Characterization of damped linear dynamical systems in free motion

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2013, 3(1): 49-62. doi: 10.3934/naco.2013.3.49

Characterization of damped linear dynamical systems in free motion

Matthias Morzfeld ^1, , Daniel T. Kawano ^2, and Fai Ma ^3,

1.	Department of Mathematics, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, United States
2.	Department of Mechanical Engineering, Rose-Hulman Institute of Technology, Terre Haute, IN 47803, United States
3.	Department of Mechanical Engineering, University of California, Berkeley, CA 94720, United States

Received October 2011 Revised November 2012 Published January 2013

Abstract
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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.

Keywords: decoupling, viscous damping, Linear systems, modal analysis., phase synchronization.

Mathematics Subject Classification: Primary: 70J30; Secondary: 34A3.

Citation: Matthias Morzfeld, Daniel T. Kawano, Fai Ma. Characterization of damped linear dynamical systems in free motion. Numerical Algebra, Control and Optimization, 2013, 3 (1) : 49-62. doi: 10.3934/naco.2013.3.49

References:

[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," 2^nd 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," 2^nd 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," 2^nd 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.

show all references

References:

[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," 2^nd 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," 2^nd 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," 2^nd 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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