Rapid Computation of Resonant Frequencies for Nonproportionally Damped Systems Using Dual Oscillators

Sanders, John W.; Inman, Daniel J.

doi:10.1115/1.4056796

Cited by 3 publications

(11 citation statements)

References 30 publications

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“…By extending that result [3], Sanders [4] was able to perform direct modal analysis of non-proportionally damped linear systems with arbitrary degrees of freedom, and at the same time opened the door to direct modal analysis of damped nonlinear oscillators with viscous damping and power-law hardening [4]. Sanders and Inman [6] then showed that the resonant frequency computation provided by the dual oscillator approach is generally more efficient than the traditional method of Foss [18] and Traill-Nash [19].…”

Section: Introductionmentioning

confidence: 92%

“…Although the extraneous solutions have no physical meaning, they can nevertheless be exploited to achieve practical ends. Indeed, Sanders's dual oscillators [3][4][5][6], which enable direct modal analysis of non-proportionally damped systems, are ultimately just extraneous solutions to the fourth-order equations. Those solutions are therefore of practical importance, even from a purely classical perspective, and in what follows we will be interested not only in the physical solutions but also in the complete set of solutions to the fourth-order equations.…”

Section: Variational Structure Of the Fourth-order Formulationmentioning

confidence: 99%

“…The fact that the governing equation ( 17) has no odd-order derivatives was exploited by Sanders and Inman [6] to perform rapid computations of resonant frequencies for non-proportionally damped systems. By writing the fourth-order system as an augmented second-order system, Sanders and Inman [6] obtained the resonant frequencies with fewer computations than the previous standard approach [18,19]. This was a direct application of the intrinsic variational structure of the fourthorder formulation.…”

Section: Case Study: the Classical Damped Harmonic Oscillatormentioning

confidence: 99%

“…Ever since the advent of classical mechanics, a clear line has been drawn between Hamiltonian and non-Hamiltonian systems [1,2]. However, a recent discovery [3][4][5][6] has blurred that line. Evidently it is always possible to treat a non-Hamiltonian system as if it were Hamiltonian simply by considering the equations of motion at the fourth-order level in time.…”

Section: Introductionmentioning

confidence: 99%

“…At the fourth-order level, the classical equations are intrinsically variational, in that they are always derivable from a stationary action integral, even when the system is non-conservative or non-holonomic [5]. This has already led to two classical applications: the direct modal analysis of damped dynamical systems [4] and subsequently a new and more efficient algorithm for computing a damped system's resonant frequencies [6]. In light of these recent applications, it is the present author's view that higher-derivative formulations of classical mechanics merit further investigation, and the central purpose of the present work is to investigate the geometric structure of such formulations.…”

Section: Introductionmentioning

confidence: 99%

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Intrinsic variational structure of higher-derivative formulations of classical mechanics

Sanders¹

2023

Preprint

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This paper investigates the geometric structure of higher-derivative formulations of classical mechanics. It is shown that every even-order formulation of classical mechanics higher than the second order is intrinsically variational, in the sense that the equations of motion are always derivable from a minimum action principle, even when the system is non-Hamiltonian. Particular emphasis is placed on the fourth-order formulation, as that is shown to be the lowest order for which the governing equations are intrinsically variational. The Noether symmetries and associated conservation laws of the fourth-order formulation, including its Hamiltonian, are derived along with the natural auxiliary conditions. The intrinsic variational structure of higher-derivative formulations makes it possible to treat non-Hamiltonian systems as if they were Hamiltonian, with immediate classical applications. A case study of the classical damped harmonic oscillator is presented for illustration, and an action is formulated for a higher-order Navier-Stokes equation.

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Section: Introductionmentioning

confidence: 92%

Section: Variational Structure Of the Fourth-order Formulationmentioning

confidence: 99%

Section: Case Study: the Classical Damped Harmonic Oscillatormentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Intrinsic variational structure of higher-derivative formulations of classical mechanics

Sanders¹

2023

Preprint

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A canonical Hamiltonian formulation of the Navier–Stokes problem

Sanders,

DeVoria,

Washuta

et al. 2024

J. Fluid Mech.

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This paper presents a novel Hamiltonian formulation of the isotropic Navier–Stokes problem based on a minimum-action principle derived from the principle of least squares. This formulation uses the velocities $u_{i}(x_{j},t)$ and pressure $p(x_{j},t)$ as the field quantities to be varied, along with canonically conjugate momenta deduced from the analysis. From these, a conserved Hamiltonian functional $H^{*}$ satisfying Hamilton's canonical equations is constructed, and the associated Hamilton–Jacobi equation is formulated for both compressible and incompressible flows. This Hamilton–Jacobi equation reduces the problem of finding four separate field quantities ( $u_{i}$ , $p$ ) to that of finding a single scalar functional in those fields – Hamilton's principal functional ${S}^{*}[u_{i},p,t]$ . Moreover, the transformation theory of Hamilton and Jacobi now provides a prescribed recipe for solving the Navier–Stokes problem: find ${S}^{*}$ . If an analytical expression for ${S}^{*}$ can be obtained, it will lead via canonical transformation to a new set of fields which are simply equal to their initial values, giving analytical expressions for the original velocity and pressure fields. Failing that, if one can only show that a complete solution to this Hamilton–Jacobi equation does or does not exist, that will also resolve the question of existence of solutions. The method employed here is not specific to the Navier–Stokes problem or even to classical mechanics, and can be applied to any traditionally non-Hamiltonian problem.

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Extension of Hamiltonian Mechanics to Non-Conservative Systems Via Higher-Order Dynamics

Sanders,

Becker,

DeVoria

2024

Journal of Vibration and Acoustics

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This paper presents a detailed review of the emerging topic of higher-order dynamics and their intrinsic variational structure, which has enabled—for the very first time in history—the general application of Hamiltonian formalism to non-conservative systems. Here the general theory is presented alongside several interesting applications that have been discovered to date. These include the direct modal analysis of non-proportionally damped dynamical systems, a new and more efficient algorithm for computing the resonant frequencies of damped systems with many degrees of freedom, and a canonical Hamiltonian formulation of the Navier-Stokes problem. A significant merit of the Hamiltonian formalism is that it leads to the transformation theory of Hamilton and Jacobi, and specifically the Hamilton-Jacobi equation, which reduces even the most complicated of problems to the search for a single scalar function (or functional, for problems in continuum mechanics). With the extension of the Hamiltonian framework to non-conservative systems, now every problem in classical mechanics can be reduced to the search for a single scalar. This discovery provides abundant opportunities for further research, and here we list just a few potential ideas. Indeed, the present authors believe there may be many more applications of higher-order dynamics waiting to be discovered.

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Rapid Computation of Resonant Frequencies for Nonproportionally Damped Systems Using Dual Oscillators

Cited by 3 publications

References 30 publications

Intrinsic variational structure of higher-derivative formulations of classical mechanics

Intrinsic variational structure of higher-derivative formulations of classical mechanics

A canonical Hamiltonian formulation of the Navier–Stokes problem

Extension of Hamiltonian Mechanics to Non-Conservative Systems Via Higher-Order Dynamics

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