Closed-form estimator for the matrix-variate Gamma distribution

2021 ◽  
pp. 1
Author(s):  
Gustav Alfelt
Keyword(s):  
Closed Form ◽  
Gamma Distribution ◽  
The Matrix

The Elastic Field of an Elliptic Inclusion With a Slipping Interface

1986 ◽  
Vol 53 (1) ◽  
pp. 103-107 ◽  
Author(s):  
E. Tsuchida ◽  
T. Mura ◽  
J. Dundurs
Keyword(s):  
Closed Form ◽  
Infinite Series ◽  
Elastic Field ◽  
Elliptic Inclusion ◽  
Numerical Examples ◽  
Elastic Fields ◽  
Bonded Interface ◽  
The Matrix ◽  
Displacement Potentials ◽  
Uniform Expansion

The paper analyzes the elastic fields caused by an elliptic inclusion which undergoes a uniform expansion. The interface between the inclusion and the matrix cannot sustain shear tractions and is free to slip. Papkovich–Neuber displacement potentials are used to solve the problem. In contrast to the perfectly bonded interface, the solution cannot be expressed in closed form and involves infinite series. The results are illustrated by numerical examples.

scholarly journals Properties of Matrix Variate Confluent Hypergeometric Function Distribution

2016 ◽  
Vol 2016 ◽  
pp. 1-12 ◽  
Author(s):  
Arjun K. Gupta ◽  
Daya K. Nagar ◽  
Luz Estela Sánchez
Keyword(s):  
Hypergeometric Function ◽  
Random Matrices ◽  
Gamma Distribution ◽  
Confluent Hypergeometric Function ◽  
Density Functions ◽  
Function Distribution ◽  
Gamma Distributions ◽  
The Matrix

We study matrix variate confluent hypergeometric function kind 1 distribution which is a generalization of the matrix variate gamma distribution. We give several properties of this distribution. We also derive density functions ofX2-1/2X1X2-1/2,(X1+X2)-1/2X1(X1+X2)-1/2, andX1+X2, wherem×mindependent random matricesX1andX2follow confluent hypergeometric function kind 1 and gamma distributions, respectively.

Modal Analysis of Multi-Stepped Distributed-Parameter Rotor-Bearing Systems Using Exact Dynamic Elements

2001 ◽  
Vol 123 (3) ◽  
pp. 401-403 ◽  
Author(s):  
Seong-Wook Hong ◽  
Jong-Heuck Park
Keyword(s):  
Modal Analysis ◽  
Closed Form ◽  
Complete Analysis ◽  
Distributed Parameter ◽  
Analysis Method ◽  
Closed Form Solutions ◽  
Analysis Scheme ◽  
Rotor Bearing ◽  
The Matrix ◽  
Transcendental Functions

Although the exact dynamic elements have been suggested by the authors [1] and proved to be useful for the dynamic analysis of distributed-parameter rotor-bearing systems, difficulty remains in computation because of the presence of transcendental functions in the matrix. This paper proposes a complete analysis scheme for the exact dynamic elements, a generalized modal analysis method, to obtain exact and closed form solutions of time and frequency domain responses for multi-stepped distributed-parameter rotor-bearing systems. A numerical example is provided for validating the proposed method.

Modified Anisotropic Gurson Yield Criterion for Porous Ductile Sheet Metals

2000 ◽  
Vol 123 (4) ◽  
pp. 409-416 ◽  
Author(s):  
W. Y. Chien ◽  
J. Pan ◽  
S. C. Tang
Keyword(s):  
Finite Element ◽  
Closed Form ◽  
Yield Criterion ◽  
Plastic Anisotropy ◽  
Matrix Material ◽  
Plastic Behavior ◽  
Computational Results ◽  
Sheet Metals ◽  
Element Analysis ◽  
The Matrix

The influence of plastic anisotropy on the plastic behavior of porous ductile materials is investigated by a three-dimensional finite element analysis. A unit cell of cube containing a spherical void is modeled. The Hill quadratic anisotropic yield criterion is used to describe the matrix normal anisotropy and planar isotropy. The matrix material is first assumed to be elastic perfectly plastic. Macroscopically uniform displacements are applied to the faces of the cube. The finite element computational results are compared with those based on the closed-form anisotropic Gurson yield criterion suggested in Liao et al. 1997, “Approximate Yield Criteria for Anisotropic Porous Ductile Sheet Metals,” Mech. Mater., pp. 213–226. Three fitting parameters are suggested for the closed-form yield criterion to fit the results based on the modified yield criterion to those of finite element computations. When the strain hardening of the matrix is considered, the computational results of the macroscopic stress-strain behavior are in agreement with those based on the modified anisotropic Gurson’s yield criterion under uniaxial and equal biaxial tensile loading conditions.

scholarly journals Matrix basis for plane and modal waves in a Timoshenko beam

2016 ◽  
Vol 3 (11) ◽  
pp. 160825 ◽  
Author(s):  
Julio Cesar Ruiz Claeyssen ◽  
Daniela de Rosso Tolfo ◽  
Leticia Tonetto
Keyword(s):  
Closed Form ◽  
Timoshenko Beam ◽  
Plane Waves ◽  
Crack Problem ◽  
Wave Interaction ◽  
Beam Model ◽  
High Frequencies ◽  
Transmitted Waves ◽  
The Matrix ◽  
Non Local

Plane waves and modal waves of the Timoshenko beam model are characterized in closed form by introducing robust matrix basis that behave according to the nature of frequency and wave or modal numbers. These new characterizations are given in terms of a finite number of coupling matrices and closed form generating scalar functions. Through Liouville’s technique, these latter are well behaved at critical or static situations. Eigenanalysis is formulated for exponential and modal waves. Modal waves are superposition of four plane waves, but there are plane waves that cannot be modal waves. Reflected and transmitted waves at an interface point are formulated in matrix terms, regardless of having a conservative or a dissipative situation. The matrix representation of modal waves is used in a crack problem for determining the reflected and transmitted matrices. Their euclidean norms are seen to be dominated by certain components at low and high frequencies. The matrix basis technique is also used with a non-local Timoshenko model and with the wave interaction with a boundary. The matrix basis allows to characterize reflected and transmitted waves in spectral and non-spectral form.

Analysis and Computation of Two Body Impact in Three Dimensions

2017 ◽  
Vol 12 (4) ◽  
Author(s):  
Yan-Bin Jia ◽  
Feifei Wang
Keyword(s):  
Closed Form ◽  
Contact Point ◽  
Positive Definiteness ◽  
Rigid Bodies ◽  
Three Dimensions ◽  
Sliding Velocity ◽  
Form Analysis ◽  
Coulomb’S Friction ◽  
The Matrix ◽  
Single Contact Point

A formal impulse-based analysis is presented for the collision of two rigid bodies at single contact point under Coulomb's friction in three dimensions (3D). The tangential impulse at the contact is known to be linear in the sliding velocity whose trajectory, parametrized with the normal impulse and referred to as the hodograph, is governed by a generally nonintegrable ordinary differential equation (ODE). Evolution of the hodograph is bounded by rays in several invariant directions of sliding in the contact plane. Exact lower and upper bounds are derived for the number of such invariant directions, utilizing the established positive definiteness of the matrix defining the governing ODE. If the hodograph reaches the origin, it either terminates (i.e., the contact sticks) or continues in a new direction (i.e., the contact resumes sliding) whose existence and uniqueness, only assumed in the literature, are proven. Closed-form integration of the ODE becomes possible as soon as the sliding velocity turns zero or takes on an invariant direction. Assuming Stronge's energy-based restitution, a complete algorithm is described to combine fast numerical integration (NI) with a case-by-case closed-form analysis. A number of solved collision instances are presented. It remains open whether the modeled impact process will always terminate under Coulomb's friction and Stronge's (or Poisson's) restitution hypothesis.

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