COSMOS Chapter 9 Solution 184

COSMOS: Complete Online Solutions Manual Organization System Chapter 9, Solution 184. (a) From the solution of Problem 9.169 have Ix = 1W 2 a 2 g I xy = 1W 2 a 4 g Iy = W 2 a g I yz = 1W 2 a 8 g Iz = 5W 2 a 6 g I zx = 3W 2 a 8 g Substituting into Equation (9.56) 2 2 2 2  1 5   W 2   2  1   5   5  1   1  1  3   W 2  K −  + 1 +   a   K +   (1) + (1)   +    −   −   −     a  K 6   g   6   6  2   4  8  8    g   2   2 3 3 2 2  1   5   1  1  2  3  5  1   1  1  3    W 2  −   (1)   −    − (1)   −    − 2       a  = 0 8  6  4   4  8  8    g  2   6   2  8   Simplifying and letting K = W 2 a ζ yields g ζ 3 − 2.33333ζ 2 + 1.53125ζ − 0.192708 = 0 Solving yields ζ 1 = 0.163917 ζ 2 = 1.05402 ζ 3 = 1.11539 The principal moments of inertia are then K1 = 0.1639 W 2 a
g K 2 = 1.054 W 2 a
g K3 = 1.115 W 2 a
g continued COSMOS: Complete Online Solutions Manual Organization System (b) To determine the direction cosines λx , λ y , λz of each principal axis, use two of the equations of Equations (9.54) and (9.57). Then K1 : Begin with Equations (9.54a) and (9.54b). ( I x − K1 )( λx )1 − I xy ( λ y )1 − I zx ( λz )1 = 0 ( )( )1 − I yz ( λz )1 = 0 − I xy ( λx )1 + I y − K 2 λ y Substituting  1 1W 2   W 2  a  λy  − 0.163917   a   ( λx )1 −    g  4 g   2   1W 2  W  − a  ( λx )1 + (1 − 0.163917 )  a 2   λ y 4 g   g    ( )1 −  83 Wg a2  ( λz )1 = 0  ( )1 −  81 Wg a2  ( λz )1 = 0  Simplifying yields ( )1 − 1.5 ( λz )1 = 0 1.34433 ( λx )1 − λ y ( )1 − 0.149507 ( λz )1 = 0 − 0.299013 ( λx )1 + λ y Adding and solving for ( λz )1 ( λz )1 = 0.633715 ( λx )1 ( λ y )1 = 1.34433 − 1.5 ( 0.633715) ( λx )1 and then = 0.393758 ( λx )1 Now substitute into Equation (9.57) ( λx )12 + 0.393758 ( λx )1  2 + 0.633715 ( λx )  = 1 ( λx )1 = 0.801504 or ( λ y )1 = 0.315599 and ∴ 2 (θ x )1 = 36.7° (θ y )1 = 71.6° (θ z )1 = 59.5° ( λz )1 = 0.507925
K 2 : Begin with Equations (9.54a) and (9.54b). ( I x − K 2 )( λx )2 − I xy ( λ y )2 − I zx ( λz )2 ( =0 )( )2 − I yz ( λz )2 = 0 − I xy ( λx )2 + I y − K 2 λ y continued COSMOS: Complete Online Solutions Manual Organization System Substituting  1 1W 2   W 2  a  λy  − 1.05402   a   ( λx )2 −  2 g   4 g     1W 2  W  − a  ( λx )2 + (1 − 1.05402 )  a 2   λ y 4 g   g    ( )2 −  83 Wg a2  ( λz )2 = 0  ( )2 −  81 Wg a2  ( λz )2 = 0  Simplifying yields ( )2 − 1.5 ( λz )2 = 0 − 2.21608 ( λx )2 − λ y ( )2 + 2.31396 ( λz )2 = 0 4.62792 ( λx )2 + λ y Adding and solving for ( λz )2 ( λz )2 = − 2.96309 ( λx )2 ( λ y )2 = − 2.21608 − 1.5 ( − 2.96309 ) ( λx )2 and then = 2.22856 ( λx )2 Now substitute into Equation (9.57) ( λx )22 + 2.22856 ( λx )2  2 +  − 2.96309 ( λx )  = 1 ( λx )2 or and 2 = 0.260410 ( λ y )2 = 0.580339 ∴ (θ x )2 ( λz )2 = − 0.771618 (θ y )2 = 54.5° = 74.9° (θ z )2 = 140.5°
K 3 : Begin with Equations (9.54a) and (9.54b). ( I x − K3 )( λx )3 − I xy ( λ y )3 − I zx ( λz )3 ( =0 )( )3 − I yz ( λz )3 = 0 − I xy ( λx )3 + I y − K 3 λ y Substituting  1 1W 2   W 2  a  λy  − 1.11539   a   ( λx )3 −    g  4 g   2   1W 2  W  a  ( λx )3 + (1 − 1.11539 )  a 2   λ y − 4 g   g    ( )3 −  83 Wg a2  ( λz )3 = 0  ( )3 −  81 Wg a2  ( λz )3 = 0  continued COSMOS: Complete Online Solutions Manual Organization System Simplifying yields ( )3 − 1.5 ( λz )3 = 0 − 2.46156 ( λx )3 − λ y ( )3 + 1.08328 ( λz )3 = 0 2.16657 ( λx )3 + λ y Adding and solving for ( λz )3 ( λz )3 = − 0.707885 ( λx )3 ( λ y )3 = − 2.46156 − 1.5 ( − 0.707885) ( λx )3 and then = −1.39973 ( λx )3 Now substitute into Equation (9.57) ( λx )32 +  −1.39973 ( λx )3  2 2 +  − 0.707885 ( λx )  = 1 ( λ x )3 or and (i) = 0.537577 ( λ y )3 = − 0.752463 ∴ (θ x )3 ( λz )3 = 57.5° = − 0.380543 (θ y )3 = 138.8° (θ z )3 = 112.4°
(c) Note: Principal axis 3 has been labeled so that the principal axes form a right-handed set. To obtain the direction cosines corresponding to the labeled axis, the negative root of Equation (i) must be chosen; that is, ( λx )3 = − 0.537577 Then (θ x )3 = 122.5° (θ y )3 = 41.2° (θ z )3 = 67.6°