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

Characterization of damped linear dynamical systems in free motion

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

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.
Citation: Matthias Morzfeld, Daniel T. Kawano, Fai Ma. Characterization of damped linear dynamical systems in free motion. Numerical Algebra, Control & 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. doi: 10.1115/1.2899425. Google Scholar

[2]

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

[3]

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

[4]

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

[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. doi: 10.1115/1.3627262. Google Scholar

[6]

R. M. Chalasani, Ride performance potential of active suspension systems - part I: simplified analysis based on a quarter-car model,, in, (1986), 187. Google Scholar

[7]

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

[8]

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

[9]

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

[10]

R. Fletscher, "Practical Methods of Optimization,", 2nd edition, (2000). Google Scholar

[11]

I. Gohberg, P. Lancaster and L. Rodman, "Matrix Polynomials,", Academic Press, (1982). Google Scholar

[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. doi: [10.1016/0093-6413(82)90035-0. Google Scholar

[13]

P. Hagedorn and S. Otterbein, "Technische Schwingungslehre,", Springer, (1987). doi: 10.1007/978-3-642-83164-5. Google Scholar

[14]

K. Huseyin, "Vibrations and Stability of Multiple Parameter Systems,", Noordhoff, (1978). Google Scholar

[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. doi: 10.1115/1.3153815. Google Scholar

[16]

D. J. Inman, "Vibration with Control,", Wiley, (2006). Google Scholar

[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. doi: 10.1016/j.jsv.2011.05.013. Google Scholar

[18]

P. Lancaster, "Lambda-Matrices and Vibrating Systems,", Pergamon Press, (1966). Google Scholar

[19]

P. Lancaster and M. Tismenetsky, "The Theory of Matrices,", 2nd edition, (1985). Google Scholar

[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. doi: 10.1016/j.jsv.2009.02.005. Google Scholar

[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. doi: 10.1016/j.jsv.2010.02.017. Google Scholar

[22]

L. Meirovitch, "Methods of Analytical Dynamics,", McGraw-Hill, (1970). Google Scholar

[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. doi: 10.1137/100818637. Google Scholar

[24]

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

[25]

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

[26]

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

[27]

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

[28]

J. Nocedal and S. T. Wright, "Numerical Optimization,", 2nd edition, (2006). Google Scholar

[29]

J. W. Strutt (Lord Rayleigh), "The Theory of Sound, Vol. I,", Dover, (1945). Google Scholar

[30]

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

[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. Google Scholar

show all references

References:
[1]

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

[2]

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

[3]

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

[4]

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

[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. doi: 10.1115/1.3627262. Google Scholar

[6]

R. M. Chalasani, Ride performance potential of active suspension systems - part I: simplified analysis based on a quarter-car model,, in, (1986), 187. Google Scholar

[7]

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

[8]

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

[9]

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

[10]

R. Fletscher, "Practical Methods of Optimization,", 2nd edition, (2000). Google Scholar

[11]

I. Gohberg, P. Lancaster and L. Rodman, "Matrix Polynomials,", Academic Press, (1982). Google Scholar

[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. doi: [10.1016/0093-6413(82)90035-0. Google Scholar

[13]

P. Hagedorn and S. Otterbein, "Technische Schwingungslehre,", Springer, (1987). doi: 10.1007/978-3-642-83164-5. Google Scholar

[14]

K. Huseyin, "Vibrations and Stability of Multiple Parameter Systems,", Noordhoff, (1978). Google Scholar

[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. doi: 10.1115/1.3153815. Google Scholar

[16]

D. J. Inman, "Vibration with Control,", Wiley, (2006). Google Scholar

[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. doi: 10.1016/j.jsv.2011.05.013. Google Scholar

[18]

P. Lancaster, "Lambda-Matrices and Vibrating Systems,", Pergamon Press, (1966). Google Scholar

[19]

P. Lancaster and M. Tismenetsky, "The Theory of Matrices,", 2nd edition, (1985). Google Scholar

[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. doi: 10.1016/j.jsv.2009.02.005. Google Scholar

[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. doi: 10.1016/j.jsv.2010.02.017. Google Scholar

[22]

L. Meirovitch, "Methods of Analytical Dynamics,", McGraw-Hill, (1970). Google Scholar

[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. doi: 10.1137/100818637. Google Scholar

[24]

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

[25]

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

[26]

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

[27]

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

[28]

J. Nocedal and S. T. Wright, "Numerical Optimization,", 2nd edition, (2006). Google Scholar

[29]

J. W. Strutt (Lord Rayleigh), "The Theory of Sound, Vol. I,", Dover, (1945). Google Scholar

[30]

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

[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. Google Scholar

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