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Piezo-electric nano-positioning stages are being widely used in applications in which precision and accuracy in the order of nano, and high scanning speeds are paramount. This paper presents a Finite Element Analysis (FEA) of the parallel piezo-flexural nano-positioning (PPNP) stages to investigate motion interference between their different axes. Cross-coupling is one of the significant contributors to undesirable runouts in the precision positioning of PPNP actuators. Using ABAQUS/CAE 2018 software, a 3D model of a PPNP stage was developed. The model consists of a central elastic body connected to a fixed frame through four flexural hinges. A cylindrical stack of multiple piezoelectric disks is placed between the moving central body and the fixed frame. Extensive simulations were carried out for three different friction coefficients in the piezoelectric disks’ contact surfaces, different frame materials, and different geometrical configurations of the stage and the hinges. As a result, it was observed that the primary root cause of the mechanical cross-coupling effect could be realized in the combination of the slip and rotation of the piezoelectric disks due to their frictional behavior with the stage moving in the tangential direction, concurrent with changes in the geometry of the stage.

Parallel piezo-flexural nano-positioning (PPNP) stages are used in a variety of advanced technologies such as precision metrology, scanning probe microscopy [

The nano-positioning systems’ literature embodies a number of finite element analysis (FEA) efforts to understand the mechanics of piezo-flexural actuators. Elmustafa and Lagally [

The literature lacks investigating the effects of Friction between the piezoelectric disks and the stacks and the frame in piezo-flexural systems. One of this study’s main motivations is to investigate the effects of frictional behavior on the mechanical cross-coupling effect in PPNP stages. A comprehensive model of a PPNP stage was simulated in ABAQUS finite element software, containing the model’s central elastic body, which is movable in the center, connected to a fixed frame through four flexural hinges. In this model, seven separate cylindrical piezoelectric disks, one upon the other, are placed between the moving central body and the fixed frame. Simulations for various friction coefficients in the contact surfaces of the piezoelectric disks, different frame materials, and different perturbations in the stage’s geometry and the hinges were carried out.

The remainder of the paper is organized as follows: In Section 2, the experimental observation of the cross-coupling effect in the PPNP stage from Bashash et al. [

According to Bashash et al., the coupling of perpendicular axes in the PPNP stages imposes a significant limitation on their positioning accuracy [

An earlier study has shown by moving the stage in one direction; the corresponding position sensor picks a slight displacement in the other direction [

In this paper, ABAQUS FEA software is used to carry out the process of mechanical cross-coupling simulation. In the simulation, the loading steps are modeled based on the closed-curve of the experimental data. This curve is mainly due to hysteresis and is modeled virtually in this study.

Due to the piezoelectric actuators’ asymmetrical arrangements, the simulation is applied in a 3D geometry instead of the 2D geometry.

Poisson’s ratio | Young’s modulus (GPa) | Material type | Component |
---|---|---|---|

0.33 | 68.9 | Aluminium | Frame |

0.34 | 36 | PICMA^{®} P-885 Ceramic | Piezoelectric disk |

The surface contact between different piezoelectric elements is modeled through Coulomb’s law of Friction. This model can prevent the piezoelectric surfaces from unwanted penetration by selecting the hard contact penetration option in ABAQUS. These surface contacts include contacts of the piezoelectric elements with each other, the contact of piezoelectric disk #1 with the moving part of the frame, and the contact of the piezoelectric disk #7 with the fixed part of the frame. Friction coefficients for the three different cases used in this study are tabulated in

The first, second, and third cases are related to the different friction coefficients between the piezoelectric disks, whereas the friction coefficients between the top and bottom piezoelectric disks and the frame are kept the same for all cases.

Finite element analysis can be used to investigate the effects of different materials for the frame and the piezoelectric ceramics on the cross-coupling effect. The effects of geometrical perturbations, such as changes in the square hole’s dimensions inside the frame and the flexural hinges, on the cross-coupling effect, could be investigated.

Therefore, the following cases to this investigation are added:

Fourth case: Changing the material of the frame from Aluminum to Steel with Young’s modulus of 210 GPa and Poisson’s ratio of 0.3. In this case, the frictional conditions are similar to the first case.

Between piezo disk #1 and moving part of the frame | Between piezo disk #7 and the fixed frame | Between piezo disks | Friction case |
---|---|---|---|

0.7 | 0.7 | 0.1 | First |

0.7 | 0.7 | 0.2 | Second |

0.7 | 0.7 | 0.3 | Third |

Fifth case: Changing the piezoelectric ceramics material with Young’s modulus of 11 GPa and Poisson’s ratio of 0.34. In this case, the frictional conditions are similar to the first case.

Sixth case: Changing the geometry of the frame by changing the dimensions of the inner square hole. These changes include changing the length from 35 mm to 26.25 mm and the radius corner from 4 mm to 3 mm, as shown in

Seventh case: Changing the frame’s geometry by decreasing the thickness of one of the hinges in the y-direction from 1.5 mm to 1.25 mm, as shown in

In all cases, all the six degrees of freedom of the frame edges are constrained. Actuation by piezoelectric stacks for the moving part of the frame is modeled with a left-side pressure and pre-pressures in the x and y directions, respectively. These pressures are applied in lieu of the voltage applied to the piezoelectric actuators (see

This simulation includes the following four important steps:

1) Pre-pressure loading step;

2) Left side pressure loading step;

3) Pre-pressure unloading step;

4) Left side pressure unloading step.

In other words, the loading process consists of two primary steps, and the third and fourth steps are the unloading process of the model. At the first step, only a pre-pressure with a magnitude of 25 KPa is applied. This pressure is applied smoothly and statically within 0.05 seconds. At the second step, another pressure is applied to the moving frame’s left side without changing the pre-pressure applied in the first step. This pressure is equal to 148 KPa, which is applied smoothly and statically within 1 second. So, the loading process takes 1.05 seconds to be completed. In the third step, the pre-pressure is unloaded within 0.05 seconds while no changes are applied to the left side pressure. In the fourth step, the left side pressure is unloaded at 1 second. So, the unloading process, which is completed in the third and fourth steps, takes 1.05 seconds, the same as the loading process, resulting in a total simulation time of 2.1 seconds. Simulation parameters are tabulated in

The type of element used for meshing is C3D8R, an eight-node linear brick element with 24 DOFs. This type of element is capable of traversing shear deformation. The meshing of different parts of the frame, hinges, and the piezoelectric stack is shown in

The simulation results for the seven cases mentioned in the previous section are discussed here. The simulation results of the first case for the displacement and stress distributions in the stage at the end of loading are shown in

Step number | Step | Step time | Pre-pressure magnitude (KPa) | Left side pressure magnitude (KPa) |
---|---|---|---|---|

1 | Pre-pressure loading | 0.05 | 25 | 0 |

2 | Left side pressure loading | 1 | 25 | 148 |

3 | Pre-pressure unloading | 0.05 | 0 | 148 |

4 | Left side pressure unloading | 1 | 0 | 0 |

The stress analysis result shows that the flexural hinges bear the highest stress levels, especially at the connection points to the fixed part of the frame. The results of the first case for the mechanical cross-coupling effect are shown in

Note that the mismatch between the two diagrams in

The convergence of simulation is a key factor in ensuring the correctness of numerical simulation. To end this, the simulations with different mesh sizes (coarse and fine mesh size) are run and the results are shown in

In order to evaluate the effect of different friction coefficients, i.e., 0.1, 0.2, and 0.3, between the piezoelectric disks, the numerical simulations are re-run and the results are plotted in

In order to understand the effects of the frame’s material stiffness, the FEA simulations for two different frames Young’s modulus values are run and the results are plotted in

As can be seen from

K s t a g e ∝ K h i n g e ∝ E d 3 (1)

where E is Young’s modulus and d is the thickness of the hinge in the XY plane.

Therefore, due to an identical increase of the stage stiffness in both directions, the cross-coupling effect does not undergo any changes. This result is in good agreement with the experimental results.

To evaluate the effects of the piezoelectric material’s stiffness on the mechanical cross-coupling effect, the FEA simulation for two different piezoelectric Young’s modulus values are rerun and the results are plotted in

It can be seen from

In order to investigate the impacts of changing the stage’s dimension on the mechanical cross-coupling effect, the FEA simulation for two cases with different frame and hinge geometries (sixth and seventh case) are re-run and the results are plotted in

It can be seen from

It can be concluded from

By analyzing

In this study, FEA simulations are carried out to investigate the effects of different mechanical properties, frictional behavior, and geometry of parallel piezo-flexural nanopositioning (PPNP) stages on the mechanical cross-coupling effect between their different axes. In all simulation and experimental trials, the effects of inertia and damping at low frequencies were ignored. The stress distribution in the stage shows that the flexural hinges have the highest stress level, especially at the connection points to the fixed part of the frame. Also, surveying the influence of friction coefficient between piezoelectric disks shows that the piezoelectric disks slip on each other at low friction coefficients, resulting in a linear cross-coupling behavior. Increasing the friction coefficient leads to a sudden change in the displacement of the stage in the y-direction when the excitation forces are present in the x-direction only, and the piezoelectric disks undergo a stick-slip frictional behavior. This sticking leads to some degree of rotation in the piezoelectric disks, resulting in a nonlinear cross-coupling behavior. Another observation shows that changing the frame’s material causes the stage’s stiffness to be altered in both directions. Hence, changing the material of the frame has no impact on the cross-coupling effect.

Moreover, changing the piezoelectric disks’ material does not cause the cross-coupling effect to be changed, as it does not affect the stage’s stiffness. Different geometries of the stage are also considered by changing both the dimensions of the inner hole of the stage and the hinges’ thickness. The result is that changing the dimensions of the inner hole of the stage only causes the stiffness to be changed in the y-direction, and consequently, the cross-coupling effect is changed. FEA simulations also show that the proportion of stiffness between two axes is changed by choosing a different thickness for the hinge. This consequently causes the mechanical cross-coupling effect to be changed. Finally, the main parameters that affect the mechanical cross-coupling are the combination of slip and rotation of the piezoelectric disks concurrent with a change in the stage’s geometry. The results obtained in this paper can provide a better insight into the mechanics of the PPNP stages for potential improvement of the cross-coupling effect in the product design and development phase.

The first author would like to sincerely thank Dr. Peiman Mosaddegh, Dr. Saeed Bashash, and Dr. Nader Jalili. This work would not have been possible without their scientific supports.

The authors declare no conflicts of interest regarding the publication of this paper.

Shafiee, A., Ahmadian, A. and Akbari, A. (2021) A Parametric Study of Mechanical Cross-Coupling in Parallel-Kinematics Piezo-Flexural Nano-Positioning Systems. Open Journal of Applied Sciences, 11, 596-613. https://doi.org/10.4236/ojapps.2021.115043