PSI - Issue 24

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

942

4

Author name / Structural Integrity Procedia 00 (2019) 000–000

3. Theoretical background: modal superposition method

In case of a n-DOFs system, the dynamic equation of motion is:

[ M ] { ¨ x } + [ C ] { ˙ x } + [ K ] { x } = { f ( t ) }

(1)

where [ M ] is the inertial matrix, [ K ] is the sti ff ness matrix, [ C ] is the damping matrix, { x } is the displacement vector for the n-DOFs dynamic and { f ( t ) } is the force vector for the n-DOFs: { f ( t ) } T = f 1 ( t ) f 2 ( t ) ... f n ( t ) T (2) By considering the natural coordinates { η } , it can be obtained: [ M ] [ Φ ] { ¨ η } + [ C ] [ Φ ] { ˙ η } + [ K ] [ Φ ] { η } = { f ( t ) } (3) where [ Φ ] is the modal matrix, which has n-degrees of freedom as rows and a number of columns equal to the considerated vibration modes. By multiplicating both the equation members of (3) by [ Φ ] T , it states:

[ Φ ] T [ M ] [ Φ ] { ¨ η } + [ Φ ] T [ C ] [ Φ ] { ˙ η } + [ Φ ] T [ K ] [ Φ ] { η } = [ Φ ] T { f ( t ) }

(4)

Under the proportional damping hypothesis, from (4) derives:

T { f ( t ) }

[ M ] C { ¨ η } + [ C ] C { ˙ η } + [ K ] C { η } = [ Φ ]

(5)

where [ M ] c = [ Φ ] T [ K ] [ Φ ]. The subscript C denotes matrices deriving from the mass normalisation process. (5) represents a decoupled equations system where vector [ Φ ] T f ( t ) members are the modal forces: [ Φ ] T { f ( t ) } =    V 11 f 1 ( t ) + V 21 f 2 ( t ) + ... + V n 1 f n ( t ) V 12 f 1 ( t ) + V 22 f 2 ( t ) + ... + V n 2 f n ( t ) V 1 n f 1 ( t ) + V 2 n f 2 ( t ) + ... + V nn f n ( t )    (6) V terms in equation (6) are the modal matrix components. Relationship between natural coordinates and principal, or modal, coordinates can be expressed by the modal matrix as follows: { x ( t ) } = [ Φ ] { η ( t ) } (7) The equations of motion can be released by substituting in the system equations of motion (1) real coordinates with modal coordinates. The equations system (1), because of the orthogonality of vectors associated to vibration modes, can be reduced to n decoupled equations of motion for each degree of freedom. For this reason, indicated with ω the pulsation, the respective equation of motion for each i degree can be obtained as (Botis (2012)): { ¨ η i ( t ) } + 2 ξ i ω i { ˙ η i ( t ) } + ω 2 i { η i ( t ) } = F C i M C i (8) where F C i = { Φ } T i { f ( t ) } (9) T [ M ] [ Φ ], [ C ] c = [ Φ ] T [ C ] [ Φ ] and [ K ] c = [ Φ ]