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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 & 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).   Google Scholar [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.  doi: 10.1142/S0218202502002069.  Google Scholar [3] M. Camar-Eddine and P. Seppecher, Determination of the closure of the set of elasticity functionals,, Arch. Ration. Mech. Anal., 170 (2003), 211.  doi: 10.1007/s00205-003-0272-7.  Google Scholar [4] M. T. Chu and S.-F. Xu, Spectral decomposition of real symmetric quadratic $\lambda$-matrices and its applications,, Math. Comp., 78 (2009), 293.  doi: 10.1090/S0025-5718-08-02128-5.  Google Scholar [5] T. J. Chung, General Continuum Mechanics,, Cambridge University Press, (2007).   Google Scholar [6] E. B. Curtis, D. Ingerman and J. A. Morrow, Circular planar graphs and resistor networks,, Linear Algebra Appl., 283 (1998), 115.  doi: 10.1016/S0024-3795(98)10087-3.  Google Scholar [7] R. M. Foster, A reactance theorem,, The Bell System Technical Journal, 3 (1924), 259.  doi: 10.1002/j.1538-7305.1924.tb01358.x.  Google Scholar [8] R. M. Foster, Theorems regarding the driving-point impedance of two-mesh circuits,, The Bell System Technical Journal, 3 (1924), 651.  doi: 10.1002/j.1538-7305.1924.tb00944.x.  Google Scholar [9] I. Gohberg, P. Lancaster and L. Rodman, Matrix Polynomials,, Computer Science and Applied Mathematics, (1982).   Google Scholar [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.  doi: 10.1007/s10659-010-9260-y.  Google Scholar [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.  doi: 10.1098/rspa.2007.0345.  Google Scholar [12] G. W. Milton and P. Seppecher, Electromagnetic circuits,, Netw. Heterog. Media, 5 (2010), 335.  doi: 10.3934/nhm.2010.5.335.  Google Scholar [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, 405 (2010), 2935.   Google Scholar [14] F. Tisseur and K. Meerbergen, The quadratic eigenvalue problem,, SIAM Rev., 43 (2001), 235.  doi: 10.1137/S0036144500381988.  Google Scholar

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##### References:
 [1] R. Bott and R. J. Duffin, Impedance synthesis without use of transformers,, Journal of Applied Physics, 20 (1949).   Google Scholar [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.  doi: 10.1142/S0218202502002069.  Google Scholar [3] M. Camar-Eddine and P. Seppecher, Determination of the closure of the set of elasticity functionals,, Arch. Ration. Mech. Anal., 170 (2003), 211.  doi: 10.1007/s00205-003-0272-7.  Google Scholar [4] M. T. Chu and S.-F. Xu, Spectral decomposition of real symmetric quadratic $\lambda$-matrices and its applications,, Math. Comp., 78 (2009), 293.  doi: 10.1090/S0025-5718-08-02128-5.  Google Scholar [5] T. J. Chung, General Continuum Mechanics,, Cambridge University Press, (2007).   Google Scholar [6] E. B. Curtis, D. Ingerman and J. A. Morrow, Circular planar graphs and resistor networks,, Linear Algebra Appl., 283 (1998), 115.  doi: 10.1016/S0024-3795(98)10087-3.  Google Scholar [7] R. M. Foster, A reactance theorem,, The Bell System Technical Journal, 3 (1924), 259.  doi: 10.1002/j.1538-7305.1924.tb01358.x.  Google Scholar [8] R. M. Foster, Theorems regarding the driving-point impedance of two-mesh circuits,, The Bell System Technical Journal, 3 (1924), 651.  doi: 10.1002/j.1538-7305.1924.tb00944.x.  Google Scholar [9] I. Gohberg, P. Lancaster and L. Rodman, Matrix Polynomials,, Computer Science and Applied Mathematics, (1982).   Google Scholar [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.  doi: 10.1007/s10659-010-9260-y.  Google Scholar [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.  doi: 10.1098/rspa.2007.0345.  Google Scholar [12] G. W. Milton and P. Seppecher, Electromagnetic circuits,, Netw. Heterog. Media, 5 (2010), 335.  doi: 10.3934/nhm.2010.5.335.  Google Scholar [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, 405 (2010), 2935.   Google Scholar [14] F. Tisseur and K. Meerbergen, The quadratic eigenvalue problem,, SIAM Rev., 43 (2001), 235.  doi: 10.1137/S0036144500381988.  Google Scholar
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