1 of 38

Non-linear Dynamic Analysis of Spur Gear pair with Rotor Bearing Clearances

Shubham Wasnik, Vikash Kumar, Shubhranshu Ranjan Sharma, Somnath Sarangi

Department of Mechanical Engineering, IIT Patna, India

  • Presence of backlash - as it provide clearance to prevent binding of mating gear.
  • Role of bearing clearance - influences the distribution of load and inadequate clearance can cause binding and mechanical failure.
  • Both the factors are source of nonlinearity in gear. Hence, these factors are important for nonlinear gear dynamics study

Research Motivation

2 of 38

3 Degree of Freedom Model

Paper ID: 171 Session: BEN IMSD-ACMD (16-20 Oct, 2022) New Delhi

2

14-10-2022

3 of 38

Literature Review

Paper ID: 171 Session: BEN IMSD-ACMD (16-20 Oct, 2022) New Delhi

3

14-10-2022

  • Ozguven and Houser [1] surveyed various linear geared models, which were predominantly dynamic models without taking into account nonlinear factors such as backlash, time-varying mesh stiffness (TVMS)
  • Wang et al. [2] surveyed nonlinear dynamic models of gear systems and identified some of the major non-linear factors like TVMS of gear teeth, backlash, static transmission error, friction between gear teeth, bearing clearance, and gear tooth profile imperfection.
  • Galhoud et al. [3] took into account a transnational system with two degrees of freedom (DOF) and a gap, and used the piece-wise linear method to discover the forced harmonic response.
  • Kahraman, A. and Singh, R. [4] given a simplified purely torsional mechanical model of a gear pair with backlash and static transmission error is modelled assuming shaft and bearing perfectly rigid.

4 of 38

Objective/Motivation

Paper ID: 171 Session: BEN IMSD-ACMD (16-20 Oct, 2022) New Delhi

4

14-10-2022

  • Most of the previous work is focused on linear model without considering some of the major nonlinear factors like gear backlash, bearing clearance and static transmission error.
  • Analysis and simulation of 3 DOF spur gear model with nonlinear factors like gear backlash, bearing clearance and static transmission error.
  • Model backlash function and bearing clearance function using piecewise linear function.
  • To reduce number of equation of motion to reduce the complexity using difference of dynamic transmission error and static transmission error.

5 of 38

Methodology

Paper ID: 171 Session: BEN IMSD-ACMD (16-20 Oct, 2022) New Delhi

5

14-10-2022

Figure 1: 3 DOF spur gear model with backlash and rotor bearing clearances

6 of 38

Methodology

Paper ID: 171 Session: BEN IMSD-ACMD (16-20 Oct, 2022) New Delhi

6

14-10-2022

Equation of motion

​

​

Combining equation of motion of pinion and gear in terms of

Here e(t) is static transmission error

7 of 38

Methodology

Paper ID: 171 Session: BEN IMSD-ACMD (16-20 Oct, 2022) New Delhi

7

14-10-2022

Combined equation is as follows

 

Final equation of motion is as follows

8 of 38

Methodology

Paper ID: 171 Session: BEN IMSD-ACMD (16-20 Oct, 2022) New Delhi

8

14-10-2022

Backlash function

Bearing clearance function

Dimensionless form of equation is as follows

9 of 38

Methodology

Paper ID: 171 Session: BEN IMSD-ACMD (16-20 Oct, 2022) New Delhi

9

14-10-2022

Where,

Parameter for simulation

10 of 38

Results

Paper ID: 171 Session: BEN IMSD-ACMD (16-20 Oct, 2022) New Delhi

10

14-10-2022

Figure 2: Period one response at 𝛺=1.5

(a) Time response plot

(b) Phase plot

(c) Poincare plot

11 of 38

Results

Paper ID: 171 Session: BEN IMSD-ACMD (16-20 Oct, 2022) New Delhi

11

14-10-2022

Figure 3: Period two response at 𝛺=1.48

(a) Time response plot

(b) Phase plot

(c) Poincare plot

12 of 38

Results

Paper ID: 171 Session: BEN IMSD-ACMD (16-20 Oct, 2022) New Delhi

12

14-10-2022

Figure 4: Period four response at 𝛺=1.44

(a) Time response plot

(b) Phase plot

(c) Poincare plot

13 of 38

Results

Paper ID: 171 Session: BEN IMSD-ACMD (16-20 Oct, 2022) New Delhi

13

14-10-2022

Figure 5: Period eight response at 𝛺=1.402

(a) Time response plot

(b) Phase plot

(c) Poincare plot

14 of 38

Results

Paper ID: 171 Session: BEN IMSD-ACMD (16-20 Oct, 2022) New Delhi

14

14-10-2022

Figure 6: Chaotic response at 𝛺=1.4

(a) Time response plot

(b) Phase plot

(c) Poincare plot

15 of 38

Discussion/Conclusion

Paper ID: 171 Session: BEN IMSD-ACMD (16-20 Oct, 2022) New Delhi

15

14-10-2022

​

  • Various non-linear factors like backlash, bearing clearance, static transmission error is incorporated in the model and their effects are analyzed on system's dynamic responses.
  • With the decrease in dimension less excitation frequency the response bifurcate from period one solution, to period two, four, eight and finally lead to chaotic response.

​

16 of 38

8 Degree of Freedom Model

Paper ID: 171 Session: BEN IMSD-ACMD (16-20 Oct, 2022) New Delhi

16

14-10-2022

17 of 38

Objective/Motivation

Paper ID: 171 Session: BEN IMSD-ACMD (16-20 Oct, 2022) New Delhi

17

14-10-2022

  • Most of the previous work is focused on linear model without considering some of the major nonlinear factors like gear backlash, bearing clearance and static transmission error, time-varying mesh stiffness (TVMS).
  • A eight degree of freedom spur gear model is formulated by incorporating improved TVMS and improved crack model with various crack depths.
  • This work aims to study effect of various nonlinear factors like gear backlash, static transmission error and bearing clearance with effect of various crack depth.
  • Piecewise linear function is used to model backlash function and bearing clearance function.

Motivation:

Contribution:

18 of 38

TVMS Model

Paper ID: 171 Session: BEN IMSD-ACMD (16-20 Oct, 2022) New Delhi

18

14-10-2022

 

Basic TVMS model [5]

Major flaws:

​

  • Misalignment between base and root circle
  • Exclusion of transition curve for the fillet portion of the tooth
  • Consideration of linear Hertzian contact stiffness
  • Calculation of fillet foundation stiffness without considering structural coupling effect of nearby loaded tooth

​

Figure 7: Base model [5]

19 of 38

Paper ID: 171 Session: BEN IMSD-ACMD (16-20 Oct, 2022) New Delhi

19

14-10-2022

Improved TVMS model

Figure 8: Gear model at root circle [6,7]

Figure 9: Structural coupling effect due to nearby loaded tooth [6,7]

20 of 38

Paper ID: 171 Session: BEN IMSD-ACMD (16-20 Oct, 2022) New Delhi

20

14-10-2022

 

21 of 38

Cracked Tooth Model

Paper ID: 171 Session: BEN IMSD-ACMD (16-20 Oct, 2022) New Delhi

21

14-10-2022

Basic crack model [5]

Limitation:

  • Limiting line is taken as straight line
  • Formulation is not accurate when crack propagates after central axis

​

Formulation:

 

 

Figure 10: Base crack model [5]

Limiting Line

22 of 38

Paper ID: 171 Session: BEN IMSD-ACMD (16-20 Oct, 2022) New Delhi

22

14-10-2022

Improved crack model [6,7]

Formulation:

 

 

 

 

q≤q1max

Limiting line

q>q1max

Limiting line

Figure 11: Improved crack model [6]

23 of 38

Paper ID: 171 Session: BEN IMSD-ACMD (16-20 Oct, 2022) New Delhi

23

14-10-2022

Figure 12: Eight DOF model with backlash, bearing clearance and static transmission error

Improved 8 DOF Model

24 of 38

Paper ID: 171 Session: BEN IMSD-ACMD (16-20 Oct, 2022) New Delhi

24

14-10-2022

Equation of motion pinion

Equation of motion of gear

Equation of motion of motor

Equation of motion of load

25 of 38

Paper ID: 171 Session: BEN IMSD-ACMD (16-20 Oct, 2022) New Delhi

25

14-10-2022

Bearing clearance function

Backlash function

 

Static transmission error

26 of 38

Paper ID: 171 Session: BEN IMSD-ACMD (16-20 Oct, 2022) New Delhi

26

14-10-2022

Simulation Parameter [6,7]

  • Fault is imparted on pinion
  • Pinion is rotating at 100/3 Hz
  • Sample rate – 100 KHz
  • Simulation time – 1.5 s
  • Last 1 sec steady state data is used

27 of 38

Results

Paper ID: 171 Session: BEN IMSD-ACMD (16-20 Oct, 2022) New Delhi

27

14-10-2022

Figure 13: (a) Location of fault (zp/2), (b) TVMS with respect to time

(a)

(b)

28 of 38

Paper ID: 171 Session: BEN IMSD-ACMD (16-20 Oct, 2022) New Delhi

28

14-10-2022

Figure 14: (a) Response, (b) Cepstrum at different crack depth

(a)

(b)

29 of 38

Paper ID: 171 Session: BEN IMSD-ACMD (16-20 Oct, 2022) New Delhi

29

14-10-2022

Figure 15: Dynamic response (a) without bearing clearance, (b) with bearing clearance

(a)

(b)

30 of 38

Paper ID: 171 Session: BEN IMSD-ACMD (16-20 Oct, 2022) New Delhi

30

14-10-2022

Figure 16: Plot at healthy tooth condition (a) Response, (b) FFT, (c) Cepstrum, (d) Phase portrait, (e) Poincare map

(b)

(a)

(c)

(d)

(e)

31 of 38

Paper ID: 171 Session: BEN IMSD-ACMD (16-20 Oct, 2022) New Delhi

31

14-10-2022

Figure 17: Plot at 20 % cracked tooth condition (a) Response, (b) FFT, (c) Cepstrum, (d) Phase portrait, (e) Poincare map

(b)

(a)

(c)

(e)

(d)

32 of 38

Paper ID: 171 Session: BEN IMSD-ACMD (16-20 Oct, 2022) New Delhi

32

14-10-2022

Figure 18: Plot at 40 % cracked tooth condition (a) Response, (b) FFT, (c) Cepstrum, (d) Phase portrait, (e) Poincare map

(a)

(b)

(c)

(d)

(e)

33 of 38

Paper ID: 171 Session: BEN IMSD-ACMD (16-20 Oct, 2022) New Delhi

33

14-10-2022

Figure 19: Plot at 60 % cracked tooth condition (a) Response, (b) FFT, (c) Cepstrum, (d) Phase portrait, (e) Poincare map

(d)

(e)

(a)

(b)

(c)

34 of 38

Discussion/Conclusion

Paper ID: 171 Session: BEN IMSD-ACMD (16-20 Oct, 2022) New Delhi

34

14-10-2022

  • The developed model is lashed with improved TVMS model and improved crack model with various crack depth.
  • Various non-linear factors like backlash, bearing clearance, static transmission error is incorporated in the model to makes the model more realistic.
  • Magnitude of vibration is more in case of bearing clearance.
  • Studied the non-linear characteristics of the system at different crack depth through Poincare map and phase portrait.
  • The Poincare map unveiled that the system dynamics changes significantly as the crack depth increases.

​

35 of 38

References

Paper ID: 171 Session: BEN IMSD-ACMD (16-20 Oct, 2022) New Delhi

35

14-10-2022

  1. Özgüven, H. N., & Houser, D. R. (1988). Mathematical models used in gear dynamics—a review. Journal of sound and vibration, 121(3), 383-411.
  2. Wang, J., Li and, R., & Peng, X. (2003). Survey of nonlinear vibration of gear transmission systems. Appl. Mech. Rev., 56(3), 309-329.
  3. Galhoud, L. E., Masri, S. F., & Anderson, J. C. (1987). Transfer function of a class of nonlinear multidegree of freedom oscillators.
  4. Kahraman, A., & Singh, R. (1990). Non-linear dynamics of a spur gear pair. Journal of sound and vibration, 142(1), 49-75.
  5. Mohammed, O. D., Rantatalo, M., & Aidanpää, J. O. (2015). Dynamic modelling of a one-stage spur gear system and vibration-based tooth crack detection analysis. Mechanical Systems and Signal Processing, 54, 293-305.
  6. Kumar, V., Rai, A., Mukherjee, S., & Sarangi, S. (2021). A Lagrangian approach for the electromechanical model of single-stage spur gear with tooth root cracks. Engineering Failure Analysis, 129, 105662.
  7. Kumar, V., Kumar, A., Kumar, S., & Sarangi, S. (2022). TVMS calculation and dynamic analysis of carburized spur gear pair. Mechanical Systems and Signal Processing, 166, 108436.

​

​

​

36 of 38

DoM Lab, IIT Patna, India

Paper ID: 171 Session: BEN IMSD-ACMD (16-20 Oct, 2022) New Delhi

36

14-10-2022

Figure 20: (a) Machine Fault Simulator, (b) Wind Turbine Drive Simulator

(a)

(b)

37 of 38

Research Scholar

Paper ID: 171 Session: BEN IMSD-ACMD (16-20 Oct, 2022) New Delhi

37

14-10-2022

Vikash Kumar

Research Area: Mathematical Modelling and Fault Diagnosis of Motor-Gearbox System

Subrata Kumar Behera

Research Area:

Continuum Mechanics

Sanjeev Kumar

Research Area:

Gearbox Fault Diagnosis

Rashi Aditi Ranjan

Research Area:

Non-linear Dynamics and Control

38 of 38

Thank you

Paper ID: 171 Session: BEN IMSD-ACMD (16-20 Oct, 2022) New Delhi

38

14-10-2022

Contact Us:

Dr. Somnath Sarangi, somsara@iitp.ac.in, +91 9470836481

Department of Mechanical Engineering

Indian Institute of Technology Patna, Bihar, India

​