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.
*9.181 Probs. 9.145 and 9.149
(a)
Find the principal mass moment of inertia at the origin.
Answer to Problem 9.181P
The principal mass moment of inertia at the origin is
Explanation of Solution
Calculation:
Refer to problem 9.145 and 9.149.
Substitute the values of
Solve the above Equation.
Thus, the principal mass moment of inertia at the origin is
(b)
Find the principal axis of inertia at the origin.
Answer to Problem 9.181P
The principal axis of inertia at the origin for
The principal axis of inertia at the origin for
The principal axis of inertia at the origin for
Explanation of Solution
Calculation:
Use Equation 9.54 and 9.57 to find the direction cosines
For
Use Equation 9.54a and 9.54b.
Substitute
Simplifying,
Solving the Equations (1) and (2) for
Substitute into
Substitute into Equation 9.57:
Thus, the principal axis of inertia at the origin for
For
Use Equation 9.54a and 9.54b.
Substitute
Simplifying
Solving Equations (3) and (4) for
Substitute
Substitute into Equation 9.57:
Thus, the principal axis of inertia at the origin for
For
Use Equation 9.54a and 9.54b.
Substitute
Simplifying
Solving Equations (5) and (6) for
Substitute
Substitute into Equation 9.57:
Find the direction cosines corresponding to the labelled axis, take the negative root of
Thus, the principal axis of inertia at the origin is
Sketch the orientation of the principal axis to the
Refer to Figure 1.
Principal axis 3 has been labelled so that the principle axes form a right handed set.
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Chapter 9 Solutions
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