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Soft-impact dynamics of deformable bodies

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Abstract

Systems constituted by impacting beams and rods of non-negligible mass are often encountered in many applications of engineering practice. The impact between two rigid bodies is an intrinsically indeterminate problem due to the arbitrariness of the velocities after the instantaneous impact and implicates an infinite value of the contact force. The arbitrariness of after-impact velocities is solved by releasing the impenetrability condition as an internal constraint of the bodies and by allowing for elastic deformations at contact during an impact of finite duration. In this paper, the latter goal is achieved by interposing a concentrate spring between a beam and a rod at their contact point, simulating the deformability of impacting bodies at the interaction zones. A reliable and convenient method for determining impact forces is also presented. An example of engineering interest is carried out: a flexible beam that impacts on an axially deformable strut. The solution of motion under a harmonic excitation of the beam built-in base is found in terms of transverse and axial displacements of the beam and rod, respectively, by superimposition of a finite number of modal contributions. Numerical investigations are performed in order to examine the influence of the rigidity of the contact spring and of the ratio between the first natural frequencies of the beam and the rod, respectively, on the system response, namely impact velocity, maximum displacement, spring stretching and contact force. Impact velocity diagrams, nonlinear resonance curves and phase portraits are presented to determine regions of periodic motion with impacts and the appearance of chaotic solutions, and parameter ranges where the functionality of the non-structural element is at risk.

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References

  1. Abdul Azeez M.F., Vakakis A.F.: Numerical and experimental analysis of a continuous overhung rotor undergoing vibro-impacts. Int. J. Non-linear Mech. 34(3), 415–435 (1999)

    Article  MATH  Google Scholar 

  2. Aidanpaa J.O, Gupta R.B.: Periodic and chaotic behaviour of a threshold-limited two-degree-of-freedom system. J. Sound Vib. 165, 305–307 (1993)

    Article  ADS  Google Scholar 

  3. Alibert J.-J., Seppecher P., dell’isola F.: Truss modular beams with deformation energy depending on higher displacement gradients. Math. Mech. Solids 8(1), 51–73 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  4. Andreaus U., Dell’isola F., Porfiri M.: Piezoelectric passive distributed controllers for beam flexural vibrations. JVC J. Vib. Control 10(5), 625–659 (2004)

    Article  MATH  Google Scholar 

  5. Andreaus U., Placidi L., Rega G.: Numerical simulation of the soft contact dynamics of an impacting bilinear oscillator. Commun. Nonlinear Sci. Numer. Simul. 15, 2603–2616 (2010)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  6. Andreaus U., Placidi L., Rega G.: Soft impact dynamics of a cantilever beam: equivalent SDOF model versus infinite-dimensional system. J. Mech. Eng. Sci. 225(10), 2444–2456 (2011)

    Google Scholar 

  7. Babitsky, V.I.: Theory of Vibro-Impact Systems and Applications. Springer, Berlin (Revised Translation from Russian, Moscow, Nauka) (1978)

  8. Balachandran B.: Dynamics of an elastic structure excited by harmonic and aharmonic impactor motions. J. Vib. Control 9, 265–279 (2003)

    Article  MATH  Google Scholar 

  9. Bishop, S.R., Thompson, M.G., Foale, S.: Prediction of period-1 impact in a driven beam. Proc. R. Soc. Lond. A 452(1954), 2579–2592 (1996)

  10. Blazejczyk-Okolewska B., Kapitaniak T.: Co-existing attractors of impact oscillator. Chaos Solitons Fractals 9, 1439–1443 (1998)

    Article  ADS  MATH  Google Scholar 

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

    Article  ADS  MATH  Google Scholar 

  12. Brach R.: Mechanical Impact Dynamics: Rigid Body Collisions. Wiley, New York (1991)

    Google Scholar 

  13. Carcaterra A., Ciappi E.: Prediction of the compressible stage slamming force on rigid and elastic system impacting over the water surface. Nonlinear Dyn. 21(2), 193–220 (2000)

    Article  MATH  Google Scholar 

  14. Carcaterra A., Ciappi E., Iafrati A., Campana E.F.: Shock Spectral analysis of elastic systems impacting on the water surface. J. Sound Vib. 229(3), 579–605 (2000)

    Article  ADS  Google Scholar 

  15. Czołczyński K., Kapitaniak T.: Influence of the mass and stiffness ratio on a periodic motion of two impacting oscillators. Chaos Solitons Fractals 17, 1–10 (2003)

    Article  ADS  MATH  Google Scholar 

  16. Czołczyński K., Kapitaniak T.: On the existence of a stable periodic solution of two impacting oscillators with damping. Int. J. Bifurcat. Chaos 14, 3931–3947 (2004)

    Article  Google Scholar 

  17. Czołczyński K., Kapitaniak T.: On the influence of the resonant frequencies ratio on stable periodic solutions of two impacting oscillators. Int. J. Bifurcat. Chaos 16, 3707–3715 (2006)

    Article  MATH  Google Scholar 

  18. de los Santos M.A., Cardona S., Sanchez-Reyes J.: A global simulation model for hermetic reciprocating compressor. ASME J. Vib. Acoust. 113(3), 395–400 (1991)

    Article  Google Scholar 

  19. de Souza S.L.T., Caldas I.L., Viana R.L., Balthazar J.M., Brasil R.M.L.R.F.: Basins of attraction changes by amplitude constraining of oscillators with limited power supply. Chaos Solitons Fractals 26, 1211–1220 (2005)

    Article  ADS  MATH  Google Scholar 

  20. dell’Isola F., Batra R.C.: Saint-Venant’s problem for porous linear elastic materials. J. Elast. 47(1), 73–81 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  21. dell’Isola F., Guarascio M., Hutter K.: A variational approach for the deformation of a saturated porous solid. A second-gradient theory extending Terzaghi’s effective stress principle. Arch. Appl. Mech. 70(5), 323–337 (2000)

    Article  ADS  MATH  Google Scholar 

  22. dell’Isola F., Kosinski W.: Deduction of thermodynamic balance laws for bidimensional nonmaterial directed continua modelling interphase layers. Arch. Mech. 45, 333–359 (1993)

    MathSciNet  MATH  Google Scholar 

  23. dell’Isola F., Madeo A., Placidi L.: Linear plane wave propagation and normal transmission and reflection at discontinuity surfaces in second gradient 3D continua. Z. Angew. Math. Mech. 92(1), 52–71 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  24. dell’Isola F., Madeo A., Seppecher P., Boundary conditions at fluid-permeable interfaces in porous media: a variational approach. Int. J. Solids Struct. 46(17), 3150–3164 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  25. dell’Isola F., Romano A.: On a general balance law for continua with an interface. Ricerche Mat. 35, 325–337 (1986)

    MathSciNet  MATH  Google Scholar 

  26. dell’Isola F., Romano A.: On the derivation of thermomechanical balance equations for continuous systems with a nonmaterial interface. Int. J. Eng. Sci. 25, 1459–1468 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  27. dell’Isola F., Seppecher P.: Edge contact forces and quasi-balanced power. Meccanica 32, 33–52 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  28. dell’Isola F., Seppecher P.: The relationship between edge contact forces, double forces and interstitial working allowed by the principle of virtual power. Comptes Rendus de lAcademie de Sciences Serie IIb: Mecanique, Physique, Chimie, Astronomie 321, 303–308 (1995)

    MATH  Google Scholar 

  29. dell’Isola F., Vidoli S.: Continuum modelling of piezoelectromechanical truss beams: an application to vibration damping. Arch. Appl. Mech. 68(1), 1–19 (1998)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  30. dell’Isola F., Vidoli S.: Damping of bending waves in truss beams by electrical transmission lines with PZT actuators. Arch. Appl. Mech. 68, 626–636 (1998)

    Article  ADS  MATH  Google Scholar 

  31. Dick A.J., Balachandran B., Yabuno H., Numatsu M., Hayashi K., Kuroda M., Ashida K.: Utilizing nonlinear phenomena to locate grazing in the constrained motion of a cantilever beam. Nonlinear Dyn. 57, 335–349 (2009)

    Article  MATH  Google Scholar 

  32. Di Egidio A., Luongo A., Vestroni F.: A non-linear model for the dynamics of open cross-section thin-walled beams—Part I: formulation. Int. J. Non-Linear Mech. 38(7), 1067–1081 (2003)

    Article  MATH  Google Scholar 

  33. Di Egidio A., Luongo A., Vestroni F.: A non-linear model for the dynamics of open cross-section thin-walled beams—Part II: forced motion. Int. J. Non-Linear Mech. 38(7), 1083–1094 (2003)

    Article  MATH  Google Scholar 

  34. Fathi A., Popplewell N.: Improved approximations for a beam impacting a stop. J. Sound Vib. 170(3), 365–375 (1994)

    Article  ADS  MATH  Google Scholar 

  35. Fegelman K.J.L., Grosh K.: Dynamics of a flexible beam contacting a linear spring at low frequency excitation: experiment and analysis. ASME J. Vib. Acoust. 124(2), 237–249 (2002)

    Article  Google Scholar 

  36. Goldsmith W.: Impact. The Theory and Physical Behaviour of Colliding Solids. Dover, New York (1960)

    MATH  Google Scholar 

  37. Iafrati A., Carcaterra A., Ciappi A., Campana E.F.: Hydroelastic analysis of a simple oscillator impacting the free surface. J. Ship Res. 44(4), 278–289 (2000)

    Google Scholar 

  38. Jerrelind J., Stensson A.: Non-linear dynamics of parts in engineering systems. Chaos Solitons Fractals 11, 2413–2428 (2000)

    Article  ADS  MATH  Google Scholar 

  39. Knudsen J., Massih A.R.: Vibro-impact dynamics of a periodically forced beam. ASME J. Press. Vessel Technol. 122(2), 210–221 (2000)

    Article  Google Scholar 

  40. Lee K.: Dynamic contact analysis for the valvetrain dynamics of an internal combustion engine by finite element techniques. Proc. Inst. Mech. Eng. D J. Automobile Eng. 218(3), 353–358 (2004)

    Google Scholar 

  41. Leine R.I., van Campen D.H., Keultjes W.J.G.: Stick-slip whirl interaction in drillstring dynamics. ASME J. Vibr. Acoust. 124(2), 209–220 (2002)

    Article  Google Scholar 

  42. Lin J.-H., Weng C.-C.: Probability analysis of seismic pounding of adjacent buildings. Earthquake Eng. Struct. Dyn. 30(10), 1539–1557 (2001)

    Article  Google Scholar 

  43. Luongo A.: Mode localization in dynamics and buckling of linear imperfect continuous structures. Nonlinear Dyn. 25(1–3), 133–156 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  44. Luongo A., Di Egidio A.: Divergence, Hopf and double-zero bifurcations of a nonlinear planar beam. Comput. Struct. 84(24–25), 1596–1605 (2006)

    Article  Google Scholar 

  45. Luongo A., Di Egidio A., Paolone A.: On the proper form of the amplitude modulation equations for resonant systems. Nonlinear Dyn. 27(3), 237–254 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  46. Luongo A., Paolone A.: Multiple scale analysis for divergence-hopf bifurcation of imperfect symmetric systems. J. Sound Vib. 218(3), 527–539 (1998)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  47. Luongo A., Paolone A., Di Egidio A.: Multiple timescales analysis for 1:2 and 1:3 Resonant Hopf bifurcations. Nonlinear Dyn. 34(3–4), 269–291 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  48. Luongo A., Paolone A., Piccardo G.: Postcritical behavior of cables undergoing two simultaneous galloping modes. Meccanica 33(3), 229–242 (1998)

    Article  MATH  Google Scholar 

  49. Luongo A., Piccardo G.: Linear instability mechanisms for coupled translational galloping. J. Sound Vib. 288(4–5), 1027–1047 (2005)

    Article  ADS  Google Scholar 

  50. Luongo A., Romeo F.: Real wave vectors for dynamic analysis of periodic structures. J. Sound Vib. 279(1–2), 309–325 (2005)

    Article  ADS  Google Scholar 

  51. Maragakis E.A., Jennings P.C.: Analytical modals for the rigid body motions of skew bridges. Earthquake Eng. Struct. Dyn. 15(8), 923–944 (1987)

    Article  Google Scholar 

  52. Maurini C., dell’Isola F., Del Vescovo D.: Comparison of piezoelectronic networks acting as distributed vibration absorbers. Mech. Syst. Signal Process. 18, 1243–1271 (2004)

    Article  ADS  Google Scholar 

  53. Maurini C., Pouget J., dell’Isola F.: Extension of the Euler-Bernoulli model of piezoelectric laminates to include 3D effects via a mixed approach. Comput. Struct. 84(22–23), 1438–1458 (2006)

    Article  Google Scholar 

  54. Maurini C., Pouget J.F., dell’Isola F.: On a model of layered piezoelectric beams including transverse stress effect. Int. J. Solids Struct. 41, 4473–4502 (2004)

    Article  MATH  Google Scholar 

  55. Oppenheimer C.H., Dubowsky S.: A methodology for predicting impact-induced acoustic noise in machine systems. J. Sound Vib. 266(5), 1025–1051 (2003)

    Article  ADS  Google Scholar 

  56. Paolone A., Vasta M., Luongo A.: Flexural-torsional bifurcations of a cantilever beam under potential and circulatory forces I: non-linear model and stability analysis. Int. J. Non-Linear Mech. 41(4), 586–594 (2006)

    Article  ADS  Google Scholar 

  57. Peterka F.: Bifurcations and transition phenomena in an impact oscillator. Chaos Solitons Fractals 7, 1635–1647 (1996)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  58. Placidi L., dellIsola F., Ianiro N., Sciarra G.: Variational formulation of pre-stressed solid fluid mixture theory, with an application to wave phenomena. Eur. J. Mech. A/Solids 27, 582–606 (2008)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  59. Porfiri M., dellIsola F., Frattale Mascioli F.M.: Circuit analog of a beam and its application to multimodal vibration damping, using piezoelectric transducers. Int. J. Circuit Theory Appl. 32, 167–198 (2004)

    Article  MATH  Google Scholar 

  60. Porfiri M., dellIsola F., Santini E.: Modeling and design of passive electric networks interconnecting piezoelectric transducers for distributed vibration control. Int. J. Appl. Electromagnet. Mech. 21, 69–87 (2005)

    Google Scholar 

  61. Quiligotti S., Maugin G.A., Dell’Isola F.: An Eshelbian approach to the nonlinear mechanics of constrained solid-fluid mixtures. Acta Mech. 160(1–2), 45–60 (2003)

    Article  MATH  Google Scholar 

  62. Romeo F., Luongo A.: Vibration reduction in piecewise bi-coupled periodic structures. J. Sound Vib. 268(3), 601–615 (2003)

    Article  ADS  Google Scholar 

  63. Romeo F., Luongo A.: Invariant representation of propagation properties for bi-coupled periodic structures. J. Sound Vib. 257(5), 869–886 (2002)

    Article  ADS  Google Scholar 

  64. Sciarra G., dell’Isola F., Hutter K.: A solid-fluid mixture model allowing for solid dilatation under external pressure. Continuum Mech. Thermodyn. 13(5), 287–306 (2001)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  65. van de Vorst E.L.B., van Campen D.H., de Kraker A., Fey R.H.B.: Periodic solutions of a multi-DOF beam system with impact. J. Sound Vib. 192, 913–925 (1996)

    Article  MATH  Google Scholar 

  66. Wagg D.J., Bishop S.R.: Application of non-smooth modelling techniques to the dynamics of a flexible impacting beam. J. Sound Vib. 256(5), 803–820 (2002)

    Article  ADS  Google Scholar 

  67. Wang C., Kim J.: New analysis method for a thin beam impacting against a stop based on the full continuous model. J. Sound Vib. 191(5), 809–823 (1996)

    Article  ADS  Google Scholar 

  68. Wang C., Kim J.: The dynamic analysis of a thin beam impacting against a stop of general three-dimensional geometry. J. Sound Vib. 203(2), 237–249 (1997)

    Article  ADS  Google Scholar 

  69. Wu T.X., Thompson D.J.: The effects of track non-linearity on wheel/rail impact. Proc. Inst. Mech. Eng. F J. Rail Rapid Trans. 218(1), 1–15 (2004)

    Article  MathSciNet  Google Scholar 

  70. Yin X.C., Qin Y., Zou H.: Transient responses of repeated impact of a beam against a stop. Int. J. Solids Struct. 44, 7323–7339 (2007)

    Article  MATH  Google Scholar 

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Author notes

  1. Ugo Andreaus

    Present address: , Via Eudossiana, 18 00184, Rome, Italy

  2. Bernardino Chiaia

    Present address: , Corso Duca degli Abruzzi, 24, 10129, Turin, Italy

  3. Luca Placidi

    Present address: , C.so Vittorio Emanuele II, 39, 00186, Rome, Italy

Authors and Affiliations

  1. Dipartimento di Ingegneria Strutturale e Geotecnica, Universita di Roma La Sapienza, Rome, Italy

    Ugo Andreaus

  2. Dipartimento di Ingegneria Strutturale e Geotecnica, Politecnico di Torino, Turin, Italy

    Bernardino Chiaia

  3. International Telematic University Uninettuno, Rome, Italy

    Luca Placidi

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  1. Ugo Andreaus
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  2. Bernardino Chiaia
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  3. Luca Placidi
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Correspondence to Luca Placidi.

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Communicated by Francesco dell’Isola and Samuel Forest.

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Andreaus, U., Chiaia, B. & Placidi, L. Soft-impact dynamics of deformable bodies. Continuum Mech. Thermodyn. 25, 375–398 (2013). https://doi.org/10.1007/s00161-012-0266-5

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