1 At present, domestic and foreign scholars have done a lot of research on the dynamic characteristics of planetary and star gear transmissions. For the planetary gear train and the star train, a linear model including eccentricity error and time-varying mesh stiffness is used; equivalent mass; Ms, Mpi and Mr are the masses of each gear; mi is the prime mover on r. Effective mass; mi is the equivalent mass of the load on rrb; Td is the input torque; Ki is the torsional stiffness of the input shaft; K is the torsional stiffness of the output shaft; Ks, Kr is the bending stiffness of each gear support shaft; Tl For load torque.
The differential equations (6) are semi-positive timing variable parameters and nonlinear second-order differential equations. There are both nonlinear resilience terms and linear resilience terms. At the same time, the system coordinates contain rigid body displacements, so they cannot be directly written. The second-order differential equation in the form of a matrix, and the solution analysis method of the multi-degree-of-freedom system cannot be performed based on the matrix form. Therefore, a set of generalized coordinates of the system is introduced: the degree of freedom of the system is The original (3N+8) is added to 6), and the linear restoring force corresponding to Hs, Vs, Yi, Zi, (i=1, 2, N), Hr, Vr is expressed as a special gap of 0. The gap nonlinear function, equation (5) is expressed as matrix form: mass matrix, damping matrix, stiffness matrix is ​​load column vector, f(x) is gap nonlinear function column vector, with the following unified form: 4 gears Sub-meshing parameter analysis The various types of errors caused by gear manufacturing and installation are the main factors that cause gear vibration. The main factors that consider the unevenness of the base section error of the gear are studied. Consider the gear eccentricity error caused by installation and manufacturing.
The periodically changing meshing stiffness is the main source of vibration for the gear vibration. The Ishikawa method is used to calculate the meshing stiffness of the gear pair.
In the star train, when the number of teeth of the sun gear and the ring gear is not an integral multiple of the number of the star wheel, the sun gear and the ring gear are no longer in synchronous engagement with the star wheel, but maintain a certain meshing phase. In the kinetic analysis, the effective meshing damping coefficient of each gear pair corresponding meshing gear pair is determined according to the number of gears in the system: the torsional damping coefficient of the input shaft and the output shaft: the bending damping coefficient of the supporting shaft: respectively The equivalent mass of the meshing gear; the damping ratio Y ranges from Q 03 to 0.1, where Q07.Kt is the torsional stiffness of the shaft, io is the moment of inertia of the prime mover or load, and g is the sun gear or The moment of inertia of the ring gear; the damping ratio Y ranges from Q005 to 0.075, which is taken as 0. 01Ks is the bending stiffness of the bearing shaft, ms is the sum of the masses of the shaft and the gear on the shaft, and the damping ratio is Y. The value range is about 0.003-0.1. In this paper, we take the solution of 0.015 equation and the result analysis for the multi-frequency excitation and parametric Coupling gap-type nonlinear differential equations (7). This paper uses the variable step size Gill integral method. Get the numerical solution of the equation and finally For a reducer (N=3), the parameters of the reducer are calculated by the above method as follows: the number of teeth: zs=15; and the tooth profile error, which is taken as positive zpi=18 which changes with the gear meshing period; zr=string function . The major period of each gear is eccentrically misplaced; the 呗 呗 star type 呃 0° tooth width e = eight 1 running gear eccentric misreading WESd.net 15Mm; initial phase angle of eccentricity error: U=U=U=0, tooth Frequency error: 6pi=10Mm; the initial phase (h) of the tooth frequency error is the dynamic performance curve of the system under different parameters of n=5000r/min, which is the system without the eccentricity error excitation. The dynamic response is the dynamic response of the system under the flank clearance b=50m and the tooth frequency error without eccentricity error excitation. It is the dynamic response of the system to the tooth frequency error and the eccentricity error excitation at the flank clearance b=100m. In the figure, the coordinate f is the meshing period f= of the gear teeth, and k is the meshing tooth frequency; Figures (a) and (4) are respectively x'p! And the time history of x; Figures (b) and (e) are the FFT spectra of x and x, respectively; Figures (c) and (f) are zs and sum, respectively. The time-domain history of the meshing dynamic load coefficient between the two, in the figure p = visible, the system's displacement response curve ((a) and (d)) fluctuates in the tooth frequency, and contains high frequency in each tooth frequency cycle. vibration. The spectroscopy of the displacement response ((b) and (e)) can be used to find a discrete spectrum with a frequency of mk (m is a positive integer), and the spectral value of the tooth frequency k is much larger than the submaximum spectrum (5k). The values ​​have a ratio of 7.5 and 2.8. Therefore, in the case of single-frequency excitation, the response of the system mainly appears as an integer multiple of the tooth frequency, and the response of the tooth frequency is the largest. The main reason is that the meshing stiffness mainly fluctuates at the fundamental frequency of the tooth frequency, and also includes the frequency doubling component of the fundamental frequency. Because the system is a linear time-varying system and is excited by single frequency, the time-varying meshing stiffness of the gear pair with the fundamental frequency fluctuation has a frequency doubling component. The high frequency component in the stiffness makes the superharmonic response appear in the system response. In this case, as shown in (c) and (f), the dynamic load factor fluctuates in a cycle of the tooth frequency, and the load distribution of each star wheel does not appear uneven. This is consistent with the calculation result in the system when there is a gap in the system. In the eccentricity error, the low-frequency modulation waveform that does not exist in the displacement response time history of (a) and (d) can be observed. The frequency of the modulating waveform is 0.02k in the FFT spectrum analysis and can still be observed in the spectrogram. The superharmonic response differs in that the number of frequencies corresponding to the superharmonic response is more. Although the spectral value of the fundamental frequency remains the largest, the spectral values ​​of the superharmonic components increase to varying degrees. The ratio of the spectral value to the sub-maximum spectrum is increased in (b) and (e) respectively. The dynamic load factor of the gear teeth in the joints of 3. 4 and 21 (c) and (f) is increased, but the amplitude is not large; There is no uneven load distribution on each star wheel Since the system does not have dislocation and back impact, the dynamic load of the system has not changed greatly. When there is gear eccentricity error, the fluctuation of the dynamic response of the system increases ((a) and (d) )), and the axial frequency excitation due to eccentricity error causes the system to produce a distinct low-frequency response, a series of low-frequency spectra are densely distributed on the response spectrum ((b) and (e)), and the spectrum is not like the sum of the main discrete The spectrum is composed, but a number of spectrums that are not multiples of the fundamental frequency appear. At the same time, in the dynamic load coefficient curves of (C) and (1), it can be seen that during the meshing process of the teeth, the repeated impact phenomenon of meshing tooth surface separation and combination occurs. The tooth surface separation is expressed as the meshing dynamic load coefficient between the teeth. It is 0, and no dynamic load coefficient is found in the sum. In the middle, it can be clearly found that the tooth surface separation (g) and (h) are the meshing between zs and zp2, zs and ZP3, respectively. The time-domain history of the dynamic load coefficient, comparing (g), (h) with the figure (C Xiang, can be concluded that due to the existence of eccentricity error, the load of the star wheel is obviously uneven, and the gear teeth are the largest. The dynamic load is also added to 4.5 from 1.51 due to the existence of eccentricity error. It can be seen that the eccentricity error exacerbates the nonlinear characteristics of the system, which not only causes uneven distribution of load in the system, but also makes the dynamic load of the system large. The dynamic characteristics deteriorate, so every effort should be made to avoid the existence of various types of eccentricity errors during manufacturing and assembly.
5 Conclusion Under the influence of time-varying mesh stiffness, even if there is no gap, the star gear transmission system will generate multi-frequency response gap for single-frequency error excitation, so that the system responds with low-frequency modulation frequency under single-frequency excitation. When there is no eccentricity error excitation and the gear meshing does not cause the tooth surface separation shock, the influence of the clearance on the dynamic load of the gear and the load uniformity of each star wheel is small. The eccentricity error of the system can cause the dynamic load coefficient of the gear teeth to be significantly large and Load unevenness occurs between the teeth. Under the same vibration parameters, the eccentricity error can cause the tooth surface separation shock.
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