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.

A Time-Varying Stiffness Rotor Active Magnetic Bearings Under Combined Resonance

2008 ◽  
Vol 75 (1) ◽  
Author(s):  
U. H. Hegazy ◽  
M. H. Eissa ◽  
Y. A. Amer
Keyword(s):  
Multiple Scales ◽  
Nonlinear Oscillations ◽  
Equations Of Motion ◽  
Multiple Scale ◽  
Magnetic Bearings ◽  
Time Varying ◽  
Active Magnetic Bearings ◽  
Gyroscopic Effects ◽  
Nonlinear Equations Of Motion ◽  
Combined Resonance

This paper is concerned with the nonlinear oscillations and dynamic behavior of a rigid disk-rotor supported by active magnetic bearings (AMB), without gyroscopic effects. The nonlinear equations of motion are derived considering a periodically time-varying stiffness. The method of multiple scales is applied to obtain four first-order differential equations that describe the modulation of the amplitudes and the phases of the vibrations in the horizontal and vertical directions. The stability and the steady-state response of the system at a combination resonance for various parameters are studied numerically, applying the frequency response function method. It is shown that the system exhibits many typical nonlinear behaviors, including multiple-valued solutions, jump phenomenon, hardening, and softening nonlinearity. A numerical simulation using a fourth-order Runge-Kutta algorithm is carried out, where different effects of the system parameters on the nonlinear response of the rotor are reported and compared to the results from the multiple scale analysis. Results are compared to available published work.

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.

Nonlinear Dynamic Response of a Tension Leg Platform

1999 ◽  
Vol 121 (4) ◽  
pp. 219-226 ◽  
Author(s):  
P. Bar-Avi
Keyword(s):  
Environmental Conditions ◽  
Nonlinear Dynamic ◽  
Equations Of Motion ◽  
Continuous System ◽  
Offshore Structures ◽  
Physical Parameters ◽  
Wave Forces ◽  
Tension Leg Platform ◽  
Nonlinear Dynamic Response ◽  
Nonlinear Equations Of Motion

Of the classes of offshore structures, the tension leg platform (TLP) is particularly well suited for deepwater operation. The structure investigated in this paper is assumed to consist of a flexible cable attached to a buoyant deck at the top. The cable is modeled as a beamlike continuous system subjected to wave, current, and wind forces. The derivation of the nonlinear equations of motion include nonlinearities due to geometry as well as due to wave forces. The equations of motion are solved and the TLP’s response to various environmental conditions and other physical parameters is evaluated.

Coupled Lateral and Torsional Nonlinear Transient Rotor–Bearing System Analysis With Applications

2015 ◽  
Vol 137 (9) ◽  
Author(s):  
Jianming Cao ◽  
Paul Allaire ◽  
Timothy Dimond
Keyword(s):  
Rotational Speed ◽  
System Analysis ◽  
Equations Of Motion ◽  
Squeeze Film ◽  
Rotor Bearing ◽  
Nonlinear Transient ◽  
Gyroscopic Effects ◽  
Nonlinear Equations Of Motion ◽  
Torsional Analysis ◽  
Bearing System

This paper provides a time transient method for solving coupled lateral and torsional analysis of a flexible rotor–bearing system including gyroscopic effects, nonlinear short journal bearings, nonlinear short squeeze film dampers (SFDs), and external nonlinear forces/torques. The rotor is modeled as linear, and the supporting components, including bearings and dampers, are modeled as nonlinear. An implicit Runge–Kutta method is developed to solve the nonlinear equations of motion with nonconstant operating speed since the unbalance force and the gyroscopic effect are related to both the rotational speed and the acceleration. The developed method is compared with a previous torsional analysis first to verify the nonlinear transient solver. Then the coupled lateral and torsional analysis of an example flexible three-disk rotor, perhaps representing a compressor, with nonlinear bearings and nonlinear dampers driven by a synchronous motor is approached. The acceleration effects on lateral and torsional amplitudes of vibration are presented in the analysis. The developed method can be used to study the rotor motion with nonconstant rotational speed such as during startup, shutdown, going through critical speeds, blade loss force, or other sudden loading.

Linear Dynamic Coupling in Geared Rotor Systems

1986 ◽  
Vol 108 (2) ◽  
pp. 171-176 ◽  
Author(s):  
J. W. David ◽  
L. D. Mitchell
Keyword(s):  
Gear Tooth ◽  
Equations Of Motion ◽  
Dynamic Coupling ◽  
Rotating Shaft ◽  
Rotor Systems ◽  
Linear Dynamic ◽  
Heavy Lift ◽  
Reaction Forces ◽  
Coupling Terms ◽  
Nonlinear Equations Of Motion

The ability to analyze accurately the torsional-axial-lateral coupled response of geared systems is the key to the prediction of dynamic gear forces, shaft moments and torques, dynamic reaction forces, and moments at all bearing points. These predictions can, in turn, be used to estimate gear-tooth lives, shaft lives, housing vibrational response, and noise generation. Typical applications would be the design and analysis of gear drives in heavy-lift helicopters, industrial speed reducers, Naval propulsion systems, and heavy, off-road equipment. In this paper, the importance of certain linear dynamic coupling terms on the predicted response of geared rotor systems is addressed. The coupling terms investigated are associated with those components of a geared system that can be modeled as rigid disks. First, the coupled, nonlinear equations of motion for a disk attached to a rotating shaft are presented. The conventional argument for ignoring these dynamic coupling terms is presented and the error in this argument is revealed. It is shown that in a geared system containing gears with more than about 50 teeth, the magnitude of some of the dynamic-coupling terms is potentially as large as the magnitude of the linear terms that are included in most rotor analyses. In addition, it is shown that the dynamic coupling terms produce the multi-frequency responses seen in geared systems. To quantitatively determine the effects of the linear dynamic-coupling terms on the predicted response of geared rotor systems, a trial problem is formulated in which these effects are included. The results of this trial problem shows that the inclusion of the linear dynamic-coupling terms changed the predicted response up to eight orders of magnitude, depending on the response frequency. In addition, these terms are shown to produce sideband responses greater than the unbalanced response of the system.

Nonsmooth Bifurcations in a Cracked Rotor-Active Magnetic Bearings System

2014 ◽  
Vol 989-994 ◽  
pp. 2825-2828 ◽  
Author(s):  
Feng Hong Yang ◽  
Hong Zhi Tong
Keyword(s):  
Equations Of Motion ◽  
Magnetic Bearings ◽  
Active Magnetic Bearings ◽  
Periodic Motions ◽  
Cracked Rotor ◽  
Rotating Shaft ◽  
Chaotic Motions ◽  
Amb System ◽  
Nonlinear Equations Of Motion ◽  
Smooth System

A cracked rotor-active magnetic bearings (AMB) system with the time-varying stiffness is modeled by a piecewise smooth system due to the breath of crack in a rotating shaft. The governing nonlinear equations of motion for the nonsmooth system are established and solved with the numerical method. The simulation results show that a grazing bifurcation, period-double bifurcation and chaotic motions exist in the response. These nonsmooth bifurcations can give rise to jumps between periodic motions, quasi-periodic motions and chaos.

Vibration control of a continuous rotating shaft employing high-static low-dynamic stiffness isolators

2016 ◽  
Vol 24 (4) ◽  
pp. 760-783 ◽  
Author(s):  
Amirhassan Abbasi ◽  
SE Khadem ◽  
Saeed Bab
Keyword(s):  
Vibration Control ◽  
Dynamic Stiffness ◽  
Equations Of Motion ◽  
Time History ◽  
Beam Theory ◽  
Multiple Scale ◽  
Resonant Peak ◽  
Rotating Shaft ◽  
Mass Eccentricity ◽  
Linear And Nonlinear

In this paper, the effects of high-static low-dynamic stiffness (HSLDS) isolators on the supports of a continuous rotating shaft for vibration control of a rotary system under mass eccentricity force are investigated. The rotating shaft is modeled using the Euler–Bernoulli beam theory. HSLDS isolators have a linear damping and linear and nonlinear (cubic) equivalent stiffness. Isolators are positioned on the supports of the rotating shaft, so that their forces are applied in radial directions. Equations of motion are extracted using the extended Hamilton principle and they are analyzed using the multiple scale method; then, the steady-state solutions and stability are studied. The effects of variations in linear and nonlinear parameters of the isolators on the static load bearing, resonant peak, frequency band of isolation and hardening nonlinearity are considered, in order to design an appropriate HSLDS isolator and to set its parameters in an optimal way. Investigating the effects of the cubic stiffness and damping values on bifurcations of the system, one may observe that inappropriate setting of these parameters causes strong or weak nonlinearity in the system and, consequently, HSLDS isolators perform less effectively than a linear one does. Then, the results are verified through analyzing the time history of the rotary system under study.

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.

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