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

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.

Download Full-text

Dynamics of Gaussian spikes on Gaussian laser beam in relativistic plasma

Laser and Particle Beams ◽

10.1017/s0263034609990309 ◽

2009 ◽

Vol 27 (4) ◽

pp. 587-593 ◽

Cited By ~ 9

Author(s):

A. Singh ◽

M. Aggarwal ◽

T.S. Gill

Keyword(s):

Differential Equations ◽

Laser Beam ◽

Nonlinear Differential Equations ◽

Beam Width ◽

Main Beam ◽

Gaussian Laser Beam ◽

Runge Kutta Method ◽

Self Focusing ◽

Gaussian Perturbation ◽

Paraxial Ray

AbstractIn the present paper, we have investigated the growth of a Gaussian perturbation superimposed on a Gaussian laser beam. The nonlinearity we have considered is of relativistic type. We have setup the nonlinear differential equations for beam width parameter of the main beam, growth and width of the laser spike by using the WKB and paraxial ray approximation. These are coupled ordinary differential equations and therefore these are simultaneously solved numerically using the Runge Kutta method. It has been observed from the analysis that self-focusing/defocusing of the main beam and the spike determine the growth dynamic of the spike.

Download Full-text

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

Journal of Engineering for Power ◽

10.1115/1.3446339 ◽

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.

Download Full-text

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

Mathematical Problems in Engineering ◽

10.1155/2013/149046 ◽

2013 ◽

Vol 2013 ◽

pp. 1-12 ◽

Cited By ~ 1

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.

Download Full-text

A New Dynamic Model of a Two-Wheeled Two-Flexible-Beam Inverted Pendulum Robot

Volume 7B: Dynamics, Vibration, and Control ◽

10.1115/imece2020-24078 ◽

2020 ◽

Author(s):

Amin Mehrvarz ◽

Mohammad Javad Khodaei ◽

William Clark ◽

Nader Jalili

Keyword(s):

Differential Equations ◽

High Performance ◽

Inverted Pendulum ◽

Equations Of Motion ◽

Flexible Beam ◽

Governing Equations ◽

Natural Modes ◽

Wheeled Inverted Pendulum ◽

Highly Nonlinear ◽

New Type

Abstract Inverted pendulums are traditional dynamic problems. If an inverted pendulum is used in a moving cart, a new type of exciting issues will appear. One of these problems is two-wheeled inverted pendulum systems. Because of their small size, high performance in quick driving, and their stability with controller, researchers and engineers are interested in them. In this paper, a new configuration of one specific robot is modeled, and its dynamic behavior is analyzed. The proposed model can move in two directions, and with a proper controller can keep its stability during the operation. In this robot, two cantilever beams are on the two-wheeled base, and they are excited by voltages to the attached piezoelectric actuators. The mathematical model of this system is obtained using the extended Hamilton’s Principle. The results show that the governing equations of motion are highly nonlinear and contain several coupled partial differential equations (PDEs). In order to extract the natural modes of the beams, the undamped, unforced equations of motion and boundary conditions of the beams are used. If a limited number of modes (N1 and N2) are selected for each beam, the coupled PDEs will be changed to N1 + N2 + 5 ordinary differential equations (ODEs). These complex equations are solved numerically, and the natural frequencies of the system are extracted. The system is then simulated in both lateral and horizontal plane movements. The simulation shows that the governing equations are correct, and the system is ready for designing a proper controller. It should be mentioned that in the future works, the derived equations will be validated experimentally, and a suitable control strategy will be applied to the system to make it automated and more applicable.

Download Full-text

Dynamic Analysis of a Tapered Composite Thin-Walled Rotating Shaft Using the Generalized Differential Quadrature Method

Mathematical Problems in Engineering ◽

10.1155/2020/1695430 ◽

2020 ◽

Vol 2020 ◽

pp. 1-14

Author(s):

Weiyan Zhong ◽

Feng Gao ◽

Yongsheng Ren ◽

Xiaoxiao Wu ◽

Hongcan Ma

Keyword(s):

Differential Equations ◽

Dynamic Model ◽

Differential Quadrature Method ◽

Equations Of Motion ◽

Differential Quadrature ◽

Quadrature Method ◽

Rotating Shaft ◽

Generalized Differential Quadrature Method ◽

Thin Walled ◽

Generalized Differential

A dynamic model of a tapered composite thin-walled rotating shaft is presented. In this model, the transverse shear deformation, rotary inertia, and gyroscopic effects have been incorporated. The equations of motion are derived based on a refined variational asymptotic method (VAM) and Hamilton’s principle. The partial differential equations of motion are reduced to the ordinary differential equations of motion by using the generalized differential quadrature method (GDQM). The validity of the dynamic model is proved by comparing the numerical results with those obtained in the literature and by using ANSYS. The effects of taper ratio, boundary conditions, ply angle, length to mean radius ratios, and mean radius to thickness ratios on the natural frequencies and critical rotating speeds are investigated.

Download Full-text

Dynamical analysis of a 2-degrees of freedom spatial pendulum

MATEC Web of Conferences ◽

10.1051/matecconf/201818401003 ◽

2018 ◽

Vol 184 ◽

pp. 01003 ◽

Cited By ~ 1

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.

Download Full-text

Analysis of the Oscillating Motion of a Solid Body on Vibrating Bearers

Machines ◽

10.3390/machines7030058 ◽

2019 ◽

Vol 7 (3) ◽

pp. 58 ◽

Cited By ~ 1

Author(s):

Bissembayev ◽

Jomartov ◽

Tuleshov ◽

Dikambay

Keyword(s):

Solid Body ◽

Quantitative Methods ◽

Nonlinear Differential Equations ◽

Equations Of Motion ◽

Horizontal Displacement ◽

Oscillatory Process ◽

Highly Nonlinear ◽

Cumulative Curve ◽

The Stability ◽

Oscillating Motion

This article considers the oscillation of a solid body on kinematic foundations, the main elements of which are rolling bearers bounded by high-order surfaces of rotation at horizontal displacement of the foundation. Equations of motion of the vibro-protected body have been obtained. It is ascertained that the obtained equations of motion are highly nonlinear differential equations. Stationary and transitional modes of the oscillatory process of the system have been investigated. It is determined that several stationary regimes of the oscillatory process exist. Equations of motion have been investigated also by quantitative methods. In this paper the cumulative curves in the phase plane are plotted, a qualitative analysis for singular points and a study of them for stability are performed. In the Hayashi plane a cumulative curve of a body protected against vibration forms a closed path which does not tend to the stability of a singular point. This means that the vibration amplitude of a body protected against vibration does not remain constant in a steady state, but changes periodically.

Download Full-text

On the Successive Linearisation Approach to the Flow of Reactive Third-Grade Liquid in a Channel with Isothermal Walls

Mathematical Problems in Engineering ◽

10.1155/2013/635392 ◽

2013 ◽

Vol 2013 ◽

pp. 1-7 ◽

Cited By ~ 2

Author(s):

S. S. Motsa ◽

O. D. Makinde ◽

S. Shateyi

Keyword(s):

Differential Equations ◽

Nonlinear Differential Equations ◽

Boundary Value ◽

Linearization Method ◽

Third Grade ◽

Efficient Technique ◽

Analytical Scheme ◽

Highly Nonlinear ◽

Successive Linearization Method ◽

Modeling Flow

The nonlinear differential equations modeling flow of a reactive third-grade liquid between two parallel isothermal plates is investigated using a novel hybrid of numerical-analytical scheme known as the successive linearization method (SLM). Numerical and graphical results obtained show excellence in agreement with the earlier results reported in the literature. A comparison with numerical results generated using the inbuilt MATLAB boundary value solverbvp4cdemonstrates that the new SLM approach is a very efficient technique for tackling highly nonlinear differential equations of the type discussed in this paper.

Download Full-text