Sideway
output.to from Sideway
Draft for Information Only

Content

Centroid of Volume by Geometric Decomposition
  Centroids of Composite 3D Body
   Centroid of composite 3D body 1
   Centroid of composite 3D body 2

Centroid of Volume by Geometric Decomposition

By denoting the position of the element of a 3D body relative to the coordinate system as xi,yi,zk, the coordinates x, y, z , of the center of gravity of the plane body where the resultant force W applied can be determined by equating the moment of the resultant force W about the yz plane, zx plane and xy plane and the corresponding mements of elements of the 3D body about the coordinate axis respectively. Imply.

image

Therefore the coordinates x , y , and  z of the center of gravity of the plane body is.

image

If a 3D body can be divided into some common 3D bodys, the 3D body can be considered as a composite 3D body. For a composite 3D body, the cooresponding mements of elements of the plane body can also be grouped according to the composition of the 3D body. Imply

image

For a homogeneous 3D body, the first moment of the 3D body volume is

image

Centroids of Composite 3D Body

Centroid of composite 3D body 1

image

For example, the signed volume of a composite 3D body bounded by surfaces in rectangular form , Imply

image

The composite 3D volume can be divided into two common shapes. Therefore, centroid of the bounded volume by concept of composite 3D body is

image

The volume of the 3D body is

image

Coordinate x of centroid

The first moment of area can be either positive and negative. The centroid of volume 1 is located to the positive side of the yz plane, the first moment of volume is positive.  Since the centroid of volume 2 is located to the negative side of the yz plane, the first moment of area is negative and the negative sign is usually assigned to the x coordinate of the centroid of volume 2. Imply

image

Similarly if the coordinate x of centroid of 3D volume is positive, the coordinate x of the centroid is located to the positive side of yz plane.

Coordinate y and z of  centroid

By symmetry, the coordinate y and  z of centroid lie on the x axis, the line of symmetry.

Centroid of composite 3D body 2

Since the first moment of volume of a hole can be considered as the negative sense of the first moment of volume by assuming the first moment of volume of the same volume of negative sense. Therefore the first moment of volume of a hole is negative. The negative sign is usually assigned to the volume of the hole and the volume of a hole is called negative volume.

image

For example, the signed volume of a composite 3D volume bounded by surfaces in rectangular form , Imply

image

The 3D volume body can be divided into common shapes. Therefore, centroid of the bounded voulme by concept of composite 3D volume body is

image

The volume of 3D body is

image

Coordinate x of centroid

The first moment of volume can be either positive and negative. The centroid of volume 1 plus volume 3 is located to the positive side of the yz planes, the first moment of volume is positive.  Since the centroid of volume 2 is located to the negative of the yz planes, the first moment of volume is negative and the negative sign is usually assigned to the x coordinate of the centroid of volume 1. And for the centroid of hole, negative volume 3 is located to the positive side of the yz planes, the coordinate of the centroid of volume 3 is positive but the sense of the volume is negative. Imply

image

If c is equal to a, the coordinate x of centroid of volume of the 3D body is located on the origin 0 of x axis.

Coordinate y and z of  centroid

By symmetry, the coordinate y and  z of centroid lie on the x axis, the line of symmetry.

 


©sideway

ID: 120700004 Last Updated: 12/7/2012 Revision: 0 Ref:

close

References

  1. I.C. Jong; B.G. rogers, 1991
  2. F.P. Beer; E.R. Johnston,Jr.; E.R. Eisenberg, 2004
close

Latest Updated LinksValid XHTML 1.0 Transitional Valid CSS!Nu Html Checker Firefox53 Chromena IExplorerna
IMAGE

Home 5

Business

Management

HBR 3

Information

Recreation

Hobbies 7

Culture

Chinese 1097

English 337

Reference 67

Computer

Hardware 149

Software

Application 187

Digitization 24

Numeric 19

Programming

Web 757

CSS 1

ASP.NET 1

Regular Expression 1

HTML

Knowledge Base

Common Color 1

Html Entity (Unicode) 1

Html 401 Special 1

OS 389

MS Windows

Windows10 1

.NET Framework 1

DeskTop 7

Knowledge

Mathematics

Formulas 8

Algebra 20

Number Theory 206

Trigonometry 18

Geometry 18

Calculus 67

Complex Analysis 21

Engineering

Tables 8

Mechanical

Mechanics 1

Rigid Bodies

Statics 92

Dynamics 37

Fluid 5

Fluid Kinematics 5

Control

Process Control 1

Acoustics 19

FiniteElement 2

Physics

Electric 10

Biology 1

Geography 1


Copyright © 2000-2019 Sideway . All rights reserved Disclaimers last modified on 06 September 2019