Skip to main content
SpringerLink
Log in
Book cover

Recent Trends in Wave Mechanics and Vibrations pp 127–137Cite as

Transverse Vibrations of an Axially Travelling String

  • Conference paper
  • First Online:

  • 526 Accesses

Part of the book series: Lecture Notes in Mechanical Engineering ((LNME))

Abstract

In this paper, the transverse vibration of axially travelling string is analysed. The axial velocity of the string is periodically varying about an average value. Applying direct perturbation method (MMS), an analytical solution is found. An analysis of principal parametric resonances is carried out when changing frequency of the axial velocity is zero, close to zero and twice the natural frequency. Mathematical analysis is carried out to determine the stability and instability zones. The results show that instability occurs when changing frequency of the axial velocity is close to two times the natural frequency, whereas no instability occurs when changing frequency is close to zero. A case study of bandsaw is discussed. The stability and instability zones are plotted for the first five natural frequencies.

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Abbreviations

A:

Cross-sectional area of the string

L:

Length of the string

ρ:

Mass density of the string

P:

Tension force in the string

P0:

Initial tension force in the string

κ:

Pulley support parameter

v0:

Dimensionless mean velocity of the string

\( y = \frac{{y^{*} }}{L} \) :

Dimensionless transverse displacement of the string

\( x = \frac{{x^{*} }}{L} \) :

Dimensionless spatial variable

\( t = (1/{\text{L}})\sqrt {(P_{0} /\rho A} )t^{ * } \) :

Dimensionless time

\( \Omega ^{*} = \frac{1}{L}\sqrt {\frac{{P_{0} }}{\rho A}}\Omega \) :

Dimensional frequency of velocity variation

\( v = v^{*} /\sqrt {P_{0} /\rho A} \) :

Dimensionless axial velocity of the string

ε:

Dimensionless parameter <<1

σ:

Detuning parameter

ψn:

Mode shapes of the travelling string

ωn:

Natural frequency of the travelling string

References

  1. Swope RD, Ames WF (1963) Vibrations of a moving thread line. J Franklin Inst 275:36–55

    Article  Google Scholar 

  2. Mote CDJ (1965) A study of band-saw vibrations. J Franklin Inst 279(6):431–444

    Article  Google Scholar 

  3. Ulsoy AG, Mote CDJ (1980) Analysis of band-saw vibration. Wood Sci 13:1–10

    Google Scholar 

  4. Mote CDJ (1966) On the nonlinear oscillation of an axially moving string. J Appl Mech 33:463–464

    Article  Google Scholar 

  5. Thurman AL, Mote CDJ (1969) Free, periodic, nonlinear oscillation of an axially moving strip. J Appl Mech 36:83–91

    Article  Google Scholar 

  6. Wickert JA, Mote CDJ (1990) Classical vibration analysis of axially moving continua. J Appl Mech ASME 57:738–744

    Article  Google Scholar 

  7. Wickert JA (1994) Response solutions for the vibration of a travelling string on an elastic foundation. J Vib Acoust 116:137–199

    Article  Google Scholar 

  8. Wicket JA (1992) Non-linear vibration of a travelling tensioned beam. Int J Non-Linear Mech 27:503–517

    Article  Google Scholar 

  9. Pakdemirli M, Ulsoy AG, Ceranoglu A (1994) Transverse vibration of an axially accelerating string. J Sound Vib 169(2):179–196

    Article  Google Scholar 

  10. Pakdemirli M, Ulsoy AG (1997) Stability analysis of an axially accelerating string. J Sound Vib 203:815–832

    Article  Google Scholar 

  11. Chen L-Q, Zu JW, Wu J, Yang X-D (2004) Transverse vibration of an axially accelerating viscoelastic string with geometric nonlinearity. J Eng Math 48:171–182

    Article  Google Scholar 

  12. Kesimili A, Ozkaya E, Bagdatli SM (2015) Nonlinear vibrations of spring supported axially moving string. Nonlinear Dyn 81:1523–1534

    Article  MathSciNet  Google Scholar 

  13. Ghayesh MH (2009) Stability characteristics of an axially accelerating string supported by an elastic foundation. Mech Mach Theory 44:1964–1979

    Article  Google Scholar 

  14. Ghayesh MH (2008) Nonlinear transversal vibration and stability of an axially moving string supported by a partial viscoelastic guide. J Sound Vib 314:757–774

    Google Scholar 

  15. Ghayesh MH, Moradian N (2011) Nonlinear dynamic response of axially moving, stretched viscoelasic string. Arch Appl Mech 81:781–799

    Article  Google Scholar 

  16. Oz HR, Pakdermili M (1999) Vibrations of axially moving beam with time-dependent velocity. J Sound Vib 227(2):239–257

    Google Scholar 

  17. Oz HR, Pakdermili M, Boyaci H (2001) Non-linear vibrations of an axially moving beam with time-dependent velocity. Int J Non-linear Mech 36:107–115

    Article  Google Scholar 

  18. Ghayesh MH (2012) Stability and bifurcations of an axially moving beam with an intermediate spring support. Nonlinear Dyn 69:193–210

    Article  MathSciNet  Google Scholar 

  19. Ghayesh MH, Ambali M, Paidouissis MP (2012) Nonlinear vibrations and stability of an axially moving beam with an intermediate spring support: two-dimensional analysis. Nonlinear Dyn 70(1):335–354

    Article  MathSciNet  Google Scholar 

  20. Sahoo B, Panda LN, Pohit G (2013) Parametric and internal resonances of an axially moving beam with time dependent velocity. Model Simul Eng 18 p. Article ID 919517

    Google Scholar 

  21. Chakraborty G, Mallik AK, Hatwal H (1999) Non-linear vibration of a travelling beam. Int J Non-Linear Mech 34:655–670

    Article  Google Scholar 

  22. Chakraborty G, Mallick AK (1999) Non-linear vibration of travelling beam having an intermediate guide. Nonlinear Dyn 20:247–265

    Article  Google Scholar 

  23. Mao X-Y, Ding H, Chen L-Q (2019) Internal resonance of a super critically axially moving beam subjected to the pulsating speed. Nonlinear Dyn 95:631–651

    Article  Google Scholar 

  24. Panda LN, Kar RC (2007) Nonlinear dynamics of a pipe conveying pulsating fluid with parametric and internal resonances. Nonlinear Dyn 49:9–30

    Article  Google Scholar 

  25. Panda LN, Kar RC (2008) Nonlinear dynamics of a pipe conveying pulsating fluid with combination, principal parametric and internal resonances. J Sound Vib 309:375–406

    Article  Google Scholar 

  26. Oz HR (2001) Non-linear vibrations and stability analysis of tensioned pipes conveying fluid with variable velocity. Int J Non-Linear Mech 36(1):1031–1039

    Article  Google Scholar 

  27. Oz HR, Boyaci H (2000) Transverse vibrations of tensioned pipe conveying fluid with time-dependent velocity. J Sound Vib 236(2):259–276

    Article  Google Scholar 

  28. Ghayesh MH, Paidouissis MP, Modarres-Sadeghi Y (2011) Three-dimensional dynamics of a fluid conveying cantilevered pipe with an additional spring support and an end mass. J Sound Vib 330:2869–2899

    Article  Google Scholar 

  29. Ghayesh MH, Paidouissis MP, Ambali Marco (2013) Nonlinear dynamics of cantilevered extensible pipes conveying fluid. J Sound Vib 332:6405–6418

    Article  Google Scholar 

  30. Zhou S, Yu T-J, Yang X-D, Zhang W (2017) Global dynamics of pipes conveying pulsating fluid in the supercritical regime. Int J Appl Mech 9:1750029 (29 p)

    Article  Google Scholar 

  31. Javadi M, Noorian MA, Irani S (2019) Stability analysis of pipes conveying fluid with fractional viscoelastic model. Mechanica 1–12

    Google Scholar 

  32. Nayfeh AH (1998) Nonlinear interactions. Wiley, New York

    Google Scholar 

  33. Nayfeh AH, Mook DT (1979) Nonlinear oscillations. Wiley, New York

    MATH  Google Scholar 

  34. Nayfeh AH (1981) Introduction to perturbation techniques. Wiley, New York

    MATH  Google Scholar 

  35. Nayfeh AH, Balachandran B (1995) Applied nonlinear dynamics: analytical computational and experimental methods. Wiley-Interscience, New York

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, Institute of Technical Education and Research, SIKSHA ‘O’ ANUSANDHAN Deemed to be University, Bhubaneswar, 751030, India

    Shashendra Kumar Sahoo

  2. Department of Mechanical Engineering, N.I.T, Shillong, Meghalaya, India

    H. C. Das

  3. Department of Mechanical Engineering, C.E.T, BPUT, Bhubaneswar, India

    L. N. Panda

Authors
  1. Shashendra Kumar Sahoo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. H. C. Das
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. L. N. Panda
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shashendra Kumar Sahoo .

Editor information

Editors and Affiliations

  1. Department of Mathematics, National Institute of Technology Rourkela, Rourkela, Odisha, India

    S. Chakraverty

  2. Vibration Research Group, Von karman society, Jalpaiguri, India

    Paritosh Biswas

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Singapore Pte Ltd.

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Sahoo, S.K., Das, H.C., Panda, L.N. (2020). Transverse Vibrations of an Axially Travelling String. In: Chakraverty, S., Biswas, P. (eds) Recent Trends in Wave Mechanics and Vibrations. Lecture Notes in Mechanical Engineering. Springer, Singapore. https://doi.org/10.1007/978-981-15-0287-3_11

Download citation

  • DOI: https://doi.org/10.1007/978-981-15-0287-3_11

  • Published:

  • Publisher Name: Springer, Singapore

  • Print ISBN: 978-981-15-0286-6

  • Online ISBN: 978-981-15-0287-3

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation