Reduction in the order of systems of nonlinear differential equations of motion by the elimination of rapidly decaying solutions

1975 ◽  
Vol 11 (8) ◽  
pp. 867-872
Author(s):  
V. A. Lazaryan ◽  
L. A. Dlugach ◽  
I. A. Zil'berman ◽  
M. L. Korotenko
Keyword(s):  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Equations Of Motion

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.

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 Dynamical analysis of a 2-degrees of freedom spatial pendulum

2018 ◽  
Vol 184 ◽  
pp. 01003 ◽  
Author(s):  
Stelian Alaci ◽  
Florina-Carmen Ciornei ◽  
Sorinel-Toderas Siretean ◽  
Mariana-Catalina Ciornei ◽  
Gabriel Andrei Ţibu
Keyword(s):  
Differential Equations ◽  
Degrees Of Freedom ◽  
Nonlinear Differential Equations ◽  
Equations Of Motion ◽  
Dynamical Analysis ◽  
Analysis Software ◽  
Lagrange Equations ◽  
Moments Of Inertia ◽  
Dynamical Simulation ◽  
Principal Moments Of Inertia

A spatial pendulum with the vertical immobile axis and horizontal mobile axis is studied and the differential equations of motion are obtained applying the method of Lagrange equations. The equations of motion were obtained for the general case; the only simplifying hypothesis consists in neglecting the principal moments of inertia about the axes normal to the oscillation axes. The system of nonlinear differential equations was numerically integrated. The correctness of the obtained solutions was corroborated to the dynamical simulation of the motion via dynamical analysis software. The perfect concordance between the two solutions proves the rightness of the equations obtained.

Modeling, Simulation, and Modal Analysis of a Hydraulic Valve Lifter With Oil Compressibility Effects

1991 ◽  
Vol 113 (1) ◽  
pp. 46-54 ◽  
Author(s):  
C. T. Hatch ◽  
A. P. Pisano
Keyword(s):  
Differential Equations ◽  
Modal Analysis ◽  
Nonlinear Differential Equations ◽  
Equations Of Motion ◽  
Check Valve ◽  
Third Order ◽  
Bulk Oil ◽  
Special Cases ◽  
Linear Differential ◽  
Hydraulic Valve

A two-degree-of-freedom (2-DOF), analytical model of a hydraulic valve lifter is derived. Special features of the model include the effects of bulk oil compressibility, multimode behavior due to plunger check valve modeling, and provision for the inclusion of third and fourth body displacements to aid in the use of the model in extended, multi-DOF systems. It is shown that motion of the lifter plunger and body must satisfy a coupled system of third-order, nonlinear differential equations of motion. It is also shown that the special cases of zero oil compressibility and/or 1-DOF motion of lifter plunger can be obtained from the general third-order equations. For the case of zero oil compressibility, using Newtonian fluid assumptions, the equations of motion are shown to reduce to a system of second-order, linear differential equations. The differential equations are numerically integrated in five scenarios designed to test various aspects of the model. A modal analysis of the 2-DOF, compressible model with an external contact spring is performed and is shown to be in excellent agreement with simulation results.

scholarly journals Differential equations of motion of a material point in the perpendicular plane to the plane of the gravitating disk

2021 ◽  
Vol 24 (3) ◽  
pp. 1307
Author(s):  
Zhenisgul Rakhmetullina ◽  
Indira Uvaliyeva ◽  
Farida Amenova
Keyword(s):  
Analytical Solution ◽  
Differential Equations ◽  
Nonlinear Equations ◽  
Nonlinear Differential Equations ◽  
Equations Of Motion ◽  
Material Point ◽  
Point System ◽  
Disk Material ◽  
Analytical Review ◽  
Celestial Bodies

This paper presents an analytical solution of the differential equations of motion of a material point in the plane perpendicular to the plane of the gravitating disk. The differential equations of the problem under study and the applied Gilden's method are described in the works of A. Poincaré. Differential equations refer to nonlinear equations. The analysis of methods for solving nonlinear differential equations was carried out. The methodology of applying the Gilden method to the solution of the differential equations under consideration can be applied in studies of the problem of the motion of celestial bodies in the “disk-material point” system in perpendicular planes. To identify the various properties of the gravitating disk, an analytical review of the state of the problem of the motion of a material point in the field of a gravitating disk is carried out. Summing up the presented review on the problem under study, a conclusion is made. The substantive formulation of the problem is described, which is formulated as follows: the study of the influence of disk-shaped bodies on the motion of a material point and methods for their solution.

Nonlinear Dynamic Model and Equations of RV Transmission System

2012 ◽  
Vol 510 ◽  
pp. 536-540 ◽  
Author(s):  
Li Jun Shan ◽  
Xue Fang ◽  
Wei Dong He
Keyword(s):  
Differential Equations ◽  
Lagrange Equation ◽  
Nonlinear Differential Equations ◽  
Equations Of Motion ◽  
Transmission System ◽  
Dynamics Model ◽  
Nonlinear Dynamic Model ◽  
Restoring Force ◽  
Meshing Stiffness ◽  
Nonlinear Dynamics Model

The nonlinear dynamics model of gearing system is developed based on RV transmission system. The influence of the nonlinear factors as time-varying meshing stiffness, backlash of the gear pairs and errors is considered. By means of the Lagrange equation the multi-degree-of-freedom differential equations of motion are derived. The differential equations are very hard to solve for which are characterized by positive semi-definition, time-variation and backlash-type nonlinearity. And linear and nonlinear restoring force are coexist in the equations. In order to solve easily, the differential equations are transformed to identical dimensionless nonlinear differential equations in matrix form. The establishment of the nonlinear differential equations laid a foundation for The Solution of differential equations and the analysis of the nonlinearity characteristics.

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.

scholarly journals Analysis of the brachistochronic motion of a variable mass nonholonomic mechanical system

2016 ◽  
Vol 43 (1) ◽  
pp. 19-32 ◽  
Author(s):  
Bojan Jeremic ◽  
Radoslav Radulovic ◽  
Aleksandar Obradovic
Keyword(s):  
Differential Equations ◽  
Mechanical System ◽  
Nonlinear Differential Equations ◽  
Equations Of Motion ◽  
Variable Mass ◽  
Nonholonomic Constraints ◽  
Initial Position ◽  
Mass Particle ◽  
Control Forces ◽  
And Control

The paper considers the brachistochronic motion of a variable mass nonholonomic mechanical system [3] in a horizontal plane, between two specified positions. Variable mass particles are interconnected by a lightweight mechanism of the ?pitchfork? type. The law of the time-rate of mass variation of the particles, as well as relative velocities of the expelled particles, as a function of time, are known. Differential equations of motion, where the reactions of nonholonomic constraints and control forces figure, are created based on the general theorems of dynamics of a variable mass mechanical system [5]. The formulated brachistochrone problem, with adequately chosen quantities of state, is solved, in this case, as the simplest task of optimal control by applying Pontryagin?s maximum principle [1]. A corresponding two-point boundary value problem (TPBVP) of the system of ordinary nonlinear differential equations is obtained, which, in a general case, has to be numerically solved [2]. On the basis of thus obtained brachistochronic motion, the active control forces, along with the reactions of nonholonomic constraints, are determined. The analysis of the brachistochronic motion for different values of the initial position of a variable mass particle B is presented. Also, the interval of values of the initial position of a variable mass particle B, for which there are the TPBVP solutions, is determined.

scholarly journals Bertolino-Baksa stability at nonlinear vibrations of motor vehicles

2017 ◽  
Vol 44 (2) ◽  
pp. 271-291 ◽  
Author(s):  
Ljudmila Kudrjavceva ◽  
Milan Micunovic ◽  
Danijela Miloradovic ◽  
Aleksandar Obradovic
Keyword(s):  
Differential Equations ◽  
Degrees Of Freedom ◽  
Nonlinear Differential Equations ◽  
Equations Of Motion ◽  
Motor Vehicles ◽  
Numerical Code ◽  
Parameter Method ◽  
Spatial Stability ◽  
Linear Vibrations ◽  
Linear Differential

Research of vehicle response to road roughness is particularly important when solving problems related to dynamic vehicle stability. In this paper, unevenness of roads is considered as the source of non-linear vibrations of motor vehicles. The vehicle is represented by an equivalent spatial model with seven degrees of freedom. In addition to solving the response by simulating it within a numerical code, quasi-linearization of nonlinear differential equations of motion is carried out. Solutions of quasi-linear differential equations of forced vibrations are determined using the small parameter method and are indispensable for the study of spatial stability of the vehicle. An optimal stabilization for a simplified two-dimensional model was performed. Spatial stability and internal resonance are considered briefly.

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