Polar Orthotropic Inhomogeneous Circular Plates: Vibration Tailoring

2010 ◽  
Vol 77 (3) ◽  
Author(s):  
Demetris Pentaras ◽  
Isaac Elishakoff
Keyword(s):  
Eigenvalue Problem ◽  
Closed Form ◽  
Circular Plate ◽  
Mode Shape ◽  
Fundamental Mode ◽  
Inverse Eigenvalue Problem ◽  
Static Deflection ◽  
Circular Plates ◽  
Inverse Eigenvalue ◽  
Polar Orthotropic

Problem of matching a desired fundamental natural frequency is solved in the closed form for the polar-orthotropic inhomogeneous circular plate, which is clamped along its circumference. The vibration tailoring is performed by posing a semi-inverse eigenvalue problem. To do this, the fundamental mode shape is postulated. Namely, the analytical expression due to Lekhnitskii, and pertaining to the static deflection of the homogeneous circular plate is demanded to serve as an exact mode shape of the inhomogeneous plate. The analytical and numerical results are reported for several ratios of orthotropic coefficient.

Vibration Tailoring of a Polar Orthotropic Circular Plate With a Translational Spring

2008 ◽  
Vol 75 (3) ◽  
Author(s):  
Isaac Elishakoff ◽  
Demetris Pentaras
Keyword(s):  
Closed Form ◽  
Circular Plate ◽  
Mode Shape ◽  
Inverse Method ◽  
Closed Form Solution ◽  
Form Solution ◽  
Translational Spring ◽  
Orthotropic Circular Plate ◽  
Polar Orthotropic

In this study, the vibration tailoring problem is analytically solved for the polar orthotropic circular plate with translational spring along its circumference. By using the semi-inverse method and postulating the mode shape as a polynomial, we derive a closed-form solution.

scholarly journals Toeplitz nonnegative realization of spectra via companion matrices

2019 ◽  
Vol 7 (1) ◽  
pp. 230-245
Author(s):  
Macarena Collao ◽  
Mario Salas ◽  
Ricardo L. Soto
Keyword(s):  
Eigenvalue Problem ◽  
Half Plane ◽  
Toeplitz Matrix ◽  
Toeplitz Matrices ◽  
Nonnegative Matrix ◽  
Inverse Eigenvalue Problem ◽  
Complex Numbers ◽  
Inverse Eigenvalue ◽  
Companion Matrices ◽  
Solution Matrix

Abstract The nonnegative inverse eigenvalue problem (NIEP) is the problem of finding conditions for the existence of an n × n entrywise nonnegative matrix A with prescribed spectrum Λ = {λ1, . . ., λn}. If the problem has a solution, we say that Λ is realizable and that A is a realizing matrix. In this paper we consider the NIEP for a Toeplitz realizing matrix A, and as far as we know, this is the first work which addresses the Toeplitz nonnegative realization of spectra. We show that nonnegative companion matrices are similar to nonnegative Toeplitz ones. We note that, as a consequence, a realizable list Λ= {λ1, . . ., λn} of complex numbers in the left-half plane, that is, with Re λi≤ 0, i = 2, . . ., n, is in particular realizable by a Toeplitz matrix. Moreover, we show how to construct symmetric nonnegative block Toeplitz matrices with prescribed spectrum and we explore the universal realizability of lists, which are realizable by this kind of matrices. We also propose a Matlab Toeplitz routine to compute a Toeplitz solution matrix.

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