Estimation of impact forces during multi-point collisions involving small deformations

Abstract

Collision between hard objects causes abrupt changes in the velocities of the system, which are characterized by very large contact forces over very small time durations. A common approach in the analysis of such collisions is to describe the system velocities using an impulse–momentum based relationship. The time duration of impact and the deformations at the contact points are usually assumed to be negligible for such an impact models, and the system velocities are evolved in terms of the impulses on the system. This type of impact models are usually relevant for hard (rigid) impacts, where deformations at the contact points can be considered negligible. However, these models cannot determine the forces during the impact process. The main objective of this work is to reformulate the impulse–momentum based model to determine the forces during an impact event, by relaxing the rigidity assumption to allow small deformations at the contact points.

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

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
Fig. 25
Fig. 26
Fig. 27
Fig. 28
Fig. 29
Fig. 30
Fig. 31
Fig. 32
Fig. 33
Fig. 34
Fig. 35

References

  1. 1.

    Anitescu, M., Potra, F.A., Stewart, D.E.: Time-stepping for three-dimensional rigid body dynamics. Comput. Methods Appl. Mech. Eng. 177(3), 183–197 (1999)

    MathSciNet  MATH  Article  Google Scholar 

  2. 2.

    Awrejcewicz, J., Kudra, G.: Rolling resistance modelling in the celtic stone dynamics. Multibody Syst. Dyn. 45(2), 155–167 (2019)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Boulanger, G.: Sur le choc avec frottement des corps non parfaitement élastiques. Rev. Sci. 77, 325–327 (1939)

    MATH  Google Scholar 

  4. 4.

    Bowling, A.: Dynamic performance, mobility, and agility of multi-legged robots. J. Dyn. Syst. Meas. Control 128(4), 765–777 (2006)

    Article  Google Scholar 

  5. 5.

    Brake, M.: An analytical elastic-perfectly plastic contact model. Int. J. Solids Struct. 49(22), 3129–3141 (2012)

    Article  Google Scholar 

  6. 6.

    Brogliato, B.: Nonsmooth Mechanics: Models, Dynamics and Control, 2nd edn. Springer, London (1999)

    Google Scholar 

  7. 7.

    Brogliato, B., Ten Dam, A., et al.: Numerical simulation of finite dimensional multibody nonsmooth mechanical systems. Appl. Mech. Rev. 55(2), 107–149 (2002)

    Article  Google Scholar 

  8. 8.

    Chakraborty, N., Berard, S., Akella, S., Trinkle, J.C.: An implicit time-stepping method for multibody systems with intermittent contact. In: Robotics: Science and Systems (2007)

    Google Scholar 

  9. 9.

    Chatterjee, A., Bowling, A.: Resolving the unique invariant slip-direction in rigid three-dimensional multi-point impacts at stick–slip transitions. In: ASME 2018 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, pp. V006T09A008–V006T09A008. American Society of Mechanical Engineers, New York (2018)

    Google Scholar 

  10. 10.

    Chatterjee, A., Bowling, A.: Modeling three-dimensional surface-to-surface rigid contact and impact. Multibody Syst. Dyn. 46, 1–40 (2019)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Chatterjee, A., Rodriguez, A., Bowling, A.: Analytic solution for planar indeterminate impact problems using an energy constraint. Multibody Syst. Dyn. 42(3), 347–379 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  12. 12.

    Craig, J.: Introduction to Robotics: Mechanics and Control. Addison-Wesley, Reading (1989)

    Google Scholar 

  13. 13.

    Darboux, G.: Etude géométrique sur les percussions et le choc des corps. Bull. Sci. Math. Astron. 4(1), 126–160 (1880)

    MathSciNet  MATH  Google Scholar 

  14. 14.

    Djerassi, S.: Collision with friction; Part A: Newton’s hypothesis. Multibody Syst. Dyn. 21(1), 37–54 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  15. 15.

    Djerassi, S.: Collision with friction; Part B: Poisson’s and Stronge’s hypotheses. Multibody Syst. Dyn. 21(1), 55–70 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  16. 16.

    Djerassi, S.: Stronge’s hypothesis-based solution to the planar collision-with-friction problem. Multibody Syst. Dyn. 24(4), 493–515 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  17. 17.

    Djerassi, S.: Three-dimensional, one-point collision with friction. Multibody Syst. Dyn. 27(2), 173–195 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  18. 18.

    Dormand, J., Prince, P.: A family of embedded Runge–Kutta formulae. J. Comput. Appl. Math. 6(1), 19–26 (1980)

    MathSciNet  MATH  Article  Google Scholar 

  19. 19.

    Flickinger, D., Bowling, A.: Simultaneous oblique impacts and contacts in multibody systems with friction. Multibody Syst. Dyn. 23(3), 249–261 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  20. 20.

    Flores, P., Machado, M., Silva, M.T., Martins, J.M.: On the continuous contact force models for soft materials in multibody dynamics. Multibody Syst. Dyn. 25(3), 357–375 (2011)

    MATH  Article  Google Scholar 

  21. 21.

    Ghaednia, H., Brake, M.R., Berryhill, M., Jackson, R.L.: Strain hardening from elastic–perfectly plastic to perfectly elastic flattening single asperity contact. J. Tribol. 141(3), 031402 (2019)

    Article  Google Scholar 

  22. 22.

    Ghaednia, H., Cermik, O., Marghitu, D.B.: Experimental and theoretical study of the oblique impact of a tennis ball with a racket. Proc. Inst. Mech. Eng., Part P: J. Sports Eng. Technol. 229(3), 149–158 (2015)

    Article  Google Scholar 

  23. 23.

    Gharib, M., Hurmuzlu, Y.: A new contact force model for low coefficient of restitution impact. J. Appl. Mech. 79(6), 064506 (2012)

    Article  Google Scholar 

  24. 24.

    Gheadnia, H., Cermik, O., Marghitu, D.B.: Experimental and theoretical analysis of the elasto-plastic oblique impact of a rod with a flat. Int. J. Impact Eng. 86, 307–317 (2015)

    Article  Google Scholar 

  25. 25.

    Gholami, F., Nasri, M., Kövecses, J., Teichmann, M.: A linear complementarity formulation for contact problems with regularized friction. Mech. Mach. Theory 105, 568–582 (2016)

    Article  Google Scholar 

  26. 26.

    Gilardi, G., Sharf, I.: Literature survey of contact dynamics modeling. Mech. Mach. Theory 37(10), 1213–1239 (2002)

    MathSciNet  MATH  Article  Google Scholar 

  27. 27.

    Giouvanidis, A., Dimitrakopoulos, I.: Modelling contact in rocking structures with a nonsmooth dynamics approach. In: ECCOMAS Congress 2016-Proceedings of the 7th European Congress on Computational Methods in Applied Sciences and Engineering, p. 0000 (2016)

    Google Scholar 

  28. 28.

    Glocker, C., Studer, C.: Formulation and preparation for numerical evaluation of linear complementarity systems in dynamics. Multibody Syst. Dyn. 13(4), 447–463 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  29. 29.

    Goldsmith, W.: Impact. Courier Corporation, New York (2001)

    Google Scholar 

  30. 30.

    Gonthier, Y., McPhee, J., Lange, C., Piedboeuf, J.-C.: A regularized contact model with asymmetric damping and dwell-time dependent friction. Multibody Syst. Dyn. 11(3), 209–233 (2004)

    MATH  Article  Google Scholar 

  31. 31.

    Hertz, H.: Über die Berührung fester elastischer Körper (On the Contact of Elastic Solids). Reine und Angewandte Mathematik. London: (Instruction anglaise dans miscellaneous papers by H. Hertz) Eds. Jones et Schaott, 1896

  32. 32.

    Huněk, I.: On a penalty formulation for contact-impact problems. Comput. Struct. 48(2), 193–203 (1993)

    MATH  Article  Google Scholar 

  33. 33.

    Hunt, K., Crossley, F.R.E.: Coefficient of restitution interpreted as damping in vibroimpact. J. Appl. Mech. 42(2), 440–445 (1975)

    Article  Google Scholar 

  34. 34.

    Jackson, R.L., Green, I.: A finite element study of elasto-plastic hemispherical contact against a rigid flat. J. Tribol. 127(2), 343–354 (2005)

    Article  Google Scholar 

  35. 35.

    Jaeger, J.: New Solutions in Contact Mechanics, Wit Press, Southampton (2005)

    Google Scholar 

  36. 36.

    Jia, Y.-B.: Energy-based modeling of tangential compliance in 3-dimensional impact. In: Algorithmic Foundations of Robotics IX, pp. 267–284. Springer, Berlin (2011)

    Google Scholar 

  37. 37.

    Johnson, K.L., Johnson, K.L.: Contact Mechanics. Cambridge University Press, Cambridge (1987)

    Google Scholar 

  38. 38.

    Kane, T., Levinson, D.: Dynamics: Theory and Applications. McGraw-Hill, New York (1985)

    Google Scholar 

  39. 39.

    Kardel, K., Ghaednia, H., Carrano, A.L., Marghitu, D.B.: Experimental and theoretical modeling of behavior of 3d-printed polymers under collision with a rigid rod. Addit. Manuf. 14, 87–94 (2017)

    Google Scholar 

  40. 40.

    Keller, J.: Impact with friction. J. Appl. Mech. 53(1), 1–4 (1986)

    MathSciNet  MATH  Article  Google Scholar 

  41. 41.

    Kireenkov, A.: Coupled models of sliding and rolling friction. Dokl. Phys. 53(4), 233–236. (2008)

    MATH  Article  Google Scholar 

  42. 42.

    Kireenkov, A.: Coulomb law in generalized differential form in problems of dynamics of rigid bodies with combined kinematics. Mech. Solids 45(2), 166–175 (2010)

    Article  Google Scholar 

  43. 43.

    Kosenko, I., Aleksandrov, E.: Implementation of the Contensou-Erismann model of friction in frame of the hertz contact problem on modelica. In: Proceedings of the 7th International Modelica Conference, No. 043, Como; Italy, 20–22 September 2009, pp. 288–298. Linköping University Electronic Press, Linköping (2009)

    Google Scholar 

  44. 44.

    Kraus, P.R., Kumar, V.: Compliant contact models for rigid body collisions. In: Robotics and Automation, 1997. Proceedings., 1997 IEEE International Conference on, vol. 2, pp. 1382–1387. IEEE, New York (1997)

    Google Scholar 

  45. 45.

    Kudra, G., Szewc, M., Ludwicki, M., Awrejcewicz, J.: Modeling and simulation of bifurcation dynamics of a double spatial pendulum excited by a rotating obstacle. Int. J. Struct. Stab. Dyn. 19(12), 1950145 (2019)

    MathSciNet  Article  Google Scholar 

  46. 46.

    Kudra, G., Szewc, M., Wojtunik, I., Awrejcewicz, J.: Shaping the trajectory of the billiard ball with approximations of the resultant contact forces. Mechatronics 37, 54–62 (2016)

    Article  Google Scholar 

  47. 47.

    Lankarani, H.: Contact force model with hysteresis damping for impact analysis of multibody systems. J. Mech. Des. 112(3), 369–376 (1990)

    Google Scholar 

  48. 48.

    Lankarani, H., Nikravesh, P.: A contact force model with hysteresis damping for impact analysis of multibody systems. J. Mech. Des. 112(3), 369–376 (1990)

    Google Scholar 

  49. 49.

    Leine, R.I., Glocker, C.: A set-valued force law for spatial Coulomb–Contensou friction. Eur. J. Mech. A, Solids 22(2), 193–216 (2003)

    MathSciNet  MATH  Article  Google Scholar 

  50. 50.

    Liu, C., Zhao, Z., Brogliato, B.: Frictionless multiple impacts in multibody systems. I. Theoretical framework. Proc. R. Soc. A, Math. Phys. Eng. Sci. 464(2100), 3193–3211 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  51. 51.

    Liu, C., Zhao, Z., Brogliato, B.: Variable structure dynamics in a bouncing dimer. Ph.D. dissertation, INRIA (2008)

  52. 52.

    Liu, C., Zhao, Z., Brogliato, B.: Frictionless multiple impacts in multibody systems. II. Numerical algorithm and simulation results. Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 465, 1–23. (2009)

    MathSciNet  MATH  Article  Google Scholar 

  53. 53.

    Liu, T.: Non-jamming conditions in multi-contact rigid-body dynamics. Multibody Syst. Dyn. 22(3), 269–295 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  54. 54.

    Liu, T., Wang, M.Y.: Computation of three-dimensional rigid-body dynamics with multiple unilateral contacts using time-stepping and Gauss-Seidel methods. IEEE Trans. Autom. Sci. Eng. 2(1), 19–31 (2005)

    Article  Google Scholar 

  55. 55.

    Marghitu, D., Stoenescu, E.: Rigid body impact with moment of rolling friction. Nonlinear Dyn. 50(3), 597–608 (2007)

    MathSciNet  MATH  Article  Google Scholar 

  56. 56.

    Marhefka, D.W., Orin, D.E.: A compliant contact model with nonlinear damping for simulation of robotic systems. IEEE Trans. Syst. Man Cybern., Part A, Syst. Hum. 29(6), 566–572 (1999)

    Article  Google Scholar 

  57. 57.

    Moreau, J.: Numerical aspects of the sweeping process. Comput. Methods Appl. Mech. Eng. 177(3), 329–349 (1999)

    MathSciNet  MATH  Article  Google Scholar 

  58. 58.

    Papadopoulos, P., Solberg, J.: A Lagrange multiplier method for the finite element solution of frictionless contact problems. Math. Comput. Model. 28(4), 373–384 (1998)

    MATH  Article  Google Scholar 

  59. 59.

    Peña, F., Lourenço, P.B., Campos-Costa, A.: Experimental dynamic behavior of free-standing multi-block structures under seismic loadings. J. Earthq. Eng. 12(6), 953–979 (2008)

    Article  Google Scholar 

  60. 60.

    Peña, F., Prieto, F., Lourenço, P.B., Campos Costa, A., Lemos, J.V.: On the dynamics of rocking motion of single rigid-block structures. Earthq. Eng. Struct. Dyn. 36(15), 2383–2399 (2007)

    Article  Google Scholar 

  61. 61.

    Pfeiffer, F., Glocker, C.: Multi-Body Dynamics with Unilateral Constraints. Wiley, New York (1996)

    Google Scholar 

  62. 62.

    Putignano, C., Afferrante, L., Carbone, G., Demelio, G.: A new efficient numerical method for contact mechanics of rough surfaces. Int. J. Solids Struct. 49(2), 338–343 (2012)

    Article  Google Scholar 

  63. 63.

    Roberts, S.M., Shipman, J.S.: Two-Point Boundary Value Problems: Shooting Methods. Am. Elsevier, New York (1972)

    Google Scholar 

  64. 64.

    Rodriguez, A.: Dynamic simulation of multibody systems in simultaneous, indeterminate contact and impact with friction. Ph.D. dissertation, UTA (2014)

  65. 65.

    Rodriguez, A., Bowling, A.: Solution to indeterminate multi-point impact with frictional contact using constraints. Multibody Syst. Dyn. 28(4), 313–330 (2012)

    MathSciNet  Article  Google Scholar 

  66. 66.

    Rodriguez, A., Bowling, A.: Study of Newton’s cradle using a new discrete approach. Multibody Syst. Dyn. 33(1), 61–92 (2015)

    MathSciNet  Article  Google Scholar 

  67. 67.

    Rodriguez, A., Chatterjee, A., Bowling, A.: Solution to three-dimensional indeterminate contact and impact with friction using rigid body constraints. In: ASME 2015 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, pp. V006T10A037–V006T10A037. American Society of Mechanical Engineers, New York (2015)

    Google Scholar 

  68. 68.

    Routh, E.J., et al.: Dynamics of a System of Rigid Bodies. Dover, New York (1960)

    Google Scholar 

  69. 69.

    Shampine, L.F., Reichelt, M.W.: The Matlab ode suite. SIAM J. Sci. Comput. 18(1), 1–22 (1997)

    MathSciNet  MATH  Article  Google Scholar 

  70. 70.

    Sharf, I., Zhang, Y.: A contact force solution for non-colliding contact dynamics simulation. Multibody Syst. Dyn. 16(3), 263–290 (2006)

    MathSciNet  MATH  Article  Google Scholar 

  71. 71.

    Simo, J.C., Laursen, T.: An augmented Lagrangian treatment of contact problems involving friction. Comput. Struct. 42(1), 97–116 (1992)

    MathSciNet  MATH  Article  Google Scholar 

  72. 72.

    Skrinjar, L., Slavič, J., Boltežar, M.: A review of continuous contact-force models in multibody dynamics. Int. J. Mech. Sci. 145, 171–187 (2018)

    Article  Google Scholar 

  73. 73.

    Stewart, D.E., Trinkle, J.C.: An implicit time-stepping scheme for rigid body dynamics with inelastic collisions and Coulomb friction. Int. J. Numer. Methods Eng. 39(15), 2673–2691 (1996)

    MathSciNet  MATH  Article  Google Scholar 

  74. 74.

    Stewart, D.: Rigid-body dynamics with friction and impact. SIAM Rev. 42(1), 3–39 (2000)

    MathSciNet  MATH  Article  Google Scholar 

  75. 75.

    Stronge, W.: Impact Mechanics. Cambridge University Press, Cambridge (2000)

    Google Scholar 

  76. 76.

    Stronge, W.: Chain reaction from impact on aggregate of elasto-plastic ‘rigid’ bodies. Int. J. Impact Eng. 28(3), 291–302 (2003)

    Article  Google Scholar 

  77. 77.

    Stronge, W.: Smooth dynamics of oblique impact with friction. Int. J. Impact Eng. 51, 36–49 (2013)

    Article  Google Scholar 

  78. 78.

    Vigué, P., Vergez, C., Karkar, S., Cochelin, B.: Regularized friction and continuation: comparison with Coulomb’s law. J. Sound Vib. 389, 350–363 (2017)

    Article  Google Scholar 

  79. 79.

    Wang, Y.-T., Kumar, V., Abel, J.: Dynamics of rigid bodies undergoing multiple frictional contacts. In: Robotics and Automation, 1992. Proceedings., 1992 IEEE International Conference on, pp. 2764–2769. IEEE, New York (1992).

    Google Scholar 

  80. 80.

    Whittaker, E.: A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, 2nd edn. Cambridge University Press, Cambridge (1917)

    Google Scholar 

  81. 81.

    Wriggers, P., Zavarise, G.: Computational contact mechanics. In: Encyclopedia of Computational Mechanics (2004)

    Google Scholar 

  82. 82.

    Yastrebov, V.A., Anciaux, G., Molinari, J.-F.: From infinitesimal to full contact between rough surfaces: evolution of the contact area. Int. J. Solids Struct. 52, 83–102 (2015)

    Article  Google Scholar 

  83. 83.

    Yigit, A.S., Christoforou, A.P., Majeed, M.A.: A nonlinear visco-elastoplastic impact model and the coefficient of restitution. Nonlinear Dyn. 66(4), 509–521 (2011)

    MathSciNet  Article  Google Scholar 

  84. 84.

    Zhang, H., Brogliato, B., Liu, C.: Dynamics of planar rocking-blocks with Coulomb friction and unilateral constraints: comparisons between experimental and numerical data. Multibody Syst. Dyn. 32(1), 1–25 (2014)

    MathSciNet  Article  Google Scholar 

  85. 85.

    Zhang, H., Brogliato, B.: The planar rocking-block: analysis of kinematic restitution laws, and a new rigid-body impact model with friction. Ph.D. dissertation, INRIA (2011)

  86. 86.

    Zhao, Z., Liu, C., Brogliato, B.: Planar dynamics of a rigid body system with frictional impacts. II. Qualitative analysis and numerical simulations. Proc. R. Soc. A, Math. Phys. Eng. Sci. 465(2107), 2267–2292 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  87. 87.

    Zhao, Z., Liu, C., Brogliato, B.: Energy dissipation and dispersion effects in granular media. Phys. Rev. E 78(3), 031307 (2008)

    MathSciNet  Article  Google Scholar 

  88. 88.

    Zhao, Z., Liu, C., Brogliato, B.: Planar dynamics of a rigid body system with frictional impacts. II. Qualitative analysis and numerical simulations. Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 465, 2267–2292. (2009)

    MathSciNet  MATH  Article  Google Scholar 

  89. 89.

    Zhuravlev, V.: The model of dry friction in the problem of the rolling of rigid bodies. J. Appl. Math. Mech. 62(5), 705–710 (1998)

    MathSciNet  Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Alan Bowling.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix A: Frictional and rigid body constraints

Appendix A: Frictional and rigid body constraints

This section discusses the derivation of \({\mathbf{C}}\) in (5) as well as the energetic termination condition for the impact events. During impact events, points may stick, slip or slip-reverse. The slip-state of any given impact point is characterized in terms of the sliding speed along the impact plane,

$$ \left \{ \textstyle\begin{array}{l@{\ }c@{\ }c@{\quad }l} s_{i} =0 & \text{and} & \dot{s}_{i} =0 & {\text{sticking},} \\ s_{i} =0 & \text{and} & \dot{s}_{i} \neq 0 & \text{stick--slip transition,} \\ s_{i} \neq 0 & & & {\text{slipping},} \end{array}\displaystyle \right . $$
(52)

where the subscript \(i\) refers to the impact point. In the literature, stick–slip behavior is represented as a set of complementarity conditions [7], which establish frictional force relationships for different slip states. These complementarity conditions are defined based on Coulomb friction [4, 7, 28], and form the basis for the LCP, which are solved using optimization techniques [28, 61]. This work uses a similar set of complementarity conditions based on Coulomb friction law,

$$ \left \{ \textstyle\begin{array}{l@{\quad }l} {\text{sticking}} & dp_{t_{i}} \leq \mu _{s} \ dp_{z_{i}} , \\ {\text{stick-slip transition}} & dp_{t_{i}} =\mu _{s} \ dp_{z_{i}} , \\ {\text{slipping}} & dp_{t_{i}} =\mu _{d} \ dp_{z_{i}} , \end{array}\displaystyle \right . $$
(53)

where \(dp_{t_{i}} = \sqrt{{dp_{x_{i}}}^{2} + {dp_{y_{i}}}^{2}}\), \(\mu _{s}\) and \(\mu _{d}\) are the values of static and dynamic coefficients of friction, and \(i= \{ 1,\ldots,4 \}\). Here the complementarity conditions are expressed in terms of the forces in (5), where \(dt\) is canceled from both sides, leaving the differential impulses.

A.1 Frictional constraint for slipping

If impact point \(i\) slips, then based on (53), the tangential and normal differential impulses are related as

$$ \displaystyle \frac{dp_{x_{i}}}{dt} = -\mu _{i} \cos (\phi _{i}) \ \frac{dp_{z_{i}}}{dt}, \qquad \displaystyle \frac{dp_{y_{i}}}{dt} = -\mu _{i} \sin (\phi _{i}) \ \frac{dp_{z_{i}}}{dt}, $$
(54)

where \(\mu _{i}\) is the coefficient of friction for point \(i\). Here this value is selected as \(\mu _{i} = \mu _{d}\). The equations in (54) also govern the force relations during slip reversal. The frictional constraints in (54) can be expressed as

$$ \left \lbrack \textstyle\begin{array}{c} dp_{x_{i}} + \mu _{i} \ \cos (\phi _{i}) \ dp_{z_{i}} \\ dp_{y_{i}} + \mu _{i} \ \sin (\phi _{i}) \ dp_{z_{i}} \end{array}\displaystyle \right \rbrack =U_{i} \ d{\mathbf{p}} ={\mathbf{0}}. $$
(55)

A.2 Frictional constraint for sticking

When impact point \(i\) sticks, the tangential components of its velocity must remain equal to zero, \(s_{i} = 0\), and therefore \(v_{x_{i}} = v_{y_{i}} = 0\). The differential velocities of the sticking points are obtained from (3) as

$$ d{\boldsymbol{\vartheta }} =M \ d{\mathbf{p}} =\left [ {\mathbf{m}}_{x_{1}}^{T} \ \ {\mathbf{m}}_{y_{1}}^{T} \ \ {\mathbf{m}}_{z_{1}}^{T} \ \ \ldots \right ]^{T} d {\mathbf{p}} .$$
(56)

When an impact point sticks [75],

$$ \left \lbrack \textstyle\begin{array}{c} dv_{x_{i}} \\ dv_{y_{i}} \end{array}\displaystyle \right \rbrack =\left \lbrack \textstyle\begin{array}{c} {\mathbf{m}}_{x_{i}} \\ {\mathbf{m}}_{y_{i}} \end{array}\displaystyle \right \rbrack d{\mathbf{p}} =U_{i} \ d{\mathbf{p}} ={\mathbf{0}} .$$
(57)

A.3 Rigid-body constraints

The frictional constraints are sufficient for resolving single-point impacts, but multi-point analysis requires additional constraints. Here such constraints are derived using the rigidity properties of the impacting body. These constraints are expressed in terms of the line connecting two points on a rigid body:

$$ \hat{\boldsymbol{\eta }} =\frac{({\mathbf{P}}_{Oi} - {\mathbf{P}}_{Oj})}{||{\mathbf{P}}_{Oi} - {\mathbf{P}}_{Oj}||} =\left [ \eta _{x} \ \ \eta _{y} \ \ \eta _{z} \right ]^{T} $$
(58)

where \({\mathbf{P}}_{Oi}\) and \({\mathbf{P}}_{Oj}\) are the position vectors from the body-attached reference point \(O\) to the impact points \(i,j= \{ 1,\ldots,4 \}\), \(i \neq j\). The relative motion of two impact points on the same rigid body must be equal to zero when observed from the body-attached reference frame and point, therefore

$$ \begin{gathered} ({\mathbf{v}}_{i} - {\mathbf{v}}_{j}) \cdot \hat{\boldsymbol{\eta }} =0, \\ (v_{x_{i}}-v_{x_{j}})\eta _{x} + (v_{y_{i}}-v_{y_{j}})\eta _{y} + (v_{z_{i}}-v_{z_{j}}) \eta _{z} ={\mathbf{\mathbf{w}}}_{ij} \ {\boldsymbol{\vartheta }} =0. \end{gathered} $$
(59)

Equation (59) provides a constraint equation which relates the velocities of points \(i\) and \(j\). Now, it is possible to transform this velocity based rigid-body constraint into a constraint on the differential impulses (or forces) based on a work–energy argument, as shown in [10, 11, 65, 66]. Let us consider one way constraining the velocities using (59), such that \(v_{x_{i}}\) depends upon all other velocity components,

$$ {\boldsymbol{\vartheta }} = \left \lbrack \textstyle\begin{array}{c} v_{x_{j}} - (v_{y_{i}}-v_{y_{j}})\frac{\eta _{y}}{\eta _{x}} - (v_{z_{i}}-v_{z_{j}}) \frac{\eta _{z}}{\eta _{x}} \\ v_{y_{i}} \\ v_{z_{i}} \\ v_{x_{j}} \\ v_{y_{j}} \\ v_{z_{j}} \end{array}\displaystyle \right \rbrack = \underbrace{ \left \lbrack \textstyle\begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} -\alpha & -\beta & 1 & \alpha & \beta \\ 1 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 1 \end{array}\displaystyle \right \rbrack }_{P} {\boldsymbol{\vartheta }}_{s} $$
(60)

where \({\boldsymbol{\vartheta }}_{s}\) are independent velocities, \(P\) is a matrix representing the rigid-body constraint, and \(\alpha =\frac{\eta _{y}}{\eta _{x}}\) and \(\beta = \frac{\eta _{z}}{\eta _{x}}\). This may be also expressed as

$$ {\boldsymbol{\vartheta }} = J \dot{\mathbf{q}} = P {\boldsymbol{\vartheta }}_{s} .$$
(61)

This yields

$$ {\boldsymbol{\vartheta }}_{s} = P^{+} J \dot{\mathbf{q}} $$
(62)

where \(P^{+} = {(P^{T} P)}^{-1} P^{T}\) is the generalized inverse of the matrix \(P\). As shown in [12], the virtual work due to the contact forces yields the relation,

$$ {\mathbf{F}}^{T} {\boldsymbol{\vartheta }} = {\boldsymbol{\Gamma }}^{T} \dot{\mathbf{q}} .$$
(63)

Let \({\mathbf{F}}_{s}\) be a vector containing the constrained force terms corresponding to the independent velocities \({\boldsymbol{\vartheta }}_{s}\). Enforcement of the velocity-level rigid-body constraints should not cause any change in the virtual work of the system. Hence,

$$ {{\mathbf{F}}_{s}}^{T} {\boldsymbol{\vartheta }}_{s} = {\boldsymbol{\Gamma }}^{T} \dot{\mathbf{q}} .$$
(64)

Now substituting the relation \({\boldsymbol{\Gamma }} = J^{T} {\mathbf{F}}\) and (62) into (64) yields

$$ {{\mathbf{F}}_{s}}^{T} P^{+} J \dot{\mathbf{q}} = {\left ( J^{T} {\mathbf{F}} \right )}^{T} \dot{\mathbf{q}} .$$
(65)

This yields the constrained force relationships,

$$ {\mathbf{F}} = (P^{+})^{T} {\mathbf{F}}_{s}, \qquad {\mathbf{F}}_{s} = P^{T} {\mathbf{F}} .$$
(66)

Therefore, from (66), a projection of rigid-body constraints onto the force space can be obtained as

$$ \left ( I - {\left ( P^{+} \right )}^{T} P^{T} \right ) {\mathbf{F}} = { \mathbf{0}} .$$
(67)

Note that in (67), \({\left ( P^{+} \right )}^{T} P^{T} \neq I\). The matrix \({\left ( P^{+} \right )}^{T} P^{T} \) projects the forces \({\mathbf{F}}\) to a space orthogonal to the velocity-level constraint such that it is equal to \({\mathbf{F}}\). Now using the coefficients of \(P\) from (60), the matrix \(I - {\left ( P^{+} \right )}^{T} P^{T}\) can be symbolically obtained as

$$ I - {\left ( P^{+} \right )}^{T} P^{T} = \frac{1}{2 \left ( \alpha ^{2}+ \beta ^{2} +1 \right )} \left \lbrack \textstyle\begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} 1 & \alpha & \beta & -1 & - \alpha & - \beta \\ \alpha & \alpha ^{2} & \alpha \beta & -\alpha & -\alpha ^{2} & - \alpha \beta \\ \beta & \alpha \beta & \beta ^{2} & - \beta & - \alpha \beta & - \beta ^{2} \\ -1 & - \alpha & - \beta & 1 & \alpha & \beta \\ - \alpha & - \alpha ^{2} & - \alpha \beta & \alpha & \alpha ^{2} & \alpha \beta \\ - \beta & - \alpha \beta & -\beta ^{2} & \beta & \alpha \beta & \beta ^{2} \\ \end{array}\displaystyle \right \rbrack .$$
(68)

It can be seen that all equations obtained by substituting (68) into (67) yield the same force constraint:

$$ f_{x_{i}} + \alpha f_{y_{i}} + \beta f_{z_{i}} - f_{x_{j}} - \alpha f_{y_{j}} - \beta f_{z_{j}} = 0 .$$
(69)

Now substituting back \(\alpha = \frac{\eta _{y}}{\eta _{x}}\) and \(\beta = \frac{\eta _{z}}{\eta _{x}}\), and rearranging gives the rigid-body constraint,

$$ (f_{x_{i}}-f_{x_{j}})\eta _{x} + (f_{y_{i}}-f_{y_{j}})\eta _{y} + (f_{z_{i}}-f_{z_{j}}) \eta _{z} = {\mathbf{\mathbf{w}}}_{ij}{\mathbf{F}} = 0 $$
(70)

or in terms of differential impulses

$$ {\mathbf{\mathbf{w}}}_{ij} d{\mathbf{p}} = 0 .$$
(71)

The rigid-body constraint in (71) is valid for any two arbitrary contact points \(i\) and \(j\). Hence to resolve the differential impulses for an \(n\)-point impact in terms of a single independent parameter, \(n-1\) such rigid-body constraints would need to be solved along with \(2n\) frictional constraint equations.

A.4 Final frictional and rigid body constraints

The frictional and rigid-body constraints from (55), (57) and (71) for the example block in Fig. 1 can be expressed as

$$ H \ d{\mathbf{p}} =\left \lbrack \textstyle\begin{array}{c} U_{1} \\ \vdots \\ U_{4} \\ {\mathbf{w}}_{14} \\ {\mathbf{w}}_{24} \\ {\mathbf{w}}_{34} \end{array}\displaystyle \right \rbrack d{\mathbf{p}} =0 .$$
(72)

Note that the \({\mathbf{w}}_{ij}\) will always be linearly independent as long as \(i \neq j\). The matrix \(H \in \mathbb{R}^{11 \times 12}\) is rank deficient by one row. The linearly dependent and independent columns of \(H\) can be separated as \(H =\left [ H_{s} \ \ H_{r} \right ]\). Here \(H_{s} \in \mathbb{R}^{11 \times 11}\) is a full rank matrix, \(H_{r} \in \mathbb{R}^{11 \times 1}\) is a linearly dependent column in \(H\), such that

$$ d{\mathbf{p}} =\left \lbrack \textstyle\begin{array}{c} -H_{s}^{-1} H_{r} \\ 1 \end{array}\displaystyle \right \rbrack dp_{n} ={\mathbf{C}} \ dp_{n} $$
(73)

where \({\mathbf{C}}\) relates all of the differential impulses to the single independent impulse parameter \(dp_{n}\).

If \(s_{i} = 0\), the differential impulses must satisfy the no-slip condition,

$$ \sqrt{ dp_{x_{i}}^{2} + dp_{y_{i}}^{2} } \leq \mu _{s} \ dp_{z_{i}} \quad \implies \quad \sqrt{ C_{x_{i}}^{2} + C_{y_{i}}^{2} } \leq \mu _{s} \ C_{z_{i}} $$
(74)

where \(C_{x_{i}}\), \(C_{y_{i}}\) and \(C_{z_{i}}\) are the element of \({\mathbf{C}}\) associated with \(dp_{x_{i}}\), \(dp_{y_{i}}\), and \(dp_{z_{i}}\) respectively. Thus (74) implies that the slip state of an impact point is dependent on the friction and rigid-body constraints, rather than the value of \(p_{n}\). If the no-slip condition is not satisfied, then point \(i\) slips along a new direction \(\hat{\phi }_{i}\). This is known as slip reversal. The following section presents how this new slip direction \(\hat{\phi }_{i}\) is calculated. Since this work considers rigid bodies, if two or more impact points stick, then sticking is enforced in all other contact points. Hence there can be only three possibility during stick–slip transition: 1) all points enter sticking, 2) a single point sticks, while the rest slip-reverse or 3) all points slip-reverse in some new directions. If the slip states of any of the impact points must be altered, \(H\) must be re-evaluated and all conclusions of sticking and slipping must be rechecked.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Chatterjee, A., Ghaednia, H., Bowling, A. et al. Estimation of impact forces during multi-point collisions involving small deformations. Multibody Syst Dyn (2020). https://doi.org/10.1007/s11044-020-09743-z

Download citation

Keywords

  • Contact
  • Impact
  • Impact Forces
  • Rigid Body
  • Small Deformations