A method for predicting the levitation accuracy of near-field ultrasonic levitation objects
By constructing a vibration model of near-field ultrasonic suspended objects, predicting the vibration characteristics of suspended objects, solving the problem that the suspension accuracy cannot reach the nanoscale in the prior art, and achieving higher prediction accuracy and application range.
Patent Information
- Application Number
- CN202411291494.3
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- Filing Date
- 2024-09-14
- Publication Date
- 2025-05-02
- Estimated Expiration
- 2044-09-14
AI Technical Summary
The prior art is difficult to accurately predict the vibration characteristics of near-field ultrasonic suspended objects, resulting in the suspension accuracy not reaching the requirements of ultra-precision applications at the nanoscale.
By obtaining the resonant frequency of the near-field ultrasonic vibration system and the normalized amplitude distribution of the radiation disk, the dimensionless thickness and internal pressure distribution of the extruded air film are described, and the equivalent stiffness and damping coefficients are obtained based on the small disturbance method to construct a vibration model of the near-field ultrasonic suspended object to achieve suspension accuracy prediction.
A theoretical model for the vibration amplitude of suspended objects considering flexible radiation disks was established, which improved prediction accuracy, expanded the application range, and provided theoretical guidance for system parameter selection and precision suspension control.
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Figure CN119167459B_ABST
Abstract
Description
Technical Field
[0001] The invention relates to the field of precision suspension, and in particular to a method for predicting the suspension accuracy of a near-field ultrasonic suspended object. Background Art
[0002] Ultrasonic levitation is mainly divided into standing wave ultrasonic levitation and near-field ultrasonic levitation. In standing wave ultrasonic levitation, radiators and reflectors are needed to generate standing wave acoustic fields at the same time, so that granular objects can be suspended on the acoustic field pressure nodes. Near-field ultrasonic levitation uses the suspended object itself as a reflector, resulting in a significant reduction in the suspension height, about tens to hundreds of microns, but allowing a greater load-bearing capacity. Therefore, near-field ultrasonic levitation has considerable potential for application in non-contact positioning systems, transmission systems, squeeze film bearings and other fields. Compared with other levitation technologies such as magnetic levitation and air cushion levitation, the near-field ultrasonic levitation system has the advantages of compactness and high energy utilization, because it has no restrictions on the material of the suspended object and does not require an additional pressurized air supply.
[0003] The air pressure of the squeeze film in near-field ultrasonic suspension is in a periodic change state. The vibration of the ultrasonic frequency causes the average air pressure value in one squeeze cycle to be greater than the surrounding atmospheric pressure, thus generating a suspension force that offsets the gravity of the suspended object. Therefore, the suspended object will produce vibrations caused by the change in air film pressure instead of being completely still in a certain position. Since the vibration amplitude of the suspended object is very small relative to the thickness of the squeeze film, this vibration can usually be ignored. Therefore, in most applications, the suspended object is considered to be stationary. However, in ultra-precision applications where the suspension accuracy reaches the nanometer level, this vibration cannot be ignored. Therefore, accurately predicting this vibration is crucial for the application of near-field ultrasonic suspension in high-precision fields.
[0004] For near-field ultrasonic levitation, the vibration amplitude of the suspended object is usually used to characterize the accuracy of the system. The smaller the amplitude of the suspended object's vibration, the higher the accuracy of the suspension system. Some scholars obtained the air film pressure distribution by coupling the vibration of the suspended object with the thickness of the air film. Then, they calculated the vibration performance of the suspended object by solving the motion equation related to the suspended object and the film pressure. However, their calculated results are significantly smaller than the experimental observations. So far, there is still no model that can accurately predict the vibration characteristics of suspended objects. Summary of the invention
[0005] In view of the shortcomings and improvement needs of the existing technology, the present invention provides a method for predicting the levitation accuracy of a near-field ultrasonic levitated object. Its purpose is to effectively predict the vibration amplitude of a near-field ultrasonic levitated object during levitation, and to provide theoretical guidance for parameter optimization to improve the levitation accuracy of the system and precise levitation control of the levitated object.
[0006] In order to solve the technical problems in the above background, the present invention provides a method for predicting the levitation accuracy of a near-field ultrasonic levitation object, the steps comprising:
[0007] Obtain the resonant frequency of the near-field ultrasonic vibration system and the normalized amplitude distribution of the radiation disk;
[0008] Based on the normalized amplitude distribution, a squeezed air film of a near-field ultrasonic suspension system is described;
[0009] Based on the small perturbation method, the equivalent stiffness and damping coefficient corresponding to the squeeze film are obtained;
[0010] Based on the equivalent stiffness and the damping coefficient, construct a vibration model of a near-field ultrasonic levitated object;
[0011] Based on the vibration model, the levitation accuracy prediction of the near-field ultrasonic levitation object is completed.
[0012] Preferably, describing the squeezed air film includes describing its dimensionless thickness and internal pressure distribution, wherein the description of the dimensionless thickness includes:
[0013]
[0014] Where H is the dimensionless thickness of the squeezed air film; r is the radial coordinate of any point in the solution domain, ξ m is the maximum vibration amplitude of the radiation disk, h a is the average distance between the center of the suspended object and the center of the radiation disk in the vertical direction within one extrusion cycle, f is the resonant frequency of the near-field ultrasonic vibration system; t is time; A(r) represents the normalized amplitude distribution of the near-field ultrasonic vibration system.
[0015] Preferably, the internal pressure distribution of the squeezed air film is expressed by the Reynolds equation:
[0016]
[0017] Among them, P represents the dimensionless air film pressure; H represents the dimensionless thickness of the squeezed air film; R represents the dimensionless radial position; T represents the dimensionless time; σ represents the squeeze number.
[0018] Preferably, the Reynolds equation is solved by using the Newton-Raphson method to obtain the dimensionless pressure distribution P0 and the film thickness distribution H0 in the squeezed air film under steady state, and the solution method includes:
[0019]
[0020] Among them, F zrepresents the suspension force generated by the squeeze film; W represents the gravity of the suspended object; m and g are the mass of the suspended object and the local gravitational acceleration respectively; the mass of the suspended object can be measured by a balance.
[0021] Preferably, firstly, based on the small perturbation method, a partial differential equation describing the disturbed air pressure is obtained:
[0022]
[0023] Where P represents the dimensionless air film pressure; H represents the dimensionless thickness of the squeeze film; R represents the dimensionless radial position; P0 and H0 represent the dimensionless pressure distribution and air film thickness distribution in the squeeze film under steady state, respectively; γ represents the ratio of the perturbation frequency to the resonant frequency; θ represents the position in the circumferential direction; P H and They represent the dimensionless disturbance pressure caused by displacement disturbance and velocity disturbance respectively; σ represents the extrusion number.
[0024] Preferably, the partial differential equations of the disturbance pressure are solved simultaneously to obtain the disturbance pressure P H and distribution, and based on the disturbance pressure P H and To obtain the equivalent stiffness and damping coefficient corresponding to the squeeze film:
[0025]
[0026] Where k represents the equivalent stiffness corresponding to the squeezed air film; c represents the damping coefficient corresponding to the squeezed air film; p a represents the ambient pressure; θ represents the position in the circumferential direction; h a It is expressed as the average distance between the center of the suspended object and the center of the radiation disk within one compression cycle in the vertical direction; γ represents the ratio of the perturbation frequency to the resonant frequency; T represents the dimensionless time; f represents the resonant frequency of the near-field ultrasonic vibration system; L represents the radius of the radiation disk and the suspended object.
[0027] Preferably, the vibration model constructed includes:
[0028]
[0029] In the formula, |z o | represents the vibration amplitude of the suspended object; ξ m represents the maximum amplitude of the vibration of the radiation disk; k represents the equivalent stiffness corresponding to the squeezed air film; c represents the damping coefficient corresponding to the squeezed air film; f represents the resonant frequency of the near-field ultrasonic vibration system; and m represents the mass of the suspended object.
[0030] Compared with the prior art, the present invention has the following beneficial effects:
[0031] The present invention establishes a theoretical model of the vibration amplitude of a suspended object taking into account a flexible radiation disk, which is closer to the actual operation of a near-field ultrasonic suspension system. Therefore, the application range of the present method is wider and the calculation result is more accurate. In addition, the present method predicts the vibration amplitude of a suspended object by calculating the actual flow field distribution in the squeezed air film, so that the influence mechanism of different design and operation parameters on the suspension accuracy of the system can be theoretically analyzed, and further used to design the optimization of operation parameters and guide actual production. In addition, the present method can provide a theoretical basis for the flow field distribution control of precision suspension. BRIEF DESCRIPTION OF THE DRAWINGS
[0032] In order to more clearly illustrate the technical solution of the present invention, the following briefly introduces the drawings required for use in the embodiments. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative labor.
[0033] Figure 1 A schematic diagram of a method flow of an embodiment of the present invention;
[0034] Figure 2 It is a schematic diagram of a finite element model of a vibration system in a near-field ultrasonic suspension according to an embodiment of the present invention;
[0035] Figure 3 It is a schematic diagram of experimental measurement of the amplitude of the radiation disk in the near-field ultrasonic levitation according to an embodiment of the present invention;
[0036] Figure 4 It is a schematic diagram of a theoretical model of a suspended object in a free state in near-field ultrasonic levitation according to an embodiment of the present invention;
[0037] Figure 5 It is an equivalent mechanical model diagram of the near-field ultrasonic suspension system in an embodiment of the present invention;
[0038] Figure 6 It is a comparison diagram of the calculation result of the vibration amplitude of the suspended object of the prediction method in the embodiment of the present invention and the experimental result.
[0039] Explanation of the reference numerals in the accompanying drawings: 1. Radiating disk; 2. Amplitude transformer, 2-1. Amplitude transformer flange structure; 3. Langevin transducer; 3-1. Front cover; 3-2. Piezoelectric vibration unit; 3-3. Rear cover; 3-4. Fastening bolts; 4. Base; 5. Wires; 6. Laser displacement sensor probe; 7. Controller; 8. Computer; 9. Ultrasonic generator. DETAILED DESCRIPTION
[0040] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.
[0041] In order to make the above-mentioned objects, features and advantages of the present invention more obvious and easy to understand, the present invention is further described in detail below with reference to the accompanying drawings and specific embodiments.
[0042] like Figure 1 FIG. 1 is a schematic diagram of the method flow of this embodiment, and the steps include:
[0043] S1. Obtain the resonant frequency of the near-field ultrasonic vibration system and the normalized amplitude distribution of the radiation disk.
[0044] Among them, near-field ultrasonic vibration systems such as Figure 2 As shown, it mainly includes: Langevin transducer 3, amplitude transformer 2 and radiation plate 1. Among them, the Langevin transducer mainly includes front cover plate 3-1, piezoelectric vibration unit 3-2, rear cover plate 3-3 and fastening bolts 3-4. Finite element software is used to model this vibration system, and modal analysis is used to obtain the resonant frequency f of the vibration system. In the solution process, the vibration system is divided into free grids, and the minimum grid size is set to 5mm. Figure 3 As shown, since the horn flange structure 2 - 1 is connected to the base 4 , the displacement here is set to zero.
[0045] like Figure 3 As shown, the operating frequency of the ultrasonic generator 9 is set to f, and a sinusoidal electrical signal of the same frequency is applied to the Langevin transducer 3 through the wire 5. According to the inverse piezoelectric effect, the Langevin transducer 3 generates longitudinal vibrations of the same frequency, which are amplified by the horn 2 and then transmitted to the radiation disk 1. Therefore, the radiation disk will generate vibrations with a frequency of f, and the vibration amplitude distribution is measured by the laser displacement sensor probe 6. The signal detected by the laser displacement sensor probe 6 is processed by the controller 7 and then transmitted to the computer 8 for storage and analysis. By changing the position of the probe, the maximum amplitude ξ of the radiation disk vibration can be obtained. m and the normalized vibration distribution A(r).
[0046] S2. Description of the squeezed air film of the near-field ultrasonic levitation system based on the normalized amplitude distribution.
[0047] Describing the squeeze film includes describing its dimensionless thickness and internal pressure distribution.
[0048] like Figure 4As shown in the figure, the suspended object is freely suspended on the radiation disk, and the centers of the radiation disk and the suspended object are represented as O and O1 respectively. The radius of the suspended object and the radiation disk are both L. During the suspension process, the radiation disk and the suspended object remain concentric and parallel. At this time, the thickness H of the squeezed air film without considering the vibration of the suspended object is expressed as:
[0049]
[0050] Where H is the dimensionless thickness of the squeezed air film; r is the radial coordinate of any point in the solution domain; ξ m is the maximum vibration amplitude of the radiation disk; h a is the average distance between the center of the suspended object and the center of the radiation disk in the vertical direction within one extrusion cycle; f is the resonant frequency of the near-field ultrasonic vibration system; t is time; A(r) represents the normalized amplitude distribution of the near-field ultrasonic vibration system.
[0051] Based on the gas lubrication theory, the dimensionless Reynolds equation describing the pressure distribution in the squeeze gas film is:
[0052]
[0053] Where P represents the dimensionless air film pressure; H represents the dimensionless thickness of the squeezed air film; R represents the dimensionless radial position, where P = p / p a ; H = h / h a ; R=r / L and T=2πft; σ=12μ a ωL 2 / p a h a 2 represents the squeeze number, which characterizes the pressure generated by the squeeze effect; here, p a is the ambient pressure; θ is the position in the circumferential direction; L is the radius of the radiation disk and the suspended object; μ a represents the aerodynamic viscosity coefficient; ω represents the vibration angular frequency of the radiation disk.
[0054] like Figure 4 As shown in the figure, at the boundary of the squeeze film, due to contact with the external environment air pressure, the air pressure satisfies P(R=1,T)=1. Due to the continuous distribution of air pressure in the squeeze film, the gradient of air pressure at the center O of the squeeze film is zero, that is,
[0055] By solving the above Reynolds equation using the Newton-Raphson method, we can obtain the dimensionless pressure distribution P0 and film thickness distribution H0 in the squeeze film under steady state. Under steady state conditions, the suspension force generated by the squeeze film offsets the gravity of the suspended object, which can be expressed by the following equation:
[0056]
[0057] Among them, F z represents the suspension force generated by the squeeze film; W represents the gravity of the suspended object; m and g are the mass of the suspended object and the local gravitational acceleration respectively; the mass of the suspended object can be measured by a balance.
[0058] S3. Based on the small perturbation method, the equivalent stiffness and damping coefficient corresponding to the squeezed air film are obtained.
[0059] like Figure 5 As shown, an equivalent mechanical model of the near-field ultrasonic suspension system is established for this embodiment, in which the squeeze film can be equivalent to a spring and a damper. The coefficients of the equivalent spring and damper are solved using the small perturbation method. Assuming a displacement disturbance Δh and a velocity disturbance Applied to the suspended object, the dimensionless film thickness and pressure are re-expressed as:
[0060]
[0061] Where ΔH=Δh / h a and represent dimensionless displacement disturbance and velocity disturbance respectively; h a is the average distance between the center of the suspended object and the center of the radiation disk in a vertical compression cycle; P H and Represent the dimensionless disturbance pressure caused by displacement disturbance and velocity disturbance respectively; further, the partial differential equation describing the disturbance pressure can be expressed as:
[0062]
[0063] Where P represents the dimensionless air film pressure; H represents the dimensionless thickness of the squeeze film; R represents the dimensionless radial position; P0 and H0 represent the dimensionless pressure distribution and air film thickness distribution in the squeeze film under steady state, respectively; γ represents the ratio of the perturbation frequency to the resonant frequency; θ represents the position in the circumferential direction; P H and They represent the dimensionless disturbance pressure caused by displacement disturbance and velocity disturbance respectively; σ represents the extrusion number.
[0064] Solve the above partial differential equations describing the disturbance pressure simultaneously, and combine the corresponding boundary conditions to obtain the disturbance pressure P H and Therefore, the equivalent stiffness k and damping coefficient c corresponding to the squeeze film can be solved using the following equations:
[0065]
[0066]
[0067] Where k represents the equivalent stiffness corresponding to the squeezed air film; c represents the damping coefficient corresponding to the squeezed air film; p a represents the ambient pressure; θ represents the position in the circumferential direction; h a It is expressed as the average distance between the center of the suspended object and the center of the radiation disk within one compression cycle in the vertical direction; γ represents the ratio of the perturbation frequency to the resonant frequency; T represents the dimensionless time; f represents the resonant frequency of the near-field ultrasonic vibration system; L represents the radius of the radiation disk and the suspended object.
[0068] S4. Based on the equivalent stiffness and damping coefficient, a vibration model of the near-field ultrasonic levitated object is constructed.
[0069] Since the stiffness and damping coefficients are obtained by solving the dimensionless air film thickness H0 and pressure P0, the information of the amplitude distribution A(r) of the radiation disk is already included in the coefficients. At this time, the up and down vibration of the radiation disk is regarded as equal amplitude vibration, and its vibration amplitude is expressed as z i =ξ m sin(2πft), where ξ m is the maximum amplitude of the radiation disk vibration. The vibration displacement of the suspended object is expressed by z o According to Newton's second law, the near-field ultrasonic system satisfies the following equation:
[0070]
[0071] Therefore, the transfer function of the near-field ultrasonic system can be obtained by using Laplace transformation:
[0072]
[0073] Among them, Z o (s) and Z i (s) are z o and z i The image function of s is a complex variable. According to this transfer function, the vibration amplitude of the suspended object can be obtained by using Laplace inverse transformation:
[0074]
[0075] In the formula, |z o | represents the vibration amplitude; ξ m is the maximum amplitude of the radiation disk vibration; k is the equivalent stiffness corresponding to the squeezed air film; c is the damping coefficient corresponding to the squeezed air film; f is the resonant frequency of the near-field ultrasonic vibration system; m is the mass of the suspended object.
[0076] The vibration amplitude of the floating object is described by the above model.
[0077] S5. Based on the vibration model, the levitation accuracy prediction of near-field ultrasonic levitation objects is completed.
[0078] Based on the vibration model constructed by S4, the maximum amplitude ξ of different radiation disks is changed m , we can establish a mathematical relationship between the vibration amplitude of the suspended object and the maximum amplitude of the radiation disk, such as Figure 6 As shown. It can be found that the vibration amplitude of the suspended object increases slowly with the increase of the maximum amplitude of the radiation disk, which means that the accuracy of the suspension system decreases slowly. In addition, the calculation results of the prediction method proposed in the present invention compare well with the experimental results. Therefore, this method realizes the prediction of the vibration amplitude of the suspended object by calculating the actual flow field distribution in the squeezed air film, so it can theoretically analyze the influence mechanism of different design and operating parameters on the suspension accuracy of the system, and further be used to design the optimization of operating parameters and guide actual production. In addition, this method can provide a theoretical basis for realizing flow field distribution control for precision suspension.
[0079] The embodiments described above are only descriptions of the preferred embodiments of the present invention and are not intended to limit the scope of the present invention. Without departing from the design spirit of the present invention, various modifications and improvements made to the technical solutions of the present invention by ordinary technicians in this field should all fall within the protection scope determined by the claims of the present invention.
Claims
1. A method for predicting the levitation accuracy of a near-field ultrasonic levitation object, characterized in that the steps include: Obtain the resonant frequency of the near-field ultrasonic vibration system and the normalized amplitude distribution of the radiation disk; Based on the normalized amplitude distribution, a squeezed air film of a near-field ultrasonic suspension system is described; Based on the small perturbation method, the equivalent stiffness and damping coefficient corresponding to the squeeze film are obtained; Based on the equivalent stiffness and the damping coefficient, a vibration model of a near-field ultrasonic levitated object is constructed; the constructed vibration model includes: In the formula, |z o | represents the vibration amplitude of the suspended object; ξ m represents the maximum amplitude of the vibration of the radiation disk; k represents the equivalent stiffness corresponding to the squeezed air film; c represents the damping coefficient corresponding to the squeezed air film; f represents the resonant frequency of the near-field ultrasonic vibration system; m represents the mass of the suspended object; Based on the vibration model, the levitation accuracy prediction of the near-field ultrasonic levitation object is completed.
2. The method for predicting the levitation accuracy of a near-field ultrasonic levitation object according to claim 1, characterized in that: Describing the squeezed air film includes describing its dimensionless thickness and internal pressure distribution, wherein the description of the dimensionless thickness includes: Where H is the dimensionless thickness of the squeezed air film; r is the radial coordinate of any point in the solution domain, ξ m is the maximum vibration amplitude of the radiation disk, h a is the average distance between the center of the suspended object and the center of the radiation disk in the vertical direction within one extrusion cycle, f is the resonant frequency of the near-field ultrasonic vibration system; t is time; A(r) represents the normalized amplitude distribution of the near-field ultrasonic vibration system.
3. The method for predicting the levitation accuracy of a near-field ultrasonic levitation object according to claim 2, characterized in that: The internal pressure distribution of the squeeze film is expressed by the Reynolds equation: Among them, P represents the dimensionless air film pressure; H represents the dimensionless thickness of the squeezed air film; R represents the dimensionless radial position; T represents the dimensionless time; σ represents the squeeze number.
4. The method for predicting the levitation accuracy of a near-field ultrasonic levitation object according to claim 3, characterized in that: The Newton-Raphson method is used to solve the Reynolds equation to obtain the dimensionless pressure distribution P0 and the film thickness distribution H0 in the squeeze film under steady state. The solution method is include: Among them, F z represents the suspension force generated by the squeeze film; W represents the gravity of the suspended object; m and g are the mass of the suspended object and the local gravitational acceleration respectively; the mass of the suspended object can be measured by a balance.
5. The method for predicting the levitation accuracy of a near-field ultrasonic levitation object according to claim 1, characterized in that: First, based on the small perturbation method, the partial differential equation describing the disturbed air pressure is obtained as: Where P represents the dimensionless air film pressure; H represents the dimensionless thickness of the squeeze film; R represents the dimensionless radial position; P0 and H0 represent the dimensionless pressure distribution and air film thickness distribution in the squeeze film under steady state, respectively; γ represents the ratio of the perturbation frequency to the resonant frequency; θ represents the position in the circumferential direction; P H and They represent the dimensionless disturbance pressure caused by displacement disturbance and velocity disturbance respectively; σ represents the extrusion number.
6. The method for predicting the levitation accuracy of a near-field ultrasonic levitation object according to claim 5, characterized in that: Solve the partial differential equations of the disturbance pressure simultaneously to obtain the disturbance pressure P H and distribution, and based on the disturbance pressure P H and To obtain the equivalent stiffness and damping coefficient corresponding to the squeeze film: Where k represents the equivalent stiffness corresponding to the squeeze film; c represents the damping coefficient corresponding to the squeeze film; p a represents the ambient pressure; θ represents the position in the circumferential direction; h a It is expressed as the average distance between the center of the suspended object and the center of the radiation disk within one compression cycle in the vertical direction; γ represents the ratio of the perturbation frequency to the resonant frequency; T represents the dimensionless time; f represents the resonant frequency of the near-field ultrasonic vibration system; L represents the radius of the radiation disk and the suspended object.