A General Superelement Model on a Moving Reference Frame for Planar Multibody System Dynamics

1991 ◽  
Vol 113 (1) ◽  
pp. 43-49 ◽  
Author(s):  
O. P. Agrawal ◽  
R. Kumar
Keyword(s):  
Differential Equations ◽  
Reference Frame ◽  
Inverse Dynamics ◽  
Equations Of Motion ◽  
Mathematical Formulation ◽  
Computational Time ◽  
Single Degree Of Freedom ◽  
Generalized Force ◽  
Time Required ◽  
Energy Relations

This paper presents a mathematical formulation for a superelement on a moving reference frame. A group of constrained elements, forming a single degree-of-freedom with respect to some reference frame, is represented by one element. The configuration of a typical superelement is determined with respect to the reference frame using one parameter only. This reduces the number of generalized coordinates needed to define the configuration of the entire system. Energy relations and generalized force expressions are developed in terms of the reference coordinates and superelement parameters. An inverse dynamics approach is used to obtain the functions associated with a superelement. The differential equations of motion of a system consisting of one reference frame and several superelements are derived. Constraints internal to the superelements do not appear in the resulting differential equations. This reduces the dimensions of the problem and the computational time required to solutions. Two examples are considered to demonstrate the feasibility and the efficiency of the method.

Drop-Induced Shock Mitigation Using Adaptive Magnetorheological Energy Absorbers Incorporating a Time Lag

2015 ◽  
Vol 137 (1) ◽  
Author(s):  
Young-Tai Choi ◽  
Norman M. Wereley
Keyword(s):  
Equations Of Motion ◽  
Time Lag ◽  
Soft Landing ◽  
Single Degree Of Freedom ◽  
Bingham Number ◽  
Shock Mitigation ◽  
Payload Mass ◽  
Energy Absorbers ◽  
Time Required ◽  
The Impact

This study addresses the nondimensional analysis of drop-induced shock mitigated using magnetorheological energy absorbers (MREAs) incorporating a time lag. This time lag arises from two sources: (1) the time required to generate magnetic field in the electromagnet once current has been applied and (2) the time required for the particles in the magnetorheological fluid to form chains. To this end, the governing equations of motion for a single degree-of-freedom (SDOF) system using an MREA with a time lag were derived. Based on these equations, nondimensional stroke, velocity, and acceleration of the payload were derived, where the MREA with a time lag was used to control payload deceleration after the impact. It is established that there exists an optimal Bingham number that allows the payload mass to achieve a soft landing, that is, the payload comes to rest after utilizing the available stroke of the MREA. Finally, the shock mitigation performance when using this optimal Bingham number control strategy is analyzed, and the effects of time lag are quantified.

scholarly journals Partial Acoustic Filtering Applied to the Equations of Compressible Flow

1991 ◽  
Vol 113 (4) ◽  
pp. 709-712
Author(s):  
J. R. Torczynski
Keyword(s):  
Acoustic Wave ◽  
Equations Of Motion ◽  
Density Variation ◽  
Computational Time ◽  
Pressure Waves ◽  
Volumetric Heating ◽  
Small Width ◽  
Acoustic Filtering ◽  
Time Required ◽  
Laser Cell

Gas contained in a rectangular laser cell of large length and small width is subjected to large, transient, spatially nonuniform, volumetric heating when pumped. The heating time scale is much longer than the time required for an acoustic wave to traverse the width but can be comparable to the time required for an acoustic wave to traverse the length. Approximate equations describing the motion are derived by applying partial acoustic filtering to the equations of motion: pressure waves traversing the width are removed while pressure waves traversing the length are retained. For a simplified one-dimensional example, a significant density variation is found across the width of the laser cell; moreover, this density variation is in good agreement with a numerical solution of the unapproximated gas dynamic equations although the latter requires two orders of magnitude more computational time

The Study of a Single Degree of Freedom System with Time Dependent Parameters

1967 ◽  
Vol 71 (678) ◽  
pp. 439-440 ◽  
Author(s):  
B. V. Dasarathy ◽  
P. Srinivasan
Keyword(s):  
Differential Equation ◽  
General Solution ◽  
Differential Equations ◽  
Equations Of Motion ◽  
Degree Of Freedom ◽  
Time Dependent ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Stochastic Nature ◽  
Restoring Forces

The equations of motion of many mechanical systems reduce to ordinary differential equations with time dependent parameters. A single degree of freedom system, with the damping and restoring forces varying with time according to the law … m (t+k)nhas been studied using the WKBJ approximation. The general solution of the differential equation can be used to arrive at the response of such time dependent systems, subjected to excitations which are specified to be of either deterministic or stochastic nature.

Normal form method in search for periodic solutions of ordinary differential equations. Case of the fourth order

2018 ◽  
pp. 24-34
Author(s):  
V. F. Edneral ◽  
O. D. Timofeevskaya
Keyword(s):  
Normal Form ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Periodic Solutions ◽  
Fourth Order ◽  
Numerical Solutions ◽  
Numerical Calculations ◽  
Computational Time ◽  
Resonant Normal Form ◽  
Normal Form Method

Introduction:The method of resonant normal form is based on reducing a system of nonlinear ordinary differential equations to a simpler form, easier to explore. Moreover, for a number of autonomous nonlinear problems, it is possible to obtain explicit formulas which approximate numerical calculations of families of their periodic solutions. Replacing numerical calculations with their precalculated formulas leads to significant savings in computational time. Similar calculations were made earlier, but their accuracy was insufficient, and their complexity was very high.Purpose:Application of the resonant normal form method and a software package developed for these purposes to fourth-order systems in order to increase the calculation speed.Results:It has been shown that with the help of a single algorithm it is possible to study equations of high orders (4th and higher). Comparing the tabulation of the obtained formulas with the numerical solutions of the corresponding equations shows good quantitative agreement. Moreover, the speed of calculation by prepared approximating formulas is orders of magnitude greater than the numerical calculation speed. The obtained approximations can also be successfully applied to unstable solutions. For example, in the Henon — Heyles system, periodic solutions are surrounded by chaotic solutions and, when numerically integrated, the algorithms are often unstable on them.Practical relevance:The developed approach can be used in the simulation of physical and biological systems.

scholarly journals Green–Haar wavelets method for generalized fractional differential equations

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Mujeeb ur Rehman ◽  
Dumitru Baleanu ◽  
Jehad Alzabut ◽  
Muhammad Ismail ◽  
Umer Saeed
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Haar Wavelet ◽  
Haar Wavelets ◽  
Computational Time ◽  
Operational Matrices ◽  
Numerical Techniques ◽  
Fractional Differential ◽  
Fractional Boundary Value Problems ◽  
Better Than

Abstract The objective of this paper is to present two numerical techniques for solving generalized fractional differential equations. We develop Haar wavelets operational matrices to approximate the solution of generalized Caputo–Katugampola fractional differential equations. Moreover, we introduce Green–Haar approach for a family of generalized fractional boundary value problems and compare the method with the classical Haar wavelets technique. In the context of error analysis, an upper bound for error is established to show the convergence of the method. Results of numerical experiments have been documented in a tabular and graphical format to elaborate the accuracy and efficiency of addressed methods. Further, we conclude that accuracy-wise Green–Haar approach is better than the conventional Haar wavelets approach as it takes less computational time compared to the Haar wavelet method.

Disk Position Nonlinearity Effects on the Chaotic Behavior of Rotating Flexible Shaft-Disk Systems

2012 ◽  
Vol 28 (3) ◽  
pp. 513-522 ◽  
Author(s):  
H. M. Khanlo ◽  
M. Ghayour ◽  
S. Ziaei-Rad
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Dynamic Behavior ◽  
Chaotic Behavior ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Disk System ◽  
Partial Differential ◽  
Rayleigh Beam ◽  
Flexible Shaft

AbstractThis study investigates the effects of disk position nonlinearities on the nonlinear dynamic behavior of a rotating flexible shaft-disk system. Displacement of the disk on the shaft causes certain nonlinear terms which appears in the equations of motion, which can in turn affect the dynamic behavior of the system. The system is modeled as a continuous shaft with a rigid disk in different locations. Also, the disk gyroscopic moment is considered. The partial differential equations of motion are extracted under the Rayleigh beam theory. The assumed modes method is used to discretize partial differential equations and the resulting equations are solved via numerical methods. The analytical methods used in this work are inclusive of time series, phase plane portrait, power spectrum, Poincaré map, bifurcation diagrams, and Lyapunov exponents. The effect of disk nonlinearities is studied for some disk positions. The results confirm that when the disk is located at mid-span of the shaft, only the regular motion (period one) is observed. However, periodic, sub-harmonic, quasi-periodic, and chaotic states can be observed for situations in which the disk is located at places other than the middle of the shaft. The results show nonlinear effects are negligible in some cases.

Dynamics of Rolling-Element Bearings—Part I: Cylindrical Roller Bearing Analysis

1979 ◽  
Vol 101 (3) ◽  
pp. 293-302 ◽  
Author(s):  
P. K. Gupta
Keyword(s):  
Differential Equations ◽  
Normal Force ◽  
Equations Of Motion ◽  
Roller Bearing ◽  
Rolling Element Bearings ◽  
Analytical Formulation ◽  
Rolling Element ◽  
Cylindrical Roller Bearing ◽  
Cylindrical Roller ◽  
Bearing Analysis

An analytical formulation for the roller motion in a cylindrical roller bearing is presented in terms of the classical differential equations of motion. Roller-race interaction is analyzed in detail and the resulting normal force and moment vectors are determined. Elastohydrodynamic traction models are considered in determining the roller-race tractive forces and moments. Formulation for the roller end and race flange interaction during skewing of the roller is also considered. Roller-cage interactions are assumed to be either hydrodynamic or fully metallic. Simple relationships are used to determine the churning and drag losses.

Mulit-Pulse Orbits and Chaotic Motion of Composite Laminated Plates

2008 ◽  
Author(s):  
Xiangying Guo ◽  
Wei Zhang ◽  
Ming-Hui Yao
Keyword(s):  
Numerical Simulation ◽  
Normal Form ◽  
Differential Equations ◽  
Thin Plate ◽  
Equations Of Motion ◽  
Laminated Plates ◽  
Phase Method ◽  
Rectangular Thin Plate ◽  
Partial Differential ◽  
Chaotic Motions

This paper presents an analysis on the nonlinear dynamics and multi-pulse chaotic motions of a simply-supported symmetric cross-ply composite laminated rectangular thin plate with the parametric and forcing excitations. Firstly, based on the Reddy’s three-order shear deformation plate theory and the model of the von Karman type geometric nonlinearity, the nonlinear governing partial differential equations of motion for the composite laminated rectangular thin plate are derived by using the Hamilton’s principle. Then, using the second-order Galerkin discretization approach, the partial differential governing equations of motion are transformed to nonlinear ordinary differential equations. The case of the primary parametric resonance and 1:1 internal resonance is considered. Four-dimensional averaged equation is obtained by using the method of multiple scales. From the averaged equation obtained here, the theory of normal form is used to give the explicit expressions of normal form. Based on normal form, the energy phase method is utilized to analyze the global bifurcations and multi-pulse chaotic dynamics of the composite laminated rectangular thin plate. The results obtained above illustrate the existence of the chaos for the Smale horseshoe sense in a parametrical and forcing excited composite laminated thin plate. The chaotic motions of the composite laminated rectangular thin plate are also found by using numerical simulation. The results of numerical simulation also indicate that there exist different shapes of the multi-pulse chaotic motions for the composite laminated rectangular thin plate.

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