On a rank factorisation problem arising in vibration analysis A. - PowerPoint PPT Presentation

On a rank factorisation problem arising in vibration analysis A. Quadrat Inria Paris, Ouragan, IMJ–PRG, Sorbonne Universit´ e In collaboration with Elisa Hubert (Univ. Lyon), Axel Barrau (Safran Tech), Yacine Bouzidi (Inria Lille), Roudy Dagher (Inria Chile) Journ´ ees Nationales de Calcul Formel Luminy, 02/03/2020

Motivation: Gear box vibration analysis Elisa Hubert CIFRE PhD’s thesis ⋆ (Safran Tech) Gearbox health monitoring Up to date surveillance Reduction of damage frequency Optimization of maintenance intervention Say VRAOUMMM*! ⋆ D´ emodulation d’amplitude et de phase de signaux multi-porteuses : Application aux signaux vibratoires d’engrenage , Univ. Lyon, 2019 A. Quadrat On a rank factorisation problem arising in vibration analysis

Demodulation problem M ∈ K n × m , D 1 , . . . , D r ∈ K n × n • K : field (e.g. Q , R , C ),   0 . . . 1 . .  ∈ K n × n   . . J n = . 1 .  1 . . . 0 • Definition: M ∈ C n × m is centrohermitian if J n M J m = M CH n , m ( K ): set of all the centrohermitian matrices of K n × m � � · � Frob : Frobenius norm , i.e. � A � Frob = trace ( A ⋆ A ) • The demodulation problem: M ∈ CH n , m ( C ) Find u ∈ CH n , 1 ( C ) , v i ∈ CH 1 , m ( C ) : � r i =1 D i u v i = M arg min u ∈ CH n , 1 ( C ) , v i ∈ CH 1 , m ( C ) � � r i =1 D i u v i − M � Frob A. Quadrat On a rank factorisation problem arising in vibration analysis

Gear box vibration analysis A. Quadrat On a rank factorisation problem arising in vibration analysis

Gear box vibration analysis Gearing vibration produced by 2 wheels W 1 and W 2 ( Capdessus 92 ) A. Quadrat On a rank factorisation problem arising in vibration analysis

Gear box vibration analysis Gearing vibration produced by the teeth of 2 wheels W 1 and W 2 • N i denotes the number of teeth of the wheel W i f e = N 1 f 1 = N 2 f 2 • s static transmission error producing the vibration ( Mark 78 ) s ( t ) = s e ( t ) ( F + s 0 ( t ) + s 1 ( t ) + s 2 ( t )) s e and s 0 : periodic signals of frequency f e s i : periodic signal of frequency f i , i = 1 , 2 ⇒ Amplitude modulation ( Capdessus 92 ) ⇒ s ( t ) = s c ( t ) s m ( t ) s m : modulation f m = T − 1 (low-frequency) m s c : carrier f c = T − 1 = N f m , N ∈ N ≥ 0 (high-frequency) c A. Quadrat On a rank factorisation problem arising in vibration analysis

Gear box vibration analysis • p : R → R : T -periodic integrable function, a ∈ R : � p ( t ) , a ≤ t < a + T , p 0 ( t ) = 0 , else , � � ⇒ p ( t ) = p 0 ( t ) ⋆ δ ( t − j T ) = p 0 ( t − j T ) j ∈ Z j ∈ Z • Fourier transform & Poisson’s formula yield � j � � � � � p 0 ( ν ) 1 ν − j 1 ν − j � � p ( ν ) = ˆ ˆ δ = T ˆ p 0 δ T T T T j ∈ Z j ∈ Z • Inverse Fourier transform p ( t ) = � 2 π ijt j ∈ Z c j ( p ) e ( Fourier series ) T � j � a + T � c j ( p ) = 1 = 1 p ( t ) e − 2 π ijt T dt T ˆ p 0 T T a • c j ( p ) = c − j ( p ) , j ∈ Z . p is real ⇒ c − j ( p ) = c j ( p ), j ∈ Z A. Quadrat On a rank factorisation problem arising in vibration analysis

Gear box vibration analysis T − 1 = N T − 1 • s ( t ) = s c ( t ) s m ( t ), c m � j � � � j 1 s m ( ν ) = � ˆ j ∈ Z ( c m ) j δ ν − , ( c m ) j = ˆ s m 0 , T m T m T m � k � � � ν − k ( c c ) k = 1 s c ( ν ) = � ˆ k ∈ Z ( c c ) k δ , s c 0 ˆ , T c T c T c � l � � � l 1 s ( ν ) = � ˆ l ∈ Z ( c s ) l δ ν − , ( c s ) l = s 0 ˆ , T m T m T m � � j − k � � s ( ν ) = (ˆ ˆ s m ⋆ ˆ s c )( ν ) = ( c m ) j ( c c ) k δ ν − T m T c j ∈ Z k ∈ Z � � ν − j + k N � � = ( c m ) j ( c c ) k δ T m j ∈ Z k ∈ Z � ⇒ ( c s ) l = ( c m ) j ( c c ) k { ( j , k ) ∈ Z 2 | j + k N = l } A. Quadrat On a rank factorisation problem arising in vibration analysis

Gear box vibration analysis .   . .   ( c s ) − 1      ( c s ) 0  =     ( c s ) 1      .  . . . . . . . . . .     . . . . . . . . . . . . . . . .     ( c c ) − 1 ( c m ) 2 N − 1 ( c m ) N − 1 ( c m ) − 1 ( c m ) − N − 1 ( c m ) − 2 N − 1     . . . . . .        ( c c ) 0  ( c m ) 2 N ( c m ) N ( c m ) 0 ( c m ) − N ( c m ) − 2 N     . . . . . . .     ( c c ) 1 ( c m ) 2 N +1 ( c m ) N +1 ( c m ) 1 ( c m ) − N +1 ( c m ) − 2 N +1     . . . . . .     .  . . . . . . .    . . . . . . . . . . . . . . . . A. Quadrat On a rank factorisation problem arising in vibration analysis

Gear box vibration analysis Hypotheses on the signals • s c : finite number of non-zero harmonics � � ν − k � � s c ( ν ) = ˆ ( c c ) k δ ⇒ ( c s ) l = ( c m ) l − k N ( c c ) k T c 0 ≤| k |≤ r 0 ≤| k |≤ r • s m does not induce overlaps in ˆ s = ˆ s m ⋆ ˆ s c k N + | j | < ( k + 1) N − | j | ∀ k ∈ Z , ⇔ 2 | j | < N T m T m T m T m • • • • k N k N + | j | ( k + 1) N − | j | ( k + 1) N T m T m T m T m T m T m ( c m ) j = 0 for all j / ∈ { j ∈ Z | 2 | j | < N } = {− p , . . . , p } , where p = N − 1 , if N odd , N 2 − 1 , if N even 2 A. Quadrat On a rank factorisation problem arising in vibration analysis

Gear box vibration analysis A. Quadrat On a rank factorisation problem arising in vibration analysis

Gear box vibration analysis • | l | ≤ t := p + r N       ( c s ) − t ( c c ) − r ( c m ) − p . . .       . . .  .   .   .              ( c s ) − 1 ( c c ) − 1 ( c m ) − 1                   c s := ( c s ) 0 , c c := ( c c ) 0 , c m := ( c m ) 0             ( c s ) 1 ( c c ) 1 ( c m ) 1                   . . .  .   .   .  . . .             ( c s ) t ( c c ) r ( c m ) p   c m 0 . . . 0 . ...   .  0 .  c m   N odd : A = = c m ⊗ I 2 r +1  .  ... ... .   . 0   0 . . . 0 c m c s = A c c A. Quadrat On a rank factorisation problem arising in vibration analysis

Gear box vibration analysis • The vectorization satisfies � � W T ⊗ U vec ( U V W ) = vec ( V ) U = I 2 r +1 , W = c m T ∈ C 1 × (2 p +1) , V = c c ∈ C (2 r +1) × 1 , ⇒ c c c m T = vec − 1 ( c s ) c − j ( s • ) = c j ( s • ) ⇒ c c , c m , vec − 1 ( c s ) centrohermitian • Amplitude demodulation problem: M s ∈ CH (2 r +1) , (2 p +1) ( C ). Find c c ∈ CH (2 r +1) , 1 ( C ) and c c ∈ CH 1 , (2 p +1) ( C ) such that c c c m T = M s • Phase and amplitude demodulation problem: T + D c c c m 2 T = M s , D := 2 π i f c diag ( − r , . . . , r ) c c c m 1 A. Quadrat On a rank factorisation problem arising in vibration analysis

The rank factorization problem • The rank factorization problem: Determine u ∈ CH n , 1 ( C ) and v 1 , . . . , v r ∈ CH 1 , m ( C ) − if they exist − such that r � M = D i u v i ( bilinear polynomial system ) i =1 • A system-theoretical approach: r ) T ∈ K r × m A ( u ) := ( D 1 u . . . D r u ) ∈ K n × r , v := ( v T . . . v T 1 ⇒ A ( u ) v = M • Necessary condition: rank K ( M ) ≤ r • If rank K ( M ) = r then v has full row rank since r � � �� � � D i − λ i λ − 1 λ v = λ i v i = 0 , λ k � = 0 ⇒ M = D k u v i k i =1 1 ≤ i � = k ≤ r ∀ λ ∈ K \ { 0 } : ( u , v ) �→ ( λ u , λ − 1 v ) A. Quadrat On a rank factorisation problem arising in vibration analysis

The rank factorization problem ( D 1 u . . . D r u ) v = M , l := rank K ( M ) ≤ r � �� � A ( u ) • L ∈ K p × n : full row rank matrix defining a basis of ker K ( . M ) = { λ ∈ K 1 × n | λ M = 0 } , p := dim K (ker K ( . M )) A ( u ) v = M ⇒ L A ( u ) v = L M = 0 ⇒ L A ( u ) = 0 ( v full row rank)  L D 1 u = 0 ,   . . ⇔ ( V ) .   L D r u = 0 , ⇔ u = Z ψ, ∀ ψ ∈ K d × 1 d = dim K ( V ) & Z ∈ K n × d full column rank defining a basis of V ⇒ i = 1 , . . . , r , L D i Z = 0 A. Quadrat On a rank factorisation problem arising in vibration analysis

The rank factorization problem • X ∈ K n × l : full column rank matrix defining a basis of im K ( M . ) ⇒ ∃ ! Y ∈ K l × m full row rank : M = X Y • ker K ( . M ) = im K ( . L ) ⇒ ker K ( L . ) = im K ( M . ) = im K ( X . ) i = 1 , . . . , r , L D i Z = 0 ⇒ ∃ W i ∈ K l × d : D i Z = X W i A ( Z ψ ) = ( D 1 Z ψ . . . D r Z ψ ) = X ( W 1 ψ . . . W r ψ ) • Let u = Z ψ and B ( ψ ) := ( W 1 ψ . . . W r ψ ) ∈ K l × r A ( Z ψ ) v = X B ( ψ ) v = M = X Y ⇒ B ( ψ ) v = Y ( X full colum rank) • Example: If l := rank K ( M ) = r , then B ( ψ ) ∈ K r × r A. Quadrat On a rank factorisation problem arising in vibration analysis