Skip to main content
SpringerLink
Log in

Investigation of vibration characteristics of a Jeffcott rotor system under the influence of nonlinear restoring force, hydrodynamic effect, and gyroscopic effect

  • Technical Paper
  • Published:
Journal of the Brazilian Society of Mechanical Sciences and Engineering Aims and scope Submit manuscript

Abstract

A horizontally supported Jeffcott rotor system (JRS) is analyzed to determine the effect of several important factors on the horizontal and vertical oscillations of the JRS. These factors include the rotor eccentricity, the nonlinear restoring force due to large deformation of the rotor shaft in bending, the hydrodynamic force from the journal bearings, and the gyroscopic effect. Nonlinear analysis by the method of multiple scales is performed to obtain the autonomous amplitude-phase equations of the JRS for the simultaneous resonance condition. Both the localized and nonlocalized oscillations are analyzed. The amplitude-frequency response curves plotted by numerically simulating the amplitude-phase equations show some interesting results, including multiple solutions, jump phenomena, multiple loops, and quasiperiodic motions. Linear stability analysis is performed to verify the stability of steady-state solutions. Analytical expressions are derived for determining the critical values of different parameters corresponding to the initiation of turning points for localized oscillations. The study shows that the hydrodynamic parameter associated with the viscosity of lubricant used in journal bearings can effectively control the amplitude of vibration and unwanted jump phenomena.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18

Similar content being viewed by others

References

  1. Saito S (1985) Calculation of nonlinear unbalance response of horizontal Jeffcott rotors supported by ball bearings with radial clearances. J Vib Acoust Stress Reliab Trans ASME 107(4):416–420

    Article  Google Scholar 

  2. Shad MR, Michon G, Berlioz A (2012) Nonlinear dynamics of rotors due to large deformations and shear effects. Appl Mech Mater 110(116):3593–3599

    Google Scholar 

  3. Shad MR, Michon G, Berlioz A (2011) Analytical study of the dynamic behavior of geometrically nonlinear shaft-disk rotor systems. Mec Ind 12:433–443

    Google Scholar 

  4. Shad MR, Michon G, Berlioz A (2011) Modeling and analysis of nonlinear rotordynamics due to higher order deformations in bending. Appl Math Model 35:2145–2159

    Article  MathSciNet  Google Scholar 

  5. Choi SK, Noah ST (1994) Mode-locking and chaos in a jeffcott rotor with bearing clearances. J Appl Mech Trans ASME 61(1):131–138

    Article  Google Scholar 

  6. Kim YB, Noah ST (2015) Steady-state analysis of a nonlinear rotor-housing system. J Eng Gas Turbines Power Trans ASME 113(4):550–556

    Article  Google Scholar 

  7. Saeed NA, El-Gohary HA (2017) On the non-linear oscillations of a horizontally supported Jeffcott rotor with a non-linear restoring force. Nonlinear Dyn 88:293–314

    Article  Google Scholar 

  8. Xu Y, Luo ACJ (2020) Period-1 to period-8 motions in a nonlinear Jeffcott rotor system. J Comput Nonlinear Dyn Trans ASME 15:1–13

    Google Scholar 

  9. Kim YB, Noah ST (1990) Bifurcation analysis for a modified Jeffcott rotor with bearing clearances. Nonlinear Dyn 1(3):221–241

    Article  Google Scholar 

  10. Joseph PC, Marian W (2013) Bifurcation analysis of periodic orbits of a non-smooth Jeffcott rotor model. Commun Nonlinear Sci Numer Simul 18:2571–2580

    Article  MathSciNet  Google Scholar 

  11. Kim YB, Noah ST (1996) Quasi-periodic response and stability analysis for a non-linear Jeffcott rotor. J Sound Vib 190(2):239–253

    Article  Google Scholar 

  12. Yabuno H, Kashimura T, Inoue T, Ishida Y (2011) Non-linear normal modes and primary resonance of horizontally supported Jeffcott rotor. Nonlinear Dyn 66:377–387

    Article  Google Scholar 

  13. Ehrich FF (1988) High order sub-harmonic response of high-speed rotors in bearing clearance. J Vib Acoust Trans ASME 110(1):9–16

    Article  Google Scholar 

  14. Childs DW (1981) Fractional-frequency rotor motion due to nonsymmetric clearance effects. J Eng Power Trans ASME 104:533–541

    Article  Google Scholar 

  15. Cveticanin L (2005) Free vibration of a Jeffcott rotor with pure cubic non-linear elastic property of the shaft. Mech Mach Theory 40:1330–1344

    Article  Google Scholar 

  16. Yabuno H, Kunitho Y, Inoue T, and Ishida Y (2007) Non-linear Analysis of Rotor Dynamics by using the Method of Multiple Scales. In: Hu HY, Kreuzer E (eds) IUTAM Symposium on Dynamics and Control of Nonlinear Systems with Uncertainty, IUTAM Book Series, 2, p 167–176

  17. Wu J, Legrand M, and Pierre C (2010) Non-Synchronous Vibration of a Jeffcott Rotor due to Internal Radial Clearance in Roller Bearings. In: The 8th IFToMM International Conference on Rotor Dynamics (KIST), Sep 12–15, Seoul, South Korea, p 446–453

  18. Ganesan R (1996) Dynamic response and stability of a rotor-support system with non-symmetric bearing clearances. Mech Mach Theory 31(6):781–798

    Article  Google Scholar 

  19. Ganesan R (1997) Nonlinear vibrations and stability of a rotor-bearing system with non-symmetric clearances. J Eng Gas Turbines Power Trans ASME 119(2):418–424

    Article  Google Scholar 

  20. Adiletta G, Guido AR, Rossi C (1996) Non-periodic motions of a Jeffcott rotor with non-linear elastic restoring forces. Nonlinear Dyn 11:37–59

    Article  Google Scholar 

  21. Lahmar M, Bou-Said B (2015) Nonlinear dynamic response of an unbalanced flexible rotor supported by elastic bearings lubricated with Piezo-Viscous polar fluids. Lubricants 3:281–310

    Article  Google Scholar 

  22. Tiwari R (2017) Rotor systems: analysis and identification. CRC Press, Boca Raton

    Google Scholar 

  23. Zachariadis DC (2001) Critical speeds and unbalance response of a Jeffcott rotor on angular misaligned hydrodynamic bearings. SAE Tech. Pap. 2001-01-3912

  24. Reddy MR, Srinivas J (2016) Vibration analysis of a support excited rotor system with hydrodynamic journal bearings. Procedia Eng 144:825–832

    Article  Google Scholar 

  25. Byrtus M, Dyk S (2018) Rigid Jeffcott rotor bifurcation behaviour using different models of hydrodynamic bearings. Springer Proc Math Stat 248:75–85

    MathSciNet  MATH  Google Scholar 

  26. Boyaci A, Lu D, Schweizer B (2014) Stability and bifurcation phenomena of Laval/Jeffcott rotors in semi-floating ring bearings. Nonlinear Dyn 79:1535–1561

    Article  Google Scholar 

  27. Rajalingham C, Ganesan N, Prabhu BS (1986) Conditions for backward whirling motion of a flexible rotor supported on hydrodynamic journal bearings. J Sound Vib 111(1):29–36

    Article  Google Scholar 

  28. Rao J (1982) Conditions for backward synchronous whirl of a flexible rotor in hydrodynamic bearings. Mech Mach Theory 17(2):143–152

    Article  Google Scholar 

  29. Guo Z, Kirk RG (2003) Instability boundary for rotor-hydrodynamic bearing systems, part 1: Jeffcott rotor with external damping. J Vib Acoust Trans ASME 125:417–422

    Article  Google Scholar 

  30. Alnefaie K (2010) Start-up dynamic analysis of a rotor supported by fluid film bearings. J Vib Control 16(6):879–896

    Article  Google Scholar 

  31. Rajalingham C, Prabhu BS (1983) The effect of inlet film boundary conditions on the steady state characteristics of a hydrodynamic journal bearing. Wear 89(2):117–124

    Article  Google Scholar 

  32. Amer TS, Galal AA, Abady IM, Elkafly HF (2021) The dynamical motion of a gyrostat for the irrational frequency case. Appl Math Model 89:1235–1267

    Article  MathSciNet  Google Scholar 

  33. Duchemin M, Berlioz A, Ferraris G (2006) Dynamic behavior and stability of a rotor under base excitation. J Vib Acoust Trans ASME 128(5):576–585

    Article  Google Scholar 

  34. Carrella A, Friswell MI, Zotov A, Ewins DJ, Tichonov A (2009) Using nonlinear springs to reduce the whirling of a rotating shaft. Mech Syst Signal Process 23:2228–2235

    Article  Google Scholar 

  35. Khanlo HM, Ghayour M, Ziaei-Rad S (2012) Disk position nonlinearity effects on the chaotic behavior of rotating flexible shaft-disk systems. J Mech 28(3):513–522

    Article  Google Scholar 

  36. Samantaray AK (2009) Steady-state dynamics of a non-ideal rotor with internal damping and gyroscopic effects. Nonlinear Dyn 56:443–451

    Article  Google Scholar 

  37. Ishida Y, Inoue T (2004) Internal resonance phenomena of the Jeffcott rotor with nonlinear spring characteristics. J Vib Acoust Trans ASME 126(4):476–484

    Article  Google Scholar 

  38. Mittal RK, Kulkarni SS, Singh RK (2018) Multiple degree of freedom rotordynamics of stability modeling in high-speed micromilling of Ti-6Al-4V. Procedia Manuf 26:607–616

    Article  Google Scholar 

  39. Jahromi AF, Bhat RB, Xie WF (2015) Forward and backward whirling of a rotor with gyroscopic effect. In: Sinha JK (ed) Vibration engineering and technology of machinery, mechanisms and machine science. Springer International Publishing, Switzerland, pp 879–887 (23)

    Chapter  Google Scholar 

  40. Greenhill LM, Cornejo GA (1995) Critical speeds resulting from unbalance excitation of backward whirl modes. Am Soc Mech Eng Des Eng Div 84(2):991–1000

    Google Scholar 

  41. El-Sabaa FM, Amer TS, Gad HM, Bek MA (2020) On the motion of a damped rigid body near resonances under the influence of harmonically external force and moments. Results in Physics 19:103352

    Article  Google Scholar 

  42. Nayfeh A (2005) Resolving controversies in the application of the method of multiple scales and the generalized method of averaging. Nonlinear Dyn 40:61–102

    Article  MathSciNet  Google Scholar 

  43. Zhu X, Wu Z (2014) Equilibrium point bifurcation and singularity analysis of HH model with constraint. Abstr Appl Anal 2014(SI37):1–8

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work has been supported by DST SERB—SRG under Project File no. SRG/2019/001445 and National Institute of Technology Calicut under Faculty Research Grant.

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, National Institute of Technology Calicut, Calicut, Kerala, 673601, India

    Amit Malgol, K. P. Vineesh & Ashesh Saha

Authors
  1. Amit Malgol
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. K. P. Vineesh
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ashesh Saha
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Amit Malgol.

Additional information

Technical Editor: Samuel da Silva.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendices

Appendix 1

The time derivatives of \(A\left( {T_{1} ,T_{2} } \right)\) and \(B\left( {T_{1} ,T_{2} } \right)\) w.r.t \(T_{2}\) are given by

$$ \begin{gathered} \frac{{\partial A\left( {T_{1} ,T_{2} } \right)}}{{\partial T_{2} }} = \frac{I}{{8\left( {\omega_{2} + 2\omega_{1} } \right)\left( {2\omega_{1} - \omega_{2} } \right)\varepsilon \omega_{1}^{2} \omega_{2}^{2} }}( - 24\omega_{1} B\left( {T_{1} ,T_{2} } \right)^{2} \lambda^{2} \overline{A}\left( {T_{1} ,T_{2} } \right)\varepsilon \omega_{2}^{2} \hfill \\ e^{{ - 2iT_{0} \left( { - \omega_{2} + \omega_{1} } \right)}} + 16\omega_{1}^{3} \lambda B\left( {T_{1} ,T_{2} } \right)^{2} \overline{A}\left( {T_{1} ,T_{2} } \right)\omega_{2}^{2} \varepsilon e^{{ - 2iT_{0} \left( { - \omega_{2} + \omega_{1} } \right)}} - 32\omega_{1}^{2} B\left( {T_{1} ,T_{2} } \right)^{2} \lambda^{2} \hfill \\ \overline{A}\left( {T_{1} ,T_{2} } \right)\omega_{2} \varepsilon e^{{ - 2iT_{0} \left( { - \omega_{2} + \omega_{1} } \right)}} - 4\lambda \omega_{1} B\left( {T_{1} ,T_{2} } \right)^{2} \overline{A}\left( {T_{1} ,T_{2} } \right)\omega_{2}^{4} \varepsilon e^{{ - 2iT_{0} \left( { - \omega_{2} + \omega_{1} } \right)}} + 32\omega_{1}^{3} \hfill \\ B\left( {T_{1} ,T_{2} } \right)^{2} \lambda^{2} \overline{A}\left( {T_{1} ,T_{2} } \right)\varepsilon e^{{ - 2iT_{0} \left( { - \omega_{2} + \omega_{1} } \right)}} + 24\lambda^{2} A\left( {T_{1} ,T_{2} } \right)^{2} \overline{A}\left( {T_{1} ,T_{2} } \right)\omega_{2}^{2} \varepsilon \omega_{1} - 192\omega_{1}^{3} \hfill \\ B\left( {T_{1} ,T_{2} } \right)\lambda^{2} \overline{B}\left( {T_{1} ,T_{2} } \right)A\left( {T_{1} ,T_{2} } \right)\varepsilon + 48\omega_{1}^{3} \lambda A\left( {T_{1} ,T_{2} } \right)^{2} \overline{A}\left( {T_{1} ,T_{2} } \right)\varepsilon \omega_{2}^{2} - 12\lambda \omega_{1} A\left( {T_{1} ,T_{2} } \right)^{2} \hfill \\ \overline{A}\left( {T_{1} ,T_{2} } \right)\omega_{2}^{3} \varepsilon - 12\lambda \omega_{1} A\left( {T_{1} ,T_{2} } \right)^{2} \overline{A}\left( {T_{1} ,T_{2} } \right)\omega_{2}^{4} \varepsilon + 16I\omega_{1}^{3} G_{h} B\left( {T_{1} ,T_{2} } \right)\omega_{2}^{3} \varepsilon \hfill \\ e^{{ - iT_{0} \left( { - \omega_{2} + \omega_{1} } \right)}} - 4i\omega_{1}^{2} \mu_{h1} A\left( {T_{1} ,T_{2} } \right)\omega_{2}^{4} \varepsilon - 4i\omega_{1} G_{h} B\left( {T_{1} ,T_{2} } \right)\omega_{2}^{5} \varepsilon e^{{ - iT_{0} \left( { - \omega_{2} + \omega_{1} } \right)}} \hfill \\ + 16i\omega_{1}^{4} H_{h} A\left( {T_{1} ,T_{2} } \right)\omega_{2}^{2} \varepsilon + 16i\omega_{1}^{4} \mu_{h1} A\left( {T_{1} ,T_{2} } \right)\omega_{2}^{2} \varepsilon - 4i\omega_{1}^{2} H_{h} A\left( {T_{1} ,T_{2} } \right)\omega_{2}^{4} \varepsilon \hfill \\ - 64\omega_{1}^{3} \lambda^{2} A\left( {T_{1} ,T_{2} } \right)^{2} \overline{A}\left( {T_{1} ,T_{2} } \right)\varepsilon - 4\omega_{1}^{3} f_{h} {\Omega }^{2} \omega_{2}^{2} e^{{\frac{{iT_{1} \left( { - \omega_{1} + {\Omega }} \right)}}{\varepsilon }}} + 4\omega_{1}^{2} f_{h} {\Omega }^{3} \omega_{2}^{2} e^{{\frac{{iT_{1} \left( { - \omega_{1} + {\Omega }} \right)}}{\varepsilon }}} \hfill \\ + \omega_{1} f_{h} {\Omega }^{2} \omega_{2}^{4} e^{{\frac{{iT_{1} \left( { - \omega_{1} + {\Omega }} \right)}}{\varepsilon }}} - f_{h} {\Omega }^{3} \omega_{2}^{4} e^{{\frac{{iT_{1} \left( { - \omega_{1} + {\Omega }} \right)}}{\varepsilon }}} + 32\omega_{1}^{3} \lambda B\left( {T_{1} ,T_{2} } \right)\overline{B}\left( {T_{1} ,T_{2} } \right)A\left( {T_{1} ,T_{2} } \right) \hfill \\ \omega_{2}^{2} \varepsilon + 16\omega_{1} B\left( {T_{1} ,T_{2} } \right)\lambda^{2} \overline{B}\left( {T_{1} ,T_{2} } \right)A\left( {T_{1} ,T_{2} } \right)\omega_{2}^{2} \varepsilon - 8\lambda \omega_{1} B\left( {T_{1} ,T_{2} } \right)\overline{B}\left( {T_{1} ,T_{2} } \right)A\left( {T_{1} ,T_{2} } \right) \hfill \\ \omega_{2}^{4} \varepsilon ), \hfill \\ \end{gathered} $$
$$ \begin{gathered} \frac{{\partial B\left( {T_{1} ,T_{2} } \right)}}{{\partial T_{2} }} = \frac{1}{{8e^{{ - iT_{0} \left( { - \omega_{2} + \omega_{1} } \right)}} \left( {2\omega_{1} - \omega_{2} } \right)\left( {2\omega_{1} + \omega_{2} } \right)\varepsilon \omega_{2}^{3} }}( - 4\omega_{1}^{2} f_{h} {\Omega }^{2} \omega_{2}^{2} \hfill \\ e^{{\frac{{i\left( { - \omega_{1} T_{0} \varepsilon - T_{1} \omega_{2} + \omega_{2} \varepsilon T_{0} + T_{1} {\Omega }} \right)}}{\varepsilon }}} + 16G_{h} A\left( {T_{1} ,T_{2} } \right)\varepsilon \omega_{2}^{2} \omega_{1}^{3} + 4\mu_{h2} B\left( {T_{1} ,T_{2} } \right)\omega_{2}^{5} \varepsilon e^{{ - iT_{0} \left( { - \omega_{2} + \omega_{1} } \right)}} \hfill \\ + 8i\lambda^{2} A\left( {T_{1} ,T_{2} } \right)^{2} \overline{B}\left( {T_{1} ,T_{2} } \right)\omega_{2}^{2} \varepsilon e^{{iT_{0} \left( { - \omega_{2} + \omega_{1} } \right)}} - 4i\lambda A\left( {T_{1} ,T_{2} } \right)^{2} \overline{B}\left( {T_{1} ,T_{2} } \right)\omega_{2}^{4} \varepsilon e^{{iT_{0} \left( { - \omega_{2} + \omega_{1} } \right)}} \hfill \\ + 32i\omega_{1}^{2} \lambda A\left( {T_{1} ,T_{2} } \right)\overline{A}\left( {T_{1} ,T_{2} } \right)B\left( {T_{1} ,T_{2} } \right)\omega_{2}^{2} \varepsilon e^{{ - iT_{0} \left( { - \omega_{2} + \omega_{1} } \right)}} - f_{h} {\Omega }^{3} \omega_{2}^{3} \hfill \\ e^{{\frac{{i\left( { - \omega_{1} T_{0} \varepsilon - T_{1} \omega_{2} + \omega_{2} \varepsilon T_{0} + T_{1} {\Omega }} \right)}}{\varepsilon }}} - 12i\lambda B\left( {T_{1} ,T_{2} } \right)^{2} \overline{B}\left( {T_{1} ,T_{2} } \right)\varepsilon \omega_{2}^{4} e^{{ - iT_{0} \left( { - \omega_{2} + \omega_{1} } \right)}} + 4H_{h} B\left( {T_{1} ,T_{2} } \right) \hfill \\ \omega_{2}^{4} e^{{ - iT_{0} \left( { - \omega_{2} + \omega_{1} } \right)}} + 4H_{h} B\left( {T_{1} ,T_{2} } \right)\omega_{2}^{5} \varepsilon e^{{ - iT_{0} \left( { - \omega_{2} + \omega_{1} } \right)}} - 480i\omega_{1}^{2} B\left( {T_{1} ,T_{2} } \right)^{2} \lambda^{2} \overline{B}\left( {T_{1} ,T_{2} } \right) \hfill \\ \varepsilon e^{{ - iT_{0} \left( { - \omega_{2} + \omega_{1} } \right)}} - 16\omega_{1}^{2} H_{h} B\left( {T_{1} ,T_{2} } \right)\omega_{2}^{3} \varepsilon e^{{ - iT_{0} \left( { - \omega_{2} + \omega_{1} } \right)}} - 32i\omega_{1} A\left( {T_{1} ,T_{2} } \right)^{2} \lambda^{2} \overline{B}\left( {T_{1} ,T_{2} } \right) \hfill \\ \omega_{2} \varepsilon e^{{ - iT_{0} \left( { - \omega_{2} + \omega_{1} } \right)}} - 4\omega_{1} G_{h} A\left( {T_{1} ,T_{2} } \right)\omega_{2}^{4} \varepsilon + 16iB\left( {T_{1} ,T_{2} } \right)\lambda^{2} A\left( {T_{1} ,T_{2} } \right)\overline{A}\left( {T_{1} ,T_{2} } \right)\varepsilon \omega_{2}^{2} \hfill \\ e^{{ - iT_{0} \left( { - \omega_{2} + \omega_{1} } \right)}} + 120iB\left( {T_{1} ,T_{2} } \right)^{2} \lambda^{2} \overline{B}\left( {T_{1} ,T_{2} } \right)\varepsilon \omega_{2}^{2} e^{{ - iT_{0} \left( { - \omega_{2} + \omega_{1} } \right)}} - 192i\omega_{1}^{2} B\left( {T_{1} ,T_{2} } \right)\lambda^{2} \hfill \\ A\left( {T_{1} ,T_{2} } \right)\overline{A}\left( {T_{1} ,T_{2} } \right)\varepsilon e^{{ - iT_{0} \left( { - \omega_{2} + \omega_{1} } \right)}} + 16i\omega_{1}^{2} \lambda A\left( {T_{1} ,T_{2} } \right)^{2} \overline{B}\left( {T_{1} ,T_{2} } \right)\omega_{2}^{2} \varepsilon e^{{iT_{0} \left( { - \omega_{2} + \omega_{1} } \right)}} \hfill \\ + 4\omega_{1}^{2} f_{h} {\Omega }^{3} \omega_{2} e^{{\frac{{i\left( { - \omega_{1} T_{0} \varepsilon - T_{1} \omega_{2} + \omega_{2} \varepsilon T_{0} + T_{1} {\Omega }} \right)}}{\varepsilon }}} + 48i\omega_{1}^{2} \lambda B\left( {T_{1} ,T_{2} } \right)^{2} \overline{B}\left( {T_{1} ,T_{2} } \right)\varepsilon \omega_{2}^{2} \hfill \\ e^{{ - iT_{0} \left( { - \omega_{2} + \omega_{1} } \right)}} - 16\omega_{1}^{2} \mu_{h2} B\left( {T_{1} ,T_{2} } \right)\omega_{2}^{3} \varepsilon e^{{ - iT_{0} \left( { - \omega_{2} + \omega_{1} } \right)}} - 8i\lambda A\left( {T_{1} ,T_{2} } \right)\overline{A}\left( {T_{1} ,T_{2} } \right)B\left( {T_{1} ,T_{2} } \right) \hfill \\ \omega_{2}^{4} \varepsilon e^{{ - iT_{0} \left( { - \omega_{2} + \omega_{1} } \right)}} + f_{h} {\Omega }^{2} \omega_{2}^{4} e^{{\frac{{i\left( { - \omega_{1} T_{0} \varepsilon - T_{1} \omega_{2} + \omega_{2} \varepsilon T_{0} + T_{1} {\Omega }} \right)}}{\varepsilon }}} . \hfill \\ \end{gathered} $$

Appendix 2

The coefficients of the Jacobian matrix [\({\text{J}}\)] as shown in Eq. (27) are expressed as

$$ J_{11} = - \frac{1}{2}\mu_{1} - \frac{1}{2}H - \frac{\lambda }{{8\omega_{1} }}a_{20}^{2} \sin \left( {2\phi_{10} - 2\phi_{20} } \right) - \frac{{a_{20}^{2} \sin \left( {2\phi_{10} - 2\phi_{20} } \right)\lambda^{2} }}{{4\omega_{1} \left( {2\omega_{1} - \omega_{2} } \right)\omega_{2}^{2} }}\left( {2\omega_{1} - 3\omega_{2} } \right), $$
$$ J_{12} = \frac{1}{{\omega_{1} }}Ga_{20} \omega_{2} \sin \left( {\phi_{10} - \phi_{20} } \right) - \frac{\lambda }{{4\omega_{1} }}a_{20}^{2} a_{10} \cos \left( {2\phi_{10} - 2\phi_{20} } \right) - \frac{{a_{20}^{2} \cos \left( {2\phi_{10} - 2\phi_{20} } \right)\lambda^{2} a_{10} }}{{2\omega_{1} \left( {2\omega_{1} - \omega_{2} } \right)\omega_{2}^{2} }}\left( {2\omega_{1} - 3\omega_{2} } \right) - \frac{{f{\Omega }^{2} \left( { - 3\omega_{1} + {\Omega }} \right)}}{{4\omega_{1}^{2} }}\cos \left( {\phi_{10} } \right), $$
$$ J_{13} = - \frac{1}{{2\omega_{1} }}G\omega_{2} \cos \left( {\phi_{10} - \phi_{20} } \right) - \frac{\lambda }{{4\omega_{1} }}a_{20} a_{10} \sin \left( {2\phi_{10} - 2\phi_{20} } \right) - \frac{{a_{20} \sin \left( {4\phi_{10} - 4\phi_{20} } \right)\lambda^{2} a_{10} }}{{2\omega_{1} \left( {2\omega_{1} - \omega_{2} } \right)\omega_{2}^{2} }}\left( {2\omega_{1} - 3\omega_{2} } \right), $$
$$ J_{14} = - \frac{1}{{2\omega_{1} }}Ga_{20} \omega_{2} \sin \left( {\phi_{10} - \phi_{20} } \right) + \frac{\lambda }{{4\omega_{1} }}a_{20}^{2} a_{10} \cos \left( {2\phi_{10} - 2\phi_{20} } \right) - \frac{{4a_{20}^{2} \cos \left( {2\phi_{10} - 2\phi_{20} } \right)\lambda^{2} a_{10} }}{{\omega_{1} \left( {2\omega_{1} - \omega_{2} } \right)\omega_{2}^{2} }}\left( {2\omega_{1} - 3\omega_{2} } \right), $$
$$ J_{21} = \frac{{f{\Omega }^{2} \left( {{\Omega } - 3\omega_{1} } \right)}}{{4a_{10}^{2} \omega_{1}^{2} }}\cos \left( {\phi_{10} } \right) - \frac{{3\lambda a_{10} }}{{4\omega_{1} }} + \frac{{\lambda^{2} }}{{4\left( {4\omega_{1}^{2} - \omega_{2}^{2} } \right)\omega_{1} \omega_{2}^{2} }}\left( { - 6a_{10} \omega_{2}^{2} + 16\omega_{1}^{2} a_{10} } \right) - \frac{1}{{2a_{10}^{2} \omega_{1} }}Ga_{20} \omega_{2} \sin \left( {\phi_{10} - \phi_{20} } \right), $$
$$ J_{22} = \frac{{f{\Omega }^{2} \left( {{\Omega } - 3\omega_{1} } \right)}}{{4a_{10} \omega_{1}^{2} }}\sin \left( {2\phi_{10} } \right) + \frac{{\lambda a_{20}^{2} }}{{4\omega_{1} }}\sin \left( {2\phi_{10} - 2\phi_{10} } \right) + \frac{{a_{20}^{2} \sin \left( {2\phi_{10} - 2\phi_{20} } \right)\lambda^{2} }}{{2\omega_{1} \left( {2\omega_{1} - \omega_{2} } \right)\omega_{2}^{2} }}\left( {2\omega_{1} - 3\omega_{2} } \right) + \frac{1}{{2a_{10} \omega_{1} }}Ga_{20} \omega_{2} \cos \left( {\phi_{10} - \phi_{20} } \right), $$
$$ J_{23} = - \frac{\lambda }{{8\omega_{1} }}\left( {4a_{20} + 2a_{20} \cos \left( {2\phi_{10} - 2\phi_{20} } \right)} \right) + \frac{{\lambda^{2} }}{{4\left( {4\omega_{1}^{2} - \omega_{2}^{2} } \right)\omega_{1} \omega_{2}^{2} }}(48\omega_{1}^{2} a_{20} - 4\omega_{2}^{2} a_{20} ) - \frac{{a_{20} \cos \left( {2\phi_{10} - 2\phi_{10} } \right)\lambda^{2} }}{{2\omega_{1} \left( {2\omega_{1} - \omega_{2} } \right)\omega_{2}^{2} }}\left( {2\omega_{1} - 3\omega_{2} } \right) + \frac{1}{{2a_{10} \omega_{1} }}G\omega_{2} \sin \left( {\phi_{10} - \phi_{20} } \right), $$
$$ J_{24} = - \frac{{\lambda a_{20}^{2} }}{{4\omega_{1} }}\sin \left( {2\phi_{10} - 2\phi_{20} } \right) - \frac{{a_{20}^{2} \sin \left( {2\phi_{10} - 2\phi_{10} } \right)\lambda^{2} }}{{2\omega_{1} \left( {2\omega_{1} - \omega_{2} } \right)\omega_{2}^{2} }}\left( {2\omega_{1} - 3\omega_{2} } \right) - \frac{1}{{2a_{10} \omega_{1} }}Ga_{20} \omega_{2} \cos \left( {\phi_{10} - \phi_{20} } \right), $$
$$ J_{31} = \frac{1}{{2\omega_{2} }}G\omega_{1} \cos \left( {\phi_{10} - \phi_{20} } \right) + \frac{\lambda }{{4\omega_{2} }}a_{10} a_{20} \sin \left( {2\phi_{10} - 2\phi_{20} } \right) - \frac{{a_{10} \sin \left( {2\phi_{10} - 2\phi_{20} } \right)\lambda^{2} a_{20} }}{{2\left( {4\omega_{1}^{2} - \omega_{2}^{2} } \right)\omega_{2}^{2} }}\left( { - \omega_{2} + 4\omega_{1} } \right), $$
$$ J_{32} = - \frac{1}{{2\omega_{2} }}G\omega_{1} a_{10} \sin \left( {\phi_{10} - \phi_{20} } \right) + \frac{\lambda }{{4\omega_{2} }}a_{10}^{2} a_{20} {\text{cos}}\left( {2\phi_{10} - 2\phi_{20} } \right) - \frac{{a_{10}^{2} \cos \left( {2\phi_{10} - 2\phi_{20} } \right)\lambda^{2} a_{20} }}{{2\left( {4\omega_{1}^{2} - \omega_{2}^{2} } \right)\omega_{2}^{2} }}\left( { - \omega_{2} + 4\omega_{1} } \right), $$
$$ J_{33} = - \frac{1}{2}\mu_{2} - \frac{1}{2}H + \frac{\lambda }{{8\omega_{2} }}a_{10}^{2} \sin \left( {2\phi_{10} - 2\phi_{20} } \right) - \frac{{a_{10}^{2} \sin \left( {2\phi_{10} - 2\phi_{20} } \right)\lambda^{2} }}{{4\left( {4\omega_{1}^{2} - \omega_{2}^{2} } \right)\omega_{2}^{2} }}\left( { - \omega_{2} + 4\omega_{1} } \right), $$
$$ J_{34} = \frac{1}{{\omega_{2} }}Ga_{10} \omega_{1} \sin \left( {\phi_{10} - \phi_{20} } \right) - \frac{\lambda }{{4\omega_{2} }}a_{10}^{2} a_{20} {\text{cos}}\left( {2\phi_{10} - 2\phi_{20} } \right) - \frac{{a_{10}^{2} {\text{cos}}\left( {2\phi_{10} - 2\phi_{20} } \right)\lambda^{2} a_{20} }}{{2\left( {4\omega_{1}^{2} - \omega_{2}^{2} } \right)\omega_{2}^{2} }}\left( { - \omega_{2} + 4\omega_{1} } \right) - \frac{{f{\Omega }^{2} \left( {{\Omega } - 3\omega_{2} } \right)}}{{4\omega_{2}^{2} }}\sin \left( {2\phi_{20} } \right), $$
$$ J_{41} = - \frac{\lambda }{{8\omega_{2} }}\left( {4a_{10} + 2a_{10} \cos \left( {2\phi_{10} - 2\phi_{20} } \right)} \right) + \frac{{\lambda^{2} }}{{4\left( {4\omega_{1}^{2} - \omega_{2}^{2} } \right)\omega_{2}^{3} }}( - 4a_{10} \omega_{2}^{2} + 48\omega_{1}^{2} a_{10} ) + \frac{{a_{10} \cos \left( {2\phi_{10} - 2\phi_{20} } \right)\lambda^{2} }}{{2\omega_{2}^{2} \left( {4\omega_{1}^{2} - \omega_{2}^{2} } \right)}}\left( {4\omega_{1} - \omega_{2} } \right) + \frac{1}{{2a_{20} \omega_{2} }}G\omega_{1} \sin \left( {2\phi_{10} - 2\phi_{20} } \right), $$
$$ J_{42} = \frac{{\lambda a_{10}^{2} }}{{4\omega_{2} }}\sin \left( {2\phi_{10} - 2\phi_{20} } \right) - \frac{{\lambda^{2} a_{10}^{2} \sin \left( {2\phi_{10} - 2\phi_{20} } \right)}}{{2\omega_{2}^{2} \left( {4\omega_{1}^{2} - \omega_{2}^{2} } \right)}}\left( {4\omega_{1} - \omega_{2} } \right) + \frac{1}{{2a_{20} \omega_{2} }}Ga_{10} \omega_{1} \cos \left( {\phi_{10} - \phi_{20} } \right), $$
$$ J_{43} = \frac{{f{\Omega }^{2} \left( {{\Omega } - 3\omega_{2} } \right)}}{{4a_{20}^{2} \omega_{2}^{2} }}{\text{sin}}\left( {\phi_{20} } \right) - \frac{{3\lambda a_{20} }}{{4\omega_{2} }} + \frac{{\lambda^{2} }}{{4\left( {4\omega_{1}^{2} - \omega_{2}^{2} } \right)\omega_{2}^{3} }}\left( {120\omega_{1}^{2} a_{20} - 30\omega_{2}^{2} a_{20} } \right) - \frac{1}{{2a_{20}^{2} \omega_{2} }}Ga_{10} \omega_{1} \sin \left( {\phi_{10} - \phi_{20} } \right), $$
$$ J_{44} = - \frac{{f{\Omega }^{2} \left( {{\Omega } - 3\omega_{2} } \right)}}{{4a_{20} \omega_{2}^{2} }}{\text{cos}}\left( {\phi_{20} } \right) - \frac{\lambda }{{4\omega_{2} }}a_{10}^{2} \sin \left( {2\phi_{10} - 2\phi_{20} } \right) + \frac{{a_{10}^{2} \sin \left( {2\phi_{10} - 2\phi_{20} } \right)\lambda^{2} }}{{2\left( {4\omega_{1}^{2} - \omega_{2}^{2} } \right)\omega_{2}^{2} }}\left( {4\omega_{1} - \omega_{2} } \right) - \frac{1}{{2a_{20} \omega_{2} }}Ga_{10} \omega_{1} \cos \left( {\phi_{10} - \phi_{20} } \right). $$

Appendix 3

The coefficients of characteristic Eq. (29) are \(\alpha_{1}\), \(\alpha_{2}\), \(\alpha_{3}\) and \(\alpha_{4}\) can be written as

$$ \alpha_{1} = - J_{44} - J_{22} - J_{11} - J_{33} , $$
$$ \alpha_{2} = - J_{14} J_{41} - J_{12} J_{21} - J_{13} J_{31} + J_{11} J_{44} + J_{22} J_{44} - J_{42} J_{24} + J_{33} J_{44} - J_{34} J_{43} + J_{11} J_{33} + J_{22} J_{33} - J_{32} J_{23} + J_{11} J_{22} , $$
$$ \begin{gathered} \alpha_{3} = J_{14} J_{41} J_{22} + J_{42} J_{24} J_{33} - J_{22} J_{33} J_{44} + J_{34} J_{43} J_{11} - J_{12} J_{31} J_{23} + J_{34} J_{43} J_{22} \hfill \\ \quad - J_{42} J_{14} J_{21} + J_{13} J_{31} J_{44} - J_{22} J_{11} J_{44} + J_{12} J_{21} J_{33} - J_{11} J_{33} J_{44} - J_{14} J_{31} J_{43} \hfill \\ \quad + J_{32} J_{23} J_{11} - J_{12} J_{41} J_{24} + J_{32} J_{23} J_{44} - J_{32} J_{13} J_{21} - J_{22} J_{33} J_{11} - J_{42} J_{23} J_{34} \hfill \\ \quad + J_{34} J_{13} J_{41} + J_{14} J_{41} J_{33} + J_{42} J_{24} J_{11} + J_{12} J_{21} J_{44} - J_{32} J_{43} J_{24} + J_{13} J_{31} J_{22} , \hfill \\ \end{gathered} $$
$$ \begin{gathered} \alpha_{4} = J_{22} J_{11} J_{33} J_{44} - J_{32} J_{13} J_{41} J_{24} + J_{32} J_{23} J_{14} J_{41} - J_{12} J_{31} J_{43} J_{24} - J_{42} J_{23} J_{14} J_{31} \hfill \\ \quad - J_{34} J_{43} J_{22} J_{11} - J_{34} J_{43} J_{22} J_{11} + J_{42} J_{13} J_{31} J_{24} - J_{12} J_{41} J_{23} J_{34} + J_{32} J_{43} J_{24} J_{11} \hfill \\ \quad + J_{42} J_{14} J_{21} J_{33} - J_{32} J_{23} J_{11} J_{44} + J_{34} J_{13} J_{41} J_{22} + J_{12} J_{31} J_{23} J_{44} + J_{42} J_{23} J_{34} J_{11} \hfill \\ \quad - J_{32} J_{43} J_{14} J_{21} - J_{42} J_{24} J_{11} J_{33} + J_{12} J_{41} J_{24} J_{33} - J_{13} J_{31} J_{22} J_{44} - J_{14} J_{41} J_{22} J_{33} \hfill \\ \quad - J_{42} J_{13} J_{21} J_{34} - J_{12} J_{21} J_{33} J_{44} + J_{14} J_{31} J_{43} J_{22} + J_{32} J_{13} J_{21} J_{44} + J_{12} J_{21} J_{34} J_{43} . \hfill \\ \end{gathered} $$

Appendix 4

The coefficients of Eq. (32) are written as

$$ \begin{gathered} q_{6} = 256\omega_{1}^{6} \lambda^{4} - 36\omega_{1}^{2} \lambda^{3} \omega_{2}^{6} - 72\omega_{1}^{4} \lambda^{2} \omega_{2}^{6} + 144\omega_{1}^{6} \lambda^{2} \omega_{2}^{4} - 384\omega_{1}^{6} \lambda^{3} \omega_{2}^{2} - 192\omega_{1}^{4} \lambda^{4} \omega_{2}^{2} \hfill \\ \quad + 240\omega_{1}^{4} \lambda^{3} \omega_{2}^{4} + 36\omega_{1}^{2} \lambda^{4} \omega_{2}^{4} + 9\omega_{1}^{2} \lambda^{2} \omega_{2}^{8} . \hfill \\ \end{gathered} $$
$$ p_{6} = 900\lambda^{4} + 9\lambda^{2} \omega_{2}^{4} - 180\lambda^{3} \omega_{2}^{2} . $$
$$ q_{41} = 1024\omega_{1}^{7} \lambda^{2} \omega_{2}^{2} + 384\omega_{1}^{5} \lambda \omega_{2}^{6} + 96\omega_{1}^{3} \lambda^{2} \omega_{2}^{6} - 640\omega_{1}^{5} \lambda^{2} \omega_{2}^{4} - 48\omega_{1}^{3} \lambda \omega_{2}^{8} - 768\omega_{1}^{7} \lambda \omega_{2}^{4} . $$
$$ p_{41} = - 48\omega_{2}^{5} \lambda {\Omega } + 480\lambda^{2} \omega_{2}^{3} . $$
$$ q_{40} = - 1024\omega_{1}^{8} \lambda^{2} \omega_{2}^{2} - 384\omega_{1}^{6} \lambda \omega_{2}^{6} - 96\omega_{1}^{4} \lambda^{2} \omega_{2}^{6} + 640\omega_{1}^{6} \lambda^{2} \omega_{2}^{4} + 48\omega_{1}^{4} \lambda \omega_{2}^{8} + 768\omega_{1}^{8} \lambda \omega_{2}^{4} . $$
$$ p_{40} = 48\lambda \omega_{2}^{6} - 480\lambda^{2} \omega_{2}^{4} . $$
$$ q_{22} = 1024\omega_{1}^{8} \omega_{2}^{4} - 512\omega_{1}^{6} \omega_{2}^{6} + 64\omega_{1}^{4} \omega_{2}^{8} . $$
$$ p_{22} = 64\omega_{2}^{6} . $$
$$ q_{21} = - 2048\omega_{1}^{9} \omega_{2}^{4} - 128\omega_{1}^{5} \omega_{2}^{8} + 1024\omega_{1}^{7} \omega_{2}^{6} . $$
$$ p_{21} = - 128\omega_{2}^{7} . $$
$$ q_{20} = - 256\omega_{1}^{6} H\omega_{2}^{6} \mu_{1} + 16\omega_{1}^{4} \omega_{2}^{8} \mu_{1}^{2} + 64\omega_{1}^{6} \omega_{2}^{8} - 128\omega_{1}^{6} H^{2} \omega_{2}^{6} - 512\omega_{1}^{8} \omega_{2}^{6} + 16\omega_{1}^{4} H^{2} \omega_{2}^{8} + 512\omega_{1}^{8} H\mu_{1} \omega_{2}^{4} + 1024\omega_{1}^{10} \omega_{2}^{4} + 32\omega_{1}^{4} H\mu_{1} \omega_{2}^{8} + 256\omega_{1}^{8} H^{2} \omega_{2}^{4} - 128\omega_{1}^{6} \mu_{1}^{2} \omega_{2}^{6} + 256\omega_{1}^{8} \omega_{2}^{4} \mu_{1}^{2} . $$
$$ p_{20} = 16\omega_{2}^{6} \mu_{2}^{2} + 32\omega_{2}^{6} H\mu_{2} + 64\omega_{2}^{8} + 16\omega_{2}^{6} H^{2} . $$
$$ q_{06} = - 64\omega_{1}^{4} \omega_{2}^{4} + 32\omega_{1}^{2} \omega_{2}^{6} - 4\omega_{2}^{8} . $$
$$ p_{06} = - 4\omega_{2}^{2} . $$
$$ q_{05} = - 192\omega_{2}^{6} \omega_{1}^{3} + 24\omega_{1} \omega_{2}^{8} + 384\omega_{2}^{4} \omega_{1}^{5} . $$
$$ p_{05} = 24\omega_{2}^{3} . $$
$$ q_{04} = - 576\omega_{1}^{6} \omega_{2}^{4} + 288\omega_{1}^{4} \omega_{2}^{6} - 36\omega_{2}^{8} \omega_{1}^{2} . $$
$$ p_{04} = - 36\omega_{2}^{4} . $$

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Malgol, A., Vineesh, K.P. & Saha, A. Investigation of vibration characteristics of a Jeffcott rotor system under the influence of nonlinear restoring force, hydrodynamic effect, and gyroscopic effect. J Braz. Soc. Mech. Sci. Eng. 44, 105 (2022). https://doi.org/10.1007/s40430-021-03277-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40430-021-03277-x

Keywords

Search

Navigation