An example relating to the lectures January 30 in the course about stability
P
P
L/2
k
k
α
L/2
A
A
A one degree of freedom structure is made up of a rigid bar with length L and a spring with
stiffness k as indicated in the above figure. The structure and the vertical load P constitutes a
mechanical system. The spring force is zero when angel α is zero.
a) Determine by means of moment equilibrium with respect to hinge A for all α, 0≤ α≤180o,
the value/values of P that give equilibrium. Sketch (or plot by use of Matlab) the result in a P
versus α diagram!
b) Determine the potential energy U of the mechanical system. (U = strain energy in spring +
potential energy of load P.)
c) Determine by means of the equilibrium condition dU/dα=0 for all α, 0≤ α≤180o, the
value/values of P that give equilibrium. Compare with the result obtained in a).
d) Determine by study of d2U/dα2 for all α, 0≤ α≤180o, if the equilibrium values of P from a)
or c) give stable, neutral or unstable equilibrium of the system.
e) Plot (eg by use of Matlab) U, dU/dα and d2U/dα2 versus α for 0< α<180o, for a few
different values of P, including P=0, P=0.5 (kL/4), P= (kL/4) and P=1.5 (kL/4), and assuming
L=1. See from the curves when the system is in equilibrium and then if stable equilibrium or
not.
/PJG, 2015-02-02