, Volume 48, Issue 5, pp 1081–1091

Development, modeling and deflection analysis of hybrid micro actuator with integrated thermal and piezoelectric actuation

  • Hasan Pourrostami
  • Hassan Zohoor
  • Mohamad H. Kargarnovin

DOI: 10.1007/s11012-012-9653-z

Cite this article as:
Pourrostami, H., Zohoor, H. & Kargarnovin, M.H. Meccanica (2013) 48: 1081. doi:10.1007/s11012-012-9653-z


Micro actuators are irreplaceable part of motion control in minimized systems.

The current study presents an analytical model for a new Hybrid Thermo Piezoelectric micro actuator based on the combination of piezoelectric and thermal actuation mechanisms. The micro actuator structure is a double PZT cantilever beam consisting of two arms with different lengths. The presented micro actuator uses the structure of electrothermal micro actuator in which polysilicon material is replaced by PZT. Also the voltage and poling directions are considered in the lengthwise of PZT beams. As a result, the piezoelectric actuation mechanism is based on d33 strain coefficient.

The tip deflection of micro actuator is obtained using Timoshenko beam theory. Analytical results are compared with FEM results along with other reported results in the literature. The effects of geometrical parameters and PZT material constants on actuator tip deflection are studied to provide an efficient optimization of HTP micro actuator.


Hybrid Micro actuator Thermo piezoelectric Deflection Timoshenko beam theory 

1 Introduction

Nowadays, micro actuators have a key role in the development of minimized systems. There is an immense growth in evolution and design of micro actuators in the precision field of micromanipulation and micro assembly. A variety of applications have been envisaged for micro actuators including cell manipulation [1], micro assembly of optical MEMS devices [2] and flexible micro assembly stations [3]. To fulfill the required tasks, the micro actuators need high performances such as high resolution, high accuracy, quick response time and high range of positioning.

Different types of actuation mechanisms can be classified into thermal/electrothermal, electrostatic, magnetic/electromagnetic, piezoelectric and other categories. Electrothermal and piezoelectric micro actuators are the most prized actuators due to their remarkable characteristics. They are widely used in variety of applications including micro gripper [4], scanning probe arrays [5], micro positioning applications [6], liquid drop ejectors [7], control valves [8] and acoustic control [9].

High positioning accuracy, rapid response and high mechanical power can be achieved using the piezoelectric micro actuators, in spite of low deformation range capability [10]. Taking the thermal micro actuators into account, they exhibit advantages including large output force with reasonable displacement, easy fabrication process and operation at small drive voltages [11]. Despite consuming a lot of power and showing slow response time, their disadvantages are far outweighed by remarkable advantages. Combination of these two classes of micro actuator as a hybrid micro actuator overcomes their limitations. Rakotondrabe and Ivan have attempted to integrate electrothermal and piezoelectric actuation mechanisms [12]. They presented a new hybrid thermo piezoelectric micro actuator which utilized the thermal bimorph principle and piezocantilever micro actuator. The micro structure was a unimorph piezoelectric cantilever making up of a piezoelectric layer and a passive copper layer. While electrical voltage was used to control the piezoelectric actuation, a Peltier module (as an external source of heat generation) was applied to provide uniform temperature variation. By using an electrical field, the piezoelectric layer expanded/contracted and led to bending the cantilever. In addition, by varying the temperature, a secondary bending was occurred due to the different thermal expansions in both layers.

Although the actuator behavior under input voltage and temperature has been studied, the effects of PZT material constant have not been investigated yet. In fact, PZT material constants including piezoelectric stain and dielectric constant are strongly dependent on operation temperature. Therefore, temperature has a double effect by causing different thermal expansions and changing piezoelectric material constants.

In the current study, a new model of thermo piezoelectric micro actuator is proposed. The presented micro actuator uses the structure of electrothermal micro actuator in which polysilicon material is replaced by PZT. Also the piezoelectric actuation mechanism is based on d33 strain coefficient which means the voltage and poling directions are considered in the lengthwise of PZT beams.

The effects of geometrical and material constants on actuator tip deflection were studied to provide an efficient optimization of HTP micro actuator. Moreover, Timoshenko beam theory was employed to accurately model the interaction between different physical fields including mechanical piezoelectric and thermal.

2 Modeling

Figure 1 shows the configuration of new Hybrid Thermo Piezoelectric (HTP) micro actuator. This micro actuator has a double cantilever beam structure which is made up of PZT.
Fig. 1

The structure of the hybrid thermopiezoelectric micro actuator

The structure of the micro actuator consists of a pair of adjacent arms with different lengths and a small conjunction beam [13, 14]. The actuator has an integrated structure with no hinges or joints. Moreover, both arms and conjunction beam have same cross-sectional areas and poling direction.

In electrothermal micro actuators with the same structure, passing the electric current through the arms, led to an enhancement in temperature and expansion of the longer arm, as compared with the shorter arm, which is due to larger electrical resistance of the longer arm. As a result, different expansions cause the deflection of the actuator. This type of actuation mechanism occurs when the actuators are made up with low electrical resistivity materials such as polysilicon (10−4 to 10−3 Ω m) [13].

Contrarily to polysilicon, PZT materials have high electrical resistivity (more than 109 Ω m at room temperature). By applying voltage, low heat will be generated in the micro actuator. Although, it is possible to generate more heat by increasing input voltage, it can also depolarize or destroy the micro actuator. Therefore, in order to have any types of thermal deformation it is necessary to provide an external source for applying temperature. Considering micro size of actuator and the reported experimental work, a uniform temperature variation is applied on the surface of micro beams [12], which results in higher elongation of the longer arm than the shorter arm.

If the applied voltage direction is along the polarization direction in PZT beam, it expands, otherwise it contracts (extensional mode) [15].

In order to achieve more deflection in presented micro actuator, the applied voltage direction must be along the polarization direction in long arm and in the opposite direction in short arm. In this case, the long arm expands not only due to temperature change but also due to piezoelectric actuation whereas in the short arm thermal effect increases the length but piezoelectric actuation decreases the length.

The actuator can be considered in the state of plane stress because the beam’s width and heights are much shorter than the beam’s length. Furthermore, it is assumed to deflect only laterally and no external mechanical load is applied on the actuator.

2.1 Beam model and field approximations

According to Timoshenko beam theory (TBT), the related displacement field in the beam is assumed as follows [16]: Here ux and w are the beam neutral axis displacement in the x and z directions and ψ is the bending rotations of the vertical lines perpendicular to the neutral axis. Furthermore, based on small deformation theory the axial strain (εx) and shear strain (γxz) associated with this displacement field in xz plane are [16]:

2.2 Constitutive relations

Prior to poling, the PZT is an isotropic material but behaves as transversely isotropic after the poling process [17]. When the poling is along other than thickness directions (z direction), the material matrices can be obtained by tensor transformations [15]. The geometrical and material coordinate systems for a PZT element are shown in Fig. 2. For poling along the x axis, the geometrical and material coordinate systems are shown in Fig. 2(b).
Fig. 2

A PZT element (a) the poling in z direction, (b) the poling in x direction [17]

The general 3D constitutive relations for the piezoelectric materials written in terms of non-zero stress and strain components are [15]: in which {σ}, {ε}, {E} and {D} represent the columns of stress components, strain components, electrical field components and electrical displacement components respectively. Also [C], [e] and [ϵ] are the stiffness matrix, the piezoelectric matrix and the dielectric permittivity matrix.
The linear constitutive equations of piezoelectricity under plane assumptions when the temperature is applied and the poling is along the x axis are written as [17]: where dij are piezoelectric strain constants and are defined by [e]=[C][d].
It should be noted that because of small aspect ratio (height/length) in PZT beams, it is considered that the thermal strain in z direction are negligible and there is only thermal strain in lengthwise direction. Moreover α is the thermal expansion coefficient of the PZT and \(\bar{c}_{11}\), \(\bar{c}_{13}\), \(\bar{c}_{33}\), \(\bar{e}_{13}\), \(\bar{e}_{33}\) and \(\bar{ \epsilon}_{33}\) in Eqs. (5a)–(5c) are the reduced material constants of the piezoelectric under plane stress assumptions and they are given by: For transversely isentropic piezoelectric beam with small aspect ratio, the thickness in z-direction is stress free. Therefore, it is plausible to set σz=τyz=τxy=γyz=γxy=0. Moreover, there are no electric displacements in z-direction on beam’s top and bottom surfaces which it means Dz=0 when zh/2 [17, 18].
By setting σz=0 in Eq. (5a), εz yields to:
$$ \varepsilon_{z} = - \frac{\bar{c}_{13}}{\bar{c}_{11}} ( \varepsilon_{x} - \alpha\Delta T ) + \bar{e}_{31} E_{x} $$
Also by setting Dz=0 when zh/2, the following equations are obtained: According to Eq. (3) the shear strain is constant through the cross section. On the other hand, the voltage is applied only in the x direction. Then it is assumed that the z component of electric field Ez is constant along the z direction within the beam [18]. Therefore, due to no applied voltage in z direction it can be assumed that the z component of electrical displacement field Dz is considered constant and negligible through the cross section. The shear strain can be obtained from Eq. (5c) by setting Dz=0:
$$ \gamma_{x z} = - \frac{ \epsilon_{11}}{e_{15}} E_{z} $$
By substituting Eq. (9) into Eq. (3), the bending rotation ψ can be defined as:
$$ \psi ( x ) = \frac{d w}{d x} - \gamma_{x z} = \frac{d w}{d x} + \frac{ \epsilon_{11}}{e_{15}} E_{z} $$
By substituting Eq. (7) into Eqs. (5a)–(5c) and using mentioned assumptions, the constitutive equations reduce to: where \(\tilde{c}_{33} = \bar{c}_{33} - \frac{\bar{c}_{13}^{2}}{\bar{c}_{11}}\), \(\tilde{e}_{33} = \bar{e}_{33} - \frac{\bar{c}_{13}}{\bar{c}_{11}} \bar{e}_{31}\), \(\tilde{ \epsilon}_{33} = \bar{ \epsilon}_{33} + \frac{\bar{e}_{31}^{2}}{\bar{c}_{11}}\) and \(\tilde{c}_{55} = c_{55}\).
For the piezoelectric beam under-consideration the Maxwell equation is [18]:
$$ \frac{\partial D_{x}}{\partial x} + \frac{\partial D_{z}}{\partial z} = 0 $$
Since the voltage is applied only longitudinally i.e. in the x direction, then it is assumed that the z component of electric displacement field Dz remains constant and negligible along the z direction within the beam. Hence Eq. (12) reduces to:
$$ \frac{d D_{x}}{d x} = 0 $$
By substituting Eq. (11b) into Eq. (13), the following equation is obtained:
$$ \frac{\partial E_{x}}{\partial x} = - \frac{\tilde{e}_{33}}{\tilde{ \epsilon}_{33}} \frac{\partial}{\partial x} ( \varepsilon_{x} + \alpha\Delta T ) $$
For a uniform temperature distribution, ΔT/∂x becomes zero. After recalling εx from Eq. (2) and by integrating Eq. (14), the electric field component in x direction can be obtained:
$$ E_{x} = \frac{\tilde{e}_{33}}{\tilde{ \epsilon}_{33}} z \frac{\partial \psi}{ \partial x} + c $$
where c is integration constant and can be obtained using electrical boundary condition. For an undeformed beam under applied voltage the electric field is V/L. Therefore, the value of c in expression (15) is V/L.
$$ E_{x} = \frac{\tilde{e}_{33}}{\tilde{ \epsilon}_{33}} z \frac{\partial \psi}{ \partial x} + \frac{V}{L} $$
where V is applied voltage and L is the total length of actuator.

3 Solution method

The structure of the actuator is a plane-frame structure with two fixed bases (see Fig. 3). To analyze the displacement and rotation of the beam at the fixed end (point G), first of all the expansion of each arm due to the thermal and piezoelectric effects must be determined separately. Note that the positive temperature gradient causes both arms to expand, where the voltage change enlarges the upper arms and contracts the lower arm due to polarization and voltage direction. It should be mentioned that the piezoelectric effect along the arm MGN is assumed to be negligible (see Fig. 3(a)). Based on the stated information, the expansion of each arm due to simultaneous effects of temperature and piezoelectricity can be given by: where ψM and ψN are the bending rotations of points M and N at the end of arms (see Fig. 3(a)). Using Eq. (10), ψM and ψN are: Noted that θM and θN are the slopes of points M and N. In this case since Lg is too small in comparison with the arms length, it is assumed that the slopes of points M, N and G are equal.
Fig. 3

(a) Double arm structure model along with considered system of coordinate. (b) The free body diagram of the structure with 3 redundant systems of forces. (c) The bending moment distribution on the short arm due to X1 and X3 force components. (d) The applied virtual force and moment at point G of micro actuator

The actuator under consideration is a statically indeterminate structure with the degree of the indeterminacy of 3. Each support can be released and replaced by two forces and one moment components. The three redundant systems of forces (X1, X2 and X3) are shown in Fig. 3(b).

The three redundant systems of forces X can be obtained by solving a set of simultaneous equations [13]:
$$ \left [ \begin{array}{c@{\quad}c@{\quad}c} f_{11} & f_{12} & f_{13} \\ f_{21} & f_{22} & f_{23} \\ f_{31} & f_{32} & f_{33} \end{array} \right ] \left [ \begin{array}{c} X_{1} \\ X_{2} \\ X_{3} \end{array} \right ] = \left [ \begin{array}{c} \Delta L_{g} \\ \Delta L_{2} - \Delta L_{1} \\ 0 \end{array} \right ] $$
where terms fij represent flexibility coefficients, which can be obtained by the diagram product of the bending moments and given by [13]: where E is the beam Young’s modulus, I(bh3/12) is the second cross-section moment, EI represents the beam flexural rigidity, b and h are the width and heights of the beams cross-section. The bending moment distribution on the short arm by redundant system of forces is shown in Fig. 3(c). In order to obtain the beam free end deflection at point G, the Castiglione theorem is employed. To conduct calculation for this purpose, we apply a virtual system of loads (P,Q) at point G of the beam as it is shown in Fig. 3(d). The bending moment distribution in the short arm due to the virtual and redundant systems forces is:
$$ M_{2} = X_{3} - x X_{1} + \tfrac{ ( L_{1} ( - 2 x_{2}L_{2}L_{1}^{2} P + x_{2}L_{1}^{3} P + 3 x_{2}L_{2}^{2}L_{1} P + 6 x_{2}L_{2}L_{1} Q - L_{1}^{3} Q + L_{2}^{2}L_{1}^{2} P - 3 L_{2}^{2}L_{1} Q - L_{1} L_{2}^{3} P + 2 L_{2}^{3} Q ) )}{L_{1}^{4} - 2L_{1} L_{2}^{3} - 2L_{2} L_{1}^{3} + L_{2}^{4} + 6L_{1}^{2} L_{2}^{2}} $$
According to the Castiglione theorem, the deflection and slope in the free end of the actuator (point G) can be obtained as follows: By solving simultaneously above equations, the deflection of actuator tip can be obtained.

4 Results and discussion

In this section, the derived governing equations are solved for beam’s tip deflection. Primarily, the obtained analytical results are compared with the results of FEM model along with the reported results in the literature [12, 13]. Then, the effects of geometrical parameters and PZT material constants on actuator tip deflection are studied to provide an efficient optimization of HTP micro actuator.

In order to obtain numerical results, the data listed in Tables 1 and 2 are used for geometrical and material properties, respectively. Since there is no voltage in z-direction, the z component of electric field (Ez) is ignored and set Ez=0.
Table 1

Geometrical data for actuator

Length of the long arm, L1 (μm)


Length of the gap, Lg (μm)


Width of beams, w (μm)


Height of beams, h (μm)


Table 2

Material properties [15, 19]

Material properties

(25 °C)

(125 °C)

Stiffness matrix component, c11 (GPa)



Stiffness matrix component, c33 (GPa)



Stiffness matrix component, c12 (GPa)



Stiffness matrix component, c13 (GPa)



Piezoelectric Transverse strain constant, d31 (C/N)



Piezoelectric Extensional strain constant, d33 (C/N)



Piezoelectric strain constant, d15 (C/N)



Expansion coefficient, α (1/°C)



Dielectric constant, ϵ11 (C/V m)



Dielectric constant, ϵ33 (C/V m)



In current analysis, PZT-5H is used because it owns the largest piezoelectric coefficients at room temperature among other PZT ceramics; however it has the low Curie point temperature [19].

Curie point is a temperature which if a piezoelectric element is heated to it, the element becomes completely depolarized and loses its piezoelectric effect. Therefore a piezoelectric element can function for long period without marked depolarization only at temperature below the Curie point. For PZT-5H, its low curie point (170 °C) limits the working temperature range. It is considered that the working temperature range for safe function is between 25 °C (room temperature) and 125 °C.

Several researches have been done about temperature dependence of PZT material properties which are shown that for PZT-5H: (i) piezoelectric strain constants d31 and d33 increase by increasing temperature while d15 almost remains constant. (ii) Piezoelectric dielectric constants ϵ11 and ϵ33 increase by increasing temperature. (iii) For mentioned temperature range the elastic constants do not change remarkably by increasing temperature [19, 20].

4.1 Verifications

In the first step, variation of non-dimensional deflection (w/L1) of point G versus non-dimensional length ratio (L2/L1) under a constant voltage (V=10 V) and uniform temperature gradient (ΔT=100 °C) are plotted. The analytical result is compared with the similar result from the FEM analysis (see Fig. 4). From this figure, it is evident that the analytical result is in a good agreement with those of FEM. Since in this analysis the applied uniform temperature distribution is not altered with the voltage, then the effects of ΔT and voltage on beam’s deflection can be superposed. Therefore, the obtained result is similar to reported result by Huang and Lee [13]. To discuss further, note that when L2=L1, the temperature gradient has no effect on the actuator tip deflection, however, referred to Fig. 4 it is seen that the tip deflection is not zero because of piezoelectric effect. Also it has to be mentioned that for a single arm beam actuator (L2=0), no lateral deflection is produced and beam only expands (contracts).
Fig. 4

The variation of non-dimensional deflection vs. beam arms ratio

In the next step of result verification, the actuator tip deflection (point G) is obtained for different ΔT at constant voltage (V=10 V) when L1 and L2 are 1000 μm, 500 μm, respectively. As it is seen from Fig. 5, when a uniform temperature distribution is applied, the deflection variation is a linear function of the applied temperature both in the analytical and FEM results. On the other hand, on view of experimentation, a similar result is reported by Rakotondrabe and Ivan [12].
Fig. 5

The deflection versus applied temperature

Finally, the variation of the actuator tip deflection is plotted versus voltage change at ΔT=100 °C when L1=1000 μm and L2=500 μm, again using the analytical and FEM approaches (see Fig. 6). From this figure it is evident that the variations of both models follow a linear trend as it is reported in experimental work [12]. Furthermore, at V=10 (V) the maximum differences between analytical and FEM results is 8 %.
Fig. 6

The deflection versus applied voltage

It is seen that the electrical field component in z direction (Ez) has no tangible effect in actuator tip deflection.

4.2 Geometry effects

In this section, the variations of geometrical parameters i.e. the length of each arm, gap length and cross sectional dimensions are studied. Figure 7 illustrates the variation of actuator tip deflection versus L2/L1 at ΔT=100 °C, V=10 V, L1=1000 μm. Referred to this figure one can say that maximum deflection can be achieved at the range of 0.4<L2/L1<0.5.
Fig. 7

The variation of tip deflection vs. actuator arms ratio

The deflection of actuator tip can be changed by varying the gap length between the long and short beams. Figure 8 shows the actuator tip deflection versus L2/L1 ratio for different gaps lengths when ΔT=100 °C, V=10 V. From this figure it is evident that the smaller gap length enables the actuator to operate more efficiently.
Fig. 8

The deflection as a function of the actuator arms ratio for two different gaps lengths

As it is expressed by Huang and Lee [13], the obtained results show that the deflection of the actuator is independent of the beam height and the beam width if the dimensions of arms cross section do not change along the arms length. However, it is shown by FEM that reduction of beam height would increase the deflection a little.

4.3 Material property effects

Based on derived formulations, it is clear that the actuator tip deflection will be a function of material properties of d33, d31, d15, ϵ11 and ϵ33. In general, all of these material properties for PZT materials are function of temperature [20]. Based on foregoing discussion on Sect. 4.2, the best choices for the arm lengths will be L1=1000 μm and L2=500 μm.

Figure 9 illustrate the variation of actuator tip deflection as a function of piezoelectric strain constants d33, d31 and d15 when ΔT=100 °C, V=10 V.
Fig. 9

(a) The deflection versus piezoelectric Extensional strain constant d33. (b) The deflection versus piezoelectric Transverse strain constant d31. (c) The deflection versus piezoelectric strain constant d15

As it seen in Figs. 9(a), (b) and (c), a nonlinear variation trend exists when piezoelectric strain constants increase. By increasing the piezoelectric Extensional strain constant d33, deflection decreases although by increasing the absolute value of d31, actuator tip deflection increases in a nonlinear fashion. Also as it is seen there is no tangible variation in deflection when d15 increases. The obtained results show that among PZT strain constants, d33 is the most sensitive strain constant that deflection depends strongly on it.

Finally the deflection of actuator tip as a function of dielectric constants ϵ11 and ϵ33 are plotted in Fig. 10. It is evident from this figure that the actuator tip deflection increases by increasing the value of dielectric constant ϵ33 and does not change remarkably in scale of micrometer when dielectric constant ϵ11 increases.
Fig. 10

(a) The deflection as a function piezoelectric dielectric constant ϵ11. (b) The deflection as a function piezoelectric dielectric constant ϵ33

Based on the presented results on Figs. 9 and 10, it becomes clear that the highly sensitive material properties, will have significant influence on the deformation behavior of structures made of this materials. On the other hand, since in presented model the voltage and PZT polarization directions are in x direction and no voltage is applied in z direction, based on constitutive governing equation we can come to conclusion that in scale of micrometer there is a negligible shear strain in micro actuator and those parameters that cause shear strain (ϵ11, d15 and Ez) have no tangible effect on tip deflection.

5 Conclusion

In current study, a new model for the hybrid thermopiezoelectric actuator through merging piezoelectric and thermal effects is proposed. The suggested model for the actuator possesses the higher range for deflection owning the thermal actuation coupled with higher resolution from the piezoelectric actuation. The most important conclusion arising from this study are as follows:
  1. (a)

    By increasing the applied voltage and temperature gradient, the actuator tip deflection was linearly increased.

  2. (b)

    The actuator tip deflection was decreased upon increasing the gap length.

  3. (c)

    The maximum deflection of HTP micro actuator can be obtained when the arms length ratio is at the range of 0.4<L2/L1<0.5.

  4. (d)

    Increasing the piezoelectric transverse strain constant (d31) and piezoelectric dielectric constant (ϵ33) gave rise to a nonlinear increasing trend in actuator tip deflection.

  5. (e)

    Increasing the piezoelectric extensional strain constant (d33) decreased the actuator tip deflection.

  6. (f)

    Increasing or decreasing the piezoelectric strain constant (d15) and piezoelectric dielectric constant (ϵ11), had no effect in actuator tip deflection.

  7. (g)

    Temperature affected the deflection of hybrid actuator by causing different thermal expansions in arms and increasing the PZT material constants.

  8. (h)

    To achieve more deflection, PZTs with lower extensional strain constant (d33) and higher piezoelectric dielectric constant (ϵ33) were optimized for presented model of HTP micro actuator.


Copyright information

© Springer Science+Business Media Dordrecht 2012

Authors and Affiliations

  • Hasan Pourrostami
    • 1
  • Hassan Zohoor
    • 2
  • Mohamad H. Kargarnovin
    • 3
  1. 1.Department of Mechanical Engineering and Aerospace, Science and Research BranchIslamic Azad UniversityTehranIran
  2. 2.Center of Excellence in Design, Robotics, and AutomationSharif University of TechnologyTehranIran
  3. 3.School of Mechanical EngineeringSharif University of TechnologyTehranIran

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