Milling chatter monitoring method based on voltage signal of ultrasonic vibration auxiliary milling device

Abstract

The invention discloses a milling chatter monitoring method based on a voltage signal of an ultrasonic vibration auxiliary milling device, and the method comprises the steps: building a proportional relation between the voltage signal of the device and an external milling load through building a kinetic model of the ultrasonic vibration auxiliary milling device; designing an acquisition circuit to extract a voltage signal of the ultrasonic vibration auxiliary milling device; a frequency domain method and a frequency elimination algorithm are used for carrying out noise reduction processing on signals, frequency band energy comparison is adopted for representing flutter characteristics in voltage signals of the device, and ultrasonic vibration auxiliary milling flutter monitoring is achieved under the condition that a sensor is not used.

Description

Technical Field

[0001] The present invention relates to the field of ultrasonic vibration assisted machining, and particularly to a milling chatter monitoring method based on the voltage signal of an ultrasonic vibration assisted milling device. Background Art

[0002] Complex thin-walled parts are widely used in the fields of aerospace, energy, precision instruments, automotive manufacturing, etc. due to their light weight and compact structure. For example, non-general thin-walled parts such as integral frames, integral panels, turbine disks, and turbine blades of aero-engines are mostly processed by milling. In the milling of thin-walled parts, chatter is easily caused, resulting in a decrease in the accuracy and surface quality of the machined workpiece. The chatter phenomenon is a self-excited vibration between the cutting tool and the workpiece. This kind of bad vibration will cause serious damage to the workpiece surface. In order to reduce chatter, high-stability cutting parameters are usually pre-selected. However, this conservative method often leads to a reduction in machining efficiency. Therefore, it is necessary to study chatter monitoring technology to identify chatter during the machining process. Summary of the Invention

[0003] The technical problem to be solved by the present invention is to provide a milling chatter monitoring method based on the voltage signal of an ultrasonic vibration assisted milling device to monitor it only through the voltage signal of the ultrasonic vibration assisted milling device without relying on other sensors in view of the defects involved in the background art.

[0004] The present invention adopts the following technical solutions to solve the above technical problems:

[0005] A milling chatter monitoring method based on the voltage signal of an ultrasonic vibration assisted milling device includes the following steps:

[0006] Step 1), establish a first-order dynamic model of the ultrasonic vibration assisted milling device, and characterize the load of the ultrasonic vibration assisted milling device according to the first-order dynamic model of the ultrasonic vibration assisted milling device;

[0007] Step 2), use the Mason equivalent circuit to transform the first-order dynamic model of the ultrasonic vibration assisted milling device into an electrical model, and obtain the proportional change relationship between the voltage signal of the ultrasonic vibration assisted milling device and the milling chatter characteristics;

[0008] Step 3), use an ultrasonic power supply to excite the ultrasonic vibration assisted milling device, and use an acquisition circuit to acquire the voltage signal of the ultrasonic vibration assisted milling device;

[0009] Step 4), process the acquired voltage signal by the short-time Fourier transform method to obtain the spectrum of the frequency domain information of the voltage signal, and use the frequency cancellation algorithm to denoise the spectrum of the frequency domain information of the voltage signal;

[0010] Step 5), based on the positive proportional change relationship between the voltage signal and the milling chatter characteristics, the frequency information spectrum of the noise-reduced voltage signal is extracted by using the band energy ratio to obtain the milling chatter characteristics.

[0011] As a further optimization scheme of the milling chatter monitoring method based on the voltage signal of the ultrasonic vibration-assisted milling device of the present invention, in step 1), the load of the ultrasonic vibration-assisted milling device is characterized according to the following formula:

[0012]

[0013] In the formula, F t (t) is the milling force suffered by the ultrasonic vibration-assisted milling device when it is unloaded, M system is the system modal mass of the first-order dynamic model of the ultrasonic vibration-assisted milling device, b system is the system modal damping of the first-order dynamic model of the ultrasonic vibration-assisted milling device, k system is the system stiffness damping of the first-order dynamic model of the ultrasonic vibration-assisted milling device, and x2 is the vibration displacement of the first-order dynamic model of the ultrasonic vibration-assisted milling device.

[0014] As a further optimization scheme of the milling chatter monitoring method based on the voltage signal of the ultrasonic vibration-assisted milling device of the present invention, step 2) specifically includes the following steps:

[0015] Step 2.1), the equivalent circuit model of the ultrasonic vibration-assisted milling device includes a static capacitor C0, a static resistor R0, a dynamic inductor L1, a dynamic capacitor C1 and a dynamic resistor. One end of C0 is electrically connected to one end of R0 and one end of L1 respectively, and the other end is electrically connected to the other end of R0 and one end of R1 respectively; one end of C1 is electrically connected to the other end of L1, and the other end is electrically connected to the other end of R1; both ends of C0 are connected to the voltage U of the ultrasonic vibration-assisted milling device t ;

[0016] In the case of no load, the voltage equation expression of the equivalent circuit model of the ultrasonic vibration-assisted milling device is:

[0017]

[0018] In the formula, U tis the voltage of the equivalent circuit model of the ultrasonic vibration assisted milling device under no-load condition, L1 is the dynamic inductance of the equivalent circuit model of the ultrasonic vibration assisted milling device, R1 is the dynamic resistance of the equivalent circuit model of the ultrasonic vibration assisted milling device, C1 is the dynamic capacitance of the equivalent circuit model of the ultrasonic vibration assisted milling device, and I is the current of the equivalent circuit model of the ultrasonic vibration assisted milling device;

[0019] Step 2.2), under the condition of having a load, the load characterization of the ultrasonic vibration assisted milling device and the voltage equation expression of the equivalent circuit model are changed as follows:

[0020]

[0021] In the formula, F ext is the load external force received by the ultrasonic vibration assisted milling device, U2 is the load voltage of the equivalent electrical model of the ultrasonic vibration assisted milling device, and their relationship is expressed as U t ∝F ext .

[0022] As a further optimization scheme of the milling chatter monitoring method based on the voltage signal of the ultrasonic vibration assisted milling device of the present invention, the specific steps of step 3) are as follows:

[0023] The voltammetry method is used to collect the voltage signal of the ultrasonic vibration assisted milling device; the collection circuit includes two parallel branches, one of which is composed of an ultrasonic vibration assisted milling device and a current collection resistor R I connected in series, and the other branch is composed of a voltage collection resistor R U and a protection resistor R k connected in series, and both ends of the collection circuit are connected to the ultrasonic power supply;

[0024] The voltage at both ends of the ultrasonic vibration assisted milling device In the formula, U m is the voltage at both ends of the voltage collection resistor R U .

[0025] As a further optimization scheme of the milling chatter monitoring method based on the voltage signal of the ultrasonic vibration assisted milling device of the present invention, step 4) includes the following specific steps

[0026] Step 4.1), the voltage signal of the ultrasonic vibration assisted milling device collected is a discrete time-domain signal, denoted as {x(n), n = 1, 2,..., N}, and the fast Fourier transform is performed on the voltage signal to obtain the spectrum of the voltage signal In the formula, is the complex unit root, and j is the imaginary unit;

[0027] Step 4.2), determine the harmonic frequencies to be removed, i.e., the notch frequency set F re ; the notch frequency set F re includes the tooth passing frequency set F resp and the alternating current frequency set F reAC ; the expression of the tooth passing frequency set F resp is F resp (k1) = {f sp , 2f sp ,..., k1f sp}}, the expression of the alternating current frequency set F reAC is F reAC (k2) = {f AC , 2f AC ,..., k2f AC}}, where k1 and k2 are the preset first threshold and second threshold respectively; f sp is the spindle fundamental frequency, Ω is the spindle speed of the machine tool; f AC is the alternating current fundamental frequency, f AC = 50Hz;

[0028] Step 4.3), assuming the notch frequency set F re has k elements, set the transfer function where the frequency resolution f S is the sampling rate, t sample is the sampling time;

[0029] Step 4.4), after filtering the harmonic noise frequencies, obtain the noise-reduced spectrum according to the following formula

[0030] Compared with the prior art, the present invention adopts the above technical solutions and has the following technical effects:

[0031] The present invention designs a milling chatter monitoring method based on the voltage signal of the ultrasonic vibration-assisted milling device, establishes a dynamic model of the ultrasonic vibration-assisted milling device, reveals the relationship between the device voltage signal and the milling load, designs an acquisition circuit for collecting the voltage signal of the ultrasonic vibration-assisted milling device, constructs a signal processing method for the device voltage signal on this basis, and extracts the chatter characteristics of the device voltage signal by using the band energy ratio, which can accurately reflect the chatter phenomenon in milling. Description of the Drawings

[0032] Figure 1 is the flowchart of the chatter monitoring method based on the voltage signal of the ultrasonic vibration-assisted milling device;

[0033] Figure 2 is the schematic diagram of the ultrasonic vibration device;

[0034] Figure 3 is the first-order dynamic model of the device;

[0035] Figure 4 is the equivalent circuit of the ultrasonic vibration device;

[0036] Figure 5 is the principle of voltage signal acquisition of the ultrasonic vibration device;

[0037] Figure 6 is the schematic diagram of the acquisition circuit;

[0038] Figure 7 is the extraction process of the flutter characteristics of the device voltage signal. Specific implementation manners

[0039] The technical solutions of the present invention will be further described in detail below with reference to the accompanying drawings:

[0040] The present invention can be implemented in many different forms and should not be considered limited to the embodiments described herein. On the contrary, these embodiments are provided so that this disclosure will be thorough and complete, and will fully convey the scope of the present invention to those skilled in the art. In the drawings, components are enlarged for clarity.

[0041] The present invention and its implementation manners have been described above, and this description is not restrictive. The actual implementation manners are not limited thereto. Generally speaking, if those of ordinary skill in the art are inspired by it and design similar structural manners and embodiments without creative efforts without departing from the gist of the present invention, they should fall within the protection scope of the present invention.

[0042] Figure 1 is the flowchart of the flutter monitoring method based on the voltage signal of the ultrasonic vibration assisted milling device, which specifically includes the following steps:

[0043] Step 1), establish the first-order dynamic model of the ultrasonic vibration assisted milling device, and characterize the load of the ultrasonic device according to the dynamic model of the ultrasonic vibration device;

[0044] The transducer structure of the ultrasonic vibration device is as Figure 2 shown. The device includes a rear cover plate, a piezoelectric sheet, a flange plate, a horn, a compression cap, and a milling cutter from back to front. The milling cutter is connected to the horn through a chuck and a compression cap; the horn and the flange plate are an integral structure and are fixed on the tool shank through the flange plate; the piezoelectric transducer is formed by connecting the piezoelectric sheet and the rear cover plate with bolts. The presence of the piezoelectric ceramic sheet makes the entire device form a complex electro-mechanical-displacement system, and the dynamic modeling analysis of the ultrasonic device is carried out.

[0045] During machining, the axial milling force is transmitted to the piezoelectric ceramic through the tool and the horn. Since the horn has the effect of amplifying displacement, the model is divided at the flange, and the device is divided into the transducer end and the tool end, corresponding to the two mass blocks of the second-order equivalent mass-spring-damping model. Since the piezoelectric transducer is fixed through the flange, its corresponding mass block model M t can be regarded as fixed support; the transducer is excited by alternating current to generate an exciting force F t , and the displacement generated by its corresponding mass block is x1; the modal damping and modal stiffness of the transducer are denoted as b t and k t . Similarly, the modal mass, modal damping, and modal stiffness of the tool end are expressed as M d , b d and k d , the bottom of the tool is free, and the displacement generated at the tip of the tool is x2. For milling machining, the axial feed speed of the tool during the machining process is relatively low compared with the vibration speed and can be ignored. Only the axial vibration of the device during milling is considered. Therefore, the displacements x1 and x2 are both periodic functions. Assuming F ext is the axial milling force received by the milling cutter during the milling process, the dynamic expression of the displacement x1 of the transducer can be obtained according to the established dynamic model as follows:

[0046] Similarly, the dynamic expression of the milling cutter tip x2 is:

[0047] The displacement of the tool end is N times the displacement of the piezoelectric ceramic sheet, expressed as:

[0048] Combining the above three formulas, we get: Then the dynamic model of the ultrasonic device can be simplified to a first-order MSD system, as Figure 3 shown. The modal mass of the system can be obtained as M system = M d + M t / N, the modal stiffness is k system = k t / N, the modal damping is b system = b t / N, and the overall force of the system is F system = F t - F ext .

[0049] Step 2), use the Mason equivalent circuit to transform the dynamic model of the device into an electrical model, and point out the relationship between the voltage signal of the device and the milling load;

[0050] According to the first-order dynamic model of the simplified transducer, when the transducer is not subject to external loads, the load of the ultrasonic vibration-assisted milling device is characterized by the following formula:

[0051]

[0052] According to Mason's rule, the equivalent circuit model of the ultrasonic transducer can be expressed as Figure 4 the circuit shown. The equivalent circuit model can relate mechanical components such as mass, spring, and damping in the dynamic model to electrical components such as inductance, capacitance, and resistance, and qualitatively describe the relationship between the milling force and the electrical signal of the device. The equivalent circuit is divided into two parts: the electrical end and the mechanical end. The electrical end consists of the static capacitance C0 and the static resistance R0, which remain constant when the structure of the ultrasonic transducer remains unchanged and can be regarded as constants. The mechanical end consists of the dynamic inductance L1, the dynamic capacitance C1, and the dynamic resistance R1, which reflect the dynamic characteristics of the device. The LC oscillation circuit formed by the dynamic inductance and the dynamic capacitance exhibits dynamic characteristics similar to those of the mechanical vibration system, and the resistance R1 reflects the mechanical dissipation of the transducer itself. The voltage equation of the mechanical end under no-load conditions is expressed as:

[0053]

[0054] There is a corresponding relationship between the mechanical characteristics of the ultrasonic transducer and the electrical parameters of each part of the mechanical end of the equivalent circuit, satisfying: L1∝M system , C1∝1 / k system , R1∝b system , U t ∝F system ,

[0055] During the milling process, the device changes from the no-load state to the load state and is subjected to an external load F ext acting. The vibration equation of the device under load is expressed as

[0056] According to electromechanical analogy, the voltage equation of the mechanical end of the equivalent circuit in the electrical model of the device also changes accordingly, expressed as:

[0057] Then

[0058]

[0059] The external force acting on the device during processing is positively correlated with the voltage change in its equivalent circuit. Observing the voltage change in the circuit can reflect the change in the milling force during the processing, and thus monitor the chatter of the workpiece.

[0060] Step 3), use the ultrasonic power supply to excite the ultrasonic vibration assisted milling device, and use the acquisition circuit to collect the voltage signal of the ultrasonic vibration assisted milling device; specifically, it includes the following steps:

[0061] As Figure 5 described, the signal acquisition system of the ultrasonic vibration device includes: machine tool spindle, ultrasonic device, acquisition circuit, ultrasonic power supply, data acquisition card, upper computer, workpiece, fixture and workbench. During processing, the ultrasonic power supply outputs a high-frequency voltage for driving the device. On the one hand, the acquisition circuit inputs the voltage output by the power supply to the device to drive the device, and on the other hand, inputs the voltage and current signals of the device into the data acquisition card. The data acquisition card transmits the collected signals to the computer, and finally processes the signals in the computer.

[0062] As Figure 6 shown, the voltammetry method is used to collect the device signals. The principle of voltammetry for measuring voltage and current is based on Ohm's law, that is, the relationship between resistance, voltage and current can be expressed as R = U / I. In the circuit, a small-value current sampling resistor R I is connected in series with the transducer to measure the current. The current flowing through the transducer is I = U I / R I . The voltage across the transducer is measured by connecting a large resistor in parallel with the transducer. To ensure the safety of the acquisition board, a large resistor R k and a small resistor R U are connected in series and then connected in parallel across the transducer, and the voltage across the small resistor is collected. Then the voltage across the ultrasonic transducer is U t =(R k +R U )·U m / R U .

[0063] Step 4), process the voltage signal by the frequency domain method to obtain the frequency domain information of the voltage signal, and use the frequency cancellation algorithm to denoise the device voltage signal; specifically, it includes the following steps:

[0064] The voltage signal of the device collected by the digital acquisition card is a discrete time-domain signal, which can be recorded as {x(n), n = 1, 2,..., N}. Perform a fast Fourier transform on the signal to obtain the spectrum of this section of the signal In the formula, is the complex unit root, and j is the imaginary unit.

[0065] After obtaining the spectrum of the signal, it is necessary to determine the fundamental frequency of the harmonic frequency, that is, the spindle fundamental frequency f sp and the alternating current fundamental frequency f AC .

[0066] The spindle fundamental frequency f spRelated to the spindle speed, The fundamental frequency of the alternating current is f AC = 50Hz.

[0067] The notch frequency set F re is the harmonic frequency to be removed, including the tooth passing frequency set F resp and the alternating current frequency set F reAC ; among them, F resp (k1) = {f sp , 2f sp ,..., k1f sp}, F reAC (k2) = {f AC , 2f AC ,..., k2f AC}, k1 and k2 are the preset first threshold and second threshold respectively.

[0068] To facilitate the setting of k1 and k2, a cut-off frequency f max can be set, and make

[0069] Introduce a very narrow stopband to reduce the error caused by the frequency resolution. The actual frequency difference between adjacent scales in the frequency domain reflects the frequency resolution of the signal frequency components, and the expression is In the formula, df is the frequency resolution, f S is the sampling rate, and t sample is the sampling time.

[0070] Set the half bandwidth of the stopband of the spectrum to df. If the notch frequency set F re has k elements, then the stopband frequency range of each element in the notch frequency set is [F re (k) - df, F re (k) + df], and set the transfer function In the passband of the spectrum, the gain of the transfer function is 1, and the frequencies in the passband are allowed to pass; the gain of the stopband is 0, and the harmonic frequencies in the stopband are removed.

[0071] After filtering out the harmonic noise frequencies, the noise-reduced spectrum is obtained according to the following formula

[0072] Step 5), adopt the frequency band energy ratio to extract the flutter characteristics of the voltage signal processed by the frequency domain method.

[0073] For a section of spectrum signal, the ratio of the energy in a certain frequency band to the entire frequency band is usually called the frequency band energy ratio (BER), and its formula is: In the formula, f down and f upare the lower and upper boundaries of the signal frequency band, and f1 and f2 are the lower and upper boundaries of the target frequency band, respectively.

[0074] When extracting the BER, it is necessary to determine the center and bandwidth of the frequency band. When chatter occurs in thin-walled parts, the chatter frequency of the workpiece will approach the natural frequency of the workpiece. Therefore, the first two natural frequencies of the workpiece are used as the energy centers of the frequency band, and the bandwidth of each frequency band accounts for 15% of the bandwidth of the analyzed signal.

[0075] To ensure the real-time monitoring, the data processing flow of chatter monitoring is as Figure 7 shown.

[0076] Those skilled in the art of this technology can understand that, unless otherwise defined, all terms (including technical terms and scientific terms) used here have the same meaning as the general understanding of those of ordinary skill in the art to which this invention belongs. It should also be understood that terms defined in general dictionaries should be understood to have meanings consistent with the meanings in the context of the prior art, and will not be interpreted with idealized or overly formal meanings unless defined as here.

[0077] The specific embodiments described above have further elaborated on the purpose, technical solutions, and beneficial effects of the present invention. It should be understood that the above description is only the specific embodiments of the present invention and is not used to limit the present invention. Any modifications, equivalent replacements, improvements, etc. made within the spirit and principles of the present invention shall be included within the protection scope of the present invention.

Claims

1. A milling chatter monitoring method based on the voltage signal of an ultrasonic vibration-assisted milling device, characterized in that It includes the following steps: Step 1), establish a first-order dynamic model of the ultrasonic vibration-assisted milling device, and characterize the load of the ultrasonic vibration-assisted milling device according to the first-order dynamic model of the ultrasonic vibration-assisted milling device; Step 2), use the Mason equivalent circuit to transform the first-order dynamic model of the ultrasonic vibration-assisted milling device into an electrical model, and obtain the proportional change relationship between the voltage signal of the ultrasonic vibration-assisted milling device and the milling chatter characteristics; Step 3), use an ultrasonic power supply to excite the ultrasonic vibration-assisted milling device, and use an acquisition circuit to collect the voltage signal of the ultrasonic vibration-assisted milling device; Step 4), process the collected voltage signal by the short-time Fourier transform method to obtain the spectrum of the frequency-domain information of the voltage signal, and use a frequency elimination algorithm to denoise the spectrum of the frequency-domain information of the voltage signal; Step 5), based on the proportional change relationship between the voltage signal and the milling chatter characteristics, extract the milling chatter characteristics from the spectrum of the frequency information of the denoised voltage signal by using the band energy ratio, and obtain the chatter characteristics of the milling.

2. The milling chatter monitoring method based on the voltage signal of the ultrasonic vibration assisted milling device according to claim 1, characterized in that, In Step 1), the load of the ultrasonic vibration-assisted milling device is characterized according to the following formula: In the formula, F t (t) is the milling force received by the ultrasonic vibration assisted milling device during no-load operation, M system is the system modal mass of the first-order dynamic model of the ultrasonic vibration assisted milling device, b system is the system modal damping of the first-order dynamic model of the ultrasonic vibration assisted milling device, k system is the system stiffness damping of the first-order dynamic model of the ultrasonic vibration assisted milling device, and x2 is the vibration displacement of the first-order dynamic model of the ultrasonic vibration assisted milling device.

3. The milling chatter monitoring method based on the voltage signal of the ultrasonic vibration-assisted milling device according to claim 2, wherein, The specific steps of Step 2) include the following steps: Step 2.1), the equivalent circuit model of the ultrasonic vibration assisted milling device includes a static capacitance C0, a static resistance R0, a dynamic inductance L1, a dynamic capacitance C1 and a dynamic resistance. One end of C0 is electrically connected to one end of R0 and one end of L1 respectively, and the other end is electrically connected to the other end of R0 and one end of R1 respectively; One end of C1 is electrically connected to the other end of L1, and the other end is electrically connected to the other end of R1; Both ends of C0 are connected to the voltage U of the ultrasonic vibration assisted milling device t ; In the case of no load, the voltage equation expression of the equivalent circuit model of the ultrasonic vibration-assisted milling device is: where U t is the voltage of the equivalent circuit model of the ultrasonic vibration assisted milling device under no-load condition, L1 is the dynamic inductance of the equivalent circuit model of the ultrasonic vibration assisted milling device, R1 is the dynamic resistance of the equivalent circuit model of the ultrasonic vibration assisted milling device, C1 is the dynamic capacitance of the equivalent circuit model of the ultrasonic vibration assisted milling device, and I is the current of the equivalent circuit model of the ultrasonic vibration assisted milling device; Step 2.2), in the case of having a load, the load characterization of the ultrasonic vibration-assisted milling device and the voltage equation expression of the equivalent circuit model are changed as follows: Where, F ext is the external load force on the ultrasonic vibration assisted milling device, and U2 is the load voltage of the equivalent electrical model of the ultrasonic vibration assisted milling device. The relationship between the two is expressed as U t ∝F ext .

4. The milling chatter monitoring method based on the voltage signal of the ultrasonic vibration assisted milling device according to claim 3, characterized in that, The specific steps of Step 3) are as follows: The voltammetry method is used to collect the voltage signal of the ultrasonic vibration assisted milling device; the acquisition circuit includes two parallel branches, one of which consists of an ultrasonic vibration assisted milling device and a current acquisition resistor R I connected in series, and the other branch consists of a voltage acquisition resistor R U and a protection resistor R k connected in series, and both ends of the acquisition circuit are connected to the ultrasonic power supply; Voltage at both ends of the ultrasonic vibration assisted milling device In the formula, U m is the voltage across the voltage acquisition resistor R U at both ends.

5. The milling chatter monitoring method for the voltage signal of the ultrasonic vibration assisted milling device according to claim 4, characterized in that, Step 4) includes the following specific steps, Step 4.1), the voltage signal of the ultrasonic vibration assisted milling device collected is a discrete time domain signal, denoted as {x(n), n = 1, 2,..., N}, and the fast Fourier transform is performed on the voltage signal to obtain the frequency spectrum of the voltage signal wherein is the complex unit root and j is the imaginary unit; Step 4.2), determine the harmonic frequencies to be removed, i.e., the notch frequency set F re ; the notch frequency set F re includes the tooth passing frequency set F resp and the alternating current frequency set F reAC ; the expression of the tooth passing frequency set F resp is F resp (k1) = {f sp , 2f sp ,..., k1f sp}}, the expression of the alternating current frequency set F reAC is F reAC (k2) = {f AC , 2f AC ,..., k2f AC}}, where k1 and k2 are the preset first threshold and second threshold respectively; f sp is the spindle fundamental frequency, Ω is the spindle speed of the machine tool; f AC is the alternating current fundamental frequency, f AC = 50Hz; Step 4.3), let the notch frequency set F re have k elements, and set the transfer function where the frequency resolution f S is the sampling rate, and t sample is the sampling time; Step 4.4), after filtering out the harmonic noise frequency, obtain the noise-reduced spectrum according to the following formula

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