Mass Properties

1. Mass Properties Mass property calculation was one of the first features implemented in CAD/CAM systems. • Curve length • Mass • Cross-sectional area • Center of mass • Centroid of a cross-sectional area • First moment of inertia • Surface area • Second moment of inertia • Centroid of a surface area • Products of inertia • Volume • Centroid of a volume Mechanical Engineering dept.

2. x* = x + dx y* = y + dy z* = z + dz 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 Mechanical Engineering dept.

3. 1 0 0 0 [ Rx] cos(θ) = -sin(θ) sin(θ) cos(θ) 0 cos(θ) 0 sin(θ) [ Ry] 0 1 = 0 0 cos(θ) -sin(θ) Transformations - Rotation Rewriting in a matrix form Rotation – This operation requires an entity, a center of rotation, and axis of rotation x* x 0 cos(θ) -sin(θ) y y* sin(θ) = 0 cos(θ) 1 z 0 0 z* P* = [ Rz] P Point P rotates about the z axis x* = x cos(θ) – y sin(θ) y* = x sin(θ) + y scos(θ) z* = z P* = [ R] P Mechanical Engineering dept.

4. 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. 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 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. The length of the contour is given by the sum of the lengths of C1, C2,…..Cn. Mechanical Engineering dept.

6. 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. Mechanical 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 vector rc. The volume Vm of a multiply connected object with holes is given by, Mechanical Engineering dept.

8. ∫ ∫ ∫ m =ρ dV = ρV V ∫ ∫ ∫ r dm m rc= m Mass & Centroid Mass 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, ∫ ∫ ∫ m = ρdV m Assuming the density ρremains constant through out the object we have, Centroid Same formulation as for volume, replace volume by mass. 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, 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. 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. 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 SolidWorks Determine the mass properties Mechanical Engineering dept.

13. Option button allows you to set the proper units Mass Properties - SolidWorks 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 2D Analysis 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. 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. 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. Mechanical Engineering dept.

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