Taxicab geometry in table of higher-order elements

Biolek, Zdeněk; Biolek, Dalibor; Biolková, Viera; Kolka, Zdenek

doi:10.1007/s11071-019-05218-9

Cited by 8 publications

(13 citation statements)

References 41 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…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 Lagrangian is set up via the constitutive relations of the individual elements.…”

Section: Discussionmentioning

confidence: 99%

“…The Lagrangian is set up via the constitutive relations of the individual elements. The individual momenta are arranged according to (24) as sums of the generalized voltages of the corresponding orders across selected elements of the circuit. The Lagrangian is given as a sum of the state functions of the individual elements provided with the appropriate sign.…”

Section: Discussionmentioning

confidence: 99%

“…Consider a circuit comprised of general HOEs. The element ε h = (α max ,β min ) is the hidden element of the current representation of the circuit [24]. Let us introduce new variables u = v (αmax) and x = i (βmin) .…”

Section: Tellegen's Theorem For Circuits With Higher-order Elementsmentioning

confidence: 99%

“…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 duality principle, the two last mentioned elements can be replaced by their linear versions of FDNC and FDPC (Frequency Dependent Positive Conductance). The coordinates of the hidden element (FDPC) indicate that the generalized current x is a threefold time integral of the current, or i (−3) .…”

Section: Lagrange's Formalismmentioning

confidence: 99%

See 3 more Smart Citations

Higher-Order Hamiltonian for Circuits with (α,β) Elements

Biolek

Biolková

et al. 2020

Entropy

Self Cite

View full text Add to dashboard Cite

The paper studies the construction of the Hamiltonian for circuits built from the (α,β) elements of Chua’s periodic table. It starts from the Lagrange function, whose existence is limited to Σ-circuits, i.e., circuits built exclusively from elements located on a common Σ-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 Σ-circuits is designed and examined with the example of a nonlinear Pais–Uhlenbeck oscillator.

show abstract

Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Section: Tellegen's Theorem For Circuits With Higher-order Elementsmentioning

confidence: 99%

Section: Lagrange's Formalismmentioning

confidence: 99%

See 2 more Smart Citations

Higher-Order Hamiltonian for Circuits with (α,β) Elements

Biolek

Biolková

et al. 2020

Entropy

Self Cite

View full text Add to dashboard Cite

show abstract

“…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.…”

Section: Introductionmentioning

confidence: 99%

Lagrangian for Circuits with Higher-Order Elements

Biolek

Biolková

2019

Entropy

Self Cite

View full text Add to dashboard Cite

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.

show abstract

A review of the mechanical inerter: historical context, physical realisations and nonlinear applications

Wagg

2021

Nonlinear Dyn

View full text Add to dashboard Cite

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.

show abstract

Taxicab geometry in table of higher-order elements

Cited by 8 publications

References 41 publications

Higher-Order Hamiltonian for Circuits with (α,β) Elements

Higher-Order Hamiltonian for Circuits with (α,β) Elements

Lagrangian for Circuits with Higher-Order Elements

A review of the mechanical inerter: historical context, physical realisations and nonlinear applications

Contact Info

Product

Resources

About