PSI - Issue 17

Antonino Morassi et al. / Procedia Structural Integrity 17 (2019) 98–104 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

100

3

0  . Suppose that the mass density 0  changes, and denote by ( ) x

0 r x 

with respect to the mass density

the

( )

  = +

mass density per unit length of the perturbed nanobeam. The mass variation ( −1 ∫ ( ) ) 0 1/2 = 0 , ( ) ∈ ∞ (0, ) , 0 < − ≤ ( ) ≤ + in [0,L], where 0 < ≤ , for a given small number , and − , + are given constant. Moreover, we assume that support of the mass variation ( ) r x  is a subset compactly contained in [0,L/2]. We denote by ( ( ), ( )) =1 , n≥1, the nth eigenpair of the problem (1) with 0  replaced by ( ) x  . Our goal is the determination of an approximation to ( ) x  , or, equivalently, to ( ) r x  , using a finite amount of spectral data { ( )} =1 , where N is a given integer. The proposed reconstruction method is based on an iterative procedure which, at every step, uses a linearized Taylor approximation of the eigenvalue in terms of the unknown mass variation. We first present the linearization in a neighborhood of the unperturbed nanobeam, next we shall introduce the iteration. A key mathematical tool is the explicit expression of the first order change of an eigenvalue with respect to the smallness parameter ε. With reference to the initial uniform nanobeam, we have ( )   = = − / 2 0 ( ) ( ) 1 L n n n n r x x dx      , with ( ) ( ) 2 ( ) x u x n n  = , (2) n=1,…,N. The family    =  1 ( ) n n x is a basis of the square integrable functions defined on the interval [0, L/2]. Therefore, we look for an N-dimensional approximation of the mass variation of the type ( ) r x  is such that

N

1  = k

( ) 0 r x 

( ) 0

( ) x

( )

=

k 







,

(3)

k

[0, / 2] L

where [0, / 2] L  is the characteristic function of the interval [0, L/2], and the numbers ( )   N k k 1 0 = 

play the role of the

Generalized Fourier Coefficients of the mass variation ( ) ( ) 0 r x 

. Replacing the expression (3) in (2), we obtain the

NxN linear system

N

 =   / 2 0 L n

 = k 1

( )

0 

A

( ) ( ) x x dx

A

n 

=

, n=1,…, N, with

, n,k=1,…,N.

(4)

nk

k

nk

k

A direct calculation shows that the linear system (4) admits a unique solution, which implies the following closed form expression for the first order mass variation:

   

   

2 1 2 1 N N + −

L k x 

2 1 2 N +

N 

1, =  N 

  

   

( ) 0 r x

2

( ) 8 =

sin

0 

k 





,

−

(5)

j

[0, / 2] L



k

j

j k

1

=

see Dilena et al. (2019c). A better estimation of the mass variation ( ) r x  can be obtained by iterating the above procedure. Let us denote by   N n EXP n 1 =  the target values of the eigenvalues ( )   N n n 1 =   . The mass coefficient ( ) x  is estimated by the iteration ( ) ( ) ( ) ( ) ( ) ( ) x r x x j j j + = +   1 , j≥0, where the increment ( ) ( ) r x j at the jth step is evaluated by solving a linear system analogous to (4), e.g., ( ) ( ) ( )  = = N k j j nk j n k A 1   , n=1,…,N, where the coefficients ( ) j nk A are