Stabilizing and destabilizing effects of damping in non-conservative systems: Some new results

Abdullatif, Mahmoud; Mukherjee, Ranjan; Hellum, Aren

doi:10.1016/j.jsv.2017.09.018

Cited by 15 publications

(12 citation statements)

References 23 publications

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“…The analytically derived stability boundary of the 2-DoF system is analyzed in [6] regarding infinitesimally small, incomplete, and indefinite damping matrices. Systems with two neutrally-stable damping levels, whereby the system initially gains stability but then loses stability as the level of damping is increased, were observed in [7]. In paper [8], Brouwer's problem with a heavy particle in a rotating vessel is considered.…”

Section: Literature Reviewmentioning

confidence: 99%

Neutralization of the Destabilization Effect Caused by Small Damping Force in Non-conservative System

Puzyrov

Acho

Pujol

et al. 2021

Lecture Notes in Mechanical Engineering

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For the first time, the destabilizing effect of small dissipative forces was marked by Thomson and Tait in 1879. They showed that a statically unstable conservative system stabilized by gyroscopic forces could be destabilized again by introducing small damping forces. Later on, in 1924, Kimball noticed the paradoxical effect of damping on dynamic stability for rotor systems that have stable steady motions for a certain range of speed but become unstable when the speed is changed to a value outside the range. Further progress in engineering and technology has witnessed that such phenomenon arises in numerous mechanical applications and may cause equipment malfunction. In the present article, this problem is studied by comparing the stability conditions for undamped and weakly damped systems. The aim is to find the interrelation between the matrix elements characterizing the damping force (assuming that the stiffness matrix is known), which allows to avoid or minimize the negative effect of small damping in the presence of non-conservative positional forces. For the case study of two Degrees-of-Freedom system, a simple analytical technique is suggested for this purpose. This technique is demonstrated on two mechanical systems: a system with friction-induced vibrations and a double pendulum under the action of the follower force.

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Section: Literature Reviewmentioning

confidence: 99%

Neutralization of the Destabilization Effect Caused by Small Damping Force in Non-conservative System

Puzyrov

Acho

Pujol

et al. 2021

Lecture Notes in Mechanical Engineering

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“…Algorithm 1 Fractional Model Construction: FSEM based Iterative Algorithm (Model Error type-I) 1: Initial guess q 0 = {α 0 , β 0 } 2: Do i = 0, 1, · · · 3:…”

Section: Fractional Model Construction: Fsem-based Iterative Algorithmmentioning

confidence: 99%

“…We conclude the paper with a summary and conclusion. 1]. The left-and right-sided fractional derivative of order σ, n − 1 < σ ≤ n, n ∈ N, are defined as (see e.g., [42,45])…”

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confidence: 99%

Fractional Sensitivity Equation Method: Application to Fractional Model Construction

Kharazmi

Zayernouri

2019

J Sci Comput

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Fractional differential equations provide a tractable mathematical framework to describe anomalous behavior in complex physical systems, yet they introduce new sensitive model parameters, i.e. derivative orders, in addition to model coefficients. We formulate a sensitivity analysis of fractional models by developing a fractional sensitivity equation method. We obtain the adjoint fractional sensitivity equations, in which we present a fractional operator associated with logarithmic-power law kernel. We further construct a gradient-based optimization algorithm to compute an accurate parameter estimation in fractional model construction. We develop a fast, stable, and convergent Petrov-Galerkin spectral method to numerically solve the coupled system of original fractional model and its corresponding adjoint fractional sensitivity equations.Key words. sensitive fractional orders, model error, logarithmic-power law kernel, Petrov-Galerkin spectral method, iterative algorithm, parameter estimation.Fractional Sensitivity Analysis. Sensitivity assessment approaches are commonly categorized as, finite difference, continuum and discrete derivatives, and computational or automatic differentiation, where the sensitivity coefficients are generally defined as partial derivative of corresponding functions (model output) with respect to design/analysis parameters of inter- †

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“…This property is often referred to as Ziegler's paradox [14], since adding a viscous damping to the joints of a double pendulum subjected to the follower force destabilizes the system. After the publication of Ziegler in 1952, investigations of the destabilizing effect of damping have attracted the attention of numerous researchers [15][16][17][18][19][20] and are continued to this day [5,21,22]. Furthermore, an additional interesting feature of the "reciprocal" damping effect follows: when adding a damping to unstable non-conservative system, this yields stabilization of an unstable system (in the Lyapunov sense), and the further increasing of damping leads back to instability [23].…”

Section: Introductionmentioning

confidence: 99%

Stability and Boundedness of the Solutions of Multi-Parameter Dynamical Systems with Circulatory Forces

2020

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We consider a linear dynamical system under the action of potential and circulatory forces. The matrix of potential forces is positive definite, and the main question is when the circulatory forces induce instability to the system. Different approaches to studying the problem are discussed and illustrated by examples. The case of multiple eigenvalues also is considered, and sufficient conditions of instability are obtained. Some issues of the dynamics of a nonlinear system with an unstable linear approximation are discussed. The behavior of trajectories in the case of unstable equilibrium is investigated, and an example of the chaotic behavior versus the case of bounded solutions is presented and discussed.

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Stabilizing and destabilizing effects of damping in non-conservative systems: Some new results

Cited by 15 publications

References 23 publications

Neutralization of the Destabilization Effect Caused by Small Damping Force in Non-conservative System

Neutralization of the Destabilization Effect Caused by Small Damping Force in Non-conservative System

Fractional Sensitivity Equation Method: Application to Fractional Model Construction

Stability and Boundedness of the Solutions of Multi-Parameter Dynamical Systems with Circulatory Forces

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