Three-Dimensional Eigenvalue Analysis of Resonators

2006 ◽  
pp. 295-319
Keyword(s):  
Three Dimensional ◽  
Eigenvalue Analysis

Evaluation of Joining Strength of Silicon-Resin Interface at a Vertex in 3D Joint Structure

2011 ◽  
Author(s):  
Hideo Koguchi ◽  
Kazuhisa Hoshi
Keyword(s):  
Stress Distribution ◽  
Boundary Element ◽  
Three Dimensional ◽  
Stress Singularity ◽  
Singularity Analysis ◽  
Eigenvalue Analysis ◽  
Singular Stress ◽  
Two Phase ◽  
Joining Strength ◽  
Element Method

Portable electric devices such as mobile phone and portable music player become compact and improve their performance. High-density packaging technology such as CSP (Chip Size Package) and Stacked-CSP is used for improving the performance of devices. CSP has a bonded structure composed of materials with different properties. A mismatch of material properties may cause stress singularity, which lead to the failure of bonding part in structures. In the present paper, stress analysis using boundary element method and an eigenvalue analysis using finite element method are used for evaluating the intensity of singularity at a vertex in three-dimensional joints. Three-dimensional boundary element program based on the fundamental solution for two-phase isotropic materials is used for calculating the stress distribution in a three-dimensional joint. Angular function in the singular stress field at the vertex in the three-dimensional joint is calculated using eigen vector determined from the eigenvalue analysis. The joining strength of interface in several kinds of sillicon-resin specimen with different triangular bonding areas is investigated analytically and experimentally. Experiment for debonding the interface in the joints is firstly carried out. Stress singularity analysis for the three-dimensional joints subjected to an external force for debonding the joints is secondly conducted. Combining results of the experiment and the analysis yields a final stress distribution for evaluating the strength of interface. Finally, a relationship of force for delamination in joints with different bonding areas is derived, and a critical value of the 3D intensity of singularity is determined.

Evaluation of Joining Strength of Silicon-Resin Interface at a Vertex in a Three-Dimensional Joint Structure

2012 ◽  
Vol 134 (2) ◽  
Author(s):  
Hideo Koguchi ◽  
Kazuhisa Hoshi
Keyword(s):  
Stress Distribution ◽  
Boundary Element ◽  
Three Dimensional ◽  
Stress Singularity ◽  
Singularity Analysis ◽  
Eigenvalue Analysis ◽  
Singular Stress ◽  
Two Phase ◽  
Joining Strength ◽  
Element Method

Portable electric devices such as mobile phones and portable music players have become compact and improved their performance. High-density packaging technology such as chip size package (CSP) and stacked-CSP is used for improving the performance of devices. CSP has a bonded structure composed of materials with different properties. A mismatch of material properties may cause a stress singularity, which leads to the failure of the bonding part in structures. In the present paper, stress analysis using the boundary element method and an eigenvalue analysis using the finite element method are used for evaluating the intensity of a singularity at a vertex in three-dimensional joints. A three-dimensional boundary element program based on the fundamental solution for two-phase isotropic materials is used for calculating the stress distribution in a three-dimensional joint. Angular function in the singular stress field at the vertex in the three-dimensional joint is calculated using an eigenvector determined from the eigenvalue analysis. The joining strength of interface in several kinds of sillicon-resin specimen with different triangular bonding areas is investigated analytically and experimentally. An experiment for debonding the interface in the joints is firstly carried out. Stress singularity analysis for the three-dimensional joints subjected to an external force for debonding the joints is secondly conducted. Combining results of the experiment and the analysis yields a final stress distribution for evaluating the strength of interface. Finally, a relationship of force for delamination in joints with different bonding areas is derived, and a critical value of the 3D intensity of the singularity is determined.

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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