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 ൌ ∑ ஶ௡ୀଵ ௡ 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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