978-0073398242 Appendix B Solution Manual Part 14

subject Type Homework Help
subject Pages 9
subject Words 1274
subject Authors Brian Self, David Mazurek, E. Johnston, Ferdinand Beer, Phillip Cornwell

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page-pf1
PROBLEM B.71* (Continued)
(b) To determine the direction cosines , ,
x
yz

of each principal axis, use two of the equations of
Eqs. (9.54) and Eq. (9.57). Then
1111
xy x y y yz z
Substituting:

11 1
11 1
0.0925( ) ( ) 0.1503( ) 0
xy z


1
0.45357( )
x
Now substitute into Eq. (9.57):
222
11 1
( ) [0.45357( ) ] [2.4022( ) ] 1
xx x


or 1
( ) 0.37861
x
2222
xy x y y yz z
222
xyz
333
222
(2.5002 10 )( ) [(31.1726 29.7840)(10 )]( ) (4.0627 10 ) ) 0
xyz



page-pf2
PROBLEM B.71* (Continued)
Simplifying
22 2
1.3405( ) ( ) 3.5222( ) 0
xy z


z
22
( ) 0.48713( )
zx

and then 22
2
( ) [ 1.3405 3.5222( 0.48713)]( )
0.37527( )
yx
x

Now substitute into Eq. (9.57):
y
z
122
( ) 31.6 ( ) 71.4 ( ) 114.5
xy z


3:K Begin with Eqs. (9.54a) and (9.54b).
3333
xxxyyzxz
xy x y y yz z
Substituting:
333
333
[(26.4325 32.2541)(10 )]( ) (2.5002 10 )( ) (8.8062 10 )( ) 0
xyz




33 3
2.3118( ) ( ) 3.7565( ) 0
xy z


Adding and solving for 3
():
z
page-pf3
PROBLEM B.71* (Continued)
Now substitute into Eq. (9.57):
22 2
33 3
( ) [ 2.5795( ) ] [0.071276( ) ] 1
xx x


(i)
333
xyz
page-pf4
PROBLEM B.72*
For the component described in Problem 9.167, 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.
page-pf5
PROBLEM B.72* (Continued)
(b) To determine the direction cosines , ,
x
yz

of each principal axis, use two of the equations of
Eq. (9.54) and Eq. (9.57). Then
1:K Begin with Eqs. (9.54a) and (9.54b).
1211
xy x y y yz z
Substituting
113
WWW



222
111
( ) (1 0.163917) ( ) ( ) 0
xyz
aaa




xy z
11 1
0.299013( ) ( ) 0.149507( ) 0
xy z


Adding and solving for 1
():
z
222
11 1
( ) [0.393758( ) ] (0.633715( ) ] 1
xx x


or 1
( ) 0.801504
x
and 1
( ) 0.315599
y
1
( ) 0.507925
z
22 2 2
222 2
()()()()0
xxxyyzxz
xy x y y yz z
Ik I I


page-pf6
PROBLEM B.72* (Continued)
Substituting
113
WWW



xy z
22 2
4.62792( ) ( ) 2.31396( ) 0
xy z


Adding and solving for 2
()
z
22
( ) 2.96309( )
zx

22 2
xx x
or 2
( ) 0.260410
x
and 2
( ) 0.580339
y
2
( ) 0.771618
z

33 3 3
3333
() ( )() () 0
xy x y y yz z
IIKI


Substituting
222
333
222
333
113
1.11539 ( ) ( ) ( ) 0
248
11
( ) (1 1.11539) ( ) ( ) 0
48
xyz
xyz
WWW
aaa
ggg
WWW
aaa
ggg



















Simplifying
33 3
2.46156( ) ( ) 1.5( ) 0
xy z


page-pf7
PROBLEM B.72* (Continued)
Adding and solving for
3
()
z
the direction cosines corresponding to the labeled axis, the negative root of Eq. (i) must be chosen;
that is,
3
( ) 0.537577.
x

page-pf8
PROBLEM B.73*
For the component described in Problem 9.168, 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.
page-pf9
PROBLEM B.73* (Continued)
Solving yields
12 3
2.25890 17.27274 19.08046KK K
page-pfa
PROBLEM B.73* (Continued)
2:K Begin with Eqs. (9.54a) and (9.54b):
()()()()0
xxxyyzxz
IK I I


Substituting
44 4
tt t


22 2
22 2
0.13913( ) ( ) 0.74522( ) 0
xy z


2
4.02304 ( )
x
Now substitute into Eq. (9.57):
22 2
22 2
( ) [4.02304( ) ] [ 5.58515( ) ] 1
xx x


or 2
( ) 0.14377
x
33 3 3
3333
xxxyyzxz
xy x y y yz y
44 4
333
[(18.91335 19.08046) ( ) 1.33333 ( ) 0.66667 ( ) 0
xy z
aa a




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