Mass Property Analysis

Mass Property Analysis

The size, volume, surface area, and other properties available from a solid model are most often part of the design constraints your design must satisfy.The following are mass property calculations available in today’s solid modeling programs:

Volume Density Mass

Surface area Centroid Moment of Inertia

Product of Inertia Radii of Gyration Principal Axes

Principal Moments

Mass Properties

• Volume

• Surface Area

• Density

• Mass

In this lesson, you will investigate the following mass properties:

Volume

• Volume is the amount of three-dimensional space contained within an object.

• Design engineers use volume to determine the amount of material needed to produce a part.

• Different formulas for different shapes

V = H x W x L

V = 4” x 4” x 8”

V = 128 in3

8

4

4

Rectangular Prism

Volume in Cubic Units

It is imperative to keep your units the same when measuring and calculating volume.

• Cubic inches (in3)• Cubic feet (ft3)• Cubic yards (yds3)• Cubic centimeters

(cm3)• Cubic meters (m3)

Measure volume using cubic units:

Volume Formulas for Prisms, Cylinders, Pyramids, or Cones

If B is the area of the base of a prism, cylinder, pyramid, or cone and H is the height of the solid, then the formula for the volume is

V = BH

Note: You will need to calculate the area of the shape for the base of the prism. For example: If the solid is a triangular prism, then you will need to calculate the area of the triangle for the base and then calculate the volume.

Area Formulas for Bases of Prisms, Cylinders, and Pyramids

Rectangular Prism – base is rectangle, therefore A = length * width or A = lw

Cylinder – base is a circle, therefore A = pi * radius of circle squared or A = πr2

Square Pyramid – base is a square, therefore A = length * width or A = lw or side squared since the sides are the same on a square or A = s2.

Volume of a ConeA Special Case

• A cone is 1/3 of a cylinder.

• The base of a cylinder is a circle.

• The area of a circle is A=πr2

• Therefore, the formula for the volume of a cone is

V= 1/3Ah

where A=πr2

and h is the height of the cone.

Density

• Density is defined as mass per unit volume.

• Density is different for every material and can be found in a machinist handbook.

Mass

• Mass is the amount of matter in an object or the quantity of the inertia of the object.

• Many materials are purchased by weight; to find weight, you need to know the mass.

Mass = Volume x Density

Using the volume from the previous example:V = 128 in3

Mass = 128 in3 x .035 lbs/in3

Mass = 4.48 lbs

Polypropylene has a density of .035 lbs/in3 and

Surface Area

• Surface area is the squared

dimensions of the exterior surface.

• Surface area is important when determining coatings and heat transfer of a part.

B

C

D

E

FA

A= 4in x 4in = 16 in2

B= 4in x 8in = 32 in2

C= 4in x 8in = 32 in2

D= 4in x 8in = 32 in2

E= 4in x 8in = 32 in2

F= 4in x 4in = 16 in2

A + B+ C + D+ E + F = 160 in2

To start the Mass Property function, right click the solid model name in the Browser.

Pick Properties

Mass Property values will be used for predicting material

quantity needed for production, finishing, packaging and

shipping.

Additional Mass Properties

Centroid

• A 3D point defining the geometric center of a solid.

• Do not confuse centroid with the center of gravity.

• The two only exist at the same 3D point when the part has uniform geometry and density.

Moments of Inertia

• An object’s opposition

to changing its motion about an axis.

• This property is most often used when calculating the deflection of beams.

pdVrI 2

= Integral (Calculus)

I = Moments of Inertia

r = Distance of all points in an element from the axis

p = Density of the material

dV= Division of the entire

body into small volume units.

Products of Inertia

• Is similar to moments of inertia only that products of inertia are relative to two axes instead of one.

• You will notice an XY, YZ, or ZX after the I symbol when defining products of inertia compared to moments of inertia.

Radii of Gyration

• A dimension from the axis where all mass is concentrated,and will produce the same moment of inertia.

m

Ixkx

K = Radius of gyration about an axis

M = Mass

I = Moments of inertia

Principal Axes

• The lines of intersection

created from three mutually

perpendicular planes, with the three planes’ point of intersection at the centroid of the part.

The X, Y, and Z axes show the principal axes of the ellipsoid.

Principal Moments

• Principal moments are the moments of inertia related to the principal axes of the part.

• Brodinski, K. G. (1989). Engineering materials properties and selection. Prentice Hall, Inc., ISBN 0-13-277997-8.

• Budinski, K. G. (1992). Engineering materials, 4th Edition. Prentice Hall, Inc., ISBN 0-13-276635-3.

• Gere, J. M., & Timoshenko, S. P. (1997). Mechanics of materials, PWS Publishing Company, ISBN 0-534-93429-3.

• Lockhart, S. D., & Johnson, C. M. (1999). Engineering design communication: Conveying design through graphics, Preliminary Edition, Addison Wesley Longman, Inc., ISBN 0-201-38042-0.

• Madsen, D. A., Shumaker, T. M., Turpin, J. L., & Stark, C. (1994). Engineering design and drawing, 2nd Edition, Delmar Publishers Inc., ISBN 0-8273-6720-1.

Sources