Identification of Nonlinear Bolted Lap Joint Paraeters using Force State Mapping International Journal of Solids and Structures, 44 (007) 8087 808 Hassan Jalali, Haed Ahadian and John E Mottershead
_ 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.
Static tangential loading leads to a softening stiffness effect: F( x) 3 5 x k3x k5 k x... 3
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
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
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
.5 N Preload0N 3 N 6 N 3 Excitation levels Preload conditions Preload540N 7
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
Displaceent dependent daping c( q c c q c q ax ) 0 ax ax L Energy constraint U 5 3.85 0 F.39 U 5 7.4 0 F.56 Preload 0N Preload 540N Excitation at 6N 9
Full line experiental Dashed line analytical Hysteresis Loops Preload0N Area and orientation approxiately correct Possible underestiation of the cubic stiffening ter 0
Perturbation Methods for the Estiation of Paraeter Variability in Stochastic Model Updating Mechanical Systes and Signal Processing (008) 75 773 Haed Haddad Khodaparast, John E Mottershead and Michael I Friswell
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 )
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
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
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
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
Known deterinistic paraeters.0 kg (i,,3).0 N/ ( i 3,4) i k i k 6 3.0 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
Convergence of natural frequency distributions 8
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 ) 00.9.43..88 6.40 00 -.45 -.70 -.68 -.6 36.56 00 0.56 0.6 0.6.3 58.58 50 0.4 0.45.50-89.88-4.8 50-0.68.96 0.75-89.96-3.07 50. 0.6 0.34-90.07-59.87 Methods:.Siplified perturbation ethod.full perturbation ethod 3.Perturbation ethod by Hua 4.Miniu variance Collins et al. 5.Miniu variance Friswell 9
Effect of saple sie 60 Error nor in SD of paraeters (0 runs) % 50 40 30 0 0 0 0 0 50 00 00 300 400 500 600 700 800 900 000 Nuber of Saples for siulating easured data 0
Experiental Case Study Plate hickness Variability Arrangeent of acceleroeters (A, B, C, D) and driving point (F) 5 Nuber of observations 4 3 0 3.60-3.75 3.75-3.90 3.90-4.05 4.05-4.0 4.0-4.35 hicknesses () Plate thickness distribution
Measureent Distribution fro 0 plates Mode Nuber Plate No. 3 4 5 9.774 84.83 33.970 589.404 656.359.65 9.9 337.86 605.60 665.854 3 3.56 9.440 340.84 60.603 673.357 4 8.048 98.63 355.0 60.39 700.798 5 8.533 303.809 357.0 630.809 704.505 6 8.596 30.00 36.488 635.533 73.07 7 9.796 3.76 36.4 646.765 7.79 8 35.058 35.393 374.368 653.584 738.395 9 34.478 3.5 374.406 649.30 737.56 0 38.4 3.8 38.93 667.03 755.89 Mean 8.70 303.77 357.597 630.033 705.77 SD 6.0.03 7.048 5.35 3.854 Mode Nuber Plate No. 6 7 8 9 0 93.576 09.603 343.097 68.879 85.5 953.666 06.86 37.890 650.395 860.5 3 955.55 9.445 376.98 669.899 868.07 4 980.403 65.77 44.8 736.74 94.60 5 995.88 69.660 433.00 743.750 946.55 6 999.48 84.455 440.34 765.45 957.58 7 09.05 84.608 467.366 766.36 987.556 8 03.837 5.375 487.5 85.60 0.640 9 03.9 4.40 479.68 84. 03.354 0 053.974 53.60 59.0 866.665 03.377 Mean 994.469 7.5 433.78 747.780 943.543 SD 38.877 53.840 56.77 79.3 7.908
Paraeterisation into Four Regions Initial paraeters t i 4, std( t ) 0.8, i,...,4. i 3
Convergence of Paraeter Estiates 4. x 0-3 4.5 t t 0. 0. COV (t ) COV (t ) Change in ean value of paraeters 4. 4.05 4 3.95 3.9 3.85 t 3 t 4 Change in COV of paraeters 0.8 0.6 0.4 0. 0. 0.08 0.06 COV (t 3 ) COV (t 4 ) 3.8 0.04 3.75 0 5 0 5 0 5 30 35 40 45 50 Iterations 0.0 0 5 0 5 0 5 30 35 40 45 50 Iterations 4
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 3.978 4.000 4.40 0.553 4.07 0.59 0.8 0.9 403.45-8.868 3.969 4.000 4.00 0.78 0.83 0.6 0.8 0.04 396.894 6.708 3.98 4.000 3.986 0.45 0.00 0.64 0.8 0.66 387.805.9 3.98 4.000 3.80 0.477-4.044 0.67 0.8 0.06 379.04 3.353 5
Measured, initial and updated ean natural frequencies Measured (H) Initial FE (H) Updated FE (H) Initial FE % error Updated FE % error Mode () 8.70 8.3 8. -0.30-0.473 Mode () 303.77 307.47 306.339.30.043 Mode (3) 357.597 356.645 355.85-0.66-0.675 Mode (4) 630.033 637.433 633.88.75 0.50 Mode (5) 705.77 705.467 70.777-0.043-0.566 Mode (6) 994.469 00.9 996.865 0.780 0.4 Measured, initial and updated std of natural frequencies Measured (H) Initial FE (H) Updated FE (H) Initial FE % error Updated FE % error Mode () 6.0 0.943 5.750 48.4-4.34 Mode ().03 47.385 3.777 93.85 4.503 Mode (3) 7.048 39.3 5.80 30. -0.957 Mode (4) 5.35 65.655 6.797 60.75 6.90 Mode (5) 3.854 7.379 8.644 7.6 -.84 Mode (6) 38.877 08.445 40.66 78.944 3.36 6