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.

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.

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.

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.

Dynamics of a hydropower generator subjected to unbalanced magnetic pull

2011 ◽  
Vol 225 (9) ◽  
pp. 2076-2088 ◽  
Author(s):  
Y Calleecharan ◽  
J-O Aidanpää
Keyword(s):  
Periodic Motion ◽  
Critical Speed ◽  
Radial Component ◽  
Electrical Machines ◽  
External Forcing ◽  
New Model ◽  
Unbalanced Magnetic Pull ◽  
Wide Range ◽  
Jeffcott Rotor ◽  
Rotor Model

Eccentricity leading to unbalanced magnetic pull (UMP) in electrical machines is a significant concern in industry. The UMP is known to be composed of two components: a radial component and a tangential one. Models that are used in industry tend to include the radial component alone. In this article, a Jeffcott rotor model together with a new UMP model that incorporates both radial and tangential UMP constituents is studied for an industrial hydropower generator. Characterizing the UMP as springs permits the model to inherit UMP stiffness contribution. Interesting dynamics are observed with the new model for a wide range of external forcing frequencies. It is shown firstly that the new UMP model is sensitive to forcing frequency in the rotor movements. Secondly, it is found that this sensitivity to forcing frequency increases with decreasing rotor system stiffness. Moreover, quasi-periodic motion in the rotor displacements is observed and it is noted that the rotor does not need to be forced by frequencies above its critical speed for this less desirable motion to occur. Thus, it becomes interesting to be able to account for the UMP stiffness contribution in order to curb machine malfunction which might result from these UMP forces.

scholarly journals New Approach to the Planetary Theory

1979 ◽  
Vol 81 ◽  
pp. 69-72 ◽  
Author(s):  
Manabu Yuasa ◽  
Gen'ichiro Hori
Keyword(s):  
Perturbation Method ◽  
Canonical Form ◽  
Equations Of Motion ◽  
Disturbing Function ◽  
Averaging Principle ◽  
The Sun ◽  
Planetary Theory ◽  
New Approach ◽  
Canonical Perturbation ◽  
Relative Coordinates

A new approach to the planetary theory is examined under the following procedure: 1) we use a canonical perturbation method based on the averaging principle; 2) we adopt Charlier's canonical relative coordinates fixed to the Sun, and the equations of motion of planets can be written in the canonical form; 3) we adopt some devices concerning the development of the disturbing function. Our development can be applied formally in the case of nearly intersecting orbits as the Neptune-Pluto system. Procedure 1) has been adopted by Message (1976).

A New Approach to the Rigid Finite Element Method in Modeling Spatial Slender Systems

2018 ◽  
Vol 18 (02) ◽  
pp. 1850017 ◽  
Author(s):  
Iwona Adamiec-Wójcik ◽  
Łukasz Drąg ◽  
Stanisław Wojciech
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Dynamic Analysis ◽  
Equations Of Motion ◽  
Nonlinear Dynamic Analysis ◽  
New Approach ◽  
Static And Dynamic Analysis ◽  
Rigid Finite Element Method ◽  
Longitudinal Flexibility ◽  
Element Method

The static and dynamic analysis of slender systems, which in this paper comprise lines and flexible links of manipulators, requires large deformations to be taken into consideration. This paper presents a modification of the rigid finite element method which enables modeling of such systems to include bending, torsional and longitudinal flexibility. In the formulation used, the elements into which the link is divided have seven DOFs. These describe the position of a chosen point, the extension of the element, and its orientation by means of the Euler angles Z[Formula: see text]Y[Formula: see text]X[Formula: see text]. Elements are connected by means of geometrical constraint equations. A compact algorithm for formulating and integrating the equations of motion is given. Models and programs are verified by comparing the results to those obtained by analytical solution and those from the finite element method. Finally, they are used to solve a benchmark problem encountered in nonlinear dynamic analysis of multibody systems.

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.

scholarly journals Stabilizing Ship Motion With a Dual System Inertial Disk

2019 ◽  
Author(s):  
Torstein R. Storaas ◽  
Kasper Virkesdal ◽  
Gitle S. Brekke ◽  
Thorstein Rykkje ◽  
Thomas Impelluso
Keyword(s):  
Oil And Gas ◽  
New Technologies ◽  
Equations Of Motion ◽  
Moving Frame ◽  
Moving Frames ◽  
New Approach ◽  
Lie Group Theory ◽  
Moving Frame Method ◽  
Frame Method ◽  
Modern Mathematics

Abstract Norwegian industries are constantly assessing new technologies and methods for more efficient and safer maintenance in the aqua cultural, renewable energy, and oil and gas industries. These Norwegian offshore industries share a common challenge: to install new equipment and transport personnel in a safe and controllable way between ships, farms and platforms. This paper deploys the Moving Frame Method (MFM) to analyze ship stability moderated by a dual gyroscopic inertial device. The MFM describes the dynamics of the system using modern mathematics. Lie group theory and Cartan’s moving frames are the foundation of this new approach to engineering dynamics. This, together with a restriction on the variation of the angular velocity used in Hamilton’s principle, enables an effective way of extracting the equations of motion. This project extends previous work. It accounts for the dual effect of two inertial disk devices, it accounts for the prescribed spin of the disks. It separates out the prescribed variables. This work displays the results in 3D on cell phones. It represents a prelude to testing in a wave tank.

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