scholarly journals Stabilization and Observability of a Rotating Timoshenko Beam Model

2007 ◽  
Vol 2007 ◽  
pp. 1-19 ◽  
Author(s):  
Alexander Zuyev ◽  
Oliver Sawodny
Keyword(s):  
Timoshenko Beam ◽  
Galerkin Approximation ◽  
Sufficient Conditions ◽  
Beam Equation ◽  
Distributed Parameter ◽  
Dimensional System ◽  
Beam Model ◽  
Finite Dimensional ◽  
Galerkin Approximations ◽  
Rotating Timoshenko Beam

A control system describing the dynamics of a rotating Timoshenko beam is considered. We assume that the beam is driven by a control torque at one of its ends, and the other end carries a rigid body as a load. The model considered takes into account the longitudinal, vertical, and shear motions of the beam. For this distributed parameter system, we construct a family of Galerkin approximations based on solutions of the homogeneous Timoshenko beam equation. We derive sufficient conditions for stabilizability of such finite dimensional system. In addition, the equilibrium of the Galerkin approximation considered is proved to be stabilizable by an observer-based feedback law, and an explicit control design is proposed.

Timoshenko Beam Modeling for Parameter Estimation of NASA Mini-Mast Truss

1993 ◽  
Vol 115 (1) ◽  
pp. 19-24 ◽  
Author(s):  
Ji Yao Shen ◽  
Jen-Kuang Huang ◽  
L. W. Taylor
Keyword(s):  
Timoshenko Beam ◽  
Closed Form Solution ◽  
Computational Effort ◽  
Form Solution ◽  
Beam Equation ◽  
Distributed Parameter ◽  
Beam Model ◽  
Timoshenko Beam Model ◽  
Modal Characteristics ◽  
Cantilevered Beam

In this paper a distributed parameter model for the estimation of modal characteristics of NASA Mini-Mast truss is proposed. A closed-form solution of the Timoshenko beam equation, for a uniform cantilevered beam with two concentrated masses, is derived so that the procedure and the computational effort for the estimation of modal characteristics are improved. A maximum likelihood estimator for the Timoshenko beam model is also developed. The resulting estimates from test data by using Timoshenko beam model are found to be comparable to those derived from other approaches.

scholarly journals An integrodifferential approach to modeling, control, state estimation and optimization for heat transfer systems

2016 ◽  
Vol 26 (1) ◽  
pp. 15-30 ◽  
Author(s):  
Andreas Rauh ◽  
Luise Senkel ◽  
Harald Aschemann ◽  
Vasily V. Saurin ◽  
Georgy V. Kostin
Keyword(s):  
Heat Transfer ◽  
Finite Element ◽  
Control Strategies ◽  
Open Loop ◽  
Distributed Parameter ◽  
Dimensional System ◽  
State Equations ◽  
Large Space ◽  
Finite Dimensional ◽  
Spatially Distributed

Abstract In this paper, control-oriented modeling approaches are presented for distributed parameter systems. These systems, which are in the focus of this contribution, are assumed to be described by suitable partial differential equations. They arise naturally during the modeling of dynamic heat transfer processes. The presented approaches aim at developing finite-dimensional system descriptions for the design of various open-loop, closed-loop, and optimal control strategies as well as state, disturbance, and parameter estimation techniques. Here, the modeling is based on the method of integrodifferential relations, which can be employed to determine accurate, finite-dimensional sets of state equations by using projection techniques. These lead to a finite element representation of the distributed parameter system. Where applicable, these finite element models are combined with finite volume representations to describe storage variables that are—with good accuracy—homogeneous over sufficiently large space domains. The advantage of this combination is keeping the computational complexity as low as possible. Under these prerequisites, real-time applicable control algorithms are derived and validated via simulation and experiment for a laboratory-scale heat transfer system at the Chair of Mechatronics at the University of Rostock. This benchmark system consists of a metallic rod that is equipped with a finite number of Peltier elements which are used either as distributed control inputs, allowing active cooling and heating, or as spatially distributed disturbance inputs.

Active Control of Flexible Structures Subject to Distributed and Seismic Disturbances

1993 ◽  
Vol 115 (4) ◽  
pp. 649-657 ◽  
Author(s):  
Akira Ohsumi ◽  
Yuichi Sawada
Keyword(s):  
Active Control ◽  
Flexible Structures ◽  
Distributed Parameter ◽  
Dimensional System ◽  
Random Disturbance ◽  
Modal Expansion ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Real Earthquake ◽  
The Mathematical Model

The purpose of this paper is to present a method of active control for suppressing the vibration of a mechanically flexible cantilever beam which is subject to a distributed random disturbance and also a seismic input at the clamped end. First, the mathematical model of the flexible structure is established by a stochastic partial differential equation which describes the Euler-Bernoulli type distributed parameter system with internal viscous damping and subject to the seismic and distributed random inputs. Second, the distributed parameter model, which is considered as an infinite-dimensional system, is reduced to a finite-dimensional one by using the modal expansion, and split into the controlled part and the uncontrolled (residual) one. The principal approach is to regard the observation spillover due to uncontrolled part as a colored observation noise and construct an estimator, and then we construct the optimal control system. Finally, simulation studies are presented by using a real earthquake accelerogram data.

Automatic Generation of Component Modes for Rotordynamic Substructures

1989 ◽  
Vol 111 (1) ◽  
pp. 6-10 ◽  
Author(s):  
S. H. Crandall ◽  
N. A. Yeh
Keyword(s):  
Timoshenko Beam ◽  
Dynamic Models ◽  
Dynamic Properties ◽  
Automatic Generation ◽  
Component Mode Synthesis ◽  
Beam Equation ◽  
Beam Model ◽  
Exact Integration ◽  
Beam Elements ◽  
Loading Patterns

Dynamic analysis models are customarily employed in turbomachinery design to predict critical whirling speeds and estimate dynamic response due to loads imposed by unbalance, misalignment, maneuvers, etc., Traditionally these models have been assembled from beam elements and been analyzed by transfer matrix methods. Recently there has been an upsurge of interest in the development of improved dynamic models making use of finite element analysis and/or component mode synthesis. We are currently developing a procedure for modelling and analyzing multi-rotor systems [1] which employs component mode synthesis applied to rotor and stator substructures. A novel feature of our procedure is a program for the automatic generation of the component modes for substructures modelled as Timoshenko beam elements connected to other substructures by bearings, couplings, and localized structural joints. The component modes for such substructures consist of constraint modes and internal modes. The former are static deflection shapes resulting from removing the constraints one at a time and imposing unit deflections at the constraint locations. The latter have traditionally been taken to be a subset of the natural modes of free vibration of the substructure with all constraints imposed. It has however been pointed out [2] that any independent set of geometrically admissible modes may be used. We take advantage of this and employ static deflections under systematically selected loading patterns as internal modes. All component modes are thus obtained as static deflections of a simplified beam model which has the same span and same constraints as the actual substructure but which has piecewise uniform dynamic properties. With the loading patterns we employ, all modes are represented by fourth order polynomials with piecewise constant coefficients. We have developed an algorithm for the automatic calculation of these coefficients based on exact integration of the Timoshenko beam equation using singularity functions. The procedure is illustrated by applying it to a simplified system with a single rotor structure and a single stator structure. The accuracy of the procedure is examined by comparing its results with an exact analytical solution and with a component mode synthesis using true eigenfunctions as internal modes.

scholarly journals Exact boundary controllability of Galerkin approximations of Navier-Stokes system for soret convection

2015 ◽  
Vol 5 (2) ◽  
pp. 41-49
Author(s):  
Sivaguru S. Ravindran
Keyword(s):  
Soret Effect ◽  
Galerkin Approximation ◽  
Type System ◽  
Exact Controllability ◽  
Navier Stokes ◽  
Finite Dimensional ◽  
Galerkin Approximations ◽  
Diffusive Convection ◽  
Hilbert Uniqueness Method ◽  
Bounded Smooth

We study the exact controllability of finite dimensional Galerkin approximation of a Navier-Stokes type system describing doubly diffusive convection with Soret effect in a bounded smooth domain in Rd (d = 2, 3) with controls on the boundary. The doubly diffusive convection system with Soret effect involves a difficult coupling through second order terms. The Galerkin approximations are introduced undercertain assumptions on the Galerkin basis related to the linear independence of suitable traces of its elements over the boundary. By Using Hilbert uniqueness method in combination with a fixed point argument, we prove that the finite dimensional Galerkin approximations are exactly controllable.

Frequency Domain Approach to Solve the Time-Optimal Control of a Distributed Parameter System

1995 ◽  
Author(s):  
Hasan Alli ◽  
Tarunraj Singh
Keyword(s):  
Optimal Control ◽  
Wave Equation ◽  
Time Optimal Control ◽  
Distributed Parameter ◽  
Dimensional System ◽  
Distributed Parameter System ◽  
Parameter System ◽  
Finite Dimensional ◽  
Time Optimal ◽  
Frequency Domain Approach

Abstract In this paper, the time-optimal control of the wave equation is derived in closed form. A frequency domain approach is used to obtain the time-optimal solution which is bang-off-bang. The system studied in this paper is a uniform flexible rod with a control input at each end, whose dynamics in axial vibration is represented by the wave equation. In order to verify the optimality of the control profile derived for the distributed parameter system, the system is discretized in space and a series of time-optimal control problems are solved for the finite dimensional model, with increasing number of flexible modes. In the limit, the controllers show the convergence of the first and final switch of the bang-bang controller of the finite dimensional system to the first and final switch of the bang-off-bang controller of the distributed parameter system, in addition to the convergence of the maneuver time. The number of switches in between the first and final switch is a function of the order of the finite dimensional system. The maneuver time of the distributed parameter system is compared to that of an equivalent rigid system and the coincidence of the time-optimal controller for the flexible and rigidized systems is illustrated for certain maneuvers.

Beyond fractional derivatives: local approximation of other convolution integrals

2009 ◽  
Vol 466 (2114) ◽  
pp. 563-581 ◽  
Author(s):  
Satwinder Jit Singh ◽  
Anindya Chatterjee
Keyword(s):  
Numerical Solution ◽  
Fractional Order ◽  
Galerkin Approximation ◽  
Local Approximation ◽  
Dimensional System ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Convolution Integrals ◽  
Damped Systems ◽  
Special Case

Dynamic systems involving convolution integrals with decaying kernels, of which fractionally damped systems form a special case, are non-local in time and hence infinite dimensional. Straightforward numerical solution of such systems up to time  t needs computations owing to the repeated evaluation of integrals over intervals that grow like  t . Finite-dimensional and local approximations are thus desirable. We present here an approximation method which first rewrites the evolution equation as a coupled infinite-dimensional system with no convolution, and then uses Galerkin approximation with finite elements to obtain linear, finite-dimensional, constant coefficient approximations for the convolution. This paper is a broad generalization, based on a new insight, of our prior work with fractional order derivatives ( Singh & Chatterjee 2006 Nonlinear Dyn. 45 , 183–206). In particular, the decaying kernels we can address are now generalized to the Laplace transforms of known functions; of these, the power law kernel of fractional order differentiation is a special case. The approximation can be refined easily. The local nature of the approximation allows numerical solution up to time t with computations. Examples with several different kernels show excellent performance. A key feature of our approach is that the dynamic system in which the convolution integral appears is itself approximated using another system, as distinct from numerically approximating just the solution for the given initial values; this allows non-standard uses of the approximation, e.g. in stability analyses.

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