Algorithms for synthesizing mechanical systems with maximal natural frequencies
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We consider a simpler version of an open problem in system realization theory, which has relevance to several important problems in biomedicine, altering the dynamic response of discrete and continuous systems, connectivity of Very Large Scale Integrated circuits, as well as the co-ordination of Unmanned vehicles. The fundamental question this article tries to answer is the following one: Given all the components of a system, how do we put these components together in order to obtain a desired response? In the simplest form, this basic question arises in mechanical systems where, the objective is to connect the masses with springs in a suitable way, and in the most general form, it arises in biomedicine where one is interested in engineering and achieving a desired output by either allowing certain new interactions or disallowing some interactions to take place between the proteins, nucleic acids and other cellular components. We formulate a simpler version of this problem in one dimension (i.e., all the masses and springs are arranged along a line), where the objective is to choose a set of springs to connect the masses so that the resulting "graph" structure is as stiff as possible. The system considered corresponds to an ungrounded structure and will always admit a rigid body mode; for that reason, the smallest natural frequency is zero and we use the smallest non-zero natural frequency as a metric for stiffness of the structure and we maximize this objective. Maximizing the smallest non-zero frequency increases all the natural frequencies thereby making the system stiffer. We develop an iterative primal-dual algorithm and a cutting plane algorithm to solve the problem and provide preliminary computational results on a network up to nine masses. 2012 Elsevier Ltd. All rights reserved.
