Descriptor: Time-varying meshing stiffness of the flank clearance. Chaos 0刖 In the gear-coupled rotor-bearing system, the nonlinear factors are various, such as: flank clearance, bearing clearance (rolling bearing), nonlinear oil film Force (sliding bearing) and time-varying meshing stiffness of the gear pair. Many nonlinear dynamics problems have been studied in single-degree-of-freedom or multi-degree-of-freedom systems. For example, A.Kahraman considers a three-degree-of-freedom system and applies linear segmentation techniques to study the nonlinear frequency response of the system. The variable mesh stiffness is developed into Fourier series, and the interaction between time-varying meshing stiffness and gap nonlinearity is studied. HNZgven establishes a semi-definite 6-degree-of-freedom nonlinear model with time-varying mesh stiffness; more nonlinear models See also. In these models, the drive shaft is mounted on a rolling bearing, and the lateral vibration of the system along the line of engagement and its vertical direction is uncoupled, and only the one-sided contact, separation or impact between the gear pairs is considered. However, in some dynamic systems, sliding bearings are often used, and the lateral vibrations in the above two directions are coupled to each other due to the influence of the oil film force. Chin-shong Chen and TN. Shiau studied the effects of squeeze film dampers on the dynamic characteristics of the system. At the same time, the method used to study this nonlinear system has always been a concern. A. Raghothama used the incremental harmonic balance method to study the bifurcation and chaos of the system.
1 Dynamics model As shown, the driving gear is mounted on the rigid shaft, the two ends of the shaft are just supported, and the driven gear z2 is mounted on the elastic shaft with bending rigidity. The two ends of the shaft are supported on the sliding bearing, and the whole system is geared. The center plane is symmetrical. The following force balance equations can be listed for the center displacement of the journal-to-gear plate driven gear e', e2 - the driving gear, the driven gear angular displacement, the mltm2 driving gear, the driven gear mass P1, P2 - the driving gear, the driven Gear Moment of Inertia Fe The unbalanced force caused by the eccentricity of the driven gear < is <1, <2—the angular velocity T of the driving gear and the driven gear, and the dynamic torque T, T20 generated by the meshing force of the T2 gear—primer and load Static torque rt applied to the driving gear and driven gear respectively, r2 gear base circle radius kxx,k,kxy,kyx——sliding bearing oil film stiffness coefficient Sxx,dyy,dxy,dyx——sliding bearing oil film damping coefficient National Natural Science Foundation of China (19990511). 20001121 received the first draft, and 20010528 received the revised draft. The formula (3)~(9) generation A controls the stability of the system. When A=1, the system will generate the bifurcation. When A=-1, a double-cycle bifurcation is generated; when 1=1, a saddle-node bifurcation is generated; I is a complex number, and when IAI=1, a Hopf bifurcation is generated. It is a Floquet multiplier curve calculated according to Floquet theory, and the obtained instability speed is 4 573.22r/min, at which time A=0.9999+0.0142i, so when the system is unstable, Hopf bifurcation is generated. i is a complex unit.
The 11134 cross section and phase diagram at /111 is as shown by a, b, c, d and a, b, c, d. There is no extrusion and tooth removal, but the system has generated Hopf bifurcation, and the 7-cycle solution is divided. The newly generated 6-period solution has a very small amplitude relative to the power frequency component, so it is difficult to observe in the middle. .
3 The chaotic motion caused by the change of the rotational speed of the book4 speed is 4573.1r/min. The deformation of the meshing tooth pair, the spectrum, the Poinca technique cross section and the phase diagram are as a, b, c, d and a, b, c, d respectively. Do not. The gear meshing is normal. Under the unbalanced mass excitation, the system performs periodic motion, and only the power frequency and the meshing frequency (not shown in the spectrogram) are included in the response. In the figure, the amplitude of the spectral curve is logarithmic, C = -) 12 / (2cs is the response frequency.
When the response speed is 4573.25r/min and the spectrum is 573.25r/min, the time response of the system, the deformation of the meshing pair, the spectrum, the PoincaA cross section, and the phase diagram are respectively PoliccaA cross section and phase at the speed of 4573.25r/min. . 3 The response when the rotational speed is 4573.275r/min, the deformation amount of the meshing pair, the spectrum, the PoincaA cross section, and the phase diagram are as shown in a, b, c, d and a, b, c, d, respectively. After the system further generates Hopf bifurcation, the quasi-periodic motion is performed, and the Poincarfe cross section forms a loop diagram. The time response of the test shows that only at a certain point in time, there is a very slight toothing, but there is no toothing, and the other figures do not change much.
When the speed is 4 580r/min, the time response of the system, the deformation of the pair of teeth, the spectrum, the Poincak cross section, and the phase diagram are as follows: a, b, c, d and 0a, b, c, d are 7K. Or do quasi-periodic motion, but the circular Poincar 纟 cross section is more dense, and its surrounding area is significantly increased. The phenomenon of squeezing teeth is slightly aggravated. Occasionally, the response time at 4 580 r/min and the Poincar cross-section and phase diagram of the spectrum 0 at 4580 r/min have slight de-coking. Compared with the case of n,=4573.275r/min, in addition to the power frequency component, other frequency components increase significantly on the spectrum curve, and at 0/%=18, the newly appearing frequency components are more prominent. From the phase diagram, there are also signs of an increase in the amplitude of other frequency components.
3. When the response is 600r/min at 4600r/min, the time response of the system, the deformation of the meshing pair, the spectrum, the PoincaA cross section, and the phase diagram are shown in Figures 11a, b, c, d and 2a, b, respectively. c, d 7K. The system further generates Hopf bifurcation. When doing quasi-periodic motion, the circular pattern composed of dense points of Poincak cross section becomes a circular pattern composed of 7~8 segments. The phenomenon of toothing is very serious and can be observed from the phase diagram.
In some time periods, the de-toothing phenomenon is also very serious, while the time intervals of the brief breaks are not equal. The spectrum is no longer a completely discrete line. In the range of 0/0 = 12-30, the line has obvious continuous features and its amplitude is relatively increased. The phase diagram is transformed into a quasi-periodic orbit by a series of Hopf bifurcations from the original periodic elliptical orbit. The amplitude of the other frequency components is also increased from the phase diagram.
The response of 1 speed is 4 600r/min and the response of the spectrum 3.6 when the speed is 4620r/min is 620r/min. The time response of the system, the bite is a displacement, a/io-(b) is a displacement. 1(d) The displacement, spectrum, PoincaA cross section and phase diagram of the Poincar section and the phase diagram of the pair of teeth at a displacement of 2 at a speed of 4 600 r/min are 3a, b, c, d and 4a, b, respectively. c, d does not. After further Hopf bifurcation, the system generates chaotic motion, and the Poinca technique cross section changes from the front ring to a myriad of points within a limited range. The spectrum is continuous, and at individual frequency points, its value is equivalent to the amplitude corresponding to the power frequency. It can be seen from the time response that when the chaotic motion occurs in the system, the rotor will have a large deformation along the r2 direction, which will increase the gear center distance.
There is no toothing phenomenon, the phenomenon of tooth removal is serious, and there is tooth back contact after tooth removal. Therefore, slap will be inevitable. The phase diagram also loses the characteristics of quasi-periodic motion, showing the chaotic features of infinite self-similarity.
3 Response and spectrum when the speed is 4 620r/min 4 Conclusion When the speed increases, the time-varying meshing stiffness of the gear will first cause the system to generate Hopf bifurcation. When the fork is to a certain extent, the tooth pair will produce teeth and teeth. Phenomenon, the system develops from periodic motion to quasi-periodic motion. When the speed exceeds a certain value, the system will generate chaotic motion, accompanied by strong tooth removal and back contact, the amplitude is obviously increased, and the vibration is severe.
When the system develops from chaotic motion to periodic or quasi-periodic motion, the deformation of the rotor along the centerline of the gear is large enough to cause an arcing or broken shaft accident of the rotor of an actual unit.
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