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![a Characteristic quarter-circle with a radius of 3 for the parallel (solid line) and series (dashed line) connection of six fundamental electrical elements. To decrease the order to two via removing one parallel element, this element must be the meminductor ML. Then the hidden element moves to the position of the inductor (-1,0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(-1,0)$$\end{document}. For the same task for the series connection, we must remove the memcapacitor MC. b The parallel (solid line) and series (dashed line) connections of the meminductor ML and inerter I lead to third-order one-ports](publication/335522357/figure/fig7/AS:941819818962946@1601558751808/a-Characteristic-quarter-circle-with-a-radius-of-3-for-the-parallel-solid-line-and.png)
a Characteristic quarter-circle with a radius of 3 for the parallel (solid line) and series (dashed line) connection of six fundamental electrical elements. To decrease the order to two via removing one parallel element, this element must be the meminductor ML. Then the hidden element moves to the position of the inductor (-1,0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(-1,0)$$\end{document}. For the same task for the series connection, we must remove the memcapacitor MC. b The parallel (solid line) and series (dashed line) connections of the meminductor ML and inerter I lead to third-order one-ports
Source publication
The paper deals with the analysis of the order of the differential equation of motion describing the dynamics of a one-port network compounded of series or parallel connections of arbitrary elements from Chua’s table. It takes advantage of the fact that the elements in the table are arranged in a square graticule, which conforms to the so-called ta...
Citations
... In general, Hamilton's principle applies in circuits composed exclusively of subsets of the e 0 to e m elements. In the case of current representation of the circuit, the (a max , b min ) element is the so-called hidden element e h [24] ...
... where d is the distance of theẽ element from the diagonal according to (22), and its sign is given by the location of the element below (?) or above (-) the R-diagonal. The variation of the terms in the integrands (24) can be expressed as variations of the state functions. It leads to the compact result ...
In this work, the definition of the constitutive relation of a classical higher-order two-terminal element from Chua's table is extended to the coupled element. The way and the conditions of introducing the corresponding potential function are shown. The forms of the Lagrangian and Hamiltonian of circuits containing coupled elements are derived. The modeling techniques using coupled elements are demonstrated.
... where h is the pitch of the helix. As a result, b hel can be designed using the geometry of the cylinder and helix using Equation (9). Fluid inerters have a significant level of inherent damping due to the fluid dynamic effects. ...
... [37]) are linear, and so do not translate to nonlinear systems, and (ii) there is a strong coupling between the inertance and damping (i.e. see Equations (9) and (12)) making it very difficult to design and specify separate inertance and damping values. ...
... This idea was further extended, using data from experimental tests, to include memory effects in both inertance and damping by Wagg and Pei [101], and then friction as well by Zhang et al. [124]. These studies also included some comparison between mem-and nonlinear models for the inerter, an idea that was also considered by Biolek et al. [8,9] in the context of higher-order electrical elements. ...
In this paper, a review of the nonlinear aspects of the mechanical inerter will be presented. The historical context goes back to the development of isolators and absorbers in the first half of the twentieth century. Both mechanical and fluid-based nonlinear inerter devices were developed in the mid- and late twentieth century. However, interest in the inerter really accelerated in the early 2000s following the work of Smith [87], who coined the term ‘inerter’ in the context of a force–current analogy between electrical and mechanical networks. Following the historical context, both fluid and mechanical inerter devices will be reviewed. Then, the application of nonlinear inerter-based isolators and absorbers is discussed. These include different types of nonlinear energy sinks, nonlinear inerter isolators and geometrically nonlinear inerter devices, many relying on concepts such as quasi-zero-stiffness springs. Finally, rocking structures with inerters attached are considered, before conclusions and some future directions for research are presented.
... Consider a circuit comprised of general HOEs. The element εh = (αmax,βmin) is the hidden element of the current representation of the circuit [24]. Let us introduce new variables u = v (αmax) and x = i (βmin) . ...
... The oscillator is built from the memristor MR and the linear (0,−2) and (1,−3) elements. Utilizing the MOVE transformation [24] in Chua's table and the Consider linear FDPC and FDNR elements in series with the HP (Hewlett-Packard) memristor [7], with the nonlinear dopant drift being modeled via the Joglekar window function with the parameter p = 1 [28]. The constitutive relations will be in the form [29] ( ) ( ) ( ) ( ) ...
... The Hamiltonian of the circuit, containing only the HOEs from one diagonal, can be constructed via Equation (24). Equation (24) represents the known structure of a higher-order Hamiltonian, in which the generalized coordinates, generalized momenta, and higher-order Lagrangian appear. ...
: The paper studies the construction of the Hamiltonian for circuits built from the (,) elements of Chua’s periodic table. It starts from the Lagrange function, whose existence is limited to -circuits, i.e., circuits built exclusively from elements located on a common -diagonal of the table. We show that the Hamiltonian can also be constructed via the generalized Tellegen’s theorem. According to the ideas of predictive modeling, the resulting Hamiltonian is made up exclusively of the constitutive relations of the elements in the circuit. Within the frame of Ostrogradsky’s formalism, the simulation scheme of -circuits is designed and examined with the example of a nonlinear Pais–Uhlenbeck oscillator.
... e inner sum in (12) gives the total contribution of the FDNR elements to KV (1) L law along the i-th loop. ...
... Since this is a variational problem with fixed endpoints (the coordinates q i do not change at instants t 1 and t 2 ), the first element on the right side of (14) is zero. Integrating (12) with respect to time and subtracting from (14) yields ...
... In this case, equations describing an L-C circuit will become equations of R-FDNR just by changing the voltage v (0) to its derivative, v (1) . e transition from the L-C to the dual R-FDNR circuit by increasing α is a special case of transformation, described in [12] as the MOVE transformation. Via this transformation, the original circuit can be modified such that the topology remains unchanged but each element is replaced by another element which is shifted in Chua's table by an offset (Δα, Δβ), where Δα and Δβ can be As can be seen in Figure 3, dual L-C and R-FDNR circuits share the same generalized coordinates and velocities, i.e., the loop charges and currents. ...
The classic form of Hamilton’s variational principle does not hold for circuits with dissipative elements. It is shown in the paper that this may not be true in the case of systems consisting of the so-called higher-order elements. Hamilton’s principle is then extended to circuits containing the classical resistors and Frequency Dependent Negative Resistors (FDNRs). The extension is also made to any pair of elements which are the nearest neighbours on any Σ -diagonal of Chua’s table.
... The MOVE transformation relocates the group of elements by m / n increments along the voltage / current axis, thus the (α,β) element is changed to the (α+m,β+n) element [10]. Observing the dynamics of the original circuit in the (v (k) , i (l) ) space, then the corresponding dual circuit will exhibit identical dynamics in the (v (k+m) , i (l+n) ) space. ...
... Section 2 summarizes the current classification of the fundamental elements according to their positions on the ∆-diagonals of the table and the definitions of their state functions according to [27]. Recalled therein are some rules of the taxicab geometry, which hold for the table of elements, recently published in [28]. The following key Section 3 describes the derivation of Hamilton's variational principle for circuits containing elements from an arbitrary Σ-diagonal of the table. ...
... In such a type of geometry, the circle drawn around a central point has the form of a square whose diagonals occupy the horizontal and vertical positions. According to [28], the order of the differential equation describing the behavior of one-port consisting of serial and parallel combinations of HOEs is equal to the radius of the smallest quarter-circle that incepts all the elements of the one-port. The center of the quarter-circle is occupied by the so-called hidden element with the coordinates (αMAX,βMIN) or (αMIN,βMAX) for a series or a parallel connection of the elements. ...
... In such a type of geometry, the circle drawn around a central point has the form of a square whose diagonals occupy the horizontal and vertical positions. According to [28], the order O of the differential equation describing the behavior of one-port consisting of serial and parallel combinations of HOEs is equal to the radius of the smallest quarter-circle that incepts all the elements of the one-port. The center of the quarter-circle is occupied by the so-called hidden element with the coordinates (α MAX ,β MIN ) or (α MIN ,β MAX ) for a series or a parallel connection of the elements. ...
The necessary and sufficient conditions of the validity of Hamilton’s variational principle for circuits consisting of (α,β) elements from Chua’s periodical table are derived. It is shown that the principle holds if and only if all the circuit elements lie on the so-called Σ-diagonal with a constant sum of the indices α and β. In this case, the Lagrangian is the sum of the state functions of the elements of the L or +R types minus the sum of the state functions of the elements of the C or −R types. The equations of motion generated by this Lagrangian are always of even-order. If all the elements are linear, the equations of motion contain only even-order derivatives of the independent variable. Conclusions are illustrated on an example of the synthesis of the Pais–Uhlenbeck oscillator via the elements from Chua’s table.
It is shown for the first time that a large class of generic meminductors with a specific form of the state equation can be constructed from elements of Chua’s periodical table and a multiport inductor. The electrical port of this inductor is identical to the physical port of the meminductor. The other ports, which may be of general, i.e. not electrical nature, interconnect the processes that make up the overall dynamics of the element. This result allows creating predictive models of a wide range of existing MEMS (Micro-Electro-Mechanical Systems) exhibiting the behavior of generic meminductors. A model building procedure is described using the example of an elastic meminductive system. The model is then implemented in SPICE and verified by computer simulations.
