Calculation:

Refer to problem 9.145 and 9.149.

Ix=26.4325×10−3 kg⋅m2Iy=31.1726×10−3 kg⋅m2Iz=8.5773×10−3 kg⋅m2

Ixy=2.5002×10−3 kg⋅m2Iyz=4.0627×10−3 kg⋅m2Izx=8.8062×10−3 kg⋅m2

Substitute the values of 26.4325×10−3 kg⋅m2 for Ix, 31.1726×10−3 kg⋅m2 for Iy, 8.5773×10−3 kg⋅m2 for Iz, 2.5002×10−3 kg⋅m2 for Ixy, 4.0627×10−3 kg⋅m2 for Iyz and 8.8062×10−3 kg⋅m2 for Izx into Equation 9.56.

{K3−[(26.4325+31.1726+8.5773)10−3]K2+[26.4325(31.1726)+(31.1726)(8.5773)+(8.5773)26.4325]−(2.5002)2−(4.0627)2−(8.8062)2(10−6)K−[(26.4325)(31.1726)(8.5773)−(26.4325)(4.0627)2−31.1726(8.8062)2−(8.5773)(2.5002)2−2(2.5002)(4.0627)(8.8062)10−9]}=0K3−(66.1824×10−3)K2+(1217.76×10−6)K−(3981.23×10−9)=0

Solve the above Equation.

K1=4.1443×10−3 kg⋅m2K2=29.7840×10−3 kg⋅m2K3=32.2541×10−3 kg⋅m2

Thus, the principal mass moment of inertia at the origin is K1=414×10−3 kg⋅m2, K2=29.8×10−3 kg⋅m2, and K3=32.3×10−3 kg⋅m2_.

Calculation:

Use Equation 9.54 and 9.57 to find the direction cosines λx,λy,λz of each principal axis.

For K1:

Use Equation 9.54a and 9.54b.

(Ix−K1)(λx)1−Ixy(λ)1−Izx(λz)1=0Ixy(λx)1+(Iy−K1)(λ)1−Iyz(λz)1=0

Substitute 4.1443×10−3 kg⋅m2 for K1, 29.7840×10−3 kg⋅m2 for K2, 32.2541×10−3 kg⋅m2 for K3, 26.4325×10−3 kg⋅m2 for Ix, 31.1726×10−3 kg⋅m2 for Iy, 2.5002×10−3 kg⋅m2 for Ixy, 4.0627×10−3 kg⋅m2 for Iyz, and 8.8062×10−3 kg⋅m2 for Izx.

[(26.4325−4.1443)(10−3)](λx)1−(2.5002×10−3)(λy)1−(8.8062×10−3)(λz)1=0−(2.5002×10−3)(λx)1+[31.1726−4.1443](10−3)[(λy)1−(4.0627×10−3)](λz)1=0

Simplifying,

8.9146(λx)1−(λy)1−3.5222(λz)1=0 (1)

−0.0925(λx)1−(λy)1−0.1503(λz)1=0 (2)

Solving the Equations (1) and (2) for (λz)1.

(λz)1=2.4022(λx)1

Substitute into 2.4022(λx)1 for (λz)1 in Equation (1).

(λy)1=[8.9146−3.5222(2.4022)(λx)1]=0.45357(λx)1

Substitute into Equation 9.57:

(λx)12+[0.45357(λx)1]2+[2.4022(λx)1]2=1(λx)1=0.37861(λz)1=0.90950(λx)1=0.17173

(θx)1=67.8°(θy)1=80.1°(θz)1=24.6°

Thus, the principal axis of inertia at the origin for K1 is (θx)1=67.8°, (θy)1=80.1°, and (θz)1=24.6°_.

For K2:

Use Equation 9.54a and 9.54b.

(Ix−K2)(λx)2−Ixy(λy)2−Izx(λz)2=0−Ixy(λx)2+(Iy−K2)(λy)2−Iyz(λz)2=0

Substitute 4.1443×10−3 kg⋅m2 for K1, 29.7840×10−3 kg⋅m2 for K2, 32.2541×10−3 kg⋅m2 for K3, 26.4325×10−3 kg⋅m2 for Ix, 31.1726×10−3 kg⋅m2 for Iy, 2.5002×10−3 kg⋅m2 for Ixy, 4.0627×10−3 kg⋅m2 for Iyz, and 8.8062×10−3 kg⋅m2 for Izx.

[26.4325−29.7840×10−3][(λx)2−(2.5002×10−3)(λy)2−(8.8062×10−3)(λz)2]=0−(2.5002×10−3)(λx)2+[(31.1726−29.7840)(10−3)](λy)2−(4.0627×10−3)(λz)2=0

Simplifying

−1.3405(λx)2−(λy)2−3.5222(λz)2=0 (3)

−1.8005(λx)2+(λy)2−2.9258(λz)2=0 (4)

Solving Equations (3) and (4) for (λz)2:

(λz)2=−0.48713(λx)2

Substitute −0.48713(λx)2 for (λz)2 in Equation (3).

(λy)2=[−1.3405−3.5222(−0.48713)](λx)2=0.37527(λx)2

Substitute into Equation 9.57:

(λx)22+[0.37527(λx)2]2+[−0.48713(λx)2]2=1(λx)2=0.85184(λz)2=−0.41496(λy)2=0.31967

(θx)2=31.6°(θy)2=71.4°(θz)2=114.5°

Thus, the principal axis of inertia at the origin for K2 is (θx)2=31.6°, (θy)2=71.4°, and (θz)2=114.5°_.

For K3:

Use Equation 9.54a and 9.54b.

(Ix−K3)(λx)3−Ixy(λy)3−Izx(λz)3=0−Ixy(λx)3+(Iy−K3)(λy)3−Iyz(λz)3=0

Substitute 4.1443×10−3 kg⋅m2 for K1, 29.7840×10−3 kg⋅m2 for K2, 32.2541×10−3 kg⋅m2 for K3, 26.4325×10−3 kg⋅m2 for Ix, 31.1726×10−3 kg⋅m2 for Iy, 2.5002×10−3 kg⋅m2 for Ixy, 4.0627×10−3 kg⋅m2 for Iyz, and 8.8062×10−3 kg⋅m2 for Izx.

[(26.4325−32.2541)10−3](λx)3−(2.5002×10−3)(λy)3−(8.8062×10−3)(λz)3=0[−(2.5002×10−3)(λx)3+[31.1726−32.2541(10−3)](λ3)+[31.1726−32.2541(10−3)][(λy)3−(4.0627×10−3)(λz)3]=0]

Simplifying

−2.3285(λx)3−(λy)3−3.5222(λz)3=0 (5)

−2.3118(λx)3−(λy)3−3.7565(λz)3=0 (6)

Solving Equations (5) and (6) for (λz)3.

(λz)3=0.071276(λx)3

Substitute 0.071276(λx)3 for (λz)3 in Equation (5).

(λy)3=[−2.3285−3.5222(0.071276)(λx)3]=−2.5795(λx)3

Substitute into Equation 9.57:

(λx)32+[−2.5795(λx)3]2+[0.07127(λx)3]2=1

(λx)3=0.36134(λz)3=−0.025755(λy)3=0.93208

(θx)3=68.8°(θy)3=158.8°(θz)3=88.5°

Find the direction cosines corresponding to the labelled axis, take the negative root of (λx)3.

(λx)3=−0.36134

(θx)3=111.2°(θy)3=21.2°(θz)3=91.5°

Thus, the principal axis of inertia at the origin is (θx)3=111.2°, (θy)3=21.2°, and (θz)3=91.5°_.

Sketch the orientation of the principal axis to the x,y,z axis as shown in Figure 1.

Refer to Figure 1.

Principal axis 3 has been labelled so that the principle axes form a right handed set.