A Simplified Nonlinear Transient Analysis Method for Gas Bearings

2012 ◽  
Vol 134 (1) ◽  
Author(s):  
M. Amine Hassini ◽  
Mihai Arghir
Keyword(s):  
Thin Film ◽  
Equations Of Motion ◽  
Time Step ◽  
Gas Bearings ◽  
Rational Function Approximation ◽  
Nonlinear Transient ◽  
Reaction Forces ◽  
Small Perturbation Method ◽  
Nonlinear Aerodynamic ◽  
Good Agreement

The traditional small perturbation method is successfully used for linear dynamic analysis of gas bearings but excludes any nonlinear study. Investigating large displacements requires the evaluation of the nonlinear aerodynamic forces in the thin film. To avoid solving the unsteady compressible thin film fluid equations, we propose a method based on the use of frequency dependent dynamic coefficients and on the rational function approximation of the resulting impedances. Calculating impedances for several eccentricities enables mapping the full dynamic behavior of the bearing. A set of ordinary differential equations is then developed by using the inverse of Laplace transform. The equations of motion of the rotor are subsequently solved numerically with local linearization at each time step. The numerical results obtained by using impedances are in good agreement with the reaction forces obtained by solving the full nonlinear transient Reynolds equation.

Torsional Vibrations of Synchronous Motor Driven Trains Using p-Method

1995 ◽  
Vol 117 (1) ◽  
pp. 152-160 ◽  
Author(s):  
W. J. Chen
Keyword(s):  
Dynamic Equilibrium ◽  
Dynamic Stiffness ◽  
Equations Of Motion ◽  
Synchronous Motor ◽  
Time Step ◽  
Integration Algorithm ◽  
Damping Matrix ◽  
Nonlinear Transient ◽  
Direct Numerical Integration ◽  
Stiffness And Damping

Synchronous motors produce damaging oscillating torques during startup. If the system is not properly analyzed and designed, the torsional excitation can be very destructive. This paper presents a systematic approach to the dynamic analysis of synchronous motor driven rotating machinery. The p-version of the finite element method is used in the formulation of the equations of motion which provides a great deal of simplicity in the modeling process. The convergence is achieved by increasing the polynomial order of the basis functions of the geometric elements. The system damping matrix can be constructed from the element level or can be calculated by specifying the critical damping factors for a given number of modes of interest. A modified Newmark integration method is employed in the nonlinear transient response calculation. The nonlinearity of the flexible resilient couplings can be easily implemented into this direct numerical integration algorithm. The dynamic stiffness and damping of the resilient couplings are updated at each time step to ensure the dynamic equilibrium. Two examples have been employed to illustrate the validity of the proposed algorithm. The effectiveness, accuracy, and simplicity of the use of p-method on the torsional vibration of synchronous motor driven trains are demonstrated in this paper.

Experimental/Analytical Analysis of Human Locomotion Using Bondgraphs

2003 ◽  
Vol 125 (4) ◽  
pp. 490-498 ◽  
Author(s):  
Cristian Pop ◽  
Amir Khajepour ◽  
Jan P. Huissoon ◽  
Aftab E. Patla
Keyword(s):  
Equations Of Motion ◽  
Force Plate ◽  
Human Locomotion ◽  
Ground Reaction Forces ◽  
Phase Detection ◽  
Body Structure ◽  
Double Support ◽  
Model Predictions ◽  
Reaction Forces ◽  
Good Agreement

A new vectorial bondgraph approach for modeling and simulation of human locomotion is introduced. The vectorial bondgraph is applied to an eight-segment gait model to derive the equations of motion for studying ground reaction forces (GRFs) and centers of pressure (COPs) in single and double support phases of ground and treadmill walking. A phase detection technique and accompanying transition equation is proposed with which the GRFs and COPs may be calculated for the transitions from double-to-single and single-to-double support phases. Good agreement is found between model predictions and experimental data obtained from force plate measurements. The bondgraph modeling approach is shown to be both informative and adaptable, in the sense that the model resembles the human body structure, and that modeled body segments can be easily added or removed.

Thermal interactions in rolling bearings

2019 ◽  
Vol 234 (8) ◽  
pp. 1233-1253 ◽  
Author(s):  
Pradeep K Gupta ◽  
Jared I Taketa ◽  
Craig M Price
Keyword(s):  
Shear Stress ◽  
Heat Generation ◽  
Equations Of Motion ◽  
Rolling Bearing ◽  
Temperature Fields ◽  
Time Step ◽  
Rolling Bearings ◽  
Bearing Materials ◽  
Very High ◽  
Good Agreement

Rolling bearing dynamics model, based on classical differential equations of motion of bearing elements coupled with thermal interactions, is presented. While churning and drag effects are based on classical laminar and turbulent flow theories, independently measured lubricant rheology, including shear dependence of viscosity, is used to model lubricant traction. The energy equation is integrated through the lubricant film to first compute Newtonian traction with thermal effects. Viscosity dependence on shear stress is then applied to model “shear-thinning” effects. At very high contact pressure and very low slide-to-roll ratios material creep effects, where the behavior of lubricated and dry contacts is similar, are implemented, while a shear stress limit is applied at very high slide-to-roll ratios. Traction predictions for a typical contact in a traction rig show good agreement with experimental traction data. Transient heat generations are time-averaged over thermal time step to compute time-varying temperature fields in the bearing, which alter properties of bearing materials, operating bearing geometry, and rheology of the lubricant. As the transient solutions converge to stable operating temperatures, bearing heat generation approaches the expected steady-state value. Heat generation predictions for both ball and rolling bearings are in good agreement with measured experimental data.

scholarly journals Horizon radiation reaction forces

2020 ◽  
Vol 2020 (10) ◽  
Author(s):  
Walter D. Goldberger ◽  
Ira Z. Rothstein
Keyword(s):  
Black Hole ◽  
Equations Of Motion ◽  
Finite Size ◽  
Radiation Reaction ◽  
Effective Field ◽  
Compact Objects ◽  
Finite Size Effect ◽  
Binary Black Holes ◽  
Dissipative Effects ◽  
Reaction Forces

Abstract Using Effective Field Theory (EFT) methods, we compute the effects of horizon dissipation on the gravitational interactions of relativistic binary black hole systems. We assume that the dynamics is perturbative, i.e it admits an expansion in powers of Newton’s constant (post-Minkowskian, or PM, approximation). As applications, we compute corrections to the scattering angle in a black hole collision due to dissipative effects to leading PM order, as well as the post-Newtonian (PN) corrections to the equations of motion of binary black holes in non-relativistic orbits, which represents the leading order finite size effect in the equations of motion. The methods developed here are also applicable to the case of more general compact objects, eg. neutron stars, where the magnitude of the dissipative effects depends on non-gravitational physics (e.g, the equation of state for nuclear matter).

scholarly journals Adaptive Shell Spherical Reflector Actuated with PVDF-TrFE Thin Film Strain Actuators

Actuators ◽  
2020 ◽  
Vol 10 (1) ◽  
pp. 7
Author(s):  
Kainan Wang ◽  
Thomas Godfroid ◽  
Damien Robert ◽  
André Preumont
Keyword(s):  
Thin Film ◽  
Active Material ◽  
Quadratic Model ◽  
Influence Functions ◽  
Regularization Technique ◽  
Thin Polymer ◽  
Optical Modes ◽  
Electrostrictive Material ◽  
Ill Conditioning ◽  
Good Agreement

This paper discusses the design and manufacturing of a thin polymer spherical adaptive reflector of diameter D=200 mm, controlled by an array of 25 independent electrodes arranged in a keystone configuration actuating a thin film of PVDF-TrFE in d31-mode. The 5 μm layer of electrostrictive material is spray-coated. The results of the present study confirm that the active material can be modelled by a unidirectional quadratic model and that excellent properties can be achieved if the material is properly annealed. The experimental influence functions of the control electrodes are determined by a quasi-static harmonic technique; they are in good agreement with the numerical simulations and their better circular symmetry indicates a clear improvement in the manufacturing process, as compared to a previous study. The low order optical modes can be reconstructed by combining the 25 influence functions; a regularization technique is used to alleviate the ill-conditioning of the Jacobian and allow to approximate the optical modes with reasonable voltages.

Analytical Modelling of the Dynamics of the Four-Bar Mechanism: A Comparison of Some Methods

2018 ◽  
Author(s):  
J. P. Meijaard ◽  
V. van der Wijk
Keyword(s):  
Virtual Work ◽  
Equations Of Motion ◽  
Differential Algebraic Equations ◽  
Force Balance ◽  
Dynamic Force ◽  
Algebraic Equations ◽  
Principle Of Virtual Work ◽  
Principal Vector ◽  
Reaction Forces ◽  
Principal Points

Some thoughts about different ways of formulating the equations of motion of a four-bar mechanism are communicated. Four analytic methods to derive the equations of motion are compared. In the first method, Lagrange’s equations in the traditional form are used, and in a second method, the principle of virtual work is used, which leads to equivalent equations. In the third method, the loop is opened, principal points and a principal vector linkage are introduced, and the equations are formulated in terms of these principal vectors, which leads, with the introduced reaction forces, to a system of differential-algebraic equations. In the fourth method, equivalent masses are introduced, which leads to a simpler system of principal points and principal vectors. By considering the links as pseudorigid bodies that can have a uniform planar dilatation, a compact form of the equations of motion is obtained. The conditions for dynamic force balance become almost trivial. Also the equations for the resulting reaction moment are considered for all four methods.

scholarly journals A high-fidelity realization of the Euclid code comparison N-body simulation with Abacus

2019 ◽  
Vol 485 (3) ◽  
pp. 3370-3377 ◽  
Author(s):  
Lehman H Garrison ◽  
Daniel J Eisenstein ◽  
Philip A Pinto
Keyword(s):  
Power Spectrum ◽  
High Performance ◽  
High Fidelity ◽  
Step Size ◽  
Time Step ◽  
Code Comparison ◽  
Force Accuracy ◽  
Integration Errors ◽  
Better Than ◽  
Good Agreement

Abstract We present a high-fidelity realization of the cosmological N-body simulation from the Schneider et al. code comparison project. The simulation was performed with our AbacusN-body code, which offers high-force accuracy, high performance, and minimal particle integration errors. The simulation consists of 20483 particles in a $500\ h^{-1}\, \mathrm{Mpc}$ box for a particle mass of $1.2\times 10^9\ h^{-1}\, \mathrm{M}_\odot$ with $10\ h^{-1}\, \mathrm{kpc}$ spline softening. Abacus executed 1052 global time-steps to z = 0 in 107 h on one dual-Xeon, dual-GPU node, for a mean rate of 23 million particles per second per step. We find Abacus is in good agreement with Ramses and Pkdgrav3 and less so with Gadget3. We validate our choice of time-step by halving the step size and find sub-percent differences in the power spectrum and 2PCF at nearly all measured scales, with ${\lt }0.3{{\ \rm per\ cent}}$ errors at $k\lt 10\ \mathrm{Mpc}^{-1}\, h$. On large scales, Abacus reproduces linear theory better than 0.01 per cent. Simulation snapshots are available at http://nbody.rc.fas.harvard.edu/public/S2016.

Structural Analysis of an Offshore Jacket Model Under Seismic Excitation

2008 ◽  
Author(s):  
Helder J. D. Correia ◽  
Anto´nio C. Mendes ◽  
Carlos A. F. S. Oliveira
Keyword(s):  
Point Of View ◽  
Shaking Table ◽  
Seismic Excitation ◽  
Model Structure ◽  
Structural Members ◽  
Jacket Structures ◽  
Reaction Forces ◽  
Nodal Displacements ◽  
Good Agreement

In the present work the action of earthquakes upon offshore jacket structures is analysed by means of ADINA software. Our case-study refers to an existing model structure, previously constructed at the Laboratory of Fluid Mechanics of UBI, which has been analysed from the hydrodynamic point of view — Mendes et al. [1, 2]. The seismic excitation will be imposed at the base of this model structure, with frequencies and amplitudes corresponding to actual earthquake conditions transposed to the model scale of 1:45. The FEM software is utilised to calculate the natural frequencies of the model and to obtain stresses at selected members, as well as their nodal displacements. Our purpose is to quantify maximum stresses occurring in critical structural members and to verify the survivability criterion. The predictions of the numerical model, in terms of the reaction forces at the base and acceleration at the top of the structure, are then correlated with the experimental measurements performed when the model structure is excited in an especially designed shaking table (Correia [3]), revealing a good agreement between both results.

Vibration analysis of multiple degrees of freedom mechanical system with dry friction

2012 ◽  
Vol 227 (7) ◽  
pp. 1505-1514 ◽  
Author(s):  
SD Yu ◽  
BC Wen
Keyword(s):  
Mechanical System ◽  
Dry Friction ◽  
Degrees Of Freedom ◽  
Simple Procedure ◽  
Mathematical Problem ◽  
Equations Of Motion ◽  
Inequality Constraints ◽  
Integration Scheme ◽  
Time Step ◽  
Multiple Degrees Of Freedom

This article presents a simple procedure for predicting time-domain vibrational behaviors of a multiple degrees of freedom mechanical system with dry friction. The system equations of motion are discretized by means of the implicit Bozzak–Newmark integration scheme. At each time step, the discontinuous frictional force problem involving both the equality and inequality constraints is successfully reduced to a quadratic mathematical problem or the linear complementary problem with the introduction of non-negative and complementary variable pairs (supremum velocities and slack forces). The so-obtained complementary equations in the complementary pairs can be solved efficiently using the Lemke algorithm. Results for several single degree of freedom and multiple degrees of freedom problems with one-dimensional frictional constraints and the classical Coulomb frictional model are obtained using the proposed procedure and compared with those obtained using other approaches. The proposed procedure is found to be accurate, efficient, and robust in solving non-smooth vibration problems of multiple degrees of freedom systems with dry friction. The proposed procedure can also be applied to systems with two-dimensional frictional constraints and more sophisticated frictional models.

A Strategy to Accelerate the Numerical Integration in the Dynamic Simulation of Multibody Systems

1997 ◽  
Author(s):  
Jesús Cardenal ◽  
Javier Cuadrado ◽  
Eduardo Bayo
Keyword(s):  
Multibody Systems ◽  
Equations Of Motion ◽  
Stiff Systems ◽  
Step Size ◽  
Step Method ◽  
Integral Of Motion ◽  
Time Step ◽  
Time Step Size ◽  
Variable Time ◽  
Numerical Integrator

Abstract This paper presents a multi-index variable time step method for the integration of the equations of motion of constrained multibody systems in descriptor form. The basis of the method is the augmented Lagrangian formulation with projections in index-3 and index-1. The method takes advantage of the better performance of the index-3 formulation for large time steps and of the stability of the index-1 for low time steps, and automatically switches from one method to the other depending on the required accuracy and values of the time step. The variable time stepping is accomplished through the use of an integral of motion, which in the case of conservative systems becomes the total energy. The error introduced by the numerical integrator in the integral of motion during consecutive time steps provides a good measure of the local integration error, and permits a simple and reliable strategy for varying the time step. Overall, the method is efficient and powerful; it is suitable for stiff and non-stiff systems, robust for all time step sizes, and it works for singular configurations, redundant constraints and topology changes. Also, the constraints in positions, velocities and accelerations are satisfied during the simulation process. The method is robust in the sense that becomes more accurate as the time step size decreases.

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