Effect of Backlash on Vibration Spectrum in Spur Gearboxes Incorporating Sleeve Bearings

1991 ◽  
Author(s):  
Fawzi M. A. El-Saeidy
Keyword(s):  
Fourier Transform ◽  
Time Domain ◽  
Vibration Spectrum ◽  
Equations Of Motion ◽  
Analysis Procedure ◽  
Radial Clearance ◽  
Time Varying ◽  
Mesh Stiffness ◽  
Gearbox Vibration ◽  
Center Distance

Abstract An Analytical model is presented to simulate effect of tooth backlash on vibration spectrum of spur gearboxes incorporating sleeve bearings. Included in the model are: elasticity of shafts, friction between meshing teeth, interaction between gearbox casing and internals, and time-varying tooth backlash (backlash is a function of operating center distance), mesh stiffness (stiffness is calculated based on the strain energies of the tooth that is treated as a cantilever beam of involute shape) and Hertzian mesh damping. The bearing forces are calculated with consideration of bearing radial clearance and system vibrations. The analysis is applied to a single stage gearbox and equations of motion are numerically integrated to obtain system reponse in time domain. This response is transformed into frequency domain (vibration spectrum) using Fast Fourier Transform (FFT) algorithm and samples of the results are shown for different values of tooth backlash. The results show that backlash has a pronounced effect on gearbox vibration and study provides an analysis procedure for predicting such effects.

A Probabilistic Tolerance Allocation Method for Dynamic Mechanical Systems With Periodic Response and Discontinuous Forcing Functions

1995 ◽  
Author(s):  
F. Zhang ◽  
B. J. Gilmore ◽  
A. Sinha
Keyword(s):  
Combustion Engine ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Optimization Procedure ◽  
Tolerance Allocation ◽  
Radial Clearance ◽  
Time Varying ◽  
Link Length ◽  
Design Variables ◽  
Forcing Functions

Abstract Tolerance allocation standards do not exist for mechanical systems whose response are time varying and are subjected to discontinuous forcing functions. Previous approaches based on optimization and numerical integration of the dynamic equations of motion encounter difficulty with determining sensitivities around the force discontinuity. The Alternating Frequency/Time approach is applied here to capture the effect of the discontinuity. The effective link length model is used to model the system and to account for the uncertainties in the link length, radial clearance and pin location. Since the effective link length model is applied, the equations of motion for the nominal system can be applied for the entire analysis. Optimization procedure is applied to the problem where the objective is to minimize the manufacturing costs and satisfy the constraints imposed on mechanical errors and design variables. Examples of tolerance allocation are presented for a single cylinder internal combustion engine.

scholarly journals Frequency and Vibration Characteristics of High-Speed Gear-Rotor-Bearing System with Tooth Root Crack considering Compound Dynamic Backlash

2019 ◽  
Vol 2019 ◽  
pp. 1-19 ◽  
Author(s):  
Jie Liu ◽  
Weiqiang Zhao ◽  
Weiwei Liu
Keyword(s):  
Time Domain ◽  
Crack Depth ◽  
Transmission System ◽  
Gear System ◽  
Time Varying ◽  
Tooth Root ◽  
Meshing Stiffness ◽  
The Time Domain ◽  
Center Distance ◽  
Root Crack

Considering the microstructure of tooth surface and the dynamic characteristics of the vibration responses, a compound dynamic backlash model is employed for the gear transmission system. Based on the fractal theory and dynamic center distance, respectively, the dynamic backlash is presented, and the potential energy method is applied to compute the time-varying meshing stiffness, including the healthy gear system and the crack fault gear system. Then, a 16-DOF coupled lateral-torsional gear-rotor-bearing transmission system with the crack fault is established. The fault characteristics in the time-domain waveform and frequency response and statistics data are described. The effect of crack on the time-varying meshing stiffness is analyzed. The vibration response of three backlash models is compared. The dynamic response of the system is explored with the increase in crack depth in detail. The results show that the fault features of countershaft are more obvious. Obvious fluctuations are presented in the time-domain waveform, and sidebands can be found in the frequency domain responses when the tooth root crack appears. The effect of compound dynamic backlash on the system is more obvious than fixed backlash and backlash with changing center distance. The vibration displacement along meshing direction and dynamic meshing force increases with the increase in crack depth. Backlash and variation of center distance show different tendencies with increasing crack depth under different rotational speeds. Amplitude of the sidebands increases with crack depth increasing. The amplitude of multiplication frequency of rotational frequency has an obvious variation with growing crack depth. The sidebands of the multiplication frequency of meshing frequency show more details on the system with complex backlash and crack fault.

scholarly journals Dynamic Modeling and Vibration Characteristics Analysis of Deep-Groove Ball Bearing, Considering Sliding Effect

Mathematics ◽  
2021 ◽  
Vol 9 (19) ◽  
pp. 2408
Author(s):  
Fanjie Li ◽  
Xiaopeng Li ◽  
Dongyang Shang
Keyword(s):  
Ball Bearing ◽  
Equations Of Motion ◽  
Constant Load ◽  
Vibration Characteristics ◽  
Vibration Response ◽  
System Model ◽  
Radial Clearance ◽  
Time Varying ◽  
Deep Groove ◽  
Deep Groove Ball Bearing

To study the vibration characteristics of deep-groove ball bearing, considering the influence of sliding, the dynamic model of the DGB 6205 system is established in this paper. The DGB 6205 system model includes the movement of the bearing inner ring in the X and Y directions, the rotation of the cage, the rotation movement of each ball, the revolution movement of each ball and the movement along the radial direction of each ball. Based on the system model, the differential equations of motion of the system are established, and the correctness of the model is verified by experiment. The slip characteristics of the DGB 6205 system are studied by numerical simulation. At the same time, the influence of time-varying load on the vibration characteristics of the system is studied. Then, the sensitivity of system parameters is analyzed. The results show that the sliding speed between the ball and the inner raceway is greater than that between the ball and the outer raceway. The radial vibration response of DGB 6205 system under time-varying load is less than that under constant load. The increase of radial clearance will increase the vibration response of DGB 6205 system.

Dynamic Behavior of Helical Gears with Effects of Shaft and Bearing Flexibilities

2011 ◽  
Vol 86 ◽  
pp. 26-29
Author(s):  
Kai Feng ◽  
Shigeki Matsumura ◽  
Haruo Houjoh
Keyword(s):  
Experimental Tests ◽  
Time Varying ◽  
Gear Pair ◽  
Contact Ratio ◽  
Mesh Stiffness ◽  
Helical Gears ◽  
Shaft System ◽  
Proposed Model ◽  
Center Distance ◽  
Good Agreement

This study presents a numerical model of helical gears to consider the effects of shaft and bearing flexibility. A primary feature of this study is that the time-varying mesh stiffness is not just determined by the geometry of gear pair but also updated for each iteration according to the change of center distance. The effects of shaft and bearing flexibilities are discussed by comparing the dynamic response of gear pairs supported with a rigid and a flexible bearing-shaft system. The results show that the pressure angle and contact ratio are significantly changed due to the center-distance variation of gears and the gear pair with a flexible bearing-shaft system has much larger vibration. Finally, experimental tests are conducted to validate the proposed model. The predicted results show good agreement with the experimental data.

Discrete Time, Discrete Frequency

2021 ◽  
pp. 106-155
Author(s):  
Victor Lazzarini
Keyword(s):  
Signal Processing ◽  
Fourier Transform ◽  
Discrete Time ◽  
Time Domain ◽  
Continuous Time ◽  
Time Varying ◽  
Discrete Frequency ◽  
The Fourier Transform ◽  
Definition Of ◽  
The Time Domain

This chapter is dedicated to exploring a form of the Fourier transform that can be applied to digital waveforms, the discrete Fourier transform (DFT). The theory is introduced and discussed as a modification to the continuous-time transform, alongside the concept of windowing in the time domain. The fast Fourier transform is explored as an efficient algorithm for the computation of the DFT. The operation of discrete-time convolution is presented as a straight application of the DFT in musical signal processing. The chapter closes with a detailed look at time-varying convolution, which extends the principles developed earlier. The conclusion expands the definition of spectrum once more.

Fault features analysis of a compound planetary gear set with damaged planet gears

2017 ◽  
Vol 232 (9) ◽  
pp. 1586-1604 ◽  
Author(s):  
Guoyan Li ◽  
Fangyi Li ◽  
Haohua Liu ◽  
Dehao Dong
Keyword(s):  
Time Domain ◽  
Planetary Gear ◽  
Significant Feature ◽  
Time Interval ◽  
Time Varying ◽  
Mesh Stiffness ◽  
Vibration Signals ◽  
Fault Features ◽  
Fault Properties ◽  
The Time Domain

The fault properties of compound planetary gear set are much more complicated than the simple planetary gear set. A damaged planet will induce two periodic transient impulses in the raw signals and generates modulation sidebands around the mesh harmonics. This paper aims to investigate the fault properties of a compound planetary gear set in damaged planet conditions. A dynamic model is proposed to simulate the vibration signals. The time interval between the fault-induced close impulses in the time domain is used as a significant feature to locate the faulty planet. Considering the phase relations, the time-varying mesh stiffness is obtained. Then, the fault properties are demonstrated in the simulation, and the theoretical derivations are experimentally verified.

Analysis of Resonances and Instabilities in Elliptical Gear Systems

1997 ◽  
Author(s):  
Ming-Jun Lai ◽  
Ali A. Seireg
Keyword(s):  
Dynamic Performance ◽  
Transmission Function ◽  
Primary Resonance ◽  
Time Varying ◽  
Mesh Stiffness ◽  
Kinematic Constraint ◽  
First Order ◽  
Average Value ◽  
Center Distance ◽  
Input Torque

Abstract Two types of elliptical gears, namely first-order and second-order oval elliptical gears, are studied in this paper. The dynamic behavior of such gears is investigated by considering the effect of the eccentricity ratio and the corresponding transmission function, the time-varying mesh stiffness, time-varying mesh inertia, mesh damping as well as the operating speed and external load. The influence of the system parameters on the different resonance and instability conditions is also investigated. The results are compared with those for the circular gears with the same center distance, number of teeth and mesh stiffness and show that the elliptical gears can exhibit better dynamic performance than the circular gears under such conditions. Within the kinematic constraint for the angle of obliquity of the elliptical gears, it is shown that the larger the eccentricity is in their useful range of operation, the more advantageous such gears can be. It is found from the analysis of the dynamic magnification of the tooth load at the primary resonance that a zero magnification occurs for the circular gears with sinusoidal mesh stiffness fluctuation when the ratio of the amplitude of input torque fluctuation to its average value equals the fluctuation ratio of the mesh stiffness. A minimum magnification is also found for the two types of elliptical gears when the mesh stiffness fluctuation is similarly tuned to the torque fluctuation.

Study of the Modeling of the Gear Dynamics Considering Mesh Stiffness and Sliding Friction

2010 ◽  
Vol 29-32 ◽  
pp. 618-623
Author(s):  
Cheng Zhong Gu ◽  
Xin Yue Wu
Keyword(s):  
Fourier Transform ◽  
Sliding Friction ◽  
Approximate Analytical Solution ◽  
Friction Torque ◽  
Time Varying ◽  
Gear Pair ◽  
Mesh Stiffness ◽  
Friction Modeling ◽  
Period Function ◽  
Stiffness Variation

Time- varying mesh stiffness and sliding friction between teeth are the great excitation for vibration and noise in gears system. But, there are rarely studies on this topic. This paper proposes a new dynamic modeling of gear system, which is effect of mesh stiffness variation, sliding friction and distribution of load. Firstly, the expression of time-varying mesh stiffness is gained, which is a period function. Secondly, a new friction modeling has the same period as mesh stiffness, is proposed. Thirdly, friction torque of each gear pair is calculated respectively, which is considering the distribution of load and time-varying friction arm. Finally, because all parameter have the same cycle, it is easy to get the approximate analytical solution to non-line model of gear dynamic by fourier transform.

JOINT INTERPRETATION OF AIRBORNE ELECTROMAGNETIC DATA IN THE TIME DOMAIN AND FREQUENCY DOMAIN

2018 ◽  
Vol 12 (7-8) ◽  
pp. 76-83
Author(s):  
E. V. KARSHAKOV ◽  
J. MOILANEN
Keyword(s):  
Fourier Transform ◽  
Frequency Domain ◽  
Time Domain ◽  
Frequency Interval ◽  
Single Spectrum ◽  
Simultaneous Inversion ◽  
Homogeneous Half Space ◽  
Time Domain Data ◽  
The Time Domain

Тhe advantage of combine processing of frequency domain and time domain data provided by the EQUATOR system is discussed. The heliborne complex has a towed transmitter, and, raised above it on the same cable a towed receiver. The excitation signal contains both pulsed and harmonic components. In fact, there are two independent transmitters operate in the system: one of them is a normal pulsed domain transmitter, with a half-sinusoidal pulse and a small "cut" on the falling edge, and the other one is a classical frequency domain transmitter at several specially selected frequencies. The received signal is first processed to a direct Fourier transform with high Q-factor detection at all significant frequencies. After that, in the spectral region, operations of converting the spectra of two sounding signals to a single spectrum of an ideal transmitter are performed. Than we do an inverse Fourier transform and return to the time domain. The detection of spectral components is done at a frequency band of several Hz, the receiver has the ability to perfectly suppress all sorts of extra-band noise. The detection bandwidth is several dozen times less the frequency interval between the harmonics, it turns out thatto achieve the same measurement quality of ground response without using out-of-band suppression you need several dozen times higher moment of airborne transmitting system. The data obtained from the model of a homogeneous half-space, a two-layered model, and a model of a horizontally layered medium is considered. A time-domain data makes it easier to detect a conductor in a relative insulator at greater depths. The data in the frequency domain gives more detailed information about subsurface. These conclusions are illustrated by the example of processing the survey data of the Republic of Rwanda in 2017. The simultaneous inversion of data in frequency domain and time domain can significantly improve the quality of interpretation.

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