CN116644644B - Restoring force prediction method based on eccentric and inclined states of suspended object - Google Patents
Restoring force prediction method based on eccentric and inclined states of suspended object Download PDFInfo
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Abstract
The invention belongs to the technical field of non-contact suspension, and particularly relates to a restoring force prediction method based on eccentric and inclined states of a suspended object, which comprises the steps of obtaining resonant frequency and amplitude distribution of an ultrasonic transducer; establishing a restoring force prediction theoretical model taking eccentricity and inclination of a suspended object into consideration; deducing a fluid motion control equation in the extrusion film based on a gas lubrication theory; obtaining the inclination angle and the restoring force of the suspended object by using a finite difference method and a spline interpolation method; and establishing a mathematical relationship between the restoring force and the eccentric distance, and verifying with an experimental result. The invention discloses a mechanism for generating restoring force under the near-field acoustic levitation eccentricity from the angle of extrusion fluid movement, and provides a restoring force prediction method which simultaneously considers the levitation object eccentricity and inclination, so that the magnitude of the restoring force when the levitation object is eccentric can be accurately predicted, and a theoretical basis is provided for the application of the near-field acoustic levitation in the field of non-contact precise levitation.
Description
Technical Field
The invention belongs to the technical field of non-contact suspension, and particularly relates to a restoring force prediction method based on an eccentric and inclined state of a suspended object.
Background
The principle of near-field ultrasonic levitation is to utilize vibration at ultrasonic frequencies to generate a nonlinear extrusion flow field. Under the excitation of sine alternating voltage, piezoelectric ceramics contained in the ultrasonic transducer converts electric energy into mechanical energy by utilizing inverse piezoelectric effect, so that the ultrasonic transducer periodically vibrates at a certain frequency and amplitude. The ultrasonic vibration of the radiating disk in the ultrasonic transducer can generate an extrusion flow field between the radiating disk and a suspended object. In the extrusion flow field, the average air pressure value in one period is larger than the ambient air pressure value, so that levitation force is generated to realize non-contact levitation.
Because the near-field ultrasonic levitation has the advantages of compact structure, easy control, no physical contact and the like, the positioning and transmission system based on the near-field ultrasonic levitation can be applied to levitation and wafer transfer in semiconductor manufacturing. In addition, the extrusion air film bearing based on near field ultrasonic suspension has the advantages of no friction, low maintenance cost, high speed and the like.
For near-field ultrasonic levitation, the flow field distribution generated by vibration is symmetrical in an ideal situation, so that the levitated object can be stabilized in an equilibrium position. This symmetry breaks when the levitated object is eccentric with respect to the radiating disk. This asymmetric flow field distribution produces a restoring force to urge the suspended objects back to an equilibrium position due to the presence of gas viscous forces.
The mechanism and prediction of the generation of the restoring force is critical for applications of near-field acoustic levitation in the field of precision levitation. In the aspect of theoretical research, some scholars propose a spring vibrator model based on the near boundary acoustic flow theory to estimate the stability and restoring force of a suspended object. The surface restoring force has a large relationship with the amplitude distribution of the radiating disc, the mass of the suspended object and the offset position as a result of the study. However, due to its linear assumption, the spring vibrator model can only handle small amplitude oscillations. Other models for studying restoring forces are based on time-averaged local potential energy methods or gas lubrication theory. It is not to be neglected that the deflection and tilting of the suspended mass are symbiotic, so that the model is not accurate enough in either case of analysis alone, especially for larger deflection displacements, the influence of the tilting on the gas film pressure distribution and the magnitude of the restoring force is greater. Up to now, no model is established that takes both cases into consideration at the same time. Therefore, establishing a mathematical model that takes into account the offset and tilt of the levitated object is of great importance for accurately predicting the stability of the levitated system.
Disclosure of Invention
Aiming at the defects and improvement demands of the prior art, the invention provides a restoring force prediction method which simultaneously considers the eccentricity and the inclination of a suspended object. The method aims to effectively predict the restoring force generated by the system when the suspended object deflects, and provide theoretical guidance for parameter optimization for improving the stability of the system and precise motion control of the suspended object.
The invention realizes the above purpose through the following technical scheme:
the restoring force prediction method based on the eccentric and inclined states of the suspended object is applied to predicting the restoring force of an extrusion air film generated in near-field ultrasonic suspension established by an ultrasonic transducer on the suspended object, and comprises the following steps:
s1: obtaining the resonant frequency of an ultrasonic transducer, and carrying out measurement experiments according to the resonant frequency to obtain the amplitude distribution of a radiation disc in the ultrasonic transducer;
s2: constructing a restoring force prediction theoretical model based on the amplitude distribution and the eccentricity of the suspended object in the eccentric and inclined states, and obtaining a dimensionless expression of the extrusion air film under the prediction theoretical model;
s3: constructing a fluid motion control equation in the extrusion gas film based on a gas lubrication theory and a dimensionless expression of the extrusion gas film;
s4: introducing an eight-point discrete method to solve the fluid motion control equation to obtain dimensionless pressure distribution in the extrusion air film, constructing a moment balance equation and a suspension force balance equation when a suspended object is suspended above the extrusion air film based on the pressure distribution, and processing the moment balance equation and the suspension force balance equation by adopting a finite difference method and a spline interpolation method to obtain an inclination angle and a restoring force expression of the suspended object;
s5: and calculating the restoring force of the suspended object under different eccentricities by adopting the restoring force expression, generating a nonlinear relation between the restoring force and the corresponding eccentricity, using the actual restoring force and the corresponding eccentricity, which are tested and measured by the suspended object under different eccentricities, as a comparison group, and comparing the nonlinear relation with the comparison group to obtain the prediction precision.
As a further optimization scheme of the present invention, in step S1, the steps include:
s1.1: measuring the resonant frequency f of the ultrasonic transducer by adopting a precise impedance analyzer;
s1.2: setting an ultrasonic generator to output a sinusoidal voltage signal with the same resonant frequency f to the ultrasonic transducer;
s1.3: and measuring the amplitude distribution V (r) of the radiation disc of the ultrasonic transducer by adopting a scanning laser vibration meter.
As a further optimization scheme of the invention, in the step S2, when the eccentricity of the suspended object relative to the radiation disk is e, the overlapped part of the suspended object and the radiation disk is provided with a squeezing domain omega s The rest is set as a non-extrusion domain omega n In order to obtain an expression solving the intra-domain squeeze film, it is assumed that the suspended object has grooves of a size not to squeeze domain Ω n The dimensionless air film thickness expression is as follows:
wherein r and θ are coordinates of any point in the solution domain, α is an inclination angle of the floating object, h a For the distance between the center of the suspended object and the center of the radiation disk in the vertical direction, T=2pi ft is dimensionless time, h g For suspending the depth of the slot in the object.
As a further optimization scheme of the present invention, in step S3, the equation of fluid motion control in the squeeze film is:
in the formula, σ=12 μ a ωL 2 /p a h a 2 To characterize the number of extrusions, the pressure, μ, generated by the extrusion effect a Is the dynamic viscosity coefficient of air, Λ s ＝6μ a L 2 /p a h a 2 Sum lambda a ＝-ρ s L 2 /2p a Representing the degree of influence of the speed and acceleration of the relative movement of the suspended object on the pressure distribution, p=p/P a ,H＝h/h a r=r/L and u=u/L are the dimensionless gas film pressure, gas film thickness, radial position and relative motion displacement, respectively, wherein ρ s To squeeze the density distribution of the film, p a For the ambient pressure, L is the radius of the radiating disk and the suspended object, p and h represent the actual air pressure and thickness in the squeeze film respectively,and->The speed and acceleration of the relative motion of the levitated object, respectively.
In step S4, based on the discontinuous thickness of the extruded air film caused by the offset of the suspended object, eight-point discrete method is introduced to solve the formula (2), so as to obtain the dimensionless air film pressure P in the extruded air film, the inclination caused by the offset satisfies the moment balance of the air film force distribution relative to the center of the suspended object, and the moment balance equation is:
in addition, the balance of the gravity of the suspended object and the levitation force generated by the squeeze film is expressed as:
wherein m and g are the mass of the suspended object and the local gravitational acceleration, respectively, and the position of the suspended object satisfying the formulas (3) and (4) is defined as a local equilibrium position, and the restoring force generated by the squeeze film is expressed as:
in the formula, the shear stress tau zθ And τ zr Gas flow from the θ and r directions, θ 1 And obtaining the inclination angle and the restoring force of the suspended object by using a spline interpolation method for the angle of the connecting line of the micro-element rdrd theta and the center of the suspended object.
The invention has the beneficial effects that:
(1) In the invention, a theoretical calculation model which simultaneously considers the eccentricity and the inclination of the suspended object is established, and the model is closer to the actual motion condition, so that the method can accurately predict the return force values under different eccentricities.
(2) The prediction method provided by the invention realizes the prediction of restoring force by calculating the actual flow field distribution in the extrusion gas film, so that the influence mechanism of different designs and operation parameters on the stability of the suspension system can be analyzed theoretically, and the prediction method is further used for optimizing the design operation parameters and guiding the actual production.
(3) The prediction method provided by the invention can provide a theoretical basis for realizing flow field distribution control of accurate positioning.
Drawings
FIG. 1 is a flow chart of a near-field ultrasonic levitation restoring force prediction method in the present invention;
FIG. 2 is a schematic diagram of the structure of an ultrasonic transducer of the present invention;
FIG. 3 is a schematic diagram of near-field acoustic levitation in the present invention taking both levitating object eccentricity and tilt into account;
FIG. 4 is a schematic illustration of the structure of an imaginary suspension tray with slots in an embodiment of the invention;
FIG. 5 is a flow chart of the restoring force calculation in the present invention;
fig. 6 is a graph comparing the calculation results with the experimental results of the prediction method proposed in the present invention.
In fig. 2: 1. a radiating plate; 2. a horn; 3. a front cover plate; 4. a piezoelectric ceramic sheet; 5. a back cover plate; 6. an ultrasonic generator; 7. and (5) conducting wires.
Detailed Description
The invention will now be described in further detail with reference to the accompanying drawings, wherein it is to be understood that the following detailed description is for the purpose of illustration only and is not to be construed as limiting the scope of the invention, as various insubstantial modifications and adaptations of the invention to those skilled in the art may be made in light of the foregoing disclosure.
Example 1
As shown in fig. 1, the present application provides a restoring force prediction method based on eccentric and oblique states of a suspended object, where the method is applied to restoring force prediction of an extrusion air film generated in near-field ultrasonic suspension established by an ultrasonic transducer on the suspended object, and includes the following steps:
s1: obtaining the resonant frequency of an ultrasonic transducer, and carrying out measurement experiments according to the resonant frequency to obtain the amplitude distribution of a radiation disc in the ultrasonic transducer;
s2: constructing a restoring force prediction theoretical model based on amplitude distribution and the eccentricity of a suspended object in eccentric and inclined states, and obtaining a dimensionless expression of an extrusion air film under the prediction theoretical model;
s3: based on a gas lubrication theory and a dimensionless expression of the extrusion gas film, constructing a fluid motion control equation in the extrusion gas film;
s4: introducing an eight-point discrete method to solve a fluid motion control equation to obtain dimensionless pressure distribution in an extrusion air film, constructing a moment balance equation and a suspension force balance equation when a suspended object is suspended above the extrusion air film based on the pressure distribution, and processing the moment balance equation and the suspension force balance equation by adopting a finite difference method and a spline interpolation method to obtain an inclination angle and a restoring force expression of the suspended object;
s5: and calculating the restoring forces of a plurality of groups of different eccentricities by adopting a restoring force expression, generating nonlinear relations between the plurality of groups of restoring forces and corresponding eccentricities, using the actual restoring forces and the corresponding eccentricities which are tested under the different eccentricities as a comparison group, and comparing the nonlinear relations with the comparison group to obtain the prediction precision.
Further, in step S1, the steps include:
s1.1: measuring the resonant frequency f of the ultrasonic transducer by adopting a precise impedance analyzer;
s1.2: setting an ultrasonic generator to output a sine voltage signal with the same resonant frequency f to an ultrasonic transducer;
s1.3: the amplitude distribution V (r) of the radiating disk of the ultrasonic transducer is measured using a scanning laser vibrometer.
In this embodiment, as shown in fig. 2, the ultrasonic transducer mainly includes a piezoelectric ceramic plate 4, a back cover plate 5, a front cover plate 3, a horn 2, and a radiation disk 1. The resonant frequency f of the ultrasonic transducer is measured using a precision impedance analyzer. The ultrasonic generator 6 is arranged to output a sine voltage signal with the same frequency f, the sine voltage signal is applied to the piezoelectric ceramic plate 4 through a lead 7, the piezoelectric ceramic plate 4 generates vibration with the same frequency due to the inverse piezoelectric effect, the amplitude of the vibration is amplified by the amplitude transformer 2 and is transmitted to the radiation disc 1, and the amplitude distribution V (r) of the radiation disc is measured by the scanning laser vibration meter.
Further, as shown in FIG. 3, in step S2, the levitated object is levitated freely on the radiation tray 1, centers of the radiation tray 1 and the levitated object are denoted by O and O, respectively r When the offset distance of the levitated object relative to the radiation tray 1 is e, the levitated object is tilted by an angle α, and since the angle of tilt is very small, the projected area of the levitated object on the radiation tray is approximately equal to the area of the levitated object. Since the area of the radiating disk serves as a solving domain, it can be divided into a squeeze domain Ω s (portion where the levitated object and the radiating disk overlap) and non-extrusion domain Ω n (the remainder); to allow the film thickness to be expressed across the solution domain, a suspended object with grooves as shown in FIG. 4 was introduced. The size of the groove is the same as the size of the non-extrusion domain, and the depth of the groove is defined as h g . Then, the non-dimensional expression of the squeeze film H is:
wherein r and θ are coordinates of any point in the solution domain, α is an inclination angle of the floating object, h a For the distance between the center of the suspended object and the center of the radiation disk in the vertical direction, T=2pi ft is dimensionless time, h g For suspending the depth of the slot in the object.
In the present embodiment, since the inclination of the suspended object is considered, the squeeze film H satisfies at a minimum value near the center OWhere r=r/L is the dimensionless radial position and L is the radius of the radiating disc and the suspended object. Therefore, in this position the squeezing effect is most pronounced, with the air pressure variation satisfying +.>Wherein p=p/P a Is the pressure of a dimensionless air film, p a Is at ambient atmospheric pressure. Further, as shown in fig. 3, at the boundary of the squeeze film, the air pressure value should satisfy P (r=l) =1.
Further, based on the gas lubrication theory, in step S3, when the suspended object is considered to generate motion due to the restoring force, the fluid motion control equation in the squeeze film is as follows:
in the formula, σ=12 μ a ωL 2 /p a h a 2 To characterize the number of extrusions, the pressure, μ, generated by the extrusion effect a Is the dynamic viscosity coefficient of air, Λ s ＝6μ a L 2 /p a h a 2 Sum lambda a ＝-ρ s L 2 /2p a Representing the degree of influence of the velocity and acceleration of the suspended object movement on the pressure distribution, p=p/P a ,H＝h/h a r=r/L and u=u/L are each a dimensionless gasFilm pressure, film thickness, radial position and relative motion displacement, where ρ s To squeeze the density distribution of the film, p a For the ambient pressure, L is the radius of the radiating disk and the suspended object, p and h represent the actual air pressure and thickness in the squeeze film respectively,and->The speed and acceleration of the relative motion of the levitated object, respectively.
Further, in step S4, based on the discontinuous thickness of the extruded air film caused by the offset of the suspended object, an eight-point discrete method and a newton-lavson method are introduced to solve the formula (2), so as to obtain the dimensionless air film pressure P in the extruded air film, the inclination caused by the offset satisfies the moment balance of the air film force distribution relative to the center of the suspended object, and the moment balance equation is:
in addition, the balance of the gravity of the suspended object and the levitation force generated by the squeeze film is expressed as:
wherein m and g are the mass of the suspended object and the local gravitational acceleration, respectively, and the position of the suspended object satisfying the formulas (3) and (4) is defined as a local equilibrium position, and the restoring force generated by the squeeze film is expressed as:
in the formula, the shear stress tau zθ And τ zr Gas flow from squeeze film θ and r directions, θ 1 Is connected with the center of the suspended object by the infinitesimal rdrdrd thetaAngle of line.
Since the mass of the suspended object is measured by a precision balance instrument, the magnitude of the suspended force is known, and the suspended height and the inclination angle are unknown, the magnitude of the restoring force of the suspended object under the inclination condition can be obtained by using a two-time spline interpolation method as shown in fig. 5.
By varying the eccentricity e, a mathematical relationship between the restoring force and the eccentric distance can be established, and as shown in fig. 6, a nonlinear increase in restoring force can be found with an increase in eccentricity. When the eccentricity is smaller, the calculated result of the prediction method provided by the invention is relatively close to the calculated result without considering the inclination because the inclination angle is smaller at the moment. When the eccentricity is larger, the inclination angle is larger, and compared with the calculation result without considering inclination, the calculation result of the prediction method provided by the invention is more approximate to the experimental result. The closeness of the calculation result to the experimental result is described by R-square (determination coefficient), the value of which is approximately close to 1, which means that the higher the closeness is. The R-square value of the calculation result and the experimental result proposed by the invention is 0.9374, and the R-square value of the calculation result and the experimental result under no consideration of inclination is 0.8769. Therefore, the method realizes the prediction of restoring force by calculating the actual flow field distribution in the extrusion gas film, so that the influence mechanism of different designs and operation parameters on the stability of the suspension system can be analyzed theoretically, and the method is further used for optimizing the design operation parameters and guiding the actual production. In addition, the method can provide a theoretical basis for realizing flow field distribution control of accurate positioning.
The above embodiments are only for illustrating the technical solution of the present application, and are not limiting thereof; although the present application has been described in detail with reference to the foregoing embodiments, one of ordinary skill in the art will appreciate that: the technical scheme described in the foregoing embodiments can be modified or some technical features thereof can be replaced by equivalents; such modifications and substitutions do not drive the essence of the corresponding technical solutions to depart from the spirit and scope of the technical solutions of the embodiments of the present application.
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