Research Papers

An Accurate Spatial Discretization and Substructure Method With Application to Moving Elevator Cable-Car Systems—Part I: Methodology

[+] Author and Article Information
W. D. Zhu

e-mail: wzhu@umbc.edu

H. Ren

e-mail: hui.ren@mscsoftware.com
Department of Mechanical Engineering,
University of Maryland, Baltimore County,
1000 Hilltop Circle,
Baltimore, MD 21250

1Corresponding author.

2Currently a development engineer at the MSC Software Corporation.

Contributed by the Design Engineering Division of ASME for publication in the Journal of Vibration and Acoustics. Manuscript received October 8, 2011; final manuscript received May 6, 2013; published online July 23, 2013. Assoc. Editor: Jean Zu.

J. Vib. Acoust 135(5), 051036 (Jul 23, 2013) (10 pages) Paper No: VIB-11-1232; doi: 10.1115/1.4024557 History: Received October 08, 2011; Revised May 06, 2013

A spatial discretization and substructure method is developed to accurately calculate dynamic responses of one-dimensional structural systems, which consist of length-variant distributed-parameter components, such as strings, rods, and beams, and lumped-parameter components, such as point masses and rigid bodies. The dependent variable of a distributed-parameter component is decomposed into boundary-induced terms and internal terms. The boundary-induced terms are interpolated from boundary motions, and the internal terms are approximated by an expansion of trial functions that satisfy the corresponding homogeneous boundary conditions. All the matching conditions at the interfaces of the components are satisfied, and the expansions of the dependent variables of the distributed-parameter components absolutely and uniformly converge if the dependent variables are smooth enough. Spatial derivatives of the dependent variables, which are related to internal forces/moments of the distributed-parameter components, such as axial forces, bending moments, and shear forces, can be accurately calculated. Combining component equations that are derived from Lagrange's equations and geometric matching conditions that arise from continuity relations leads to a system of differential algebraic equations (DAEs). When the geometric matching conditions are linear, the DAEs can be transformed to a system of ordinary differential equations (ODEs), which can be solved by an ODE solver. The methodology is applied to several moving elevator cable-car systems in Part II of this work.

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Zhu, W. D., and Chen, Y., 2005, “Forced Response of Translating Media With Variable Length and Tension: Application to High-Speed Elevators,” Proc. Inst. Mech. Eng., Part K, 219, pp. 35–52. [CrossRef]
Meirovitch, L., 1997, Principles and Techniques of Vibrations, Prentice-Hall, Upper Saddle River, NJ.
Sansone, G., 1991, Orthogonal Functions, revised English ed., Dover, New York.
Lanczos, C., 1956, Applied Analysis, Van Nostrand, Princeton, NJ.
Orszag, S. A., 1969, “Numerical Methods for the Simulation of Turbulence,” Phys. Fluids, 12, pp. 250–257. [CrossRef]
Kreiss, H. O., and Oliger, J., 1972, “Comparison of Accurate Methods for the Integration of Hyperbolic Equations,” Tellus, 24, pp. 199–215. [CrossRef]
Gottlieb, D., and Orszag, S. A., 1977, Numerical Analysis of Spectral Methods: Theory and Applications, SIAM, Philadelphia.
Fornberg, B., 1998, A Practical Guide to Pseudospectral Methods, Cambridge University, Cambridge, UK.
Zienkiewicz, O. C., Taylor, R. L., and Zhu, J. Z., 2005, The Finite Element Method: Its Basis and Fundamentals, 6th ed., Elsevier Butterworth-Heinemann, New York.
Mao, K. M., and Sun, C. T., 2005, “A Refined Global-Local Finite Element Analysis Method,” Int. J. Numer. Methods Eng., 32(1), pp. 29–43. [CrossRef]
Šolín, P., Segeth, K., and Doleel, I., 2003, Higher-Order Finite Element Methods, Chapman & Hall/CRC, London.
Wilson, E. L., Taylor, R. L., Doherty, W. P., and Ghabouss, J., 1973, “Incompatible Displacement Models,” S. J.Fenves, ed., Numerical and Computer Methods in Structural Mechanics, Academic, New York, pp. 43–57.
Mote, C. D., Jr., 1971, “Global-Local Finite Element,” Int. J. Numer. Methods Eng., 3(4), pp. 565–574. [CrossRef]
Simo, J. C., and Rafai, M. S., 1990, “A Class of Mixed Assumed Strain Methods and the Method of Incompatible Modes,” Int. J. Numer. Methods Eng., 29, pp. 1595–1638. [CrossRef]
Patera, A. T., 1984, “A Spectral Element Method for Fluid Dynamics: Laminar Flow in a Channel Expansion,” J. Comput. Phys., 54, pp. 468–488. [CrossRef]
Karniadakis, G. E., and Sherwin, S., 2005, Spectral/Hp Element Methods for Computational Fluid Dynamics, 2nd ed., Oxford University, London, pp. 7–21.
Canuto, C., Hussaini, M. Y., Quarteroni, A., and Zang, T. A., 2007, Spectral Methods: Evolution to Complex Geometries and Applications to Fluid Dynamics, Springer, Berlin-Heidelberg.
Toselli, A., and Widlund, O., 2005, Domain Decomposition Methods: Algorithms and Theory, Springer, Berlin.
Li, W. L., 2000, “Free Vibrations of Beams With General Boundary Conditions,” J. Sound Vib., 237(4), pp. 709–725. [CrossRef]
Xu, H., and Li, W. L., 2008, “Dynamic Behavior of Multi-Span Bridges Under Moving Loads With Focusing on the Effect of the Coupling Conditions Between Spans,” J. Sound Vib., 312(4), pp. 736–753. [CrossRef]
Li, W. L., Zhang, X., Du, J., and Liu, Z., 2009, “An Exact Series Solution for the Transverse Vibration of Rectangular Plates With General Elastic Boundary Supports,” J. Sound Vib., 321(4), pp. 254–269. [CrossRef]
Hesthaven, J. S., and Warburton, T., 2008, Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications, Springer Verlag, New York.
Hurty, W. C., 1960, “Vibrations of Structural Systems by Component Mode Synthesis,” Proc. Am. Soc. Civ. Eng., 85(EM4), pp. 51–69.
Hurty, W. C., 1965, “Dynamic Analysis of Structural System Using Component Modes,” AIAA J., 3(4), pp. 678–685. [CrossRef]
Craig, R. R., Jr., and Bampton, M. C., 1968, “Coupling of Substructures for Dynamic Analyses,” AIAA J., 6(7), pp. 1313–1319. [CrossRef]
Bamford, R. M., 1967, “A Modal Combination Program for Dynamic Analysis of Structures,” Jet Propulsion Laboratory Technical Memorandum No. 33–290.
Bajan, R. L., Feng, C. C., and Jaszlics, J. J., 1968, “Vibration Analysis of Complex Structural Systems by Modal Substitution,” Proceedings of the 39th Shock and Vibration Symposium, Pacific Grove, CA, October 22–24.
Hintz, R. M., 1975, “Analytical Methods in Component Mode Analysis,” AIAA J., 13, pp. 1007–1016. [CrossRef]
MacNeal, R. H., 1971, “A Hybrid Method of Component Mode Synthesis,” Comput. Struct., 1(4), pp. 581–601. [CrossRef]
Craig, R. R., Jr., and Chang, C.-J., 1975, “Substructure Coupling With Reduction of Interface Coordinates by Fixed-Interface Methods,” TICOM Report No. 75-1.
Dowell, E. H., 1972, “Free Vibrations of an Arbitrary Structures in Terms of Component Modes,” ASME J. Appl. Mech., 39, pp. 727–732. [CrossRef]
Dowell, E. H., 1979, “On Some General Properties of Combined Dynamical Systems,” ASME J. Appl. Mech., 46, pp. 206–209. [CrossRef]
Rudin, W., 1976, Principles of Mathematical Analysis, 3rd ed., McGraw-Hill, New York, pp. 147–158.
Courant, R., and Hilbert, D., 1953, Methods of Mathematical Physics, Vol. I, Interscience, New York.
Kreyszig, E., 1989, Introductory Functional Analysis with Applications, Wiley, New York.
Funaro, D., 1992, Polynomial Approximation of Differential Equations, Springer-Verlag, Berlin.
Hesthaven, J. S., Gottlieb, S., and Gottlieb, D., 2007, Spectral Methods for Time-Dependent Problems, Cambridge University, Cambridge, UK, pp. 67–69.
Deitmar, A., 2005, A First Course in Harmonic Analysis, 2nd ed., Springer, New York.
Lanczos, C., 1966, Discourse on Fourier Series, Hafner, New York.
Goldstein, H., Poole, C., and Safko, J., 2002, Classical Mechanics, Addison Wesley, New York, pp. 45–49.
Gear, C. W., Leimkuhler, B., and Gupta, C. K., 1985, “Automatic Integration of Euler-Lagrange Equations With Constraints,” J. Comput. Appl. Math., 12–13, pp. 77–90. [CrossRef]
Brenan, K. E., Campbell, S. L., and Petzold, L. R., 1989, Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations, SIAM, Philadelphia.
Hairer, E., and Wanner, G., 1996, Solving Ordinary Differential Equations II: Stiffness and Differential Algebraic Problems, Springer-Verlag, Berlin Heidelberg, p. 470.





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