Exponential stabilization of an axially moving string with geometrical nonlinearity by linear boundary feedback

In this paper, we consider the system modeled by an axially moving string and a mass-damper-spring (MDS) controller, applied at the right-hand side (RHS) boundary of the string. We are concerned with the nonlinear string and the effect of the control mechanism. We stabilize the system through a proposed boundary velocity feedback control law. Linear and nonlinear control laws through this controller are proposed. In this paper, we find that a linear boundary feedback caused the total mechanical energy of the system to decay an asymptotically, but it fails for an exponential decay. However, a nonlinear boundary feedback controller can stabilize the system exponentially. The asymptotic and exponential stability are verified.

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A generalized exponential stabilization for a class of semilinear parabolic equations: Linear boundary feedback control approach

Mathematical Methods in the Applied Sciences ◽

10.1002/mma.7735 ◽

2021 ◽

Author(s):

Chengzhou Wei ◽

Junmin Li

Keyword(s):

Feedback Control ◽

Parabolic Equations ◽

Exponential Stabilization ◽

Semilinear Parabolic Equations ◽

Linear Boundary ◽

Boundary Feedback Control ◽

Control Approach ◽

Boundary Feedback ◽

Semilinear Parabolic

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Suppression of Vibration in a Nonlinear Axially Moving String by the Boundary Control

Volume 1D: 16th Biennial Conference on Mechanical Vibration and Noise ◽

10.1115/detc97/vib-4077 ◽

1997 ◽

Author(s):

Shahram M. Shahruz

Keyword(s):

Boundary Control ◽

Negative Feedback ◽

Linear Boundary ◽

Axially Moving String ◽

Axially Moving ◽

Nonlinear String

Abstract In this note, a nonlinear axially moving string is considered. It is proved that the nonlinear string can be stabilized by the linear boundary control, which is the negative feedback of the transversal velocity of the string at one end.

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Complex Eigenvalue Analysis and Nonlinear Transient Vibration of an Axially Moving String With an Attached Travelling Oscillator Under Wind Excitations

Volume 5: 22nd International Conference on Design Theory and Methodology; Special Conference on Mechanical Vibration and Noise ◽

10.1115/detc2010-28414 ◽

2010 ◽

Author(s):

Yuefang Wang ◽

Lefeng Lu¨ ◽

Lihua Huang

Keyword(s):

Green’S Function ◽

Green's Function ◽

Geometrical Nonlinearity ◽

Eigenvalue Analysis ◽

Complex Eigenvalue ◽

First Eigenvalue ◽

Transient Vibration ◽

Axially Moving String ◽

Nonlinear Transient ◽

Axially Moving

The eigenvalue analysis and nonlinear dynamics of an axially moving string with an attached traveling oscillator excited by wind loadings are presented in this paper. The couplings in both of mass and stiffness between the string and the oscillator are considered. The Green’s function in an explicit form is obtained by the theorem of Green’s function construction, and the analytical transcendental equation is directly obtained. The maximum variance rate is defined to analyze the coupling strength of the subsystems. It is demonstrated that the first eigenvalue changes significantly when the natural frequency of the oscillator is close to that of the string’s first frequency. The equation of motion of nonlinear transient vibration of the coupled system is obtained by considering aerodynamic loading and geometric nonlinearity of the string. The equation is solved numerically for the transient response of the vibration. The effect of the geometrical nonlinearity due to large deformation and the interaction between the oscillator and the string are also presented.

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Boundary Control of an Axially Moving String Via Lyapunov Method

Journal of Dynamic Systems Measurement and Control ◽

10.1115/1.2802425 ◽

1999 ◽

Vol 121 (1) ◽

pp. 105-110 ◽

Cited By ~ 66

Author(s):

Rong-Fong Fung ◽

Chun-Chang Tseng

Keyword(s):

Boundary Control ◽

Lyapunov Method ◽

Sliding Mode ◽

Semigroup Theory ◽

Active Vibration Control ◽

Nonlinear Partial Differential Equation ◽

Distributed Parameter ◽

Axially Moving String ◽

Active Vibration ◽

Axially Moving

This paper presents the active vibration control of an axially moving string system through a mass-damper-spring (MDS) controller at its right-hand side (RHS) boundary. A nonlinear partial differential equation (PDE) describes a distributed parameter system (DPS) and directly selected as the object to be controlled. A new boundary control law is designed by sliding mode associated with Lyapunov method. It is shown that the boundary feedback states only include the displacement, velocity, and slope of the string at RHS boundary. Asymptotical stability of the control system is proved by the semigroup theory. Finally, finite difference scheme is used to validate the theoretical results.

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