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The fundamental frequencies of longitudinally vibrating rods carrying tip mass and transversally vibrating beams carrying tip mass by using several methods

Year 2021, Volume: 42 Issue: 1, 209 - 217, 29.03.2021

Aydın Demir

https://doi.org/10.17776/csj.789526

Abstract

The present paper is concerned with the determination of the frequency equation and sensitivity of the eigenfrequencies of a fixed-free longitudinally vibrating rod and transversally vibrating beam carrying a tip mass by using several methods. First, the exact frequency equations of the such systems are established, and then approximate formulas are given for the fundamental frequency using several methods which contain the equivalent system, Rayleigh quotient, Dunkerley’s formula and continuous system model. The applicability and proximity of these methods versus exact solutions reviewed. The results are compared in a wide range of relevant parameters to give a clear idea about the validity of the proposed formulas. These new derived equations can be very useful for a design engineer who is interested in the eigencharacteristics of similar systems and their sensitivity.

Keywords

Axial vibrations longitudinal vibrations rod beam tip mass

References

  • [1] Humar J. L., Dynamics of Structures, 3rd ed. NJ: Prentice-Hall, Englewood Cliffs, (1990).
  • [2] Meirovitch L., Analytical Methods in Vibrations 1st ed. London: Collier-Macmillan Limited, (1967).
  • [3] Timoshenko S., Young D. H., Weaver W. JR., Vibration Problems in Engineering, 4th ed. New York: Wiley, (1974).
  • [4] Tse F. S., Morse I. E., Hinkle R. T., Mechanical Vibrations, Theory and Applications, 2nd ed. Boston MA: Allyn and Bacon, (1978).
  • [5] Gürgöze M., On the eigenfrequencies of longitudinally vibrating rods carrying a tip mass and spring–mass in-span, Journal of Sound and Vibration, 216(2) (1998) 295-308.
  • [6] Turhan Ö., On the eigencharacteristics of longitudinally vibratıng rods with a cross-section discontinuity, Journal of Sound and Vibration, 248(1) (2001) 167-177.
  • [7] Gürgöze M., Erol H., On the eigencharacteristics of multi-step rods carrying a tip mass subjected to non-homogeneous external viscous damping, Journal of Sound and Vibration, 267 (2003) 355-365.
  • [8] Lin H.P., Chang S.C., Free vibrations of two rods connected by multi-spring–mass systems, Journal of Sound and Vibration, 330(11) (2011) 2509-2519.
  • [9] Gu U. C., Cheng C. C., Vibration analysis of a high-speed spindle under the action of a moving mass, Journal of Sound and Vibration, 278 (2004) 1131-1146.
  • [10] Chang C. H., Free vibration of a simply supported beam carrying a rigid mass at the middle, Journal of Sound and Vibration, 237(4) (2000) 733-744.
  • [11] Turhan Ö., On the fundamental frequency of beams carrying a point mass: Rayleigh approximations versus exact solutions, Journal of Sound and Vibration, 230(2) (2000) 449-459.
  • [12] Low K. H., A modified Dunkerley formula for eigenfrequencies of beams carrying concentrated masses, International Journal of Mechanical Sciences, 42 (2000) 1287-1305.
  • [13] Li Q. S., Vibratory characteristics of multi-step beams with an arbitrary number of crack and concentrated masses, Applied Acoustics, 62 (2001) 691-706.
  • [14] Kirk C. L., Wiedemann S. M., Natural frequencies and mode shapes of a free-free beam with large end masses, Journal of Sound and Vibration, 254(5) (2002) 939-949.
  • [15] Özkaya E., Non-linear transverse vibrations of a simply supported beam carrying concentrated masses, Journal of Sound and Vibration, 257(3) (2002) 413-424.
  • [16] Low K. H., Natural frequencies of a beam–mass system in transverse vibration: Rayleigh estimation versus eigenanalysis solutions, International Journal of Mechanical Sciences, 45 (6-7) (2003) 981–993.
  • [17] Banerjee J. R., Free vibration of beams carrying spring-mass systems -A dynamic stiffness approach, Computers and Structures, 104–105 (2012) 21–26.
  • [18] Li X. F., Tang A.Y., Xi L.Y., Vibration of a Rayleigh cantilever beam with axial force and tip mass, Journal of Constructional Steel Research, 80 (2013) 15-22.
  • [19] Matt Carlos Frederico T., Simulation of the transverse vibrations of a cantilever beam with an eccentric tip mass in the axial direction using integral transforms, Applied Mathematical Modelling, 37(22) (2013) 9338–9354.
  • [20] Labȩdzki P., Pawlikowski R., Radowicz A., Transverse vibration of a cantilever beam under base excitation using fractional rheological model, AIP Conference, (2018).
  • [21] Şakar G., The effect of axial force on the free vibration of an Euler-Bernoulli beam carrying a number of various concentrated elements, Shock and Vibration, 20(3) (2013) 357–367.
  • [22] Edwards P. , Transverse Vibration of Euler Beam, (2018) 247–266.

Year 2021, Volume: 42 Issue: 1, 209 - 217, 29.03.2021

Aydın Demir

https://doi.org/10.17776/csj.789526

Abstract

References

  • [1] Humar J. L., Dynamics of Structures, 3rd ed. NJ: Prentice-Hall, Englewood Cliffs, (1990).
  • [2] Meirovitch L., Analytical Methods in Vibrations 1st ed. London: Collier-Macmillan Limited, (1967).
  • [3] Timoshenko S., Young D. H., Weaver W. JR., Vibration Problems in Engineering, 4th ed. New York: Wiley, (1974).
  • [4] Tse F. S., Morse I. E., Hinkle R. T., Mechanical Vibrations, Theory and Applications, 2nd ed. Boston MA: Allyn and Bacon, (1978).
  • [5] Gürgöze M., On the eigenfrequencies of longitudinally vibrating rods carrying a tip mass and spring–mass in-span, Journal of Sound and Vibration, 216(2) (1998) 295-308.
  • [6] Turhan Ö., On the eigencharacteristics of longitudinally vibratıng rods with a cross-section discontinuity, Journal of Sound and Vibration, 248(1) (2001) 167-177.
  • [7] Gürgöze M., Erol H., On the eigencharacteristics of multi-step rods carrying a tip mass subjected to non-homogeneous external viscous damping, Journal of Sound and Vibration, 267 (2003) 355-365.
  • [8] Lin H.P., Chang S.C., Free vibrations of two rods connected by multi-spring–mass systems, Journal of Sound and Vibration, 330(11) (2011) 2509-2519.
  • [9] Gu U. C., Cheng C. C., Vibration analysis of a high-speed spindle under the action of a moving mass, Journal of Sound and Vibration, 278 (2004) 1131-1146.
  • [10] Chang C. H., Free vibration of a simply supported beam carrying a rigid mass at the middle, Journal of Sound and Vibration, 237(4) (2000) 733-744.
  • [11] Turhan Ö., On the fundamental frequency of beams carrying a point mass: Rayleigh approximations versus exact solutions, Journal of Sound and Vibration, 230(2) (2000) 449-459.
  • [12] Low K. H., A modified Dunkerley formula for eigenfrequencies of beams carrying concentrated masses, International Journal of Mechanical Sciences, 42 (2000) 1287-1305.
  • [13] Li Q. S., Vibratory characteristics of multi-step beams with an arbitrary number of crack and concentrated masses, Applied Acoustics, 62 (2001) 691-706.
  • [14] Kirk C. L., Wiedemann S. M., Natural frequencies and mode shapes of a free-free beam with large end masses, Journal of Sound and Vibration, 254(5) (2002) 939-949.
  • [15] Özkaya E., Non-linear transverse vibrations of a simply supported beam carrying concentrated masses, Journal of Sound and Vibration, 257(3) (2002) 413-424.
  • [16] Low K. H., Natural frequencies of a beam–mass system in transverse vibration: Rayleigh estimation versus eigenanalysis solutions, International Journal of Mechanical Sciences, 45 (6-7) (2003) 981–993.
  • [17] Banerjee J. R., Free vibration of beams carrying spring-mass systems -A dynamic stiffness approach, Computers and Structures, 104–105 (2012) 21–26.
  • [18] Li X. F., Tang A.Y., Xi L.Y., Vibration of a Rayleigh cantilever beam with axial force and tip mass, Journal of Constructional Steel Research, 80 (2013) 15-22.
  • [19] Matt Carlos Frederico T., Simulation of the transverse vibrations of a cantilever beam with an eccentric tip mass in the axial direction using integral transforms, Applied Mathematical Modelling, 37(22) (2013) 9338–9354.
  • [20] Labȩdzki P., Pawlikowski R., Radowicz A., Transverse vibration of a cantilever beam under base excitation using fractional rheological model, AIP Conference, (2018).
  • [21] Şakar G., The effect of axial force on the free vibration of an Euler-Bernoulli beam carrying a number of various concentrated elements, Shock and Vibration, 20(3) (2013) 357–367.
  • [22] Edwards P. , Transverse Vibration of Euler Beam, (2018) 247–266.
There are 22 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Engineering Sciences
Authors

Aydın Demir 0000-0002-8514-2301

Publication Date March 29, 2021
Submission Date September 2, 2020
Acceptance Date March 2, 2021
Published in Issue Year 2021Volume: 42 Issue: 1

Cite

APA Demir, A. (2021). The fundamental frequencies of longitudinally vibrating rods carrying tip mass and transversally vibrating beams carrying tip mass by using several methods. Cumhuriyet Science Journal, 42(1), 209-217. https://doi.org/10.17776/csj.789526