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finite-element analysis

FEA is used in various fields of engineering, such as

mechanical, civil, and aerospace engineering, to simulate the

behavior of a structure, system, or material. By analyzing the

structure in smaller segments, FEA allows engineers to examine

the stress, strain, deformation, and other important parameters

at different points of the design. This helps engineers predict

how the structure or system will behave under different loads or

conditions, which can help refine the design to optimize its

performance and safety.

FEA involves creating a 3D virtual model of the structure

and applying a set of mathematical equations to each small

element. These equations relate the movement of each element to

the forces acting upon it in its location. With this information,

engineers can create a visual representation of the structure,

showing the areas of the design where stresses and deformations

occur.

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Analysis of Sheet Metal Extrusion Process Using Finite Element Method

Analysis of Sheet Metal Extrusion Process Using Finite Element Method

Ｉｎｔｅｒｎａｔｉｏｎａｌ　Ｊｏｕｒｎａｌ　ｏｆ　Ａｕｔｏｍａｔｉｏｎ　ａｎｄ　Ｃｏｍｐｕｔｉｎｇ　７（３），Ａｕｇｕｓｔ　２０１０，２９５—３０２　

ＤＯＩ：１０．１００７／ｓ１　１６３３—０１０—０５０６—８　

Ａｎａｌｙｓｉｓ　ｏｆ　Ｓｈｅｅｔ　Ｍｅｔａｌ　Ｅｘｔｒｕｓｉｏｎ　Ｐｒｏｃｅｓｓ　Ｕｓｉｎｇ　Ｆｉｎｉｔｅ　

Ｅｌｅｍｅｎｔ　Ｍｅｔｈｏｄ　

Ｘｉｎ—Ｃｕｎ　Ｚｈｕａｎｇ　Ｈｕａ　Ｘｉａｎｇ　Ｚｈｅｎ　Ｚｈａｏ　

Ｄｅｐａｒｔｍｅｎｔ　ｏｆ　Ｐｌａｓｔｉｃｉｔｙ　Ｔｅｃｈｎｏｌｏｇｙ，Ｓｈａｎｇｈａｉ　Ｊｉａｏ　Ｔｏｎｇ　Ｕｎｉｖｅｒｓｉｔｙ）Ｓｈａｎｇｈａｉ　２０００３０，ＰＲＣ　

Ａｂｓｔｒａｃｔ：　Ｓｈｅｅｔ　ｂｕｌｋ　ｍｅｔａｌ　ｆｏｒｍｉｎｇ　ｐｒｏｃｅｓｓｅｓ　ｈａｖｅ　ｂｅｅｎ　ｗｉｄｅｌｙ　ｄｅｖｅｌｏｐｅｄ　ｔｏ　ｔｈｅ　ｆａｃｉｌｉｔａｔｅ　ｍａｎｕｆａｃｔｕｒｅ　ｏｆ　ｃｏｍｐｌｉｃａｔｅｄ　３Ｄ　ｐａｒｔｓ．　Ｈｏｗｅｖｅｒ，ｔｈｅｒｅ　ｉｓ　ｓｔｉｌｌ　ｎｏｔ　ｅｎｏｕｇｈ　ｋｎｏｗ—ｈｏｗ　ａｖａｉｌａｂｌｅ．Ｉｎ　ｔｈｉｓ　ｐａｐｅｒ，ａｓ　ｏｎｅ　ｏｆ　ｔｈｅ　ｔｙｐｉｃａｌ　ｓｈｅｅｔ　ｂｕｌｋ　ｍｅｔａ１　ｆｏｒｍｉｎｇ　ｐｒｏｃｅｓｓｅｓ，ｔｈｅ　ｓｈｅｅｔ　ｍｅｔａｌ　ｅｘｔｒｕｓｉｏｎ　ｐｒｏｃｅｓｓ　ｗａｓ　ｓｔｕｄｉｅｄ．Ａ　ｒｅａｓｏｎａｂｌｅ　ｆｉｎｉｔｅ　ｅｌｅｍｅｎｔ　ｍｅｔｈｏｄ（ＦＥＭ）ｍｏｄｅｌ　ｏｆ　ｓｈｅｅｔ　ｍｅｔａｌ　ｅｘｔｒｕｓｉｏｎ　ｐｒｏｃｅｓｓ　ｔａｋｉｎｇ　ｔｈｅ　ｉｎｆｌｕｅｎｃｅ　ｏｆ　ｆｌｏｗ—ｓｔｒｅｓｓ　ｃｕｒｖｅ　ｗｉｔｈ　ｗｉｄｅ　ｒａｎｇｅ　ｏｆ　ｐｌａｓｔｉｃ　ｓｔｒａｉｎ　ａｎｄ　ｄｕｃｔｉｌｅ　ｄａｍａｇｅ　ｉｎｔｏ　ｃｏｎｓｉｄｅｒａｔｉｏｎ　ｗａｓ　ｅｓｔａｂｌｉｓｈｅｄ　ａｎｄ　ｓｉｍｕｌａｔｅｄ　ｂｙ　ａｎ　ａｒｂｉｔｒａｒｙ　Ｌａｇｒａｎｇｉａｎ—Ｅｕｌｅｒｉａｎ（ＡＬＥ）ＦＥＭ　ｉｍｐｌｅｍｅｎｔｅｄ　ｉｎ　ＭＳＣ．Ｍａｒｃ．Ｖａｌｉｄａｔｅｄ　ｂｙ　ｃｏｍｐａｒｉｎｇ　ｔｈｅ　ｒｅｓｕｌｔｓ　ｗｉｔｈ　ｅｘｐｅｒｉｍｅｎｔ，　ｓｏｍｅ　ｐｈｅｎｏｍｅｎｏｌｏｇｉｃａｌ　ｃｈａｒａｃｔｅｒｉｓｔｉｃｓ，ｓｕｃｈ　ａｓ　ｍｅｔａｌ　ｆｌｏｗ　ｂｅｈａｖｉｏｒ，ｓｈｒｉｎｋａｇｅ　ｃａｖｉｔｙ，ａｎｄ　ｔｈｅ　ｉｎｆｌｕｅｎｃｅ　ｏｆ　ｄｉｆｉｅｒｅｎｔ　ｃｏｍｂｉｎａｔｉｏｎｓ　ｏｆ　ｄｉａｍｅｔｅｒ　ｏｆ　ｐｕｎｃｈ，ｄｉａｍｅｔｅｒ　ｏｆ　ｅｘｔｒｕｓｉｏｎ　ｏｕｔｌｅｔ，ａｎｄ　ｄｉａｍｅｔｅｒ　ｏｆ　ｐｒｅ—ｐｕｎｃｈｅｄ　ｈｏｌｅ　ｗｅｒｅ　ａｎａｌｙｚｅｄ　ａｎｄ　ｃｏｎｃｌｕｄｅｄ，ｗｈｉｃｈ　ｃａｎ　ｂｅ　ｕｓｅｄ　ａｓ　ｔｈｅｏｒｅｔｉｃａｌ　ｆｕｎｄａｍｅｎｔａｌ　ｆｏｒ　ｔｈｅ　ｄｅｓｉｇｎ　ｏｆ　ｔｈｅ　ｓｈｅｅｔ　ｍｅｔａｌ　ｅｘｔｒｕｓｉｏｎ　ｐｒｏｃｅｓｓ．　

Finite Element Model Updating of Bridge Structures Based on Sensitivity Analysis and Optimizatio

Finite Element Model Updating of Bridge Structures Based on Sensitivity Analysis and Optimizatio

Ｗｕｈａｎ　Ｕｎｉｖｅｒｓｉｔｙ　

Ｊｏｕｒｎａｌ　ｏｆＮａｔｕｒａｌ　

Ａｒｔｉｃｌｅ　Ｉ　Ｄ　１　００７－－１　２０２（２００８）０　１　・００８７－－０６　

ＤＯＩ　１０．１００７／ｓ１　１８５９　００８—０ｌ１７　１　２００８，Ｖｏ１．１３　Ｎｏ．１，０８７—０９２　

Ｆｉｎｉｔｅ　Ｅｌｅｍｅｎｔ　Ｍｏｄｅｌ　Ｕｐｄａｔｉｎｇ　ｏｆ　

Ｂｒｉｄｇｅ　Ｓｔｒｕｃｔｕ　ｒｅｓ　Ｂａｓｅｄ　ｏｎ　Ｓｅｎｓｉｔｉｖｉｔｙ　

Ａｎａｌｙｓｉｓ　ａｎｄ　Ｏｐｔｉｍｉｚａｔｉｏｎ　Ａｌｇｏｒｉｔｈｍ　

口ＨＵＡＮＧ　Ｍｉｎｓｈｕｉ’，ＺＨＵ　Ｈｏｎｇｐｉｎｇ’，　十　

Ｉ．Ｓｃｈｏｏｌ　ｏｆ　Ｃｉｖｉｌ　Ｅｎｇｉｎｅｅｒｉｎｇ　ａｎｄ　Ｍｅｃｈａｎｉｃｓ，Ｈｕａｚｈｏｎｇ　

ｏｆ　Ｓｃｉｅｎｃｅ　ａｎｄ　Ｔｅｃｈｎｏｌｏｇｙ，Ｗｕｈａｎ　４３００７４，Ｈｕｂｅｉ，Ｃｈｉｎａ；　

２．Ｈｕｂｅｉ　Ｋｅｙ　Ｌａｂｏｒａｔｏｒｙ　ｏｆ　Ｃｏｎｔｒｏｌ　Ｓｔｒｕｃｔｕｒｅ，Ｈｕａｚｈｏｎｇ　

ｏｆ　Ｓｃｉｅｎｃｅ　ａｎｄ　Ｔｅｃｈｎｏｌｏｇｙ，Ｗｕｈａｎ　４３００７４，Ｈｕｂｅｉ，Ｃｈｉｎａ　Ｕｎｉｖｅｒｓｉｔｙ　

Ｕｎｉｖｅｒｓｉｔｙ　

Ａｂｓｔｒａｃｔ：Ｔｈｅ　ｄｙｎａｍｉｃ　ｃｈａｒａｃｔｅｒｉｓｔｉｃｓ　ｏｆ　ｂｒｉｄｇｅ　ｓｔｒｕｃｔｕｒｅｓ，ｓｕｃｈ　ａｓ　ｔｈｅ　ｎａｔｕｒａｌ　ｆｒｅｑｕｅｎｃｉｅｓ，ｍｏｄｅ　ｓｈａｐｅｓ　ａｎｄ　ｍｏｄｅｌ　ｄａｍｐｉｎｇ　ｒａｔｉｏ，ａｒｅ　ｔｈｅ　ｂａｓｉｓ　ｏｆ　ｓｔｒｕｃｔｕｒａ１　ｄｙｎａｍｉｃ　ｃｏｍｐｕｔａｔｉｏｎ．ｓｅｉｓｍｉｃ　ａｎａｌｙｓｉｓ．ｖｉｂｒａｔｉｏｎ　ｃｏｎｔｒｏ１　ａｎｄ　ｓｔｒｕｃｔｕｒａ１　ｈｅａｌｔｈ　ｃｏｎｄｉｔｉｏｎ　ｍｏｎｉｔｏｒｉｎｇ．１ｎ　ｔｈｉｓ　Ｐ，ｄＤｅ　ａ　ｔｈｒｅｅ　ｄｉｍｅｎｓｉｏｎａ１　ｆｉｎｉｔｅ—ｅｌｅｍｅｎｔ　ｍｏｄｅｌｉｓ　ｅｓｔａｂｌｉｓｈｅｄ　ｆｏｒ　ａ　ｈｉｇｈｗａｙ　ｂｒｉｄｇｅ　ｏｖｅｒ　ａ　ｒａｉｌｗａｙ　ｏｎ　Ｎｏ．３　１　２　Ｎａｔｉｏｎａ１　Ｈｉｇｈｗａｙ　ａｎｄ　ｔｈｅ　ａｍｂｉｅｎｔ　ｔｅｓｔ　ｉｓ　ｃａｒｒｉｅｄ　ｏｕｔ　ｉｎ　ｓｉｔｅ．ｔｈｅ　ｄｙｎａｍｉｃ　ｃｈａｒａｃｔｅｒｉｓｔｉｃｓ　ｏｆ　ｔｈｅ　ｂｒｉｄｇｅ　ａｒｅ　ｓｔｕｄｉｅｄ　ｕｓｉｎｇ　ｔｈｅ　ｆｉｎｉｔｅ—ｅｌｅｍｅｎｔ　ａｎａｌｙｓｉｓ　ａｎｄ　ａｍｂｉｅｎｔ　ｖｉｂｒａｔｉｏｎ　ｍｅａｓｕｒｅｍｅｎｔｓ．Ｃｏｍｐａｒｉｓｏｎ　ｂｅｔｗｅｅｎ　ｔｈｅ　ｔｈｅｏｒｅｔｉｃａ１　ａｎｄ　ｅｘｐｅｒｉｍｅｎｔａ１　ｒｅｓｕｌｔｓ　ｓｈｏｗｓ　ｔｈａｔ　ｔｈｅ　ｆｒｅｑｕｅｎｃｙ　ｄｉｆｆｅＦｅｎｃｅｓ　ｏｆ　ｔｈｅ　ｍｏｄｅｓ　ｒａｎｇｅ　ｂｅｔｗｅｅｎ　０．４４％ａｎｄ　８．７７％．１ｆ　ｔｈｅ　ｍｅａｓｕｒｅｍｅｎｔ　ｉｓ　ｍｏｒｅ　ｒｅｌｉａｂｌｅ．ｔｈｅ　ｆｉｎｉｔｅ　ｅｌｅｍｅｎｔ　ｍｏｄｅ１　ｕｐｄａｔｉｎｇ　ｉｓ　ｎｅｃｅｓｓａｒｙ．Ｔｈｕｓ．ａ　ｓｅｔ　ｏｆ　ｄｅ　ｓｉｇｎ　ｖａｒｉａｂｌｅｓ　ｉｓ　ｓｅｌｅｃｔｅｄ　ｂａｓｅｄ　ｏｎ　ｓｅｎｓｉｔｉｖｉｔｙ　ａｎａｌｙｓｉｓ　ｔｈｅｎ　ｔｈｅ　ｆｉｎｉｔｅ　ｅｌｅ—　ｍｅｎｔ　ｍｏｄｅ１　ｏｆ　ｔｈｅ　ｂｒｉｄｇｅ　ｉｓ　ｕｐｄａｔｅｄ　ｂａｓｅｄ　ｏｎ　ｏｐｔｉｍｉｚａｔｉｏｎ　ａｌｇｏｒｉｔｈｍ．Ｔｈｅ　ｒｅｓｕｌｔｓ　ｏｆ　ｍｏｄｅｌ　ｕｐｄａｔｉｎｇ　ｓｈｏｗ　ｔｈａｔ　ｔｈｅ　ｐｒｏｐｏｓｅｄ　ｕｐｄａｔｉｎｇ　ｍｅｔｈｏｄ　ｉｎ　ｔｈｉｓ　ｐａｐｅｒ　ｉｓ　ｍｏｒｅ　ｓｉｍｐｌｅ　ａｎｄ　ｅｆｆｅｃｔｉｖｅ，ｔｈｅ　ｕｐｄａｔｅｄ　ｆｉｎｉｔｅ　ｅｌｅｍｅｎｔ　ｍｏｄｅｌ　ｃａｎ　ｒｅｆｌｅｃｔ　ｔｈｅ　ｄｙｎａｍｉｃ　ｃｈａｒａｃｔｅ　ｒｉｓｔｉｃｓ　ｏｆ　ｔｈｅ　ｂｒｉｄｇｅ　ｂｅｔｔｅｒ．ｔｈｅ　ａｎａｌｙｔｉｃａ１　ｒｅｓｕｌｔｓ　ｃａｎ　ｐｒｏｖｉｄｅ　ｔｈｅ　ｔｈｅｏｒｅｔｉｃａ１　ｂａｓｉｓ　ｆｏｒ　ｄａｍａｇｅ　ｉｄｅｎｔｉｆｉｃａｔｉｏｎ　ａｎｄ　ｈｅａｌｔｈ　ｃｏｎ—　ｄｉｔｉｏｎ　ｍｏｎｉｔｏｒｉｎｇ　ｏｆ　ｔｈｅ　ｂｒｉｄｇｅ．　Ｋｅｙ　ｗｏｒｄｓ：ｓｅｎｓｉｔｉｖｉｔｙ　ａｎａｌｙｓｉｓ；ｏｐｔｉｍｉｚａｔｉｏｎ　ａｌｇｏｒｉｔｈｍ；ｍｏｄｅｌ　ｕｐｄａｔｉｎｇ；　ｂｒｉｄｇｅ　ｓｔｒｕｃｔｕｒｅ　ＣＬＣ　ｎｕｍｂｅｒ：Ｕ　４４６．３　

[USACE] Geotechnical Analysis by the Finite Element Method

[USACE] Geotechnical Analysis by the Finite Element Method

DEPARTMENT OF THE ARMY

U.S. Army Corps of Engineers

CECW-EGWashington, DC 20314-1000ETL 1110-2-544

Technical Letter

No. 1110-2-54431 July 1995

Engineering and Design

GEOTECHNICAL ANALYSIS BY THE FINITE

ELEMENT METHOD

Distribution Restriction Statement

Approved for public release; distribution is unlimited.DEPARTMENT OF THE ARMYETL 1110-2-544

U.S. Army Corps of Engineers

CECW-EGWashington, DC 20314-1000

Technical Letter31 July 1995

No. 1110-2-544

Engineering and Design

GEOTECHNICAL ANALYSIS BY THE FINITE

ELEMENT METHOD

1. Purpose

This engineer technical letter (ETL) provides guid-complex geometries, a variety of loading conditions,

ance on the use of the finite element method in thenonlinear material behavior, nonhomogeneous mate-

analysis of problems in geotechnical engineering. rial distribution, and soil-structure interaction effects

Dynamic Characteristics of Inflated Torus Using Finite Element Analysis

Dynamic Characteristics of Inflated Torus Using Finite Element Analysis

、ｂｌ＿２０　Ｎｏ．１　ＣＡＤＤＭ　Ｊｕｎｅ　２０１Ｏ　Ｄｙｎａｍｉｃ　Ｃｈａｒａｃｔｅｒｉｓｔｉｃｓ　ｏｆ　Ｉｎｆｌａｔｅｄ　Ｔｏｒｕｓ　Ｕｓｉｎｇ　Ｆｉｎｉｔｅ　Ｅｌｅｍｅｎｔ　Ａｎａｌｙｓｉｓ　ＫＡＭＥＬ　Ｍ　Ａ，ＨＥ　Ｗａｉ－ｌｉａｎｇ　（Ｓｃｈｏｏｌ　ｏｆＡｓｔｒｏｎａｕｔｉｃｓ，Ｂｅｉｊｉｎｇ　Ｕｎｉｖｅｒｓｉｔｙ　ｏｆＡｅｒｏｎａｕｔｉｃｓ　ａｎｄ　Ａｓｔｒｏｎａｕｔｉｃｓ，Ｂｅｉｊｉｎｇ　１００１９１，Ｃｈｉｎａ）　Ａｂｓｔｒａｃｔ：Ｉｎ　ｒｅｃｅｎｔ　ｙｅａｒｓ，ｉｎｆｌａｔａｂｌｅ　ｓｔｒｕｃｔｕｒｅｓ　ｈａｖｅ　ｂｅｅｎ　ａ　ｓｕｂｉｅｃｔ　ｏｆ　ｉｎｔｅｒｅｓｔ　ｆｏｒ　ｓｐａｃｅ　ａｐｐｌｉｃａｔｉｏｎｓ　ｓｕｃｈ　ａｓ　ｃｏｍｍｕｎｉｃａｔｉｏｎ　ａｎｔｅｎｎａ，ｓｏｌａｒ　ｔｈｅｒｍａｌ　ｐｒｏｐｕｌｓｉｏｎ　ａｎｄ　ｅｎｔｒｙ／ｌａｎｄｉｎｇ　ｓｙｓｔｅｍｓ．Ｔｈｅ　ｉｎｆｌａｔａｂｌｅ　ｓｔｒｕｃｔｕｒｅｓ　ｃｈａｒａｃｔｅｒｉｚｅｄ　ｂｙ　ｈｉｇｈ　ｓｔｒｅｎｇｔｈ—ｔｏ—ｍａｓｓ　ｒａｔｉｏｓ．ｍｉｎｉｍａｌ　ｓｔｏｗａｇｅ　ｖｏｌｕｍｅ，ｗｈｉｃｈ　ｍａｋｅｓ　ｔｈｅｍ　ｓｕｉｔａｂｌｅ　ｆｏｒ　ｃｏｓｔ—ｅｆｆｅｃｔｉｖｅ　ｌａｒｇｅ　ｓｐａｃｅ　ｓｔｒｕｃｔｕｒｅｓ．Ａ　ｔｙｐｉｃａｌ　ｅｘａｍｐｌｅ　ｆｏｒ　ｔｈｅ　ｉｎｆｌａｔａｂｌｅ　ｓｔｒｕｃｔｕｒｅ　ｉＳ　ｔｈｅ　ｉｎｆｌａｔｅｄ　ｔｏｒｕｓ　ｗｈｉｃｈ　ｏｆｔｅｎ　ｕｓｅｄ　ｉｎ　ｏｒｄｅｒ　ｔｏ　ｐｒｏｖｉｄｅ　ｓｔｒｕｃｔｕｒｅ　ｓｕｐｐｏｒｔ．Ｉｎ　ｔｈｉｓ　ｓｔｕｄｙ，ｏｕｒ　ｍａｉｎ　ｆｏＣＵＳ　ｉｓ　ｔｏ　ｕｎｄｅｒｓｔａｎｄ　ｔｈｅ　ｄｙｎａｍｉｃ　ｃｈａｒａｃｔｅｒｉｓｔｉｃｓ　ｏｆ　ａｎ　ｉｎｆｌａｔｅｄ　ｔｏｒｕｓ　ｉｎ　ｏｒｄｅｒ　ｔｏ　ｆｏｒｍｕｌａｔｅ　ａｎ　ａｃｃｕｒａｔｅ　ｍａｔｈｅｍａｔｉｃａｌ　ｍｏｄｅｌ　ｓｕｉｔａｂｌｅ　ｆｏｒ　ａｃｔｉｖｅ　ｖｉｂｒａｔｉｏｎ　ｃｏｎｔｒｏｌ　ａｐｐｌｉｃａｔｉｏｎｓ．Ａ　ｃｏｍｍｅｒｃｉａｌ　ｆｉｎｉｔｅ　ｅｌｅｍｅｎｔ　ｐａｃｋａｇｅ，ＡＮＳＹＳ，ｉｓ　ｕｓｅｄ　ｔｏ　ｍｏｄｅｌ　ｔｈｅ　ｉｎｆｌａｔｅｄ　ｔｏｍｓ．Ｔｏ　ｖｅｒｉｆｙ　ｔｈｅ　ｍｏｄｅｌ　ｔｈｅ　ｏｂｔａｉｎｅｄ　ｆｒｅｑｕｅｎｃｉｅｓ　ａｎｄ　ｍｏｄｅ　ｓｈａｐｅｓ　ａｒｅ　ｃｏｍｐａｒｅｄ　ｗｉｔｈ　ｔｈｅ　ｐｕｂｌｉｓｈｅｄ　ｒｅｓｕｌｔｓ，ｗｈｉｃｈ　ａｒｅ　ｄｅｒｉｖｅｄ　ｕｓｉｎｇ　ａｎａｌｙｔｉｃａｌ　ａｐｐｒｏａｃｈ，ｔｈｅ　ｖｅｒｉｆｉｃａｔｉｏｎ　ｓｈｏｗｓ　ａ　ｇｏｏｄ　ａｇｒｅｅｍｅｎｔ　ｂｅｔｗｅｅｎ　ｔｈｅ　ＦＥＭ　ａｎｄ　ｔｈｅ　ａｎａｌｙｔｉｃａｌ　ｒｅｓｕｌｔｓ．Ｂａｓｅｄ　ｏｎ　ｔｈｅ　ｖｅｒｉｆｉｅｄ　ｍｏｄｅ１．ｐａｒａｍｅｔｒｉｃ　ｓｔｕｄｙ　ｗａｓ　ｉｎｖｅｓｔｉｇａｔｅｄ；ｔｈｅ　ｍａｔｅｒｉａｌ　ｔｈｉｃｋｎｅｓｓ　ｉｎｃｒｅａｓｅ　ｃａｕｓｅｓ　ｔｈｅ　ｎａｔｕｒａｌ　ｆｒｅｑｕｅｎｃｉｅｓ　ｄｅｃｒｅａｓｅ，　ｗｈｉｌｅ　ｔｈｅ　ｉｎｃｒｅａｓｅ　ｏｆ　ｔｈｅ　ｉｎｆｌａｔｉｏｎ　ｐｒｅｓｓｕｒｅ　ｓｉｍｐｌｙ　ｒｅｓｕｌｔｓ　ｉｎ　ｓｔｉｆｅｎｉｎｇ　ｔｈｅ　ｒｉｎｇ，ｗｈｉｃｈ　ｍｅａｎｓ　ｔｈａｔ　ｔｈｅ　ｎａｔｕｒａｌ　ｆｒｅｑｕｅｎｃｙ　ｉｎｃｒｅａｓｅｄ．Ｔｈｅ　ＦＥＭ　ａｎａｌｙｓｉｓ　ｇｉｖｅｓ　ａｎ　ｅａｓｙ　ａｎｄ　ｆａｓｔ　ｗａｙ　ｆｏｒ　ｔｈｅ　ｖｉｂｒａｔｉｏｎ　ａｎａｌｙｓｉｓ　ｏｆ　ｔｈｅ　ｓｔｒｕｃｔｕｒｅｓ　ｃｏｍｐａｒｅｄ　ｗｉｔｈ　ｔｈｅ　ｃｏｍｐｌｉｃａｔｅｄ　ａｎａｌｙｔｉｃａｌ　ｓｏｌｕｔｉｏｎｓ．　Ｋｅｙｗｏｒｄｓ：ｄｙｎａｍｉｃ　ｃｈａｒａｃｔｅｒｉｓｔｉｃｓ；ｉｎｆｌａｔａｂｌｅ　ｔｏｍｓ；ｆｉｎｉｔｅ　ｅｌｅｍｅｎｔ　ｍｏｄｅｌｌｉｎｇ（ＦＥＭ）；ｍｏｄａｌ　ａｎａｌｙｓｉｓ　ＣＬＣ　ｎｕｍｂｅｒ：ＴＢ　１ｌ５　Ｄｏｃｕｍｅｎｔ　ｃｏｄｅ：Ａ　Ａｒｔｉｃｌｅ　ＩＤ：１００３・－４９５１（２０１０）０１・・００１５－・０６　ｌ　ＩｎｔｒＯｄｕｃｔｉＯｎ　Ｉｎｆｌａｔａｂｌｅ　ｓｔｒｕｃｔｕｒｅｓ　ｓｈｏｗ　ｓｉｇｎｉｆｉｃａｎｔ　ｐｒｏｍｉｓｅｓ　ｆｏｒ　ｆｕｔｕｒｅ　ｓｐａｃｅ　ａｐｐｌｉｃａｔｉｏｎｓ．Ｔｈｅ　ｄｙｎａｍｉｃ　ｂｅｈａｖｉｏｒ　ｏｆ　ｉｎｆｌａｔｅｄ　ｓｐａｃｅ—ｂａｓｅｄ　ｄｅｖｉｃｅｓ　ｉｓ　ｐａｒｔｉｃｕｌａｒｌｙ　ｉｍｐｏｒｔａｎｔ　ｆｏｒ　ｓａｔｅｌｌｉｔｅ　ｓｔｒｕｃｔｕｒｅｓ　ｓｉｎｃｅ　ｔｈｅｙ　ａｒｅ　ｓｕｂｊｅｃｔｅｄ　ｔｏ　ａ　ｖａｒｉｅｔｙ　ｏｆ　ｄｙｎａｍｉｃ　ｌｏａｄｉｎｇｓＩｌＪ．　Ｓｐａｃｅ　ｓｔｒｕｃｔｕｒｅｓ　ｈａｖｅ　ｖｉｂｒａｔｉｏｎ　ｐｒｏｂｌｅｍｓ　ｄｕｅ　ｔｏ　ｌｏｗ　ｓｔｒｕｃｔｕｒａｌ　ｓｔｉｆｆｎｅｓｓ　ａｎｄ　ｔｈｅ　ｌａｃｋ　ｏｆ　ａｉｒ　ｄａｍｐｉｎｇ　ｉｎ　ｔｈｅ　ｖａｃｕｕｍ　ｏｆ　ｓｐａｃｅ．Ｉｎｆｌａｔａｂｌｅ　ｓｐａｃｅ　ｓｔｒｕｃｔｕｒｅｓ　ａｒｅ　ｌｉｋｅｌｙ　ｔｏ　ｂｅ　ｅｖｅｎ　ｍｏｒｅ　ｆａｃｉｎｇ　ｔｏ　ｖｉｂｒａｔｉｏｎ　ｐｒｏｂｌｅｍｓ　ｂｅｃａｕｓｅ　ｏｆ　ｌｏｗ　ｓｔｒｕｃｔｕｒａｌ　ｓｔｉｆｆｎｅｓｓ　ａｎｄ　ｍａｔｅｒｉａｌ　ｄａｍｐｉｎｇ．　Ａ　ｔｏｒｕｓ，ａｌｔｈｏｕｇｈ　ａ　ｓｉｍｐｌｅ　ｓｔｒｕｃｔｕｒｅ　ｉｎ　ｃｏｎｃｅｐｔ，　ｉｓ　ａｃｔｕａｌｌｙ　ｎｕｍｅｒｉｃａｌｌｙ　ｖｅｒｙ　ｃｏｍｐｌｉｃａｔｅｄ．Ｆｅｗ　ｆｕｌｌｙ　ａｎａｌｙｔｉｃａｌ　ｓｏｌｕｔｉｏｎｓ　ｆｏｒ　ｔｈｅ　ｖｉｂｒａｔｉｏｎｓ　ｏｆ　ｔｏｒｏｉｄｓ　ｅｘｉｓｔ　ｄｕｅ　ｔｏ　ｔｈｅ　ｃｏｍｐｌｅｘｉｔｙ　ｏｆ　ｔｈｅ　ｐｒｏｂｌｅｍ．　Ｉｎ　ｔｈｅ　ｅａｒｌｙ　１　９７３’Ｓ，Ｌｅｉｓｓａ　ｉｎｖｅｓｔｉｇａｔｅｄ　ｔｈｅ　ｆｕｎｄａｍｅｎｔａｌ　ｅｑｕａｔｉｏｎｓ　ｏｆ　ｔｈｉｎ　ｓｈｅｌｌ　ｔｈｅｏｒｙ　ｗｈｉｃｈ　ａｒｅ　ｂａｓｅｄ　ｏｎ　ｔｈｅ　Ｓａｎｄｅｒｓ　ｓｈｅｌｌ　ｔｈｅｏｒｙ　ｉｎ　ｈｉｓ　ｐｕｂｌｉｓｈｅｄ　Ｒｅｃｅｉｖｅｄ　Ｄａｔｅ：Ｍａｒｃｈ　１１，２０１０　Ｂｉｏｇｒａｐｈｙ：ＫＡＭＥＬＭＡ（１９７１一），ｍａｌｅ，ｂｏｒｎｉｎ　ＳｈａｒｋｉａＣｉｔｙ，Ｅｇｙｐｔ，Ｐｈ．Ｄ　ｃａｎｄｉｄａｔｅ，ｍａｉｎ　ｒｅｓｅａｒｃｈｆｉｅｌｄｉｓ　ｓｔｒｕｃｔｕｒｅｖｉｂｒａｔｉｏｎ　ｃｏｎｔｒｏｌ　ｋａｍｅｌ１９７１２００７＠ｙａｈｏｏ．