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June  2014, 9(2): 299-314. doi: 10.3934/nhm.2014.9.299

Characterization and synthesis of Rayleigh damped elastodynamic networks

1. 

Mathematics Department, University of Utah, 155 S 1400 E RM 233, Salt Lake City, UT 84112-0090, United States

2. 

Department of Mathematics, University of Utah, 155 S 1400 E RM 233, Salt Lake City, UT 84112-0090

Received  May 2013 Revised  March 2014 Published  July 2014

We consider damped elastodynamic networks where the damping matrix is assumed to be a non-negative linear combination of the stiffness and mass matrices (also known as Rayleigh or proportional damping). We give here a characterization of the frequency response of such networks. We also answer the synthesis question for such networks, i.e., how to construct a Rayleigh damped elastodynamic network with a given frequency response. Our analysis shows that not all damped elastodynamic networks can be realized when the proportionality constants between the damping matrix and the mass and stiffness matrices are fixed.
Citation: Alessandro Gondolo, Fernando Guevara Vasquez. Characterization and synthesis of Rayleigh damped elastodynamic networks. Networks and Heterogeneous Media, 2014, 9 (2) : 299-314. doi: 10.3934/nhm.2014.9.299
References:
[1]

R. Bott and R. J. Duffin, Impedance synthesis without use of transformers, Journal of Applied Physics, 20 (1949), 816.

[2]

M. Camar-Eddine and P. Seppecher, Closure of the set of diffusion functionals with respect to the Mosco-convergence, Math. Models Methods Appl. Sci., 12 (2002), 1153-1176. doi: 10.1142/S0218202502002069.

[3]

M. Camar-Eddine and P. Seppecher, Determination of the closure of the set of elasticity functionals, Arch. Ration. Mech. Anal., 170 (2003), 211-245. doi: 10.1007/s00205-003-0272-7.

[4]

M. T. Chu and S.-F. Xu, Spectral decomposition of real symmetric quadratic $\lambda$-matrices and its applications, Math. Comp., 78 (2009), 293-313. doi: 10.1090/S0025-5718-08-02128-5.

[5]

T. J. Chung, General Continuum Mechanics, Cambridge University Press, Cambridge, 2007.

[6]

E. B. Curtis, D. Ingerman and J. A. Morrow, Circular planar graphs and resistor networks, Linear Algebra Appl., 283 (1998), 115-150. doi: 10.1016/S0024-3795(98)10087-3.

[7]

R. M. Foster, A reactance theorem, The Bell System Technical Journal, 3 (1924), 259-267. doi: 10.1002/j.1538-7305.1924.tb01358.x.

[8]

R. M. Foster, Theorems regarding the driving-point impedance of two-mesh circuits, The Bell System Technical Journal, 3 (1924), 651-685. doi: 10.1002/j.1538-7305.1924.tb00944.x.

[9]

I. Gohberg, P. Lancaster and L. Rodman, Matrix Polynomials, Computer Science and Applied Mathematics, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1982.

[10]

F. Guevara Vasquez, G. W. Milton and D. Onofrei, Complete characterization and synthesis of the response function of elastodynamic networks, J. Elasticity, 102 (2011), 31-54. doi: 10.1007/s10659-010-9260-y.

[11]

G. W. Milton and P. Seppecher, Realizable response matrices of multi-terminal electrical, acoustic and elastodynamic networks at a given frequency, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 464 (2008), 967-986. doi: 10.1098/rspa.2007.0345.

[12]

G. W. Milton and P. Seppecher, Electromagnetic circuits, Netw. Heterog. Media, 5 (2010), 335-360. doi: 10.3934/nhm.2010.5.335.

[13]

G. W. Milton and P. Seppecher, Hybrid electromagnetic circuits, Proceedings of the Eighth International Conference on Electrical Transport and Optical Properties of Inhomogeneous Media, ETOPIM-8, Physica B: Condensed Matter, 405 (2010), 2935-2937.

[14]

F. Tisseur and K. Meerbergen, The quadratic eigenvalue problem, SIAM Rev., 43 (2001), 235-286. doi: 10.1137/S0036144500381988.

show all references

References:
[1]

R. Bott and R. J. Duffin, Impedance synthesis without use of transformers, Journal of Applied Physics, 20 (1949), 816.

[2]

M. Camar-Eddine and P. Seppecher, Closure of the set of diffusion functionals with respect to the Mosco-convergence, Math. Models Methods Appl. Sci., 12 (2002), 1153-1176. doi: 10.1142/S0218202502002069.

[3]

M. Camar-Eddine and P. Seppecher, Determination of the closure of the set of elasticity functionals, Arch. Ration. Mech. Anal., 170 (2003), 211-245. doi: 10.1007/s00205-003-0272-7.

[4]

M. T. Chu and S.-F. Xu, Spectral decomposition of real symmetric quadratic $\lambda$-matrices and its applications, Math. Comp., 78 (2009), 293-313. doi: 10.1090/S0025-5718-08-02128-5.

[5]

T. J. Chung, General Continuum Mechanics, Cambridge University Press, Cambridge, 2007.

[6]

E. B. Curtis, D. Ingerman and J. A. Morrow, Circular planar graphs and resistor networks, Linear Algebra Appl., 283 (1998), 115-150. doi: 10.1016/S0024-3795(98)10087-3.

[7]

R. M. Foster, A reactance theorem, The Bell System Technical Journal, 3 (1924), 259-267. doi: 10.1002/j.1538-7305.1924.tb01358.x.

[8]

R. M. Foster, Theorems regarding the driving-point impedance of two-mesh circuits, The Bell System Technical Journal, 3 (1924), 651-685. doi: 10.1002/j.1538-7305.1924.tb00944.x.

[9]

I. Gohberg, P. Lancaster and L. Rodman, Matrix Polynomials, Computer Science and Applied Mathematics, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1982.

[10]

F. Guevara Vasquez, G. W. Milton and D. Onofrei, Complete characterization and synthesis of the response function of elastodynamic networks, J. Elasticity, 102 (2011), 31-54. doi: 10.1007/s10659-010-9260-y.

[11]

G. W. Milton and P. Seppecher, Realizable response matrices of multi-terminal electrical, acoustic and elastodynamic networks at a given frequency, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 464 (2008), 967-986. doi: 10.1098/rspa.2007.0345.

[12]

G. W. Milton and P. Seppecher, Electromagnetic circuits, Netw. Heterog. Media, 5 (2010), 335-360. doi: 10.3934/nhm.2010.5.335.

[13]

G. W. Milton and P. Seppecher, Hybrid electromagnetic circuits, Proceedings of the Eighth International Conference on Electrical Transport and Optical Properties of Inhomogeneous Media, ETOPIM-8, Physica B: Condensed Matter, 405 (2010), 2935-2937.

[14]

F. Tisseur and K. Meerbergen, The quadratic eigenvalue problem, SIAM Rev., 43 (2001), 235-286. doi: 10.1137/S0036144500381988.

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