Abstract
The response function of a network of springs and masses, an elastodynamic network, is the matrix valued function W(ω), depending on the frequency ω, mapping the displacements of some accessible or terminal nodes to the net forces at the terminals. We give necessary and sufficient conditions for a given function W(ω) to be the response function of an elastodynamic network, assuming there is no damping. In particular we construct an elastodynamic network that can mimic a suitable response in the frequency or time domain. Our characterization is valid for networks in three dimensions and also for planar networks, which are networks where all the elements, displacements and forces are in a plane. The network we design can fit within an arbitrarily small neighborhood of the convex hull of the terminal nodes, provided the springs and masses occupy an arbitrarily small volume. Additionally, we prove stability of the network response to small changes in the spring constants and/or addition of springs with small spring constants.
Similar content being viewed by others
References
Bott, R., Duffin, R.J.: Impedance synthesis without use of transformers. J. Appl. Phys. 20, 804 (1949). doi:10.1063/l.1698532
Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)
Camar-Eddine, M., Seppecher, P.: Closure of the set of diffusion functionals with respect to the Mosco-convergence. Math. Models Methods Appl. Sci. 12(8), 1153–1176 (2002)
Camar-Eddine, M., Seppecher, P.: Determination of the closure of the set of elasticity functionals. Arch. Ration Mech. Anal. 170(3), 211–245 (2003)
Curtis, E.B., Ingerman, D., Morrow, J.A.: Circular planar graphs and resistor networks. Linear Algebra Appl. 283(1–3), 115–150 (1998). doi:10.1016/S0024-3795(98)10087-3
Foster, R.M.: A reactance theorem. Bell Syst. Tech. J. 3, 259–267 (1924)
Foster, R.M.: Theorems regarding the driving-point impedance of two-mesh circuits. Bell Syst. Tech. J. 3, 651–685 (1924)
Golub, G.H., Van Loan, C.F.: Matrix Computations, 3rd edn. Johns Hopkins Studies in the Mathematical Sciences. Johns Hopkins University Press, Baltimore (1996)
Milton, G.W., Seppecher, P.: 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(2092), 967–986 (2008)
Milton, G.W., Seppecher, P.: (2009) Electromagnetic circuits. Networks and Heterogeneous Media (2009, submitted). See also arXiv:0805.1079v2 [physics.class-ph] (2008)
Milton, G.W., Seppecher, P.: Hybrid electromagnetic circuits. Physica B (2009, submitted). See also arXiv:0910.0798v1 [physics.optics] (2009)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Guevara Vasquez, F., Milton, G.W. & Onofrei, D. Complete Characterization and Synthesis of the Response Function of Elastodynamic Networks. J Elast 102, 31–54 (2011). https://doi.org/10.1007/s10659-010-9260-y
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10659-010-9260-y