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.

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.

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.

Multiple regression and nonlinear models

2017 ◽  
Author(s):  
Andrew Gelman ◽  
Deborah Nolan
Keyword(s):  
Regression Analysis ◽  
Social Science ◽  
Multiple Regression ◽  
Nonlinear Model ◽  
Nonlinear Models ◽  
Good Strategy ◽  
Science Journal ◽  
Golf Putting ◽  
Social Science Journal ◽  
To Come

This chapter covers multiple regression and links statistical inference to general topics such as lurking variables that arose earlier. Many examples can be used to illustrate multiple regression, but we have found it useful to come to class prepared with a specific example, with computer output (since our students learn to run the regressions on the computer). We have found it is a good strategy to simply use a regression analysis from some published source (e.g., a social science journal) and go through the model and its interpretation with the class, asking students how the regression results would have to differ in order for the study’s conclusions to change. The chapter includes examples that revisit the simple linear model of height and income, involve the class in models of exam scores, and fit a nonlinear model (for more advanced classes) for golf putting.

Free Vibration Energy and Period of a String With Time-Varying Length

2000 ◽  
Author(s):  
Seung-Yop Lee
Keyword(s):  
Free Vibration ◽  
Traveling Waves ◽  
Moving Boundary ◽  
String Tension ◽  
New Technique ◽  
Vibration Energy ◽  
Time Varying ◽  
Constant Rate ◽  
Varying Length ◽  
A New Technique

Abstract We introduce a new technique to analyze free vibration energy and natural frequencies of a string with time-varying length by dealing with traveling waves. One of boundaries is vertically fixed but it moves axially at a constant rate. When the string length is varied, the free vibration energy and period also change with time. String tension and non-zero instantaneous vertical velocity at the moving boundary results in the energy variation. When the string undergoes retraction, the vibration energy increases with time. The new technique using traveling waves gives the exact amount of energy transferred into the vibrating system at the moving boundary. The free vibration energy are compared with previous results using the perturbation method and Kotera’s approach.

scholarly journals Oscillatory behavior of two nonlinear microbial models of soil carbon decomposition

2014 ◽  
Vol 11 (7) ◽  
pp. 1817-1831 ◽  
Author(s):  
Y. P. Wang ◽  
B. C. Chen ◽  
W. R. Wieder ◽  
M. Leite ◽  
B. E. Medlyn ◽  
...  
Keyword(s):  
Stability Analysis ◽  
Microbial Biomass ◽  
Soil Carbon ◽  
Nonlinear Model ◽  
Linear Models ◽  
Nonlinear Models ◽  
Oscillatory Behavior ◽  
Carbon Pools ◽  
Carbon Decomposition ◽  
Pool Model

Abstract. A number of nonlinear models have recently been proposed for simulating soil carbon decomposition. Their predictions of soil carbon responses to fresh litter input and warming differ significantly from conventional linear models. Using both stability analysis and numerical simulations, we showed that two of those nonlinear models (a two-pool model and a three-pool model) exhibit damped oscillatory responses to small perturbations. Stability analysis showed the frequency of oscillation is proportional to √(ϵ−1−1) Ks/Vs in the two-pool model, and to √(ϵ−1−1) Kl/Vl in the three-pool model, where ϵ is microbial growth efficiency, Ks and Kl are the half saturation constants of soil and litter carbon, respectively, and /Vs and /Vl are the maximal rates of carbon decomposition per unit of microbial biomass for soil and litter carbon, respectively. For both models, the oscillation has a period of between 5 and 15 years depending on other parameter values, and has smaller amplitude at soil temperatures between 0 and 15 °C. In addition, the equilibrium pool sizes of litter or soil carbon are insensitive to carbon inputs in the nonlinear model, but are proportional to carbon input in the conventional linear model. Under warming, the microbial biomass and litter carbon pools simulated by the nonlinear models can increase or decrease, depending whether ϵ varies with temperature. In contrast, the conventional linear models always simulate a decrease in both microbial and litter carbon pools with warming. Based on the evidence available, we concluded that the oscillatory behavior and insensitivity of soil carbon to carbon input are notable features in these nonlinear models that are somewhat unrealistic. We recommend that a better model for capturing the soil carbon dynamics over decadal to centennial timescales would combine the sensitivity of the conventional models to carbon influx with the flexible response to warming of the nonlinear model.

Nonlinear Modeling of a Rigid Rotor Supported on Rolling Element Bearings With Localized Defects

2010 ◽  
Author(s):  
Karthik Kappaganthu ◽  
C. Nataraj
Keyword(s):  
Nonlinear Model ◽  
Nonlinear Models ◽  
Nonlinear Modeling ◽  
Rolling Element Bearings ◽  
Rigid Rotor ◽  
Rolling Element ◽  
Localized Defects ◽  
Bearing Race ◽  
Overall Performance ◽  
Bearing System

In this paper a nonlinear model for defects in rolling element bearings is developed. Detailed nonlinear models are useful to detect, estimate and predict failure in rotating machines. Also, accurate modeling of the defect provides parameters that can be estimated to determine the health of the machine. In this paper the rotor-bearing system is modeled as a rigid rotor and the defects are modeled as pits in the bearing race. Unlike the previous models, the motion of the rolling element thorough the defect is not modeled as a predetermined function; instead, it is dynamically determined since it depends on the clearance and the position of the shaft. Using this nonlinear model, the motion of the shaft is simulated and the effect of the rolling element passing through the defect is studied. The effect of shaft parameters and the defect parameters on the precision of the shaft and the overall performance of the system is studied. Finally, suitable measures for health monitoring and defect tracking are suggested.

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