scholarly journals Limit Cycle Predictions of a Nonlinear Journal-Bearing System

1990 ◽  
Vol 112 (2) ◽  
pp. 168-171 ◽  
Author(s):  
M. T. M. Crooijmans ◽  
H. J. H. Brouwers ◽  
D. H. van Campen ◽  
A. de Kraker
Keyword(s):  
Journal Bearing ◽  
Equations Of Motion ◽  
Continuation Method ◽  
Time Discretization ◽  
Linear Stability Theory ◽  
Algebraic Equations ◽  
System Parameters ◽  
Nonlinear Algebraic Equations ◽  
System Equations ◽  
Nonlinear Equations Of Motion

An analysis is presented of the self-excited vibrations of a journal carried in a cylindrical fluid film bearing. Using linear stability theory, the values of the system parameters at the point of loss of stability are determined. These values agree well with those of previous investigators. Solutions of the nonlinear system equations are obtained by time discretization and by an arc-continuation method for solving the obtained nonlinear algebraic equations. In this way periodic solutions of the nonlinear equations of motion are calculated as a function of the system parameters. The behavior of the journal can be explained by the results of these calculations.

Stability Analysis of a Hydrodynamic Journal Bearing With Rotating Herringbone Grooves

2003 ◽  
Vol 125 (2) ◽  
pp. 291-300 ◽  
Author(s):  
G. H. Jang ◽  
J. W. Yoon
Keyword(s):  
Journal Bearing ◽  
Equations Of Motion ◽  
Fourier Series Expansion ◽  
Algebraic Equations ◽  
High Rotational Speed ◽  
Dynamic Coefficients ◽  
Stiffness Coefficients ◽  
Hydrodynamic Journal Bearing ◽  
The Stability ◽  
Herringbone Grooves

This paper presents an analytical method to investigate the stability of a hydrodynamic journal bearing with rotating herringbone grooves. The dynamic coefficients of the hydrodynamic journal bearing are calculated using the FEM and the perturbation method. The linear equations of motion can be represented as a parametrically excited system because the dynamic coefficients have time-varying components due to the rotating grooves, even in the steady state. Their solution can be assumed as a Fourier series expansion so that the equations of motion can be rewritten as simultaneous algebraic equations with respect to the Fourier coefficients. Then, stability can be determined by solving Hill’s infinite determinant of these algebraic equations. The validity of this research is proved by the comparison of the stability chart with the time response of the whirl radius obtained from the equations of motion. This research shows that the instability of the hydrodynamic journal bearing with rotating herringbone grooves increases with increasing eccentricity and with decreasing groove number, which play the major roles in increasing the average and variation of stiffness coefficients, respectively. It also shows that a high rotational speed is another source of instability by increasing the stiffness coefficients without changing the damping coefficients.

Influence of Wheel-Rail Profiles on the Hunting Vibrations of Rail Vehicle Trucks

1985 ◽  
Vol 107 (2) ◽  
pp. 167-174 ◽  
Author(s):  
A. F. D’Souza ◽  
W-J. Tsung
Keyword(s):  
Dynamic Performance ◽  
Equations Of Motion ◽  
Algebraic Equations ◽  
Wheel Wear ◽  
Hunting Behavior ◽  
Describing Functions ◽  
High Acceleration ◽  
Nonlinear Algebraic Equations ◽  
Wheel Profile ◽  
Freight Trucks

The effect of several wheel and rail profiles on the hunting behavior of three-piece North American freight truck is investigated by the method of describing functions. After replacing the nonlinear terms by their equivalent describing functions, the differential equations of motion are converted to a set of coupled nonlinear algebraic equations which are then solved by the Newton-Raphson method. It is shown that the wearing of the rail profile has a significant adverse effect on the dynamic behavior. It greatly lowers the critical speed for the onset of hunting and raises the frequency, thereby causing high acceleration levels. It is also shown that the modified Heumann wheel profile exhibits a superior dynamic performance for freight trucks than the standard new wheel profile used in North America. The effects of wheel wear and loads on hunting are also investigated.

Dynamic analysis of a 2-lobe geometrically imperfect journal bearing system

2016 ◽  
Vol 231 (7) ◽  
pp. 934-950 ◽  
Author(s):  
Dharmendra Jain ◽  
Satish C Sharma
Keyword(s):  
Journal Bearing ◽  
Equations Of Motion ◽  
Constant Flow ◽  
Kutta Method ◽  
Transient Motion ◽  
Nonlinear Transient ◽  
Linear And Nonlinear ◽  
Nonlinear Equations Of Motion ◽  
Bearing System ◽  
Flow Valve

The present study is concerned with the linear and nonlinear transient motion analysis of a 2-lobe geometrically imperfect hybrid journal bearing system compensated with constant flow valve restrictor. The trajectories of journal center motion for a geometrically imperfect rotating journal (barrel, bellmouth and undulation type journal) have been numerically simulated by solving the linear and nonlinear equations of motion of journal center using a fourth order Runga–Kutta method. The numerically computed results for the journal center trajectories indicate that the 2-lobe bearing [Formula: see text] is more stable with geometrically imperfect journal as compared to the circular bearing with imperfect journal.

Symbolic Linearized Equations for Nonholonomic Multibody Systems With Closed-Loop Kinematics

2018 ◽  
Author(s):  
Márton Kuslits ◽  
Dieter Bestle
Keyword(s):  
Lagrange Multipliers ◽  
Multibody Systems ◽  
Closed Loop ◽  
Equations Of Motion ◽  
Linear Equations ◽  
Algebraic Equations ◽  
Computationally Efficient ◽  
Linearized Equations ◽  
Nonlinear Equations Of Motion ◽  
Loop Constraints

Multibody systems and associated equations of motion may be distinguished in many ways: holonomic and nonholonomic, linear and nonlinear, tree-structured and closed-loop kinematics, symbolic and numeric equations of motion. The present paper deals with a symbolic derivation of nonlinear equations of motion for nonholonomic multibody systems with closed-loop kinematics, where any generalized coordinates and velocities may be used for describing their kinematics. Loop constraints are taken into account by algebraic equations and Lagrange multipliers. The paper then focuses on the derivation of the corresponding linear equations of motion by eliminating the Lagrange multipliers and applying a computationally efficient symbolic linearization procedure. As demonstration example, a vehicle model with differential steering is used where validity of the approach is shown by comparing the behavior of the linearized equations with their nonlinear counterpart via simulations.

Hopf Bifurcation in Gas Journal Bearings

1993 ◽  
Vol 46 (7) ◽  
pp. 392-398 ◽  
Author(s):  
K. Czołczyn´ski
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Bifurcation Theory ◽  
Journal Bearing ◽  
Equations Of Motion ◽  
Journal Bearings ◽  
Fredholm Alternative ◽  
Stiffness Coefficients ◽  
Nonlinear Equations Of Motion ◽  
Gas Journal Bearing

This paper reviews a numerical investigation of the problem of small self-excited vibrations in gas journal bearings. The method of analysis is based on the Hopf bifurcation theory, in which the approximate periodic solutions of nonlinear equations of motion are computed using the Fredholm alternative. This theory enables us to construct the bifurcating periodic solutions and to determine their stability. The equations of motion of the investigated gas journal bearing have been formulated after estimating the damping and stiffness coefficients of a gas film. For this purpose, a new method of identification has been proposed.

scholarly journals Speed Oscillations of a Vehicle Rolling on a Wavy Road

2021 ◽  
Vol 11 (21) ◽  
pp. 10431
Author(s):  
Walter V. Wedig
Keyword(s):  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Travel Speed ◽  
Differential Algebraic Equations ◽  
Polar Coordinates ◽  
Algebraic Equations ◽  
Traffic Lights ◽  
Analytical Approximations ◽  
Vertical Vibrations ◽  
Nonlinear Equations Of Motion

Every driver knows that his car is slowing down or accelerating when driving up or down, respectively. The same happens on uneven roads with plastic wave deformations, e.g., in front of traffic lights or on nonpaved desert roads. This paper investigates the resulting travel speed oscillations of a quarter car model rolling in contact on a sinusoidal and stochastic road surface. The nonlinear equations of motion of the vehicle road system leads to ill-conditioned differential–algebraic equations. They are solved introducing polar coordinates into the sinusoidal road model. Numerical simulations show the Sommerfeld effect, in which the vehicle becomes stuck before the resonance speed, exhibiting limit cycles of oscillating acceleration and speed, which bifurcate from one-periodic limit cycle to one that is double periodic. Analytical approximations are derived by means of nonlinear Fourier expansions. Extensions to more realistic road models by means of noise perturbation show limit flows as bundles of nonperiodic trajectories with periodic side limits. Vehicles with higher degrees of freedom become stuck before the first speed resonance, as well as in between further resonance speeds with strong vertical vibrations and longitudinal speed oscillations. They need more power supply in order to overcome the resonance peak. For small damping, the speeds after resonance are unstable. They migrate to lower or supercritical speeds of operation. Stability in mean is investigated.

Analysis of Nonlinear Forced Vibrations of Shallow Shells With Cut-Outs by Using the R-Function Method

2011 ◽  
Author(s):  
Galyna Pilgun ◽  
Marco Amabili
Keyword(s):  
Equations Of Motion ◽  
Continuation Method ◽  
Forced Vibrations ◽  
Second Step ◽  
Geometrically Nonlinear ◽  
Shallow Shells ◽  
Natural Modes ◽  
Mesh Free ◽  
Nonlinear Equations Of Motion ◽  
Admissible Functions

Geometrically nonlinear forced vibrations of shells based on the domains with cut-outs are investigated. Classical nonlinear shallow-shell theories retaining in-plane inertia is used to calculate the strain energy; the shear deformation is neglected. A mesh-free technique based on classic approximate functions and the R-function theory is used to build the discrete model of the nonlinear vibrations. This allowed for constructing the sequences of admissible functions that satisfy given boundary conditions in domains with complex geometries. Shell displacements are expanded by using Chebyshev orthogonal polynomials. A two-step approach is implemented to solve the problem: first a linear analysis is conducted to identify natural frequencies and corresponding natural modes to be used in the second step as a basis for nonlinear displacements. The system of ordinary differential equations is obtained by using Lagrange approach on both steps. The convergence of the solution is studied by using different multimodal expansions. The pseudo-arclength continuation method and bifurcation analysis are used to study the nonlinear equations of motion. Numerical responses are obtained in the spectral neighbourhood of the lowest natural frequency. When possible, obtained results are compared to those available in the literature.

scholarly journals A Dynamical Study of a Rotor Supported by a Radial Journal Bearing Incorporating a Spherical Squeeze Film Damper

2021 ◽  
Vol 31 (5) ◽  
pp. 10-16
Keyword(s):  
Journal Bearing ◽  
Equations Of Motion ◽  
Spherical Shape ◽  
Squeeze Film ◽  
Dynamical Study ◽  
Squeeze Film Damper ◽  
Mechanical Waves ◽  
Squeeze Film Dampers ◽  
The Stability ◽  
Nonlinear Equations Of Motion

The reduction of noises, vibration, and mechanical waves transmitting through water from the shells of submarines is essential to their safe operation and travelling. Vibrations from the rotors of the engines are widely deemed as one of the main sources to which engineers have tried to attenuate with various designs. Squeeze-film dampers can be easily integrated into rotor-bearing structures in order to lower the level of vibrations caused by rotors out of balance. For this advantage, squeeze-film dampers are widely used in air-turbine engines. This paper presents preliminary results of a numerical simulation of a shaft running on a journal bearing integrated with a squeeze-film damper and evaluates the capacity in reducing vibrations concerning the stability of static equilibrium of the shaft journal center. The proposed damper is designed in spherical shape with self-aligning capacity. The results were obtained using finite difference method and numerical integration of the full nonlinear equations of motion.

Nonlinear Interactions in Systems of Multiple Order Centrifugal Pendulum Vibration Absorbers

2013 ◽  
Vol 135 (6) ◽  
Author(s):  
Brendan J. Vidmar ◽  
Steven W. Shaw ◽  
Brian F. Feeny ◽  
Bruce K. Geist
Keyword(s):  
Equations Of Motion ◽  
System Stability ◽  
System Response ◽  
Nonlinear Interactions ◽  
Vibration Absorbers ◽  
Centrifugal Pendulum Vibration Absorbers ◽  
System Equations ◽  
Improved Performance ◽  
Nonlinear Equations Of Motion ◽  
Selection Of

We consider nonlinear interactions in systems of order-tuned torsional vibration absorbers with sets of absorbers tuned to different orders. In all current applications, absorber systems are designed to reduce torsional vibrations at a single order. However, when two or more excitation orders are present and absorbers are introduced to address different orders, nonlinear interactions become possible under certain resonance conditions. Under these conditions, a common example of which occurs for orders n and 2n, crosstalk between the absorbers, acting through the rotor inertia, can result in instabilities that are detrimental to system response. In order to design absorber systems that avoid these interactions, and to explore possible improved performance with sets of absorbers tuned to different orders, we develop predictive models that allow one to examine the effects of absorber mass distribution and tuning. These models are based on perturbation methods applied to the system equations of motion, and they yield system response features, including absorber and rotor response amplitudes and stability, as a function of parameters of interest. The model-based analytical results are compared against numerical simulations of the complete nonlinear equations of motion, and are shown to be in good agreement. These results are useful for the selection of absorber parameters to achieve desired performance. For example, they allow for approximate closed form expressions for the ratio of absorber masses at the two orders that yield optimal performance. It is also found that utilizing multiple order absorber systems can be beneficial for system stability, even when only a single excitation order is present.

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