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Rigid multibody system dynamics with uncertain rigid bodies

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Abstract

This paper is devoted to the construction of a probabilistic model of uncertain rigid bodies for multibody system dynamics. We first construct a stochastic model of an uncertain rigid body by replacing the mass, the center of mass, and the tensor of inertia by random variables. The prior probability distributions of the stochastic model are constructed using the maximum entropy principle under the constraints defined by the available information. The generators of independent realizations corresponding to the prior probability distribution of these random quantities are further developed. Then several uncertain rigid bodies can be linked to each other in order to calculate the random response of a multibody dynamical system. An application is proposed to illustrate the theoretical development.

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Abbreviations

h :

Vector of the parameters describing the domain \(\mathcal{D}_{i}\)

k :

Vector of the Coriolis forces (N m)

m i :

Mass of \({{\hbox {{RB}}}}_{i}\) (kg)

\(\underline{m}_{i}\) :

Nominal value of the mass of \({{\hbox {{RB}}}}_{i}\) (kg)

n b :

Number of rigid bodies

n c :

Number of holonomic constraints

q :

Vector of the applied forces and torques (N and N m)

r i :

Position vector of \(\mathcal{G}_{i}\) (m)

r 0,i :

Initial position of \({\mathcal{G}}_{i}\) (m)

\(\underline{\mathbf{r}}_{0,i}\) :

Initial position of \(\underline {\mathcal{G}}_{i}\) (m)

r b,i :

Position vector of the barycenter of the domain \(\mathcal{D}_{i}\) (m)

s i :

Rotation vector of \({{\hbox {{RB}}}}_{i}\) (rad)

s 0,i :

Initial angular position of \({{\hbox {{RB}}}}_{i}\) (rad)

u :

Vector of the position and angle of the centers of mass (m and rad)

v 0,i :

Initial velocity of \({\mathcal{G}}_{i}\) (m/s)

x :

Position vector in the inertial frame (m)

x′:

Position vector in the local frame (m)

C 0 :

Normalisation constant relative to R 0,i

\(C_{G_{i}}\) :

Normalisation constant relative to [G i ]

\(C_{G_{0,i}}\) :

Normalisation constant relative to [G 0,i ]

\(\mathcal{D}_{i}\) :

Admissible domain of R 0,i

\(\mathcal{G}_{i}\) :

Center of mass of \({{\hbox {{RB}}}}_{i}\)

\(\underline{\mathcal{G}}_{i}\) :

Center of mass of the nominal model of \({{\hbox {{RB}}}}_{i}\)

\(\boldsymbol{\mathcal{G}}_{i}\) :

Random center of mass of the probabilistic model of \({{\hbox {{RB}}}}_{i}\)

[G i ]:

Normalized positive definite bounded (3×3) random matrix

\({[}{G}_{i}^{\mathrm{max}}{]}\) :

Upper Bound for random matrix [G i ]

[G 0,i ]:

Normalized positive definite (3×3) random matrix

[H i ]:

Second order moment of inertia of \({{\hbox {{RB}}}}_{i}\) (kg m2)

[J i ]:

Tensor of inertia of \({{\hbox {{RB}}}}_{i}\) (kg m2)

\({[}\widetilde{J}_{i}{]}\) :

Tensor of inertia of \({{\hbox {{RB}}}}_{i}\) with unit mass (m2)

\({[}\underline{J}_{i}{]}\) :

Nominal value of the tensor of inertia of \({{\hbox {{RB}}}}_{i}\) (kg m2)

[J i ]:

Random tensor of inertia of the probabilistic model of \({{\hbox {{RB}}}}_{i}\) (kg m2)

\({[}\mathbf{J}_{i}^{\mathrm{max}}{]}\) :

Random upper Bound for random matrix [J i ] (kg m2)

K :

Random vector of the Coriolis forces (N m)

\({[}\underline{L}_{{Z}_{i}}{]}\) :

Upper triangular matrix relative to the Cholesky factorisation of \([{\underline{Z}}_{i}{]}\) (m)

[M]:

Mass matrix (kg)

[M]:

Random mass matrix (kg)

M i :

Random mass of the probabilistic model of \({{\hbox {{RB}}}}_{i}\) (kg)

R 0,i :

Random initial position of \(\boldsymbol {\mathcal{G}}_{i}\) (m)

\({{\hbox {{RB}}}}_{i}\) :

Rigid body i

U :

Random vector of the position and angle of the centers of mass (m and rad)

[Z i ]:

Second order moment of inertia of \({{\hbox {{RB}}}}_{i}\) with unit mass (m2)

\({[}{\underline{Z}}_{i}{]}\) :

Nominal value of [Z i ]

[Z i ]:

Random second order moment of inertia of \({\hbox {{RB}}}_{i}\) with unit mass (m2)

\({[}{Z}_{i}^{\mathrm{max}}{]}\) :

Upper Bound for random matrix [Z i ] (m2)

\(\delta_{M_{i}}\) :

Coefficient of variation for M i

λ :

Real-valued Lagrange multiplier relative to [G 0,i ]

λ :

Vector of the Lagrange multipliers of the constraints (N)

λ l , λ u :

Real-valued Lagrange multipliers relative to [G i ]

[μ]:

Matrix-valued Lagrange multiplier relative to [G i ]

[μ 0]:

Matrix-valued Lagrange multiplier relative to [G 0,i ]

λ r :

Lagrange multipliers relative to R 0,i (m−1)

ρ :

Mass density (kg/m3)

φ :

Constraint function

ω i :

Angular velocity of \({{\hbox {{RB}}}}_{i}\) (rad/s)

ω 0,i :

Initial angular velocity of \({{\hbox {{RB}}}}_{i}\) (rad/s)

Γ :

Gamma function

Ω i :

Domain of \({{\hbox {{RB}}}}_{i}\)

References

  1. Batou, A., Soize, C.: Experimental identification of turbulent fluid forces applied to fuel assemblies using an uncertain model and fretting-wear estimation. Mech. Syst. Signal Process. 23, 2141–2153 (2009)

    Article  Google Scholar 

  2. Baumgarte, J.: Stabilization of constraints and integrals of motion in dynamical systems. Comput. Methods Appl. Mech. Eng. 1(1), 1–16 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  3. Carrarini, A.: Reliability based analysis of the crosswind stability of railway vehicles. J. Wind Eng. Ind. Aerodyn. 95, 493–509 (2007)

    Article  Google Scholar 

  4. Chebli, H., Soize, C.: Experimental validation of a nonparametric probabilistic model of non homogeneous uncertainties for dynamical systems. J. Acoust. Soc. Am. 115(2), 697–705 (2004)

    Article  Google Scholar 

  5. Chen, C., Duhamel, D., Soize, C.: Probabilistic approach for model and data uncertainties and its experimental identification in structural dynamics: Case of composite sandwich panels. J. Sound Vib. 294(12), 64–81 (2006)

    Article  Google Scholar 

  6. Das, S., Ghanem, R.: A bounded random matrix approach for stochastic upscaling. Multiscale Model. Simul. 8(1), 296–325 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Durand, J.-F., Soize, C., Gagliardini, L.: Structural-acoustic modeling of automotive vehicles in presence of uncertainties and experimental identification and validation. J. Acoust. Soc. Am. 124(3), 1513–1525 (2008)

    Article  Google Scholar 

  8. Guilleminot, J., Noshadravanb, A., Soize, C., Ghanem, R.: A probabilistic model for bounded elasticity tensor random fields with application to polycrystalline microstructures. Comput. Methods Appl. Mech. Eng. 200(17–20), 1637–1648 (2011)

    Article  MATH  Google Scholar 

  9. Hastings, W.K.: Monte Carlo sampling methods using Markov chains and their applications. Biometrika 109, 57–97 (1970)

    Google Scholar 

  10. Haug, E.J.: Computer Aided Kinematics and Dynamics. Vol. I: Basic Methods. Allyn and Bacon, Boston (1989)

    Google Scholar 

  11. Jaynes, E.T.: Information theory and statistical mechanics. Phys. Rev. 106(4), 620–630 (1957)

    Article  MathSciNet  Google Scholar 

  12. Jaynes, E.T.: Information theory and statistical mechanics. Phys. Rev. 108(2), 171–190 (1957)

    Article  MathSciNet  Google Scholar 

  13. Li, L., Corina Sandu, C.: On the impact of cargo weight vehicle parameters, and terrain characteristics on the prediction of traction for off-road vehicles. J. Terramech. 44, 221–238 (2007)

    Article  Google Scholar 

  14. Murthy, R., Mignolet, M.P., El-Shafei, A.: Nonparametric stochastic modeling of uncertainty in rotordynamics—Part I: Formulation. J. Eng. Gas Turbine Power 132(9), 092501 (2010)

    Article  Google Scholar 

  15. Murthy, R., Mignolet, M.P., El-Shafei, A.: Nonparametric stochastic modeling of uncertainty in rotordynamics—Part II: Applications. J. Eng. Gas Turbine Power 132(9), 092502 (2010)

    Article  Google Scholar 

  16. Nagar, D.K., Gupta, A.K.: Matrix-variate Kummer–Beta distribution. J. Aust. Math. Soc. 73, 11–25 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  17. Negrut, D., Datar, M., Gorsich, D., Lamb, D.: A framework for uncertainty quantification in nonlinear multi-body system dynamics. In: Proceedings of the 26th Army Science Conference, Orlando, FL (2008)

    Google Scholar 

  18. Rubinstein, R.Y.: Simulation and the Monte Carlo Method. Wiley, New York (1980)

    Google Scholar 

  19. Sandu, A., Sandu, C., Ahmadian, M.: Modeling multibody dynamic systems with uncertainties. Part I: theoretical and computational aspects. Multibody Syst. Dyn. 23, 375–395 (2006)

    MathSciNet  Google Scholar 

  20. Sandu, A., Sandu, C., Ahmadian, M.: Modeling multibody dynamic systems with uncertainties. Part II: numerical applications. Multibody Syst. Dyn. 22, 241–262 (2006)

    Article  MathSciNet  Google Scholar 

  21. Schiehlen, W.: Multibody Systems Handbook. Springer, Berlin (1990)

    MATH  Google Scholar 

  22. Schiehlen, W.: Multibody system dynamics: roots and perspectives. Multibody Syst. Dyn. 1, 149–188 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  23. Schmitt, K.P., Anitescu, M., Negrut, D.: Efficient sampling for spatial uncertainty quantification in multibody system dynamics applications. Int. J. Numer. Methods Eng. 80, 537–564 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  24. Schueller, G.I.: Computational methods in stochastic mechanics and reliability analysis. Comput. Methods Appl. Mech. Eng. 194(12–16), 1251–1795 (2005)

    Google Scholar 

  25. Serfling, R.J.: Approximation Theorems of Mathematical Statistics. Wiley, New York (1980)

    Book  MATH  Google Scholar 

  26. Shannon, C.E.: A mathematical theory of communication. Bell Syst. Tech. J. 27, 379–423 and 623–659 (1948)

    MathSciNet  MATH  Google Scholar 

  27. Soize, C.: A nonparametric model of random uncertainties on reduced matrix model in structural dynamics. Probab. Eng. Mech. 15(3), 277–294 (2000)

    Article  Google Scholar 

  28. Soize, C.: Maximum entropy approach for modeling random uncertainties in transient elastodynamics. J. Acoust. Soc. Am. 109(5), 1979–1996 (2001)

    Article  Google Scholar 

  29. Soize, C.: Random matrix theory for modeling random uncertainties in computational mechanics. Comput. Methods Appl. Mech. Eng. 194(1216), 1333–1366 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  30. Soize, C.: Construction of probability distributions in high dimension using the maximum entropy principle. Applications to stochastic processes, random fields and random matrices. Int. J. Numer. Methods Eng. 76(10), 1583–1611 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  31. Soize, C., Capiez-Lernout, E., Durand, J.F., Fernandez, C., Gagliardini, L.: Probabilistic model identification of uncertainties in computational models for dynamical systems and experimental validation. Comput. Methods Appl. Mech. Eng. 198(1), 150–163 (2008)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Laboratoire Modélisation et Simulation Multi Echelle, MSME UMR 8208 CNRS, Université Paris-Est, 5 bd Descartes, 77454, Marne-la-Vallee, France

    A. Batou & C. Soize

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  1. A. Batou
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  2. C. Soize
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Batou, A., Soize, C. Rigid multibody system dynamics with uncertain rigid bodies. Multibody Syst Dyn 27, 285–319 (2012). https://doi.org/10.1007/s11044-011-9279-2

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