Analytical Modeling for the Shape Control of an Adaptive Composite Satellite Dish

2008 ◽  
Author(s):  
Su Yan ◽  
Mehrdad N. Ghasemi-Nejhad
Keyword(s):  
Shape Control ◽  
Separation Of Variables ◽  
Three Dimensional ◽  
Mode Shapes ◽  
Legendre Functions ◽  
Spherical Coordinate ◽  
Associated Legendre Functions ◽  
Satellite Dish ◽  
Active Composite ◽  
Adaptive Composite

A three-dimensional dynamic analysis of a satellite dish in spherical coordinate system is investigated. The method of separation of variables is employed to obtain the explicit 3D solution of the partial differential governing equation of the active composite satellite dish (ACSD). Then, the mode shape functions are expanded as a combination of periodic functions, associated Legendre functions, and spherical Bessel functions. The validation of the theoretical model is performed by comparing the developed analytical mode shapes with finite element method (FEM) mode shapes. Also, using the developed analytical model, the disturbance observer (DOB) controller is employed for the ACSD shape control. The numerical results show that, by employing the DOB controller, more accurate shape control of the satellite dish is achieved and less control energy is consumed, when piezoelectric actuator patches are placed on their optimal locations.

THREE-DIMENSIONAL INFINITE ELEMENTS BASED ON A TREFFTZ FORMULATION

2001 ◽  
Vol 09 (02) ◽  
pp. 381-394 ◽  
Author(s):  
ISAAC HARARI ◽  
PARAMA BARAI ◽  
PAUL E. BARBONE ◽  
MICHAEL SLAVUTIN
Keyword(s):  
Separation Of Variables ◽  
Outer Boundary ◽  
Three Dimensional ◽  
Legendre Functions ◽  
Spherical Coordinates ◽  
Singular Behavior ◽  
Associated Legendre Functions ◽  
Exterior Problems ◽  
Infinite Elements ◽  
Time Harmonic

Three-dimensional infinite elements for exterior problems of time-harmonic acoustics are developed. The infinite elements mesh only the outer boundary of the finite element domain and need not match the finite elements on the interface. A four-noded infinite element, based on separation of variables in spherical coordinates, is presented. Singular behavior of associated Legendre functions at the poles is circumvented. Numerical results validate the good performance of this approach.

Net Mass Flow Components in a Three-Dimensional Unsteady Rotor Inflow Model

2003 ◽  
Author(s):  
Ke Yu ◽  
David A. Peters
Keyword(s):  
Mass Flow ◽  
Three Dimensional ◽  
Legendre Functions ◽  
Skew Angle ◽  
Pressure Potential ◽  
Associated Legendre Functions ◽  
Galerkin Approach ◽  
Flow Equations ◽  
Closed Form Solutions ◽  
Flow Components

Potential flow equations are converted to ordinary differential equations by the Galerkin approach in which velocity and pressure potential functions are expanded in terms of closed-form solutions to Laplace’s Equation. The reduced number of generalized coordinates in a Galerkin approach gives advantages in real-time simulations, preliminary design, and dynamic eigenvalue analysis for aeroelasticity. Net mass injection from rotor sources is expected to occur in some situations, but cannot be treated by previous models. It is included in the present formulation. In this paper, frequency response due to pressure distributions corresponding to net mass flow in both axial and skew-angle flight are given. These results are compared with exact solutions obtained by the approach of a convolution integral. A brief analysis is also included with respect to numerical simulations of the Associated Legendre Functions, in which it demonstrates that net mass flow components are extremely sensitive to the recursive process of seeking Associated Legendre Functions.

Three-Dimensional Stress Singularities at Conical Notches and Inclusions in Transversely Isotropic Materials

1986 ◽  
Vol 53 (1) ◽  
pp. 89-96 ◽  
Author(s):  
Nihal Somaratna ◽  
T. C. T. Ting
Keyword(s):  
General Solution ◽  
Separation Of Variables ◽  
Three Dimensional ◽  
Transversely Isotropic ◽  
Solution Procedure ◽  
Legendre Functions ◽  
Spherical Coordinates ◽  
Stress Singularities ◽  
Isotropic Materials ◽  
Transversely Isotropic Materials

This study examines analytically the possible existence of stress singularities of the form σ = ρδf(θ,φ) at the apex of axisymmetric conical boundaries in transversely isotropic materials. (ρ, θ, φ) refer to spherical coordinates with the origin at the apex. The problems of one conical boundary and of two conical boundaries with a common apex are considered. The boundaries are either rigidly clamped or traction free. Separation of variables enables the general solution to be expressed in terms of Legendre functions of the first and second kind. Imposition of boundary conditions leads to an eigenequation that would determine possible values of δ. The degenerate case that arises when the eigenvalues of the elasiticity constants are identical is also discussed. Isotropic materials constitute only a particular case in this class of degenerate materials and previously reported eigenequations corresponding to isotropic materials are shown to be recoverable from the present results. Numerical results corresponding to a few selected cases are also presented to illustrate the solution procedure.

Analytical Solution for Three-Dimensional, Unsteady Heat Conduction in a Multilayer Sphere

2016 ◽  
Vol 138 (10) ◽  
Author(s):  
Suneet Singh ◽  
Prashant K. Jain ◽  
Rizwan-uddin
Keyword(s):  
Heat Conduction ◽  
Analytical Solution ◽  
Coordinate System ◽  
Three Dimensional ◽  
Solution Procedure ◽  
Polar Coordinate System ◽  
Legendre Functions ◽  
Spherical Coordinates ◽  
Azimuthal Direction ◽  
Spherical Coordinate

An analytical solution has been obtained for the transient problem of three-dimensional multilayer heat conduction in a sphere with layers in the radial direction. The solution procedure can be applied to a hollow sphere or a solid sphere composed of several layers of various materials. In general, the separation of variables applied to 3D spherical coordinates has unique characteristics due to the presence of associated Legendre functions as the eigenfunctions. Moreover, an eigenvalue problem in the azimuthal direction also requires solution; again, its properties are unique owing to periodicity in the azimuthal direction. Therefore, extending existing solutions in 2D spherical coordinates to 3D spherical coordinates is not straightforward. In a spherical coordinate system, one can solve a 3D transient multilayer heat conduction problem without the presence of imaginary eigenvalues. A 2D cylindrical polar coordinate system is the only other case in which such multidimensional problems can be solved without the use of imaginary eigenvalues. The absence of imaginary eigenvalues renders the solution methodology significantly more useful for practical applications. The methodology described can be used for all the three types of boundary conditions in the outer and inner surfaces of the sphere. The solution procedure is demonstrated on an illustrative problem for which results are obtained.

Axisymmetric Stokes Flow Past Cones Including the Case of the Needle

1975 ◽  
Vol 42 (3) ◽  
pp. 569-574
Author(s):  
K. N. Ghia ◽  
A. G. Mikhail
Keyword(s):  
Coordinate System ◽  
Skin Friction ◽  
Stokes Flow ◽  
Small Neighborhood ◽  
Legendre Functions ◽  
Spherical Coordinate ◽  
Associated Legendre Functions ◽  
First Order ◽  
Flow Past ◽  
Limiting Case

The Stokes flow past sharp axisymmetric cones of acute semivertex angle has been studied using an axisymmetric spherical coordinate system. The Stokes solution consists of associated Legendre functions, of the first kind, of the first-order and fractional degree related to the eigenvalues of the problem. These Legendre functions as well as the lowest eigenvalues of the Stokes solution have been accurately evaluated using two different approaches. The present results for the eigenvalues appear to be more accurate than those obtained earlier by Schwiderski, Lugt and Ugincius [3]. An important limiting case with semivertex angle δ → 0, i.e., the needle has been correctly analyzed and the results show that, as δ → 0, the Stokes flow is valid in a vanishingly small neighborhood of the needle with the skin friction being infinite at the “surface” of the needle.

Free Vibration of a Point-Supported Spherical Shell

1985 ◽  
Vol 52 (4) ◽  
pp. 890-896 ◽  
Author(s):  
T. Irie ◽  
G. Yamada ◽  
Y. Muramoto
Keyword(s):  
Free Vibration ◽  
Spherical Shell ◽  
Natural Frequencies ◽  
Ritz Method ◽  
Frequency Equation ◽  
Mode Shapes ◽  
Lagrangian Multiplier ◽  
Legendre Functions ◽  
Associated Legendre Functions ◽  
Lagrangian Multiplier Method

An analysis is presented for the free vibration of an elastically or a rigidly point-supported spherical shell. For this purpose, the deflection displacements of the shell are written in a series of the products of the associated Legendre functions and the trigonometric functions. The dynamical energies of the shell are evaluated, and the frequency equation is derived by the Ritz method. For a rigidly point-supported shell, the Lagrangian multiplier method is conveniently employed. The method is applied to a closed spherical shell supported at equispaced four points located along a parallel of latitude; the natural frequencies and the mode shapes are calculated numerically, and the effects of the point supports on the vibration are studied.

Center of Mass Theorem and the Separation of Variables for Two-Body System on a Three-Dimensional Sphere

2020 ◽  
Vol 23 (3) ◽  
pp. 306-311
Author(s):  
Yu. Kurochkin ◽  
Dz. Shoukavy ◽  
I. Boyarina
Keyword(s):  
Constant Curvature ◽  
Jacobi Equation ◽  
Separation Of Variables ◽  
Center Of Mass ◽  
Three Dimensional ◽  
Relative Distance ◽  
Dimensional Sphere ◽  
Body System ◽  
Spaces Of Constant Curvature ◽  
Hamilton Jacobi Equation

The immobility of the center of mass in spaces of constant curvature is postulated based on its definition obtained in [1]. The system of two particles which interact through a potential depending only on the distance between particles on a three-dimensional sphere is considered. The Hamilton-Jacobi equation is formulated and its solutions and trajectory equations are found. It was established that the reduced mass of the system depends on the relative distance.

scholarly journals UAV Formation Shape Control via Decentralized Markov Decision Processes

Algorithms ◽  
2021 ◽  
Vol 14 (3) ◽  
pp. 91
Author(s):  
Md Ali Azam ◽  
Hans D. Mittelmann ◽  
Shankarachary Ragi
Keyword(s):  
Control Problem ◽  
Shape Control ◽  
Formation Control ◽  
Three Dimensional ◽  
Geographical Region ◽  
Dynamic Programming Method ◽  
Theoretic Approach ◽  
Control Approach ◽  
Uav Swarm ◽  
Markov Decision

In this paper, we present a decentralized unmanned aerial vehicle (UAV) swarm formation control approach based on a decision theoretic approach. Specifically, we pose the UAV swarm motion control problem as a decentralized Markov decision process (Dec-MDP). Here, the goal is to drive the UAV swarm from an initial geographical region to another geographical region where the swarm must form a three-dimensional shape (e.g., surface of a sphere). As most decision-theoretic formulations suffer from the curse of dimensionality, we adapt an existing fast approximate dynamic programming method called nominal belief-state optimization (NBO) to approximately solve the formation control problem. We perform numerical studies in MATLAB to validate the performance of the above control algorithms.

scholarly journals Identification of Mode Shapes of a Composite Cylinder Using Convolutional Neural Networks

Materials ◽  
2021 ◽  
Vol 14 (11) ◽  
pp. 2801
Author(s):  
Bartosz Miller ◽  
Leonard Ziemiański
Keyword(s):  
Neural Networks ◽  
Convolutional Neural Networks ◽  
Composite Structures ◽  
Three Dimensional ◽  
Mode Shapes ◽  
Identification Algorithm ◽  
Material Degradation ◽  
Local Material ◽  
Noisy Input ◽  
Shape Vector

The aim of the following paper is to discuss a newly developed approach for the identification of vibration mode shapes of multilayer composite structures. To overcome the limitations of the approaches based on image analysis (two-dimensional structures, high spatial resolution of mode shapes description), convolutional neural networks (CNNs) are applied to create a three-dimensional mode shapes identification algorithm with a significantly reduced number of mode shape vector coordinates. The CNN-based procedure is accurate, effective, and robust to noisy input data. The appearance of local damage is not an obstacle. The change of the material and the occurrence of local material degradation do not affect the accuracy of the method. Moreover, the application of the proposed identification method allows identifying the material degradation occurrence.

Comparison Considerations Toward Investigating the Factors of Load and Age Group on the Maximum Reach Envelope

2020 ◽  
pp. 001872082096501
Author(s):  
Heather Johnston ◽  
Colleen Dewis ◽  
John Kozey
Keyword(s):  
Coordinate System ◽  
Three Dimensional ◽  
Spherical Coordinate System ◽  
Upper Body ◽  
Group Differences ◽  
Anthropometric Measures ◽  
Younger Adults ◽  
Present Measurement ◽  
Spherical Coordinate ◽  
Cylindrical Coordinate

Objective The objectives were to compare cylindrical and spherical coordinate representations of the maximum reach envelope (MRE) and apply these to a comparison of age and load on the MRE. Background The MRE is a useful measurement in the design of workstations and quantifying functional capability of the upper body. As a dynamic measure, there are human factors that impact the size, shape, and boundaries of the MRE. Method Three-dimensional reach measures were recorded using a computerized potentiometric system for anthropometric measures (CPSAM) on two adult groups (aged 18–25 years and 35–70 years). Reach trials were performed holding .0, .5, and 1 kg. Results Three-dimensional Cartesian coordinates were transformed into cylindrical ( r, θ , Z) and spherical ( r, θ, ϕ) coordinates. Median reach distance vectors were calculated for 54 panels within the MRE as created by incremented banding of the respective coordinate systems. Reach distance and reach area were compared between the two groups and the loaded conditions using a spherical coordinate system. Both younger adults and unloaded condition produced greater reach distances and reach areas. Conclusions Where a cylindrical coordinate system may reflect absolute reference for design, a normalized spherical coordinate system may better reflect functional range of motion and better compare individual and group differences. Age and load are both factors that impact the MRE. Application These findings present measurement considerations for use in human reach investigation and design.

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