Eigenvalues of Two- and Three-Dimensional Systems by Matrix Methods

1970 ◽  
Vol 38 (10) ◽  
pp. 1225-1229
Author(s):  
Michel Nicola
Keyword(s):  
Three Dimensional ◽  
Matrix Methods

scholarly journals Solving for three-dimensional central potentials using numerical matrix methods

2013 ◽  
Vol 81 (5) ◽  
pp. 343-350 ◽  
Author(s):  
B. A. Jugdutt ◽  
F. Marsiglio
Keyword(s):  
Three Dimensional ◽  
Matrix Methods

Quasi-3D Stability and Vibration Analyses of Sandwich Piezoelectric Plates with an Embedded CNT-Reinforced Composite Core

2016 ◽  
Vol 16 (02) ◽  
pp. 1450097 ◽  
Author(s):  
Chih-Ping Wu ◽  
Wei-Chen Li
Keyword(s):  
Free Vibration ◽  
Numerical Models ◽  
Three Dimensional ◽  
Functionally Graded ◽  
Matrix Methods ◽  
Reinforced Composite ◽  
System Equations ◽  
The Stability ◽  
Benchmark Solutions ◽  
Piezoelectric Plates

Quasi-three-dimensional (3D) stability and free vibration analyses of bi-axially loaded, simply-supported, sandwich piezoelectric plates with an embedded either a functionally graded (FG) carbon nanotube-reinforced composite (CNTRC) core or a multilayered fiber-reinforced composite (FRC) one are presented. Three different distributions of carbon nanotubes (CNTs) through the thickness of the CNTRC core, i.e. uniformly distributed and FG V-, rhombus- and X-type variations, are considered, and the effective material properties of the CNTRC core are estimated using the rule of mixtures. The Pagano method, which is conventionally used for the analysis of multilayered FRC plates, is modified to be feasible for the study of sandwich hybrid CNTRC and piezoelectric ones, in which Reissner mixed variational theorem, the successive approximation and transfer matrix methods, and the transformed real-valued solutions of the system equations are used. The modified Pagano solutions for the stability and free vibration of multilayered hybrid FRC and piezoelectric plates are in excellent agreement with the exact 3D ones available in the literature, and those for sandwich hybrid CNTRC and piezoelectric plates may be used as the benchmark solutions to assess the ones obtained by using various 2D theories and numerical models.

scholarly journals Analytical Solution for Free Vibration of Multilayered Circular Plate

2011 ◽  
Vol 2-3 ◽  
pp. 948-953 ◽  
Author(s):  
Hong Jun Ren ◽  
Xu Yuan Song ◽  
Qing Kai Han
Keyword(s):  
Analytical Solution ◽  
Circular Plate ◽  
Three Dimensional ◽  
Transversely Isotropic ◽  
Hankel Transform ◽  
Matrix Methods ◽  
Full Field ◽  
Lateral Boundary Conditions ◽  
Finite Hankel Transform ◽  
Propagation Matrix

An analytical solution is deduced for a three-dimensional transversely isotropic axisymmetric multilayered circular plate made of piezoelectric (PE) and piezomagnetic (PM) under simply supported boundary condition. The state space vector, finite Hankel transform and propagation matrix methods are utilized together to obtain the full-field solutions for the multilayered circle plate. Numerical examples for five-layered PE/PM composites with dimensionless frequencies of the multilayered plate under simple-supported lateral boundary conditions are presented. The frequencies increase with the ratio of the thickness to radius.

The orbit of Pluto over a long interval of time

1966 ◽  
Vol 25 ◽  
pp. 227-229 ◽  
Author(s):  
D. Brouwer
Keyword(s):  
Periodic Orbit ◽  
Numerical Integration ◽  
Restricted Problem ◽  
Three Dimensional ◽  
The Third ◽  
Circular Restricted Problem ◽  
Resonance Relation

The paper presents a summary of the results obtained by C. J. Cohen and E. C. Hubbard, who established by numerical integration that a resonance relation exists between the orbits of Neptune and Pluto. The problem may be explored further by approximating the motion of Pluto by that of a particle with negligible mass in the three-dimensional (circular) restricted problem. The mass of Pluto and the eccentricity of Neptune's orbit are ignored in this approximation. Significant features of the problem appear to be the presence of two critical arguments and the possibility that the orbit may be related to a periodic orbit of the third kind.

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