1. Introduction
Piezoelectric actuators (PZTs) are widely used in the field of micropositioning/nanopositioning, which has the advantages of ultra-high resolution, high response speed, high stiffness, and high thrust [
1,
2,
3]. However, the relative stroke displacement generated by PZTs is only about 10
, which severely limits their application range [
4,
5]. In view of this problem, displacement amplification mechanisms are often used to amplify the travel range of PZTs [
6]. Due to the advantages of no friction, no backlash, no wear, and a compact structure, the flexure mechanism is usually used for the displacement amplification mechanism. In various amplification mechanisms, the lever-type flexure mechanisms are commonly used for the displacement amplification, but they usually have larger dimensions, low amplification ratios, and low resonant frequencies [
7]. Different from the traditional lever-type mechanisms, the bridge-type mechanisms have the advantages of high amplification ratios, high resonance frequencies, and a small size, which have been used widely in modern industry [
8,
9,
10].
The structure configurations and modeling methods of bridge-type flexure mechanisms have been widely studied. Lobontiu et al. used the Castigliano second theorem to obtain an analytical model for the displacement amplification ratio and input/output stiffness calculations for the bridge-type mechanism [
11]. Ma et al. derived the theoretic displacement amplification ratio of the bridge-type mechanism by using the elastic beam theory and work–energy theorem [
12]. Xu et al. formulated an analytical model for the amplification ratio, input stiffness, and natural frequency calculations of a compound bridge-type amplifier based on the Euler–Bernoulli beam bending theory [
13]. Liang et al. used two compound bridge-type mechanisms to design a novel monolithic two-degree-of-freedom (DOF) rotation decoupled platform [
14]. Chen et al. designed and analyzed a three-dimensional bridge-type mechanism based on the stiffness distribution and the screw theory [
15]. Some other bridge-type mechanisms and analytical modeling methods have also been reported [
9,
16,
17,
18].
Most of the bridge-type amplification mechanisms proposed in the previous studies used the notch flexure hinges for lumped-compliance. However, the large mass of rigid bodies in the traditional bridge-type mechanism with lumped-compliance has a serious impact on its dynamic performance. Instead of the notch flexure hinges used in the lumped-compliance mechanism, the distributed-compliance mechanism uses beam flexures to significantly reduce the mass of the mechanism, which can increase the natural frequency of the mechanism effectively [
19,
20]. In addition, different from the flexure hinge, the stress generated by the deformation of the mechanism is evenly distributed on the flexure beams; therefore, the distributed-compliance mechanism has better reliability and dynamic characteristics than the traditional lumped-compliance mechanism, especially in high-speed applications. Some amplified PZTs with the distributed-compliance mechanism have been commercialized by a number of companies including Cedrat Technologies, Core Tomorrow, etc. Even so, the static and dynamic characteristics of the bridge-type mechanism with distributed-compliance have still not been analytically modeled. Finite element analysis (FEA) is the conventional method adopted to design the mechanism’s structure, and the analytical model, especially the dynamic model, has not been established in the literature. Currently, the topology-optimization-based methods for the designing of distributed compliant mechanisms are also very popular and have attracted more and more attention, because they can solve the optimal distribution of structural materials [
21,
22]. Therefore, in the design process, a simple and accurate analytical model is required, which can predict the performances of the flexure mechanism and determine its structural parameters according to the design specifications.
In addition, for the dynamic analysis of the flexure mechanism, the existing modeling methods are basically based on the Lagrange method, that is the kinetic energy and potential energy of the system need to be calculated, respectively. Ling et al. also proposed an extended dynamic stiffness modeling method to analyze the kinetostatic and dynamic characteristics of lumped-compliance flexure mechanisms based on d’Alembert’s principle [
23]. For the modeling of the lumped-compliance mechanism, the mass or the kinetic energy of the flexure hinges having a light weight and volume is usually reasonably ignored to simplify the dynamic model. Different from the lumped-compliance mechanism, in the distributed-compliance mechanism, when the beam flexure has a large thickness, its mass and kinetic energy cannot be reasonably ignored; otherwise, the modeling accuracy will be seriously reduced, which is also the difficulty in the dynamic modeling of distributed-compliance mechanisms.
For the above problems, the static and dynamic characteristics of the bridge-type displacement amplification mechanism with distributed-compliance are deeply analyzed in this paper. By using the stiffness matrix displacement method, an analytical model for the calculations of the displacement amplification ratio and input stiffness of the distributed-compliance bridge-type mechanism is established. More importantly, in order to accurately obtain the dynamic performance of the mechanism, the velocity of any point on the vibrating beam flexure is calculated to obtain the expression of the kinetic energy by solving the derivatives of the deformation curves of the beams versus time, and the natural frequency in the working direction can be obtained using the Lagrange method, which is validated by FEA and experimental testing results. The comparisons among the analytical model, FEA results, and testing results demonstrate the high accuracy of the proposed analytical model.
The main contribution of this paper is to establish an analytical model for calculating the statics and dynamic characteristics for a class of distributed-compliance bridge-type amplification mechanisms. The static model is firstly established by using the matrix displacement method, while an improved dynamic model is formulated by considering the kinetic energy of the flexure beams in the mechanism. The modeling method and corresponding theoretical formulas of the displacement amplification ratio, input stiffness, and natural frequency proposed in this paper can accurately reveal the static and dynamic characteristics of the distributed-compliance bridge-type amplification mechanism, which provides a useful and accurate reference for the optimal designing and manufacturing of such kinds of bridge-type mechanisms.
3. Improved Dynamic Model
The dynamic characteristic is an important specification in the design of compliant mechanisms. The high natural frequency of the compliant mechanism can effectively suppress external interference. Different from the lumped-compliance bridge-type mechanism, the beam flexures are used to replace the flexure hinges in the distributed-compliance mechanism. Therefore, if the mass of the beam flexures reaches a certain degree, their mass and vibration kinetic energy cannot be ignored in the process of dynamic modeling.
In order to establish the differential equation of vibration by using the Lagrange equation, the kinetic energy and potential energy of the system should be calculated first. According to the selected generalized coordinates, the kinetic energy of the rigid body can be easily obtained; therefore, this section mainly discusses the kinetic energy calculation of beam flexures. For the convenience of calculation, the beam flexure with a constant cross-section is taken as the analysis object, and its deformation is shown in
Figure 4. It is assumed that the deflection curve
of the beam flexures is the cubic equation of the abscissa in its local coordinate system:
where
and
are undetermined coefficients of the deflection curve. According to the deflections and the angles at both ends of the beam flexure,
and
, which are expressed by the generalized coordinates of the mechanism, the coefficients of the deflection curve can be obtained as
By deriving the deflection equation from time, the velocity of each point on the beam flexure can be obtained as
. Therefore, after integrating the kinetic energy of each point on the beam, the kinetic energy generated by movement in the deflection direction of the beam flexure can be calculated as
In addition to the deflection direction, each point on the beam also moves along the axial direction. As shown in
Figure 4, according to the selected generalized coordinates, the axial displacements at both ends of the beam flexure are
and
. Assuming that the axial displacement of each point after the deformation of the beam is linearly distributed along the axis, the axial displacement of each point can be calculated as
Therefore, the kinetic energy generated by the axial motion of the beam flexure can be expressed as
Considering the kinetic energy generated by the rotation of the beam flexure, as shown in
Figure 4, the rotation angle
and moment of inertia
of each infinitesimal point on the beam flexure can be calculated as
Therefore, the rotational kinetic energy generated by the rotation of each point on the beam can be calculated as
Therefore, the total kinetic energy generated by the deformation movement of one beam flexure is
Plus the kinetic energy of the rigid bodies, the total kinetic energy of the mechanism is
where
is the kinetic energy of the
ith rigid body,
represents the kinetic energy of the
jth beam flexure, and
and
are, respectively, the number of rigid bodies and beam flexures in the mechanism. Both of the two kinds of kinetic energy are represented by the selected generalized coordinates in the mechanism.
According to the deformation
caused by axial tension or compression and section moment
of the beam flexures, the potential energy of one beam flexure and the whole compliant mechanism can be obtained as
The expressions of kinetic energy and potential energy are substituted into the Lagrange equation:
where
and
are the vectors of generalized coordinates and generalized forces, respectively. By simplifying Equation (
24), the vibration differential equation of the system can be obtained as
where
and
are the equivalent mass and stiffness matrices of the improved dynamic model, respectively. When the generalized force vector
, the mechanism vibrates freely. According to the improved dynamic model of Equation (
25), the improved natural frequency can also be obtained by using Equations (
12) and (
13).
5. Conclusions
In this paper, the static and dynamic performances of a kind of bridge-type flexure mechanism with distributed-compliance were calculated by using the matrix displacement model and the improved dynamic model. Due to distributed stresses and low mass, this kind of bridge-type distributed-compliance displacement amplification mechanism has much better reliability and dynamic characteristics than the traditional amplification mechanism based on a lumped flexure hinge, especially in high-speed applications. The matrix displacement method was first deduced for the displacement amplification ratio and input stiffness calculations of the bridge-type mechanism. In the modeling method, the stiffness matrix for a free–free flexure beam element was first obtained from its theoretical compliance matrix based on the Timoshenko beam theory. According to the elemental stiffness matrix, the updated stiffness matrix of two rigid bodies connected by the beam flexures can be deduced. The mass centers of two rigid bodies and the forces applied to them are regarded as the node displacement and node forces. By expanding and superimposing each updated elemental stiffness matrix, the global stiffness matrix for the distributed-compliance mechanism can be obtained.
Furthermore, in order to improve the accuracy of the natural frequency calculation by the matrix displacement method, an improved dynamic model was constructed by examining the vibration kinetic energy of beam flexures in the bridge-type distributed-compliance mechanism. Firstly, three generalized coordinates of the distributed-compliance bridge-type mechanism were selected, and the deflection curve, axial displacement, and rotation angle of the beam flexures were then represented by the generalized coordinates. Then, by deriving the above deformation curves from time, the deflectional, axial, and rotational velocities of each point on the beam flexure can be obtained to calculate the kinetic energy and potential energy in the vibrating beams. Finally, by using the Lagrange method, a three-DOF dynamic equation for the bridge-type mechanism was established to calculate the natural frequency in the working direction.
To verify the accuracy of the analytical model, the FEA and experimental tests for two kinds of distributed-compliance bridge-type mechanism were carried out. The comparisons results showed that the analytical, FEA, and experimental results had high agreement with each other, and the maximum error was less than 8%. It was also shown that, compared with the matrix displacement model, the improved dynamic model can greatly improve the prediction accuracy of the natural frequency, from to , which has a reference value for the design of the distributed-compliance flexure mechanism and amplified piezoelectric actuator.