Skip to main content
SpringerLink
Log in

The non-dimensional response spectra of impact oscillators subjected to pulse-type base excitation

  • Published:
International Journal of Dynamics and Control Aims and scope Submit manuscript

Abstract

The oscillation due to vibration results in impact of the structures if the at-rest separation is insufficient to accommodate the relative displacements. The impact between the floors of the two adjacent buildings or the impact between adjacent decks and abutments is modeled by an impact oscillator subjected to closed-form mathematical formulations of near-fault earthquake motions as pulse-type excitation, such as cosine (±) and sinusoidal pulses (±). This study investigates the impacting responses of an impact oscillator against a barrier from an analytical perspective. The momentum-based stereo-mechanical method is used as a contact force-based model to perform the impact analysis. The equation of the motion of the impact oscillator under lateral pulse-type base excitation is non-dimensionalized by two dimensionless parameters using the standard process of non-dimensionalization of a differential equation. The system has been analyzed for the complete forced vibration part as it was excited by a single cycle of cosine and sinusoidal pulses. However, the transient solutions are also derived for the analysis. Therefore, the free vibration phase is determined until the impactor comes to a non-impacting state. The steady-state with the transient solutions for the complete impacting responses are provided for the exact solutions of the dynamic responses of the impact oscillator, such as displacement response spectra, velocity response spectra. The governing system parameters, such as damping ratio, frequency ratio of the impact oscillator, the coefficient of restitution, and the direction of the base excitation ( ±) are the governing system parameters to control the impact between the oscillator and barrier. A frequency point under resonance region from each response spectra of impact oscillator subjected to cosine ( ±) and sinusoidal pulses ( ±) is subtracted to define the non-dimensional time domain responses. The exact impact determines in out-of-phase state and the dynamic responses of impact oscillator are attenuated. The non-dimensional relative distance between the impacting oscillator and the barrier varies from 0.5 to 3.0. However, for \(\tilde{\delta } > 3\), no successive impact obtains. Accordingly, the impacting oscillator’s dynamic response spectra closely match the linear oscillator’s dynamic response spectra. All the results are mathematically derived and accurate for practical applications of a damped impact oscillator (i.e., precisely for single-degree-of-freedom systems) to determine the non-dimensional dynamic responses.

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
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24

Similar content being viewed by others

Data availability

All data, models, and code generated or used during the study appear in the submitted article.

Code availability

All data generated or analysed during this study are included in this published article (and its supplementary information files).

References

  1. Miari M, Jankowski R (2022) Analysis of pounding between adjacent buildings founded on different soil types. Soil Dyn Earthq Eng 154:107156

    Article  Google Scholar 

  2. Miari M, Jankowski R (2022) Incremental dynamic analysis and fragility assessment of buildings founded on different soil types experiencing structural pounding during earthquakes. Eng Struct 252:113118

    Article  Google Scholar 

  3. Sha B, Tao T, Xing C, Wang H, Li A (2020) Pounding analysis of isolated girder bridge under nonpulse and pulse-like earthquakes. J Perform Constr Facil 34(4):04020062

    Article  Google Scholar 

  4. Wu Q, Wang T, Ge H, Zhu H (2019) Dimensional analysis of pounding response of an oscillator based on modified kelvin pounding model. J Aerosp Eng 32(4):04019039

    Article  Google Scholar 

  5. Bruzzone F, Maggi T, Marcellini C, Rosso C (2021) 2d nonlinear and non-hertzian gear teeth deflection model for static transmission error calculation. Mech Mach Theory 166:104471

    Article  Google Scholar 

  6. Tao H, Gibert J (2019) Periodic orbits of a conservative 2-dof vibro-impact system by piecewise continuation: bifurcations and fractals. Nonlinear Dyn 95:2963–2993

    Article  MATH  Google Scholar 

  7. Al-Shudeifat MA, Saeed AS (2022) Periodic motion and frequency energy plots of dynamical systems coupled with piecewise nonlinear energy sink. J Comput Nonlinear Dyn 17(4):041005

    Article  Google Scholar 

  8. Dou C, Fan J, Li C, Cao J, Gao M (2020) On discontinuous dynamics of a class of friction-influenced oscillators with nonlinear damping under bilateral rigid constraints. Mech Mach Theory 147:103750

    Article  Google Scholar 

  9. Bruzzone F, Maggi T, Marcellini C, Rosso C (2021) Gear teeth deflection model for spur gears: proposal of a 3d nonlinear and non-hertzian approach. Machines 9(10):223

    Article  Google Scholar 

  10. Xia Y, Pang J, Yang L, Chu Z (2021) Investigation on clearance-induced vibro-impacts of torsional system based on hertz contact nonlinearity. Mech Mach Theory 162:104342

    Article  Google Scholar 

  11. Chen X, Xiao X, Bai X, Wu Q (2021) Dimensional pounding response analysis for adjacent inelastic mdof structures based on modified kelvin model. Struct Eng Mech An Intl J 79(3):347–358

    Google Scholar 

  12. Zhang J, Zhu X, Yang X, Zhang W (2019) Transient nonlinear responses of an auxetic honeycomb sandwich plate under impact loads. Int J Impact Eng 134:103383

    Article  Google Scholar 

  13. Zhang C, Gholipour G, Mousavi AA (2019) Nonlinear dynamic behavior of simply-supported rc beams subjected to combined impact-blast loading. Eng Struct 181:124–142

    Article  Google Scholar 

  14. Bureau E, Schilder F, Elmegård M, Santos IF, Thomsen JJ, Starke J (2014) Experimental bifurcation analysis of an impact oscillator—determining stability. J Sound Vib 333(21):5464–5474

    Article  Google Scholar 

  15. Ing J, Pavlovskaia E, Wiercigroch M, Banerjee S (2008) Experimental study of impact oscillator with one-sided elastic constraint. Philos Trans R Soc A Math Phys Eng Sci 366(1866):679–705

    Article  MATH  Google Scholar 

  16. Jiang S, Zhang C, Mou B (2019) Dimensional analysis of pounding effect between adjacent inelastic oscillators. In: Mechanics of structures and materials XXIV. CRC Press, pp 674–679

  17. Ambiel JHK, Brun M, Thibon A, Gravouil A (2022) Three-dimensional analysis of eccentric pounding between two-storey structures using explicit non-smooth dynamics. Eng Struct 251:113385

    Article  Google Scholar 

  18. Miari M, Choong KK, Jankowski R (2019) Seismic pounding between adjacent buildings: Identification of parameters, soil interaction issues and mitigation measures. Soil Dyn Earthq Eng 121:135–150

    Article  Google Scholar 

  19. Komodromos P (2008) Simulation of the earthquake-induced pounding of seismically isolated buildings. Comput Struct 86(7–8):618–626

    Article  Google Scholar 

  20. Burns SJ, Piiroinen PT, Hanley KJ (2019) Critical time step for dem simulations of dynamic systems using a hertzian contact model. Int J Numer Meth Eng 119(5):432–451

    Article  MathSciNet  MATH  Google Scholar 

  21. Dai W, Yang J, Shi B (2020) Vibration transmission and power flow in impact oscillators with linear and nonlinear constraints. Int J Mech Sci 168:105234

    Article  Google Scholar 

  22. Banerjee A, Das R, Calius EP (2017) Vibration transmission through an impacting mass-in-mass unit, an analytical investigation. Int J Non-Linear Mech 90:137–146

    Article  Google Scholar 

  23. Khaniki HB, Ghayesh MH, Chin R (2023) Theory and experiment for dynamics of hyperelastic plates with modal interactions. Int J Eng Sci 182:103769

    Article  MathSciNet  MATH  Google Scholar 

  24. Khaniki HB, Ghayesh MH, Chin R, Hussain S (2022) Nonlinear continuum mechanics of thick hyperelastic sandwich beams using various shear deformable beam theories. Contin Mech Thermodyn 34(3):781–827

    Article  MathSciNet  Google Scholar 

  25. Khaniki HB, Ghayesh MH, Chin R, Chen L-Q (2022) Experimental characteristics and coupled nonlinear forced vibrations of axially travelling hyperelastic beams. Thin-walled Struct 170:108526

    Article  Google Scholar 

  26. Khaniki HB, Ghayesh MH, Chin R, Amabili M (2021) Large amplitude vibrations of imperfect porous-hyperelastic beams via a modified strain energy. J Sound Vib 513:116416

    Article  Google Scholar 

  27. Khaniki HB, Ghayesh MH, Hussain S, Amabili M (2021) Effects of geometric nonlinearities on the coupled dynamics of cnt strengthened composite beams with porosity, mass and geometric imperfections. Eng Comput 38:3463–9488

    Article  Google Scholar 

  28. Khaniki HB, Ghayesh MH (2020) On the dynamics of axially functionally graded cnt strengthened deformable beams. Eur Phys J Plus 135(5):1–24

    Article  Google Scholar 

  29. Jia H-Y, Lan X-L, Luo N, Yang J, Zheng S-X, Zhang C (2019) Nonlinear pounding analysis of multispan and simply supported beam bridges subjected to strong ground motions. Shock Vib 2019:1–11

    Google Scholar 

  30. Xia C, Wang B, Luo T, Min Q, Sekulic D, Li Y (2022) Dynamic amplification factor of multi-span simply supported beam bridge under traffic flow. Adv Struct Eng 25(8):1829–1847

    Article  Google Scholar 

  31. Yu J, Jiang L, Zhou W, Liu X, Nie L, Zhang Y, Feng Y, Cao S (2021) Running test on high-speed railway track-simply supported girder bridge systems under seismic action. Bull Earthq Eng 19(9):3779–3802

    Article  Google Scholar 

  32. Ma L, Zhang W, Han W, Liu J (2019) Determining the dynamic amplification factor of multi-span continuous box girder bridges in highways using vehicle-bridge interaction analyses. Eng Struct 181:47–59

    Article  Google Scholar 

  33. Mazza F, Labernarda R (2022) Internal pounding between structural parts of seismically isolated buildings. J Earthq Eng 26(10):5175–5203

    Article  Google Scholar 

  34. Rezaei H, Moayyedi SA, Jankowski R (2020) Probabilistic seismic assessment of rc box-girder highway bridges with unequal-height piers subjected to earthquake-induced pounding. Bull Earthq Eng 18:1547–1578

    Article  Google Scholar 

  35. Banerjee A (2020) Non-dimensional analysis of the elastic beam having periodic linear spring mass resonators. Meccanica 55(5):1181–1191

    Article  MathSciNet  Google Scholar 

  36. Yang D, Guo G, Liu Y, Zhang J (2019) Dimensional response analysis of bilinear sdof systems under near-fault ground motions with intrinsic length scale. Soil Dyn Earthq Eng 116:397–408

    Article  Google Scholar 

  37. Zhai C, Jiang S, Chen Z (2015) Dimensional analysis of the pounding response of an oscillator considering contact duration. J Eng Mech 141(4):04014138

    Article  Google Scholar 

  38. Dimitrakopoulos E, Makris N, Kappos AJ (2010) Dimensional analysis of the earthquake response of a pounding oscillator. J Eng Mech 136(3):299–310

    Article  Google Scholar 

  39. Dimitrakopoulos E, Makris N, Kappos AJ (2009) Dimensional analysis of the earthquake-induced pounding between adjacent structures. Earthq Eng Struct Dyn 38(7):867–886

    Article  Google Scholar 

Download references

Acknowledgments

The authors would like to acknowledge DST/Inspire/Faculty award, reference number DST/INSPIRE/04/2018/000052 for sponsoring the research. SC would like to acknowledge the MHRD grant received from IIT Delhi during the period of this research work.

Funding

The authors would like to acknowledge DST/Inspire/Faculty award, reference number DST/INSPIRE/04/2018/000052 for sponsoring the research. SC would like to acknowledge the MHRD grant received from IIT Delhi during the period of this research work.

Author information

Authors and Affiliations

  1. Department of Civil Engineering, Indian Institute of Technology Delhi, New Delhi, India

    Sudip Chowdhury & Arnab Banerjee

Authors
  1. Sudip Chowdhury
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Arnab Banerjee
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

SC contributed to conceptualization, methodology, software, data curation, writing original draft, visualization, investigation, critical discussion, securing the fund, editing. AB contributed to conceptualization, visualization, supervision, securing the fund, reviewing, critical discussion.

Corresponding author

Correspondence to Sudip Chowdhury.

Ethics declarations

Conflict of interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chowdhury, S., Banerjee, A. The non-dimensional response spectra of impact oscillators subjected to pulse-type base excitation. Int. J. Dynam. Control 11, 2036–2057 (2023). https://doi.org/10.1007/s40435-023-01152-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40435-023-01152-2

Keywords

Search

Navigation