Concept explainers
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.
Prob. B.58
(a)
The principal mass moments of inertia at origin.
Answer to Problem B.73P
The principal mass moments of inertia at origin are
Explanation of Solution
Given information:
The thickness of steel sheet is
The following figure illustrates different centroidal axis.
Figure-(1).
Write the expression for the mass of section 1.
Here, the mass of the square section 1 is
Write the expression for the mass of section 2.
Here, the mass of the square section 2 is
Write the expression for hypotenuse side
Here, the hypotenuse side in right angled triangle
Write the expression for hypotenuse side
Here, the hypotenuse side in right angled triangle
Write the expression of position vector of the line joining
Here, the vector along the line joining the points
Write the expression for unit vector along
Here, the unit vector along
Write the expression of mass moment of inertia of section 1 about
Here, the mass moment of inertia of section 1 about
Write the expression of mass moment of inertia of section 2 about
Here, the mass moment of inertia of section 2 about
Write the expression of total mass moment of inertia about
Here, the total mass moment of inertia about
Write the expression of mass moment of inertia of section 1 about
Here, the mass moment of inertia of section 1 about
Write the expression of mass moment of inertia of section 2 about
Here, the mass moment of inertia of section 2 about
Write the expression of total mass moment of inertia about
Write the expression of mass moment of inertia of section 1 about
Here, the mass moment of inertia of section 1 about
Write the expression of mass moment of inertia of section 2 about
Here, the mass moment of inertia of section 2 about
Write the expression of total mass moment of inertia about
Here, the total mass moment of inertia about
From, the symmetry in above figure about
Write the expression for product of mass moment of inertia in
Here, the product mass moment of inertia is
Write the expression for product mass moment of inertia in
Here, the product mass moment of inertia is
Write the expression for product mass moment of inertia in
Here, the product mass moment of inertia in
Write the expression of mass moments of inertia at origin.
Here, the principle mass moment of inertia at origin is
Calculation:
Substitute
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Conclusion:
The principal mass moments of inertia at origin are
(b)
The principal axes of inertia at the origin.
Answer to Problem B.73P
The principal axes of inertia at the origin at
The principal axes of inertia at the origin at
The principal axes of inertia at the origin at
Explanation of Solution
Write the expression for direction cosine in terms of
Write the expression for direction cosine in terms of
Write the expression for the direction cosines at x, y and z axis.
Write the expression for the principal axes of the inertia at x-axis.
Write the expression for the principal axes of the inertia at y-axis.
Write the expression for the principal axes of the inertia at z-axis.
Write the expression for direction cosine in terms of
Write the expression for direction cosine in terms of
Write the expression for the direction cosines at x, y and z axis.
Write the expression for the principal axes of the inertia at x-axis.
Write the expression for the principal axes of the inertia at y-axis.
Write the expression for the principal axes of the inertia at z-axis.
Write the expression for direction cosine in terms of
Write the expression for direction cosine in terms of
Write the expression for the direction cosines at x, y and z axis.
Write the expression for the principal axes of the inertia at x-axis.
Write the expression for the principal axes of the inertia at y-axis.
Write the expression for the principal axes of the inertia at z-axis.
Calculation:
Substitute
Substitute
Add Equation (XXXIX) and Equation (XL).
Substitute
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Add Equation (XLIII) and Equation (XLIV).
Substitute
Substitute
Substitute
Substitute
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Substitute
Add Equation (XLVII) and Equation (XLVIII).
Substitute
Substitute
Substitute
Substitute
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Substitute
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Conclusion:
The principal axes of inertia at the origin at
The principal axes of inertia at the origin at
The principal axes of inertia at the origin at
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Chapter B Solutions
Vector Mechanics For Engineers
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