scholarly journals Global Dynamics of a Compressor Blade with Resonances

2016 ◽  
Vol 2016 ◽  
pp. 1-12
Author(s):  
Xiaoxia Bian ◽  
Fangqi Chen ◽  
Fengxian An
Keyword(s):  
Internal Resonance ◽  
Chaotic Dynamics ◽  
Primary Resonance ◽  
Compressor Blade ◽  
Invariant Sets ◽  
Global Dynamics ◽  
Resonance Case ◽  
Double Zero ◽  
Chaotic Motions ◽  
Pure Imaginary Eigenvalues

The global bifurcations and chaotic dynamics of a thin-walled compressor blade for the resonant case of 2 : 1 internal resonance and primary resonance are investigated. With the aid of the normal theory, the desired form associated with a double zero and a pair of pure imaginary eigenvalues for the global perturbation method is obtained. Based on the simpler form, the method developed by Kovacic and Wiggins is used to find the existence of a Shilnikov-type homoclinic orbit. The results obtained here indicate that the orbit homoclinic to certain invariant sets for the resonance case which may lead to chaos in the sense of Smale horseshoes for the system. The chaotic motions of the rotating compressor blade are also found by using numerical simulation.

Multi-Pulse Chaotic Dynamics of Eccentric Rotating Composite Laminated Circular Cylindrical Shell by Using the Extended Melnikov Method

2019 ◽  
Author(s):  
Yan Zheng ◽  
Wei Zhang ◽  
Tao Liu
Keyword(s):  
Cylindrical Shell ◽  
Normal Form ◽  
Chaotic Dynamics ◽  
Multiple Scales ◽  
Circular Cylindrical Shell ◽  
Melnikov Method ◽  
Global Dynamics ◽  
Chaotic Motions ◽  
Extended Melnikov Method ◽  
Averaged Equations

Abstract The researches of global bifurcations and chaotic dynamics for high-dimensional nonlinear systems are extremely challenging. In this paper, we study the multi-pulse orbits and chaotic dynamics of an eccentric rotating composite laminated circular cylindrical shell. The four-dimensional averaged equations are obtained by directly using the multiple scales method under the case of the 1:2 internal resonance and principal parametric resonance-1/2 subharmonic resonance. The system is transformed to the averaged equations. From the averaged equation, the theory of normal form is used to find the explicit formulas of normal form. Based on the normal form obtained, the extended Melnikov method is utilized to analyze the multi-pulse global homoclinic bifurcations and chaotic dynamics for the eccentric rotating composite laminated circular cylindrical shell. The analysis of global dynamics indicates that there exist the multi-pulse jumping orbits in the perturbed phase space of the averaged equation. From the averaged equations obtained, the chaotic motions and the Shilnikov type multi-pulse orbits of the eccentric rotating composite laminated circular cylindrical shell are found by using numerical simulation. The results obtained above mean the existence of the chaos for the Smale horseshoe sense for the eccentric rotating composite laminated circular cylindrical shell.

Multipulse Orbits and Chaotic Dynamics of an Aero-Elastic FGP Plate Under Parametric and Primary Excitations

2017 ◽  
Vol 27 (04) ◽  
pp. 1750050 ◽  
Author(s):  
F. X. An ◽  
F. Q. Chen
Keyword(s):  
Normal Form ◽  
Numerical Simulations ◽  
Chaotic Dynamics ◽  
Melnikov Method ◽  
Functionally Graded ◽  
Chaotic Motions ◽  
Averaged Equations ◽  
Functionally Graded Piezoelectric ◽  
Pure Imaginary Eigenvalues ◽  
Explicit Expressions

The multipulse global bifurcations and chaotic dynamics of a simply supported Functionally Graded Piezoelectric (FGP) rectangular plate with bonded piezoelectric layer are investigated with the case of 1:2 internal resonance and primary parametric resonance. Based on the averaged equations obtained, the theory of normal form is utilized to obtain the explicit expressions of normal form with a double zero and a pair of pure imaginary eigenvalues. According to the explicit expressions of normal form, the extended Melnikov method developed by Camassa et al. is employed to study the Shilnikov-type multipulse homoclinic bifurcations and chaotic dynamics of the aero-elastic FGP plate. The analytical results indicate that there exists the Shilnikov-type multipulse chaotic dynamics for the FGP plate. Numerical simulations are presented to show that for the FGP plate, the Shilnikov-type multipulse chaotic motions can occur. The influence of the in-plane excitation and the piezoelectric voltage excitation to the system dynamic behaviors is also discussed by numerical simulations. The results obtained here imply the existence of chaos in the sense of the Smale horseshoes for the FGP plate.

Multi-Pulse Chaotic Dynamics of the Cantilevered Pipe Conveying Pulsating Fluid

2009 ◽  
Author(s):  
Ming-Hui Yao ◽  
Wei Zhang ◽  
Dong-Xing Cao
Keyword(s):  
Normal Form ◽  
Chaotic Dynamics ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Phase Method ◽  
Global Dynamics ◽  
Pipe Material ◽  
Chaotic Motions ◽  
Cantilevered Pipe ◽  
Pulsating Fluid

The multi-pulse orbits and chaotic dynamics of the cantilevered pipe conveying pulsating fluid with harmonic external force are studied in detail. The nonlinear geometric deformation of the pipe and the Kelvin constitutive relation of the pipe material are considered. The nonlinear governing equations of motion for the cantilevered pipe conveying pulsating fluid are determined by using Hamilton principle. The four-dimensional averaged equation under the case of principle parameter resonance, 1/2 subharmonic resonance and 1:2 internal resonance and primary parametric resonance is obtained by directly using the method of multiple scales and Galerkin approach to the partial differential governing equation of motion for the cantilevered pipe. The system is transformed to the averaged equation. From the averaged equation, the theory of normal form is used to find the explicit formulas of normal form. Based on normal form obtained, the energy phase method is utilized to analyze the multi-pulse global bifurcations and chaotic dynamics for the cantilevered pipe conveying pulsating fluid. The analysis of global dynamics indicates that there exist the multi-pulse jumping orbits in the perturbed phase space of the averaged equation. From the averaged equations obtained, the chaotic motions and the Shilnikov type multi-pulse orbits of the cantilevered pipe are found by using numerical simulation. The results obtained above mean the existence of the chaos for the Smale horseshoe sense for the pulsating fluid conveying cantilevered pipe.

Multi-Pulse Heteroclinic Orbits and Chaotic Dynamics of Laminated Composite Piezoelectric Rectangular Plate

2009 ◽  
Author(s):  
Ming-Hui Yao ◽  
Wei Zhang ◽  
Dong-Xing Cao
Keyword(s):  
Normal Form ◽  
Chaotic Dynamics ◽  
Multiple Scales ◽  
Laminated Composite ◽  
Heteroclinic Orbits ◽  
Global Dynamics ◽  
Rectangular Plates ◽  
Internal Resonances ◽  
Chaotic Motions ◽  
Simply Supported

The multi-pulse orbits and chaotic dynamics of the simply supported laminated composite piezoelectric rectangular plates under combined parametric excitation and transverse loads are studied in detail. It is assumed that different layers are perfectly bonded to each other with piezoelectric actuator patches embedded. The nonlinear equations of motions for the laminated composite piezoelectric rectangular plates are derived from von Karman-type equation and third-order shear deformation laminate theory of Reddy. The four-dimensional averaged equation under the case of primary parametric resonance and 1:2 internal resonances is obtained by directly using the method of multiple scales and Galerkin approach to the partial differential governing equation of motion for the laminated composite piezoelectric rectangular plates. The system is transformed to the averaged equation. From the averaged equation, the theory of normal form is used to find the explicit formulas of normal form. Based on normal form obtained, the extended Melnikov method is utilized to analyze the multi-pulse global bifurcations and chaotic dynamics for the laminated composite piezoelectric rectangular plates. The analysis of global dynamics indicates that there exist the multi-pulse jumping orbits in the perturbed phase space of the averaged equation. From the averaged equations obtained, the chaotic motions and the Shilnikov type multi-pulse orbits of the laminated composite piezoelectric rectangular plates are found by using numerical simulation. The results obtained above mean the existence of the chaos for the Smale horseshoe sense for the simply supported laminated composite piezoelectric rectangular plates.

Multi-Pulse Chaotic Dynamics of Circular Mesh Antenna with 1:2 Internal Resonance

2017 ◽  
Vol 09 (04) ◽  
pp. 1750060 ◽  
Author(s):  
Y. Sun ◽  
W. Zhang ◽  
M. H. Yao
Keyword(s):  
Internal Resonance ◽  
Chaotic Dynamics ◽  
Circular Cylindrical Shell ◽  
Domain Of Attraction ◽  
Homoclinic Orbits ◽  
Energy Difference ◽  
Phase Method ◽  
Chaotic Motions ◽  
Difference Function ◽  
First Time

The multi-pulse homoclinic orbits and chaotic dynamics of an equivalent circular cylindrical shell for the circular mesh antenna are investigated in the case of 1:2 internal resonance in this paper for the first time. Applying the method of averaging, the four-dimensional averaged equation in the Cartesian form is obtained. The theory of normal form is used to reduce the averaged equation to a simpler form. Based on the simplified system, the energy phase method is employed to investigate the homoclinic bifurcations and the Shilnikov type multi-pulse chaotic dynamics. First, the energy difference function and the zeroes of the energy difference function are obtained. Then, the existence of the Shilnikov type multi-pulse orbits is determined. The homoclinic trees are depicted to describe the relationship among the layers diameter, the pulse numbers and the phase shift. Finally, we need to verify the condition which makes sure that any multi-pulse orbit departing from a slow sink comes back to the domain of attraction of one of the sinks. The results obtained here show the existence of the Shilnikov type multi-pulse chaotic motions of the circular mesh antenna. Numerical simulations are used to find multi-pulse chaotic motions. The results of the theoretical analysis are in qualitative agreement with the results obtained using numerical simulation.

Multi-Pulse Heteroclinic Orbits With a Melnikov Method and Chaotic Dynamics in Motion of Parametrically Excited Viscoelastic Moving Belt

2007 ◽  
Author(s):  
Ming-Hui Yao ◽  
Wei Zhang ◽  
Dong-Xing Cao
Keyword(s):  
Normal Form ◽  
Chaotic Dynamics ◽  
Multiple Scales ◽  
Melnikov Method ◽  
Equation Of Motion ◽  
Constitutive Law ◽  
Heteroclinic Orbits ◽  
Global Dynamics ◽  
Chaotic Motions ◽  
Parametrically Excited

The multi-pulse heteroclinic orbits and chaotic dynamics of a parametrically excited viscoelastic moving belt are studied in detail. Using Kelvin-type viscoelastic constitutive law, the equation of motion for viscoelastic moving belt with the external damping and parametric excitation are determined. The four-dimensional averaged equation under the case of 1:1 internal resonance and primary parametric resonance is obtained by directly using the method of multiple scales and Galerkin’s approach to the partial differential governing equation of motion for viscoelastic moving belt. The system is transformed to the averaged equation. From the averaged equation, the theory of normal form is used to find the explicit formulas of normal form. Based on normal form obtained, an extension of the Melnikov method is utilized to analyze the multi-pulse global bifurcations and chaotic dynamics for a parametrically excited viscoelastic moving belt. The analysis of global dynamics indicates that there exist the multi-pulse jumping orbits in the perturbed phase space of the averaged equation. From the averaged equations obtained, the chaotic motions and the Shilnikov type multi-pulse heteroclinic orbits of viscoelastic moving belts are found by using numerical simulation. The results obtained above mean the existence of the chaos for the Smale horseshoe sense for a parametrically excited viscoelastic moving belt.

GLOBAL BIFURCATIONS AND CHAOTIC DYNAMICS IN SUSPENDED CABLES

2009 ◽  
Vol 19 (11) ◽  
pp. 3753-3776 ◽  
Author(s):  
HONGKUI CHEN ◽  
ZHAOHUA ZHANG ◽  
JILONG WANG ◽  
QINGYU XU
Keyword(s):  
Numerical Simulations ◽  
Chaotic Dynamics ◽  
Multiple Scales ◽  
Global Bifurcation ◽  
Homoclinic Orbits ◽  
Invariant Sets ◽  
Global Bifurcations ◽  
Governing Equations ◽  
Chaotic Motions ◽  
Suspended Cables

The global bifurcations and chaotic dynamics of parametrically and externally excited suspended cables are investigated in this paper. The governing equations are obtained to describe the nonlinear transverse vibrations of suspended cables. The Galerkin procedure is introduced to simplify the governing equations of motion to ordinary differential equations with two-degrees-of-freedom. The case of one-to-one internal resonance between the modes of suspended cables, primary and principal parametric resonance of suspended cables is considered. With the method of multiple scales, parametrically and externally excited system is transformed to the averaged equation, based on which, the recently developed global bifurcation method is employed to detect the presence of orbits which are homoclinic to certain invariant sets for the resonant case. The analysis of the global bifurcations indicates that there exist the generalized Šhilnikov type multipulse homoclinic orbits in the averaged equation of suspended cables. The results obtained here mean that chaotic motions can occur in suspended cables. Numerical simulations also verify the analytical predictions. It is found, according to the results of numerical simulations, that the Šhilnikov type multipulse homoclinic orbits exist in the nonlinear motion of the cables.

NONLINEAR RESPONSES OF A STRING-BEAM COUPLED SYSTEM WITH FOUR-DEGREES-OF-FREEDOM AND MULTIPLE PARAMETRIC EXCITATIONS

2009 ◽  
Vol 19 (01) ◽  
pp. 225-243 ◽  
Author(s):  
D. X. CAO ◽  
W. ZHANG
Keyword(s):  
Nonlinear Dynamic ◽  
Internal Resonance ◽  
Degrees Of Freedom ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Primary Resonance ◽  
Coupled System ◽  
Dynamic Responses ◽  
Governing Equations ◽  
Chaotic Motions

The nonlinear dynamic responses of a string-beam coupled system subjected to harmonic external and parametric excitations are studied in this work in the case of 1:2 internal resonance between the modes of the beam and string. First, the nonlinear governing equations of motion for the string-beam coupled system are established. Then, the Galerkin's method is used to simplify the nonlinear governing equations to a set of ordinary differential equations with four-degrees-of-freedom. Utilizing the method of multiple scales, the eight-dimensional averaged equation is obtained. The case of 1:2 internal resonance between the modes of the beam and string — principal parametric resonance-1/2 subharmonic resonance for the beam and primary resonance for the string — is considered. Finally, nonlinear dynamic characteristics of the string-beam coupled system are studied through a numerical method based on the averaged equation. The phase portrait, Poincare map and power spectrum are plotted to demonstrate that the periodic and chaotic motions exist in the string-beam coupled system under certain conditions.

Multi-Pulse Chaotic Dynamics of Laminated Composite Piezoelectric Rectangular Plate

2009 ◽  
Author(s):  
Wei Zhang ◽  
Ming-Hui Yao ◽  
Dong-Xing Cao
Keyword(s):  
Normal Form ◽  
Chaotic Dynamics ◽  
Multiple Scales ◽  
Laminated Composite ◽  
Type Equation ◽  
Global Dynamics ◽  
Rectangular Plates ◽  
Internal Resonances ◽  
Chaotic Motions ◽  
Simply Supported

The multi-pulse orbits and chaotic dynamics of the simply supported laminated composite piezoelectric rectangular plates under combined parametric excitation and transverse loads are studied in detail. It is assumed that different layers are perfectly bonded to each other with piezoelectric actuator patches embedded. The nonlinear equations of motions for the laminated composite piezoelectric rectangular plates are derived from von Karman-type equation and third-order shear deformation laminate theory of Reddy. The four-dimensional averaged equation under the case of primary parametric resonance and 1:2 internal resonances is obtained by directly using the method of multiple scales and Galerkin approach to the partial differential governing equation of motion for the laminated composite piezoelectric rectangular plates. The system is transformed to the averaged equation. From the averaged equation, the theory of normal form is used to find the explicit formulas of normal form. Based on normal form obtained, the extended Melnikov method is utilized to analyze the multi-pulse global bifurcations and chaotic dynamics for the laminated composite piezoelectric rectangular plates. The analysis of global dynamics indicates that there exist the multi-pulse jumping orbits in the perturbed phase space of the averaged equation. From the averaged equations obtained, the chaotic motions and the Shilnikov type multi-pulse orbits of the laminated composite piezoelectric rectangular plates are found by using numerical simulation. The results obtained above mean the existence of the chaos for the Smale horseshoe sense for the simply supported laminated composite piezoelectric rectangular plates.

Shilnikov Type Multi-Pulse Orbits of Functionally Graded Materials Rectangular Plate

2010 ◽  
Author(s):  
Wei Zhang ◽  
Ming-Hui Yao ◽  
Dong-Xing Cao
Keyword(s):  
Normal Form ◽  
Rectangular Plate ◽  
Functionally Graded Materials ◽  
Chaotic Dynamics ◽  
Functionally Graded ◽  
Phase Method ◽  
Global Dynamics ◽  
Graded Materials ◽  
Chaotic Motions ◽  
Simply Supported

Multi-pulse chaotic dynamics of a simply supported functionally graded materials (FGMs) rectangular plate is investigated in this paper. The FGMs rectangular plate is subjected to the transversal and in-plane excitations. The properties of material are graded in the direction of thickness. Based on Reddy’s third-order shear deformation plate theory, the nonlinear governing equations of motion for the FGMs plate are derived by using the Hamilton’s principle. The four-dimensional averaged equation under the case of 1:2 internal resonance, primary parametric resonance and 1/2-subharmonic resonance is obtained by directly using the asymptotic perturbation method and Galerkin approach to the partial differential governing equation of motion for the FGMs rectangular plate. The system is transformed to the averaged equation. From the averaged equation, the theory of normal form is used to find the explicit formulas of normal form. Based on normal form obtained, the energy phase method is utilized to analyze the multi-pulse global bifurcations and chaotic dynamics for the FGMs rectangular plate. The analysis of global dynamics indicates that there exist the multi-pulse jumping orbits in the perturbed phase space of the averaged equation. From the averaged equations obtained, the chaotic motions and the Shilnikov type multi-pulse orbits of the FGMs rectangular plate are found by using numerical simulation. The results obtained above mean the existence of the chaos for the Smale horseshoe sense for the simply supported FGMs rectangular plate.

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