Engineering Optimization: Theory and Practice, Fourth Edition

72 Classical Optimization Techniques

Figure 2.4 Spring–cart system.

SOLUTION According to the principle of minimum potential energy, the system will

be in equilibrium under the loadPif the potential energy is a minimum. The potential

energy of the system is given by

potential energy(U )

=strain energy of springs−work done by external forces

=[^12 k 2 x 12 +^12 k 3 (x 2 −x 1 )^2 +^12 k 1 x 22 ] −Px 2

The necessary conditions for the minimum ofUare

∂U

∂x 1

=k 2 x 1 −k 3 (x 2 −x 1 )= 0 (E 1 )

∂U

∂x 2

=k 3 (x 2 −x 1 )+k 1 x 2 −P= 0 (E 2 )

The values ofx 1 andx 2 corresponding to the equilibrium state, obtained by solving

Eqs. (E 1 ) and (E 2 ), are given by

x 1 ∗=

Pk 3

k 1 k 2 +k 1 k 3 +k 2 k 3

x 2 ∗=

P(k 2 +k 3 )

k 1 k 2 +k 1 k 3 +k 2 k 3

The sufficiency conditions for the minimum at(x∗ 1 , x∗ 2 ) an also be verified by testingc

the positive definiteness of the Hessian matrix ofU. The Hessian matrix ofUevaluated

at(x 1 ∗, x 2 ∗) si

J

∣

∣

∣(x 1 ∗,x 2 ∗)=

     

∂^2 U

∂x 12

∂^2 U

∂x 1 ∂x 2

∂^2 U

∂x 1 ∂x 2

∂^2 U

∂x 22

     

(x 1 ∗,x∗ 2 )

=

[

k 2 +k 3 −k 3

−k 3 k 1 +k 3

]