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Characterization of damped linear dynamical systems in free motion

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  • It is well known that the free motion of a single-degree-of-freedom damped linear dynamical system can be characterized as overdamped, underdamped, or critically damped. Using the methodology of phase synchronization, which transforms any system of linear second-order differential equations into independent second-order equations, this characterization of free motion is generalized to multi-degree-of-freedom damped linear systems. A real scalar function, termed the viscous damping function, is introduced as an extension of the classical damping ratio. It is demonstrated that the free motion of a multi-degree-of-freedom system is characterized by its viscous damping function, and sometimes the characterization may be conducted with ease by examining the extrema of the viscous damping function.
    Mathematics Subject Classification: Primary: 70J30; Secondary: 34A30.

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  • [1]

    L. Barkwell and P. Lancaster, Overdamped and gyroscopic vibrating systems, ASME Journal of Applied Mechanics, 59 (1992), 176-181.doi: 10.1115/1.2899425.

    [2]

    A. Bhaskar, Criticality of damping in multi-degree-of-freedom systems, ASME Journal of Applied Mechanics, 64 (1997), 387-393.doi: 10.1115/1.2787320.

    [3]

    R. M. Bulatović, Non-oscillatory damped multi-degree-of-freedom systems, Acta Mechanica, 151 (2001), 235-244.doi: 10.1007/BF01246920.

    [4]

    R. M. Bulatović, On the heavily damped response in viscously damped dynamic systems, ASME Journal of Applied Mechanics, 71 (2004), 131-134.doi: 10.1115/1.1629108.

    [5]

    T. K. Caughey and M. E. J. Okelly, Classical normal modes in damped linear dynamic systems, ASME Journal of Applied Mechanics, 32 (1965), 583-588.doi: 10.1115/1.3627262.

    [6]

    R. M. Chalasani, Ride performance potential of active suspension systems - part I: simplified analysis based on a quarter-car model, in "ASME Symposium on Simulation and Control of Ground Vehicles and Transportation Systems," AMD-Vol. 80, DSC-Vol. 2, ASME, (1986), 187-204.

    [7]

    G. M. Connell, Asymptotic stability of second-order linear systems with semi-definite damping, AIAA Journal, 7 (1969), 1185-1187.doi: 10.2514/3.5307.

    [8]

    J. W. Demmel, "Applied Numerical Linear Algebra," Society for Industrial and Applied Mathematics, Philadelphia, 1997.doi: 10.1137/1.9781611971446.

    [9]

    R. J. Duffin, A minimax theory for overdamped networks, Journal of Rational Mechanics and Analysis, 4 (1955), 221-233.

    [10]

    R. Fletscher, "Practical Methods of Optimization," 2nd edition, Wiley, Hoboken, New Jersey, 2000.

    [11]

    I. Gohberg, P. Lancaster and L. Rodman, "Matrix Polynomials," Academic Press, New York, 1982.

    [12]

    A. J. Gray and A. N. Andry, A simple calculation of the critical damping matrix of a linear multi-degree-of-freedom system, Mechanics Research Communications, 9 (1982), 379-380.doi: [10.1016/0093-6413(82)90035-0.

    [13]

    P. Hagedorn and S. Otterbein, "Technische Schwingungslehre," Springer, Berlin, Germany, 1987.doi: 10.1007/978-3-642-83164-5.

    [14]

    K. Huseyin, "Vibrations and Stability of Multiple Parameter Systems," Noordhoff, Leiden, 1978.

    [15]

    D. J. Inman and A. N. Andry, Jr., Some results on the nature of eigenvalues of discrete damped linear systems, ASME Journal of Applied Mechanics, 47 (1980), 927-930.doi: 10.1115/1.3153815.

    [16]

    D. J. Inman, "Vibration with Control," Wiley, Hoboken, New Jersey, 2006.

    [17]

    D. T. Kawano, M. Morzfeld and F. Ma, The decoupling of defective linear dynamical systems in free motion, Journal of Sound and Vibration, 330 (2011), 5165-5183.doi: 10.1016/j.jsv.2011.05.013.

    [18]

    P. Lancaster, "Lambda-Matrices and Vibrating Systems," Pergamon Press, Oxford, United Kingdom, 1966.

    [19]

    P. Lancaster and M. Tismenetsky, "The Theory of Matrices," 2nd edition, Academic Press, New York, 1985.

    [20]

    F. Ma, A. Imam and M. Morzfeld, The decoupling of damped linear systems in oscillatory free vibration, Journal of Sound and Vibration, 324 (2009), 408-428.doi: 10.1016/j.jsv.2009.02.005.

    [21]

    F. Ma, M. Morzfeld and A. Imam, The decoupling of damped linear systems in free or forced vibration, Journal of Sound and Vibration, 329 (2010), 3182-3202.doi: 10.1016/j.jsv.2010.02.017.

    [22]

    L. Meirovitch, "Methods of Analytical Dynamics," McGraw-Hill, New York, 1970.

    [23]

    M. Morzfeld, F. Ma and B. N. Parlett, The transformation of second-order linear systems into independent equations, SIAM Journal on Applied Mathematics, 71 (2011), 1026-1043.doi: 10.1137/100818637.

    [24]

    P. C. Müller, Oscillatory damped linear systems, Mechanics Research Communications, 6 (1979), 81-86.

    [25]

    D. W. Nicholson, Eigenvalue bounds for damped linear systems, Mechanics Research Communications, 5 (1978), 147-152.

    [26]

    D. W. Nicholson, Eigenvalue bounds for linear mechanical systems with nonmodal damping, Mechanics Research Communications, 14 (1978), 115-122.

    [27]

    D. W. Nicholson, Overdamping of a linear mechanical system, Mechanics Research Communications, 10 (1983), 67-76.

    [28]

    J. Nocedal and S. T. Wright, "Numerical Optimization," 2nd edition, Springer, New York, 2006.

    [29]

    J. W. Strutt (Lord Rayleigh), "The Theory of Sound, Vol. I," Dover, New York, 1945 (reprint of the 1894 edition).

    [30]

    F. Tisseur and K. Meerbergen, The quadratic eigenvalue problem, SIAM Review, 43 (2001), 235-286.

    [31]

    S. Türkay and H. Akçay, A study of random vibration characteristics of a quarter car model, Journal of Sound and Vibration, 282 (2005), 111-124.

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