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 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.

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.

Research on Hydro-Pneumatic Balanced Suspension of Multi-Spindled Vehicle

2011 ◽  
Vol 66-68 ◽  
pp. 855-861 ◽  
Author(s):  
Lin Yang ◽  
Jun Wei Zhang ◽  
Si Zhong Chen
Keyword(s):  
Differential Equations ◽  
Lagrange Equation ◽  
Equations Of Motion ◽  
Structural Features ◽  
Pneumatic Suspension

A hydro-pneumatic balanced suspension is proposed based on the structural features of general hydro-pneumatic suspension and balanced suspension. Two types of suspensions mathematics models are built and differential equations of motion are derived with Lagrange-Equation. Performance of the two suspensions is simulated with the software of MATLAB. The results show that ride performance of vehicles is improved using hydro-pneumatic balanced suspension. Therefore, hydro-pneumatic balanced suspension is more suitable for multi-spindled vehicles.

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 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.

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

Modeling and Simulation of Motorcycle Ride Comfort Based on Bump Road

2010 ◽  
Vol 139-141 ◽  
pp. 2643-2647 ◽  
Author(s):  
Dong Mei Yuan ◽  
Xiao Mei Zheng ◽  
Ying Yang
Keyword(s):  
Degrees Of Freedom ◽  
Ride Comfort ◽  
Lagrange Equation ◽  
Equations Of Motion ◽  
Vertical Displacement ◽  
Dynamics Model ◽  
Body Dynamics ◽  
Space Formulation ◽  
Simulation Results ◽  
Multi Body

Through analyzing the motion when motorcycle runs on the bump road, the 5-DOF multi-body dynamics model of motorcycle is developed, the degrees of freedom include vertical displacement of sprung mass, rotation of sprung mass, vertical displacement of driver, and vertical displacement of front and rear suspension under sprung mass. According to Lagrange Equation, the differential equations of motion and state-space formulation are derived. Then bump road is simulated by triangle bump, and input displacement is programmed by MATLAB. With the input of bump road, motorcycle ride comfort is simulated, and the simulation results are verified by experiment results combined with two channels tire-coupling road simulator. It indicates that the simulation results and experiment results match well; the 5-DOF model has guidance for development of motorcycle ride comfort.

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.

Investigation on Nonlinear Dynamic Characteristics of a New Rigid-Flexible Gear Transmission With Wear

2019 ◽  
Vol 141 (5) ◽  
Author(s):  
Zhibo Geng ◽  
Ke Xiao ◽  
Jiaxu Wang ◽  
Junyang Li
Keyword(s):  
Nonlinear Dynamic ◽  
Operating Time ◽  
Transmission System ◽  
Gear Transmission ◽  
Time Varying ◽  
Control Parameters ◽  
Nonlinear Dynamic Model ◽  
Meshing Stiffness ◽  
Gear Backlash ◽  
Gear Transmission System

Abstract At present, the mean value of the meshing stiffness and the gear backlash is a fixed value in the nonlinear dynamic model. In this study, wear is considered in the model of the gear backlash and time-varying stiffness. With the increase of the operating time, the meshing stiffness decreases and the gear backlash increases. A six degrees-of-freedom nonlinear dynamic model of a new rigid-flexible gear pair is established with time-varying stiffness and time-varying gear backlash. The dynamic behaviors of the gear transmission system are studied through bifurcation diagrams with the operating time as control parameters. Then, the dynamic characteristics of the gear transmission system are analyzed using excitation frequency as control parameters at four operating time points. The bifurcation diagrams, Poincaré maps, fast Fourier transform (FFT) spectra, phase diagrams, and time series are used to investigate the state of motion. The results can provide a reference for the gear transmission system with wear.

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