Modeling and Boundary Control of a Hanging Cable Immersed in Water

2013 ◽  
Vol 136 (1) ◽  
Author(s):  
Michael Böhm ◽  
Miroslav Krstic ◽  
Sebastian Küchler ◽  
Oliver Sawodny
Keyword(s):  
Differential Equation ◽  
Wave Equations ◽  
Equations Of Motion ◽  
Distributed Parameter ◽  
Stream Velocity ◽  
Water Stream ◽  
Varying Coefficients ◽  
Feedforward Controller ◽  
Spatially Varying ◽  
Two Degree Of Freedom

A nonlinear distributed parameter system model governing the motion of a cable with an attached payload immersed in water is derived. The payload is subject to a drag force due to a constant water stream velocity. Such a system is found, for example, in deep sea oil exploration, where a crane mounted on a ship is used for construction and thus positioning of underwater parts of an offshore drilling platform. The equations of motion are linearized, resulting in two coupled, one-dimensional wave equations with spatially varying coefficients and dynamic boundary conditions of second order in time. The wave equations model the normal and tangential displacements of cable elements, respectively. A two degree of freedom controller is designed for this system with a Dirichlet input at the boundary opposite to the payload. A feedforward controller is designed by inverting the system using a Taylor-series, which is then truncated. The coupling is ignored for the feedback design, allowing for a separate design for each direction of motion. Transformations are introduced, in order to transform the system into a cascade of a partial differential equation (PDE) and an ordinary differential equation (ODE), and PDE backstepping is applied. Closed-loop stability is proven. This is supported by simulation results for different cable lengths and payload masses. These simulations also illustrate the performance of the feedforward controller.

Transient Vibration of Gyroscopic Systems With Unsteady Superposed Motion

1995 ◽  
Author(s):  
J. A. Wickert
Keyword(s):  
Equations Of Motion ◽  
Mathematical Structure ◽  
Time Dependent ◽  
Distributed Parameter ◽  
Rotating Shaft ◽  
Transient Vibration ◽  
Modal Amplitude ◽  
Technical Interest ◽  
Gyroscopic Systems ◽  
Two Degree Of Freedom

Abstract The equations of motion for a gyroscopic system with unsteady superposed motion are derived for the prototypical problem in which motion of an oscillating particle is measured relative to a non-inertial frame. The resulting coefficient matrices are time-dependent, and skew-symmetric acceleration terms are present both as Coriolis acceleration and as a component of net stiffness. Such mathematical structure is also demonstrated in the context of other unsteady gyroscopic systems, including flexible media that translate with time-dependent speed. Following the asymptotic approach of Krylov, Bogoliubov and Mitropolsky, a perturbation method is developed for the case in which the superposed motion varies slowly when viewed on the time scale of the natural periods of oscillation. First-order approximations for the modal amplitude and phase are obtained in closed form. The method is illustrated through two examples of technical interest: a two degree-of-freedom model of a rotating shaft, and a distributed parameter model of a moving tape.

Identification of Systems Described by Partial Differential Equations

1966 ◽  
Vol 88 (2) ◽  
pp. 463-468 ◽  
Author(s):  
F. J. Perdreauville ◽  
R. E. Goodson
Keyword(s):  
Differential Equation ◽  
Experimental Data ◽  
Equation Model ◽  
Distributed Parameter ◽  
Differential Equation Model ◽  
Partial Differential ◽  
Varying Coefficients ◽  
Partial Differential Equation Model ◽  
Linear And Nonlinear

A method is given for the identification of distributed parameter systems. Normal operating records or experimental data may be used. The method involves the determination of arbitrary parameters in an assumed partial differential-equation model of the system. The method applies equally well to linear and nonlinear equations, and equations with varying coefficients. The accuracy of the results depends upon the exactness of the model, the amount of data used, the error in numerical integration, and the amount of noise which is present in the data. Examples are given which illustrate the application of the method. Results using the method for the identification of a physical system are given.

Approximate Response of Beam-Type Resonant Biosensors

2018 ◽  
Author(s):  
Pezhman A. Hassanpour
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Exponential Function ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Impact Force ◽  
Equations Of Motion ◽  
Varying Coefficients ◽  
Time Varying Coefficients ◽  
The Impact

In this paper, the effect of absorption of antigens to the functionalized surface of a biosensor is modeled using a single degree-of-freedom mass-spring-damper system. The change in the mass of the system due to absorption is modeled with an exponential function. The governing equations of motion is derived considering the change in the mass of the system as well as the impact force due to absorption. It has been demonstrated that this equation is a linear second-order ordinary differential equation with time-varying coefficients. The solution of this differential equation is approximated by expanding the exponential function with a Taylor series and applying the method of multiple scales. The advantage of using the method of multiple scales to derive an approximate solution is in the insight it provides on the effect of each parameter on the response of the system. The free vibration response of the biosensor is derived using the approximate solution under different conditions, namely, with and without viscous damping, with and without considering the impact force, and for different binding rates.

Simulation of a MEMS RF Filter

2005 ◽  
Author(s):  
Eihab M. Abdel-Rahman ◽  
Bashar K. Hammad ◽  
Ali H. Nayfeh
Keyword(s):  
Equations Of Motion ◽  
Distributed Parameter ◽  
Parameter System ◽  
Transmission Characteristics ◽  
Coupled Resonators ◽  
Lagrange Equations ◽  
Filter Model ◽  
Rf Filter ◽  
Two Degree Of Freedom ◽  
Filter Response

We simulate the motions in a MEMS bandpass Radio-Frequency (RF) filter. The filter model is obtained by discretizing the Lagrangian of the distributed-parameter system using a Galerkin procedure. The Euler-Lagrange equations are then used to obtain a two-degree-of-freedom model consisting of two non-linearly coupled ordinary-differential equations of motion. We use the model to study the transmission characteristics of a bandpass filter made up of two coupled resonators. Three distinct response regimes, separated by two critical amplification levels Vcr1 and Vcr2, are identified in the filter response. For amplification levels up to Vcr1, the pass signal is artifact free. Two types of artifacts due to the filter dynamics appear and distort the signal for amplification levels beyond Vcr1.

scholarly journals Boundary Control for a Kind of Coupled PDE-ODE System

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Yuanting Wang ◽  
Fucheng Liao ◽  
Yonglong Liao ◽  
Zhengwei Shen
Keyword(s):  
Differential Equation ◽  
State Feedback ◽  
Coupled System ◽  
Backstepping Method ◽  
Closed Loop System ◽  
Stabilizing Controller ◽  
Varying Coefficients ◽  
Ode System ◽  
Exponentially Stable ◽  
Spatially Varying

A coupled system of an ordinary differential equation (ODE) and a heat partial differential equation (PDE) with spatially varying coefficients is discussed. By using the PDE backstepping method, the state-feedback stabilizing controller is explicitly constructed with the assumptionsλ(x)∈P[x]nandλ(x)∈C[0, l]∞, respectively. The closed-loop system is proved to be exponentially stable by this controller. A simulation example is presented to illustrate the effectiveness of the proposed method.

The effects of nonlinear energy sink and piezoelectric energy harvester on aeroelastic instability of an airfoil

2021 ◽  
pp. 107754632199358
Author(s):  
Ali Fasihi ◽  
Majid Shahgholi ◽  
Saeed Ghahremani
Keyword(s):  
Energy Harvester ◽  
Equations Of Motion ◽  
Continuation Method ◽  
Dynamical Behavior ◽  
Harmonic Balance Method ◽  
Nonlinear Energy Sink ◽  
Piezoelectric Energy ◽  
Energy Sink ◽  
Two Degree Of Freedom ◽  
Nonlinear Energy

The potential of absorbing and harvesting energy from a two-degree-of-freedom airfoil using an attachment of a nonlinear energy sink and a piezoelectric energy harvester is investigated. The equations of motion of the airfoil coupled with the attachment are solved using the harmonic balance method. Solutions obtained by this method are compared to the numerical ones of the pseudo-arclength continuation method. The effects of parameters of the integrated nonlinear energy sink-piezoelectric attachment, namely, the attachment location, nonlinear energy sink mass, nonlinear energy sink damping, and nonlinear energy sink stiffness on the dynamical behavior of the airfoil system are studied for both subcritical and supercritical Hopf bifurcation cases. Analyses demonstrate that absorbing vibration and harvesting energy are profoundly affected by the nonlinear energy sink parameters and the location of the attachment.

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