On the subject of influence of dissipative and permanent momentums on stability of uniform rotations of two elastically connected free gyroscopes of Lagrange

2019 ◽  
Vol 33 ◽  
pp. 132-141
Author(s):  
Yuri Kononov ◽  
Yaroslav Sviatenko
Keyword(s):  
Asymptotic Stability ◽  
Characteristic Equation ◽  
Joint Stiffness ◽  
Quadratic Equation ◽  
Stability Conditions ◽  
Spherical Joint ◽  
Fourth Degree ◽  
Left Hand ◽  
Resisting Medium ◽  
Complex Coefficients

In many works, there are studies of the asymptotic stability of rotation of a free Lagrange gyroscope in a resisting medium. This article generalizes this problem to the case of uniform rotations of two free Lagrange gyroscopes connected by an elastic restoring spherical hinge. The rotation of each gyroscope is maintained by a constant moment in an inertial coordinate system. The characteristic equation of the perturbed motion is presented in the form of an algebraic equation of the fourth degree with complex coefficients. Based on the innor approach, conditions of asymptotic stability are obtained in the form of a system of three inequalities. The left-hand side of these inequalities is represented, respectively, in the form of determinants of the third, fifth, and seventh orders. Up to first-order values of smallness, relative to the reciprocal of the stiffness coefficient, a study is made of the effect of the joint stiffness on stability conditions. From the conditions of positivity of the highest coefficients in three inequalities, it is shown that for a sufficiently large rigidity, the stability conditions are determined by only one inequality. Cases of degeneration of an elastic spherical joint into a spherical inelastic, cylindrical, and universal elastic joint (Hooke's joint) are considered. In the case of an inelastic spherical joint, the system of three inequalities is slightly simplified. The greatest simplification arises in the case of a cylindrical hinge. In this case, the characteristic equation is represented as a quadratic equation with complex coefficients. According to the innoric approach, the conditions of asymptotic stability are written in the form of a single inequality, the left side of which is presented in the form of third-order determinants. It is shown that this inequality coincides with the inequality obtained earlier for the case of a sufficiently large rigidity of the hinge. If the angular velocities of the proper rotations of the gyroscopes coincide, the inequality obtained for the cylindrical hinge coincides with the well-known inequality for one gyroscope. In the case of a universal elastic hinge (Hooke's hinge), the first inequality is represented as a square inequality with respect to the angular velocity of proper rotation.

On the subject of influence of dissipative and constant of moments on the stability of uniform rotations of non-free two elastically connected gyroscopes of Lagrange

2021 ◽  
Vol 34 ◽  
pp. 50-61
Author(s):  
Yurii Kononov ◽  
Yaroslav Sviatenko
Keyword(s):  
Fixed Point ◽  
Asymptotic Stability ◽  
Stability Conditions ◽  
Inertial Coordinate System ◽  
Resisting Medium ◽  
Angular Velocities ◽  
The Subject ◽  
The Stability ◽  
Spherical Hinge ◽  
Asymptotic Stability Conditions

The conditions for asymptotic stability of uniform rotations in a resisting medium of two heavy Lagrange gyroscopes connected by an elastic spherical hinge are obtained in the form of a system of three inequalities. The bottom gyroscope has a fixed point. The rotation of the gyroscopes is maintained by constant moments in the inertial coordinate system. The influence of the elasticity of the hinge on the stability conditions is estimated. It is shown that for a sufficiently high rigidity of the hinge, the asymptotic stability conditions are determined by only one inequality, which coincides with the inequality obtained for the case of a cylindrical hinge. When the angular velocities of the gyroscopes' own rotations coincide, this inequality coincides with the well--known condition for one gyroscope. Cases of degeneration of an elastic spherical hinge into a spherical inelastic, cylindrical and universal elastic hinge (Hooke's hinge) are considered. For the Hooke hinge, it is shown that there is no asymptotic stability at a sufficiently high angular velocity of gyroscopes rotation.

Asymptotic stability of fractional difference equations with bounded time delays

2020 ◽  
Vol 23 (2) ◽  
pp. 571-590
Author(s):  
Mei Wang ◽  
Baoguo Jia ◽  
Feifei Du ◽  
Xiang Liu
Keyword(s):  
Asymptotic Stability ◽  
Difference Equation ◽  
Difference Equations ◽  
Integral Inequality ◽  
Time Delays ◽  
Asymptotical Stability ◽  
Stability Conditions ◽  
Fractional Difference ◽  
Fractional Difference Equations ◽  
Fractional Difference Equation

AbstractIn this paper, an integral inequality and the fractional Halanay inequalities with bounded time delays in fractional difference are investigated. By these inequalities, the asymptotical stability conditions of Caputo and Riemann-Liouville fractional difference equation with bounded time delays are obtained. Several examples are presented to illustrate the results.

scholarly journals COMPARISON OF OPERATIVE FIXATION OF ACUTE SCAPHOID FRACTURES WITH THE NON-OPERATIVE TREATMENT: A PROSPECTIVE CLINICAL STUDY.

2020 ◽  
Vol 4 (4) ◽  
Author(s):  
Kanhaiya Lal Gupta
Keyword(s):  
Clinical Study ◽  
Operative Treatment ◽  
Scaphoid Fracture ◽  
Joint Stiffness ◽  
Prospective Clinical Study ◽  
Left Hand ◽  
Scaphoid Fractures ◽  
Common Location ◽  
Operative Fixation ◽  
Group B

Aim: to compare the results of operative fixation of acute scaphoid fractures with those of non-operative treatment. Materials & Method: This clinical study was carried out among 26 patients with an acute nondisplaced or minimally displaced scaphoid fracture reported to the OPD of Orthopedics, Jawaharlal Nehru Medical College and Hospital, Bhagalpur, Bihar, India. Patients were non-randomly allocated to group A (non-operative treatment with a cast) and group B (internal fixation with a Herbert screw). Results: mean age of the study population was 41.23 years. Majority of them were male 21 (80.7%) and rest 5 (19.3%) were female. Out of total 26 scaphoid fracture cases 16 (61.5%) were of right hand and rest 10 (38.5%) found on the left hand. Most common location of fracture was waist fracture (B2) 10 cases. 100% union was observed in group B. Conclusions: study proves that Cast treatment has the disadvantages of longer immobilisation time, joint stiffness, reduced grip strength, and longer time to return to work whereas the operative fixation of acute scaphoid fractures results in predictable satisfactory union rate and good functional outcome. Keywords: cast treatment, scaphoid fracture

scholarly journals State Observation for Lipschitz Nonlinear Dynamical Systems Based on Lyapunov Functions and Functionals

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1424 ◽  
Author(s):  
Angelo Alessandri ◽  
Patrizia Bagnerini ◽  
Roberto Cianci
Keyword(s):  
Dynamical Systems ◽  
Asymptotic Stability ◽  
Lyapunov Functions ◽  
Nonlinear Dynamical Systems ◽  
Estimation Error ◽  
Stability Conditions ◽  
Nonlinear Dynamical ◽  
State Observation ◽  
State Observers ◽  
The Stability

State observers for systems having Lipschitz nonlinearities are considered for what concerns the stability of the estimation error by means of a decomposition of the dynamics of the error into the cascade of two systems. First, conditions are established in order to guarantee the asymptotic stability of the estimation error in a noise-free setting. Second, under the effect of system and measurement disturbances regarded as unknown inputs affecting the dynamics of the error, the proposed observers provide an estimation error that is input-to-state stable with respect to these disturbances. Lyapunov functions and functionals are adopted to prove such results. Third, simulations are shown to confirm the theoretical achievements and the effectiveness of the stability conditions we have established.

ASYMPTOTIC STABILITY OF DELAY-DIFFERENCE CONTROL SYSTEM VIA MATRIX INEQUALITIES AND APPLICATION

2010 ◽  
Vol 03 (02) ◽  
pp. 347-355 ◽  
Author(s):  
K. Ratchagit
Keyword(s):  
Control System ◽  
Asymptotic Stability ◽  
Zero Solution ◽  
Discrete Version ◽  
Stability Conditions ◽  
Multiple Delays ◽  
Matrix Inequalities ◽  
Delay Difference ◽  
Lyapunov Second Method

In this paper, we obtain some criteria for determining the asymptotic stability of the zero solution of delay-difference control system in terms of certain matrix inequalities by using a discrete version of the Lyapunov second method. The result has been applied to obtain new stability conditions for some classes of delay-difference control system such as delay-difference control system with multiple delays in terms of certain matrix inequalities. Our results can be well suited for computational purposes.

scholarly journals Study of Samuelson’s General Model of National Economy and Definition of Asymptotic-Stability Conditions

2018 ◽  
Vol 10 (5) ◽  
pp. 129
Author(s):  
Athanasios D. Karageorgos ◽  
Grigoris I Kalogeropoulos
Keyword(s):  
Asymptotic Stability ◽  
Difference Equation ◽  
Saddle Point ◽  
National Economy ◽  
National Income ◽  
Stability Conditions ◽  
The Difference ◽  
Definition Of ◽  
S Model ◽  
Asymptotic Stability Conditions

In this particular paper we firstly deal with Samuelson’s model of national economy. We create a difference equation which reflects Samuelson’s model for the national income of a country taking into consideration the expenditure and the investments of the two previous years and not only the immediately previous one. Later we find the saddle-point and deal with its stability giving conditions concerning the coefficient of the difference equation and which are able (sufficient) and necessary in order for the saddle-point to be stable.

scholarly journals On the Universal Regularity of the Numbers of Generalized Recurrence Sequence and Solutions to Its Characteristic Equation of Second Order

2020 ◽  
pp. 27-33
Author(s):  
P. Kosobutskyy
Keyword(s):  
Characteristic Equation ◽  
Quadratic Equation ◽  
Second Order ◽  
Recurrence Sequence ◽  
Phase Coordinates ◽  
Fibonacci Sequences

In this work shows that the classical oscillations of the ratio of neighboring members of the Fibonacci sequences are valid for arbitrary directions on the plane of the phase coordinates, approaching, to a maximum, the solutions to the characteristic quadratic equation at a given point. The values of the solutions to the characteristic equation along the satellites are asymptotically close to their integer values of the corresponding root lines.

scholarly journals Stability investigation of quadratic systems with delay

2000 ◽  
Vol 13 (1) ◽  
pp. 85-92 ◽  
Author(s):  
Vladimir Davydov ◽  
Denis Khusainov
Keyword(s):  
Asymptotic Stability ◽  
Quadratic Form ◽  
Stability Region ◽  
Zero Solution ◽  
Stability Conditions ◽  
Quadratic Systems ◽  
Systems Of Differential Equations ◽  
Asymptotic Stability Region ◽  
Lyapunov’S Second Method ◽  
Form Stability

Systems of differential equations with quadratic right-hand sides with delay are considered in the paper. Compact matrix notation form is proposed for the systems of such type. Stability investigations are performed by Lyapunov's second method with functions of quadratic form. Stability conditions of quadratic systems with delay, uniformly by argument deviation, and with delay depending on the system's parameters are derived. A guaranteed radius of the ball of asymptotic stability region for zero solution is obtained.

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