Dry-Friction Whip and Whirl Predictions for a Rotor-Stator Model With Rubbing Contact at Two Locations

2011 ◽  
Author(s):  
Dara W. Childs ◽  
Dhruv Kumar
Keyword(s):  
Analytical Solutions ◽  
Dry Friction ◽  
Supported Housing ◽  
Precession Frequency ◽  
Rigid Rotor ◽  
Clearance Ratio ◽  
Running Speed ◽  
Nonlinear Simulation ◽  
Friction Forces ◽  
Nonlinear Connections

The present work investigates the phenomena of whip and whirl for a rigid rotor contacting at two bearing locations. The idea originated with a paper by Clark et al. in 2009 on an anemometer undergoing dry friction whip and whirl. The anemometer rotor was supported by two Teflon® bushings within an elastically supported housing. The dry-friction forces arose at the bushings. Prior models for dry friction whirl and whip have considered rub at one non-support location. The present analytical model consists of a rigid rotor connected to a rigid stator at two rubbing contact locations. Analytical solutions are developed for the following normal reaction forces at the contact locations: (1) In phase, and (2) 180 degrees out of phase. Analytical solutions are only possible for the same RCl (Radius to Clearance ratio) at the two rub locations and define regions where dry-friction whirl is possible plus indication possible boundaries between whirl and whip. These solutions are similar to Black’s in 1968. A flexible-rotor/flexible-stator model with nonlinear connections at the bearings was developed to more correctly establish the range of possible solutions. The nonlinear connections at the rub surface are modeled using Hunt and Crossley’s 1975 contact model with coulomb friction. Dry friction simulations are performed for the following rotor center of gravity (C.G.) configurations: (1) Centered, (2) 3/4 contact-span location and (3) Overhang location outside the contacts. Results from the in-phase analytical solutions and the nonlinear simulations agree to some extent with the rotor mass centered and at 3/4 location in that whirl-to-whip transitions occur near the pinned rotor-stator bounce frequency. For the overhung mass case, the nonlinear simulation predicts whip at different frequencies for the two contact locations. Neither analytical solution modes predicts this outcome. No out-of-phase solutions could be obtained via time-transient simulations. Dry-friction whirling is normally characterized as supersynchronous precession with a precession frequency equal to running speed times RCl. Simulation predictions for models with different RCl mimic whirling. Simulation predictions show increasing backward precessional (BP) frequency with increasing rotor speeds. However, individual contact velocities show slipping at all conditions. Slipping is greater at one location than the other, netting a “whirl-like” motion. For the overhung model with different RCl ratios, apart from whipping at different frequency the two contacts also whirl at different frequencies corresponding to the RCl at the respective contacts. Simulations predict a different running speed for the “jump up” in precession frequency associated with a transition from whirl-to-whip with increasing running speed than for the jump-down in precession frequency for whirl-to-whip in a speed-decreasing mode.

Dry-Friction Whip and Whirl Predictions for a Rotor-Stator Model With Rubbing Contact at Two Locations

2012 ◽  
Vol 134 (7) ◽  
Author(s):  
Dara W. Childs ◽  
Dhruv Kumar
Keyword(s):  
Analytical Solutions ◽  
Dry Friction ◽  
Supported Housing ◽  
Precession Frequency ◽  
Rigid Rotor ◽  
Clearance Ratio ◽  
Running Speed ◽  
Nonlinear Simulation ◽  
Friction Forces ◽  
Nonlinear Connections

The present work investigates dry-friction whip and whirl phenomena for a rigid rotor contacting at two bearing locations. The idea originated with a paper by Clark et al. (2009, “Investigation of the NRG #40 Anemometer Slowdown,” American Wind Energy Association, Windpower 2009, Chicago, IL, pp. 1-16) on an anemometer undergoing dry-friction whip and whirl. The anemometer rotor was supported by two Teflon® bushings within an elastically supported housing. The dry-friction forces arose at the bushings. Prior models for dry friction whirl and whip have considered rub at one nonsupport location. The present analytical model consists of a rigid rotor connected to a rigid stator at two rubbing-contact locations. Analytical solutions are developed for the following normal reaction forces at the contact locations: (1) In phase, and (2) 180° out of phase. Analytical solutions are only possible for the same radius-to-clearance ratio (RCl) at the two rub locations and define regions where dry-friction whirl is possible in addition to indicating possible boundaries between whirl and whip. These solutions are similar to Black’s (1968, “Interaction of a Whirling Rotor with a Vibrating Stator Across a Clearance Annulus,” J. Mech. Eng. Sci., 10(1), pp. 1-12) and Crandall’s (1990, “From Whirl to Whip in Rotordynamics,” IFToMM Third Intl. Conf. on Rotordynamics, Lyon, France, pp. 19-26). A flexible-rotor/flexible-stator model with nonlinear connections at the bearings was developed to more correctly establish the range of possible solutions. The nonlinear connections at the rub surface are modeled using Hunt and Crossley’s 1975 contact model with Coulomb friction (Hunt and Crossley, F., 1975, “Coefficient of Restitution Interpreted as Damping in Vibroimpact,” ASME J. Appl. Mech., 42, pp. 440). Dry friction simulations are performed for the following rotor center of gravity (C.G.) configurations with respect to the contact locations: (1) Centered, (2) [3/4]-span location, and (3) overhung, outside the contacts. Predictions from the in-phase analytical solutions and the nonlinear simulations agree to some extent when the rotor mass is centered and at the [3/4]-span location due to the fact that whirl-to-whip transitions occur near the pinned rotor-stator bounce frequency. For the overhung mass case, the nonlinear simulation predicts whip at different frequencies for the two contact locations. Neither analytical solution modes predicts this outcome. No 180 deg out-of-phase solutions could be obtained via time-transient simulations. Dry-friction whirling is normally characterized as supersynchronous precession with a precession frequency equal to the running speed ω times RCl. Simulation predictions for models with different RCl ratio mimic whirling. Specifically, with increasing rotor speed, the backward precessional (BP) frequency increases at each contact location. However, individual contact velocities show slipping at all conditions. Slipping is greater at one location than the other, netting a “whirl-like” motion. For the overhung model with different RCl ratios: in addition to whipping at different frequencies the two contacts also whirl at different frequencies corresponding to the separate RCl ratios at the respective contacts. Simulations predict a different running speed for the “jump up” in precession frequency associated with a transition from whirl-to-whip with increasing running speed than for the jump-down in precession frequency for whirl-to-whip in a speed-decreasing mode.

scholarly journals VIBRATIONS OF AN INSTANTLY LOADED DISSIPATIVE OSCILLATOR

2021 ◽  
pp. 21-31
Author(s):  
Vasyl Olshanskiy ◽  
Maksym Slipchenko ◽  
Oleksandr Spolnik ◽  
Olena Solona
Keyword(s):  
Analytical Solutions ◽  
Dry Friction ◽  
Leaf Spring ◽  
Resistance Force ◽  
Dynamic Factor ◽  
Oscillatory Process ◽  
Total Resistance ◽  
Elementary Functions ◽  
Friction Forces ◽  
Resistance Forces

The work investigates non-stationary oscillations of dissipative oscillators. The joint influence of resistance forces of different nature in the composition of the dissipative force on the oscillations of an elastic linear oscillator caused by its instantaneous loading by a constant external force is investigated. The work was limited to the case of small displacements, when the elastic characteristic of the system can be considered linear. The problem is nonlinear due to the account of the action of the dry friction force, but it allows the construction of exact analytical solutions in elementary functions. In this work, by the method of adding solutions, formulas are derived for calculating the amplitude of oscillations and the duration of half cycles. First, a variant is considered when the resistance force consists of linear viscous and dry friction forces. Then a generalization is made to the case of three resistance forces. The third force is the force of positional friction, which arises in elastic elements such as a leaf spring. It is shown that as a result of the action of the total resistance force, the oscillatory process of a loaded oscillator has a finite number of cycles and is limited in time, which is usually observed in practice. The system dynamic factor is less than two. Examples of calculations that illustrate the possibilities of the stated theory are considered. To check the adequacy of the derived calculation formulas, numerical computer integration of the nonlinear differential equations of the oscillator motion was additionally carried out. The full agreement of the numerical results obtained using various methods is shown. In addition to direct problems of dynamics, the inverse problems of determining the characteristics of the load and resistance from the results of measuring the amplitude of oscillations are also considered. The derived calculation formulas are also suitable for identifying the characteristics of the load and resistance based on the results of experimental measurements of the oscillation ranges. Examples of identifying these characteristics are given. The study showed that the nonlinear problem of motion of an instantly loaded oscillator with the total resistance of several forces of different nature has an analytical solution in elementary functions. The presence of such resistance significantly affects the motion of the oscillator after loading. The constructed analytical solutions give results such as the numerical integration of the original nonlinear differential equation on a computer, which confirms the adequacy of the formulas obtained.

Analysis of Beam-Like Structures With Displacement-Dependent Friction Forces: Part I — Single-Degree-of-Freedom Model

1995 ◽  
Author(s):  
Wayne E. Whiteman ◽  
Aldo A. Ferri
Keyword(s):  
Dry Friction ◽  
Damping Ratio ◽  
Strong Dependence ◽  
Time Integration ◽  
Single Mode ◽  
Stick Slip ◽  
Friction Forces ◽  
Equivalent Damping ◽  
Damping Characteristics ◽  
Parameter Values

Abstract The dynamic behavior of a beam-like structure undergoing transverse vibration and subjected to a displacement-dependent dry friction force is examined. In Part I, the beam is modeled by a single mode while Part II considers multi-mode representations. The displacement dependence in each case is caused by a ramp configuration that allows the normal force across the sliding interface to increase linearly with slip displacement. The system is studied first by using first-order harmonic balance and then by using a time integration method. The stick-slip behavior of the system is also studied. Even though the only source of damping is dry friction, the system is seen to exhibit “viscous-like” damping characteristics. A strong dependence of the equivalent natural frequency and damping ratio on the displacement amplitude is an interesting result. It is shown that for a given set of parameter values, an optimal ramp angle exists that maximizes the equivalent damping ratio. The appearance of two dynamic response solutions at certain system and forcing parameter values is also seen. Results suggest that the overall characteristics of mechanical systems may be improved by properly configuring frictional interfaces to allow normal forces to vary with displacement.

The Stick Motion of a Harmonically Excited, Friction-Induced Oscillator on a Sinusoidally Traveling Surface

2006 ◽  
Author(s):  
Albert C. J. Luo ◽  
Brandon C. Gegg ◽  
Steve S. Suh
Keyword(s):  
Dynamical Systems ◽  
Numerical Simulations ◽  
Dry Friction ◽  
The Other ◽  
Linear Oscillator ◽  
Analytical Prediction ◽  
Nonlinear Friction ◽  
Friction Forces

In this paper, the methodology is presented through investigation of a periodically, forced linear oscillator with dry friction, resting on a traveling surface varying with time. The switching conditions for stick motions in non-smooth dynamical systems are obtained. From defined generic mappings, the corresponding criteria for the stick motions are presented through the force product conditions. The analytical prediction of the onset and vanishing of the stick motions is illustrated. Finally, numerical simulations of stick motions are carried out to verify the analytical prediction. The achieved force criteria can be applied to the other dynamical systems with nonlinear friction forces possessing a CO - discontinuity.

Analysis of Motion of the Impact-Dry-Friction Pair of Bodies and its Application to the Investigation of the Impact Dampers Dynamics

1999 ◽  
Author(s):  
František Peterka
Keyword(s):  
Analytical Solution ◽  
Numerical Simulations ◽  
Mechanical System ◽  
Relative Motion ◽  
Dry Friction ◽  
Friction Pair ◽  
Strong Nonlinearity ◽  
Friction Forces ◽  
The Impact ◽  
Analysis Of Motion

Abstract The motion with impacts and dry friction forces appears in some mechanical systems as mechanisms with clearances, (e.g., in gearings, pins, slots, guides, valve gears etc.), impact dampers, relays, forming and mailing machines, power pics etc. Such mechanisms include one or more pairs of impacting bodies, which introduce the strong nonlinearity into the system motion. The motion of the general pair of bodies with the both-sides impacts and dry friction forces is assumed (Fig.1). It can be the part of a more complex chain of masses in the mechanical system. Dead zones in the relative motion of bodies can be caused by assumed nonlinearities. The mathematical conditions controlling the numerical simulations or analytical solution of the motion are introduced. The application of this method is explained by the study of the influence of dry friction force on amplitude-frequency characteristics of four types of dynamical and impact dampers with optimised parameters.

Nonlinear Numerical Prediction of Gas Foil Bearing Stability and Unbalanced Response

2008 ◽  
Vol 131 (1) ◽  
Author(s):  
Sébastien Le Lez ◽  
Mihaï Arghir ◽  
Jean Frêne
Keyword(s):  
Dry Friction ◽  
Stability Limit ◽  
Time Step ◽  
Structural Dynamic ◽  
Friction Forces ◽  
Foil Bearings ◽  
Mass Unbalance ◽  
Foil Bearing ◽  
Highly Nonlinear ◽  
The Stability

One of the main interests of gas foil bearings lies in their superior rotordynamic characteristics compared with conventional bearings. A numerical investigation on the stability limit and on the unbalanced response of foil bearings is presented in this paper. The main difficulty in modeling the dynamic behavior of such bearings comes from the dry friction that occurs within the foil structure. Indeed, dry friction is highly nonlinear and is strongly influenced by the dynamic amplitude of the pressure field. To deal with these nonlinearities, a structural dynamic model has been developed in a previous work. This model considers the entire corrugated foil and the interactions between the bumps by describing the foil bearing structure as a multiple degrees of freedom system. It allows the determination of the dynamic friction forces at the top and at the bottom of the bumps by simple integration of ordinary differential equations. The dynamic displacements of the entire corrugated sheet are then easily obtained at each time step. The coupling between this structural model and a gas bearing prediction code is presented in this paper and allows performing full nonlinear analyses of a complete foil bearing. The bearing stability is the first investigated problem. The results show that the structural deflection enhances the stability of compliant surface bearings compared with rigid ones. Moreover, when friction is introduced, a new level of stability is reached, revealing the importance of this dissipation mechanism. The second investigated problem is the unbalanced response of foil bearings. The shaft trajectories depict a nonlinear jump in the response of both rigid and foil bearings when the value of the unbalance increases. Again, it is evidenced that the foil bearing can support higher mass unbalance before this undesirable step occurs.

scholarly journals STABILITY OF VIBRATIONS OF THE TRACTOR AS A TWO-MASS MODEL WITH TWO NONLINEARITIES OF THE TYPE OF DRY FRICTION IN RESONANCE MODES

2021 ◽  
Author(s):  
E. Kalinin ◽  
◽  
Y. Kolesnik ◽  
M. Myasushka
Keyword(s):  
Differential Equations ◽  
Dry Friction ◽  
System Analysis ◽  
Nonlinear Differential Equations ◽  
Support Surface ◽  
Systems Methodology ◽  
Friction Forces ◽  
Resonance Modes ◽  
At Resonance ◽  
Relative Friction

Purpose of the study is to assess the possibility of calculating the stability of tractor oscillations as a system with nonlinearities such as dry friction due to the inverse problem. Research methods. The methodological basis of the work is the generalization and analysis of known scientific results regarding the dynamics of two-mass systems in resonance modes and the use of a systematic approach. The analytical method and comparative analysis were used to form a scientific problem, determine the goal and formulate the research objectives. When creating empirical models, the main provisions of the theory of stability of systems, methodology of system analysis and research of operations were used. The results of the study. Oscillations of the system with harmonic excitation by its base are considered (for example, the movement of a tractor on an uneven supporting surface). Oscillations of this system are described by nonlinear differential equations. To solve this equation, instead of friction dampers with friction forces, linear dampers with corresponding drag coefficients are included in the system. By solving the obtained system of linear inhomogeneous differential equations for the steady-state mode of oscillation, the amplitudes of oscillations of masses and deformation of springs with certain stiffness are determined. To clarify the effect of friction forces on mass oscillations in resonance modes, the obtained expressions were analyzed. A diagram of stability of mass oscillations in resonance modes is obtained. Conclusions. It has been established that if the coefficients of relative friction have such values that the point that is determined by them lies within the region bounded by segments 1-2 and 2-3 and coordinate axes, then during oscillations in the low-frequency resonance mode, the friction forces do not limit the increase in amplitudes fluctuations of masses, but only reduce the rate of their growth. If the point, which is determined by the coefficients of relative friction, lies in the region 1-1'-2'-3 '3-2-1, then the springs have intermittent deformation, that is, during the period of oscillation, one mass of the system has stops relative to another mass, or the last has stops relative to the support surface, or both masses move part of the period as a whole with the support surface. At resonance with a high frequency, the friction forces limit the amplitudes of mass oscillations if the coefficients of relative friction have such values that the point that is determined by them does not lie in the region bounded by segments 4-5 and 5-6 and the coordinate axes. Sections 4-5 and 5-6 define the boundaries of vibration stability at resonance (lines of critical ratios of the coefficients of relative friction).

Sign in / Sign up

Export Citation Format

Share Document