Analysis of Nonlinear Vibration for the Cracked Rotors With Unsymmetrical Viscoelastic Supported Conditions

2005 ◽  
Author(s):  
Changping Chen ◽  
Liming Dai ◽  
Yiming Fu
Keyword(s):  
Nonlinear Vibration ◽  
Nonlinear Vibrations ◽  
Equations Of Motion ◽  
Rotor System ◽  
Governing Equations ◽  
Response Curves ◽  
Disc Position ◽  
On Line ◽  
Cracked Rotor System ◽  
Line Crack

The nonlinear governing equations of motion for the cracked rotor system with unsymmetrical viscoelastic supported condition are derived and the nonlinear vibrations of the system are analyzed. The effects of the cracked depth, the cracked position and the disc position on the response curves of the rotational speed-the nonlinear vibration amplitude and the response curve of the nonlinear amplitude-frequency are discussed in detail. The results can be used for the on-line crack monitor of the rotor system.

Detection of a Rotor Crack Based on the Nonlinear Vibration Diagnosis Using Periodic Excitation

2005 ◽  
Author(s):  
Yukio Ishida ◽  
Tsuyoshi Inoue
Keyword(s):  
Nonlinear Vibration ◽  
Equations Of Motion ◽  
Frequency Component ◽  
Dominant Frequency ◽  
Vibration Characteristics ◽  
Periodic Excitation ◽  
Cracked Rotor ◽  
On Line ◽  
Linear And Nonlinear ◽  
Excitation Force

Detection of a rotor crack based on the nonlinear vibration diagnosis using periodic excitation force is investigated. Due to the open-close mechanism of the crack, the equations of motion of a cracked rotor have linear and nonlinear parametric terms. When a periodic excitation force is applied to the cracked rotor, various kinds of resonances due to the unique vibration characteristics of a crack. Furthermore, types of resonances, resonance points and dominant frequency component of these resonances are clarified theoretically and experimentally. These results enable us to detect a crack on-line without stopping the system.

Negative Effective Stiffness Content in Cracked Rotors

2018 ◽  
Author(s):  
Mohammad A. AL-Shudeifat ◽  
Fatima K. Alhammadi
Keyword(s):  
Equations Of Motion ◽  
Rotor System ◽  
Effective Stiffness ◽  
Cracked Rotor ◽  
Force Vector ◽  
Rotor Systems ◽  
Wide Range ◽  
Unbalance Force ◽  
Time Periodic ◽  
Cracked Rotor System

The appearance of cracks in rotor systems affects the whirl response in the neighborhood of the critical whirl rotational speeds. The combined effect of the crack depth and the unbalance force vector angle orientation with respect to the crack opening direction on the effective stiffness content of the cracked rotor system in the neighborhood of the critical rotational speed is addressed here. The effective stiffness expression of the cracked system can be obtained from the direct integration of the equations of motion of the cracked rotor system. The cracked rotor equations of motion can be expressed by the Jeffcott rotor or the finite element models. The appearance of cracks in rotor systems converts them into parametrically excited dynamical systems with time-periodic stiffness components. The interaction between the time-periodic stiffness and the external periodic forcing function of the unbalance force significantly alters the effective stiffness content in the system at both transient and steady state operations. For wide range of crack depths and unbalance force vector angles, the effective stiffness has been found to be of negative values. This means that the cracked rotor system tends to have more resistance to deflect towards the center of its whirl orbit and less resistance to deflect away under the unbalance force excitation effect. Consequently, in the negative stiffness content zone of the unbalance force vector angles, the cracked rotor system tends to exhibit a sharp growth in the vibration whirl amplitudes. However, for positive effective stiffness values, the shaft has more resistance to deflect away from its whirl orbit center. Therefore, the cracked rotor system is at higher risk of failure in the negative effective stiffness zone of unbalance force vector angles than the positive effective stiffness zone of these angles.

Application of Variational Equation of Motion to the Nonlinear Vibration Analysis of Homogeneous and Layered Plates and Shells

1963 ◽  
Vol 30 (1) ◽  
pp. 79-86 ◽  
Author(s):  
Yi-Yuan Yu
Keyword(s):  
Nonlinear Vibration ◽  
Nonlinear Vibrations ◽  
Variational Equation ◽  
Equations Of Motion ◽  
Equation Of Motion ◽  
Free Vibrations ◽  
Layered Plates ◽  
Variational Approximations ◽  
Elastic Systems ◽  
Plates And Shells

An integrated procedure is presented for applying the variational equation of motion to the approximate analysis of nonlinear vibrations of homogeneous and layered plates and shells involving large deflections. The procedure consists of a sequence of variational approximations. The first of these involves an approximation in the thickness direction and yields a system of equations of motion and boundary conditions for the plate or shell. Subsequent variational approximations with respect to the remaining space coordinates and time, wherever needed, lead to a solution to the nonlinear vibration problem. The procedure is illustrated by a study of the nonlinear free vibrations of homogeneous and sandwich cylindrical shells, and it appears to be applicable to still many other homogeneous and composite elastic systems.

scholarly journals Experimental Study on the Nonlinear Vibrations and n× Amplitudes of a Rotor With a Transverse Crack

2009 ◽  
Vol 131 (4) ◽  
Author(s):  
Jean-Jacques Sinou
Keyword(s):  
Experimental Study ◽  
Theoretical Analysis ◽  
Monitoring System ◽  
Nonlinear Response ◽  
Transverse Crack ◽  
Nonlinear Vibrations ◽  
Rotor System ◽  
Experimental Results ◽  
Frequency Components ◽  
On Line

In this paper, the nonlinear response of a rotor system containing a transverse crack is analyzed experimentally in order to propose a nondestructive detection of cracks in the rotor. More particularly, the evolutions of the n× superharmonic frequency components at the various subcritical resonant peaks and the decrease in the subcritical resonant speeds are investigated for various crack depths. The experimental results that are presented in this study confirm the theoretical analysis of many researchers and provide a possible basis for an on-line monitoring system.

scholarly journals Rub-Impact Force Induces Periodic, Quasiperiodic, and Chaotic Motions of a Controlled Asymmetric Rotor System

2021 ◽  
Vol 2021 ◽  
pp. 1-27
Author(s):  
N. A. Saeed ◽  
Emad Mahrous Awwad ◽  
Ali Maarouf ◽  
Hassan M. H. Farh ◽  
Fahd A. Alturki ◽  
...  
Keyword(s):  
Impact Force ◽  
Equations Of Motion ◽  
Rotor System ◽  
Control Law ◽  
Slow Flow ◽  
Response Curves ◽  
Chaotic Motions ◽  
Flow Equations ◽  
Impact Forces ◽  
Asymmetric Rotor

This article aims to explore the oscillatory characteristics of a controlled asymmetric rotor system when subjected to rub and impact forces between the rotor and stator. Four electromagnetic poles are used to control the whirling motion of the rotor system through a linear proportional-derivative control law. The equations of motion that govern the whole system dynamics are derived including the rub and impact forces. The derived mathematical model is analyzed in two basic steps. Firstly, the obtained model is treated as a weakly nonlinear system using perturbation analysis to obtain the slow-flow modulating equations when neglecting the rub and impact forces. Depending on the obtained slow-flow equations, different response curves are plotted to explore the system’s periodic vibrations and determine the conditions at which the system can exhibit rub and impact force. Secondly, the whole system model including the rub and impact forces is investigated by using the bifurcation diagrams, Poincare map, frequency spectrums, and temporal oscillations. The obtained results revealed that the applied control law could mitigate the system whirling oscillations and prevent the rub and impact forces if the control gains are tuned properly. However, the system can perform period-n, quasiperiodic, or chaotic motion depending on the shaft spinning speed if the controller fails to eliminate the contact between the rotor and stator.

Research on Lateral Nonlinear Vibration Behavior of Composite Shaft-Disk Rotor System

2018 ◽  
Vol 875 ◽  
pp. 149-161 ◽  
Author(s):  
Chun Jin Zhang ◽  
Yong Sheng Ren ◽  
Shu Juan Ji
Keyword(s):  
Nonlinear Vibration ◽  
Strain Energy ◽  
Nonlinear Deformation ◽  
Lagrange Equation ◽  
Rotor System ◽  
Kutta Method ◽  
Response Curves ◽  
Runge Kutta Method ◽  
Composite Shaft ◽  
Ihb Method

The characteristics of the lateral nonlinear vibration in composite shaft-disk rotor system with nonlinear deformation are studied. Firstly, the equations of the kinetic energy of the composite shaft, the disk and the eccentric mass as well as the equations of the strain energy of the composite shaft are derived. Based on these equations, the nonlinear vibration equations are deduced by using the Lagrange equation. Then, the frequency response curves and time response curves of the system are obtained by using the IHB method and verified by the fourth order Runge-Kutta method. Experimental results show that the external damping coefficient, the size of eccentric mass only influences the nonlinear amplitude. Moreover, the ply-angle, thickness to diameter(T/D) ratio, length to diameter (L/D) ratio, and the position of disk in the shaft not only produce an effect on the nonlinear amplitude, but also influence the nonlinear vibration frequency.

Strongly Nonlinear Vibrations of a Hyperelastic Thin-walled Cylindrical Shell Based on the Modified Lindstedt–Poincaré Method

2019 ◽  
Vol 19 (12) ◽  
pp. 1950160 ◽  
Author(s):  
Jing Zhang ◽  
Jie Xu ◽  
Xuegang Yuan ◽  
Wenzheng Zhang ◽  
Datian Niu
Keyword(s):  
Cylindrical Shell ◽  
Nonlinear System ◽  
Differential Equations ◽  
Nonlinear Vibrations ◽  
Harmonic Excitation ◽  
System Of Differential Equations ◽  
Response Curves ◽  
Strongly Nonlinear ◽  
Hardening Behavior ◽  
Thin Walled

Some significant behaviors on strongly nonlinear vibrations are examined for a thin-walled cylindrical shell composed of the classical incompressible Mooney–Rivlin material and subjected to a single radial harmonic excitation at the inner surface. First, with the aid of Donnell’s nonlinear shallow-shell theory, Lagrange’s equations and the assumption of small strains, a nonlinear system of differential equations for the large deflection vibration of a thin-walled shell is obtained. Second, based on the condensation method, the nonlinear system of differential equations is reduced to a strongly nonlinear Duffing equation with a large parameter. Finally, by the appropriate parameter transformation and modified Lindstedt–Poincar[Formula: see text] method, the response curves for the amplitude-frequency and phase-frequency relations are presented. Numerical results demonstrate that the geometrically nonlinear characteristic of the shell undergoing large vibrations shows a hardening behavior, while the nonlinearity of the hyperelastic material should weak the hardening behavior to some extent.

Simulation of Engine Vibration on Nonlinear Hydraulic Engine Mounts

2005 ◽  
Author(s):  
A. R. Ohadi ◽  
G. Maghsoodi
Keyword(s):  
Equations Of Motion ◽  
Low Frequency ◽  
Governing Equations ◽  
Engine Mounts ◽  
Engine Vibration ◽  
Engine Mount ◽  
Rotational Motions ◽  
The Time Domain ◽  
Nonlinear Equations Of Motion ◽  
Four Cylinders

In this paper, vibration behavior of engine on nonlinear hydraulic engine mount including inertia track and decoupler is studied. In this regard, after introducing the nonlinear factors of this mount (i.e. inertia and decoupler resistances in turbulent region), the vibration governing equations of engine on one hydraulic engine mount are solved and the effect of nonlinearity is investigated. In order to have a comparison between rubber and hydraulic engine mounts, a 6 degree of freedom four cylinders V-shaped engine under inertia and balancing masses forces and torques is considered. By solving the time domain nonlinear equations of motion of engine on three inclined mounts, translational and rotational motions of engines body are obtained for different engine speeds. Transmitted base forces are also determined for both types of engine mount. Comparison of rubber and hydraulic mounts indicates the efficiency of hydraulic one in low frequency region.

Reduction of Noise Inside a Cavity by Piezoelectric Actuators

2003 ◽  
Vol 125 (1) ◽  
pp. 12-17 ◽  
Author(s):  
I. Hagiwara ◽  
D. W. Wang ◽  
Q. Z. Shi ◽  
R. S. Rao
Keyword(s):  
Sound Pressure ◽  
Control Mechanism ◽  
Equations Of Motion ◽  
Piezoelectric Actuators ◽  
Governing Equations ◽  
Control Laws ◽  
Modal Coupling ◽  
Modal Amplitude ◽  
Global And Local ◽  
Fully Coupled

A new analytical model is developed for the reduction of noise inside a cavity using distributed piezoelectric actuators. A modal coupling method is used to establish the governing equations of motion of the fully coupled acoustics-structure-piezoelectric patch system. Two performance functions relating “global” and “local” optimal control of sound pressure levels (SPL) respectively are applied to obtain the control laws. The discussions on associated control mechanism show that both the mechanisms of modal amplitude suppression and modal rearrangement may sometimes coexist in the implementation of optimal noise control.

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