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Determine expression for the total potential energy of the system.

INSTRUCTIONS TO CANDIDATES
ANSWER ALL QUESTIONS

For the system shown below, all connections are frictionless, and the load P remains vertical at the tip. The load acts at the centerline, the pivot acts at the centerline, and there are no eccentricities at all in the system. As can be seen, the four springs are not of equal stiffness. Use different values of k, and a to do the following:

1- Determine expression for the total potential energy of the system.

2- Determine the first variation of the total potential energy and the equations that define the equilibrium conditions for the system. Carefully analyze and explain all equilibrium solutions. You may use appropriate nondimensionalization for P if you wish.

3- Plot the non-dimensional equilibrium paths from the solutions of part 2 in p-8 space for the range possible.

4- On a separate plot, determine the stability characteristics along each of the equilibrium paths from part 2 above. Be specific as to what range or ranges of 6 represent stable equilibrium configurations and what range or ranges of 0 represent unstable equilibrium configurations. Report the relevant values of using three-digit accuracy. Identify and determine the stability of any bifurcation points and/or limit point, i.e., points where a path has a slope of zero. Illustrate the stability characteristics on a plot of the equilibrium paths. 5- For different values of 0 and the applied load is somewhere on an equilibrium path but is greater than Per. Assume that the load is monotonically decreased from this level to zero. On another plot of the equilibrium paths from part 2 above, indicate all possible load vs. paths that can be traced out for this decreasing load. Use arrows to indicate the direction of decreasing load.

6- Carry your analysis for

a. k1 = k2 =

k3= k1 = k

b. k1 = k2 =

k and k3= k4= 2k-

c. k1 = k2 =

2k and k3 = k4= k

d. k1 = k, k2 = 2k, k3= 2k, and k4= k

e.

k1 = k, k2 = 0, k3= 0, and k1 = k

f. k1 = k2 = k3 = 0, and k1 = k

7- Discuss your results in a formal report and show all your work. Use any software to derive,

plot, and solve your problem.

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