Influence of Acceleration on the Critical Speed of a Jeffcott Rotor

1981 ◽  
Vol 103 (1) ◽  
pp. 108-113 ◽  
Author(s):  
H. L. Hassenpflug ◽  
R. D. Flack ◽  
E. J. Gunter
Keyword(s):  
Numerical Integration ◽  
Critical Speed ◽  
Equations Of Motion ◽  
Beat Frequency ◽  
Angular Acceleration ◽  
Amplitude Response ◽  
Peak Response ◽  
High Acceleration ◽  
Jeffcott Rotor ◽  
Theoretical Predictions

The effects of angular acceleration on a Jeffcott rotor have been examined both theoretically and experimentally. The equations of motion were solved via numerical integration. The rotor’s response to unbalance was predicted for a number of cases of acceleration and damping. Both amplitude and phase responses were studied. In addition, techniques were developed for identifying system damping from data taken during accelerated runs. The results of the analysis indicate that for high acceleration rates the amplitude response at the critical speed may be reduced by a factor of four or more. The speed at which the peak response occurs can also be shifted by 20 percent or more. Experimentally, a small lightly damped rotor (ζ = 0.0088) was run for several acceleration rates. The peak responses typically agree within 6 percent of theoretical predictions. Also, a beat frequency was observed both theoretically and experimentally after the rotor had passed through the critical speed.

Analysis of an Accelerating Rotor-Bearing System With Flexible Damped Supports

1998 ◽  
Author(s):  
Euro L. Casanova ◽  
Luis U. Medina
Keyword(s):  
Steady State ◽  
Numerical Integration ◽  
Equations Of Motion ◽  
Peak Amplitude ◽  
Graphical Form ◽  
Jeffcott Rotor ◽  
Rotor Bearing ◽  
Flexible Supports ◽  
Steady State Predictions ◽  
Bearing System

This paper deals with the dynamics of an accelerating unbalanced Jeffcott rotor-bearing system mounted on damped, flexible supports. The general equations of motion for such a system are presented and discussed. The rotor response was predicted, via numerical integration, for various cases in runup and rundown conditions and presented in graphical form. The effects of acceleration on the rotor peak amplitude and the speed at which the peak occurs is discussed and compared to steady state predictions.

The Limited-Torque Acceleration Through Critical Speed Phenomenon in Rotating Machinery

2012 ◽  
Author(s):  
Anand Srinivasan ◽  
Trent W. Thurston
Keyword(s):  
Critical Speed ◽  
Equations Of Motion ◽  
Drive Train ◽  
Transient Solution ◽  
Acceleration Rate ◽  
Jeffcott Rotor ◽  
Full Speed ◽  
Lateral Displacements ◽  
Design Speed ◽  
Rotor Model

Rotor-bearing systems of modern day turbomachinery are generally designed to operate at speeds well above the lateral critical speed(s). Acceleration from rest to design speed of turbomachines is usually accomplished by a driver such as a motor or a turbine. The driver provides the torque required to bring the drive-train to full speed. If the torque delivered by the driver is less than the torque demanded by the driven machine, the drive-train stalls at a speed below running speed. If this speed coincides with a lateral critical speed of the turbomachine, the amplitude of vibration may increase to levels high enough to trip the machine. In extreme cases, damage due to rubs from vibration excursions may occur on the rotating components. Such a phenomenon is referred to as a limited-torque-acceleration of rotors through the critical speed. A theoretical analysis of this phenomenon requires a time-transient solution of the lateral equations of motion, with the acceleration rate determined from the torque equation. In this paper, the acceleration of the Jeffcott rotor model with a variable torque input has been studied, and the time-transient response of the shaft lateral displacements has been presented. Data recorded from a turbomachine that incurs vibration excursions during limited-torque acceleration through critical speed has also been presented. The importance of fast acceleration rates through critical speeds for rotating equipment has been stressed in this paper.

Effect of gravity and angular velocity on an automatic ball balancer

2005 ◽  
Vol 219 (1) ◽  
pp. 43-51 ◽  
Author(s):  
J Chung
Keyword(s):  
Angular Velocity ◽  
Velocity Profile ◽  
Equations Of Motion ◽  
Linear Equations ◽  
Angular Acceleration ◽  
Polar Coordinates ◽  
Lagrange’S Equation ◽  
Jeffcott Rotor ◽  
Analysis Of Results ◽  
Linear Equations Of Motion

The effects of gravity and an angular velocity profile on the performance of an automatic ball balancer (ABB) are studied in this paper. In order to investigate these effects, a physical model of a Jeffcott rotor with an ABB is adopted in this study, in which gravity as well as the angular acceleration is considered. With the polar coordinates, the non-linear equations of motion are derived by using Lagrange's equation. These equations include gravity, the angular acceleration, and the angular jerk. Based on the equations derived, time responses are computed by using the generalized α method. The effects of gravity on the balancing performance are analysed. For various angular velocity profiles, the ABB performance is also evaluated. The analysis of results shows that the balancing of the rotor with an ABB can be achieved regardless of gravity. It is also shown that a smooth velocity profile results in less vibration compared with a non-smooth velocity profile.

Nonstationary analysis of nonlinear rotating shafts passing through critical speed excited by a nonideal energy source

2016 ◽  
Vol 232 (4) ◽  
pp. 572-584 ◽  
Author(s):  
A Mahmoudi ◽  
SAA Hosseini ◽  
M Zamanian
Keyword(s):  
Energy Source ◽  
Critical Speed ◽  
Equations Of Motion ◽  
Angular Acceleration ◽  
Continuous System ◽  
Multiple Scale ◽  
Sommerfeld Effect ◽  
Rotating Shaft ◽  
Gyroscopic Effects ◽  
Nonlinear Equations Of Motion

In this paper, the effect of nonlinearity on vibration of a rotating shaft passing through critical speed excited by nonideal energy source is investigated. Here, the interaction between a nonlinear gyroscopic continuous system (i.e. rotating shaft) and the energy source is considered. In the shaft model, the rotary inertia and gyroscopic effects are included, but shear deformation is neglected. The nonlinearity is due to large deflection of the shaft. Firstly, nonlinear equations of motion governing the flexural–flexural–extensional vibrations of the rotating shaft with nonconstant spin are derived by the Hamilton principle. Then, the equations are simplified using stretching assumption. To analyze the nonstationary vibration of the nonideal system, multiple-scale method is directly applied to the equations expressed in complex coordinates. Three analytical expressions that describe variation of amplitude, phase, and angular acceleration during passage through critical speed are derived. It is shown that Sommerfeld effect in specific range of driving torque occurs. Finally, effect of damping and nonlinearity on occurrence of Sommerfeld effect is investigated. It is shown that the linear model predicts the range of Sommerfeld effect occurrence inaccurately and, therefore, nonlinear analysis is necessary in the present problem.

Backward Whirl and Its Suppression of a Squeeze Film Damper Mounted Rotor/Casing System in Passage through Critical Speed with Rubs

2004 ◽  
Vol 10 (4) ◽  
pp. 561-573 ◽  
Author(s):  
Qian Ding
Keyword(s):  
Critical Speed ◽  
Angular Acceleration ◽  
Squeeze Film ◽  
Squeeze Film Damper ◽  
Internal Support ◽  
Squeeze Film Dampers ◽  
Jeffcott Rotor ◽  
Backward Whirl ◽  
Pass Through ◽  
Second Peak

In this paper we consider a flexible Jeffcott rotor mounted at the ends by identical squeeze film dampers (SFDs). The rotative speed is supposed to increase at a constant angular acceleration. There can be one-peak and two-peak solutions for different values of SFD parameters during passage through the critical speed. Calculation shows that the rotor cannot pass through the critical speed due to the occurrence of diverging backward whirl in passage of the first or second peak, if the level of acceleration is lower than the critical ones. A flexible internal support, which can be activated or deactivated at a certain position along the rotor to change the stiffness of the system to suppress large vibration, is then applied to avoid the occurrence of backward whirl. The method is found to be effective if applied in a suitable way

Interrogating the Lead-up to a Critical Speed in Rotordynamics

2021 ◽  
pp. 1-18
Author(s):  
Lawrie Virgin
Keyword(s):  
Critical Speed ◽  
Stress Test ◽  
Equations Of Motion ◽  
Rotating Shaft ◽  
Small Perturbations ◽  
New Approach ◽  
Naturally Occurring ◽  
Jeffcott Rotor ◽  
Random Disturbances ◽  
Rotor Model

Abstract This paper presents a new approach to predicting an incipient critical speed in a rotating shaft. Based on the classical governing equations of motion for an eccentric mass on a flexible shaft (the Jeffcott rotor model), the approach is centered on examining the behavior of small perturbations or random disturbances to infer the approach of a critical speed (resonance). Such disturbances, that may be based on intentional probing, or simply the result of naturally occurring fluctuations, cause small transients. It is the changing nature of these transients (as characterized by their associated eigenvalues) that is used to assess the proximity to a critical speed. In this paper the material developed is based on analysis, but generating the data from simulations or experiments will be the next step. The approach is a kind of stress-test, conceptually not dissimilar to structural health monitoring and damage detection, but here directed toward the lead-up to resonance.

Taylor-vortex instability and annulus-length effects

1976 ◽  
Vol 75 (1) ◽  
pp. 1-15 ◽  
Author(s):  
J. A. Cole
Keyword(s):  
Critical Speed ◽  
Taylor Vortex ◽  
Vortex Instability ◽  
Critical Speeds ◽  
Taylor Vortices ◽  
Torque Measurements ◽  
Concentric Cylinders ◽  
Theoretical Predictions ◽  
Range Of Values ◽  
Visual Observations

Critical speeds for the onset of Taylor vortices and for the later development of wavy vortices have been determined from torque measurements and visual observations on concentric cylinders of radius ratios R1/R2 = 0·894–0·954 for a range of values of the clearance c and length L: c/R1 = 0·0478–0·119 and L/c = 1–107. Effectively zero variation of the Taylor critical speed with annulus length was observed. The speed at the onset of wavy vortices was found to increase considerably as the annulus length was reduced and theoretical predictions are realistic only for L/c values exceeding say 40. The results were similar for all four clearance ratios examined. Preliminary measurements on eccentrically positioned cylinders with c/R1 = 0·119 showed corresponding effects.

Rolling Bearing Parametric Excitation of a Jeffcott Rotor System

2020 ◽  
Author(s):  
Ghasem Ghannad Tehrani ◽  
Chiara Gastaldi ◽  
Teresa Maria Berruti
Keyword(s):  
Parametric Excitation ◽  
Rotational Speed ◽  
Input Parameter ◽  
Equations Of Motion ◽  
Harmonic Balance Method ◽  
Rolling Bearing ◽  
Mean Value ◽  
Hertzian Contact ◽  
Rolling Bearings ◽  
Jeffcott Rotor

Abstract Rolling bearings are still widely used in aeroengines. Whenever rotors are modeled, rolling bearing components are typically modeled using springs. In simpler models, this spring is considered to have a constant mean value. However, the rolling bearing stiffness changes with time due to the positions of the balls with respect to the load on the bearing, thus giving rise to an internal excitation known as Parametric Excitation. Due to this parametric excitation, the rotor-bearings system may become unstable for specific combinations of boundary conditions (e.g. rotational speed) and system characteristics (rotor flexibility etc.). Being able to identify these instability regions at a glance is an important tool for the designer, as it allows to discard since the early design stages those configurations which may lead to catastrophic failures. In this paper, a Jeffcott rotor supported and excited by such rolling bearings is used as a demonstrator. In the first step, the expression for the time–varying stiffness of the bearings is analytically derived by applying the Hertzian Contact Theory. Then, the equations of motion of the complete system are provided. In this study, the Harmonic Balance Method (HBM) is used to as an approximate procedure to draw a stability map, thus dividing the input parameter space, i.e. rotational speed and rotor physical characteristics, into stable and unstable regions.

Wilberforce pendulum: modelling linearly damped coupled oscillations of a spring-mass system

2021 ◽  
Author(s):  
Robert Frederik Diaz Uy ◽  
Chenghao Yuan ◽  
Zhengshan Chai ◽  
Justin Khor
Keyword(s):  
Experimental Data ◽  
Angular Displacement ◽  
Beat Frequency ◽  
Vertical Displacement ◽  
Quantitative Model ◽  
Moment Of Inertia ◽  
Lagrangian Formalism ◽  
Mass System ◽  
Coupled Oscillations ◽  
Theoretical Predictions

Abstract The Wilberforce pendulum is a coupled spring-mass system, where a mass with adjustable moment of inertia is suspended from a helical spring. Energy is converted between the translational and torsional modes, and this energy conversion is most clearly observed at resonance, which occurs when the damped natural frequencies of the two oscillation modes are equal. A theoretical model—with energy losses due to viscous damping accounted for—was formulated using the Lagrangian formalism to predict the pendulum mass’ trajectory. Theoretical predictions were compared with experimental data, showing good agreement. Fourier analysis of both theoretical predictions and experimental data further corroborate the validity of our quantitative model. The dependence of oscillation features like beat frequency and maximum conversion amplitude on relevant parameters such as the initial vertical displacement, initial angular displacement and moment of inertia was also investigated and experimentally verified.

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