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.

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 OSCILLATIONS OF A PULSE LOADED OSCILLATOR WITH A SQUARE RESISTANCE IN THE COMPOSITION OF THE DISSIPATIVE FORCE

2021 ◽  
pp. 35-45
Author(s):  
Vasyl Olshanskiy ◽  
Maksym Slipchenko ◽  
Igor Tverdokhlib ◽  
Ihor Kupchuk
Keyword(s):  
Differential Equation ◽  
Inverse Problems ◽  
Analytical Solutions ◽  
Dry Friction ◽  
Equation Of Motion ◽  
Dissipative Force ◽  
Computer Integration ◽  
Lambert Function ◽  
Elementary Functions ◽  
Numerical Computer

The unsteady oscillations of a dissipative oscillator caused by an instantaneous impulse of the force are described. The case is considered when the dissipative force consists of quadratic viscous resistance and dry friction, and the theoretical results are generalized to the case of the sum of three forces. The third is the force of positional friction. Formulas for calculating the ranges of oscillations have been constructed In this case, the Lambert function of negative and positive arguments is used. It is a tabulated special function. Its value can also be calculated using its known approximations in elementary functions. It is shown that, due to the action of the dissipative force, the process of post-pulse oscillations consists of a finite number of cycles and is limited in time. This is due to the presence of dry friction among the resistance components. Examples of calculations that illustrate the possibilities of the stated theory are given. In order to check the reliability of the derived calculation formulas, numerical computer integration of the differential equation of motion was also carried out. The convergence of the numerical results obtained by two different methods is shown. Thus, it has been confirmed that with the help of analytical solutions it is possible to find the extreme displacements of the oscillator without numerically solving its nonlinear differential equation of motion. Using Lambert function and the first integral of the equation of motion made it possible to derive precise calculation formulas for determining the range of oscillations caused by the pulsed load of the oscillator. The derived formulas are suitable for calculating the value of the instantaneous impulse applied to the oscillator, which refers to the inverse problems of mechanics. Thus, by measuring the maximum displacement of the oscillator, it is possible to identify the initial velocity or instantaneous impulse applied to the oscillator. The performed numerical computer integration of the output differential equation confirmed the adequacy of the obtained analytical solutions, which concern not only direct, but also inverse problems of dynamics.

scholarly journals Anti-torque systems of electromechanical cable-suspended drills and test results

2014 ◽  
Vol 55 (68) ◽  
pp. 207-218 ◽  
Author(s):  
Pavel Talalay ◽  
Xiaopeng Fan ◽  
Zhichuan Zheng ◽  
Jun Xue ◽  
Pinlu Cao ◽  
...  
Keyword(s):  
Borehole Wall ◽  
Leaf Spring ◽  
Resistance Force ◽  
Outer Diameter ◽  
Total Resistance ◽  
Test Results ◽  
Leaf Springs ◽  
Axial Movement ◽  
Axial Resistance

AbstractTo prevent spinning of the upper non-rotated part of the electromechanical drill, an ‘anti-torque system’ has to be included in the downhole unit. At the same time, the anti-torque must allow the drill to move up and down the borehole during drilling and tripping operations. Usually the anti-torque system has a blade form of various designs that engages with the borehole wall and counteracts the torque from the stator of the driving motor. This paper presents a review of the different anti-torque systems and test results with selected designs (leaf spring, skate and U-shaped anti-torque systems). Experiments showed that the skate anti-torque system can provide the maximal holding torque between 67 and 267 Nm−1 depending on the skates’ outer diameter and ice temperature, while the leaf spring anti-torque system can provide only 2.5–40 N m−1 (in case of straight contact between the ice and the leaf springs). The total resistance force to axial movement of the skate anti-torque system lies in the range 209–454N if the system is vibrating. For the leaf spring anti-torque system, the total axial resistance force is far less (19–243 N).

scholarly journals Estimation of resistance force at steady-state sinkage for cylindrical wheel-typed lunar/planetary exploration rovers with function of push–pull locomotion

2020 ◽  
Vol 7 (1) ◽  
Author(s):  
Daisuke Fujiwara ◽  
Naoki Tsujikawa ◽  
Tetsuya Oshima ◽  
Kojiro Iizuka
Keyword(s):  
Steady State ◽  
Estimation Method ◽  
Planetary Exploration ◽  
Experimental Result ◽  
Resistance Force ◽  
Estimation Model ◽  
Loose Soil ◽  
Key Factor ◽  
The Relationship ◽  
Resistance Forces

Abstract Planetary exploration rovers have required a high traveling performance to overcome obstacles such as loose soil and rocks. Push-pull locomotion rovers is a unique scheme, like an inchworm, and it has high traveling performance on loose soil. Push-pull locomotion uses the resistance force by keeping a locked-wheel related to the ground, whereas the conventional rotational traveling uses the shear force from loose soil. The locked-wheel is a key factor for traveling in the push-pull scheme. Understanding the sinking behavior and its resistance force is useful information for estimating the rover’s performance. Previous studies have reported the soil motion under the locked-wheel, the traction, and the traveling behavior of the rover. These studies were, however, limited to the investigation of the resistance force and amount of sinkage for the particular condition depending on the rover. Additionally, the locked-wheel sinks into the soil until it obtains the required force for supporting the other wheels’ motion. How the amount of sinkage and resistance forces are generated at different wheel sizes and mass of an individual wheel has remained unclear, and its estimation method hasn’t existed. This study, therefore, addresses the relationship between the sinkage and its resistance force, and we analyze and consider this relationship via the towing experiment and theoretical consideration. The results revealed that the sinkage reached a steady-state value and depended on the contact area and mass of each wheel, and the maximum resistance force also depends on this sinkage. Additionally, the estimation model did not capture the same trend as the experimental results when the wheel width changed, whereas, the model captured a relatively the same trend as the experimental result when the wheel mass and diameter changed.

Drifting Force and Moment on Ships in Oblique Waves

1966 ◽  
Vol 10 (01) ◽  
pp. 18-24
Author(s):  
Pung Nien Hu ◽  
King Eng
Keyword(s):  
Potential Theory ◽  
General Expression ◽  
Phase Angle ◽  
Analytical Solutions ◽  
Vertical Axis ◽  
Long Waves ◽  
Elementary Functions ◽  
Oblique Waves ◽  
Side Force ◽  
Yaw Moment

A general expression for the drifting moment about the vertical axis of an oscillating ship in regular oblique waves is derived from the potential theory, following a similar procedure developed by Maruo for drifting force. Explicit analytical solutions for the drifting side force and yaw moment on thin ships in long waves are obtained in terms of simple elementary functions. The effect of the wave frequency, the draft of the ship, the displacement, and the phase angle of the ship oscillation are discussed.

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.

scholarly journals Analytical and Experimental Analysis of a Free Link in Contact with a Granular Medium

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Dan B. Marghitu ◽  
Seung Lee
Keyword(s):  
Granular Medium ◽  
Stopping Time ◽  
The Body ◽  
Resistance Force ◽  
Time Interval ◽  
The Moment ◽  
Static Resistance ◽  
The Impact ◽  
Resistance Forces ◽  
Empirical Approaches

In this study, the experimental and the simulation results for a planar free link impacting a granular medium are analyzed. The resistance force of the granular medium on the body from the moment of the impact until the body stops is very important. Horizontal and vertical static resistance forces developed by theoretical and empirical approaches are considered. The penetrating depth of the impacting end of the free link increases with the increase of the initial impacting velocity. We define the stopping time as the time interval from the moment of impact until the vertical velocity of the link end is zero. The stopping time of the end decreases as the initial velocity increases. The faster the end of the link impacts the surface of the granular medium, the sooner it will come to a stop. This phenomenon involves how rapidly a free link strikes the granular medium and how it slows down upon contact.

scholarly journals ON OSCILLATIONS DESCRIBING A GENERALIZED DIFFERENTIAL RELAY EQUATION

2020 ◽  
pp. 53-60
Author(s):  
Vasiliy Olshansky ◽  
Stanislav Olshansky ◽  
Oleksіі Tokarchuk
Keyword(s):  
Differential Equation ◽  
Initial Conditions ◽  
Oscillatory System ◽  
Equation Of Motion ◽  
Resistance Force ◽  
Oscillatory Process ◽  
Initial Deviation ◽  
Approximate Analytical Solutions ◽  
Positive Exponent ◽  
Generalized Differential

The motion of an oscillatory system with one degree of freedom, described by the generalized Rayleigh differential equation, is considered. The generalization is achieved by replacing the cubic term, which expresses the dissipative strength of the equation of motion, by a power term with an arbitrary positive exponent. To study the oscillatory process involved the method of energy balance. Using it, an approximate differential equation of the envelope of the graph of the oscillatory process is compiled and its analytical solution is constructed from which it follows that quasilinear frictional self-oscillations are possible only when the exponent is greater than unity. The value of the amplitude of the self-oscillations in the steady state also depends on the value of the indicator. A compact formula for calculating this amplitude is derived. In the general case, the calculation involves the use of a gamma function table. In the case when the exponent is three, the amplitude turned out to be the same as in the asymptotic solution of the Rayleigh equation that Stoker constructed. The amplitude is independent of the initial conditions. Self-oscillations are impossible if the exponent is less than or equal to unity, since depending on the initial deviation of the system, oscillations either sway (instability of the movement is manifested) or the range decreases to zero with a limited number of cycles, which is usually observed with free oscillations of the oscillator with dry friction. These properties of the oscillatory system are also confirmed by numerical computer integration of the differential equation of motion for specific initial data. In the Maple environment, the oscillator trajectories are constructed for various values of the nonlinearity index in the expression of the viscous resistance force and a corresponding comparative analysis is carried out, which confirms the adequacy of approximate analytical solutions.

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.

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