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X y z 120 mm 150 mm 120 mm 150 mm Problem 9.192 A section of sheet steel 2 mm thick is cut and bent into the machine component shown. Knowing that the.

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Presentation on theme: "X y z 120 mm 150 mm 120 mm 150 mm Problem 9.192 A section of sheet steel 2 mm thick is cut and bent into the machine component shown. Knowing that the."— Presentation transcript:

1 x y z 120 mm 150 mm 120 mm 150 mm Problem A section of sheet steel 2 mm thick is cut and bent into the machine component shown. Knowing that the density of steel is 7850 kg/m 3, determine the mass moment of inertia of the component with respect to (a) the x axis, (b) the y axis, (c) the z axis.

2 Solving Problems on Your Own x y z 120 mm 150 mm 120 mm 150 mm 1. Compute the mass moments of inertia of a composite body with respect to a given axis. 1a. Divide the body into sections. The sections should have a simple shape for which the centroid and moments of inertia can be easily determined (e.g. from Fig in the book). A section of sheet steel 2 mm thick is cut and bent into the machine component shown. Knowing that the density of steel is 7850 kg/m 3, determine the mass moment of inertia of the component with respect to (a) the x axis, (b) the y axis, (c) the z axis. Problem 9.192

3 Solving Problems on Your Own x y z 120 mm 150 mm 120 mm 150 mm A section of sheet steel 2 mm thick is cut and bent into the machine component shown. Knowing that the density of steel is 7850 kg/m 3, determine the mass moment of inertia of the component with respect to (a) the x axis, (b) the y axis, (c) the z axis. 1b. Compute the mass moment of inertia of each section. The moment of inertia of a section with respect to the given axis is determined by using the parallel-axis theorem: I = I + m d 2 Where I is the moment of inertia of the section about its own centroidal axis, I is the moment of inertia of the section about the given axis, d is the distance between the two axes, and m is the sections mass. Problem 9.192

4 Solving Problems on Your Own x y z 120 mm 150 mm 120 mm 150 mm A section of sheet steel 2 mm thick is cut and bent into the machine component shown. Knowing that the density of steel is 7850 kg/m 3, determine the mass moment of inertia of the component with respect to (a) the x axis, (b) the y axis, (c) the z axis. 1c. Compute the mass moment of inertia of the whole body. The moment of inertia of the whole body is determined by adding the moments of inertia of all the sections. Problem 9.192

5 Problem Solution Divide the body into sections. x y z 120 mm 150 mm 120 mm 150 mm x y z x y z y x z 2 1 3

6 x y z 120 mm 150 mm 120 mm 150 mm x y z x y z y x z Computation of Masses: Section 1: m 1 = V 1 = (7850 kg/m 3 )(0.002 m)(0.300 m) 2 = kg Section 2: m 2 = V 2 = (7850 kg/m 3 )(0.002 m)(0.150 m)(0.120 m) = kg Section 3: m 3 = m 2 = kg Problem Solution

7 x y z 120 mm 150 mm 120 mm 150 mm x y z x y z y x z Compute the moment of inertia of each section. (a) Mass moment of inertia with respect to the x axis. Section 1: (I x ) 1 = (1.413) (0.30) 2 = 1.06 x kg. m Section 2: (I x ) 2 = (I x ) 2 + m d 2 (I x ) 2 = (0.2826) (0.120) 2 + (0.2826)( ) (I x ) 2 = 7.71 x kg. m Section 3: (I x ) 3 = (I x ) 2 = 7.71 x kg. m 2 Problem Solution

8 x y z 120 mm 150 mm 120 mm 150 mm x y z x y z y x z Compute the moment of inertia of the whole area. For the whole body: I x = (I x ) 1 + (I x ) 2 + (I x ) 3 I x = 1.06 x x x = 2.60 x kg. m 2 Problem Solution I x = 26.0 x kg. m 2

9 x y z 120 mm 150 mm 120 mm 150 mm x y z x y z y x z Compute the moment of inertia of each section. (b) Mass moment of inertia with respect to the y axis. Section 1: (I y ) 1 = (1.413) ( ) = 2.12 x kg. m Section 2: (I x ) 2 = (I x ) 2 + m d 2 (I y ) 2 = (0.2826) (0.150) 2 + (0.2826)( ) (I y ) 2 = 8.48 x kg. m Section 3: (I y ) 3 = (I y ) 2 = 8.48 x kg. m 2 Problem Solution

10 x y z 120 mm 150 mm 120 mm 150 mm x y z x y z y x z Compute the moment of inertia of the whole area. For the whole body: I y = (I y ) 1 + (I y ) 2 + (I y ) 3 I y = 2.12 x x x = 3.82 x kg. m 2 Problem Solution I y = 38.2 x kg. m 2

11 x y z 120 mm 150 mm 120 mm 150 mm x y z x y z y x z Compute the moment of inertia of each section. (c) Mass moment of inertia with respect to the z axis. Section 2: (I z ) 2 = (I z ) 2 + m d 2 (I z ) 2 = (0.2826) ( ) + (0.2826)( ) (I z ) 2 = 3.48 x kg. m Section 3: (I z ) 3 = (I z ) 2 = 3.48 x kg. m 2 Section 1: (I z ) 1 = (1.413) (0.30) 2 = x kg. m Problem Solution

12 x y z 120 mm 150 mm 120 mm 150 mm x y z x y z y x z Compute the moment of inertia of the whole area. For the whole body: I z = (I z ) 1 + (I z ) 2 + (I z ) 3 I z = 1.06 x x x = x kg. m 2 Problem Solution I z = x kg. m 2


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