mass properties

7/27/2019 Mass Properties

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Ken YoussefiMechanical Engineering dept. 2

Trans form at ion s - Trans lat ionGeometric 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.

Translat ion all points move an equaldistance in a given direction.

P* = P + d

x* = x + dx

y* = y + dy

z* = z + dz


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Trans form at ions - Rotat ion

Rewriting in a matrix form

cos() -sin()x*y*

z*

=xy

z

0cos()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

Rotat ion This operation requires anentity, a center of rotation, and axis of

rotation

Point P rotates about the z axis


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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-Sect ional A rea

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

The boundary is piecewise continuous

The length of the

contour is given by the

sum of the lengths of

C1, C2,..Cn.

To calculate the areaA of the

regionR, consider the area of

element dA of sides dxL and

dyL. Integrate over the region.

The centroid of the region is

located by vectorrc.


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Surface Area

The surface areaAs 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 equalto the sum of its individual surfaces.


7/17Ken YoussefiMechanical Engineering dept. 7

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 vectorrc.

The volume Vm of a multiply connected

object with holes is given by,


8/17Ken Youssefi

Mechanical Engineering dept. 8

Mass & Centro id

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 object

we have,

dVm =

m

dVm = = V

V

Mass

Centroid

r dmrc= m

mSame formulation as for volume,

replace volume by mass.


9/17Ken Youssefi


First Moment o f Inert ia

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 objectwith respect to the XY, XZ, and YZ planes are given,

Substituting the centroid

equation, we obtain,


10/17Ken Youssefi


Second Moments of Inert ia

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 momentof 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.


11/17Ken Youssefi


Products of Inert ia

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 forthe area as well as its moments of inertia about x, y, and z axes.


12/17Ken Youssefi


Mass Propert ies CAD/CAM Sys tems

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

Determine the mass properties

SolidWorks


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Mass Propert ies - Sol idWork s

Option button allows you

to set the proper units


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Mass Propert ies Unigraph ics NX5

Calculates volume, surface area, circumference,

mass, radius of gyration, weight, moments of area,

principal moment of inertia, product of inertia, and

principal axes.

Calculates and displays geometric properties ofplanar figures. This function analyzes figures after

projecting them onto the XC-YC plane (the work

plane). True lengths, areas, etc., are obtained.

2D Analysis


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Mass Propert ies

Unigraphics NX5

Calculates principal moment of inertia,

circumference, are and center of gravity of

Sect ions. Primarily, used for automotive

body design.


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When the software analyzes the selected bodies, the information window displays the

analysis data. The following table provides a brief explanation of the information.

Area/Volume/Mass Obtains the total face area, volume and mass of a 3D object.

Centroid/1st Mom Obtains the center of mass, or Centroid.

Moments of Inertia Obtains the moment of inertia for certain 3D objects of uniform

density about specified axes.

Products of Inertia The Products of Inertia, along with the Moments of Inertia, form

the inertia tensor, and are important in rotational dynamics.

Principal Axes/Moments The Principal Axes/Moments is an orthogonal system of three

axes through the center of mass such that the three products of

inertia relative to the system are all zero.

Radius of Gyration Calculates the radius of gyration.

Information Displays the calculated data for all of the Mass Properties

options previously discussed in the Information window.Relative Errors Are estimates of the relative tolerances achieved in calculating

the mass properties. Often the relative errors are less than the

specified relative tolerances, indicating that the mass property

values are correct to within tighter tolerances than those

specified. If only a single accuracy value is specified, then +/-

Range Errors are given.


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Measure Bodies

Output

mass properties

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