Link:http://output.to/sideway/default.asp?qno=120700001 Centroid of 3D Body Centroid of 3D BodyThe centroid of 3D Body is determined by the first moment of a three dimensional body with the method of the first moment of volume. ![]() Centroids of VolumesThe using of unit elemental volume of an object to determine the centroid of a 3D body volume can be expressed as ![]() Centroid by IntegrationA triple integation is needed to evaluate with respect to the three varables. Similar to finding the volume of a 3D object body, the centroid of a 3D volume can usually be determined by performing a single integration or a double integration also. Centroid by Double IntegrationThe unit elemental volumes of an object used to determine the centroid of a 3D body can be rearranged into grouped elemental volumes. Imply ![]() After the grouping of unit elemental volumes into one elemental volume, the coordinates of the centroid of a volume can also be determined by double integration in a similar way by considering the centroid of each elemental volume block. Imply ![]() Centroid of Volume by Double Integration![]() For example, the signed volume of the 3D ellipic cylinder is bounded by surfaces in rectangular form , Imply ![]() The unit element volume of a region can be grouped into either a small vertical rectangular block or a small horizontal rectangular block . And the elemental volume ΔV becomes ![]() Considering the small horizontal rectangular block as the element volume, the centroid of the rectangular block can be determined by a single integration through sweeping the elemental centroid of the elemental rectangular block along the rectangular coordinate axis accordingly. Imply By sweeping the element of horizontal rectangular block along x axis horizontally Centroid of horizontal rectangular block. Imply ![]() Therefore, centroid of the bounded volume is ![]() Centroid by Single IntegrationThe unit elemental volumes of an object used to determine the centroid of a 3D body can be rearranged into grouped elemental volumes. Imply ![]() After the grouping of unit elemental volumes into one elemental volume, the coordinates of the centroid of a volume can also be determined by one single integration in a similar way by considering the centroid of each elemental volume block. Imply ![]() Centroid of Volume by Double IntegrationVolume by Single Integration![]() For example, the signed volume of the 3D ellipic cylinder is bounded by surfaces in rectangular form , Imply ![]() The unit element volume of a region can be grouped into either a thin vertical plane sheet or a thin horizontal plane sheet . And the elemental volume ΔV becomes ![]() Considering the thin vertical plane sheet as the element volume, the centroid of the thin vertical plane sheet can be determined by a double integration through sweeping the elemental centroid of the elemental plane sheet along the rectangular coordinate axis accordingly. Imply By staring with sweeping the element of vertical plane sheet along y axis horizontally Centroid of vertical plane sheet. Imply ![]() Therefore, centroid of the bounded volume is ![]() |
Sideway BICK Blog 03/07 |