Machinery's Handbook, 31st Edition
Moment of Inertia 239 percussion . The center of percussion is useful in determining the position of the resultant in mechanics problems involving angular acceleration of bodies about a fixed axis. Finding the Center of Percussion when Radius of Gyration and Location of Center of Gravity are Known: The center of percussion lies on a line drawn through the center of rotation and the center of gravity. The distance from the axis of rotation to the center of percussion may be calculated from the following formula q k r o 2 ' = in which q = distance from the axis of rotation to the center of percussion; k o = radius of gyration of the body with respect to the axis of rotation; and r = distance from the axis of rotation to the center of gravity of the body. Moment of Inertia An important property of areas and solid bodies is moment of inertia. Standard for mulas are derived by multiplying elementary particles of area or mass by the squares of their distances from reference axes. Moments of inertia, therefore, depend on the loca- tion of reference axes. Values are minimum when these axes pass through the centers of gravity. Three kinds of moments of inertia occur in engineering formulas: 1) Moments of inertia of plane area , I , in which the axis is in the plane of the area are found in formulas for calculating deflections and stresses in beams. When dimensions are given in inches, the units of I are inches 4 . A table of formulas for calculating the I of com mon areas can be found beginning on page 241 . 2) Polar moments of inertia of plane areas , J , in which the axis is at right angles to the plane of the area, occur in formulas for the torsional strength of shafting. When dimensions are given in inches, the units of J are inches 4 . If moments of inertia, I , are known for a plane area with respect to both x and y axes, then the polar moment for the z axis may be calculated using the equation, J I I z x y = + A table of formulas for calculating J for common areas can be found on page 252 in this section. When metric SI units are used, the formulas referred to in (1) and (2) above are valid if the dimensions are given consistently in meters or millimeters. If meters are used, the units of I and J are in meters 4 ; if millimeters are used, these units are in millimeters 4 . 3) Polar moments of inertia of masses , J M * , appear in dynamics equations involving rotational motion. J M bears the same relationship to angular acceleration as mass does to linear acceleration. If units are in the foot-pound-second system, the units of J M are ft- lbs-sec 2 or slug-ft 2 (1 slug = 1 lb-sec 2 /ft). If units are in the inch-pound-second system, the units of J M are inch-lbs-sec 2 . If metric SI values are used, the units of J M are kilogram-meter squared (kg-m 2 ). Formulas for calculating J M for various bodies are given beginning on page 253 . If the polar moment of inertia J is known for the area of a body of constant cross section, J M may be calculated using the equation, J g L J M ρ = where ρ is the density of the material, L the length of the part, and g the gravitational constant. If dimensions are in the foot-pound-second system, ρ is in lbs/ft 3 , L is in ft, g is * In some books the symbol I denotes the polar moment of inertia of masses; J M is used in this Handbook to avoid confusion with moments of inertia of plane areas.
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