Concept explainers
9.180 through 9.184 For the component described in the problem indicated, determine (a) the principal mass moments of inertia at the origin, (b) the principal axes of inertia at the origin. Sketch the body and show the orientation of the principal axes of inertia relative to the x, y, and z axes.
*9.183 Prob. 9.168
(a)
Find the principal mass moment of inertia of the cylinder at the origin O.
Answer to Problem 9.183P
The principal moment of inertia are
Explanation of Solution
Given information:
Refer Problem 9.168.
Show the moment of inertia as follows:
Calculation:
Show the Equation 9.56 as follows:
Substitute
Consider the value of
Solve the above Equation and get the value of
The principal moment of inertia are
Thus, The principal moment of inertia are
(b)
Find the angles made by the principal axis of inertia at O with the coordinate axis.
Sketch the body and show the orientation of the principal axis of inertia relative to x, y, and z axis.
Answer to Problem 9.183P
The angles made by the principal axis of inertia at O with the coordinate axis is,
Explanation of Solution
Given information:
Consider the direction cosines of each principal axis are denoted by
Calculation:
Refer Part (a).
Consider K1.
Show the Equation 9.54 as follows:
Substitute
Solve Equation (2).
Get the value of
Show the Equation 9.57 as follows:
Solve above Equation and get the value of
Show the direction cosines
Consider K2.
Show the Equation 9.54 as follows:
Substitute
Solve Equation (4).
Get the value of
Show the Equation 9.57 as follows:
Solve above Equation and get the value of
Show the direction cosines
Consider K3.
Show the Equation 9.54 as follows:
Substitute
Solve Equation (6).
Get the value of
Show the Equation 9.57 as follows:
Solve above Equation and get the value of
Show the direction cosines
The angles made by the principal axis of inertia at O with the coordinate axis is,
Sketch the body and show the orientation of the principal axis of inertia relative to x, y, and z axis as shown in Figure 1.
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