1. Introduction
The piezoelectric actuator (PEA) is a kind of high-precision actuator that can efficiently convert electric energy into mechanical energy based on the inverse piezoelectric effect [
1]. The piezoelectric actuators contain flextensional-type piezoelectric actuators, tube-type piezoelectric actuators, shear-type piezoelectric actuators and stack-type piezoelectric actuators [
2,
3]. The flextensional-type piezoelectric actuator can produce large output displacement through bending deformation, but its output force is small [
4]. The tube-type piezoelectric actuator can realize nanoscale motion with three degrees of freedom (DOFs), but low resonance frequency limits its application in high-speed scanning fields [
5]. The shear-type piezoelectric actuator has a compact structure, high resonance frequency and is suitable for high-speed application, but its output displacement is limited [
6]. The stack-type piezoelectric actuator is formed by stacking a plurality of piezoelectric pieces, and the output displacement is the sum of all piezoelectric pieces [
2]. With the excellent benefits of nanometer resolution, high resonance frequency, large output force and fast response, stack-type piezoelectric actuators are widely used in precision engineering. However, the output displacement of the stack-type piezoelectric actuator is approximately 0.1~0.2% of its length [
7], which is difficult to meet the requirements of strokes of several hundred microns. Therefore, it is necessary to amplify the output displacement of piezoelectric actuators to meet application requirements.
Compared with rigid-body displacement amplification mechanisms, flexure-based compliant displacement amplification mechanisms transmit motion and force through the deformation of flexible components, having the benefits of an integral structure, variable rigidity, no friction, no wear and no lubrication and assembly [
8,
9]. Due to these advantages, the compliant amplification mechanism has been widely used, including in lithography manufacturing [
10,
11], multistable switches [
12], micro-electro mechanical systems [
13], precision positioning stages [
14,
15], micro/nano operations [
16], automatic dispensing [
17,
18], microvibration suppression [
19], optical alignment [
20] and so on. Flexure-based compliant displacement amplification mechanisms mainly include lever-type, triangle-type and hybrid-type [
14]. The lever-type displacement amplification mechanism has a simple structure and easy displacement amplification ratio (DAR) calculation, but it usually takes a lot of space to achieve large DAR. Triangle-type displacement amplification mechanisms include Bridge-type [
21,
22], Hybats-type [
23], Moonie-type [
24], cymbal-type [
25] and so on; the bridge-type displacement amplification mechanism is the most widely used. The bridge-type displacement amplification mechanism has a compact structure that can realize a large DAR in limited space. However, the analytical modeling of the bridge-type displacement amplification mechanism is complicated due to the intrinsic coupling of kinematic and mechanical behaviors. The hybrid-type displacement amplification mechanism can be regarded as the superposition of lever-type and triangle-type [
26,
27]. Hybrid bridge–lever mechanisms can realize larger DAR. The analytical modeling of bridge–lever displacement amplification mechanisms is more complicated due to the intrinsic coupling of kinematic and mechanical behaviors of the bridge-type mechanism. Therefore, it is significant to establish an analytical model of the bridge-type displacement amplification mechanism for performance evaluation, parameter optimization and future application of a compliant mechanism with the bridge-type mechanism or the hybrid bridge–lever mechanism.
Many researchers have proposed various mathematical models to describe the performance of the bridge-type displacement amplification mechanism. Pokines et al. [
28] derived the ideal DAR of the bridge-type mechanism using the geometric relationship. Lobontiu et al. [
29] established a mathematical model for DAR and stiffness calculations of the bridge-type mechanism based on Castigliano’s second theorem. Kim et al. [
30] proposed a matrix model of the 3D bridge-type amplifier regarding flexure hinges. Ma et al. [
31] derived the ideal DAR model of the bridge-type mechanism using kinematic theory, and then the theoretical model of DAR considering the elastic deformation of flexible hinges was established using the elastic beam theory. Xu et al. [
32] established the analytical model for DAR, input stiffness and resonance frequency predictions of a compound bridge-type (CBT) displacement amplifier. Qi et al. [
22] developed a theoretic model for DAR of the bridge-type mechanism using the elastic beam theory; then, the relationship between DAR and geometric dimensions was deeply analyzed. Based on the energy conservation law and the elastic beam theory, Ling et al. [
33] proposed an enhanced mathematical model for the bridge-type compliant mechanism considering the translational and rotational stiffness. Choi et al. [
34] established a new mathematical model for DAR of the bridge-type amplification mechanism considering the deformation of all members of the amplification mechanism. The above-mentioned analytical models only paid attention to the characteristics of the bridge-type displacement amplification mechanisms as independent components. However, the bridge-type displacement amplification mechanisms were usually connected to the displacement guiding mechanisms, which can be regarded as an external load of variable stiffness. The external load had a great influence on DAR, so it is highly significant to incorporate the effect of external load into analytical models.
To improve modeling accuracy, numerous analytical models considering the effect of external load have been proposed. Liu et al. [
35] used the pseudo-rigid body modeling approach to develop a novel analytical model for the bridge-type amplifier, which assumed that the elastic deformations only occur at the flexure hinges. Liu et al. [
21] and Pan et al. [
36] proposed nonlinear models for the bridge-type displacement amplification mechanism based on the Timoshenko Beam Constraint Model (TBCM). Li et al. [
37] established an improved DAR model for the bridge-type mechanism considering the input displacement loss. Lin et al. [
38] derived a new analytical model for the DAR of the bridge-type amplifier employing the energy conservation law and the elastic beam theory. The four above-mentioned analytical models assumed that only the flexible hinges and connecting bodies are elastically deformed. Zhang et al. [
39] proposed a novel theoretical model for DAR of the bridge-type amplification mechanism considering the deformation of all members of the mechanism. Although these advanced models markedly improve calculation accuracy, the deformation of all members of the mechanism, the effect of external load and the nonlinear shear effect need to be considered for special bridge-type displacement amplification mechanisms, such as large DAR mechanism, large driving force mechanism and large external load mechanism.
A structure load performance integrated model approach for the bridge-type displacement amplification mechanism is presented here, taking the deformations of all members of the mechanism, the effect of external load and the nonlinear shear effect into consideration. The analytical model is established based on Castigliano’s second theorem, which can precisely calculate DAR, input displacement, output displacement, input stiffness and output stiffness. Model verification and analysis reveal the relationships of DAR with driving force, external load and structural parameters, and the sensitivity of DAR to structural parameters. The vertical micro/nano-positioning mechanism with the bridge-type displacement amplification mechanism is modeled and analyzed to validate the effectiveness of the proposed model.
2. Analytical Modeling
The bridge-type displacement amplification mechanism is an integral symmetrical structure consisting of a fixed body, input bodies, connecting bodies, an output body and flexible hinges. The bridge-type displacement amplification mechanism is shown in
Figure 1. A piezoelectric actuator is installed in the bridge-type displacement amplification mechanism, and it is connected with input bodies by a certain preload. The output body is usually connected with the displacement guiding mechanism, which is equivalent to an external load for the bridge-type displacement amplification mechanism. The driving voltage is applied to the piezoelectric actuator, and the displacement and driving force generated by the piezoelectric actuator drive input bodies. The bridge-type displacement amplification mechanism elastically deforms transmitting motion and force to the output body. All parts of the bridge-type displacement amplification mechanism are compliant.
Due to the symmetrical structure of the bridge-type displacement amplification mechanism, only the 1/4 integral mechanism is used to establish analytical modeling. The mechanical analysis of the ¼ mechanism is shown in
Figure 2, where
,
,
and
represent the lengths of input body, connecting body, output body and flexible hinge, respectively;
,
,
and
represent the thicknesses of input body, connecting body, output body and flexible hinge, respectively;
represents the interval of adjacent flexible hinges; and
represents the width of the mechanism. To simplify the derivation, it is assumed that
and
represent the equivalent force applied on input and output bodies, respectively;
represents the equivalent bending moment at the end of the output body; and
represents the equivalent bending moment at the driving position. Based on force and moment balance, the following equations can be obtained:
where
and
represent the equivalent force applied on every member of the 1/4 mechanism in the
X-axis direction and the
Y-axis direction, respectively;
represents the driving force of the piezoelectric actuator; and
represents the external load applied to the output body. It can be obtained from Equations (3) and (4):
According to the mechanical balance, the inner forces of any sections for the
ith (
i = 1, 2, 3, 4, 5) flexure member of the 1/4 mechanism can be obtained; the inner forces include the inner tensile force
, shear force
and bending moment
.
The total strain energy of the 1/4 mechanism can be obtained as follows:
where
represents the cross-sectional areas of
ith member, which satisfies the following relationships:
,
,
,
,
.
represents the inertia moments of
ith member, which satisfies the following relationships:
,
,
,
,
.
represents the Young’s modulus,
represents the shear modulus and
represents the dimensionless factor related to shape of the cross-section, which is usually 6/5 for a rectangular cross-section. Based on Castiglioni’s second theorem [
9], the input displacement and the output displacement of the 1/4 mechanism can be obtained as follows:
Substituting Equations (7)–(12) into Equation (13), the relationship between displacement and force of the 1/4 mechanism can be derived:
where
is the compliance matrix of the 1/4 mechanism considering the deformation of all members, the effect of external load and the nonlinear shear effect. Every element of the compliance matrix can be expressed as follows:
According to Equation (14), the total input displacement and the total output displacement of the bridge-type displacement amplification mechanism are detailed as:
Based on Equations (16) and (17), the DAR of the bridge-type displacement amplification mechanism is expressed:
Under no external load or small external load, DAR and input stiffness of the bridge-type displacement amplification mechanism are calculated by Equations (19) and (20):
When the input force is set to zero, the output stiffness of the bridge-type displacement amplification mechanism is obtained as follows:
Once structural parameters and driving force are determined, DAR decreases as external load increases. When external load increases to a certain value, the displacement amplification ability of the mechanism decreases. The ultimate load of the mechanism is obtained with the assumption of
.
Considering piezo actuator driver properties, the driving force applied to the bridge-type displacement amplification mechanism can be expressed as follows:
where
is the driving voltage,
is the piezoelectric constant and
is the stiffness of the piezoelectric actuator. Substituting Equation (23) into Equation (16), the input displacement and output displacement of the bridge-type displacement amplification mechanism considering piezoelectric actuator driver properties are derived:
Substituting Equations (23) and (24) into Equation (17), the output displacement and output displacement of the bridge-type displacement amplification mechanism considering piezoelectric actuator driver properties are derived:
According to Equations (24) and (25), the DAR of the bridge-type displacement amplification mechanism considering piezoelectric actuator driver properties is expressed:
5. Conclusions
In this paper, we propose a structure load performance integrated model approach for the bridge-type displacement amplification mechanism, which considers the deformations of all members, the effect of external load and the nonlinear shear effect. The mathematical modeling was established based on Castigliano’s second theorem, which can precisely calculate DAR, input displacement, output displacement, input stiffness and output stiffness. The proposed model was verified with the existing analytical models and FEM, and the comparison results indicate that the established model is closest to the FEM result over the existing models. With the driving force and the external load change, the maximum differences of DAR with the FEM are 1.26% and 9.93%, respectively, proving the accuracy of the proposed model. Moreover, when the interval of the adjacent flexible hinges changes, the difference between the proposed model and FEM at the peak DAR is 1.18%, which is much lower than the existing models. The variance-based global sensitivity of structure parameters to DAR was thoroughly analyzed. The sensitivity analysis results show significant sensitivity of DAR to the changes in the interval of adjacent flexible hinges, the thickness of flexure hinges, the length of flexure hinges and the length of connecting bodies, and weak sensitivity of other structure parameters to DAR. Finally, the proposed model was applied to the modeling and analysis of the vertical micro/nano-positioning mechanism with the bridge-type displacement amplification mechanism, validating the effectiveness and expansibility of the proposed modeling method.