Skip to main content
SpringerLink
Log in

An Exact Solution Technique for Impact Oscillators

  • Chapter
  • First Online:
Nonlinear Approaches in Engineering Applications

Abstract

When the two objects come in contact, impact occurs. Impact is a highly nonlinear, non-smooth and discontinuous event, thus resulting in various forms of complex nonlinear dynamics. Impact oscillators are explored for various purposes, such as vibrational energy harvesting, vibration isolation and noise insulation. Depending on the excitation and system parameters, impacting responses reach either to a steady state or remain chaotic. The frequency of the steady-state impacting response is higher compared to the excitation frequency. Due to the presence of the multi-periodic and chaotic responses, it is impossible to determine complete analytical solution for any excitation frequency of a MDOF impacting system because equations to compute. To resolve this problem, a semi-analytical method is demonstrated to compute the full transmittance spectrum in this chapter. The periodicity coefficient, which is defined as the ratio between the resulting frequency and the excitation frequency, throughout the full excitation spectrum is plotted for single-degree-of-freedom and two-degree-of-freedom system.

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

Access this chapter

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 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
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

Institutional subscriptions

References

  • Aidanpää, J. O., & Gupta, R. B. (1993). Periodic and chaotic behaviour of a threshold-limited two-degree-of-freedom system. Journal of Sound and Vibration, 165(2), 305–327.

    Article  MATH  Google Scholar 

  • Banerjee, B. (2011). An introduction to metamaterials and waves in composites. Auckland: Taylor & Francis.

    Google Scholar 

  • Banerjee, A., Das, R., & Calius, E. (2016a). A new approach for determination of the attenuation bandwidth of a resonating metamaterial. Applied Mechanics and Materials, 846, 264–269.

    Article  Google Scholar 

  • Banerjee, A., Calius, E. P., & Das, R. (2016b). The effects of cubic stiffness nonlinearity on the attenuation bandwidth of 1D elasto-dynamic metamaterials. In IMECE 2016 (p. V013T01A020). Pheonix: ASME.

    Google Scholar 

  • Banerjee, A., Chanda, A., & Das, R. (2016c). Oblique frictional unilateral contacts perceived in curved bridges. Nonlinear Dynamics, 85(4), 2207–2231.

    Article  Google Scholar 

  • Banerjee, A., Chanda, A., & Das, R. (2016). Historical origin and recent development on normal directional impact models for rigid body contact simulation: A critical review. Archives of Computational Methods in Engineering, 24(2), 397–422. https://doi.org/10.1007/s11831-016-9164-5.

    Article  MathSciNet  MATH  Google Scholar 

  • Banerjee, A., Das, R., & Calius, E. P. (2017). Frequency graded 1d metamaterials: A study on the attenuation bands. Journal of Applied Physics, 122(7), 075101. https://doi.org/10.1063/1.4998446.

    Article  Google Scholar 

  • Banerjee, A., Das, R., & Calius, E. P. (2017b). Vibration transmission through an impacting mass-in-mass unit, an analytical investigation. International Journal of Non-Linear Mechanics, 90, 137–146.

    Article  Google Scholar 

  • Banerjee, A., Chanda, A., & Das, R. (2017c). Seismic analysis of a curved bridge considering deck-abutment pounding interaction: an analytical investigation on the post-impact response. Earthquake Engineering & Structural Dynamics, 46(2), 267–290.

    Article  Google Scholar 

  • Baughman, R. H., et al. (1998). Negative Poisson’s ratios as a common feature of cubic metals. Nature, 392(6674), 362–365.

    Article  Google Scholar 

  • Błazejczyk-Okolewska, B., & Peterka, F. (1998). An investigation of the dynamic system with impacts. Chaos, Solitons & Fractals, 9(8), 1321–1338.

    Article  MATH  Google Scholar 

  • Blazejczyk-Okolewska, B., Czolczynski, K., & Kapitaniak, T. (2009). Dynamics of a two-degree-of-freedom cantilever beam with impacts. Chaos, Solitons & Fractals, 40(4), 1991–2006.

    Article  MATH  Google Scholar 

  • Brogliato, B. (1999). Nonsmooth mechanics: models, dynamics and control. London/New York: Springer.

    Book  MATH  Google Scholar 

  • Calius, E. P., et al. (2009). Negative mass sound shielding structures: Early results. Basic Solid State Physics, 246(9), 2089–2097.

    Article  Google Scholar 

  • Ding, W. C., Xie, J. H., & Sun, Q. G. (2004). Interaction of Hopf and period doubling bifurcations of a vibro-impact system. Journal of Sound and Vibration, 275(1–2), 27–45.

    Article  MATH  Google Scholar 

  • Dongping, J., & Haiyan, H. (1997). Periodic vibro-impacts and their stability of a dual component system. Acta Mechanica Sinica, 13(4), 366–376.

    Article  Google Scholar 

  • Egle, D. M. (1967). Discussion: On the stability of the impact damper (Masri, S. F., & Caughey, T.K. (1966). ASME Journal of Applied Mechanics, 33, 586–592). Journal of Applied Mechanics, 34(1), 253–253.

    Google Scholar 

  • Pope, S. A., & Laalej, H. (2014). A multi-layer active elastic metamaterial with tuneable and simultaneously negative mass and stiffness. Smart Materials and Structures, 23(7), 075020.

    Article  Google Scholar 

  • Felippa, C. A. (2015). Modal analysis of MDOF forced damped system. Introduction to Aerospace Structures (ASEN 3112). Department of Aerospace Engineering Sciences, University of Colorado at Boulder, Epub date: 2015, Date Accessed: 5th July, 2016.

    Google Scholar 

  • Friis, E., Lakes, R., & Park, J. (1988). Negative Poisson’s ratio polymeric and metallic foams. Journal of Materials Science, 23(12), 4406–4414.

    Article  Google Scholar 

  • Grubin, C. (1956). On the theory of the acceleration damper. Journal of Applied Mechanics, 23(3), 373–378.

    MATH  Google Scholar 

  • Hertz, H. (1881). On the contact of elastic solids. Journal fur die Reine und Angewandte Mathematik, 92(156–171), 110.

    Google Scholar 

  • Hertz, H. (1882). Über die Berührung fester elastischer Körper.

    Google Scholar 

  • Hu, G., et al. (2016). Metastructure with piezoelectric element for simultaneous vibration suppression and energy harvesting. Journal of Vibration and Acoustics, 139(1), 011012.

    Article  Google Scholar 

  • Huang, H. H., & Sun, C. T. (2009). Wave attenuation mechanism in an acoustic metamaterial with negative effective mass density. New Journal of Physics, 11(1), 013003.

    Article  Google Scholar 

  • Huang, G. L., & Sun, C. T. (2010). Band gaps in a multiresonator acoustic metamaterial. Journal of Vibration and Acoustics, 132(3), 031003.

    Article  Google Scholar 

  • Huang, H. H., & Sun, C. T. (2012). Anomalous wave propagation in a one- dimensional acoustic metamaterial having simultaneously negative mass density and Young’s modulus. The Journal of the Acoustical Society of America, 132, 2887.

    Article  Google Scholar 

  • Huang, H. H., Sun, C. T., & Huang, G. L. (2009). On the negative effective mass density in acoustic metamaterials. International Journal of Engineering Science, 47(4), 610–617.

    Article  Google Scholar 

  • Jankowski, R. (2005). Non-linear viscoelastic modelling of earthquake-induced structural pounding. Earthquake Engineering & Structural Dynamics, 34(6), 595–611.

    Article  Google Scholar 

  • Kocer, C., McKenzie, D. R., & Bilek, M. M. (2009). Elastic properties of a material composed of alternating layers of negative and positive Poisson’s ratio. Materials Science and Engineering: A, 505(1–2), 111–115.

    Article  Google Scholar 

  • Lakes, R. (1987). Foam structures with a negative Poisson’s ratio. Science, 235(4792), 1038–1040.

    Article  Google Scholar 

  • Lakes, R. (1993). Advances in negative Poisson’s ratio materials. Advanced Materials, 5(4), 293–296.

    Article  Google Scholar 

  • Larsen, U. D., Sigmund, O., & Bouwstra, S. (1996). Design and fabrication of compliant micromechanisms and structures with negative Poisson’s ratio. In Micro Electro Mechanical Systems, 1996, MEMS’96, Proceedings. An Investigation of Micro Structures, Sensors, Actuators, Machines and Systems. IEEE, The Ninth Annual International Workshop on. 1996. IEEE.

    Google Scholar 

  • Lei, Z. (2008). Effective 1-D material properties of coplanar-waveguide-based EBG- and meta-materials. In Metamaterials, 2008, International Workshop on. 2008.

    Google Scholar 

  • Lu, M.-H., Feng, L., & Chen, Y.-F. (2009). Phononic crystals and acoustic metamaterials. Materials Today, 12(12), 34–42.

    Article  Google Scholar 

  • Luo, G. W. (1999). Periodic motions and transition phenomena in a two-degrees-of-freedom system with perfectly plastic impact. Physics Letters A, 263(1–2), 83–90.

    Article  MathSciNet  MATH  Google Scholar 

  • Luo, G. W., Xie, J. H., & Guo, S. H. L. (2001). Periodic motions and global bifurcations of a two-degree-of-freedom system with plastic vibro-impact. Journal of Sound and Vibration, 240(5), 837–858.

    Article  Google Scholar 

  • Luo, G. W., Yu, J. N., & Zhang, J. G. (2006). Periodic-impact motions and bifurcations of a dual component system. Nonlinear Analysis: Real World Applications, 7(4), 813–828.

    Article  MathSciNet  MATH  Google Scholar 

  • Luo, G., et al. (2007). Vibro-impact dynamics near a strong resonance point. Acta Mechanica Sinica, 23(3), 329–341.

    Article  MathSciNet  MATH  Google Scholar 

  • Luo, G., et al. (2008). Periodic-impact motions and bifurcations of vibro-impact systems near 1:4 strong resonance point. Communications in Nonlinear Science and Numerical Simulation, 13(5), 1002–1014.

    Article  MathSciNet  MATH  Google Scholar 

  • Masri, S. F. (1970). General motion of impact dampers. The Journal of the Acoustical Society of America, 47(1B), 229–237.

    Article  Google Scholar 

  • Masri, S. F. (1972). Theory of the dynamic vibration neutralizer with motion-limiting stops. Journal of Applied Mechanics, 39(2), 563–568.

    Article  MATH  Google Scholar 

  • Masri, S. F., & Caughey, T. K. (1966). On the stability of the impact damper. Journal of Applied Mechanics, 33(3), 586–592.

    Article  MathSciNet  Google Scholar 

  • Newton, I. (1999). The principia: Mathematical principles of natural philosophy. California: University of California Press.

    MATH  Google Scholar 

  • Nigm, M. M., & Shabana, A. A. (1983). Effect of an impact damper on a multi-degree of freedom system. Journal of Sound and Vibration, 89(4), 541–557.

    Article  MATH  Google Scholar 

  • Peterka, F., & Blazejczyk-Okolewska, B. (2005). Some aspects of the dynamical behavior of the impact damper. Journal of Vibration and Control, 11(4), 459–479.

    Article  MATH  Google Scholar 

  • Pfeiffer, F., & Glocker, C. (1996). Multibody dynamics with unilateral contacts (Vol. 9). John Wiley & Sons.

    Google Scholar 

  • Pope, S. A., & Laalej, H. (2014). A multi-layer active elastic metamaterial with tuneable and simultaneously negative mass and stiffness. Smart Materials and Structures, 23(7), 075020.

    Article  Google Scholar 

  • Popplewell, N., Bapat, C. N., & McLachlan, K. (1983). Stable periodic vibroimpacts of an oscillator. Journal of Sound and Vibration, 87(1), 41–59.

    Article  MATH  Google Scholar 

  • Pun, D., et al. (1998). Forced vibration analysis of a multidegree impact vibrator. Journal of Sound and Vibration, 213(3), 447–466.

    Article  Google Scholar 

  • Sheng, P., et al. (2003). Locally resonant sonic materials. Physica B: Condensed Matter, 338(1–4), 201–205.

    Article  Google Scholar 

  • Sun, H., Du, X., & Frank Pai, P. (2011a). Metamaterial broadband vibration absorbers by local resonance. In Collection of Technical Papers – AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference.

    Google Scholar 

  • Sun, H., et al. (2011b). Theory and experiment research of metamaterial beams for broadband vibration absorption. In Collection of Technical Papers – AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference.

    Google Scholar 

  • Sun, H., et al. (2013). Theory and experiment research of metamaterial panel for mechanical waves absorption. In Collection of Technical Papers – AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference.

    Google Scholar 

  • Viana, R. L., De Souza, S. L. T., & Caldas, I. L. (2009). Multistability and self-similarity in the parameter-space of a vibro-impact system. Mathematical Problems in Engineering, 2009, 1–11.

    MathSciNet  MATH  Google Scholar 

  • Warburton, G. (1957). Discussion of “on the theory of the acceleration damper”. ASME Journal of Applied Mechanics, 24, 322–324.

    Google Scholar 

  • Yao, S., Zhou, X., & Hu, G. (2008). Experimental study on negative effective mass in a 1D mass–spring system. New Journal of Physics, 10(4), 043020.

    Article  Google Scholar 

  • Yue, Y. (2016). Bifurcations of the symmetric quasi-periodic motion and Lyapunov dimension of a vibro-impact system. Nonlinear Dynamics, 84(3), 1697–1713.

    Article  MathSciNet  MATH  Google Scholar 

  • Yue, Y., Xie, J. H., & Gao, X. J. (2012). Capturing the symmetry of attractors and the transition to symmetric chaos in a vibro-impact system. International Journal of Bifurcation and Chaos, 22(05), 1250109.

    Article  MATH  Google Scholar 

  • Zhang, Y., & Fu, X. (2015). On periodic motions of an inclined impact pair. Communications in Nonlinear Science and Numerical Simulation, 20(3), 1033–1042.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, University of Auckland, Auckland, New Zealand

    A. Banerjee

  2. Callaghan Innovation, Auckland, New Zealand

    A. Banerjee & E. P. Calius

  3. Sir Lawrence Wackett Aerospace Centre, School of Engineering, RMIT University, Melbourne, VIC, Australia

    R. Das

Authors
  1. A. Banerjee
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. R. Das
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. E. P. Calius
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to R. Das .

Editor information

Editors and Affiliations

  1. Xiamen University of Technology, Xiamen, China

    Liming Dai

  2. Xiamen University of Technology, Xiamen, China

    Reza N. Jazar

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Banerjee, A., Das, R., Calius, E.P. (2018). An Exact Solution Technique for Impact Oscillators. In: Dai, L., Jazar, R. (eds) Nonlinear Approaches in Engineering Applications. Springer, Cham. https://doi.org/10.1007/978-3-319-69480-1_10

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-69480-1_10

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-69479-5

  • Online ISBN: 978-3-319-69480-1

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation