American Institute of Mathematical Sciences

Advanced
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

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 & 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.  Google Scholar

[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.  Google Scholar

[3]

R. M. Bulatović, Non-oscillatory damped multi-degree-of-freedom systems, Acta Mechanica, 151 (2001), 235-244. 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), 131-134. 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), 583-588. 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 "ASME Symposium on Simulation and Control of Ground Vehicles and Transportation Systems," AMD-Vol. 80, DSC-Vol. 2, ASME, (1986), 187-204. Google Scholar

[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.  Google Scholar

[8]

J. W. Demmel, "Applied Numerical Linear Algebra," Society for Industrial and Applied Mathematics, Philadelphia, 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), 221-233.  Google Scholar

[10]

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

[11]

I. Gohberg, P. Lancaster and L. Rodman, "Matrix Polynomials," Academic Press, New York, 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), 379-380. doi: [10.1016/0093-6413(82)90035-0.  Google Scholar

[13]

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

[14]

K. Huseyin, "Vibrations and Stability of Multiple Parameter Systems," Noordhoff, Leiden, 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), 927-930. doi: 10.1115/1.3153815.  Google Scholar

[16]

D. J. Inman, "Vibration with Control," Wiley, Hoboken, New Jersey, 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), 5165-5183. doi: 10.1016/j.jsv.2011.05.013.  Google Scholar

[18]

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

[19]

P. Lancaster and M. Tismenetsky, "The Theory of Matrices," 2nd edition, Academic Press, New York, 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), 408-428. 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), 3182-3202. doi: 10.1016/j.jsv.2010.02.017.  Google Scholar

[22]

L. Meirovitch, "Methods of Analytical Dynamics," McGraw-Hill, New York, 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), 1026-1043. doi: 10.1137/100818637.  Google Scholar

[24]

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

[25]

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

[26]

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

[27]

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

[28]

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

[29]

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

[30]

F. Tisseur and K. Meerbergen, The quadratic eigenvalue problem, SIAM Review, 43 (2001), 235-286.  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), 111-124. Google Scholar

show all references

References:
[1]

L. Barkwell and P. Lancaster, Overdamped and gyroscopic vibrating systems, ASME Journal of Applied Mechanics, 59 (1992), 176-181. 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), 387-393. doi: 10.1115/1.2787320.  Google Scholar

[3]

R. M. Bulatović, Non-oscillatory damped multi-degree-of-freedom systems, Acta Mechanica, 151 (2001), 235-244. 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), 131-134. 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), 583-588. 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 "ASME Symposium on Simulation and Control of Ground Vehicles and Transportation Systems," AMD-Vol. 80, DSC-Vol. 2, ASME, (1986), 187-204. Google Scholar

[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.  Google Scholar

[8]

J. W. Demmel, "Applied Numerical Linear Algebra," Society for Industrial and Applied Mathematics, Philadelphia, 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), 221-233.  Google Scholar

[10]

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

[11]

I. Gohberg, P. Lancaster and L. Rodman, "Matrix Polynomials," Academic Press, New York, 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), 379-380. doi: [10.1016/0093-6413(82)90035-0.  Google Scholar

[13]

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

[14]

K. Huseyin, "Vibrations and Stability of Multiple Parameter Systems," Noordhoff, Leiden, 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), 927-930. doi: 10.1115/1.3153815.  Google Scholar

[16]

D. J. Inman, "Vibration with Control," Wiley, Hoboken, New Jersey, 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), 5165-5183. doi: 10.1016/j.jsv.2011.05.013.  Google Scholar

[18]

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

[19]

P. Lancaster and M. Tismenetsky, "The Theory of Matrices," 2nd edition, Academic Press, New York, 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), 408-428. 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), 3182-3202. doi: 10.1016/j.jsv.2010.02.017.  Google Scholar

[22]

L. Meirovitch, "Methods of Analytical Dynamics," McGraw-Hill, New York, 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), 1026-1043. doi: 10.1137/100818637.  Google Scholar

[24]

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

[25]

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

[26]

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

[27]

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

[28]

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

[29]

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

[30]

F. Tisseur and K. Meerbergen, The quadratic eigenvalue problem, SIAM Review, 43 (2001), 235-286.  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), 111-124. Google Scholar

[1]

Willy Sarlet, Tom Mestdag. Compatibility aspects of the method of phase synchronization for decoupling linear second-order differential equations. Journal of Geometric Mechanics, 2021  doi: 10.3934/jgm.2021019
[2]

David L. Russell. Control via decoupling of a class of second order linear hybrid systems. Discrete & Continuous Dynamical Systems - S, 2014, 7 (6) : 1321-1334. doi: 10.3934/dcdss.2014.7.1321
[3]

Peizhao Yu, Guoshan Zhang, Yi Zhang. Decoupling of cubic polynomial matrix systems. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 13-26. doi: 10.3934/naco.2020012
[4]

Guochun Wu, Yinghui Zhang. Global analysis of strong solutions for the viscous liquid-gas two-phase flow model in a bounded domain. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1411-1429. doi: 10.3934/dcdsb.2018157
[5]

Chih-Wen Shih, Jui-Pin Tseng. From approximate synchronization to identical synchronization in coupled systems. Discrete & Continuous Dynamical Systems - B, 2020, 25 (9) : 3677-3714. doi: 10.3934/dcdsb.2020086
[6]

Alexander Pimenov, Dmitrii I. Rachinskii. Linear stability analysis of systems with Preisach memory. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 997-1018. doi: 10.3934/dcdsb.2009.11.997
[7]

Ken Shirakawa. Stability analysis for phase field systems associated with crystalline-type energies. Discrete & Continuous Dynamical Systems - S, 2011, 4 (2) : 483-504. doi: 10.3934/dcdss.2011.4.483
[8]

Olena Naboka. On synchronization of oscillations of two coupled Berger plates with nonlinear interior damping. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1933-1956. doi: 10.3934/cpaa.2009.8.1933
[9]

William F. Thompson, Rachel Kuske, Yue-Xian Li. Stochastic phase dynamics of noise driven synchronization of two conditional coherent oscillators. Discrete & Continuous Dynamical Systems, 2012, 32 (8) : 2971-2995. doi: 10.3934/dcds.2012.32.2971
[10]

Yuzhou Tian, Yulin Zhao. Global phase portraits and bifurcation diagrams for reversible equivariant Hamiltonian systems of linear plus quartic homogeneous polynomials. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2941-2956. doi: 10.3934/dcdsb.2020214
[11]

Seung-Yeal Ha, Se Eun Noh, Jinyeong Park. Practical synchronization of generalized Kuramoto systems with an intrinsic dynamics. Networks & Heterogeneous Media, 2015, 10 (4) : 787-807. doi: 10.3934/nhm.2015.10.787
[12]

Shahad Al-azzawi, Jicheng Liu, Xianming Liu. Convergence rate of synchronization of systems with additive noise. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 227-245. doi: 10.3934/dcdsb.2017012
[13]

Samuel Bowong, Jean Luc Dimi. Adaptive synchronization of a class of uncertain chaotic systems. Discrete & Continuous Dynamical Systems - B, 2008, 9 (2) : 235-248. doi: 10.3934/dcdsb.2008.9.235
[14]

V. Afraimovich, J.-R. Chazottes, A. Cordonet. Synchronization in directionally coupled systems: Some rigorous results. Discrete & Continuous Dynamical Systems - B, 2001, 1 (4) : 421-442. doi: 10.3934/dcdsb.2001.1.421
[15]

Jicheng Liu, Meiling Zhao. Normal deviation of synchronization of stochastic coupled systems. Discrete & Continuous Dynamical Systems - B, 2022, 27 (2) : 1029-1054. doi: 10.3934/dcdsb.2021079
[16]

Yuzo Hosono. Phase plane analysis of travelling waves for higher order autocatalytic reaction-diffusion systems. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 115-125. doi: 10.3934/dcdsb.2007.8.115
[17]

Hedy Attouch, Aïcha Balhag, Zaki Chbani, Hassan Riahi. Fast convex optimization via inertial dynamics combining viscous and Hessian-driven damping with time rescaling. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021010
[18]

Hedy Attouch, Alexandre Cabot, Zaki Chbani, Hassan Riahi. Rate of convergence of inertial gradient dynamics with time-dependent viscous damping coefficient. Evolution Equations & Control Theory, 2018, 7 (3) : 353-371. doi: 10.3934/eect.2018018
[19]

Mohammad Akil, Haidar Badawi. The influence of the physical coefficients of a Bresse system with one singular local viscous damping in the longitudinal displacement on its stabilization. Evolution Equations & Control Theory, 2022  doi: 10.3934/eect.2022004
[20]

Evelyn Buckwar, Girolama Notarangelo. A note on the analysis of asymptotic mean-square stability properties for systems of linear stochastic delay differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1521-1531. doi: 10.3934/dcdsb.2013.18.1521

 Impact Factor: 

Tools

Metrics

  • PDF downloads (64)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy