Specified Motion of Piezoelectrically Actuated Structures

2012 ◽  
Vol 134 (2) ◽  
Author(s):  
T. M. Seigler ◽  
A. H. Ghasemi
Keyword(s):  
Linear Models ◽  
Differential Algebraic Equations ◽  
Control Input ◽  
Nonlinear Program ◽  
Algebraic Equations ◽  
Finite Dimensional ◽  
Deformable Structure ◽  
Material Points ◽  
Euler Bernoulli Beam ◽  
Program Constraint

The problem considered is that of solving for the control input that generates partly specified motion of a deformable structure with distributed piezoelectric actuation. The motion constraint, called the program constraint, is specified as a desired relation on the motion of selected material points of the structure. The solution is based on a projection method applicable to a class of finite-dimensional dynamical systems which includes many common vibration models. For a nonlinear model with a nonlinear program constraint, the procedure in general results in a set of differential algebraic equations. It is shown that for linear models with linear periodic program constraints, the system is reduced to a set of algebraic equations. Application examples are presented for a Euler-Bernoulli beam to demonstrate the usefulness of the procedure.

Dynamics of a Thin Plate With Partly Specified Motion

2009 ◽  
Author(s):  
AmirHossein Ghasemi ◽  
T. M. Seigler
Keyword(s):  
Thin Plate ◽  
Solution Method ◽  
Small Strain ◽  
Analysis Software ◽  
Element Analysis ◽  
Active Structures ◽  
Forcing Function ◽  
Finite Dimensional ◽  
Deformable Structure ◽  
Program Constraint

The problem considered here is that of determining the precise forcing function that produces specified motion of a deformable structure. While fundamental in nature, the problem is practically important for evaluating actuation requirements for active structures. Our approach to this problem is to apply an existing solution method for finite-dimensional mechanical systems. We evaluate the direct application of this method to the spatially discretized, small strain equations of a thin plate. Several example cases are considered in which the program forces are determined, and then applied in finite element analysis software for verification of the program constraint.

scholarly journals THE INVERSE SIMULATION STUDY OF AIRCRAFT FLIGHT PATH RECONSTRUCTION

Transport ◽  
2002 ◽  
Vol 17 (3) ◽  
pp. 103-107 ◽  
Author(s):  
Wojciech Blajer ◽  
Jacek A. Goszczyński ◽  
Mariusz Krawczyk
Keyword(s):  
Differential Algebraic Equations ◽  
Velocity Variation ◽  
State Variables ◽  
Algebraic Equations ◽  
Governing Equations ◽  
Inverse Simulation ◽  
Time Variations ◽  
Aircraft Motion ◽  
Additional Constraints ◽  
Program Constraint

This paper presents a uniform approach to the modelling and simulation of aircraft prescribed trajectory flight. The aircraft motion is specified by a trajectory in space, a condition on airframe attitude with respect to the trajectory, and a desired flight velocity variation. For an aircraft controlled by aileron, elevator and rudder deflections and thrust changes a tangent realization of trajectory constraints arises which yields two additional constraints on the airframe attitude with respect to the trajectory. Combining the program constraint conditions and aircraft dynamic equations the governing equations of programmed motion are developed in the form of differential-algebraic equations. A method for solving the equations is proposed. The solution consists of time variations of the aircraft state variables and the demanded control that ensures the programmed motion realization.

Inverse Dynamic Simulation of a Hydraulic Drive With Modelica

2013 ◽  
Author(s):  
Joseph Saad ◽  
Matthias Liermann
Keyword(s):  
Dynamic Simulation ◽  
Hydraulic Drive ◽  
Differential Algebraic Equations ◽  
Control Input ◽  
Modeling Languages ◽  
Algebraic Equations ◽  
Servo Drive ◽  
Inverse Dynamic ◽  
Control Error ◽  
Hydraulic Machines

Inverse dynamic simulation of hydraulic drives is helpful in early design stages of hydraulic machines to answer the question whether the drive can meet dynamic load requirements and at the same time to predict the energy consumption for required load cycles. While a forward simulation of the hydraulic drive needs an implementation of the controller which generates the control input as a function of the control error, the inverse dynamic simulation can be implemented without control. This is because the required motion is simply defined as a constraint and therefore the control error is always zero. This paper surveys examples of successful use of inverse dynamic simulation in engineering. We use the example of a hydraulic servo-drive to explain the procedure how to generate a state space description of the inverse problem from the given system of differential algebraic equations. Equation based modeling languages such as Modelica lend themselves naturally for inverse simulation because the definitions of which variables of the model are inputs and which are outputs is not made explicit in the model itself.

Multibody Inverse Dynamics Using an Energy Preserving Direct Transcription Process

2003 ◽  
Author(s):  
Carlo L. Bottasso ◽  
Alessandro Croce
Keyword(s):  
Optimal Control Problems ◽  
Inverse Dynamics ◽  
Differential Algebraic Equations ◽  
Control Problems ◽  
Algebraic Equations ◽  
Dynamic Problems ◽  
Direct Transcription ◽  
Finite Dimensional ◽  
Transcription Process ◽  
Multibody Dynamic

We propose a procedure for the solution of inverse multibody dynamic problems, here intended as optimal control problems for dynamical systems governed by differential-algebraic equations. The numerical solution is obtained by a direct transcription process based on an energy preserving scheme that ensures nonlinear unconditional stability. The resulting finite-dimensional problem is solved by sequential quadratic programming. We test the proposed methodology with the help of representative examples.

Uncertainty Quantification of Differential Algebraic Equations Using Polynomial Chaos

2021 ◽  
Author(s):  
Premjit Saha ◽  
Tarunraj Singh ◽  
Gary F. Dargush
Keyword(s):  
Linear Models ◽  
Polynomial Chaos ◽  
Initial Conditions ◽  
Surrogate Models ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Rlc Circuit ◽  
Algebraic Constraints ◽  
Interpolation Polynomials ◽  
Time Invariant

Abstract The focus of this paper is on the use of Polynomial Chaos for developing surrogate models for Differential Algebraic Equations with time-invariant uncertainties. Intrusive and non-intrusive approaches to synthesize Polynomial Chaos surrogate models are presented including the use of Lagrange interpolation polynomials as basis functions. Unlike ordinary differential equations, if the algebraic constraints are a function of the stochastic variable, some initial conditions of the differential algebraic equations are also random. A benchmark RLC circuit which is used as a benchmark for linear models is used to illustrate the development of a Polynomial Chaos based surrogate model. A nonlinear example of a simple pendulum also serves as a benchmark to illustrate the potential of the proposed approach. Statistics of the results of the Polynomial Chaos models are validated using Monte Carlo simulations in addition to estimating the evolving PDFs of the states of the pendulum.

scholarly journals A Numerical Method of the Euler-Bernoulli Beam with Optimal Local Kelvin-Voigt Damping

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Xin Yu ◽  
Zhigang Ren ◽  
Qian Zhang ◽  
Chao Xu
Keyword(s):  
Control Problem ◽  
Numerical Approximation ◽  
Optimal Parameter ◽  
Nonlinear Coefficient ◽  
Programming Algorithm ◽  
Beam Equation ◽  
Control Input ◽  
Bernoulli Beam ◽  
Finite Dimensional ◽  
Euler Bernoulli Beam

This paper deals with the numerical approximation problem of the optimal control problem governed by the Euler-Bernoulli beam equation with local Kelvin-Voigt damping, which is a nonlinear coefficient control problem with control constraints. The goal of this problem is to design a control input numerically, which is the damping and distributes locally on a subinterval of the region occupied by the beam, such that the total energy of the beam and the control on a given time period is minimal. We firstly use the finite element method (FEM) to obtain a finite-dimensional model based on the original PDE system. Then, using the control parameterization method, we approximate the finite-dimensional problem by a standard optimal parameter selection problem, which is a suboptimal problem and can be solved numerically by nonlinear mathematical programming algorithm. At last, some simulation studies will be presented by the proposed numerical approximation method in this paper, where the damping controls act on different locations of the Euler-Bernoulli beam.

Markov Jump Linear System Analysis of a Microgrid Operating in Islanded and Grid Tied Modes

2020 ◽  
Author(s):  
Gilles Mpembele ◽  
Jonathan Kimball
Keyword(s):  
Monte Carlo ◽  
Power Systems ◽  
Power System ◽  
System Analysis ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Markov Jump ◽  
Numerical Application ◽  
Conditional Moments ◽  
Markov Jump Linear Systems

<div>The analysis of power system dynamics is usually conducted using traditional models based on the standard nonlinear differential algebraic equations (DAEs). In general, solutions to these equations can be obtained using numerical methods such as the Monte Carlo simulations. The use of methods based on the Stochastic Hybrid System (SHS) framework for power systems subject to stochastic behavior is relatively new. These methods have been successfully applied to power systems subjected to</div><div>stochastic inputs. This study discusses a class of SHSs referred to as Markov Jump Linear Systems (MJLSs), in which the entire dynamic system is jumping between distinct operating points, with different local small-signal dynamics. The numerical application is based on the analysis of the IEEE 37-bus power system switching between grid-tied and standalone operating modes. The Ordinary Differential Equations (ODEs) representing the evolution of the conditional moments are derived and a matrix representation of the system is developed. Results are compared to the averaged Monte Carlo simulation. The MJLS approach was found to have a key advantage of being far less computational expensive.</div>

Asymptotics of the eigenmodes and stability of an elastic structure with general feedback matrix

2019 ◽  
Vol 84 (5) ◽  
pp. 873-911 ◽  
Author(s):  
Marianna A Shubov ◽  
Laszlo P Kindrat
Keyword(s):  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Control Input ◽  
Vibrational Modes ◽  
Boundary Feedback ◽  
Feedback Matrix ◽  
Dissipative Boundary ◽  
Initial Boundary Value ◽  
The Right ◽  
Euler Bernoulli Beam

Abstract The distribution of natural frequencies of the Euler–Bernoulli beam subject to fully non-dissipative boundary conditions is investigated. The beam is clamped at the left end and equipped with a 4-parameter ($\alpha ,\beta ,k_1,k_2$) linear boundary feedback law at the right end. The $2 \times 2$ boundary feedback matrix relates the control input (a vector of velocity and its spatial derivative at the right end), to the output (a vector of shear and moment at the right end). The initial boundary value problem describing the dynamics of the beam has been reduced to the first order in time evolution equation in the state Hilbert space equipped with the energy norm. The dynamics generator has a purely discrete spectrum (the vibrational modes) denoted by $\{\nu _n\}_{n\in \mathbb {Z}^{\prime}}$. The role of the control parameters is examined and the following results have been proven: (i) when $\beta \neq 0$, the set of vibrational modes is asymptotically close to the vertical line on the complex $\nu$-plane given by the equation $\Re \nu = \alpha + (1-k_1k_2)/\beta$; (ii) when $\beta = 0$ and the parameter $K = (1-k_1 k_2)/(k_1+k_2)$ is such that $\left |K\right |\neq 1$ then the following relations are valid: $\Re (\nu _n/n) = O\left (1\right )$ and $\Im (\nu _n/n^2) = O\left (1\right )$ as $\left |n\right |\to \infty$; (iii) when $\beta =0$, $|K| = 1$, and $\alpha = 0$, then the following relations are valid: $\Re (\nu _n/n^2) = O\left (1\right )$ and $\Im (\nu _n/n) = O\left (1\right )$ as $\left |n\right |\to \infty$; (iv) when $\beta =0$, $|K| = 1$, and $\alpha>0$, then the following relations are valid: $\Re (\nu _n/\ln \left |n\right |) = O\left (1\right )$ and $\Im (\nu _n/n^2) = O\left (1\right )$ as $\left |n\right |\to \infty$.

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