Large and Infinite Mass–Spring–Damper Networks

Abstract

This paper introduces mechanical networks as a tool for modeling complex unidirectional vibrations. Networks of this type have branches containing massless linear springs and dampers, with masses at the nodes. Tree and ladder configurations are examples demonstrating that the overall dynamics of infinite systems can be represented using implicitly defined integro-differential operators. Results from the proposed models compare well to numerical results from finite systems, so this approach may have advantages over high-order differential equations.

Further Remarks on Irrational Systems and Their Applications

“…To compute the time-response of the multivalued transfer function in (5), we consider the following definition of the inverse Laplace transform:…”

“…[4] uses fractances in a lumped model of the cardiovascular system, leading to the description of different types of heart anomalous behaviors. In [3,5,6], ISs are introduced to describe robotic formations in the form of tree-like networks and ladder-like networks. Ref.…”

Fractional‐order controllers for irrational systems

“…Thus, by taking the real and imaginary part of Equation 12and, solving with respect to k p and k leads to Equations (10) and (11), respectively. □ Proposition 3 (CRB, = 0).…”

“…In addition, in the work of Podlubny [28], it has been shown that PI 𝜆 D 𝜇 controllers provide better results when being applied to fractional-order systems. Therefore, because ISs are a type of fractional-order systems [2,10], fractional-order controllers are the most suitable option to control ISs. In general, fractional-order systems' stability and fractional-order PID controllers's design have been already studied for different applications, for some examples, see refs.…”

“…In ref. [10], an IS model for an infinite ladder of mass‐springs and dampers is introduced towards the goal of modelling complex networks of mechanical systems. Furthermore, to describe the power‐law behaviour in soft tissue, a hierarchical fractal ladder network model is proposed in [11].…”