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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 Surface Area Density Mass

Mass Property Analysis Introduction to Engineering Design Unit 3 – Lesson 3.3 – Structural Analysis 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 8 4 Rectangular Prism The teacher may want to supplement student knowledge about the use of formulas and related mathematical properties using variables. V = H x W x L V = 4” x 4” x 8” V = 128 in3 Project Lead The Way® Copyright 2006

Volume in Cubic Units Cubic inches (in3) Measure volume using cubic units: Cubic inches (in3) Cubic feet (ft3) Cubic yards (yds3) Cubic centimeters (cm3) Cubic meters (m3) It is imperative to keep your units the same when measuring and calculating volume.

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 Cone A 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.

Mass Property Analysis Introduction to Engineering Design Unit 3 – Lesson 3.3 – Structural Analysis Density Density is defined as mass per unit volume. Density is different for every material and can be found in a machinist handbook. Density is always provided, either by looking in a machinist handbook or from tables provided in 3D modeling software. Project Lead The Way® Copyright 2006

Mass Property Analysis Introduction to Engineering Design Unit 3 – Lesson 3.3 – Structural Analysis 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. Polypropylene has a density of .035 lbs/in3 and Mass = Volume x Density Explain to students that the units for the inches cubed are cancelled out through division; since there are inches cubed in the numerator and inches cubed in the denominator they equal one. One times pounds leaves pounds. Mass = 128 in3 x .035 lbs/in3 Mass = 4.48 lbs Using the volume from the previous example: V = 128 in3 Project Lead The Way® Copyright 2006

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. 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 B C D E F A 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. = 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. 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.

Sources 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.