ken youssefi mechanical engineering dept. 1 mass properties mass property calculation was one of the...

Ken Youssefi Mechanical Engineering dept. 1

Mass PropertiesMass 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


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 equal distance in a given direction.

P* = P + d

x* = x + dx

y* = y + dy

z* = z + dz


Transformations - RotationRewriting in a matrix form

cos(θ) -sin(θ)x*

y*

z*

=

x

y

z

0

cos(θ)sin(θ) 0

0 0 1

P* = [ Rz] P

cos(θ)-sin(θ)

0

cos(θ) sin(θ)0

0

0 1[ Ry] =

cos(θ)

-sin(θ)

0

cos(θ)

sin(θ)

0

0

0

1

[ Rx] =

P* = [ R] P

x* = x cos(θ) – y sin(θ)

y* = x sin(θ) + y scos(θ)

z* = z

Rotation – This operation requires an entity, a center of rotation, and axis of rotation

Point P rotates about the z axis


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.


Cross-Sectional Area

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 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.


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.


VolumeThe 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,


Mass & Centroid

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

dm = ρdV

Integrating over the distributed mass of the object,

Assuming the density ρ remains constant through out the object we have,

∫∫∫ ρdVm =

m

∫∫∫ dVm = ρ = ρV

V

Mass

Centroid

∫∫∫ r dm

rc= mmSame formulation as for volume,

replace volume by mass.


First Moment of InertiaFirst 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,


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.


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.


Mass Properties – CAD/CAM SystemsCAD 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


Mass Properties - SolidWorks

Option button allows you to set the proper units


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.

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.

2D Analysis


Mass Properties Unigraphics NX5

Calculates principal moment of inertia, circumference, are and center of gravity of Sections. Primarily, used for automotive body design.



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.



Measure Bodies

Output

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