CASE STUDY EQUILIBRIUM AND MINIMUM POTENTIAL ENERGY

7.4 CASE STUDY continued

b a L a L b k b k a F 2 F 1 x 2 x 1 Equation 7.11 defines a parabola. Since the potential energy will be at a minimum at equilibrium, the solution for displacement can be viewed as a one-dimensional optimiza- tion problem. Because this equation is so easy to differentiate, we can solve for the dis- placement as x = Fk. For example, if k = 2 Ncm and F = 5 N, x = 5N2 Ncm = 2.5 cm. A more interesting two-dimensional case is shown in Fig. 7.11. In this system, there are two degrees of freedom in that the system can move both horizontally and vertically. In the same way that we approached the one-dimensional system, the equilibrium defor- mations are the values of x 1 and x 2 that minimize the potential energy: PEx 1 , x 2 = 0.5k a x 2 1 + L a − x 2 2 − L a 2 + 0.5k b x 2 1 + L b + x 2 2 − L b 2 − F 1 x 1 − F 2 x 2 7.12 If the parameters are k a = 9 Ncm, k b = 2 Ncm, L a = 10 cm, L b = 10 cm, F 1 = 2 N, and F 2 = 4 N, use MATLAB to solve for the displacements and the potential energy. FIGURE 7.11 A two-spring system: a unloaded and b loaded. PROBLEMS 199 PROBLEMS 7.1 Perform three iterations of the Newton-Raphson method to determine the root of Eq. E7.1.1. Use the parameter val- ues from Example 7.1 along with an initial guess of t = 3 s. 7.2 Given the formula f x = −x 2 + 8x − 12 a Determine the maximum and the corresponding value of x for this function analytically i.e., using differentiation. b Verify that Eq. 7.10 yields the same results based on initial guesses of x 1 = 0, x 2 = 2, and x 3 = 6. 7.3 Consider the following function: f x = 3 + 6x + 5x 2 + 3x 3 + 4x 4 Locate the minimum by finding the root of the derivative of this function. Use bisection with initial guesses of x l = −2 and x u = 1. 7.4 Given f x = −1.5x 6 − 2x 4 + 12x a Plot the function. b Use analytical methods to prove that the function is con- cave for all values of x. c Differentiate the function and then use a root-location method to solve for the maximum f x and the corre- sponding value of x. 7.5 Solve for the value of x that maximizes f x in Prob. 7.4 using the golden-section search. Employ initial guesses of x l = 0 and x u = 2, and perform three iterations. 7.6 Repeat Prob. 7.5, except use parabolic interpolation. Employ initial guesses of x 1 = 0, x 2 = 1, and x 3 = 2, and per- form three iterations. 7.7 Employ the following methods to find the maximum of f x = 4x − 1.8x 2 + 1.2x 3 − 0.3x 4 a Golden-section search x l = – 2, x u = 4, ε s = 1. b Parabolic interpolation x 1 = 1.75, x 2 = 2, x 3 = 2.5, iterations = 5. 7.8 Consider the following function: f x = x 4 + 2x 3 + 8x 2 + 5x Use analytical and graphical methods to show the function has a minimum for some value of x in the range −2 ≤ x ≤ 1.

7.4 CASE STUDY continued