Mass Property

8/3/2019 Mass Property

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Mass Properties

Mass property calculation was one of the first features implemented in

CAD/CAM systems.

Curve length

Cross-sectional area

Surface area

Centroid of a surface area

Centroid of a cross-sectional area

Volume

Centroid of a volume

Mass

Center of mass

First moment of inertia

Second moment of inertia

Products of inertia


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Transformations -TranslationGeometric transformations are used in modeling and viewing models.Typical CAD operations such as Rotate, Mirror, zoom, Offset, Pattern,

Revolve, Extrude, are all based on geometric transformations.

Translation all points move an equaldistance in a given direction.

P* = P + d

x* = x + dxy* = y + dyz* = z + dz


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Transformations - Rotation

Rewriting in a matrix form

cos() -sin()x*

y*

z*

=

x

y

z

0

cos()sin() 0

0 0 1

P* = [ Rz] P

cos()-sin()

0

cos() sin()0

0

0 1[ Ry] =

cos()

-sin()

0

cos()

sin()

0

0

0

1

[ Rx] =

P* = [ R] P

x* = x cos() y sin()

y* = x sin() + y scos()

z* = z

Rotation This operation requires an

entity, a center of rotation, and axis ofrotation

Point P rotates about the z axis


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Curve Length

Consider the curve connectingtwo points P

1and P

2in space.

The exact length of a curve bounded bythe parametric values u1 and u2, it appliesto open and closed curves.


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Cross-Sectional Area

A cross-sectional area is a planar region bounded by a closed boundary.

The boundary is piecewise continuous

The length of thecontour is given by thesum of the lengths ofC1, C2,..Cn.

To calculate the areaA of theregionR, consider the area ofelement dA of sides dxL anddyL. Integrate over the region.

The centroid of the region islocated by vectorr

c.


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Surface Area

The surface areaAs of a bounded surfaceis formulated the same as the cross-sectional area. The major difference is thatAs is not planar in general as in the caseof B-spline or Bezier surfaces.

For objects with multiple surfaces, the total surface area is equalto the sum of its individual surfaces.


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Volume

The volume can be expressed as a triple integral by integrating thevolume element dV

The centroid of the object islocated by the vectorr

c.

The volume Vm of a multiply connectedobject with holes is given by,


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Mass & Centroid

The mass of an object can be formulated the same as its volume by

introducing the density.dm =dV

Integrating over the distributed mass of the object,

Assuming the density remains constant through out the objectwe have,

dVm =

m

dVm = = V

V

Mass

Centroid

rdmrc=

m

mSame formulation as for volume,replace volume by mass.


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First Moment of InertiaFirst moment of an area, mass, or volume is a mathematical property that isuseful in various calculations. For a lumped mass, the first moment of the mass

about a given plane is equal to the product of the mass and its perpendiculardistance from the plane. So the first moment of a distributed mass of an objectwith respect to the XY, XZ, and YZ planes are given,

Substituting the centroidequation, we obtain,


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Second Moments of Inertia

The physical interpretation of a second mass moment of inertia of anobject about an axis is that it represents the resistance of the object toany rotation, or angular acceleration, about the axis. The area momentof inertia represents the ability of the object to resist deformation.

The second moment of inertia about a given axis is the product of themass and the square of the perpendicular distance between the massand the axis.


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Products of Inertia

In some applications of mechanical or structural design it is necessary to know

the orientation of those axis that give the maximum and minimum moments ofinertia for the area. To determine that, we need to find the product of inertia forthe area as well as its moments of inertia about x, y, and z axes.

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Mass Property