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Analysis and reanalysis of mechanical systems: concept of global near-regularity

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Abstract

Some near-regular mechanical systems involve global irregularities, wherein a large number of degrees of freedom are affected by irregularity. However, no efficient solution for such global near-regular systems has yet been developed. In this paper, methods for static and dynamic analyses/reanalyses of these systems are established using graph product rules combined with matrix analysis and linear algebra. Also, these methods are generalized to systems with nonlinear behavior. The developed formulations allow reduction in computational time and storage compared to those of conventional methods. As a practical example of a global near-regular mechanical system, a subject-specific finite element mechanical model of the human spine is developed and presented.

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References

  1. Kangwai, R.D., Guest, S., Pellegrino, S.: An introduction to the analysis of symmetric structures. Comput. Struct. 71, 671–688 (1999)

    Article  MathSciNet  Google Scholar 

  2. Kamm, J., Nagy, J.G.: Optimal Kronecker product approximation of block Toeplitz matrices. SIAM J. Matrix Anal. Appl. 22, 155–172 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  3. Claeyssen, J.C.R., dos Santos Leal, L.A.: Diagonalization and spectral decomposition of factor block circulant matrices. Linear Algebra Appl. 99, 41–61 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  4. Golub, G.H., Van Loan, C.F.: Matrix Computations, vol. 3. JHU Press, Baltimore (2012)

    MATH  Google Scholar 

  5. Temme, K., Wocjan, P.: Efficient computation of the permanent of block factorizable matrices. arXiv:1208.6589 (2012)

  6. Thomas, D.L.: Dynamics of rotationally periodic structures. Int. J. Numer. Methods Eng. 14, 81–102 (1979)

    Article  MATH  Google Scholar 

  7. Dukkipati, R.V., Rao, M.A., Bhat, R.B.: Computer Aided Analysis and Design of Machine Elements. New Age International, New Delhi (2006)

    Google Scholar 

  8. Chan, H.C., Cai, C., Cheung, Y.K.: Exact Analysis of Structures with Periodicity Using U-Transformation. World Scientific, New Jersey (1998)

    Google Scholar 

  9. Mokrani, B., Bastaits, R., Horodinca, M., Romanescu, I., Burda, I., Viguié, R., Preumont, A.: Parallel piezoelectric shunt damping of rotationally periodic structures. Adv. Mater. Sci. Eng. 2015, 1–12 (2015)

  10. Laxalde, D., Thouverez, F., Lombard, J.-P.: Dynamical analysis of multi-stage cyclic structures. Mech. Res. Commun. 34, 379–384 (2007)

    Article  MATH  Google Scholar 

  11. Náprstek, J., Fischer, C.: Static and dynamic analysis of beam assemblies using a differential system on an oriented graph. Comput. Struct. 155, 28–41 (2015)

    Article  Google Scholar 

  12. Gao, G., Wang, H., Li, E., Li, G.: An exact block-based reanalysis method for local modifications. Comput. Struct. 158, 369–380 (2015)

    Article  Google Scholar 

  13. Zingoni, A.: On group-theoretic computation of natural frequencies for spring–mass dynamic systems with rectilinear motion. Commun. Numer. Methods Eng. 24, 973–987 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kirsch, U.: Combined approximations—a general reanalysis approach for structural optimization. Struct. Multidiscip. Optim. 20, 97–106 (2000)

    Article  Google Scholar 

  15. Kirsch, U.: Design-oriented analysis of structures-Unified approach. J. Eng. Mech. 129, 264–272 (2003)

    Article  Google Scholar 

  16. Kaveh, A.: Optimal Analysis of Structures by Concepts of Symmetry and Regularity. Springer, New York (2013)

    Book  MATH  Google Scholar 

  17. Shojaei, I., Kaveh, A., Rahami, H.: Analysis of structures convertible to repeated structures using graph products. Comput. Struct. 125, 153–163 (2013)

    Article  Google Scholar 

  18. Kaveh, A., Shojaei, I., Rahami, H.: New developments in the optimal analysis of regular and near-regular structures: decomposition, graph products, force method. Acta Mech. 226, 665–681 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  19. Harth, P., Michelberger, P.: Determination of loads in quasi-symmetric structure with symmetry components. Eng. Struct. 123, 395–407 (2016)

    Article  Google Scholar 

  20. Chen, Y., Feng, J.: Group-theoretic method for efficient buckling analysis of prestressed space structures. Acta Mech. 226, 957–973 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  21. Zingoni, A.: Symmetry recognition in group-theoretic computational schemes for complex structural systems. Comput. Struct. 94, 34–44 (2012)

    Article  Google Scholar 

  22. Shojaei, I., Rahami, H., and Kaveh, A.: A numerical solution for Laplace and Poisson’s equations using geometrical transformation and graph products. Appl. Math. Model. 40(17–18),7768–7783 (2016)

  23. Shojaei, I., Arjmand, N., Bazrgari, B.: An optimization-based method for prediction of lumbar spine segmental kinematics from the measurements of thorax and pelvic kinematics. Int. J. Numer. Methods Biomed. Eng. 31, 1–14 (2015)

  24. Shojaei, I., Hendershot, B.D., Wolf, E.J., Bazrgari, B.: Persons with unilateral transfemoral amputation experience larger spinal loads during level-ground walking compared to able-bodied individuals. Clin. Biomech. 32, 157–163 (2016)

    Article  Google Scholar 

  25. Wilkinson, J.H., Wilkinson, J.H.: The Algebraic Eigenvalue Problem, vol. 87. Clarendon Press, Oxford (1965)

    MATH  Google Scholar 

  26. Yueh, W.-C.: Eigenvalues of several tridiagonal matrices. Appl. Math. E-Notes 5, 210–230 (2005)

    MathSciNet  MATH  Google Scholar 

  27. Kaveh, A., Rahami, H.: An efficient analysis of repetitive structures generated by graph products. Int. J. Numer. Methods. Eng. 84, 108–126 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  28. Arjmand, N., Shirazi-Adl, A.: Sensitivity of kinematics-based model predictions to optimization criteria in static lifting tasks. Med. Eng. Phys. 28, 504–514 (2006)

    Article  Google Scholar 

  29. Bazrgari, B., Shirazi-Adl, A., Arjmand, N.: Analysis of squat and stoop dynamic liftings: muscle forces and internal spinal loads. Eur. Spine J. 16, 687–699 (2007)

    Article  Google Scholar 

  30. Shojaei, I., Allen-Bryant, K., and Bazrgari, B.: Viscoelastic Response of the human lower back to passive flexion: the effects of age. Ann. Biomed. Eng. 44(9), 2817–2826 (2016)

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Authors and Affiliations

  1. F. Joseph Halcomb III, M.D. Department of Biomedical Engineering, University of Kentucky, 510E Robotic and Manufacturing Building, Lexington, KY, 40506, USA

    Iman Shojaei & Babak Bazrgari

  2. Centre of Excellence for Fundamental Studies in Structural Engineering, Iran University of Science and Technology, P.O. Box 16846-13114, Narmak, Tehran, Iran

    Ali Kaveh

  3. School of Engineering Science, College of Engineering, University of Tehran, Tehran, Iran

    Hossein Rahami

  4. Align Technology Inc, San Jose, CA, USA

    Reza Shirazi

Authors
  1. Iman Shojaei
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  2. Ali Kaveh
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  3. Hossein Rahami
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  4. Reza Shirazi
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  5. Babak Bazrgari
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Corresponding author

Correspondence to Iman Shojaei.

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Shojaei, I., Kaveh, A., Rahami, H. et al. Analysis and reanalysis of mechanical systems: concept of global near-regularity. Acta Mech 228, 1445–1456 (2017). https://doi.org/10.1007/s00707-016-1778-z

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  • DOI: https://doi.org/10.1007/s00707-016-1778-z

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