2nd Workshop on Joints Modelling Dartington April 2009 Identification of Nonlinear Bolted Lap Joint Parameters using Force State Mapping

Transcription

1 Identification of Nonlinear Bolted Lap Joint Paraeters using Force State Mapping International Journal of Solids and Structures, 44 (007) Hassan Jalali, Haed Ahadian and John E Mottershead

2 _ Γ _ d _ F Γ Γ Γ Γ h( x, x& ) J L NL NL Joint odeling eans identification of the oint operator both qualitatively and quantitatively.

3 Static tangential loading leads to a softening stiffness effect: F( x) 3 5 x k3x k5 k x... 3

4 Haronic force at the bolted lap oint Energy dissipated vs. the force aplitude follows a power law relationship. Experiental results show that n 3 Linear viscous daping dependent upon axiu steady state displaceent aplitude c( x c c x c x ax ) 0 ax ax L 4

5 Force State Mapping Equations in odal coordinates: q && ( t) ω q( t) h( q,q & ) Q( t) ω q ( t) h( q, q& ) Q( t) q& ( t) he oint paraeters are identified by fitting a function on the surface. 5

6 Experient Single frequency excitation close to a natural frequency Measured low level force and acceleration converted to a single odal coordinate using an updated FE odel Other (higher) odes are eliinated Modal acceleration integrated analytically to obtain velocity and displaceent runcated after the fourth haronic ( Asin( iω t) B cos( i t ) q &&( t) ω ) n i i i 6

7 .5 N Preload0N 3 N 6 N 3 Excitation levels Preload conditions Preload540N 7

8 Identification shows that k 3 is alost constant Preload0N Preload540N Viscous daping c peaks at the natural frequency it depends upon the aplitude of vibration 3 Q( t) q& ( t) ω q( t) k3q( t) c( qax ) q& ( t) c( q c c q c q ax ) 0 ax ax L 8

9 Displaceent dependent daping c( q c c q c q ax ) 0 ax ax L Energy constraint U F.39 U F.56 Preload 0N Preload 540N Excitation at 6N 9

10 Full line experiental Dashed line analytical Hysteresis Loops Preload0N Area and orientation approxiately correct Possible underestiation of the cubic stiffening ter 0

11 Perturbation Methods for the Estiation of Paraeter Variability in Stochastic Model Updating Mechanical Systes and Signal Processing (008) Haed Haddad Khodaparast, John E Mottershead and Michael I Friswell

12 he Perturbation Method Classical odel updating: ( ) Measureent: Prediction: Δ Δ Mean Variability Δ Paraeters: ransforation atrix: Δ Stochastic odel updating: Δ Δ Δ Δ n k k Δ k ( Δ )( Δ Δ )

13 he Perturbation Method 0 ( Δ ): ( ) O the ean values of the updating paraeters O(Δ ) : Δ Δ k n ( ) Δ Δ Δ ( ) k k leads to an expression for the paraeter covariances 3

14 Paraeter Covariances ( Δ, Δ ) Cov ( Δ A Δ ( Δ Δ ), Δ A Δ ( Δ Δ ) Cov Cov ( Δ, Δ ) Cov ( Δ, Δ ) A Cov ( Δ, Δ ) Cov ( Δ, Δ ) ( Δ, Δ ) A A Cov Δ, Δ A A Cov Δ, Δ ( Δ, Δ ) A Cov Δ, Δ Cov Δ, Δ ( Δ, Δ ) A Cov Δ, Δ Cov Δ, Δ ( Cov ) ( ) ( ) A Cov ( Δ, Δ ) ( Cov ) ( ( ) ) ( ) Cov ( Δ, Δ ) ( Cov ) ( ( ) ) ( ( ) ) ( ) Cov Δ, Δ A ( ) ( ) L ( ) n Τ Τ Τ ( S W S W ) S W Cov(Δ,Δ )and Cov(Δ,Δ ) are deterined by forward propagation 4

15 5 ( ) ( ) ( ) ( ) ( ) ( ) ( ), A A Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Cov, Cov, Δ Cov, Δ Cov, Δ Cov ( ) ( ) ( ) n n n A L n k k k k k k i i i k,,, ; ), ( ), ( K ( ) ( ) ( ) ), ( ), ( ), ( ), ( W S W W S S S W S W S S W W S S W S W W S S Τ Τ i Τ i Τ Τ i Τ Τ i ( ) ( ) Δ Δ S Δ Δ, Cov, Cov nd order sensitivity needed

16 Siplification If the easureents and paraeters are assued to be uncorrelated then, Cov(Δ,Δ )0, Cov(Δ, Δ )0 he paraeter covariances becoe; ( Δ, Δ ) Cov ( Δ, Δ ) Cov ( Δ, Δ ) Cov ( Δ, Δ ) ( ) ( ) Cov Δ, Δ Cov Δ, Δ Cov No requireent for the second order sensitivities. 6

17 Known deterinistic paraeters.0 kg (i,,3).0 N/ ( i 3,4) i k i k N/ Unknown Gaussian rando variables with ean values and standard deviations given by μ k.0 N/, μ k.0n/, σk 0.0N/, σk 0.0N/, σk5 μ k 5.0 N/ 0.0N/ he easured data are obtained by using Monte Carlo siulation. he initial estiates of the unknown rando paraeters are, k k k 0 N/ Cov( ) ( 0.3 ) N / i,, 5 5. k i 7

18 Convergence of natural frequency distributions 8

19 Converged Distributions 0000 Saples Paraeters Initial % Error % Error () % Error () % Error (3) % Error (4) % Error (5) k k k 5 ( k ) k std std ( ) std ( k 5 ) Methods:.Siplified perturbation ethod.full perturbation ethod 3.Perturbation ethod by Hua 4.Miniu variance Collins et al. 5.Miniu variance Friswell 9

20 Effect of saple sie 60 Error nor in SD of paraeters (0 runs) % Nuber of Saples for siulating easured data 0

21 Experiental Case Study Plate hickness Variability Arrangeent of acceleroeters (A, B, C, D) and driving point (F) 5 Nuber of observations hicknesses () Plate thickness distribution

22 Measureent Distribution fro 0 plates Mode Nuber Plate No Mean SD Mode Nuber Plate No Mean SD

23 Paraeterisation into Four Regions Initial paraeters t i 4, std( t ) 0.8, i,...,4. i 3

24 Convergence of Paraeter Estiates 4. x t t COV (t ) COV (t ) Change in ean value of paraeters t 3 t 4 Change in COV of paraeters COV (t 3 ) COV (t 4 ) Iterations Iterations 4

25 Measured, Initial and Updated Mean and Standard Deviation of Paraeters t std( t ) t std( t ) t 3 std( t 3 ) t 4 std( t 4 ) Measured Paraeters Initial Paraeters Updated Paraeters Initial FE % error Updated FE % error

26 Measured, initial and updated ean natural frequencies Measured (H) Initial FE (H) Updated FE (H) Initial FE % error Updated FE % error Mode () Mode () Mode (3) Mode (4) Mode (5) Mode (6) Measured, initial and updated std of natural frequencies Measured (H) Initial FE (H) Updated FE (H) Initial FE % error Updated FE % error Mode () Mode () Mode (3) Mode (4) Mode (5) Mode (6)