Mass Properties

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Mass Properties
Mass property calculation was one of the first
features implemented in CAD/CAM systems.

• Curve length

• Mass

• Cross-sectional area

• Center of mass

• Centroid of a cross-sectional area

• First moment of inertia

• Surface area

• Second moment of inertia

• Centroid of a surface area

• Products of inertia

• Volume

• Centroid of a volume

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Transformations - Translation
Geometric 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 equal distance
in a given direction.
P P d
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Transformations - Rotation
Rewriting in a matrix form
Rotation This operation requires an entity, a
center of rotation, and axis of rotation
x
x
cos(?)
-sin(?)
0
y
y

sin(?)
0
cos(?)
1
z
z
0
0
P Rz P
Point P rotates about the z axis
x x cos(?) y sin(?)
y x sin(?) y scos(?)
z z
P R P
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Curve Length
Consider the curve connecting two points P1 and
P2 in space.
The exact length of a curve bounded by the
parametric values u1 and u2, it applies to 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
To calculate the area A of the region R, consider
the area of element dA of sides dxL and dyL.
Integrate over the region.
The centroid of the region is located by vector
rc.
The length of the contour is given by the sum of
the lengths of C1, C2,..Cn.
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Surface Area
The surface area As of a bounded surface is
formulated the same as the cross-sectional area.
The major difference is that As is not planar in
general as in the case of B-spline or Bezier
surfaces.
For objects with multiple surfaces, the total
surface area is equal to the sum of its
individual surfaces.
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Volume
The volume can be expressed as a triple integral
by integrating the volume element dV
The centroid of the object is located by the
vector rc.
The volume Vm of a multiply connected object with
holes is given by,
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Mass Centroid
Mass
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,
?
?
?
?dV
m
m
Assuming the density ? remains constant through
out the object we have,
Centroid
Same formulation as for volume, replace volume by
mass.
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First Moment of Inertia
First moment of an area, mass, or volume is a
mathematical property that is useful 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 perpendicular
distance from the plane. So the first moment of a
distributed mass of an object with respect to the
XY, XZ, and YZ planes are given,
Substituting the centroid equation, we obtain,
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Second Moments of Inertia
The physical interpretation of a second mass
moment of inertia of an object about an axis is
that it represents the resistance of the object
to any rotation, or angular acceleration, about
the axis. The area moment of inertia represents
the ability of the object to resist deformation.
The second moment of inertia about a given axis
is the product of the mass and the square of the
perpendicular distance between the mass and 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 of inertia for the area. To determine
that, we need to find the product of inertia for
the area as well as its moments of inertia about
x, y, and z axes.
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Mass Properties CAD/CAM Systems
CAD systems typically calculate the mass
properties discussed so far. Even a 2D package
(AutoCAD) calculates some of the mass properties.
You are responsible for setting up the correct
and units for length, angles and density
SolidWorks
Determine the mass properties
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Mass Properties - SolidWorks
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Mass Properties Unigraphics NX5
Calculates volume, surface area, circumference,
mass, radius of gyration, weight, moments of
area, principal moment of inertia, product of
inertia, and principal axes.
Area Using Curves
2D Analysis
Calculates and displays geometric properties of
planar figures. This function analyzes figures
after projecting them onto the XC-YC plane (the
work plane). True lengths, areas, etc., are
obtained.
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Mass Properties Unigraphics NX5
Calculates principal moment of inertia,
circumference, are and center of gravity of
Sections. Primarily, used for automotive body
design.
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Mass Properties Unigraphics NX5
When the software analyzes the selected bodies,
the information window displays the analysis
data. The following table provides a brief
explanation of the information.
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Mass Properties Unigraphics NX5
Measure Bodies
Output