Ken Youssefi Mechanical Engineering dept. 1 Mass Properties Mass 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

Ken Youssefi Mechanical Engineering dept. 2 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 x* = x + d x y* = y + d y z* = z + d z

Ken Youssefi Mechanical Engineering dept. 3 Transformations - Rotation Rewriting in a matrix form cos(θ) -sin(θ) x* y* z* = x y z 0 cos(θ) sin(θ) P* = [ R z ] P cos(θ) -sin(θ) 0 cos(θ) sin(θ) [ R y ] = cos(θ) -sin(θ) 0 cos(θ) sin(θ) [ R x ] = 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

Ken Youssefi Mechanical Engineering dept. 4 Curve Length Consider the curve connecting two points P 1 and P 2 in space. The exact length of a curve bounded by the parametric values u 1 and u 2, it applies to open and closed curves.

Ken Youssefi Mechanical Engineering dept. 5 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 C 1, C 2,…..C n. To calculate the area A of the region R, consider the area of element dA of sides dx L and dy L. Integrate over the region. The centroid of the region is located by vector r c.

Ken Youssefi Mechanical Engineering dept. 6 Surface Area The surface area A s 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.

Ken Youssefi Mechanical Engineering dept. 7 Volume The volume can be expressed as a triple integral by integrating the volume element d V The centroid of the object is located by the vector r c. The volume V m of a multiply connected object with holes is given by,

Ken Youssefi Mechanical Engineering dept. 8 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, ∫∫ ∫ ρdV m = m ∫∫∫ dV m = ρ = ρV V Mass Centroid ∫∫∫ r dm rc=rc= m m Same formulation as for volume, replace volume by mass.

Ken Youssefi Mechanical Engineering dept. 9 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,

Ken Youssefi Mechanical Engineering dept. 10 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.

Ken Youssefi Mechanical Engineering dept. 11 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.

Ken Youssefi Mechanical Engineering dept. 12 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 Determine the mass properties SolidWorks

Ken Youssefi Mechanical Engineering dept. 13 Mass Properties - SolidWorks Option button allows you to set the proper units

Ken Youssefi Mechanical Engineering dept. 14 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 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

Ken Youssefi Mechanical Engineering dept. 15 Mass Properties Unigraphics NX5 Calculates principal moment of inertia, circumference, are and center of gravity of Sections. Primarily, used for automotive body design.

Ken Youssefi Mechanical Engineering dept. 16 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. Area/Volume/MassObtains the total face area, volume and mass of a 3D object. Centroid/1st MomObtains the center of mass, or Centroid. Moments of InertiaObtains the moment of inertia for certain 3D objects of uniform density about specified axes. Products of InertiaThe Products of Inertia, along with the Moments of Inertia, form the inertia tensor, and are important in rotational dynamics. Principal Axes/MomentsThe 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 GyrationCalculates the radius of gyration. InformationDisplays the calculated data for all of the Mass Properties options previously discussed in the Information window. Relative ErrorsAre 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.

Ken Youssefi Mechanical Engineering dept. 17 Mass Properties – Unigraphics NX5 Measure Bodies Output