An Investigation of Damping Mechanisms in Translational Euler-Bernoulli Beams Using a Lyapunov-Based Stability Approach

2003 ◽  
Author(s):  
Mohsen Dadfarnia ◽  
Nader Jalili ◽  
Darren M. Dawson
Keyword(s):  
Equation Model ◽  
Bernoulli Beam ◽  
Differential Equation Model ◽  
Partial Differential Equation Model ◽  
Dissipation Mechanism ◽  
Beam Displacement ◽  
The Stability ◽  
Tip Mass ◽  
Euler Bernoulli Beam ◽  
Stability Results

A translational cantilevered Euler-Bernoulli beam with tip mass dynamics at its free end is used to study the effect of several damping mechanisms on the stabilization of the beam displacement. Specifically, a Lyapunov-based controller utilizing a partial differential equation model of the translational beam is developed to exponentially stabilize the beam displacement while the beam support is regulated to a desired set-point position. Depending on the composition of the tip mass dynamics assumption (i.e. body-mass, point-mass, or massless), it is shown that proper combination of different damping mechanisms (i.e., strain-rate, structural, or viscous damping) guarantees exponential stability of the beam displacement. This novel Lyapunov-based approach, which is based on the energy dissipation mechanism in the beam, brings new dimensions to the stabilization problem of translational beams with tip mass dynamics. The stability analysis utilizes relatively simple mathematical tools to illustrate the exponential and asymptotic stability results. The numerical results are presented to show the effectiveness of the controller.

Lyapunov-Based Vibration Control of Translational Euler-Bernoulli Beams Using the Stabilizing Effect of Beam Damping Mechanisms

2004 ◽  
Vol 10 (7) ◽  
pp. 933-961 ◽  
Author(s):  
Mohsen Dadfarnia ◽  
Nader Jalili ◽  
Bin Xian ◽  
Darren M. Dawson
Keyword(s):  
Equation Model ◽  
Bernoulli Beam ◽  
Differential Equation Model ◽  
Partial Differential Equation Model ◽  
Dissipation Mechanism ◽  
Beam Displacement ◽  
The Stability ◽  
Tip Mass ◽  
Euler Bernoulli Beam ◽  
Stability Results

A translational cantilevered Euler-Bernoulli beam with tip mass dynamics at its free end is used to study the effect of several damping mechanisms on the stabilization of the beam displacement. Specifically, a Lyapunov-based controller utilizing a partial differential equation model of the translational beam is developed to exponentially stabilize the beam displacement while the beam support is regulated to a desired set-point position. Depending on the composition of the tip mass dynamics assumption (i.e. body-mass, point-mass, or massless). it is shown that proper combination of different damping mechanisms (i.e. strain-rate, structural, or viscous damping) guarantees exponential stability of the beam displacement. This novel Lyapunov-based approach. which is based on the energy dissipation mechanism in the beam, brings new dimensions to the stabilization problem of translational beams with tip mass dynamics. The stability analysis utilizes relatively simple mathematical tools to illustrate the exponential and asymptotic stability results. The numerical results are presented to show the effectiveness of the controller.

scholarly journals A Stochastic TB Model for a Crowded Environment

2018 ◽  
Vol 2018 ◽  
pp. 1-8 ◽  
Author(s):  
Sibaliwe Maku Vyambwera ◽  
Peter Witbooi
Keyword(s):  
Compartmental Model ◽  
Deterministic Model ◽  
Reproduction Number ◽  
Stochastic Perturbation ◽  
Equation Model ◽  
Ordinary Differential Equation Model ◽  
Differential Equation Model ◽  
The Stability ◽  
Crowded Environments ◽  
Disease Free

We propose a stochastic compartmental model for the population dynamics of tuberculosis. The model is applicable to crowded environments such as for people in high density camps or in prisons. We start off with a known ordinary differential equation model, and we impose stochastic perturbation. We prove the existence and uniqueness of positive solutions of a stochastic model. We introduce an invariant generalizing the basic reproduction number and prove the stability of the disease-free equilibrium when it is below unity or slightly higher than unity and the perturbation is small. Our main theorem implies that the stochastic perturbation enhances stability of the disease-free equilibrium of the underlying deterministic model. Finally, we perform some simulations to illustrate the analytical findings and the utility of the model.

scholarly journals Non-Spiking Laser Controlled by a Delayed Feedback

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 2069
Author(s):  
Anton V. Kovalev ◽  
Evgeny A. Viktorov ◽  
Thomas Erneux
Keyword(s):  
Differential Equation ◽  
Delay Differential Equation ◽  
Optical Feedback ◽  
Laser Pulses ◽  
Equation Model ◽  
Rate Equations ◽  
Differential Equation Model ◽  
The Stability ◽  
Delay Differential ◽  
Intensity Laser

In 1965, Statz et al. (J. Appl. Phys. 30, 1510 (1965)) investigated theoretically and experimentally the conditions under which spiking in the laser output can be completely suppressed by using a delayed optical feedback. In order to explore its effects, they formulate a delay differential equation model within the framework of laser rate equations. From their numerical simulations, they concluded that the feedback is effective in controlling the intensity laser pulses provided the delay is short enough. Ten years later, Krivoshchekov et al. (Sov. J. Quant. Electron. 5394 (1975)) reconsidered the Statz et al. delay differential equation and analyzed the limit of small delays. The stability conditions for arbitrary delays, however, were not determined. In this paper, we revisit Statz et al.’s delay differential equation model by using modern mathematical tools. We determine an asymptotic approximation of both the domains of stable steady states as well as a sub-domain of purely exponential transients.

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