of the pressure generated by the transducer on the MR fluid vane-filling coefficient while changing the rotation frequency is shown in Figure 14.

Figure 14.

Dependence of pressure p generated by the transducer upon the MR fluid vane-filling coefficient while changing the rotor rotation frequency: (a) frequency changed from 670 to 2610 r/min; (b) from 3060 to 5000 r/min.

As seen from the presented charts, the value of the pressure generated by the actuator depends directly on the MR fluid vane-filling coefficient and the rotation frequency. The highest pressure value is obtained when MR fluid-filling coefficient is close to 0.9. The generated pressure increases with increasing rotation frequency (up to 13.7 kPa, when n = 4615 r/min). Figure 15 shows the dependence of the transducer’s efficiency upon MR fluid vane-filling coefficient while changing the rotation frequency of the rotor.

Figure 15.

Dependence of efficiency developed by the transducer upon MR fluid vane-filling coefficient q, when changing the rotor rotation frequency n: (a) frequency changed from 670 to 2610 r/min; (b) from 3060 to 4615 r/min.

The study results show that the transducer’s capacity also depends on the rotation frequency of the rotor and MR fluid vane-filling coefficient. However, only at low rotation frequencies of the rotor (up to 2000 r/min), the capacity rises gradually with changing MR fluid vane-filling coefficient. The maximum value of the efficiency achieved is 0.37 × 10⁻⁶ m³/s.

4. Conclusions

4.1 The performed study of the shape memory elements in pneumatic actuators leads to the following conclusions

The numerical researches have showed dependence of the maximal displacement of middle point of bended plate when both plate’s ends are fastened, on the reaction stresses. A numerical methodology has been proposed which describes the changeable volume of chamber with smart material element.

The presented engineering methodology enables to determine dependence of force generated by plate with shape memory material on mechanical stresses.

The theoretical research has shown that the change of chamber volume affects the parameters of autovibrations in vibroexciter. During the studies of the plate of the titanium-nickel alloy TN-1, it was estimated that maximum deformation of plate, whose length l = 25 × 10⁻³ m, height h = 1.5 × 10⁻³ m and width b = 20 × 10⁻³ m, changed the volume of chamber in vibroexciter by 7.5%. It caused an amplitude of vibrations to decrease in 9.1% and frequency to increase in 0.7%. Other SM alloys which have different characteristics and better deformation properties in the direction of Y axis can be used in the construction of vibroexciters. In some technological processes for precise vibroexciters, a minor change of the frequency is also important.

The methodology of calculations used in present research evaluates the deformation of the plate only approximately. For more precise calculations, it is necessary to evaluate the temperature deformations, cross stress and the fact that part of the deformation energy is transferred to the heat used to replace the material structure.

The obtained study results may be applied in designing equipment, containing SMA elements.

4.2 The performed study of the transducer with MR fluid leads to the following conclusions

The value of the pressure generated by the transducer actuator depends directly on MR fluid vane-filling coefficient and rotation frequency. The highest pressure value is obtained when MR fluid-filling coefficient is close to 0.9. The pressure increases with increasing rotation frequency (up to 13.7 kPa, when n = 4615 r/min).

The efficiency developed by the transducer also depends on the rotation frequency of the rotor and on MR fluid vane-filling coefficient. The maximum obtained efficiency value is 0.37 × 10⁻⁶ m³/s.

At low MR fluid vane-filling coefficient q values—up to 0.67—unstable work areas of the transducer caused by hermetic flaws of the system have been observed.

Findings of theoretical and experimental studies are sufficiently adequate; the obtained pressure values do not exceed 15%.

Further studies are needed to ensure hermetic properties of the system with MR fluid vanes.

Thanks

The authors thank Prof. Habil and Dr. Ramutis Bansevičius for the ideas and help in the theoretical research and experiments.