Engineering Optimization: Theory and Practice, Fourth Edition

Problems 775

Y 1

Y 2

k 1

k 2

k 3

m 1

m 2

Figure 14.13 Two-degree-of-freedom spring–mass system.

whereEis Young’s modulus,Ithe area moment of inertia,lthe length,ρthe mass

density,Athe cross-sectional area,λthe eigenvalue, andY= {Y 1 , Y 2 }T=eigenvector. If

E= 30 × 106 psi,d=2 in.,t= 0 .1 in.,l=20 in., andρg= 0 .283 lb/in^3 , determine

(a)Eigenvaluesλiand eigenvectorsYi(i= 1 , 2 )

(b)Values of∂λi/∂dand∂λi/∂t (i= 1 , 2 )

14.14 In Problem 14.13, determine the derivatives of the eigenvectors∂Yi/∂dand∂Yi/∂t
(i= 1 , 2 ).

14.15 The natural frequencies of the spring–mass system shown in Fig. 14.13 are given by
(forki=k, i= 1 , 2 ,3 andmi=m, i= 1 ,2)

λ 1 =

k

m

=ω^21 , λ 2 =

3 k

m

=ω^22

Y 1 =c 1

{

1

1

}

, Y 2 =c 2

{

1

− 1

}

whereω 1 andω 2 are the natural frequencies of vibration of the system andc 1 andc 2

are constants. The stiffness of each helical spring is given by

k=

d^4 G

8 D^3 n

wheredis the wire diameter,Dthe coil diameter,Gthe shear modulus, andnthe

number of turns of the spring. Determine the values of∂ωi/∂Dand∂Yi/∂Dfor the

following data:d= 0 .04 in.,G= 11. 5 × 106 psi,D= 0 .4 in.,n=10, andm= 32 .2 lb

• s^2 /in. The stiffness and mass matrices of the system are given by

[K]=k

[

2 − 1

−1 2

]

, [M]=m

[

1 0

0 1

]

14.16 Find the minimum volume design of the truss shown in Fig. 14.14 with constraints on
the depth of the truss (y), cross-sectional areas of the bars (A 1 andA 2 ), and the stresses