426 PROBLEMS 



PROBLEM 7. CONSTRUCTION OF AN INERTIA ELLIPSE AND AN INERTIA 



CIRCLE, 



(a) Problem. Given the following data for an angle 7" X 3^" 

 Xi"; ^ = 9-50 sq. in., I, = 7.53"*, I 2 = 4S-37" 4 ; ^ = 0.89"; r 2 = 

 2.19"; r 3 = 2.i9"; tan a = 0.241; C. G. (2.71", 0.96"), see Cambria, 

 pp. 176, 177. (i) Construct the inertia ellipse. (2) Construct the 

 inertia circle. Omit the fillets. Scale of the angle, i" = i". 



(b) Methods, (i) Inertia Ellipse. Construct angle a, tan a 

 0.241 ; and draw axes 3-3 and 4-4, which are the principal axes of the 

 inertia ellipse. Calculate r 4 f roni the relation 1^ -\- I 2 = I s + I, from 

 which Tj 2 + r 2 2 = r 3 2 + r *, and r 4 =12.25". Construct the enclosing 

 rectangle of the ellipse on the axes 3-3 and 4-4, and inscribe an ellipse 

 in this rectangle ; this ellipse is the central inertia ellipse. 



Calculate Z^_ 2 from the relation Z^_ 2 = A^ X ^i X k l + A 2 X h 2 

 X k 2 . Also calculate c and c 2 from the relation Z 1 _ 2 = Ac l r 2 = Ac 2 r 1 . 

 Compare the calculated values of c^ and c 2 with the scaled values on the 

 ellipse. Note that c t and c 2 are zero for the principal axes. 



(2) Inertia Circle. Calculate the product of inertia, Z x _ 2 = 9.67. 

 From any given point, a, lay off / x = 7.53 to the left extending to b, lay 

 off 7,3 = 45.37 to the right from b, and extending to c. At a erect a 

 perpendicular o-d = Z 1 _ 2 = 9.67. Then with center O, midway 

 between a and c, and with a radius O-d describe a circle, which will 

 be the inertia circle. A line drawn through d and e will be parallel to 

 the principal axis 4-4, and the diameter of the inertia circle will be 

 the maximum value of I 2 . 



(c) Results. (i) The inertia ellipse drawn is the central ellipse 

 of inertia, and is the smallest ellipse that can be drawn. The radii of 

 gyration about any axis can be found directly from the inertia ellipse. 

 (2) The moments of inertia about any axis can be found directly from 

 the circle of inertia. 



PROBLEM 7a. CONSTRUCTION OF AN INERTIA ELLIPSE AND AN INERTIA 



CIRCLE. 



(a) Problem. Given the data for an angle 7"X3^"X%"; 

 see Cambria, pp. 176, 177. (i) Construct the inertia ellipse. (2) 

 Construct the inertia circle. Omit the fillets. Scale of the angle, 



