Single-stage gear transmission parametrically-excited vibration response frequency spectrum prediction method based on modulation feedback

Abstract

The invention discloses a single-stage gear parametrically-excited vibration response frequency spectrum prediction method based on modulation feedback, and the method comprises the steps: building a single-degree-of-freedom kinetic equation containing time-varying meshing stiffness based on the Newton second law, and solving a main oscillation frequency through employing a modulation feedback theory and a trigonometric series approximation method; constructing a quantitative prediction model of excitation parameters and dominant side frequency orders on the basis of a first-class Bessel function, and solving system response by applying a four-order Runge-Kutta method in combination with fast Fourier transformation to verify the prediction model; a parameter sensitivity analysis framework is established through a harmonic coefficient method, and the influence of excitation parameters on the main oscillation frequency and the side frequency amplitude ratio is quantified.

Description

Technical Field

[0001] The present invention relates to the field of vibration of a single-stage gear structure, and in particular to a method for predicting a parametrically excited vibration response spectrum of a single-stage gear based on modulation feedback. Background Art

[0002] Single-stage gear structure is the core component of mechanical transmission system, which is widely used in automobile transmission, wind power gearbox, industrial reducer and aerospace power unit. [1] .

[0003] It is well known that during the meshing process of a gear pair, the mesh stiffness exhibits periodic variations due to alternating meshing and elastic deformation of the teeth, resulting in time-varying mesh stiffness. For an ideal error-free gear system, its dynamic characteristics are primarily determined by the mean and time-varying components of the mesh stiffness. This periodic variation in mesh stiffness induces parametrically excited vibrations, leading to a complex phenomenon of frequency splitting in the system's dynamic response, generating a series of sidebands near the primary meshing frequency. Frequency splitting significantly alters the system's vibration energy distribution, diffusing energy originally concentrated at the meshing frequency into the sidebands. This exacerbates vibration and noise, and may even induce resonant instability, impacting transmission accuracy and service life. Furthermore, the distribution characteristics of the sidebands are closely related to faults such as tooth root cracks and tooth spalling. If the frequency splitting patterns are unclear, fault feature extraction will be interfered with, reducing the accuracy of condition monitoring. Therefore, studying the parametrically excited vibration characteristics of gear systems excited by time-varying mesh stiffness, revealing the mechanism of frequency splitting, and proposing effective suppression methods are crucial for improving the system's dynamic performance and reliability.

[0004] Based on this, references [2] and [3] combined the idea of ​​harmonic balance with the inherent characteristics of parametric response, which can not only efficiently predict the frequency components of the system response, but also effectively determine the parameter instability domain, providing new ideas for parametric vibration analysis and prediction.

[0005] It should be pointed out that in the existing literature, most scholars study the instability domain and frequency component of parametrically excited vibration systems, and few scholars study the influence of time-varying parameter excitation on the energy distribution in the response spectrum of single-stage gears.

[0006] References

[0007] [1]Kahraman A,Singh R.Non-linear dynamics of a spur gear pair[J].Journal ofSound and Vibration,1990,142(1):49-75.

[0008] [2]Huang D, Hong L, Liu C. Computational technique to free vibration response in amulti-degree of freedom parametric system[J]. Mechanical Systems and Signal Processing, 2020,142:106777.

[0009] [3] Huang Dishan, Parametric Oscillation and Special Trigonometric Series Approximation[M]. Shanghai: Shanghai Science and Technology Press, 2023. Summary of the Invention

[0010] The present invention provides a method for predicting the parametrically excited vibration response spectrum of a single-stage gear based on modulation feedback. This method establishes a single-degree-of-freedom dynamic model that takes into account time-varying meshing stiffness. Using modulation feedback theory, the time-varying stiffness parameter excitation is converted into an equivalent feedback link. First-order Bessel functions are introduced to construct a quantitative relationship model between the excitation parameters and the sideband order. The fourth-order Runge-Kutta method is then used to solve the response, thereby revealing the energy distribution pattern of the parametrically excited vibration sidebands. Finally, a parameter sensitivity analysis framework is established using the harmonic coefficient method. The influence mechanism of the excitation parameters on the main frequency to sideband amplitude ratio is verified through theoretical derivation and numerical simulation. This method is expected to provide a quantitative basis for the dynamic design of gear systems. Details are described below.

[0011] A method for predicting the parametrically excited vibration response spectrum of a single-stage gear based on modulation feedback, the method comprising:

[0012] Based on Newton's second law, a single-degree-of-freedom dynamic equation with time-varying meshing stiffness is established, and the main oscillation frequency is solved using modulation feedback theory and trigonometric series approximation method.

[0013] A quantitative prediction model for the excitation parameters and the order of the dominant sidebands is constructed based on the first-kind Bessel function. The fourth-order Runge-Kutta method combined with the fast Fourier transform is used to solve the system response to verify the prediction model.

[0014] A parameter sensitivity analysis framework is established through the harmonic coefficient method to quantify the influence of excitation parameters on the main oscillation frequency and sideband amplitude ratio.

[0015] Wherein, the kinetic equation is:

[0016]

[0017] Where x is the generalized coordinate in the form of relative displacement, is the second-order derivative of the generalized coordinate with respect to time, m and k m(t) are the equivalent mass and time-varying meshing stiffness respectively, and:

[0018]

[0019] Where R is the base circle radius, I is the moment of inertia, and θ is the torsional angular displacement.

[0020] The time-varying meshing stiffness is expressed as:

[0021] k m =k0+A l cos(ω m t)

[0022] Where k0 is the average meshing stiffness, A l is the stiffness fluctuation excitation amplitude, ω m is the meshing frequency.

[0023] Among them, the quantitative prediction model of excitation parameters and dominant sideband order based on the first-kind Bessel function is:

[0024] k=round(κ-1)

[0025] Where k is the sideband order, κ=βω n / 2ω m is the sideband modulation coefficient.

[0026] Among them, the fourth-order Runge-Kutta method combined with fast Fourier transform is used to solve the system response:

[0027] The differential equation is solved by the fourth-order Runge-Kutta method to obtain the system response x(t) in the time domain. The time domain signal x(t) is transformed into a complex spectrum in the frequency domain by fast Fourier transform. The modulus value is taken to obtain the frequency domain graph.

[0028] By looping through the excitation frequency ω m Or excitation amplitude A l , scan the parameters, generate multiple sets of spectrum data, and finally form a three-dimensional spectrum diagram to analyze the changing rules of the main oscillation frequency and sideband under the change of excitation parameters.

[0029] Furthermore, a parameter sensitivity analysis framework is established through the harmonic coefficient method to quantify the influence of the excitation parameters on the main oscillation frequency and the sideband amplitude ratio:

[0030] By solving the harmonic coefficients of each order under different excitation parameters, the influence of excitation parameters on the main oscillation frequency and sideband amplitude ratio is analyzed, and the quantitative relationship between the dominant sideband order and its changing law are verified.

[0031] The beneficial effects of the technical solution provided by the present invention are:

[0032] 1. This paper establishes a complete analytical framework for the parametrically excited vibration frequency characteristics of a single-stage gear system under time-varying mesh stiffness through modulation feedback theory and trigonometric series approximation. This method is theoretically rigorous, computationally efficient, and can accurately predict the mapping relationship between the main frequency and sidebands.

[0033] 2. This invention innovatively introduces the first-kind Bessel function to construct a quantitative relationship model between the excitation parameters and the order of the dominant sidebands. This not only expands the dynamic characteristics analysis method of the gear system, but also provides a new theoretical basis and technical means for the design of gear vibration and noise reduction.

[0034] 3. The harmonic coefficient parameter sensitivity analysis framework proposed in this invention is universal and accurate. By quantifying the influence weight of the excitation parameters on the spectral characteristics, it realizes the accurate prediction of the dynamic response of the gear system. The calculation process is simple and accurate, which is convenient for engineering practice application and has a significant effect on improving the stability and reliability of the gear transmission system. BRIEF DESCRIPTION OF THE DRAWINGS

[0035] Figure 1 A schematic diagram of a single-stage gear structure under time-varying meshing stiffness excitation provided by the present invention;

[0036] Figure 2a According to the present invention, the excitation frequency ω m =0.1π, excitation amplitude A l = Frequency domain diagram under 50;

[0037] Figure 2b According to the present invention, the excitation frequency ω m =0.1π, excitation amplitude A l =100 frequency domain diagram;

[0038] Figure 2c According to the present invention, the excitation frequency ω m =0.1π, excitation amplitude A l = Frequency domain diagram under 200;

[0039] FIG3 is a three-dimensional spectrum diagram of parametric excitation response under different excitation amplitudes at a fixed excitation frequency provided by the present invention;

[0040] FIG4 is a three-dimensional spectrum diagram of parametric excitation response at different excitation frequencies under a fixed excitation amplitude provided by the present invention;

[0041] Figure 5a According to the present invention, the excitation frequency ω m =0.1π, the excitation amplitude A l Three-dimensional distribution diagram of harmonic coefficients in the interval [20,80];

[0042] Figure 5b According to the present invention, the excitation frequency ω m =0.1π, the excitation amplitude A l Three-dimensional distribution diagram of harmonic coefficients in the interval [80,110];

[0043] Figure 5c According to the present invention, the excitation frequency ω m =0.1π, the excitation amplitude A l Three-dimensional distribution diagram of harmonic coefficients in the interval [110,210];

[0044] Figure 6a According to the present invention, the excitation amplitude A l =100, excitation frequency ω m Three-dimensional distribution of harmonic coefficients in the interval [0.4,1];

[0045] Figure 6b According to the present invention, the excitation amplitude A l =100, excitation frequency ω m Three-dimensional distribution of harmonic coefficients in the interval [0.1, 0.4]. DETAILED DESCRIPTION

[0046] In order to make the objectives, technical solutions and advantages of the present invention more clear, the embodiments of the present invention are described in further detail below.

[0047] An embodiment of the present invention provides a method for predicting the parametrically excited vibration response spectrum of a single-stage gear structure based on modulation feedback. A dynamic model is established based on Newton's second law. A harmonic coefficient parameter sensitivity framework is constructed through modulation feedback analysis and the introduction of first-order Bessel functions. Finally, a fourth-order Runge-Kutta numerical calculation method is used, combined with a fast Fourier transform, to obtain the spectral characteristics of the system under different parameter combinations for verification. This research can be used to analyze the dynamic stability of such structures in actual engineering applications and provides a reference method for resonance control.

[0048] The embodiment of the present invention is carried out according to the following steps:

[0049] (1) Figure 1 This is a model diagram of a single-stage gear meshing system. The system consists of a pair of ideal spur gears. Only the time-varying meshing stiffness is considered, and all meshing actions of the gear pair occur on the theoretical meshing line. Nonlinear factors such as tooth side clearance are ignored, damping effects are not taken into account, and only torsional degrees of freedom are retained. In the figure, N1 and N2 represent the theoretical meshing lines of the master and slave gears, respectively; R i (i=1,2) is the base circle radius of the driving and driven wheels; I i The moment of inertia of the driving and driven wheels; Ti is the torque acting on the driving and driven wheels.

[0050] According to Newton's second law, the differential equation of torsional motion of the system can be expressed as:

[0051]

[0052] in, The second-order derivative of the torsional angular displacement of the driving and driven wheels with respect to time, θ i (i=1,2) is the torsional angular displacement of the driving and driven wheels, k m is the time-varying mesh stiffness.

[0053] In order to eliminate the rigid body displacement, the relative displacement in the direction of the gear meshing line is introduced as the generalized coordinate, which is:

[0054] x=R1θ1-R2θ2 (2)

[0055] By converting the torsional angular displacement into the linear displacement along the meshing line, the above equation can be rewritten as a single degree of freedom motion equation:

[0056]

[0057] in, is the second-order derivative of relative displacement with respect to time, m is the equivalent mass, F1 is the equivalent external excitation, k m is the time-varying meshing stiffness, and x is the relative displacement.

[0058]

[0059] make and τ=ω n t, where I0 is the reference moment of inertia, k 00 is the reference stiffness, ω n For the natural frequency, perform dimensionless calculations, ignore the forced terms, and obtain the dimensionless parametric free vibration dynamic equation. For the convenience of expression, I1, I2, k m and t respectively and τ, we finally get:

[0060]

[0061] (2) The time-varying meshing stiffness can be expressed as:

[0062] k m =k0+A l cos(ω m t) (6)

[0063] Among them, k0 is the average meshing stiffness, A l is the stiffness fluctuation excitation amplitude, ωm is the meshing frequency, the dynamic model can be equivalent to a modulation feedback model:

[0064]

[0065] in, is the stiffness modulation coefficient, is the natural frequency.

[0066] (3) Based on the modulation feedback principle, the system response can be expressed as:

[0067]

[0068] Where, ω s is the main oscillation frequency, C k is the harmonic coefficient, and k is the side frequency order.

[0069] Using Euler's formula, equation (7) can be written as:

[0070]

[0071] Substituting equation (8) into equation (9), and using the principle of harmonic balance, the above equation can be split into an infinite number of equations to obtain the harmonic coefficient C without time variables. k ,have:

[0072]

[0073] After sorting:

[0074]

[0075] Where k is an integer, ranging from negative infinity to positive infinity. Introducing the parameter ω k and b, we have:

[0076]

[0077] Combining the above recursive formulas for harmonic coefficients, formula (11) can be written in matrix form:

[0078]

[0079] Where p is the maximum number of cracking times. When it is large enough, the coefficient vector will tend to zero, that is, the right side of the equation is equal to zero. At this time, equation (13) is a homogeneous system of equations. The necessary and sufficient conditions for the system of equations to have a non-zero solution are:

[0080] det(W)=0 (14)

[0081] Where W is the coefficient matrix in formula (11).

[0082] Solving equation (14) gives 2(2p + 1) real roots, namely ±ω s ±kω m (k = -p, …, -1, 0, 1, …, p), and only one pair of roots is the main oscillation frequency ±ω of the parametric system vibration response s , and the rest are combination frequencies. The main oscillation frequency ω s and the natural frequency ω n are approximately equal numerically, that is, ω s ≈ω n , especially under low-frequency excitation conditions (ω m << ω n ), equation (11) can be simplified to:

[0083]

[0084] If the excitation frequency and the natural frequency satisfy kω m / ω n << 2, then:

[0085]

[0086] Equation (16) is the simplified recurrence formula for harmonic coefficients. Analyzing it, its mathematical structure is exactly the same as that of the first-kind Bessel function J k (κ), that is:

[0087] <000033�>

[0088] Comparing equations (16) and (17), we have κ = βω n / 2ω m , which means that the harmonic coefficient C k is a function of κ. κ reflects the modulation intensity of stiffness on the side-frequency distribution and is defined as the side-frequency modulation coefficient. When κ << 1, the side-frequency amplitude is low and decays quickly, and the main oscillation frequency amplitude is the highest; when κ >> 1, the number of side-frequency splitting increases, some amplitudes are high, and the main oscillation frequency no longer dominates in the energy distribution.

[0089] According to the properties of the first-kind Bessel function, κ directly affects the law of side-frequency energy increase and decrease. When k satisfies that is, when the side-frequency order k is close to round(κ - 1), the side-frequency amplitude reaches the maximum value. When the side-frequency order k increases from 0 to round(κ - 1), the side-frequency amplitude oscillates and increases, but the growth rate is slow. This is because when k < round(κ - 1), the Bessel function amplitude is approximately (κ / 2) k / k! Growth; when the sideband order increases from round(κ-1) to the highest order, the sideband amplitude rapidly decays to invisibility. This is because when k>round(κ-1), the Bessel function amplitude is approximately (e κ / 2k) k , exponential decay dominates, that is, high-frequency energy dissipates quickly and the system cannot maintain high-frequency oscillations.

[0090] (4) Combined with the parameters in Table 1, the fourth-order Runge-Kutta method can be used in combination with the fast Fourier transform to predict the spectral characteristics of the parametric excitation system:

[0091] Table 1 Basic parameters of single-stage gear structure

[0092]

[0093]

[0094] Figure 2 shows a fixed excitation frequency ω m =0.1π, different time-varying stiffness amplitudes A l Frequency domain response diagram under the action of . Figure 2a As shown, when A l =50,ω m =0.1π, the main oscillation frequency f s =1.233, its amplitude is the highest, and the side frequencies satisfy:

[0095]

[0096] It can be found that the sidebands strictly follow f s ±kf m The regular symmetrical distribution is between 1.03 and 1.43, and the interval between adjacent side frequencies is Δf = f m , the spectrum energy distribution shows a significant concentration characteristic, and the sideband amplitudes of more than four splits are almost invisible. At this time, κ<1, which fully complies with the attenuation law of the first-class Bessel function. l =100, such as Figure 2b As shown in the figure, the spectrum characteristics of the system have changed significantly, the sideband range has expanded to between 0.93 and 1.53, the number of frequency splits has increased to six and still strictly follows f s ±kf m The modulation rule is that the interval between adjacent side frequencies remains unchanged at 0.05. At this time, round(κ-1)=1, the amplitude of the first split side frequency exceeds the main frequency and becomes the maximum value, while the subsequent higher-order side frequencies show a monotonically decaying trend and finally become invisible. When the excitation amplitude is further increased to A l =200, such as Figure 2cAs shown in the figure, the system's spectral characteristics exhibit a more pronounced modulation effect. At this point, when round(κ-1)=4, the sidebands of the fourth split become the components with the largest energy share. Within the interval between the main frequency and the highest sideband, the energy shows an oscillatory growth characteristic, monotonically decaying to invisibility after reaching the peak, fully consistent with the amplitude prediction results of the first-kind Bessel function.

[0097] In order to further reveal the influence mechanism of the excitation amplitude on the dynamic response of the parametric system, Figure 3 shows the fixed excitation frequency ω m The three-dimensional spectrum of the system when =0.1π. Figure 3a The results of lower excitation amplitude are given. It can be seen that in this range, the amplitude of the main oscillation frequency decreases with the increase of the excitation amplitude, but it always dominates the energy distribution, while the amplitude of the side frequency component increases with A. l In addition, as the excitation amplitude increases, the sideband interval Δf remains unchanged, but the frequency distribution range becomes wider and wider. Figure 3b A quantitative relationship diagram between the excitation amplitude and the order of the dominant sideband was established. l The spectral evolution patterns of six typical cases {50, 100, 150, 200, 250, 300} in the range of κ = 50 to 300 are shown. The analysis reveals that the dominant energy peak gradually shifts from the second-order sideband to the sixth-order sideband. That is, as the excitation amplitude increases, the highest peak gradually shifts to the sideband with a higher number of splittings, and the order of the dominant sideband satisfies |k| = round(κ-1).

[0098] In order to systematically study the influence of low-frequency excitation on energy distribution, Figure 4a Shows a fixed excitation amplitude A l =100, the excitation frequency ω m The results show that as ω m As the frequency increases from 0.2π to 0.6π, the amplitude of the main oscillation frequency increases and always dominates the energy distribution, while the amplitude of the side frequency component decreases with ω. m The increase of ω shows a decaying trend, and some of them are eventually invisible, while the side frequency interval Δf increases with ω m As the frequency distribution increases linearly, the frequency distribution range becomes wider and wider, but the energy distribution will increase with ω m The increase is further concentrated towards the main oscillation frequency. Figure 4b Given the excitation frequency ω m When the excitation frequency ω is less than 0.2π m The three-dimensional relationship diagram of the highest peak side frequency order k. It can be found that as the excitation frequency decreases, the side frequency amplitude will be greater than the main frequency amplitude. When the excitation frequency ω m =0.16π and ω m = 0.12π, the sideband generated by the splitting The amplitude is the highest; when the excitation frequency ω m = 0.08π, the sideband generated by splitting twice The amplitude is the highest; when the excitation frequency ω m = 0.04π, the sideband generated by splitting 5 times The amplitude is the highest, and then the sideband amplitude drops rapidly.

[0099] In summary, the amplitude and frequency of the time-varying excitation directly alter the energy distribution of the parametrically excited free vibration response. When the time-varying stiffness amplitude is constant, the smaller the excitation frequency, the greater the number of frequency split combinations, and the more concentrated the energy distribution, which is distributed near the main oscillation frequency. The larger the excitation frequency, the greater the time-varying excitation intensity, the fewer the number of frequency split combinations, and the more dispersed the energy distribution. When the excitation frequency remains constant and the time-varying stiffness amplitude increases, the number of frequency split combinations increases, and the energy distribution becomes more dispersed. Furthermore, the spectrum of the system's steady-state response is centered on the main oscillation frequency and distributed to the left and right of the main oscillation frequency at intervals of the parameter frequency. In other words, all frequency combinations of the parametrically excited system's free vibration response can be represented by linear combinations of the main oscillation frequency and the excitation frequency. Furthermore, as the number of split combinations increases, the amplitude of the sidebands gradually decreases, eventually becoming almost invisible. This is consistent with the principles of energy finiteness and local distribution of the vibration response.

[0100] (5) Solution of harmonic coefficients:

[0101] Reorganize Equation (13) into a 2p-order linear equation, namely:

[0102] W2C=F (18)

[0103] Where W2 is the variable matrix containing the excitation frequency and system parameters, C is the harmonic coefficient vector, and:

[0104] C=[C -k … C -1 C1 … C k ] T , F=[0 … 0 -bC0 -bC0 0 … 0] T (19)

[0105]

[0106] Solve C = W2 -1 F, then we can get the harmonic coefficients C of each order k .

[0107] (6) Verification:

[0108] Assuming a fixed excitation frequency ω m =0.1π, respectively study the excitation amplitude A lThe influence of different value ranges on the harmonic coefficient is shown in Figure 5, where the horizontal axis represents the harmonic coefficient ordinal number, the vertical axis represents the excitation amplitude, and the vertical axis represents the normalized harmonic coefficient.

[0109] from Figure 5a It can be seen that the excitation amplitude A l In the range of 20 to 80, C ±4 with C ±3 The value of C0 is very small, indicating that the energy ratio of the third-order side frequency and the fourth-order side frequency is low, that is, the side frequency after the three-time split is almost invisible, and the frequency modulation caused by the time-varying stiffness almost disappears in the subsequent high-order harmonics, and the normalized harmonic coefficients C k As the excitation amplitude increases, C ±1 The increase speed is the fastest. l When it is less than 60, the main oscillation frequency amplitude is the highest, at this time the main peak energy distribution is dominant, and the excitation amplitude A l When it is greater than 60, the harmonic coefficient C ±1 The value of will be greater than 1, which means that the side frequency after the splitting is dominant, which is consistent with the phenomenon that the side frequency amplitude exceeds the main frequency obtained by Runge-Kutta method analysis.

[0110] Figure 5b is the excitation amplitude A l The evolution law of harmonic coefficients with the excitation amplitude in the range of 80 to 110. It can be found that when the excitation amplitude is less than 103, the sideband amplitude ratio of the first splitting is the maximum value, and the first-order sideband energy is dominant. When it is greater than 103, the sideband amplitude ratio of the second splitting is the maximum value. At the same time, as the excitation amplitude continues to increase, while the proportion of some sideband energies increases, the newly excited high-order sidebands also gradually appear, which verifies the conclusion that the excitation amplitude increases the number of splitting times. It is worth noting that in the critical range of 95 to 100, abnormal peaks will appear in the non-high-order harmonic coefficients, that is, the amplitude of the low-order sideband is much larger than the amplitude of the main frequency, which means that the main oscillation frequency energy is almost completely transferred to the low-order sideband. At this time, the sideband occupies a dominant position, and the system is no longer dominated by the main frequency, but is transformed into a modulated vibration dominated by pure sidebands. This is consistent with the Figure 2b The mid-frequency amplitude distribution is consistent. The subsequent decrease in amplitude relative to the peak value represents a partial recovery of the main frequency energy, indicating that there is a limit to the energy transfer of time-varying stiffness. The peak of the main oscillation frequency will not always approach 0. When the low-order sidebands are saturated, part of the energy is used to excite the higher-order sidebands, and part is re-injected into the main frequency through reverse coupling of parameter excitation.

[0111] Figure 5c The evolution law of harmonic coefficient with excitation amplitude in the range of 110~210 is shown. It can be found that when A l When the value is between 110 and 152, the second-order side frequency component C±2 The normalized amplitude ratio of A continues to dominate, indicating that the system vibration energy is still concentrated in the side frequency of the second splitting. It is worth noting that the harmonic coefficients at this stage change with A. l The increase shows an overall attenuation trend, and the smaller the distance between the sideband and the main frequency, the faster the attenuation rate, which is almost consistent with the above changes. l After crossing the critical value of 152, the dominant energy of the system shifts from the second-order sideband to the third-order sideband (C ±3 ) transfer, that is, after the second-order sideband energy reaches saturation, the modulation effect of parameter excitation will inject energy into higher-order sidebands. As the excitation amplitude continues to increase, the sideband amplitude ratio rises. When A l After increasing to 188, the recovery speed of each side frequency increases exponentially, and the recovery speed of the fourth-order side frequency is the fastest. The system enters the fourth-order side frequency dominant stage, that is, the energy continues to transfer to higher orders, accompanied by higher-order side frequencies, namely C ±7 and C ±8 excitation.

[0112] Then fix the excitation amplitude A l =100, analysis excitation frequency ω m Harmonic coefficient C k Here the excitation frequency ω m The value range is set to 0.4 to 1, and the calculated harmonic coefficients of each order are as follows: Figure 6a As shown in the figure. The horizontal axis represents the harmonic coefficient ordinal number, the vertical axis represents the excitation frequency, and the vertical axis represents the harmonic coefficient. It can be found that C ±4 The relative C0 value is very small, indicating that the energy of the fourth-order sideband component is significantly attenuated, that is, the sideband after the fourth split is almost invisible, and the frequency modulation caused by the time-varying stiffness almost disappears in the subsequent high-order harmonics, which is consistent with Figure 5a The phenomenon is similar to antisymmetry. The harmonic coefficient increases with the decrease of the excitation frequency, where C ±1 The fastest growth rate is when the excitation frequency is less than 0.54, the harmonic coefficient C ±1 The value of will be greater than 1, which is consistent with Figure 2b The Runge-Kutta method analysis shows that the side frequency amplitude is greater than the main frequency.

[0113] Figure 6b Shows the excitation frequency ω mThe dynamic evolution characteristics of the harmonic coefficient distribution in the range of 0.1 to 0.4 are shown. The results show that within this frequency range, the dominant position of the main oscillation frequency amplitude is broken, and the highest amplitude appears at a certain side frequency. The position of the dominant side frequency gradually shifts to both sides as the excitation frequency decreases, which is directly related to the modulation degree of the parameter excitation. At the same excitation frequency, the growth phase of different harmonic coefficients shows an oscillatory characteristic, and the decay phase conforms to an exponential law, which is almost consistent with the characteristics of the first-class Bessel function. At the same time, the change of harmonic coefficients shows an antisymmetric characteristic in a specific range. That is, when the excitation frequency is between 0.17 and 0.22, the harmonic coefficients decrease with increasing excitation frequency, among which the harmonics closer to the main frequency decay faster. However, when the excitation frequency is between 0.22 and 0.31, the harmonic coefficients increase with increasing excitation frequency, and the growth rate of harmonics closer to the main frequency is more significant.

[0114] Based on the above harmonic coefficient evolution analysis, the corresponding law between the parameter excitation intensity and the frequency domain energy distribution can be revealed. Table 2 and Table 3 respectively summarize the highest amplitude sideband order C under different excitation amplitude conditions and in different excitation frequency ranges. k and its corresponding sideband modulation coefficient κ range.

[0115] Table 2 Correspondence between the excitation amplitude range and the dominant sideband characteristics

[0116]

[0117] Table 3 Correspondence between excitation frequency range and dominant sideband characteristics

[0118]

[0119] In summary, an embodiment of the present invention provides a method for predicting the spectrum of the parametrically excited vibration response of a single-stage gear structure based on modulation feedback. By establishing a single-degree-of-freedom dynamic model under the action of time-varying meshing stiffness, the energy distribution law of the main oscillation frequency and sidebands in the parametrically excited vibration response is revealed through modulation feedback analysis and first-order Bessel function analysis, and the analysis results are verified by using the fourth-order Runge-Kutta method combined with fast Fourier transform. The study found that the system spectrum response presents a typical "double-peak migration" feature. The quantitative relationship model between the excitation parameters and the dominant sideband order constructed based on the first-order Bessel function can accurately predict the sideband distribution characteristics. The reliability of the quantitative relationship is further verified by the parameter sensitivity analysis framework established by the harmonic coefficient method. Compared with the traditional frequency domain analysis method, this method significantly improves the sideband prediction accuracy and computational efficiency, providing an effective theoretical tool and technical support for the dynamic performance optimization and fault diagnosis of the gear system.

[0120] Those skilled in the art will understand that the accompanying drawings are only a schematic diagram of a preferred embodiment, and the serial numbers of the embodiments of the present invention are only for description and do not represent the advantages or disadvantages of the embodiments.

[0121] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

Claims

1. A method for predicting the parametrically excited vibration response spectrum of a single-stage gear based on modulation feedback, characterized in that: The method comprises: Based on Newton's second law, a single-degree-of-freedom dynamic equation with time-varying meshing stiffness is established, and the main oscillation frequency is solved using modulation feedback theory and trigonometric series approximation method. A quantitative prediction model for the excitation parameters and the order of the dominant sidebands is constructed based on the first-kind Bessel function. The fourth-order Runge-Kutta method combined with the fast Fourier transform is used to solve the system response to verify the prediction model. A parameter sensitivity analysis framework is established through the harmonic coefficient method to quantify the influence of excitation parameters on the main oscillation frequency and sideband amplitude ratio.

2. The method for predicting the parametrically excited vibration response spectrum of a single-stage gear based on modulation feedback according to claim 1 is characterized in that: The kinetic equation is: Where x is the generalized coordinate in the form of relative displacement, is the second-order derivative of the generalized coordinate with respect to time, m and k m (t) are the equivalent mass and time-varying meshing stiffness respectively, and: Where R is the base circle radius, I is the moment of inertia, and θ is the torsional angular displacement.

3. The method for predicting the parametrically excited vibration response spectrum of a single-stage gear based on modulation feedback according to claim 1 is characterized in that: The time-varying mesh stiffness is expressed as: k m =k0+A l cos(ω m t) Where k0 is the average meshing stiffness, A l is the stiffness fluctuation excitation amplitude, ω m is the meshing frequency.

4. The method for predicting the parametrically excited vibration response spectrum of a single-stage gear based on modulation feedback according to claim 1 is characterized in that: The quantitative prediction model of excitation parameters and dominant sideband order based on the first kind of Bessel function is: k=round(κ-1) Where k is the sideband order, κ=βω n / 2ω m is the sideband modulation coefficient.

5. The method for predicting the parametrically excited vibration response spectrum of a single-stage gear based on modulation feedback according to claim 1 is characterized in that: The fourth-order Runge-Kutta method combined with fast Fourier transform is used to solve the system response: The differential equation is solved by the fourth-order Runge-Kutta method to obtain the system response x(t) in the time domain. The time domain signal x(t) is transformed into a complex spectrum in the frequency domain by fast Fourier transform. The modulus value is taken to obtain the frequency domain graph. By looping through the excitation frequency ω m Or excitation amplitude A l , scan the parameters, generate multiple sets of spectrum data, and finally form a three-dimensional spectrum diagram to analyze the changing rules of the main oscillation frequency and sideband under the change of excitation parameters.

6. The method for predicting the parametrically excited vibration response spectrum of a single-stage gear based on modulation feedback according to claim 1 is characterized in that: The parameter sensitivity analysis framework is established by the harmonic coefficient method to quantify the influence of the excitation parameters on the main oscillation frequency and the sideband amplitude ratio: By solving the harmonic coefficients of each order under different excitation parameters, the influence of excitation parameters on the main oscillation frequency and sideband amplitude ratio is analyzed, and the quantitative relationship between the dominant sideband order and its changing law are verified.

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