Abstract
In order to study the impact of periodic loads on the vibration performance of the flexible positioning platform in the key mechanisms of the ultra-high acceleration macro–micro motion platform, and to improve the performance and stability of the platform, this paper combined SolidWorks and ANSYS Workbench to conduct modal analysis and harmonic response of the flexible positioning platform. Analysis to obtain its inherent characteristics. The piezoelectric actuator provides the driving force of the micro-motion platform and achieves precise micro-motion displacement positioning, which is used to study the frequency response results of simple harmonic excitation. By analyzing the node displacement frequency response, the dynamic stiffness response characteristics of the hinge and micro-motion platform of the flexible positioning platform under actual loads were determined, and potential dangerous areas were identified. Finally, both response surface optimization and direct optimization methods were used to optimize the design of the hazard zones for the flexible positioning platform, and the results of both optimization models were validated. The findings indicate that the direct optimization results validate the accuracy of the response surface optimization. After response surface optimization, the first-order natural frequency of the flexible positioning platform increased by 3.28%, and the mass decreased by 1.84%. The maximum deformation and response peak of the first-order vibration mode decrease, and the dynamic stiffness increases. This research provides valuable reference for the structural design and vibration performance optimization of positioning platforms containing flexible hinges.
Keywords: Ultra-high acceleration macro–micro motion platform, Flexible positioning platform, Vibration performance, Harmonic response analysis, Optimization research
Subject terms: Mechanical engineering, Acoustics
Introduction
In the field of microelectronics, with the development of science and technology and the growth of demand, the market size presents a huge potential, and is expected to continue to grow rapidly in the next few years. As an important part of modern industry and scientific research, microelectronic manufacturing plays a vital role in promoting industrial competitiveness and scientific and technological progress. Micro-electronic technology has become an important foundation of modern industrial production and scientific research, and ultra-high acceleration macro–micro motion platform is a precision positioning platform, which can achieve high precision positioning of micron or even nanometer level, it can be used to manufacture micro-and nano-fabrication fields such as micro-electronic components, microelectromechanical systems, optoelectronic devices and so on, providing the key technical support for the development of the micro-electronics industry. The concept of macro–micro composite driving has been widely applied in various motion platforms. A. Sharon was among the first to introduce this concept, incorporating micro-motion positioning mechanisms into a positioning system for error compensation, thereby enhancing overall system accuracy1. Researchers at the Tokyo Institute of Technology developed a positioning system based on a ball screw and brushed DC motor, utilizing a gas hydrostatic workbench. This platform achieves a maximum travel of 400 mm and can maintain a positioning accuracy of 4 nm under certain conditions2. A motion platform developed by the National University of Singapore is driven by a linear motor, with both the macro and micro positioning stages controlled using a feedback structure. This system offers a stroke length of 5 mm and a positioning accuracy of no less than 40 nm3. Mori et al. developed a one-dimensional motion platform based on an air-buoyed support, driven by a voice coil motor (VCM), achieving an impressive positioning accuracy of 1 nm4. Gao et al. utilized four gas hydrostatic bearings to support the three-degree-of-freedom movement of a platform driven by a brushless Lorentz motor. This setup allows for a stroke of 45 mm and positioning accuracy of 20 nm5. Based on a suspension platform, Kim et al. employed a linear motor to generate both horizontal and vertical levitation forces. Their platform offers 50 mm of travel in the x and y directions, with a positioning accuracy of 5 nm and an acceleration of 1 g6. Sun Xiaoqing of the Donghua University has proposed a new parallel micro-nano positioning vibration isolation level (PVI level), which can achieve precision positioning and vibration isolation through the combined design of giant magnetostrictive actuator (GMA) and compliant amplifier7. Yang Zhixin of the Jilin University has examined the problem of backward motion, proposed an idea of forward friction to balance the reverse friction, and designed an arc-shaped flexure hinge that can recover forward friction through elasticity8. To test his idea, a positioning platform with variable initial gap between contact surfaces is designed9. In this trend, multi-physics coupling design has become a key direction for enhancing high-performance design. Under complex operating conditions, accurately developing optimization models and improving platform performance through multi-objective optimization is an effective solution10. Because of the high threshold of IC manufacturing technology, more and more researchers enter the field of precision positioning platform to improve the manufacturing technology and industry level of microelectronic manufacturing and related fields, it is of great significance to promote scientific and technological innovation and economic development11.
This paper studies the key component of the ultra-high acceleration macro–micro motion platform, the flexible positioning platform. The hinge stiffness calculation method is applied to assess its overall stiffness, and the accuracy of this method is validated through comparison with simulation results, providing a reliable foundation for dynamic stiffness research. Modal analysis and harmonic response analysis are conducted using SolidWorks and ANSYS Workbench, from which key optimization parameters are selected, and the optimal solution is found using the ANSYS Workbench optimization technique12,13. A dual-validation strategy combining response surface methodology and direct optimization is employed, and sensitivity analysis explores the relationship between optimization parameters and objectives. The aim is to improve the platform’s first-order natural frequency, reduce first-order deformation and response peaks, enhance dynamic stiffness, and ultimately achieve smooth optimization, thereby improving platform performance and ensuring design reliability. Additionally, an innovative method combining acoustic analysis and the Perfect Noise system to measure the natural frequency is proposed. This method is cost-effective, easy to operate, and novel in its experimental approach. By comparing experimental results with simulation outcomes, the accuracy of this method is validated, further confirming the reliability of the simulation analysis.
Finite element harmonic response analysis
Ultra-high acceleration macro–micro motion platform
The ultra-high acceleration macro-fretting platform is composed of many key mechanisms, such as flexible positioning platform, voice coil motor, connecting arm, piezoelectric actuator, etc. the middle region of the flexible positioning platform is composed of a hinged micro-motion platform. The flexible positioning platform is the key to realize macro–micro-ultra-precision positioning of the ultra-high acceleration macro–micro motion platform. The ultra-high acceleration macro–micro motion platform is shown in Fig. 1. The ultra-high acceleration macro–micro motion platform realizes motion positioning and micro-motion positioning on the flexible positioning platform, so when ultra-high speed macro–micro motion platform realizes ultra-precision positioning, the flexible positioning platform shown in Fig. 2 plays a key role. Table 1 is material performance, Table 2 is dimension parameter of flexible positioning platform.
Fig. 1.
Ultra-high acceleration macro–micro motion platform. Software used for analysis: Visio 2019 (https://www.microsoft.com/en-us/microsoft-365/visio).
Fig. 2.
The 3D model of the flexible positioning platform and the key dimen- sions of the model. Software used for analysis: Visio 2019 (https://www.microsoft.com/en-us/microsoft-365/visio).
Table 1.
Material properties of the flexible positioning platform.
| Name | Numerical value | Unit |
|---|---|---|
| Young’s modulus | 7.1016 × 1010 | Pa |
| Poisson’s ratio | 0.33 | |
| Density | 2793 | Kg/m3 |
Table 2.
Key dimension values of flexible positioning platform.
| Name | L1 | L2 | L3 | L4 | L5 | L6 |
|---|---|---|---|---|---|---|
| Value (mm) | 287 | 159 | 126 | 29 | 59 | 33 |
| Name | L7 | L8 | L9 | L10 | L11 | |
|---|---|---|---|---|---|---|
| Value (mm) | 80 | 29 | 18 | 30 | 18 |
The working process of the ultra-high acceleration macro–micro motion platform can be divided into several working conditions. The first is that the ultra-high acceleration macro–micro motion platform is in the beginning state, and is driven by voice coil motor and piezoelectric actuator. The second working condition is that two voice coil motors are driven forward at the same time. Working Condition Three is two voice coil motors stop driving, piezoelectric actuator forward driving. Working Condition Four is piezoelectric actuator stop driving, two voice coil motors reverse driving. Working Condition 5 is two voice coil motors stop driving, piezoelectric actuator reverse driving. In the third case, the boundary condition is obvious, and the load effect on the flexible positioning platform is great. The part of this paper mainly discusses the characteristic change of the flexible positioning platform in the third case. In the third working condition, the flexible positioning platform is only affected by the positive force of the piezoelectric actuator, and the boundary condition is that the left and right sides of the flexible positioning platform are all constrained, the piezoelectric actuator provides a driving force on the fretting platform of the flexible positioning platform14.
Finite element modal analysis
The accuracy of modal analysis directly affects the results of harmonic response analysis15,16. The mode analysis is to obtain the natural frequency and mode shape of the flexible positioning platform. By obtaining the natural frequency, we can determine whether the object will have resonance with the vibration sources around it, to avoid damage caused by resonance. There are two broad methods of modal analysis, one is the finite element method and the other is the experimental method. In this paper, the finite element method and the experimental method are used to analyze the modal of the flexible positioning platform, and the accuracy of the modal analysis is verified17.
In this paper, ANSYS Workbench is used to carry out modal analysis, in order to more comprehensive analysis of the flexible positioning platform’s own structural characteristics and actual working conditions, from the Free Mode and constrained mode analysis, first, a free modal analysis without any load and constraint is analyzed. The first six modes obtained by Free Mode analysis are all rigid body modes, the natural frequency is close to 0 Hz, Since the 7th mode is an elastic mode, the elastic mode is calculated as the first-order mode. The modal shapes and natural frequencies obtained by modal analysis are shown in Fig. 3. In figure (a) of the first-order mode, the front and rear ends of the flexible positioning platform swing back and forth along the Y-axis direction. In the second order mode diagram (b), the four left and right angles of the flexible positioning platform swing back and forth along the Y-axis direction. In the third-order mode diagram (c), the left and right ends of the flexible positioning platform swing back and forth along the Y-axis direction, The figure illustrates the mode shapes of the first six modes. When performing constrained modal analysis, fixed constraints are applied to both ends of the flexible positioning platform to simulate actual working conditions. The analysis results show that compared with the free mode, the constraint conditions significantly improve the overall stiffness of the platform, the natural frequency of the first-order mode increases from 767.2 to 1137.5 Hz, the second-order and third-order modal frequencies increase to 1549 Hz and 1712 Hz respectively, The fourth-order mode frequency increased to 2202.6 Hz, and the modal shape also changes significantly, as shown in Fig. 4. The first six orders of the free mode and the constrained mode are shown in Table 3.
Fig. 3.
Free mode mode Shape diagram. Software used for the analysis of Fig. 3: ANSYS Workbench 2020 R2 (https://www.ansys.com).
Fig. 4.
Constrained mode Shape diagram. Software used for the analysis of Fig. 4: ANSYS Workbench 2020 R2 (https://www.ansys.com).
Table 3.
The 1st to 6th order frequencies of the flexible positioning platform.
| Order | Free mode frequency (Hz) | Constraint mode frequency (Hz) |
|---|---|---|
| 1 | 767.2 | 1137.5 |
| 2 | 850.53 | 1549 |
| 3 | 1848.4 | 1712 |
| 4 | 1871.4 | 2202.6 |
| 5 | 1894.2 | 2379.2 |
| 6 | 2085 | 2565.5 |
Finite element harmonic response analysis
In order to evaluate the response of the flexible positioning platform to sinusoidal loads at specific frequencies, the dynamic characteristics of the platform were studied using harmonic response analysis. Harmonic response analysis is used to determine the steady-state response of the flexible positioning platform under a known frequency range and sinusoidal load. This analysis method is based on the frequency domain and calculates the frequency response of the structure under harmonic loads18. Figure 5 shows theHarmonic response load analysis and key node diagrams.
Fig. 5.
Harmonic response load analysis and key node diagrams. Software used for analysis: Visio 2019 (https://www.microsoft.com/en-us/microsoft-365/visio).
The harmonic load used in the swept frequency analysis is a time-varying periodic load that can be described by frequency and amplitude. This load may cause a variety of response behaviors, especially when the system is resonant or other dynamic effects occur. First, the first 12 natural frequencies and mode shapes are obtained through modal analysis, and then the modal superposition method is applied to calculate the harmonic response of the system. The sweep frequency range is 0–3000 Hz, the load step is 100 Hz, and the damping ratio is 0.01. The driving device is a piezoelectric actuator with driving force up to 900N, which is used to study the forced excitation of the flexible positioning platform. As shown in Fig. 6, the maximum deformation in this frequency range is 1.1432 mm, The maximum force is exerted on the flexure hinge instead of the micro-motion platform. Therefore, the micro-motion platform and the flexure hinge are the key research objects. Two position nodes are selected to display the dynamic characteristics of the flexure hinge and the fretting platform. Position Node 1 is located in the left region of the compliant hinge, while Position Node 2 is in the upper-right corner of the micro-motion platform. The dynamic stiffness is calculated based on the Z-direction displacement curves of the two nodes, which reflects the overall state of these two regions.
Fig. 6.
Frequency response maximum distortion cloud. Software used for the analysis: ANSYS Workbench 2020 R2 (https://www.ansys.com).
For a simple single-degree-of-freedom system, the natural frequency of the system can be expressed by the following formula:
 |
1 |
Among them 
is natural frequency, 
is system stiffness,
is system mass.
As shown in Formula (1), an increase in stiffness leads to a higher natural frequency, thereby enhancing the system’s vibration resistance. An increase in mass leads to a reduction in the natural frequency, which consequently weakens the system’s anti-vibration capability. Optimizing the stiffness-to-mass ratio to increase the natural frequency is essential for improving the anti-vibration performance of the flexible positioning platform while maintaining a lightweight structure19,20,21. To verify the stiffness of the flexible positioning platform, a combination of theoretical calculations and simulation analyses is employed.
The flexible positioning platform can be simplified as a double parallelogram mechanism consisting of eight circular flexible hinges. The stiffness of a single hinge is first calculated, and then the stiffness of the entire platform is derived using specific formulas. The dimensioning diagram of a single circular flexible hinge is shown in Fig. 7, and the parameters are listed in Table 4.
Fig. 7.
Circular-type flexible hinge. Software used for analysis: Visio 2019 (https://www.microsoft.com/en-us/microsoft-365/visio).
Table 4.
Parameters of circular-type flexible hinge.
| Name | R (mm) | T (mm) | B (mm) | H (mm) |
|---|---|---|---|---|
| Value | 3.5 | 3 | 14 | 10 |
Equation (2) represents the formula for calculating the rotational stiffness of the flexible hinge22.
 |
2 |
In the equation, 
represents the torque applied to the flexible hinge; 
is the rotation angle of the flexible hinge under the moment; 
is the material’s Young’s modulus. The coefficient 
is calculated using Eq. (3).
 |
3 |
 |
4 |
According to the energy method in material mechanics, the work done by an external force equals the total elastic strain energy of the component. The work performed by the guiding mechanism under force is transformed into the elastic potential energy of hinge deformation.
 |
5 |
The stiffness formula for a double parallel four-bar mechanism is given by23.
 |
6 |
By substituting the relevant data, the stiffness of the flexible positioning platform is obtained as:
 |
7 |
The stiffness of the flexible positioning platform is calculated using the finite element method by conducting a static analysis to determine the displacement under a unit load. The stiffness is then derived using the appropriate formula. The boundary conditions consist of fixed constraints at both ends of the platform, with a 1 N load applied in the Z-axis direction. The displacement contour map of the flexible positioning platform can be output.
The directional displacement contour map of the flexible positioning platform is presented in Fig. 8. Under the application of a unit load, the maximum displacement of the platform in the Z-axis direction is 
. Using the stiffness formula (8), the stiffness of the flexible positioning platform is calculated as follows:
 |
8 |
Fig. 8.
Displacement contour plot of the flexible positioning platform under a unit force. Software used for the analysis: ANSYS Workbench 2020 R2 (https://www.ansys.com).
As shown in Table 5, the finite element calculated stiffness differs from the theoretical calculated stiffness by 6.61%. This discrepancy arises because the theoretical formula considers only the deformation of the flexible hinge, neglecting the deformation of other parts of the flexible positioning platform. Nevertheless, the theoretical calculation provides a partial validation of the accuracy of the finite element analysis.
Table 5.
Comparison of finite element calculated stiffness and theoretical calculated stiffness.
| Name | Finite element stiffness calculation | Theoretical stiffness calculation | Error |
|---|---|---|---|
| Value | 32.05 N/μm | 34.17 N/μm | 6.61% |
Dynamic stiffness is an important parameter to measure the anti-vibration performance at different frequencies, and can reflect the dynamic response characteristics of flexible hinges and micro-motion platforms. Especially in precision positioning, the change of dynamic stiffness directly affects the positioning accuracy and stability. Dynamic stiffness is selected as the research and optimization object to improve the structural design of the flexible positioning platform and enhance its stability and positioning accuracy under dynamic load24. Figures 9 and 10 show the displacement response curve and dynamic stiffness curve of the position node respectively, reflecting the dynamic characteristics of the flexible hinge of the flexible positioning platform and the micro-motion platform in different frequency ranges. In Fig. 9, the displacement response of the two position nodes in the low frequency range of 0 to 1500 Hz is small, indicating that the hinge and the micro-motion platform have good stability to the low-frequency driving force. When the frequency is 1500 to 2000 Hz, the displacement response appears significantly. At the peak value, the dynamic characteristics at this frequency are enhanced, and node 2 is more sensitive to the dynamic driving force. Figure 10 shows the change of dynamic stiffness with frequency. The dynamic stiffness is higher in the low frequency band. When the frequency is 1500 to 2000 Hz, the dynamic stiffness decreases significantly. The dynamic response at this frequency is stronger. The dynamic stiffness increases rapidly in the high frequency band of 2000 to 3000 Hz. The rebound anti-interference ability is enhanced.
Fig. 9.
Displacement response curve of position node. Software used for the analysis of Fig. 9: ANSYS Workbench 2020 R2 https://www.ansys.com Origin 2018 www.originlab.com.
Fig. 10.
Dynamic stiffness curve of position joint. Software used for the analysis of Fig. 10: ANSYS Workbench 2020 R2 https://www.ansys.com Origin 2018 www.originlab.com.
Response surface optimization
The hinge and micro-motion platform areas of the flexible positioning platform are optimized to improve the structural performance and precision positioning capability. Through modal analysis and harmonic response analysis, the mass, natural frequency, modal vibration shape, frequency response peak and displacement response function of the flexible positioning platform are obtained25. The first-order constraint modal frequency is the lowest and is prone to resonance. Large deformation may cause positioning deviation and structural fatigue damage. Taking the first-order natural frequency, modal vibration shape, mass and frequency response peak as the optimization targets, combined with dynamic stiffness analysis, the stability and positioning accuracy of the platform under dynamic load are improved to meet the needs of high-precision positioning.
The core structure of the flexible positioning platform is composed of a micro-motion platform and a flexible hinge, and its performance is directly related to the positioning accuracy of the platform. This study focuses on optimizing several key dimensions, the horizontal value "U" of the inclination of the top of the micro-motion platform, the horizontal spacing "V" between the two arcs of the hinge, the width "L" of the middle long groove and the thickness "W" of the micro-motion platform. The optimization goal is to increase the first-order natural frequency of the platform, reduce the modal vibration shape and the peak value of the frequency response, and achieve performance optimization while reducing the mass, so as to establish a mathematical model of the platform. The response surface optimization uses the central composite design method to generate design point data, and then construct a response surface model26,27,28,29. As shown in Fig. 11, the Optimized dimension drawing of the flexible positioning platform. The relationship between the optimization objectives and the design points is shown in Table 6, which contains the design point data generated through experimental design.
Fig. 11.
Optimized dimension drawing of the flexible positioning platform. Software used for analysis: Visio 2019 (https://www.microsoft.com/en-us/microsoft-365/visio).
Table 6.
Design point of response surface.
| Order | U | V | L | W | Mass (kg) | Frequency (Hz) | Mode shape (mm) | Response peak (mm) |
|---|---|---|---|---|---|---|---|---|
| 1 | 14.0925 | 3.5 | 9.5 | 13.5 | 1.8038 | 1150.8 | 62.445 | 1.0742 |
| 2 | 13.185 | 3.5 | 9.5 | 13.5 | 1.8045 | 1149.4 | 62.7 | 1.0742 |
| 3 | 15 | 3.5 | 9.5 | 13.5 | 1.8030 | 1152.2 | 62.193 | 1.0557 |
| 4 | 14.0925 | 2 | 9.5 | 13.5 | 1.8033 | 1128.4 | 65.686 | 1.1457 |
| 5 | 14.0925 | 5 | 9.5 | 13.5 | 1.8042 | 1163.9 | 58.392 | 0.3161 |
| 6 | 14.0925 | 3.5 | 7 | 13.5 | 1.8102 | 1145.6 | 61.986 | 0.7037 |
| 7 | 14.0925 | 3.5 | 12 | 13.5 | 1.7975 | 1155.9 | 62.879 | 0.6847 |
| 8 | 14.0925 | 3.5 | 9.5 | 12 | 1.7727 | 1168.3 | 66.176 | 1.2099 |
| 9 | 14.0925 | 3.5 | 9.5 | 15 | 1.8348 | 1133.7 | 59.605 | 0.9677 |
| 10 | 13.185 | 2 | 7 | 12 | 1.7787 | 1115.9 | 64.516 | 2.1505 |
| 11 | 15 | 2 | 7 | 12 | 1.7774 | 1118.6 | 64.618 | 1.9306 |
| 12 | 13.185 | 5 | 7 | 12 | 1.7795 | 1175.1 | 61.012 | 0.4392 |
| 13 | 15 | 5 | 7 | 12 | 1.7782 | 1177.6 | 60.399 | 0.3870 |
| 14 | 13.185 | 2 | 12 | 12 | 1.7673 | 1141.4 | 67.756 | 0.9890 |
| 15 | 15 | 2 | 12 | 12 | 1.7660 | 1144.1 | 67.595 | 0.9378 |
| 16 | 13.185 | 5 | 12 | 12 | 1.7681 | 1185.2 | 61.725 | 0.6481 |
| 17 | 15 | 5 | 12 | 12 | 1.7668 | 1187.7 | 61.073 | 0.6567 |
| 18 | 13.185 | 2 | 7 | 15 | 1.8423 | 1107.5 | 63.595 | 1.6526 |
| 19 | 15 | 2 | 7 | 15 | 1.8407 | 1110.5 | 63.222 | 1.4703 |
| 20 | 13.185 | 5 | 7 | 15 | 1.8434 | 1140.5 | 56.107 | 0.4605 |
Optimized dimension drawing of the flexible positioning platform:
Design variables: X1; X2; X3; X4; X5…….Xn.
Objectives: maxδ(x), minσ(x).
Where δ(x) the frequency and σ(x) the amplitude.
Design variables:
13.185 mm ≤ X1 = U ≤ 15 mm; 2 mm ≤ X2 = V ≤ 5 mm; 7 mm ≤ X3 = L ≤ 12 mm; 12 mm ≤ X4 = W ≤ 15 mm.
As shown in Fig. 12, the mass increases with the increase of dimensions "L" and "W," ranging from 1.767 to 1.842 kg. Figure 13 indicates that the natural frequency increases with the growth of dimensions "V" and "W," ranging from 1113 to 1181 Hz, with "W" having a more significant impact on frequency improvement. Figure 14 demonstrates that the maximum deformation decreases with the increase of dimensions "V" and "W," with the maximum deformation occurring at smaller values of both dimensions. Figure 15 shows that the frequency response peak decreases with the increase of dimension "V," while the influence of dimension "W" on suppressing the peak is relatively weaker. As shown in Fig. 16, the sensitivity analysis diagram of the flexible positioning platform illustrates the positive and negative correlations between design variables and performance indicators. The positive values on the Y-axis represent positive correlations, while the negative values represent negative correlations. In the natural frequency column, “V” shows a significant positive correlation with natural frequency, while “U” and “L” exhibit slight negative correlations, with “W” having the greatest impact. “V” and “L” have a significant positive correlation with mass, making them the main factors affecting mass. In the modal shape and frequency response peak columns, both “V” and “W” demonstrate strong negative correlations, indicating that increasing “V” and “W” can effectively suppress deformation.
Fig. 12.
Quality response surface diagram. Software used for the analysis of Fig. 12: ANSYS Workbench 2020 R2 (https://www.ansys.com), Origin 2018 (www.originlab.com).
Fig. 13.
DNatural frequency response surface. Software used for the analysis of Fig. 13: ANSYS Workbench 2020 R2 (https://www.ansys.com), Origin 2018 (www.originlab.com).
Fig. 14.
Maximum deformation response surface. Software used for the analysis of Fig. 14: ANSYS Workbench 2020 R2 (https://www.ansys.com), Origin 2018 (www.originlab.com).
Fig. 15.
Response Peak response surface. Software used for the analysis of Fig. 15: ANSYS Workbench 2020 R2 (https://www.ansys.com), Origin 2018 (www.originlab.com).
Fig. 16.
Sensitivity diagram of optimal design dimensions. Software used for the analysis of Fig. 16: ANSYS Workbench 2020 R2 (https://www.ansys.com), Origin 2018 (www.originlab.com).
Through response surface optimization, the recommended dimensions are as follows: the optimized value of “U”is 13.192 mm, “V” is 4.8129 mm, “ L” is 8.2969 mm, and “W” is 12.36 mm. After adjusting the structure to these optimized dimensions, verification was performed. As shown in Fig. 17, the first-order constrained natural frequency increased from 1137.5 to 1174.8 Hz, representing a growth rate of 3.28%, while the maximum modal deformation decreased from 62.748 to 60.863 mm, a reduction of 3%. As shown in Fig. 18, the frequency response peak significantly decreased from 1.1432 to 0.60916 mm, a reduction of 46.7%. Meanwhile, the mass decreased from 1.8175 to 1.784 kg, a reduction of 1.84%. These results demonstrate that the optimized design effectively enhanced the system’s dynamic performance and anti-vibration capability, reduced vibration response, and achieved the goal of lightweight design30,31. Figure 19 shows the variation of Z-axis displacement with frequency for position nodes 1 and 2 after optimization, compared with Fig. 9. Before optimization, the Z-axis displacement of both nodes increases with the frequency, especially in the high-frequency region. After optimization, the Z-axis displacement of both nodes decreases across the frequency range, with more significant reductions in the intermediate and high-frequency regions. This indicates that the optimized structure effectively improves the dynamic stability and anti-vibration performance of the flexible positioning platform. Figure 20 compares the dynamic stiffness curve after optimization with Fig. 10, showing a significant improvement in dynamic stiffness for both position nodes across the frequency range, particularly in the intermediate and high-frequency regions. This demonstrates that structural optimization can effectively enhance the platform’s stability and anti-vibration performance.
Fig. 17.
Optimized first-order constrained mode shape diagram. Software used for the analysis of Fig. 17: ANSYS Workbench 2020 R2 (https://www.ansys.com), Origin 2018 (www.originlab.com).
Fig. 18.
Optimized frequency response maximum distortion cloud. Software used for the analysis of Fig. 18: ANSYS Workbench 2020 R2 (https://www.ansys.com), Origin 2018 (www.originlab.com).
Fig. 19.
The displacement response curve of the optimized position node. Software used for the analysis of Fig. 19: ANSYS Workbench 2020 R2 (https://www.ansys.com), Origin 2018 (www.originlab.com).
Fig. 20.
Dynamic stiffness curve of the optimized joint. Software used for the analysis of Fig. 20: ANSYS Workbench 2020 R2 (https://www.ansys.com), Origin 2018 (www.originlab.com).
Direct optimization
The direct optimization method was used to optimize the flexible positioning platform. Direct optimization is a straightforward approach that involves optimizing the design based on real computational results. The optimization method chosen is Adaptive Multiple-Objective, and the top 20 design points generated are shown in Table 7.
Table 7.
The table of details for the first 20 design points before direct optimization analysis.
| Order | U | V | L | W | Mass (kg) | Frequency (Hz) | Mode shape (mm) | Response Peak (mm) |
|---|---|---|---|---|---|---|---|---|
| 1 | 13.223 | 2.0625 | 7.1042 | 12.063 | 1.7797 | 1141.0 | 71.855 | 0.84964 |
| 2 | 13.298 | 3.5625 | 8.7708 | 12.663 | 1.7887 | 1158.5 | 64.303 | 0.67405 |
| 3 | 13.374 | 2.8125 | 10.4375 | 13.263 | 1.7968 | 1146.8 | 65.595 | 1.05359 |
| 4 | 13.450 | 4.3125 | 7.6597 | 13.863 | 1.8169 | 1149.4 | 59.424 | 0.74138 |
| 5 | 13.525 | 2.4375 | 9.3264 | 14.463 | 1.8243 | 1126.2 | 63.610 | 1.78745 |
| 6 | 13.601 | 3.9375 | 10.9931 | 12.183 | 1.7735 | 1172.8 | 64.575 | 0.42910 |
| 7 | 13.677 | 3.1875 | 8.2153 | 12.783 | 1.7922 | 1152.6 | 65.002 | 1.02107 |
| 8 | 13.752 | 4.6875 | 9.8819 | 13.383 | 1.8010 | 1163.1 | 59.534 | 0.57650 |
| 9 | 13.828 | 2.2500 | 11.5486 | 13.983 | 1.8082 | 1134.2 | 65.795 | 2.37707 |
| 10 | 13.903 | 3.7500 | 7.2894 | 14.583 | 1.8326 | 1135.8 | 59.396 | 0.77684 |
| 11 | 13.979 | 3.0000 | 8.9560 | 12.303 | 1.7801 | 1158.0 | 67.086 | 1.13128 |
| 12 | 14.055 | 4.5000 | 10.6227 | 12.903 | 1.7890 | 1169.1 | 60.947 | 0.42944 |
| 13 | 14.130 | 2.6250 | 7.8449 | 13.503 | 1.8078 | 1137.5 | 64.803 | 1.75161 |
| 14 | 14.206 | 4.1250 | 9.5116 | 14.103 | 1.8163 | 1150.0 | 59.576 | 0.59356 |
| 15 | 14.282 | 3.3750 | 11.1782 | 14.703 | 1.8239 | 1139.6 | 60.660 | 0.40302 |
| 16 | 14.357 | 4.8750 | 8.4005 | 12.423 | 1.7842 | 1173.7 | 60.381 | 0.62376 |
| 17 | 14.433 | 2.1563 | 10.0671 | 13.023 | 1.7918 | 1141.5 | 68.375 | 1.11690 |
| 18 | 14.508 | 3.6563 | 11.7338 | 13.623 | 1.8003 | 1156.2 | 61.994 | 0.92085 |
| 19 | 14.584 | 2.9063 | 7.4745 | 14.223 | 1.8237 | 1132.6 | 62.119 | 0.48497 |
| 20 | 14.660 | 4.4063 | 9.1412 | 14.823 | 1.8320 | 1144.2 | 57.629 | 0.39013 |
The recommended dimensions from the direct optimization are U = 14.375, V = 4.875, L = 8.4005, and W = 12.432, as detailed in Table 10. These dimensions were used to reconstruct the model in SolidWorks, followed by a harmonic response analysis in ANSYS Workbench. The results include the modal shapes from the direct optimization and the response peak contour plots from the harmonic response analysis, as shown in Figs. 21 and 22.
Table 10.
Result data of hammer test from measuring point 1 to measuring point 6.
| Order | Measuring point 1 | Measuring point 2 | Measuring point 3 | Measuring point 4 | Measuring point 5 | Measuring point 6 |
|---|---|---|---|---|---|---|
| 1 | 778.6 | 753.7 | 779.2 | 775.9 | 772.5 | 767.8 |
| 2 | 853.9 | 863.3 | 899.7 | 861.3 | 850.6 | 853.9 |
| 3 | 1767.7 | 1737.5 | 1749.6 | 1750.2 | 1728 | 1752.9 |
| 4 | 1851.9 | 1839.7 | 1835 | 1833 | 1822.9 | 1831.7 |
| 5 | 2034.9 | 2034.9 | 2006.6 | 2004.6 | 2001.2 | 2012 |
| 6 | 2274.4 | 2262.3 | 2270.4 | 2261.7 | 2261 | 2272.4 |
Fig. 21.
Optimized first-order constrained mode shape diagram. Optimized frequency response maximum distortion cloud. Software used for the analysis of Fig. 21: ANSYS Workbench 2020 R2 (https://www.ansys.com), Origin 2018 (www.originlab.com).
Fig. 22.
Optimized frequency response maximum distortion cloud. Optimized frequency response maximum distortion cloud. Software used for the analysis of Fig. 22: ANSYS Workbench 2020 R2 (https://www.ansys.com), Origin 2018 (www.originlab.com).
The results of response surface optimization and direct optimization are very similar, as shown in Table 8. The mass values are 1.784 kg and 1.7842 kg, with a difference of only 0.01%. The first-order constrained frequencies are 1174.8 Hz and 1173.6 Hz, differing by 0.1%. The response peaks are 0.60916 mm and 0.62229 mm, with a difference of 2.165%. These findings demonstrate that the accuracy of response surface optimization is nearly identical to that of direct optimization. Moreover, the results obtained from response surface optimization are slightly superior, further validating its accuracy and reliability.
Table 8.
Comparison between response surface optimization and direct optimization.
| Parameter | Response surface optimization | Direct optimization | Comparison |
|---|---|---|---|
| Mass (kg) | 1.784 | 1.7842 | 0.01% |
| First-order constrained frequency (Hz) | 1174.8 | 1173.6 | 0.1% |
| Response Peak (mm) | 0.60916 | 0.62229 | 2.165% |
Modal test of flexible positioning platform
For the natural frequency of the flexible positioning platform, the modal test square method based on perfect noise application is adopted in this experiment. The vibration response of the platform is collected by the built-in microphone of the flat plate, and the fast Fourier transform (FFT) in the application program is used to analyze the signal in frequency domain, so as to obtain the modal parameters of the system. The percussion excitation uses a force hammer to ensure the universality of the excitation frequency and the safety of the test platform32–34. The physical diagram of the equipment used for experimental modal analysis is shown in Fig. 23. The schematic diagram of the test setup used in the modal analysis experiment is shown in Fig. 24. In this study, nine measuring points were selected for free modal hammer testing, with the hammer impact points shown in Fig. 25. The vibration amplitudes at tapping points 3, 4, and 6 are relatively significant in the first-order mode. It can be ensured that accurate first-order modal data can be obtained at these points. Therefore, the hammer test data at tapping points 3, 4, and 6 are averaged for calculation, while data from other modes are excluded from the calculation. Similarly, the measurement points corresponding to subsequent modes will be processed in the same manner. Specific data can be found in Table 9.
Fig. 23.
Physical diagram of the experimental modal analysis system. Software used for the analysis of Fig. 23: Visio 2019 (https://www.microsoft.com/en-us/microsoft-365/visio).
Fig. 24.
Schematic diagram of experimental modal analysis images. Software used for the analysis of Fig. 24: Visio 2019 (https://www.microsoft.com/en-us/microsoft-365/visio).
Fig. 25.
Distribution of percussion points on the flexible positioning platform. Software used for the analysis of Fig. 25: Visio 2019 (https://www.microsoft.com/en-us/microsoft-365/visio).
Table 9.
Corresponding Tap points of experimental modal analysis.
| Order | Tap point |
|---|---|
| 1 | 3,4,6 |
| 2 | 1,2,8,9 |
| 3 | 4,6 |
| 4 | 3,7 |
| 5 | 3,5,7 |
| 6 | 1,2,3,8,9 |
The frequency spectrum measured by the experiment is shown in the Fig. 26. After percussion excitation, the vibration response of the platform exhibits significant modal characteristics, with the peaks in the spectrum corresponding to the natural frequencies of the platform. As shown in Fig. 22a, the first peak appears at 778.6 Hz, and the sound pressure level at this frequency is approximately 10 dB(A), indicating that this is the first natural mode frequency of the platform. Other peaks indicate the presence of higher-order modes, especially the weak signal at 1767.7 Hz, which may reflect the minimal resonance of the flexible positioning platform at this excitation point. To ensure the accuracy of the experiment, three taps were conducted at measuring points 3, 4, and 7. As shown in Fig. 27, the three fitted curves exhibit high overlap, demonstrating the accuracy of the test. The test data are presented in Tables 10 and 11. Table 10 displays the hammer test results from tapping points 1 to 6, while Table 11 presents the single-tap test data for points 7 to 9 and the identification results of the three-tap fitting for points 3, 4, and 7. This experiment verified the effectiveness and feasibility of acoustic modal analysis. Although the accuracy of this method is limited compared to traditional modal testing methods, such as acceleration sensors or laser vibrometers, its simplicity and low cost make it suitable for preliminary modal testing and analysis. The measured natural frequencies are highly consistent with the simulation values, indicating that this method can effectively capture the primary modal characteristics of the structure. Future research can further improve measurement accuracy by employing more precise instruments, such as accelerometers and laser vibration measuring devices, especially for modal identification in low-frequency and high-frequency bands.
Fig. 26.
Frequency response curve obtained by tapping at the measuring point. Software used for the analysis of Fig. 26: Perfect Noise (https://apps.apple.com/bz/app/perfect-noise/id589733856).
Fig. 27.
Fitted frequency response curve obtained by three taps at the measuring point. Software used for the analysis of Fig. 27: Perfect Noise (https://apps.apple.com/bz/app/perfect-noise/id589733856).
Table 11.
Data of single-tap tests for points 7 to 9 and three-tap fitting for points 3, 4, and 7.
| Order | Measuring point 7 | Measuring point 8 | Measuring point 9 | Measuring point 3 three-time fitted identified frequency | Measuring point 4 three-time fitted identified frequency | Measuring point 7 three-time fitted identified frequency |
|---|---|---|---|---|---|---|
| 1 | 759 | 771.8 | 777.2 | 772.5 | 763.8 | 765.1 |
| 2 | 852.6 | 855.3 | 849.9 | 848.5 | 849.9 | 853.3 |
| 3 | 1736.8 | 1748.9 | 1750.9 | 1751.6 | 1752.3 | 1737.5 |
| 4 | 1832.3 | 1853.9 | 1841.8 | 1841.8 | 1853.9 | 1844.5 |
| 5 | 1979.7 | 1997.9 | 1986.4 | 2001.2 | 1999.2 | 1997.2 |
| 6 | 2256.3 | 2261.7 | 2258.3 | 2256.3 | 2242.8 | 2244.2 |
The data shows a high level of consistency, with the differences in natural frequencies between experimental modal analysis and finite element modal analysis generally within a reasonable range, as shown in Table 12. The results for low-order modes are nearly identical to the experimental values, while high-order modes exhibit some deviations, but the errors remain within acceptable limits. Overall, the results validate the high accuracy and reliability of finite element modal analysis in predicting the modal characteristics of the structure. This experimental method provides valuable reference for testing the resonance frequencies of plate materials.
Table 12.
Comparison of natural frequency results between experimental mode and finite element mode.
| Order | Experimental mode (Hz) | Finite element mode (Hz) | Error (%) |
|---|---|---|---|
| 1 | 774.3 | 767.2 | 0.91 |
| 2 | 855.6 | 850.53 | 0.59 |
| 3 | 1751.6 | 1848.4 | 5.53 |
| 4 | 1833.7 | 1871.4 | 2.06 |
| 5 | 1995.8 | 1894.2 | 5.09 |
| 6 | 2265.4 | 2085 | 7.96 |
Conclusion
This study investigates the modal characteristics and response behavior under excitation of the flexible positioning platform in an ultra-high acceleration macro–micro motion platform through finite element modal analysis and experimental modal analysis. During the research, both response surface method and direct optimization method were used to analyze the results of the two optimization approaches. Based on these analyses, the following conclusions were drawn:
A parametric model of the flexible positioning platform was established, and finite element modal analysis and harmonic response analysis were performed in ANSYS Workbench. The natural frequency, mode shapes, frequency peak responses, frequency response functions, and dynamic stiffness of the flexible positioning platform were determined. The accuracy of the finite element model was verified through theoretical calculations of stiffness.
Multi-objective optimization was conducted to enhance platform performance. The accuracy of the optimization was verified through a comparison of response surface optimization and direct optimization methods. After optimization, the first-order natural frequency increased by 3.28%, the mass decreased by 1.84%, the first-order modal deformation was reduced by 3%, and the frequency response peak was lowered by 46.7%. These results demonstrate a significant improvement in platform performance while achieving material optimization.
In this study, a portable intelligent acoustic device, combined with an acoustic percussion method, was used to measure the natural frequency of the free mode by striking the flexible positioning platform with a force hammer. This method is easy to operate, low in cost, and particularly suitable for engineering applications with limited experimental conditions. It provides a new non-contact, non-destructive approach for measuring the natural frequency of flexible structures. Compared with the finite element modal analysis results, the overall error is within a reasonable range, verifying the reliability of the finite element modal analysis.
Author contributions
All the authors contributed extensively to the manuscript. H.Y designed the experiments and wrote the main manuscript. L.Z. revised and suggested the paper and helped with the formatting review and editing of the paper. H.S and M.H. produced the data set used in this study. All authors have read and agreed to the publication of the manuscript.
Funding
National Youth Science Foundation of China (51705132) (52405307); The fund of Henan Key Laboratory of Superhard Abrasives and Grinding Equipment, Henan University of Technology (JDKFJJ2024007).
Data availability
Datasets generated and/or analyzed during the current study are available from the corresponding author upon reasonable request. Code availability The code that support the findings of this study are available from the corresponding author upon reasonable request.
Code availability
The code that support the findings of this study are available from the corresponding author upon reasonable request.
Declarations
Competing interests
The authors declare no competing interests.
Footnotes
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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Associated Data
This section collects any data citations, data availability statements, or supplementary materials included in this article.
Data Availability Statement
Datasets generated and/or analyzed during the current study are available from the corresponding author upon reasonable request. Code availability The code that support the findings of this study are available from the corresponding author upon reasonable request.
The code that support the findings of this study are available from the corresponding author upon reasonable request.