Issue 58
M. Utzeri et alii, Frattura ed Integrità Strutturale, 58 (2021) 254-271; DOI: 10.3221/IGF-ESIS.58.19
kL kL cosh
1 cos
kL
sin kL kL
bk
bk
1
kL
kL
(18)
( cos
sinh
cosh
cos
cosh
2 cos
bk bk sinh
bk bk
kL
kL
cosh sin
cos
cosh
1 ) 0
1 b L [25]. Note that the Φ guarantees that
Φ 1 max . The Eqn.(18) can be simplified assuming
where
0 to obtain the characteristic equation of clamped-free beam case
cosh cosh 1 0 kL kL (19) We consider that the arbitrary motion of a beam undergoing free vibration may be expressed as a superposition of its free vibration mode shapes ( Φ ሺ ሻ ), each undergoing simple harmonic motion with frequency , namely we assume that: ሺ , ሻ ൌ ∑ ஶୀଵ ሺ ሻΦ ሺ ሻ (20) where Φ ሺ ሻ is a given function and ሺ ሻ is the unknown. The lagrangian of the system is defined as ℒ ൌ െ . Inserting the Eqn.(20) in ℒ and enforcing stationariety, yields to the equation of motion of each mode defined as డ డ ௧௬ ℒ ሶ െ డ డ ௬ ℒ . After some computations we get [16, 26]: ሷ ሺ ଵ ଶ ଶ ଷ ସ ሻ ሶ ଶ ሺ ଶ 2 ଷ ଷ ሻ ଶ ሺ ସ 2 ହ ଷ 3 ହ ሻ ൌ 0 (21) where ଵ ൌ ଵ Φ ଶ ሺ ሻ Φ ሺ ሻ ଶ Φ ᇱ ሺ ሻ ଶ , ଶ ൌ ଵ ቀ Φ ଶ ᇱ ሺ ሻ ቁ ଶ ൫ ఎ Φ ଶ ᇱ ሺ ሻ ൯ ଶ Φ ᇱ ሺ ሻ ସ , ଷ ൌ ଵ ቀ Φ ଶ ᇱ ሺ ሻ ቁ ቀ Φ ସ ᇱ ሺ ሻ ቁ , ൫ ఎ Φ ଶ ᇱ ሺ ሻ ൯൫ ఎ Φ ସ ᇱ ሺ ሻ ൯ Φ ᇱ ሺ ሻ , (22)
ସ ൌ ଵ Φ ଶ ᇱᇱ ሺ ሻ , ହ ൌ ଵ Φ ଶ ᇱᇱ ሺ ሻΦ ଶ ᇱ ሺ ሻ , ൌ ଵ Φ ଶ ᇱᇱ ሺ ሻΦ ସ ᇱ ሺ ሻ ଶ ൌ ா ூ ర
The nonlinear inertial terms are clearly visible. The accuracy of the proposed approximate solution is expected to diminish when the mass ratio exceeds the value of 3 [16], since the spatial shapes Φ n is only considered in linear form. The Eqn.(21) is solved by the Multiple Scale Method [19], and the approximate solution is given by
5 2
1
3
t nl n
sin 3
n u t
sin A t
A
(23)
...
n
nl
n
n
1
2
16
4
1
The nl is the nonlinear frequency, and can be written in the form
2
4
n n 0 2 nl n
4 n n A A
(24)
...
259
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