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The problem below can be regarded as a problem of structural stability. For certain values of N

lower than a critical value Ner, a perturbation A0 <1 applied to the initial rest configuration 8₁

= 0 will result in the system going back the same initial configuration, i.e. 02 = 0₁ = 0. However,

if N>Ner, a perturbation A0 <1 applied to the initial rest configuration 0₁=0 will result in a

new, permanent, rest configuration 02 #0. The scope of this exercise is to find the value of Ner for

a given generic length of the rigid bar 1, stiffness k and mass m.

Hint: In order to solve the problem and find Ner, you have to

1. apply the principle of work and energy, T₁ + U₁=T₂ + U₂, and

2. consider an angle very small so that you can assume the following Taylor series expansions

truncated to the first and second order

sin (0) = 0 +......

cos(0) =

= 1

N

m

9

02

2

k

www.x

1

Fig: 1