Chaotic Motions of a Constrained Pipe Conveying Fluid: Comparison Between Simulation, Analysis, and Experiment

1991 ◽  
Vol 58 (2) ◽  
pp. 559-565 ◽  
Author(s):  
M. P. Paidoussis ◽  
G. X. Li ◽  
R. H. Rand
Keyword(s):  
Analytical Model ◽  
Degrees Of Freedom ◽  
Simulation Analysis ◽  
Period Doubling ◽  
Critical Flow ◽  
Pipe Conveying Fluid ◽  
Chaotic Motions ◽  
Critical Flow Velocity ◽  
Period Doubling Bifurcations ◽  
Conveying Fluid

A refined analytical model is presented for the dynamics of a cantilevered pipe conveying fluid and constrained by motion limiting restraints. Calculations with the discretized form of this model with a progressively increasing number of degrees of freedom, N, show that convergence is achieved with N = 4 or 5, which agrees with previously performed fractal dimension calculations of experimental data. Theory shows that, beyond the Hopf bifurcation, as the flow is increased, a pitchfork bifurcation is followed by a cascade of period doubling bifurcations leading to chaos, which is in qualitative agreement with observation. The numerically computed theoretical critical flow velocities are in excellent quantitative agreement (5–10 percent) with experimental values for the thresholds of the Hopf and period doubling bifurcations and for the onset of chaos. An approximation for the critical flow velocity for the loss of stability of the post-Hopf limit cycle is also obtained by using center manifold concepts and normal form techniques for a simplified version of the analytical model; it is found that the values obtained in this manner are approximately within 10 percent of those computed numerically.

Two parameters affecting the dynamics characteristics of a uniform-conical assembled pipe conveying fluid

2016 ◽  
Vol 23 (3) ◽  
pp. 361-372 ◽  
Author(s):  
Zhang Zijun ◽  
Liu Yongshou ◽  
Han Tao
Keyword(s):  
Dynamic Characteristics ◽  
Natural Frequencies ◽  
Limited Range ◽  
Critical Flow ◽  
Pipe Conveying Fluid ◽  
Critical Flow Velocity ◽  
Galerkin Approach ◽  
Two Parameters ◽  
Conveying Fluid ◽  
Uniform Fluid

A modified Galerkin approach is employed to calculate the dynamic characteristics of a complex non-uniform fluid-conveying pipe assembled by a uniform and a conical segment. The effects of two geometric parameters (length ratio of the uniform part and conical truncation factor) on the dynamic stability are studied. Results prove that for the assembled fluid-conveying pipe 1) when fluid enters the pipe from the thinner end, the natural frequencies are higher than those when it enters from the wider end; 2) the critical flow velocity increases linearly with the increase of the uniform part ratio, while it decreases squarely with the increase of the conical truncation factor in a limited range and 3) the clamp-pined non-uniform pipe has a higher dimensionless critical velocity than the pin-clamped pipe when fluid enters from the wider end.

Intermittency Route to Chaos of a Cantilevered Pipe Conveying Fluid With a Mass Defect at the Free End

1995 ◽  
Vol 62 (4) ◽  
pp. 903-907 ◽  
Author(s):  
C. Semler ◽  
M. P. Pai¨doussis
Keyword(s):  
Period Doubling ◽  
Type I ◽  
Pipe Conveying Fluid ◽  
Mass Defect ◽  
Lumped Mass ◽  
Route To Chaos ◽  
Cantilevered Pipe ◽  
Planar Motions ◽  
Period Doubling Bifurcations ◽  
Conveying Fluid

The nonlinear equations for planar motions of a vertical cantilevered pipe conveying fluid are modified to take into account a small lumped mass added at the free end. The resultant equations contain nonlinear inertial terms; by discretizing the system first and inverting the inertia matrix, these terms are transferred into other matrices. In this paper, the dynamics of the system is examined when the added mass is negative (a mass defect), by means of numerical computations and by the software package AUTO. The system loses stability by a Hopf bifurcation, and the resultant limit cycle undergoes pitchfork and period-doubling bifurcations. Subsequently, as shown by the computation of Floquet multipliers, a type I intermittency route to chaos is followed—as illustrated further by a Lorenz return map, revealing the well-known normal form for this type of bifurcation. The period between “turbulent bursts” of nonperiodic oscillations is computed numerically, as well as Lyapunov exponents. Remarkable qualitative agreement, in both cases, is obtained with analytical results.

scholarly journals STABILITY OF ELASTICALLY RESTRAINED VALVE-PIPE SYSTEM WITH CRACK

2012 ◽  
Vol 06 ◽  
pp. 373-378 ◽  
Author(s):  
KWAN DO HUR ◽  
IN SOO SON ◽  
SEONG CHUL LEE
Keyword(s):  
Dynamic Stability ◽  
Critical Flow ◽  
Pipe Conveying Fluid ◽  
Attached Mass ◽  
Critical Flow Velocity ◽  
Pipe System ◽  
Local Flexibility ◽  
Stability Maps ◽  
Elastically Restrained ◽  
Conveying Fluid

The dynamic stability and natural frequency of elastically restrained pipe conveying fluid with the attached mass and crack are investigated in this paper. The pipe system with a crack is modeled by using extended Hamilton's principle with consideration of bending energy. The crack on the pipe system is represented by a local flexibility matrix and two undamaged beam segments are connected. The crack is assumed to be in the first mode of fracture and to be always opened during the vibrations. From the governing equations, the influence of attached mass, its position and crack on the dynamic stability of elastically restrained pipe system is presented. Also, the critical flow velocity for the flutter and divergence due to the variation in the position and stiffness of supported spring is studied. Finally, the critical flow velocities and stability maps of the cracked pipe conveying fluid with the attached mass are obtained by the changing parameters.

Stabilization of a flexible pipe conveying fluid with an active boundary control method

2020 ◽  
Vol 26 (19-20) ◽  
pp. 1824-1834
Author(s):  
Beiming Yu ◽  
Hiroshi Yabuno ◽  
Kiyotaka Yamashita
Keyword(s):  
Flow Velocity ◽  
Control Method ◽  
Bending Moment ◽  
The Self ◽  
Pipe Conveying Fluid ◽  
Stabilization Method ◽  
Critical Flow Velocity ◽  
Complex Eigenvalues ◽  
Excited Oscillation ◽  
Conveying Fluid

A method of stabilizing the self-excited oscillation of a cantilevered pipe conveying fluid because of non–self-adjointness is proposed theoretically and experimentally. Complex eigenvalues denoting the natural frequency and damping of the system vary with an increase in the flow velocity. When the flow velocity exceeds a critical value, the flow-generated damping becomes negative and the pipe is dynamically destabilized. The complex eigenvalues with respect to flow velocity are affected by boundary conditions. We, thus, propose a stabilization control actuating the boundary condition. The stabilization method is carried out by applying a bending moment proportional to the bottom displacement of the pipe. The effect of the proposed control method is shown by investigating the stability for the three lowest modes of the system depending on the feedback gain. It is theoretically clarified that the critical flow velocity is increased by the proposed control method. Furthermore, experiments are performed using a fluid conveying pipe with two piezoactuators at the downstream end. The piezoactuators apply a bending moment at the downstream end of the pipe according to the theoretically proposed method. Experimental results verify that the proposed stabilization method suppresses the self-excited oscillation.

EFFECT OF DISSIPATION ON THE HAMILTONIAN CHAOS IN COUPLED OSCILLATOR SYSTEMS

2002 ◽  
Vol 12 (04) ◽  
pp. 859-867 ◽  
Author(s):  
V. SHEEJA ◽  
M. SABIR
Keyword(s):  
Degrees Of Freedom ◽  
Chaotic Behavior ◽  
Period Doubling ◽  
Dissipation Function ◽  
Hamiltonian Chaos ◽  
Fixed Point Attractor ◽  
Point Attractor ◽  
Route To Chaos ◽  
Period Doubling Bifurcations ◽  
External Driving

We study the effect of linear dissipative forces on the chaotic behavior of coupled quartic oscillators with two degrees of freedom. The effect of quadratic Rayleigh dissipation functions, one with diagonal coefficients only and the other with nondiagonal coefficients as well are studied. It is found that the effect of Rayleigh Dissipation function with diagonal coefficients is to suppress chaos in the system and to lead the system to its equilibrium state. However, with a dissipation function with nondiagonal elements, other types of behaviors — including fixed point attractor, periodic attractors and even chaotic attractors — are possible even when there is no external driving. In such a system the route to chaos is through period-doubling bifurcations. This result contradicts the view that linear dissipation always causes decay of oscillations in oscillator models.

scholarly journals A Bifurcation Analysis and Sensitivity Study of Brake Creep Groan

2021 ◽  
Vol 31 (16) ◽  
Author(s):  
Shaun Smith ◽  
James Knowles ◽  
Byron Mason ◽  
Sean Biggs
Keyword(s):  
Periodic Orbits ◽  
Degrees Of Freedom ◽  
Initial Conditions ◽  
Period Doubling ◽  
Nonlinear Behavior ◽  
Sensitivity Study ◽  
Friction Model ◽  
Model Parameters ◽  
Period Doubling Bifurcations ◽  
Simple Models

Creep groan is the undesirable vibration observed in the brake pad and disc as brakes are applied during low-speed driving. The presence of friction leads to nonlinear behavior even in simple models of this phenomenon. This paper uses tools from bifurcation theory to investigate creep groan behavior in a nonlinear 3-degrees-of-freedom mathematical model. Three areas of operational interest are identified, replicating results from previous studies: region 1 contains repelling equilibria and attracting periodic orbits (creep groan); region 2 contains both attracting equilibria and periodic orbits (creep groan and no creep groan, depending on initial conditions); region 3 contains attracting equilibria (no creep groan). The influence of several friction model parameters on these regions is presented, which identify that the transition between static and dynamic friction regimes has a large influence on the existence of creep groan. Additional investigations discover the presence of several bifurcations previously unknown to exist in this model, including Hopf, torus and period-doubling bifurcations. This insight provides valuable novel information about the nature of creep groan and indicates that complex behavior can be discovered and explored in relatively simple models.

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