PSI - Issue 24

Pierluigi Fanelli et al. / Procedia Structural Integrity 24 (2019) 939–948 Author name / Structural Integrity Procedia 00 (2019) 000–000

5

943

α + βω 2 i 2 ω i

(10)

ξ i =

Duhamel convolution integral can be taking into account to solve decoupled equations of motion. Displacement re sponse obtained through the Duhamel convolution integral on each dynamic degrees of freedom is:

1 M C i ω C i

t

ξ i ω i ( t − τ ) sin ω

F C i ( τ ) e −

η i ( t ) =

C i ( t − τ ) d τ

(11)

0

where:

ω C i = ω i 1 − ξ i 2

(12)

The n-dof system dynamics analysis is reduced to calculation of the Duhamel convolution integral.

4. Numerical approach: trapezoids method

Trapezoids method is the simplest method used to integrate numerically the Duhamel convolution integral. It is based on approximating the area delimited by the function g : [ a , b ] → R with a trapezoid area. By dividing the interval [ a , b ] into two sub-ranges [ a , x 1 ] and [ x 1 , b ], the integral of g ( x ) can be approximated as: b a g ( x ) dx = x 1 a g ( x ) dx + b x 1 g ( x ) dx ≈ δ 2 g ( a ) + g ( x 1 ) + δ 2 g ( b ) + g ( x 1 ) =

b − a

δ 2

4

g ( a ) + 2 g ( x 1 ) + g ( b ) =

g ( a ) + 2 g ( x 1 ) + g ( b )

(13)

=

In order to reduce the integration error, [ a , b ] interval should be divided in n sub-ranges so that δ = ( b − a ) / n . The integral of g ( x ) can be estimated as: b a g ( x ) dx = x 1 a g ( x ) dx + ... + b x n − 1 g ( x ) dx ≈ δ 2 g ( a ) + g ( x 1 ) + ... δ 2 g ( b ) + g ( x n − 1 ) =

b − a

δ 2 g ( a ) + 2 g ( x 1 ) + ... + 2 g ( x n − 1 ) + g ( b ) =

2 n

g ( a ) + 2 g ( x 1 ) + ... + 2 g ( x n − 1 ) + g ( b )

(14)

=

The numerical error due to the method of trapezoids is linked to the integration step δ . The decrease of this value leads to an error reduction. In the here-presented work, the trapezoids method has been applied on the relation (11) for the determination of Y and Z coe ffi cients: η ( t ) = τ 0 F ( τ ) m ω e − ξω ( t − τ ) sin ω ( t − τ ) d τ = = τ 0 F ( τ ) m ω e ξωτ cos ωτ d τ e − ξω t sin ω t − τ 0 F ( τ ) m ω e ξωτ sin ωτ d τ e − ξω t cos ω t =

= Y ( t ) e − ξω t sin ω t − Z ( t ) e − ξω t cos ω t

(15)