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.

Vibration Control of Two-Degree-of-Freedom Structures Utilizing Sloshing in Nearly Square Tanks

2017 ◽  
Vol 12 (5) ◽  
Author(s):  
Takashi Ikeda ◽  
Yuji Harata ◽  
Shota Ninomiya
Keyword(s):  
Vibration Control ◽  
Equations Of Motion ◽  
Excitation Frequency ◽  
Harmonic Excitation ◽  
Degree Of Freedom ◽  
Response Curves ◽  
Chaotic Motions ◽  
Installation Angle ◽  
Two Degree Of Freedom ◽  
Theoretical Results

This paper investigates the vibration control of a towerlike structure with degrees of freedom utilizing a square or nearly square tuned liquid damper (TLD) when the structure is subjected to horizontal, harmonic excitation. In the theoretical analysis, when the two natural frequencies of the two-degree-of-freedom (2DOF) structure nearly equal those of the two predominant sloshing modes, the tuning condition, 1:1:1:1, is nearly satisfied. Galerkin's method is used to derive the modal equations of motion for sloshing. The nonlinearity of the hydrodynamic force due to sloshing is considered in the equations of motion for the 2DOF structure. Linear viscous damping terms are incorporated into the modal equations to consider the damping effect of sloshing. Van der Pol's method is employed to determine the expressions for the frequency response curves. The influences of the excitation frequency, the tank installation angle, and the aspect ratio of the tank cross section on the response curves are examined. The theoretical results show that whirling motions and amplitude-modulated motions (AMMs), including chaotic motions, may occur in the structure because swirl motions and Hopf bifurcations, followed by AMMs, appear in the tank. It is also found that a square TLD works more effectively than a conventional rectangular TLD, and its performance is further improved when the tank width is slightly increased and the installation angle is equal to zero. Experiments were conducted in order to confirm the validity of the theoretical results.

Vibration Control of 2DOF Structures Utilizing Sloshing in Square Tanks

2012 ◽  
Author(s):  
Takashi Ikeda ◽  
Yuji Harata ◽  
Shota Ninomiya
Keyword(s):  
Aspect Ratio ◽  
Vibration Control ◽  
Equations Of Motion ◽  
Excitation Frequency ◽  
Harmonic Excitation ◽  
Tuned Liquid Damper ◽  
Response Curves ◽  
Chaotic Motions ◽  
Installation Angle ◽  
Theoretical Results

This paper investigates the vibration control of a tower-like structure utilizing a square tuned liquid damper (TLD) when the structure is subjected to horizontal, harmonic excitation. In the theoretical analysis, when the tuning condition, 1:1:1:1, is satisfied, the natural frequencies of the 2DOF structure and two predominate sloshing modes are nearly equal. Galerkin’s method is used to derive the modal equations of motion for sloshing. The nonlinearity of the hydrodynamic force due to sloshing is considered in the equations of motion for the 2DOF structure. Linear viscous damping terms are incorporated into the modal equations to consider the damping effect of sloshing. Van der Pol’s method is employed to determine the frequency response curves which are compared to the numerical simulation. The influences of the excitation frequency, the tank installation angle, and the aspect ratio of the tank cross-section on the response curves are examined. The theoretical results show that whirling motions and amplitude modulated motions (AMMs) including chaotic motions may occur in the structure because swirl motions and Hopf bifurcations followed by AMMs appear in the tank. It is also found that square TLDs work more efficiently than conventional rectangular TLDs, and its performance is further improved when the aspect ratio is slightly increased and the installation angle is equal to zero. Experiments were conducted in order to confirm the validity of the theoretical results.

Nonlinear liquid sloshing in a square tank subjected to obliquely horizontal excitation

2012 ◽  
Vol 700 ◽  
pp. 304-328 ◽  
Author(s):  
Takashi Ikeda ◽  
Raouf A. Ibrahim ◽  
Yuji Harata ◽  
Tasuku Kuriyama
Keyword(s):  
Theoretical Analysis ◽  
Frequency Response ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Excitation Frequency ◽  
Frequency Range ◽  
Response Curves ◽  
Chaotic Motions ◽  
Nonlinear Responses ◽  
Good Agreement

AbstractNonlinear responses of surface waves in rigid square and nearly square tanks partially filled with liquid subjected to obliquely horizontal, sinusoidal excitation are investigated theoretically and experimentally. Two predominant modes of sloshing are significantly coupled nonlinearly because their natural frequencies are nearly identical resulting in 1:1 internal resonance. Therefore, if only one of these modes is directly excited, the other mode is indirectly excited due to the nonlinear coupling. In the nonlinear theoretical analysis, the modal equations of motion are derived for the two predominant sloshing modes as well as five higher sloshing modes. The linear viscous terms are incorporated in order to consider the damping effect of sloshing. The expressions for the frequency response curves are determined using van der Pol’s method. The influences of the excitation direction and the aspect ratio of the tank cross-section on the frequency response curves are numerically examined. Planar and swirl motions of sloshing, and Hopf bifurcations followed by amplitude modulated motions including chaotic motions, are predicted when the excitation frequency is close to one of the natural frequencies of the two predominant sloshing modes. Lyapunov exponents are calculated and reveal the excitation frequency range over which liquid chaotic motions occur. In addition, bifurcation sets are shown to clarify the influences of the parameters on the change in the structural stability. The theoretically predicted results are in good agreement with the measured data, thus the theoretical analysis was experimentally validated.

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.

scholarly journals Numerical Investigation on the Gravity Response of a Two-Pole Generator Rotor System with Interval Uncertainties

2019 ◽  
Vol 9 (15) ◽  
pp. 3036 ◽  
Author(s):  
Zhaoli Zheng ◽  
Yonghui Xie ◽  
Di Zhang
Keyword(s):  
Material Properties ◽  
Equations Of Motion ◽  
Deep Understanding ◽  
Element Model ◽  
Rotor System ◽  
Scanning Method ◽  
Rotor Systems ◽  
Generator Rotor ◽  
Gravity Response ◽  
Asymmetric Rotor

Asymmetric rotor systems widely exist in commercial plants. In the previous studies about asymmetric rotor systems, parameters such as material properties and boundary conditions are deterministic. To obtain a deep understanding of the dynamics of asymmetric rotor systems, a generator rotor system considering uncertain factors is studied in this paper. The equations of motion of the three-dimensional finite element model are solved in the rotating frame. The component mode synthesis is used to reduce the degrees of freedom. By employing the Chebyshev interval method (CIM), the uncertain gravity responses of the generator rotor system are investigated. The influences of the uncertainties in the bearing’s properties and the rotor’s material properties on the gravity response are studied in cases with a single uncertainty and double uncertainties. The accuracy and the efficiency of CIM are validated by comparing with the results of the scanning method. The results show that uncertainties have remarkable influences on the gravity response, and that these influences are different from each other. The proposed method and the results can provide guidance to the design and optimization of the rotary machinery.

Influence of Attached Masses on Impact Forces and Running Style in Heel-Toe Running

1987 ◽  
Vol 3 (3) ◽  
pp. 264-275 ◽  
Author(s):  
Alexander Bahlsen ◽  
Benno M. Nigg
Keyword(s):  
Body Mass ◽  
High Speed ◽  
Impact Force ◽  
Force Plate ◽  
The Body ◽  
Additional Mass ◽  
Impact Forces ◽  
Running Shoes ◽  
High Speed Film ◽  
Vertical Impact

Impact forces analysis in heel-toe running is often used to examine the reduction of impact forces for different running shoes and/or running techniques. Body mass is reported to be a dominant predictor of vertical impact force peaks. However, it is not evident whether this finding is only true for the real body mass or whether it is also true for additional masses attached to the body (e.g., running with additional weight or heavy shoes). The purpose of this study was to determine the effect of additional mass on vertical impact force peaks and running style. Nineteen subjects (9 males, 10 females) with a mean mass of 74.2 kg/56.2 kg (SD = 10.0 kg and 6.0 kg) volunteered to participate in this study. Additional masses were attached to the shoe (.05 and .1 kg), the tibia (.2, .4, .6 kg), and the hip (5.9 and 10.7 kg). Force plate measurements and high-speed film data were analyzed. In this study the vertical impact force peaks, Fzi, were not affected by additional masses, the vertical active force peaks, Fza, were only affected by additional masses greater than 6 kg, and the movement was only different in the knee angle at touchdown, ϵ0, for additional masses greater than .6 kg. The results of this study did not support findings reported earlier in the literature that body mass is a dominant predictor of external vertical impact force peaks.

Mulit-Pulse Orbits and Chaotic Motion of Composite Laminated Plates

2008 ◽  
Author(s):  
Xiangying Guo ◽  
Wei Zhang ◽  
Ming-Hui Yao
Keyword(s):  
Numerical Simulation ◽  
Normal Form ◽  
Differential Equations ◽  
Thin Plate ◽  
Equations Of Motion ◽  
Laminated Plates ◽  
Phase Method ◽  
Rectangular Thin Plate ◽  
Partial Differential ◽  
Chaotic Motions

This paper presents an analysis on the nonlinear dynamics and multi-pulse chaotic motions of a simply-supported symmetric cross-ply composite laminated rectangular thin plate with the parametric and forcing excitations. Firstly, based on the Reddy’s three-order shear deformation plate theory and the model of the von Karman type geometric nonlinearity, the nonlinear governing partial differential equations of motion for the composite laminated rectangular thin plate are derived by using the Hamilton’s principle. Then, using the second-order Galerkin discretization approach, the partial differential governing equations of motion are transformed to nonlinear ordinary differential equations. The case of the primary parametric resonance and 1:1 internal resonance is considered. Four-dimensional averaged equation is obtained by using the method of multiple scales. From the averaged equation obtained here, the theory of normal form is used to give the explicit expressions of normal form. Based on normal form, the energy phase method is utilized to analyze the global bifurcations and multi-pulse chaotic dynamics of the composite laminated rectangular thin plate. The results obtained above illustrate the existence of the chaos for the Smale horseshoe sense in a parametrical and forcing excited composite laminated thin plate. The chaotic motions of the composite laminated rectangular thin plate are also found by using numerical simulation. The results of numerical simulation also indicate that there exist different shapes of the multi-pulse chaotic motions for the composite laminated rectangular thin plate.

Slow flow solutions and stability analysis of single machine to infinite bus power systems

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
M. G. Suresh Kumar ◽  
C. A. Babu
Keyword(s):  
Power Systems ◽  
Power System ◽  
Single Machine ◽  
Multiple Scales ◽  
Operating Conditions ◽  
Phase Portraits ◽  
Slow Flow ◽  
Approximate Analytical Expression ◽  
Flow Equations ◽  
Load Angle

Abstract Nonlinearity is a major constraint in analysing and controlling power systems. The behaviour of the nonlinear systems will vary drastically with changing operating conditions. Hence a detailed study of the response of the power system with nonlinearities is necessary especially at frequencies closer to natural resonant frequencies of machines where the system may jump into the chaos. This paper attempt such a study of a single machine to infinite bus power system by modelling it as a Duffing equation with softening spring. Using the method of multiple scales, an approximate analytical expression which describes the variation of load angle is derived. The phase portraits generated from the slow flow equations, closer to the jump, display two stable equilibria (centers) and an unstable fixed point (saddle). From the analysis, it is observed that even for a combination of parameters for which the system exhibits jump resonance, the system will remain stable if the variation of load angle is within a bounded region.

The Effects of Upstream Flexible Barrier on the Debris Flow Entrainment and Impact Dynamics on a Terminal Barrier

2021 ◽  
Author(s):  
Hervé Vicari ◽  
C.W.W. Ng ◽  
Steinar Nordal ◽  
Vikas Thakur ◽  
W.A. Roanga K. De Silva ◽  
...  
Keyword(s):  
Debris Flow ◽  
Debris Flows ◽  
Impact Force ◽  
Dynamic Pressure ◽  
Pressure Coefficient ◽  
Flexible Barrier ◽  
Impact Forces ◽  
Flexible Barriers ◽  
The Impact ◽  
Bed Entrainment

The destructive nature of debris flows is mainly caused by flow bulking from entrainment of an erodible channel bed. To arrest these flows, multiple flexible barriers are commonly installed along the predicted flow path. Despite the importance of an erodible bed, its effects are generally ignored when designing barriers. In this study, three unique experiments were carried out in a 28 m-long flume to investigate the impact of a debris flow on both single and dual flexible barriers installed in a channel with a 6 m-long erodible soil bed. Initial debris volumes of 2.5 m<sup>3</sup> and 6 m<sup>3</sup> were modelled. For the test setting adopted, a small upstream flexible barrier before the erodible bed separates the flow into several surges via overflow. The smaller surges reduce bed entrainment by 70% and impact force on the terminal barrier by 94% compared to the case without an upstream flexible barrier. However, debris overflowing the deformed flexible upstream barrier induces a centrifugal force that results in a dynamic pressure coefficient that is up to 2.2 times higher than those recommended in guidelines. This suggests that although compact upstream flexible barriers can be effective for controlling bed entrainment, they should be carefully designed to withstand higher impact forces.

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