Dynamic Stability of a Class of Second-Order Distributed Structural Systems With Sinusoidally Varying Velocities

2013 ◽  
Vol 80 (6) ◽  
Author(s):  
W. D. Zhu ◽  
K. Wu
Keyword(s):  
Linear Model ◽  
Moving Boundary ◽  
Parametric Instability ◽  
Second Order ◽  
Moving Boundaries ◽  
Structural Systems ◽  
Dimensional Parameter ◽  
Stationary Boundary ◽  
Lumped Parameter ◽  
Instability Regions

Parametric instability in a system is caused by periodically varying coefficients in its governing differential equations. While parametric excitation of lumped-parameter systems has been extensively studied, that of distributed-parameter systems has been traditionally analyzed by applying Floquet theory to their spatially discretized equations. In this work, parametric instability regions of a second-order nondispersive distributed structural system, which consists of a translating string with a constant tension and a sinusoidally varying velocity, and two boundaries that axially move with a sinusoidal velocity relative to the string, are obtained using the wave solution and the fixed point theory without spatially discretizing the governing partial differential equation. There are five nontrivial cases that involve different combinations of string and boundary motions: (I) a translating string with a sinusoidally varying velocity and two stationary boundaries; (II) a translating string with a sinusoidally varying velocity, a sinusoidally moving boundary, and a stationary boundary; (III) a translating string with a sinusoidally varying velocity and two sinusoidally moving boundaries; (IV) a stationary string with a sinusoidally moving boundary and a stationary boundary; and (V) a stationary string with two sinusoidally moving boundaries. Unlike parametric instability regions of lumped-parameter systems that are classified as principal, secondary, and combination instability regions, the parametric instability regions of the class of distributed structural systems considered here are classified as period-1 and period-i (i>1) instability regions. Period-1 parametric instability regions are analytically obtained; an equivalent total velocity vector is introduced to express them for all the cases considered. While period-i (i>1) parametric instability regions can be numerically calculated using bifurcation diagrams, it is shown that only period-1 parametric instability regions exist in case IV, and no period-i (i>1) parametric instability regions can be numerically found in case V. Unlike parametric instability in a lumped-parameter system that is characterized by an unbounded displacement, the parametric instability phenomenon discovered here is characterized by a bounded displacement and an unbounded vibratory energy due to formation of infinitely compressed shock-like waves. There are seven independent parameters in the governing equation and boundary conditions, and the parametric instability regions in the seven-dimensional parameter space can be projected to a two-dimensional parameter plane if five parameters are specified. Period-1 parametric instability occurs in certain excitation frequency bands centered at the averaged natural frequencies of the systems in all the cases. If the parameters are chosen to be in the period-i (i≥1) parametric instability region corresponding to an integer k, an initial smooth wave will be infinitely compressed to k shock-like waves as time approaches infinity. The stable and unstable responses of the linear model in case I are compared with those of a corresponding nonlinear model that considers the coupled transverse and longitudinal vibrations of the translating string and an intermediate linear model that includes the effect of the tension change due to axial acceleration of the string on its transverse vibration. The parametric instability in the original linear model can exist in the nonlinear and intermediate linear models.

Dynamic Stability of Translating and Stationary Strings With Sinusoidally Varying Velocities

2012 ◽  
Author(s):  
W. D. Zhu ◽  
K. Wu
Keyword(s):  
Linear Model ◽  
Moving Boundary ◽  
Parametric Instability ◽  
Moving Boundaries ◽  
Dimensional Parameter ◽  
Stationary Boundary ◽  
Varying Coefficients ◽  
Lumped Parameter ◽  
Axial Acceleration ◽  
Instability Regions

Parametric instability in a system is caused by periodically varying coefficients in its governing differential equations. While parametric excitation of lumped-parameter systems has been extensively studied, that of distributed-parameter systems has been traditionally analyzed by applying Floquet theory to their spatially discretized equations. In this work, parametric instability regions of a second-order non-dispersive distributed structural system, which consists of a translating string with a constant tension and a sinusoidally varying velocity, and two boundaries that axially move with a sinusoidal velocity relative to the string, are obtained using the wave solution and the fixed point theory without spatially discretizing the governing partial differential equation. There are five cases that involve non-trivial combinations of string and boundary motions: I) a translating string with a sinusoidally varying velocity and two stationary boundaries; II) a translating string with a sinusoidally varying velocity, a sinusoidally moving boundary, and a stationary boundary; III) a translating string with a sinusoidally varying velocity and two sinusoidally moving boundaries; IV) a stationary string with a sinusoidally moving boundary and a stationary boundary; and V) a stationary string with two sinusoidally moving boundaries. Unlike parametric instability regions of lumped-parameter systems that are classified as principal, secondary, and combination instability regions, the parametric instability regions of the class of distributed structural systems considered here are classified as period-1 and period-i (i>1) instability regions. Period-1 parametric instability regions are analytically obtained; an equivalent total velocity vector is introduced to express them for all the cases considered. While period-i (i>1) parametric instability regions can be numerically calculated using bifurcation diagrams, it is shown that only period-1 parametric instability regions exist in case IV, and no period-i (i>1) parametric instability regions can be numerically found in case V. Unlike parametric instability in a lumped-parameter system that is characterized by an unbounded displacement, the parametric instability phenomenon discovered here is characterized by a bounded displacement and an unbounded vibratory energy, due to formation of infinitely compressed shock-like waves. There are seven independent parameters in the governing equation and boundary conditions, and the parametric instability regions in the seven-dimensional parameter space can be projected to a two-dimensional parameter plane if five parameters are specified. Period-1 parametric instability occurs in certain excitation frequency bands centered at the averaged natural frequencies of the systems in all the cases. If the parameters are chosen to be in the period-i (i≥1) parametric instability region corresponding to an integer k, an initial smooth wave will be infinitely compressed to k shock-like waves as time approaches infinity. The stable and unstable responses of the linear model in case I are compared with those of a corresponding nonlinear model that considers the coupled transverse and longitudinal vibrations of the translating string and an intermediate linear model that includes the effect of the tension change due to axial acceleration of the string on its transverse vibration. The parametric instability in the original linear model can exist in the nonlinear and intermediate linear models.

Parametric Instability in a Taut String With a Periodically Moving Boundary

2014 ◽  
Vol 81 (6) ◽  
Author(s):  
K. Wu ◽  
W. D. Zhu
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Linear Model ◽  
Difference Method ◽  
Periodic Boundary ◽  
Moving Boundary ◽  
Nonlinear Models ◽  
Parametric Instability ◽  
Taut String ◽  
Instability Regions

Parametric instability in a taut string with a periodically moving boundary, which is governed by a one-dimensional wave equation with a periodically varying domain, is investigated. Parametric instability usually occurs when coefficients in governing differential equations of a system periodically vary, and the system is said to be parametrically excited. Since the governing partial differential equation of the string with a periodically moving boundary can be transformed to one with a fixed domain and periodically varying coefficients, the string is parametrically excited and instability caused by the periodically moving boundary is classified as parametric instability. The free linear vibration of a taut string with a constant tension, a fixed boundary, and a periodically moving boundary is studied first. The exact response of the linear model is obtained using the wave or d'Alembert solution. The parametric instability in the string features a bounded displacement and an unbounded vibratory energy, and parametric instability regions in the parameter plane are classified as period-i (i≥1) parametric instability regions, where period-1 parametric instability regions are analytically obtained using the wave solution and the fixed point theory, and period-i (i>1) parametric instability regions are numerically calculated using bifurcation diagrams. If the periodic boundary movement profile of the string satisfies certain condition, only period-1 parametric instability regions exist. However, parametric instability regions with higher period numbers can exist for a general periodic boundary movement profile. Three corresponding nonlinear models that consider coupled transverse and longitudinal vibrations of the string, only the transverse vibration, and coupled transverse and axial vibrations are introduced next. Responses and vibratory energies of the linear and nonlinear models are calculated for both stable and unstable cases using three numerical methods: Galerkin's method, the explicit finite difference method, and the implicit finite difference method; advantages and disadvantages of each method are discussed. Numerical results for the linear model can be verified using the exact wave solution, and those for the nonlinear models are compared with each other since there are no exact solutions for them. It is shown that for parameters in the parametric instability regions of the linear model, the responses and vibratory energies of the nonlinear models are close to those of the linear model, which indicates that the parametric instability in the linear model can also exist in the nonlinear models. The mechanism of the parametric instability is explained in the linear model and through axial strains in the third nonlinear model.

Parametric Instability in a Stationary String With a Periodically Varying Length

2014 ◽  
Author(s):  
K. Wu ◽  
W. D. Zhu
Keyword(s):  
Nonlinear Model ◽  
Periodic Boundary ◽  
Moving Boundary ◽  
Nonlinear Models ◽  
Parametric Instability ◽  
Linear Vibration ◽  
Longitudinal Vibrations ◽  
Boundary Movement ◽  
Varying Length ◽  
Instability Regions

Parametric instability in a stationary string with a periodically varying length is investigated. The free linear vibration of a stationary string with a constant tension, a fixed boundary, and a periodically moving boundary is studied first. The parametric instability in the string features a bounded displacement and an unbounded vibratory energy, and parametric instability regions in the parameter plane are classified as period-i (i ≥ 1) parametric instability regions. If the periodic boundary movement profile of the string satisfies certain condition, only period-1 parametric instability regions exist. However, parametric instability regions with higher period numbers can exist for a general periodic boundary movement profile. A corresponding nonlinear model that consider coupled transverse and longitudinal vibrations of the string are introduced next. Responses and vibratory energies of the linear and nonlinear models are calculated for both stable and unstable cases using the finite difference method. The mechanism of the parametric instability is explained in the linear model and through axial strains in the nonlinear model.

Modified Reynolds Equation for Aerostatic Porous Radial Bearings With Quadratic Forchheimer Pressure-Flow Assumption

2007 ◽  
Author(s):  
Rodrigo Nicoletti ◽  
Zilda C. Silveira ◽  
Benedito M. Purquerio
Keyword(s):  
Mathematical Modeling ◽  
Porous Medium ◽  
Linear Model ◽  
Dynamic Characteristics ◽  
Reynolds Equation ◽  
Dimensional Parameter ◽  
Pressure Flow ◽  
Darcy Model ◽  
Linear Effects ◽  
Modified Reynolds Equation

The mathematical modeling of aerostatic porous bearings, represented by the Reynolds equation, depends on the assumptions for the flow in the porous medium. One proposes a modified Reynolds equation based on the quadratic Forchheimer assumption, which can be used for both linear and quadratic conditions. Numerical results are compared to those obtained with the linear Darcy model. It is shown that, the non-dimensional parameter Φ, related to non-linear effects, strongly affects the bearing dynamic characteristics, but for values of Φ > 10, the results tend to those obtained with the linear model.

On non-existence of steady periodic solutions of the Prandtl equations

2013 ◽  
Vol 717 ◽  
Author(s):  
Michael Renardy
Keyword(s):  
Velocity Profile ◽  
Periodic Solutions ◽  
Moving Boundary ◽  
Stationary Boundary ◽  
Prandtl Equations

AbstractWe prove that periodic solutions of the steady Prandtl equations do not exist on a stationary boundary. On a moving boundary, there are no solutions with a monotone velocity profile.

Parametric Instability of a Periodically Supported Pipe Without and With Vibration Absorbers

1984 ◽  
Vol 51 (1) ◽  
pp. 159-163 ◽  
Author(s):  
A. K. Mallik ◽  
S. B. Kulkarni ◽  
K. S. Ram
Keyword(s):  
Prior Knowledge ◽  
Propagation Constant ◽  
Parametric Instability ◽  
Numerical Results ◽  
Mode Shapes ◽  
Effective Control ◽  
Vibration Absorbers ◽  
Instability Regions

Parametric instability of a periodically supported pipe without and with dynamic absorbers is investigated. A method based on the notion of propagation constant is used. This method requires no prior knowledge of the mode shapes and renders the amount of computation independent of the number of spans. Numerical results are presented for single and two-span pipes. Effective control of the instability regions by means of dynamic absorbers is also demonstrated.

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