Observations of Subcritical Superharmonic and Chaotic Response in Rotordynamics

1992 ◽  
Vol 114 (1) ◽  
pp. 93-100 ◽  
Author(s):  
F. Ehrich
Keyword(s):  
Numerical Model ◽  
Nonlinear System ◽  
Natural Frequency ◽  
Transition Zone ◽  
Chaotic Behavior ◽  
Mirror Image ◽  
Wave Form ◽  
Operating Speed ◽  
Chaotic Response ◽  
Subharmonic Response

When a rotor, excited by unbalance, is operating eccentrically within a clearance and in local contact with the stator it behaves as a bilinear oscillator with a natural periodic motion that resembles bouncing. When excited by unbalance at a subcritical rotative speed which is exactly or nearly 1/Nsuper times its natural frequency, the nonlinear system will respond by bouncing at or nearly at its natural frequency, or superharmonically at a frequency exactly Nsuper times the operating speed or forcing frequency. As in supercritical subharmonic response, there is a zone with characteristics of chaotic behavior in the transition zone between any order of superharmonic response and the next highest order of superharmonic response. There is also an intricate pattern of progressive bifurcations of the orbit on entry into this characteristically chaotic region and a reverse progression on exit from this region. The response is a mirror image or reciprocal set of the more thoughly studied supercritical subharmonic response of the same bilinear oscillator system which, when excited by unbalance at a supercritical rotative speed which is exactly or nearly a whole number Nsub times its natural frequency, the nonlinear system will respond by bouncing at exactly or nearly its natural frequency at a frequency exactly 1/Nsub times the operating speed or forcing frequency. Such supercritical subharmonic response is also characterized by the appearance of characteristically chaotic behavior in the transition zone between successive orders of subharmonic response and by patterns of progressive bifurcations of the orbit on entry into and exit from each region of characteristically chaotic response. Various aspects of subcritical superharmonic response are studied in a numerical model of the nonlinear system, and are compared to data taken on the core spool of an aircraft engine gas turbine. The engine data show many of the unique characteristics of response, wave form, and spectral content predicted by the numerical model of the bilinear oscillator when operating at subcritical rotative speed.

Observations of Subcritical Superharmonic and Chaotic Response in Rotordynamics

1991 ◽  
Author(s):  
Fredric Ehrich
Keyword(s):  
Numerical Model ◽  
Nonlinear System ◽  
Natural Frequency ◽  
Transition Zone ◽  
Chaotic Behavior ◽  
Mirror Image ◽  
Wave Form ◽  
Operating Speed ◽  
Chaotic Response ◽  
Subharmonic Response

Abstract When a rotor, excited by unbalance, is operating eccentrically within a clearance and in local contact with the stator it behaves as a bilinear oscillator with a natural periodic motion that resembles bouncing. When excited by unbalance at a subcritical rotative speed which is exactly or nearly 1/Nsuper times its natural frequency, the nonlinear system will respond by bouncing at or nearly at its natural frequency, or super harmonically at a frequency exactly Nsuper times the operating speed, or forcing frequency. As in supercritical subharmonic response, there is a zone of with characteristics of chaotic behavior in the transition zone between any order of superharmonic response and the next highest order superharmonic response. There is also an intricate pattern of progressive bifurcations of the orbit on entry into this characteristically chaotic region and a reverse progression on exit from this region. The response is a mirror image or reciprocal set of the more thoughly studied supercritical subharmonic response of the same bilinear oscillator system which, when excited by unbalance at a supercritical rotative speed which is exactly or nearly a whole number Nsub times its natural frequency, the nonlinear system will respond by bouncing at exactly or nearly its natural frequency at a frequency exactly 1/Nsub times the operating speed or forcing frequency. Such supercritical subharmonic response is also characterized by the appearance of characteristically chaotic behavior in the transition zone between successive orders of subharmonic response and by patterns of progressive bifurcations of the orbit on entry into and exit from each region of characteristically chaotic response. Various aspects of subcritical superharmonic response are studied in a numerical model of the nonlinear system, and are compared to data taken on the core spool of an aircraft engine gas turbine. The engine data show many of the unique characteristics of response, wave form and spectral content predicted by the numerical model of the bilinear oscillator when operating at subcritical rotative speed.

Some Observations of Chaotic Vibration Phenomena in High-Speed Rotordynamics

1991 ◽  
Vol 113 (1) ◽  
pp. 50-57 ◽  
Author(s):  
F. F. Ehrich
Keyword(s):  
Numerical Model ◽  
High Speed ◽  
Chaotic Behavior ◽  
Narrow Region ◽  
System A ◽  
Vibratory Response ◽  
Local Contact ◽  
Jeffcott Rotor ◽  
Subharmonic Response ◽  
Static Part

Subharmonic response in rotordynamics may be encountered when a rotor is operated with its rotational centerline eccentric to that of a close clearance static part, so that local contact can take place during each orbit when the rotor is excited by residual unbalance. The rotor will tend to bounce at or near its fundamental frequency when the rotor is operated at or near a speed which is a whole number [n] times that frequency. Using a simple numerical model of a Jeffcott rotor mounted on a nonlinear spring, it is found that the vibratory response in the transition zone midway between adjacent zones of subharmonic response has all the characteristics of chaotic behavior. The transition from subharmonic to chaotic response has a complex substructure which involves a sequence of bifurcations of the orbit with variations in speed. This class of rotordynamic behavior was confirmed and illustrated by experimental observations of the vibratory response of a high-speed turbomachine, operating at a speed between 8 and 9 times its fundamental rotor frequency when in local contact across a clearance in the support system. A narrow region between zones of 8th order and 9th order subharmonic response was identified where the response had all the characteristics of the chaotic motion identified in the numerical model.

Morphology of Dianthoveus cremnophilus (Cyclanthaceae)

1989 ◽  
Vol 67 (8) ◽  
pp. 2450-2464 ◽  
Author(s):  
George J. Wilder
Keyword(s):  
Transition Zone ◽  
Primary Root ◽  
Mirror Image ◽  
Outer Cell ◽  
Wall Ingrowths ◽  
The Third ◽  
Vegetative Buds ◽  
Inflorescence Axis ◽  
Cell Layers

Dianthoveus cremnophilus exhibits terminal inflorescences and rhizomes composed of renewal shoots. A rhizome bears two kinds of vegetative buds directly, viz. renewal buds that form renewal shoots and nonrenewal buds that may develop into new rhizomes. Adult leaves consist of sheath, petiole, transition zone, and a bifid lamina that varies from unicostate to subtricostate. The inflorescence axis consists of peduncle and spadix and bears two kinds of spathes, conspicuous and inconspicuous. Seeds exhibit four outer cell layers and become split periclinally throughout the third layer, as well as longitudinally down to this layer, along one edge. Tissue outside the region of splitting may be abscinded; wall ingrowths of the ruptured third layer adhere to, and appear outermost on, the remaining portion of seed, where they directly contact the environment. Seedlings exhibit a primary root, very short hypocotyl, cotyledon, and epicotyl with plumular leaves. Shoots manifest substantial mirror-image symmetry.

Numerical Model for Collision Damping of Blades’ Tips

2011 ◽  
Vol 130-134 ◽  
pp. 2349-2353
Author(s):  
Xue Jin ◽  
Ming Zhi Zhang ◽  
Hong Wei Li ◽  
Xiao Mei Zhang
Keyword(s):  
Numerical Model ◽  
Steam Turbine ◽  
Natural Frequency ◽  
Vibration Damping ◽  
Exciting Force ◽  
Harmonic Response ◽  
Response Characteristics ◽  
Rotating Blade ◽  
Impact Damping

In order to obtain the collision damping of blades’ tips characteristics, around a blades for a 300MW steam turbine, the COMBIN40 element of ANSYS was adopted. The collision damping among blades' tips is equivalent to a damping of spring and friction with a gap. Under the conditions of exciting force and rotating blade natural frequency ,the vibration of a harmonic response was analysed, the nonlinear impact damping of blade response characteristics and the vibration damping mechanism of collision was got.

High Order Subharmonic Response of High Speed Rotors in Bearing Clearance

1988 ◽  
Vol 110 (1) ◽  
pp. 9-16 ◽  
Author(s):  
F. F. Ehrich
Keyword(s):  
Natural Frequency ◽  
High Speed ◽  
Response Curves ◽  
Whole Number ◽  
Wave Forms ◽  
Vibration Responses ◽  
Subharmonic Response ◽  
Subharmonic Vibration ◽  
Exact Computer ◽  
Very High

Subharmonic vibration refers to the response of a dynamic system to excitation at a whole-number multiple (n) of its natural frequency by vibrating asynchronously at its natural frequency, that is, at (1/n) of the excitation. The phenomenon is generally associated with asymmetry in the stiffness vs. deflection characteristic of the system. It may be characterized as the “bouncing” of the rotor on the surface of the stiff support, energized by every nth unbalance impulse prior to contact. Second, third and fourth order subharmonic vibration responses have previously been observed in high speed rotating machinery with such an asymmetry in the bearing supports. An incident is reported where 8th and 9th order subharmonic vibration responses have been observed in a high speed rotor. A simple but exact computer model of the phenomenon has been evolved based on the numerical integration of a finite difference formulation. Response curves and wave forms of rotor deflection at individual speeds are computed. It is shown that the response is a series of pseudo-critical peaks at whole-number multiples of the rotational speed. Very high orders of subharmonic vibration are found to be possible for systems with low damping and extreme nonlinearity.

A PARAMETRICALLY FORCED NONLINEAR SYSTEM WITH REVERSIBLE EQUILIBRIA

2012 ◽  
Vol 22 (06) ◽  
pp. 1230020 ◽  
Author(s):  
R. WIEBE ◽  
L. N. VIRGIN ◽  
T. P. WITELSKI
Keyword(s):  
Dynamical System ◽  
Nonlinear System ◽  
Resonance Frequency ◽  
Coordinate System ◽  
Chaotic Behavior ◽  
Nonlinear Behavior ◽  
Time Dependent ◽  
Dependent Manner ◽  
Frequency Content ◽  
The Stability

A nonlinear Duffing-type dynamical system, in which the stability of equilibria is modulated in a time-dependent manner, is investigated both experimentally and numerically. This is a low-order dynamical system with some interesting available choices in the coordinate system. The system is found to exhibit a variety of interesting nonlinear behavior including ultrasubharmonic resonance. Frequency content is used to characterize periodic and chaotic behavior and their relation to the parameter space.

Nonlinear Dynamic Analysis of an Icosahedron Frame which Exhibits Chaotic Behavior

2016 ◽  
Vol 12 (1) ◽  
Author(s):  
Lucas W. Just ◽  
Anthony M. DeLuca ◽  
Anthony N. Palazotto
Keyword(s):  
Dynamic Response ◽  
Boundary Conditions ◽  
Dynamic Analysis ◽  
Chaotic Behavior ◽  
Research Question ◽  
Nonlinear Dynamic Analysis ◽  
Surface Force ◽  
Element Analysis ◽  
Point Distribution ◽  
Chaotic Response

The research question addressed is whether a lighter than air vehicle (LTAV), which uses an internal vacuum to become positively buoyant, can be designed to provide extended loiter for U.S. Air Force applications. To achieve a vacuum, internal gases are evacuated from the vessel, which creates a dynamic response in the supporting structural frame. This paper considers the frame of an icosahedron shaped LTAV subject to external atmospheric pressure evacuated at varying rates. A static finite element analysis documented in previous research revealed a snapback phenomenon in the frame members under certain loading conditions. A nonlinear chaotic response was observed when a dynamic analysis was conducted with the same boundary conditions used in the static analysis. The chaotic response for a variety of boundary conditions, generated by varying the rate of evacuation, similar to a ramp input, is determined. An analysis of the dynamic response is determined nonlinearly using a method that relies on a reference point distribution of external pressures to distribute the surface force across the frame. A novel method of combining the power spectral density with a Lyapunov exponent was used to determine the degree of nonlinearity and chaotic response for each boundary condition examined.

scholarly journals Assessment of Effects of a Delay Block and a Nonlinear Block in Systems with Chaotic Behavior Using Lyapunov Exponents

Ingeniería ◽  
2017 ◽  
Vol 22 (2) ◽  
pp. 240
Author(s):  
Pablo César Rodríguez Gómez ◽  
Maikoll Andres Rodriguez Nieto ◽  
Jose Jairo Soriano Mendez
Keyword(s):  
Nonlinear System ◽  
Linear System ◽  
Linear Systems ◽  
Lyapunov Exponents ◽  
Chaotic Behavior ◽  
Structural Characteristics ◽  
Largest Lyapunov Exponent ◽  
Nonlinear Characteristic ◽  
Feedback Systems ◽  
Contour Plots

Context: Because feedback systems are very common and widely used, studies of the structural characteristics under which chaotic behavior is generated have been developed. These can be separated into a nonlinear system and a linear system at least of the third order. Methods such as the descriptive function have been used for analysis.Method: A feedback system is proposed comprising a linear system, a nonlinear system and a delay block, in order to assess his behavior using Lyapunov exponents. It is evaluated with three different linear systems, different delay values and different values for parameters of nonlinear characteristic, aiming to reach chaotic behavior.Results: One hundred experiments were carried out for each of the three linear systems, by changing the value of some parameters, assessing their influence on the dynamics of the system. Contour plots that relate these parameters to the Largest Lyapunov exponent were obtained and analyzed.Conclusions: In spite non-linearity is a condition for the existence of chaos, this does not imply that any nonlinear characteristic generates a chaotic system, it is reflected by the contour plots showing the transitions between chaotic and no chaotic behavior of the feedback system.Language: English  

Generating Power From Ocean Waves Using a Float With Excessive Buoyancy: Theory and Dynamic Model Results

2003 ◽  
Author(s):  
T. MacCready ◽  
T. Zambrano ◽  
B. D. Hibbs
Keyword(s):  
Numerical Model ◽  
Natural Frequency ◽  
Power Output ◽  
Ocean Surface ◽  
Ocean Waves ◽  
The Body ◽  
Energy Extraction ◽  
Ocean Energy ◽  
Wave Conditions ◽  
Rich Data

We are exploring a new approach to ocean energy extraction through a device that we refer to as the NAF (an acronym for Non-Archimedean Float). The NAF is a fully submerged body with excess buoyancy; i.e., the mass of the body is far less than the mass of the water it displaces. When such a float is tethered beneath the ocean surface the buoyancy yields a large force vector in the direction perpendicular to the isobaric surfaces that parallel the water/air interface. The constant shifting of the wave troughs provides the opportunity for energy extraction using turbines affixed to the float. We are exploring the NAF concept because its simplicity results in many inherent benefits. The device has few moving parts, gathers energy from waves coming in any direction, and exists as a non-obtrusive, completely submerged installation. A numerical model of the NAF has been created to determine the dynamic behavior and power output for various configurations and under various wave conditions. The numerical model is set up to calculate the various forces experienced by the NAF float, and from these it calculates the velocity and position of the float through time series steps. The model effectively demonstrates which variables are important and how power output relates to NAF dimensions. One early finding from the model result relates to tuning the natural frequency of the NAF to match the natural frequency of the waves. The NAF moves like an inverted pendulum, and its natural frequency is primarily dependent on the length of the pendulum. Regardless of the actual float buoyancy, the 6 to 12 second periods that typify average wave conditions dictate that the NAF tether should be between 30-m and 60-m long. Also, a scale version of this novel energy device consisting of a float tethered beneath the ocean surface was deployed off the coast of southern California. The deployment yielded rich data sequences that are sufficient for comparison with a dynamic numerical model.

Horizontal and Torsional Modes of an Ultra Large Container Ship (ULCS)

2020 ◽  
Author(s):  
P. P. Vijith ◽  
Suresh Rajendran
Keyword(s):  
Numerical Model ◽  
Numerical Method ◽  
Natural Frequency ◽  
Flexible Structures ◽  
Complex Problem ◽  
Mode Shapes ◽  
Beam Model ◽  
Torsional Vibrations ◽  
Hydrodynamic Loads ◽  
Structural Responses

Abstract The hydro elastic responses of flexible structures under fluid loading is an important concern during the design of large ocean structures. The two-way coupling between the structural responses and the hydrodynamic loads is a complex problem in large flexible floating structures since the structures can vibrate in longitudinal, vertical, horizontal, or torsional modes. The antisymmetric distortion modes may be coupled depending on the location of the centroid and the shear centre. In the case of thin walled open structures, horizontal and torsional vibrations are usually coupled due to the asymmetry of cross section as well as eccentricity between centroid of the section and shear deformation centres. The acurate estimation of dry natural frequency and modes shapes of structure is indispensable since it helps to validate the accuracy of the structural modelling. A numerical method available from one of the existing literatures is used for the estimation of dry and wet natural frequencies, and mode shapes of horizontal and torsional vibrations of an ULCS. The natural frequency and modes are essential parameters for the analysis of interaction between structural responses and hydrodynamic loads. The numerical method is based on a 1D FEM beam model. Distortion due to warping is included in the numerical model since it is well known that containerships with large hatch opening are susceptible to warping. The numerical model is subdivided into 50 stations and the mass distribution and the sectional properties are calculated in order to match the bending, shear, torsion and warping moduli of the experimental model. The dry and wet natural frequency and mode shapes for the horizontal and torsional vibrations of the ULCS is numerically calculated and compared with the experimental results.

Sign in / Sign up

Export Citation Format

Share Document