Nonlinear responses of unbalanced flexible rotating shaft passing through critical speeds

Meccanica ◽  
2021 ◽  
Author(s):  
Sadegh Amirzadegan ◽  
Mohammad Rokn-Abadi ◽  
R. D. Firouz-Abadi ◽  
Fahimeh Mehralian
Keyword(s):  
Rotating Shaft ◽  
Critical Speeds ◽  
Nonlinear Responses

Optimization of Nonlinear Unbalanced Flexible Rotating Shaft Passing Through Critical Speeds

2021 ◽  
Author(s):  
Sadegh Amirzadegan ◽  
Mohammad Rokn-Abadi ◽  
R. D. Firouz-Abadi
Keyword(s):  
Degrees Of Freedom ◽  
Nonlinear Oscillations ◽  
Nonlinear Differential Equations ◽  
Equations Of Motion ◽  
Angular Acceleration ◽  
Beam Theory ◽  
Rotor System ◽  
Rotating Shaft ◽  
Critical Speeds ◽  
Applied Torque

This work studies the nonlinear oscillations of an elastic rotating shaft with acceleration to pass through the critical speeds. A mathematical model incorporating the Von-Karman higher-order deformations in bending is developed to investigate the nonlinear dynamics of rotors. A flexible shaft on flexible bearings with springs and dampers is considered as rotor system for this work. The shaft is modeled as a beam and the Euler–Bernoulli beam theory is applied. The kinetic and strain energies of the rotor system are derived and Lagrange method is then applied to obtain the coupled nonlinear differential equations of motion for 6 degrees of freedom. In order to solve these equations numerically, the finite element method (FEM) is used. Furthermore, for different bearing properties, rotor responses are examined and curves of passing through critical speeds with angular acceleration due to applied torque are plotted. Then the optimal values of bearing stiffness and damping are calculated to achieve the minimum vibration amplitude, which causes to pass easier through critical speeds. It is concluded that the value of damping and stiffness of bearing change the rotor critical speeds and also significantly affect the dynamic behavior of the rotor system. These effects are also presented graphically and discussed.

On the Dynamical Behavior of Rotating Shafts Driven by Universal (Hooke) Couplings

1958 ◽  
Vol 25 (1) ◽  
pp. 47-51
Author(s):  
R. M. Rosenberg
Keyword(s):  
Critical Speed ◽  
Dynamical Behavior ◽  
Rotating Shaft ◽  
Critical Speeds ◽  
Elastic Shaft ◽  
Rotating Shafts

Abstract The system considered here is a massless, uniform elastic shaft carrying at its mid-point a disk (having mass) and supported at the ends by universal (Hooke) joints. The purpose of this investigation is to examine the effect of Hooke-joint angularity (as obtained by design, or from faulty alignment) on the bending stability of the rotating shaft. It is found that separate investigations are required for shafts not transmitting axial torques and for those required to transmit torques. Each gives rise to instabilities which are absent when the Hooke joint is straight. In the absence of axial torques, the shaft develops unsuspected mild critical speeds at odd integer submultiples of the “familiar” critical speed found with a straight Hooke joint. When the shaft is required to transmit moderate axial torques, the joint angularity produces true instabilities near all integer submultiples of the familiar critical speed. Surprisingly, these instabilities vanish for sufficiently large axial torques.

On Critical Speeds of a Shaft Supported by a Ball Bearing

1959 ◽  
Vol 26 (2) ◽  
pp. 199-204
Author(s):  
Toshio Yamamoto
Keyword(s):  
Ball Bearing ◽  
The Other ◽  
Ball Bearings ◽  
Rotating Shaft ◽  
Critical Speeds

Abstract The author points out that two kinds of critical speeds induced by a slight difference in the diameter of balls in a ball bearing appear in a rotating shaft supported by ball bearings. These two critical speeds have peculiar modes of vibrations which are determined by dimensions of ball bearings; one of them is motion of forward precession, the other is backward precession. The paper describes the cause of these critical speeds and the behavior of those vibrations.

Critical Speeds of Two Bearing Machines With Overhung Weight

1961 ◽  
Vol 83 (4) ◽  
pp. 377-379 ◽  
Author(s):  
Reyton F. Wojnowski ◽  
Thomas R. Faucett
Keyword(s):  
Dimensionless Form ◽  
Rotating Shaft ◽  
Critical Speeds ◽  
Rapid Calculation ◽  
Modes Of Vibration ◽  
General Frequency

The purpose of this study was to determine the general frequency expression for a rotating shaft with uniformly distributed weight, supported by two bearings, and carrying a concentrated weight at the free end. The bearing spacing and the ratio of the concentrated weight to the total distributed weight have been used as parameters. The data have thus been reduced to dimensionless form so that the results are generally applicable for this type of machine. Frequencies for the first three modes of vibration have been determined and curves plotted for rapid calculation of these frequencies.

On the Unstable Vibrations of a Shaft Carrying an Unsymmetrical Rotor

1964 ◽  
Vol 31 (3) ◽  
pp. 515-522 ◽  
Author(s):  
Toshio Yamamoto ◽  
Hiroshi O¯ta
Keyword(s):  
Natural Frequencies ◽  
Rotating Shaft ◽  
Unstable Region ◽  
Critical Speeds ◽  
Rotating Speed ◽  
Lateral Vibrations ◽  
Shaft System

In a rotating shaft system carrying an unsymmetrical rotor, there is always one unstable region in the neighborhood of the rotating speed at which the sum of two natural frequencies of the system is equal to twice the rotating speed of the shaft. In this unstable region two unstable lateral vibrations with frequencies P1 and P2 take place simultaneously and grow up steadily. Generally, frequencies P1 and P2 are not equal to the rotating speed ω of the shaft and the sum of these P1 + P2 is always equal to 2ω. Of course there are other unstable regions which appear at the major critical speeds.

A New Method for Predicting Critical Speeds in Rotordynamics

2015 ◽  
Vol 138 (2) ◽  
Author(s):  
Lawrence N. Virgin ◽  
Josiah D. Knight ◽  
Raymond H. Plaut
Keyword(s):  
Critical Speed ◽  
Industrial Applications ◽  
Rotating Shaft ◽  
Simple Method ◽  
Modeling And Analysis ◽  
Critical Speeds ◽  
Disk Storage ◽  
In Situ Conditions ◽  
Pass Through ◽  
Using Data

The prediction of critical speeds of a rotating shaft is a crucial issue in a variety of industrial applications ranging from turbomachinery to disk storage systems. The modeling and analysis of rotordynamic systems is subject to a number of complications, but perhaps the most important characteristic is to pass through a critical speed under spin-up conditions. This is associated with classical resonance phenomena and high amplitudes, and is often a highly undesirable situation. However, given uncertainties in the modeling of such systems, it can be very difficult to predict critical speeds based on purely theoretical considerations. Thus, it is clearly useful to gain knowledge of the critical speeds of rotordynamic systems under in situ conditions. The present study describes a relatively simple method to predict the first critical speed using data from low rotational speeds. The method is shown to work well for two standard rotordynamic models, and with data from experiments conducted during this study.

A Hybrid Method for the Vibration Analysis of Rotor-Bearing Systems

1991 ◽  
Vol 205 (2) ◽  
pp. 131-137 ◽  
Author(s):  
R Firoozian ◽  
H Zhu
Keyword(s):  
Finite Element ◽  
Transfer Matrix ◽  
Hybrid Method ◽  
Transfer Matrix Method ◽  
Matrix Method ◽  
Vibration Analysis ◽  
Mode Shapes ◽  
Rotating Shaft ◽  
Critical Speeds ◽  
Rotor Bearing

The transfer matrix method together with a digital computer form the foundation of the dynamic analysis of rotor-bearing systems. The properties of each segment of the rotating shaft are expressed in simple matrix form and the overall dynamic behaviour is then obtained by successive multiplication of the element matrices. The main drawback associated with this method is the numerical instability in calculating natural frequencies for complex systems. The finite element method, on the other hand, uses the element stiffness and mass matrices to form the global equation of motion for the complete system. This avoids the numerical problems of the transfer matrix method at the expense of the computer memory requirements. The new method described in this paper combines the transfer matrix and finite element techniques to form a powerful algorithm for vibration analysis of rotor-bearing systems. It is shown that the accuracy improves significantly when the transfer matrix for each shaft segment is obtained from finite element techniques. The accuracy and efficiency of the hybrid method are compared with the transfer matrix method for a simply supported uniform rotating shaft where an analytical solution for the critical speeds and mode shapes is available. The method is then applied to a flexibly supported uniform shaft and a non-uniform shaft with a large disc to show the capability of the method for finding the critical speeds of complex rotor-bearing systems.

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