scholarly journals Mixed forced, parametric, and self-oscillations with nonideal energy source and lagging forces

2021 ◽  
Vol 29 (5) ◽  
pp. 739-750
Author(s):  
Alishir Alifov ◽  
Keyword(s):  
Energy Source ◽  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Equations Of Motion ◽  
Point Of View ◽  
Ease Of Use ◽  
Direct Linearization ◽  
Plane Change ◽  
The Stability ◽  
The Right

The purpose of this study is to determine the effect of retarded forces in elasticity and damping on the dynamics of mixed forced, parametric, and self-oscillations in a system with limited excitation. A mechanical frictional self-oscillating system driven by a limited-power engine was used as a model. Methods. In this work, to solve the nonlinear differential equations of motion of the system under consideration, the method of direct linearization is used, which differs from the known methods of nonlinear mechanics in ease of use and very low labor and time costs. This is especially important from the point of view of calculations when designing real devices. Results. The characteristic of the friction force that causes self-oscillations, represented by a general polynomial function, is linearized using the method of direct linearization of nonlinearities. Using the same method, solutions of the differential equations of motion of the system are constructed, equations are obtained for determining the nonstationary values of the amplitude, phase of oscillations and the speed of the energy source. Stationary motions are considered, as well as their stability by means of the Routh–Hurwitz criteria. Performed calculations obtained information about the effect of delays on the dynamics of the system. Conclusion. Calculations have shown that delays shift the amplitude curves to the right and left, up and down on the amplitude–frequency plane, change their shape, and affect the stability of motion.

On nonlinear perturbation analysis of a structure carrying a circular cylindrical liquid tank under horizontal excitation

2018 ◽  
Vol 25 (5) ◽  
pp. 1058-1079 ◽  
Author(s):  
N. K. A. Attari ◽  
F. R. Rofooei ◽  
Z. Waezi
Keyword(s):  
Differential Equations ◽  
Perturbation Analysis ◽  
Multiple Scales ◽  
Nonlinear Differential Equations ◽  
Equations Of Motion ◽  
Structural Response ◽  
Single Mode ◽  
Excitation Amplitude ◽  
Combined System ◽  
The Stability

The lateral response of a single degree of freedom structural system containing a rigid circular cylindrical liquid tank under harmonic and earthquake excitations at a 1:2 autoparametric resonance case is considered. The governing nonlinear differential equations of motion for the combined system are solved by means of a multiple scales method considering the first three liquid sloshing modes (1,1), (0,1), and (2,1), under horizontal excitation. The fixed points of the gyroscopic type of governing differential equations are determined and their stability is investigated employing the perturbation method. The obtained results reveal an increase in the stability region for a single-mode response with respect to the excitation amplitude. The saturation phenomenon is observed in the decoupled modes of the system; however, the structural mode and the first anti-symmetric mode of liquid are a combination of the saturated mode and another mode whose scale factor is crucial for the structural response. The results of perturbation analysis are in good agreement with results obtained from numerical methods.

On the Stability of Porous Journal Bearings

1974 ◽  
Vol 96 (2) ◽  
pp. 585-590 ◽  
Author(s):  
T. F. Conry ◽  
C. Cusano
Keyword(s):  
Differential Equations ◽  
Design Variable ◽  
Nonlinear Differential Equations ◽  
Equations Of Motion ◽  
Journal Bearings ◽  
Motion Stability ◽  
Region Of Stability ◽  
Regions Of Stability ◽  
Static Eccentricity ◽  
The Stability

This investigation deals with the stability characteristics of lubricated porous journal bearings. Using the short-bearing approximation and a fully cavitated film, the regions of stability are determined by simulation of the nonlinear differential equations of motion. Stability curves are presented for values of the design variable Φ from 0.0001 to 0.02. As Φ increases, the area of the region of stability decreases. For values of Φ greater than 0.001, the value of the dimensionless speed tends to decrease with increasing static eccentricity ratios at the threshold of stability.

scholarly journals On the Lyapunov functionals method in the stability problem of Volterra integro-differential equations

2018 ◽  
Vol 20 (3) ◽  
pp. 260-272
Author(s):  
A. S. Andreev ◽  
O. A. Peregudova
Keyword(s):  
Differential Equations ◽  
Stability Problem ◽  
Infinite Delay ◽  
Zero Solution ◽  
Model Systems ◽  
Lyapunov Functionals ◽  
Structure Equations ◽  
Positive Limit Set ◽  
The Stability ◽  
The Right

In this paper, we consider the problem of applying the method of Lyapunov functionals to investigate the stability of non-linear integro-differential equations, the right-hand side of which is the sum of the components of the instantaneous action and also ones with a finite and infinite delay. The relevance of the problem is the widespread use of such complicated in structure equations in modeling the controllers using integral regulators for mechanical systems, as well as biological, physical and other processes. We develop the Lyapunov functionals method in the direction of revealing the limiting properties of solutions by means of Lyapunov functionals with a semi-definite derivative. We proved the theorems on the quasi-invariance of a positive limit set of bounded solution as well as ones on the asymptotic stability of the zero solution including a uniform one. The results are achieved by constructing a new structure of the topological dynamics of the equations under study. The theorems proved are applied in solving the stability problem of two model systems which are generalizations of a number of known models of natural science and technology.

scholarly journals Modified Variational Iteration Algorithm-II: Convergence and Applications to Diffusion Models

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-14 ◽  
Author(s):  
Hijaz Ahmad ◽  
Tufail A. Khan ◽  
Predrag S. Stanimirović ◽  
Yu-Ming Chu ◽  
Imtiaz Ahmad
Keyword(s):  
Differential Equations ◽  
Numerical Solutions ◽  
Nonlinear Models ◽  
Nonlinear Differential Equations ◽  
Variational Iteration Method ◽  
Iteration Algorithm ◽  
Compact Finite Difference Method ◽  
Variational Iteration ◽  
Linear And Nonlinear ◽  
The Stability

Variational iteration method has been extensively employed to deal with linear and nonlinear differential equations of integer and fractional order. The key property of the technique is its ability and flexibility to investigate linear and nonlinear models conveniently and accurately. The current study presents an improved algorithm to the variational iteration algorithm-II (VIA-II) for the numerical treatment of diffusion as well as convection-diffusion equations. This newly introduced modification is termed as the modified variational iteration algorithm-II (MVIA-II). The convergence of the MVIA-II is studied in the case of solving nonlinear equations. The main advantage of the MVIA-II improvement is an auxiliary parameter which makes sure a fast convergence of the standard VIA-II iteration algorithm. In order to verify the stability, accuracy, and computational speed of the method, the obtained solutions are compared numerically and graphically with the exact ones as well as with the results obtained by the previously proposed compact finite difference method and second kind Chebyshev wavelets. The comparison revealed that the modified version yields accurate results, converges rapidly, and offers better robustness in comparison with other methods used in the literature. Moreover, the basic idea depicted in this study is relied upon the possibility of the MVIA-II being utilized to handle nonlinear differential equations that arise in different fields of physical and biological sciences. A strong motivation for such applications is the fact that any discretization, transformation, or any assumptions are not required for this proposed algorithm in finding appropriate numerical solutions.

Torquewhirl—A Theory to Explain Nonsynchronous Whirling Failures of Rotors With High-Load Torque

1978 ◽  
Vol 100 (2) ◽  
pp. 235-240
Author(s):  
J. M. Vance
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
High Power ◽  
Nonlinear Differential Equations ◽  
Equations Of Motion ◽  
Rotating Machinery ◽  
High Load ◽  
Driving Mechanism ◽  
Load Torque ◽  
Driving Torque

Numerous unexplained failures of rotating machinery by nonsynchronous shaft whirling point to a possible driving mechanism or source of energy not identified by previously existing theory. A majority of these failures have been in machines characterized by overhung disks (or disks located close to one end of a bearing span) and/or high power and load torque. This paper gives exact solutions to the nonlinear differential equations of motion for a rotor having both of these characteristics and shows that high ratios of driving torque to damping can produce nonsynchronous whirling with destructively large amplitudes. Solutions are given for two cases: (1) viscous load torque and damping, and (2) load torque and damping proportional to the second power of velocity (aerodynamic case). Criteria are given for avoiding the torquewhirl condition.

Stability of the High-Speed Journal Bearing Under Steady Load: 1—The Incompressible Film

1962 ◽  
Vol 84 (3) ◽  
pp. 351-357 ◽  
Author(s):  
M. M. Reddi ◽  
P. R. Trumpler
Keyword(s):  
High Speed ◽  
Journal Bearing ◽  
Orbital Motion ◽  
Hydrodynamic Force ◽  
Nonlinear Differential Equations ◽  
Equations Of Motion ◽  
Static Equilibrium ◽  
The Stability ◽  
External Loads ◽  
Steady Load

The phenomenon of oil-film whirl in bearings subjected to steady external loads is analyzed. The journal, assumed to be a particle mass, is subjected to the action of two forces; namely, the external load acting on the bearing and the hydrodynamic force developed in the fluid film. The resulting equations of motion for a full-film bearing and a 180-deg partial-film bearing are developed as pairs of second-order nonlinear differential equations. In evaluating the hydrodynamic force, the contribution of the shear stress on the journal surface is found to be negligible for the full-film bearing, whereas for the partial-film bearing it is found to be significant at small attitude values. The equations of motion are linearized and the coefficients of the resulting characteristic equations are studied for the stability of the static-equilibrium positions. The full-film bearing is found to have no stable static-equilibrium position, whereas the 180-deg partial-film bearing is found to have stable static-equilibrium positions under certain parametric conditions. The equations of motion for the full-film bearing are integrated numerically on a digital computer. The results show that the journal center, depending on the parametric conditions, acquired either an orbital motion or a dynamical path of increasing attitude terminating in bearing failure.

Application of GDQ method in nonlinear analysis of a flexible manipulator undergoing large deformation

2013 ◽  
Vol 227 (12) ◽  
pp. 2671-2685 ◽  
Author(s):  
Mohammad R Fazel ◽  
Majid M Moghaddam ◽  
Javad Poshtan
Keyword(s):  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Equations Of Motion ◽  
Nonlinear Partial Differential Equation ◽  
Flexible Manipulator ◽  
Runge Kutta Method ◽  
Highly Nonlinear ◽  
Frechet Derivatives ◽  
Fréchet Derivatives ◽  
Generalized Differential

Analysis of a flexible manipulator as an initial value problem, due to its large deformations, involves nonlinear ordinary differential equations of motion. In the present work, these equations are solved through the general Frechet derivatives and the generalized differential quadrature (GDQ) method directly. The results so obtained are compared with those of the fourth-order Runge–Kutta method. It is seen that both the results match each other well. Further considering the same manipulator as a boundary value problem, its governing equation is a highly nonlinear partial differential equation. Again applying the general Frechet derivatives and the GDQ method, it is seen that the results are in good match with the linear theory. In both cases, the general Frechet derivatives are introduced and successfully used for linearization. The results of the present study indicate that the GDQ method combined with the general Frechet derivatives can be successfully used for the solution of nonlinear differential equations.

scholarly journals Nonlinear Dynamic Analysis and Optimization of Closed-Form Planetary Gear System

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Qilin Huang ◽  
Yong Wang ◽  
Zhipu Huo ◽  
Yudong Xie
Keyword(s):  
Differential Equations ◽  
Closed Form ◽  
Dynamic Model ◽  
Total Mass ◽  
Nonlinear Differential Equations ◽  
Equations Of Motion ◽  
Planetary Gear ◽  
Transmission System ◽  
Gear Transmission ◽  
Gear Transmission System

A nonlinear purely rotational dynamic model of a multistage closed-form planetary gear set formed by two simple planetary stages is proposed in this study. The model includes time-varying mesh stiffness, excitation fluctuation and gear backlash nonlinearities. The nonlinear differential equations of motion are solved numerically using variable step-size Runge-Kutta. In order to obtain function expression of optimization objective, the nonlinear differential equations of motion are solved analytically using harmonic balance method (HBM). Based on the analytical solution of dynamic equations, the optimization mathematical model which aims at minimizing the vibration displacement of the low-speed carrier and the total mass of the gear transmission system is established. The optimization toolbox in MATLAB program is adopted to obtain the optimal solution. A case is studied to demonstrate the effectiveness of the dynamic model and the optimization method. The results show that the dynamic properties of the closed-form planetary gear transmission system have been improved and the total mass of the gear set has been decreased significantly.

scholarly journals Analytical study of auto-balancing within the framework of the flat model of a rotor and an auto-balancer with a single cargo

2021 ◽  
Vol 2 (7 (110)) ◽  
pp. 66-73
Author(s):  
Gennadiy Filimonikhin ◽  
Lubov Olijnichenko ◽  
Guntis Strautmanis ◽  
Antonina Haleeva ◽  
Vasyl Hruban ◽  
...  
Keyword(s):  
Differential Equations ◽  
Analytical Study ◽  
Rotation Axis ◽  
Equations Of Motion ◽  
The Third ◽  
Rotor Imbalance ◽  
Small Resistance ◽  
The Stability ◽  
Resistance Forces ◽  
Rotor Model

This paper reports the analytically established conditions for the onset of auto-balancing for the case of a flat rotor model on isotropic elastic-viscous supports and an auto-balancer with a single load. The rotor is statically unbalanced, the rotation axis is vertical. The auto-balancer has a single cargo – a pendulum, a ball, or a roller. The balancing capacity of the cargo is equal to the rotor imbalance. The physical-mathematical model of the system is described. The differential equations of motion are recorded in dimensionless form relative to the coordinate system that rotates synchronously with the rotor. The so-called main movement has been found; in it, the cargo synchronously rotates with the rotor and balances it. The differential equations of motion are linearized in the neighborhood of the main movement. A characteristic equation has been constructed. It helped investigate the stability of the main movement (an auto-balancing mode) for the cases of the absence and presence of resistance forces in the system. It was established that in the absence of resistance forces in the system: – the rotor has three characteristic rotational speeds, and the first always coincides with the resonance frequency; – auto-balancing occurs when the rotor rotates at speeds between the first and second ones, and above the third characteristic speed; – the value of the second and third characteristic speeds is significantly influenced by the ratio of weight to the mass of the system; – the second and third characteristic speeds monotonously increase with an increase in the ratio of cargo weight to the mass of the system. Resistance forces significantly affect both the values of the second and third characteristic speeds and the conditions of their existence. Small resistance forces do not change the quality behavior of the system. With high resistance forces, the number of characteristic speeds decreases to one. The paper reports the results applicable to an auto-balancer with many cargoes when it balances the imbalance that equals the balancing capacity of the auto-balancer

scholarly journals Application of M-Matrices for the Study of Mathematical Models of Living Systems

2018 ◽  
Vol 13 (1) ◽  
pp. 208-237 ◽  
Author(s):  
N.V. Pertsev ◽  
B.Yu. Pichugin ◽  
A.N. Pichugina
Keyword(s):  
Differential Equations ◽  
Stability Problem ◽  
Distributed Delay ◽  
Linearization Method ◽  
Living Systems ◽  
High Dimensional ◽  
Linear Differential ◽  
The Stability ◽  
Several Delays ◽  
The Right

Some results are presented of application of M-matrices to the study the stability problem of the equilibriums of differential equations used in models of living systems. The models studied are described by differential equations with several delays, including distributed delay, and by high-dimensional systems of differential equations. To study the stability of the equilibriums the linearization method is used. Emerging systems of linear differential equations have a specific structure of the right-hand parts, which allows to effectively use the properties of M-matrices. As examples, the results of studies of models arising in immunology, epidemiology and ecology are presented.

Sign in / Sign up

Export Citation Format

Share Document