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

Awrejcewicz, Jan; Losyeva, Nataliya; Puzyrov, Volodymyr

doi:10.3390/sym12081210

Cited by 11 publications

(7 citation statements)

References 40 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…(In the domains Ib, Ic the expression dis 2 ( , p) is negative.) Now, bringing together these results, we can make a conclusion about hierarchy of all six roots of polynomials U j ( ) with the aim to find the solution of system ( 18)- (20). Results are presented in Table 1 and Fig.…”

Section: Lemma 3 For Any Positive Values Of P the Polynomial U 3 ( )...mentioning

confidence: 95%

See 1 more Smart Citation

Analytical Scheme of Stability Analysis for 4-DoF Mechanical System Subjected to Friction-Induced Vibrations

Puzyrov

Pujol

Acho

2022

J. Vib. Eng. Technol.

Self Cite

View full text Add to dashboard Cite

Purpose: The stability problem for non-conservative multi-parameter dynamical system is usually associated with labor-intensive calculations, and numerical methods do not always allow one to obtain the desired information. The presence of circulatory forces often leads to the so-called ”destabilization effect” of the system under the influence of small dissipative forces. In this regard, it seems important to develop analytical approaches that make it possible to use a simplified scheme for checking the stability conditions. Methods: When obtaining and analyzing stability conditions, the algebra of polynomials and elements of mathematical analysis are applied. To obtain a simplified scheme for checking the stability conditions, an asymptotic method is used. Results and Conclusion: A mechanical system with four degrees of freedom which is under the action of dissipative, potential and non-conservative potential (circulatory) forces is considered. The stability problem of friction-induced vibrations is studying. In the case of weak damping an analytical approach is proposed that makes it possible to simplify the analysis of stability conditions, which, due to the presence of many uncertain parameters, are very cumbersome. With the help of numerical testing, the adequacy of the results obtained for the reduced conditions and full stability conditions was established. The results of the analysis make it possible to single out the ”advantageous” regions in the space of dimensionless parameters, which makes it possible to improve the design of the system to increase its reliability.

show abstract

Section: Lemma 3 For Any Positive Values Of P the Polynomial U 3 ( )...mentioning

confidence: 95%

“…This problem highlights some important effects that arise in the study of the dynamics of systems with dissipationinduced instabilities. In paper [20] some optimized algorithms for stability analysis of systems with circulatory forces were suggested.…”

Section: Introductionmentioning

confidence: 99%

Analytical Scheme of Stability Analysis for 4-DoF Mechanical System Subjected to Friction-Induced Vibrations

Puzyrov

Pujol

Acho

2022

J. Vib. Eng. Technol.

Self Cite

View full text Add to dashboard Cite

show abstract

“…As one can see from formula (9), normally, the first term on the right-hand side is negative, thus in a case when n is slightly less, then ffiffiffiffi ffi m 2 p The appearance of a small damping force brings instability to a stable undamped system. However, if the equality k d ¼ 2k d holds, then the system (1) is stable.…”

Section: Methodsmentioning

confidence: 99%

“…This problem highlights some crucial effects that arise in the field of dissipation-induced instabilities. In paper [9], some optimized algorithms were suggested to reduce the computation time and obtain stability conditions in a simpler form. Regarding various mechanical applications, we mention the friction-induced vibrations [10,11], squeal vibration in drum brakes [12,13], the Levitron system [14], aeroelastic systems [15], the effect of rocket propulsion as a source of a follower load acting on a beam [16] and others.…”

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

Self Cite

View full text Add to dashboard Cite

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.

show abstract

“…Then, if the following conditions hold TpTNTp≠0,1emTpTNTr=0, the system (1.4) is unstable by flutter [4]. This result contains, as a special case, the famous Merkin's theorem [1], which assumes the commutativity of the matrices K and N [2,3,12]. It should be noted that in applications, the condition given in (1.6) relies on an analysis of the eigen structure of the matrix K , and an appropriate choice of the orthonormal eigenvectors that are to be included in the submatrix T p [5].…”

Section: Introductionmentioning

confidence: 99%

On the stability and instability of linear potential systems subjected to infinitesimal circulatory forces

Bulatović

Udwadia

2021

Proc. R. Soc. A.

View full text Add to dashboard Cite

The stability of linear multi-degree-of-freedom stable potential systems with multiple natural frequencies under the action of infinitesimal circulatory forces is considered. Contrary to the received view that such systems are inherently unstable, a careful study shows that such systems have a much more complex behaviour than previously recognized and could exhibit an alternation of stability and instability that depends on the structure of the potential system and its interaction with the circulatory forces. The conditions under which stability or instability ensues and the nature of this alternation in stability are explicitly obtained. In low-dimensional stable potential systems, when the coefficients of the circulatory forces are proportional to an arbitrarily small scalar parameter, all the circulatory forces that cause flutter instability are described.

show abstract

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

Cited by 11 publications

References 40 publications

Analytical Scheme of Stability Analysis for 4-DoF Mechanical System Subjected to Friction-Induced Vibrations

Analytical Scheme of Stability Analysis for 4-DoF Mechanical System Subjected to Friction-Induced Vibrations

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

On the stability and instability of linear potential systems subjected to infinitesimal circulatory forces

Contact Info

Product

Resources

About