Optimizing the spring constants of forced, damped and circular spring-mass systems—characterization of the discrete and periodic bi-Laplacian operator

We optimize the spring constants $k^{i,j}$ (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 $1$ s and $-1$ s; 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, $0> k^{i,i+2} \, \, = \, \, - \tfrac{1}{4} \, k^{i,i+1}$ 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.