Please use this identifier to cite or link to this item: http://repositorio.ufla.br/jspui/handle/1/49635
Title: Optimizing the spring constants of forced, damped and circular spring-mass systems-characterization of the discrete and periodic bi-Laplacian operator
Keywords: Periodically forced oscillators
Force transmissibility
Circular spring-mass systems
Stiffness matrix
Negative stiffness
Inverse vibration problem
Minimization problem involving matrices
Circulant matrices
Asymptotic limit for matrices
One-dimensional discrete bi-harmonic operator
Eigenpair
Trace Jansen inequality
Issue Date: 26-Jul-2021
Publisher: Oxford University Press (OUP)
Citation: OLIVEIRA, L. L. A. de; TRAVAGLIA, M. V. Optimizing the spring constants of forced, damped and circular spring-mass systems-characterization of the discrete and periodic bi-Laplacian operator. Ima Journal of Applied Mathematics, [S.l.], v. 86, n. 4, p. 785-807, Aug. 2021. DOI: 10.1093/imamat/hxab021.
Abstract: We optimize the spring constants ki,j (stiffness) of circular spring-mass systems with nearest-neighbour (NN) and next-nearest-neighbour (NNN) springs only. In this optimization problem, such systems are also subjected to damping and periodic external forces. The function to be minimized is the average ratio of the square norm of the on-site internal forces (response) to the square norm of the external on-site forces (input). Under the average of this response/input ratio is meant the average over time and over all configurations of external forces. As main result, it is established that the optimum stiffness matrix converges to the discrete and periodic bi-Laplacian operator as the size n of the system increases. Such a result is obtained under the following assumptions: (a) the system has the natural mode shape (eigenvector) of alternating 1s and −1s; and (b) the (external) forcing frequency is at least 1.095 times higher than the highest natural frequency. It is remarkable that this optimum stiffness matrix exhibits negative stiffness for the springs linking NNN point masses. More specifically, as n increases, 0>ki,i+2=−14ki,i+1 is the relation between the optimum NNN spring constant and the optimum NN spring constant. Such systems illustrate that the introduction of negative stiffness springs in some specific positions does in fact reduce the average response/input ratio. Numerical tables illustrating the main result are given.
URI: https://academic.oup.com/imamat/article/86/4/785/6314452
http://repositorio.ufla.br/jspui/handle/1/49635
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